Mathematics For Economics And Finance Pdf

The primary scope of our Journal is to provide a forum to exchange ideas in economic theory which expresses economic and financial concepts and laws using formal and well-constructed mathematical reasoning, Mathematical Decision Theory, Game Theory, Functional Analysis, Differential Geometry and so on. Our Journal covers and publishes original researches and new significant results and methods of Mathematical Economics, Finance, Game Theory and applications, mathematical methods of economics, finance and management, Quantitative Decision theory and Risk Theory. The mathematical form of economic and financial laws appears of fundamental importance to the developments and deep understanding of Economics and Finance themselves. Such a translation in mathematical terms can determine whether an economic or financial intuition shows a coherent and logical meaning. Also, a full rational and mathematical development of economic ideas can itself suggest new economic concepts and deeper economic intuitions.

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Volume I

Issue 1(1)

Winter 2015

ISSN-L: 2458-0813

eISSN: 2458-0813

Journal's DOI: https://doi.org/10.14505/jmef

Journal's Issue DOI: https://doi.org/10.14505/jmef.v1.1(1).00

Journal of Mathematical Economics

and F inance

Contents:

Journal of

Advanced Research

in Law and

Economics is

designed to provide

Bounded Rational

Speculative and Hedging

Interaction Model in Oil

and U.S. Dollar Markets

David CARFÌ

University of California

Riverside, USA

Michael CAMPBELL

California State University,

Fullerton, USA … 4

Effectiveness and

Efficiency Trade-Off in the

Demerit Goods Taxation: a

Non-Standard Approach

Francesco MUSOLINO

Banco Popolare, Catania,

Italy … 34

The Easterlin threshold:

a first glimpse

Francesco STRATI

University of Siena,

Italy … 29

A model for coopetitive

games

David CARFÌ

University of California

Riverside, USA …46

Winter 2015

Volume I, Issue 1(1)

Editor in Chief

D. Carfì, University of California Riverside, USA

University of Messina, Italy

Co-Editors

M. Campbell, Aurislink, Israel / USA

Chapman University, USA

M. Gualdani, George Washington University, USA

Assistant-Editors

A. Agnew, California State University Fullerton, USA

A. Donato, University of Messina, Italy

Editorial Coordinators

A. Kushner, Russian Academy of Sciences, Russia

M. Maroun, University of California Riverside, USA

Editorial Advisory Board

T. Arthanari, University of Auckland, New Zealand

V. Balan, University of Bucharest, Romania

B. Blandina, Ernst & Young, Belgium

M. T. Calapso, University of Messina, Italy

K. Drachal, Warsaw Technology University, Poland

S. Federico, University of Calgary, Canada

G. Fontana, Leeds University Business School, UK

G. Giaquinta, University of Catania, Italy

S. Haroutunian, Armenian State University, Armenia

Z. Ibragimov, California State University Fullerton, USA

J. Martinez-Moreno, University of Jaén, Spain

R. Michaels, California State University Fullerton, USA

J. Mikes, University of Olomouc, Czech Republic

F. Musolino, Banco Popolare, Italy

R. Niemeyer, University of California Riverside, USA

M. Okura, Doshisha Women's College of Liberal Arts, Japan

K. Oliveri, Tor Vergata University, Rome, Italy

D. Panuccio, University of Messina, Italy

A. Pintaudi, LUISS Guido Carli University, Italy

R. Pincak, Slovak Academy of Sciences, Bratislava

A. Ricciardello, University of Enna, Italy

D. Schilirò, University of Messina, Italy

A. Shelekov, Lomonosov University, Moscow

M. Squillante, University of Sannio, Italy

A. Trunfio, University of Padua, Italy

L. Ungureanu, Spiru Haret University, Romania

A. Ventre, University of Naples, Italy

L. Verstraelen, Katholieke Universiteit, Belgium

http://www.asers.eu/asers-publishing

ISSN-L: 2458 -0813

eISSN: 2458-0813

Journal's DOI: https://doi.org/10.14505/jmef

jmef.asers@gmail.com

davidcarfi@gmail.com

Journal of Mathematical Economics and Finance (J MEF) is a biannually peer-reviewed journal of

Association for Sustainable Education, Research and Science .

Aims and Scope . The primary scope of our Journal is to provide a forum to exchange ideas in

economic theory which expresses economic and financial concepts and laws using formal and well-

constructed mathematical reasoning, Mathematical Decision Theory, Game Theory, Functional Analysis,

Differential Geometry and so on.

Our Journal covers and publishes original researches and new significant results and methods of

Mathematical Economics, Finance, Game Theory and applications, mathematical methods of economics,

finance and management, Quantitative Decision theory and Risk Theory. The mathematical form of economic

and financial laws appears of fundamental importance to the developments and deep understanding of

Economics and Finance themselves. Such a translation in mathematical terms can determine whether an

economic or financial intuition shows a coherent and logical meaning. Also, a full rational and mathematical

development of economic ideas can itself suggest new economic concepts and deeper economic intuitions.

Editor invitation. The editors encourage the submission of high quality, insightful, well-written papers

that explore current and new issues in Mathematical Economics, Finance, Econophysics, Game Theory and

applications, mathematical methods of economics, finance and management, Quantitative Decision theory and

Risk Theory and the common grounds between these discipline areas.

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English (consistent British or American), not under consideration by other journals.

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the first issue released.

How to cite. JMEF follows the format of the Chicago Manual of Style, 15th edition, chapter 16 (a brief

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case of LaTeX submission, JMEF cites and refers using BibTeX (a brief guide may be found at

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davidcarfi@gmail.com

Full author's guidelines are available from:

http://asers.eu/journals/jmef/instructions-for-authors

Call for Papers

Winter_Issue 2015

Journal of Mathematical Economics and Finance

DOI: https://doi.org/10.14505/jmef.v1.1(1).01

Bounded Rational Speculative and Hedging Interaction Model

in Oil and U.S. Dollar Markets

David Carf`ı

Department of Mathematics, University of California Riverside, USA

Department of Economics, California State University Fullerton, USA

davidcarfi@gmail.com

Michael Campbell

Department of Physics, California State University Fullerton, USA

michaeljcampbell@outlook.com

Abstract:

A 'bounded rational' overlay is constructed for a model of an interaction between two players

who speculate on oil and the U.S. dollar, subject to ﬁnancial transaction taxes. This model also has

two types of operators: a real economic subject (Air) and an investment bank (Bank).

Many investment operators (banks) are also considered. Their behavior equilibrates much more

quickly, as they react to the move of Air. In this sense, Air is an acting external agent (such as

with an external magnetic ﬁeld in a magnetic system), whereas the random component of the bounded

rational behavior of banks is 'annealed' (i.e., averaged out before Air makes its next transaction).

Under certain conditions for the model, the equilibrium measure for the bank agents, after Air has

played its strategy, is a Gibbs measure from statistical mechanics, as the interactions between opera-

tors are that for a Potential game.

Keywords: Airlines, Bank, Cross-hedging, Currency Markets, Financial Risk, Financial Transaction

Taxes, Game Theory, Hedging, Speculation, Potential Games, Bounded Rationality, Logit Equilib-

rium, Gibbs Equilibrium, Noisy Directional Learning, Phase Transition, Entropy.

JEL Classiﬁcation: C65, C73, C79.

1. Introduction

Certain models in game theory (Anderson et al. 1999, 2002; Binmore and Samuelson 1997;

Binmore et al. 1995; Ceccatto and Huberman 1989) analyze the dynamics of decisions made by agents

who adjust their decisions in the direction of higher payoﬀs, subject to random error and/or infor-

mation ("deviations-from-rationality" models, cf. Blume 1996). These errors, which are essentially

failures to choose the most optimal payoﬀ, are understood to be intrinsic to the agents, and can be

due to stochastic elements such as preference shocks, experimentation, actual mistakes in judgment,

or a lack of complete information about the system. In all of these contexts, the error is assumed to

be due to intrinsic properties of the agents; i.e., the error is due to the agents making decisions that

deviate from the true, optimal decision.

Some types of bounded-rational potential games have an intrinsic notion of Gibbsian equilib-

rium that can result from drift-diﬀusion dynamics, or independently, a static notion of agents having

imperfect information about the system1. For potential games, the maximum of the potential is con-

sidered to be an appropriate reﬁnement of the Nash equilibrium (c.f., Carbonell-Nicolau and McLean

2014). In this static case, agents are restricted to play mixed strategies with a ﬁxed, possibly non-

maximal potential (and hence non-maximal utilities, which immediately implies bounded rational

1c.f. "large deviation theory" in Ellis 1999

Volume I, Issue 1(1), Winter 2015

decisions) along with a condition to maximize (Shannon) information entropy, which can be inter-

preted as agents arbitraging information out of the system to gain knowledge so they can adapt and

improve. Both approaches yield the same Gibbs equilibrium measure and resulting "temperature"2.

In the static case, the constraint of a constant average potential3will specify the temperature

of the system. Alternatively, the constraint of a ﬁxed temperature will specify the average potential.

Conservation of potential can then be interpreted as specifying the temperature of the system - that

is, how much the agents deviate from rationality. In the dynamics model, the ﬂuctuation-dissipation

theorem shows the temperature is proportional to the square of the coeﬃcient of the diﬀusion variable

which generates non-rational decisions. In the static entropy model, temperature is a Lagrange

multiplier for the restriction of a speciﬁed mean potential in the minimum free energy problem of

large deviation theory. In both cases, lower temperature represents more rational behavior, and at

zero temperature, it can easily be shown the equilibrium measure for both models is exactly the

(reﬁned) Nash equilibrium found for the corresponding classical rational model of the potential game

(Monderer and Shapley 1996). An exact solution to an inﬁnite-agent Gibbsian bounded-rational

Cournot model4was found in Campbell 2005, and this model captures much of the formalism of the

model in this article. Very few Gibbsian models can be solved exactly, and the solvability of that

Cournot model allows for direct analysis of agent output in the inﬁnite-agent case.

It will be demonstrated that organization of agents' behavior depends on the value of the

variance of the random component of decisions (proportional to 'temperature' as with the 'ﬂuctuation-

dissipation' relation), as well as other parameters.

The Gibbsian approach provides a mechanism (statistical mechanics) for looking at the be-

havior of an inﬁnite number of agents in the bounded-rational case. Likewise, the Gibbs measure is

a bounded-rational generalization of the Nash equilibrium. It turns out that for a potential game

(with a unique Nash equilibrium), the Nash equilibrium is attained from the Gibbs measure at zero

temperature (rational behavior).

A dynamical system for the case of a continuum of agent decisions that also yields the Gibbs

measure is presented here. For these dynamics, the agents are myopic5 in pure strategy space; that

is they play a single pure strategy at any point in time, and can only make inﬁnitesimal adjustments

in pure strategy space over inﬁnitesimal time. The diﬀerence between this dynamical approach and

those in Anderson et al. 1999, 2002; Blume 1996; Ceccatto and Huberman 1989 is that a 'global'

approach is used6. The dynamics in this paper are purely local in space and time: agents adjust

their present pure strategy based on the present pure strategy of the other agents. This is a realistic

assumption for large numbers of agents, since the amount of information in space (i.e., the strategies

of all of the other agents at a given moment in time) and past time is too much for an agent to process

in a practical way. Since agents follow the gradient of a potential function perturbed by noise, the

term "noisy directional learning" (Anderson et al. 1999) aptly describes the agents in a dynamical

sense.

2Even for a small number of agents, signiﬁcant components of the information inﬂuencing agents is qual-

itative, and not quantiﬁable. Such information is typically processed intuitively, and the outcome will likely

depend on each agent's beliefs. Therefore a bounded-rational approach is justiﬁed here. The temperature

parameter can be used to tune the inﬂuence of bounded rationality anywhere from zero (standard rational

game theory) to as high as is needed to try to ﬁt data more accurately.

3This condition, in statistical mechanical models, is "conservation of energy". We'll call this "conservation

of potential", which indirectly quantiﬁes the degree of agent's deviation from rationality ("temperature" in

statistical mechanics).

4This solvable Cournot model has a uniform distribution of goods which have a non-discrete range of

output; i.e. an interval of real numbers.

5Agents are myopic in their rational decisions. For example, a bank will change its investment by a

relatively small amount over a small time, with a purely rational decision. An agent's decision has a non-

rational component (stochastic summand in the dynamics), which may allow a large deviation (non-myopic)

with a certain probability.

6For a potential game, a single dynamical equation can track all agents.

Journal of Mathematical Economics and Finance

2. Literature review

In this paper, we shall use literature from many diﬀerent ﬁelds, as we shall show in the

following survey.

2.1 Complete study of diﬀerentiable games literature

For what concerns the complete study of diﬀerentiable games and related mathematical back-

grounds, introduced and applied to economic theories since 2008 by Carf`ı, the interested reader could

see Carf`ı 2008a, 2009b,c,g,a,f, 2010a,b,e, 2012 and the papers by Carf`ı et al., such as Baglieri et al.

2010, 2012a; Carf`ı and Magaudda 2009; Carf`ı et al. 2010a,b; Carf`ı and Ricciardello 2009, 2010, 2011,

2012a,c,e,g,d,h,b,i,f,j, 2013b,a; Agreste et al. 2012; Carf`ı and Fici 2012a,b; Carf`ı and Perrone 2012b,a,

2013; Carf`ı and Pintaudi 2012a,b; Carf`ı and Schilir`o 2012d,b,c,e,a,f, 2013a,b; Carf`ı et al. 2011, 2013;

Carf`ı and Lanzafame 2013.

2.2 Econophysics literature

Moreover, we shall consider several papers connecting physical and economic theories (see

Anderson et al. 1999, 2002; Beightler and Wilde 1966; Binmore and Samuelson 1997; Binmore et al.

1995; Blume 1996; Blume and Durlauf 2002; Brock 1993; Campbell 2005; Cannas et al. 2004; Cav-

agna et al. 1999; Ceccatto and Huberman 1989; Carbonell-Nicolau and McLean 2014; Durlauf 1996;

Dai 1990; Ellis 1999; Kac 1968; Lavis 2005; MacIsaac et al. 1995; Milotti 2002; Mu and Ma 2003;

Mohammed and Scheutzow 2003; Monderer and Shapley 1996; Nagle 1970; Plerou et al. 2002; Reif

1965; Simon 1993; Smith and Foley 2004; Smith et al. 2003; Vieira and Gon¸calves 1999; Dupuis and

Williams 1994; Dai and Williams 1995; Harrison and Williams 1987; SD 2015).

A thorough treatment of many of the techniques used hereon, can be found in the classic text

Chorin and Hald 2009.

2.3 Financial market games literature

Speciﬁc applications of the previous methodologies, also strictly related to the present model,

have been illustrated by Carf`ı and Musolino (Carf`ı and Musolino 2011b,a, 2012a,c,e,b,d, 2013a,b,c,

2014b,a, 2015a,b).

2.4 Applications of the game complete study literature

Other important applications of the complete examination methodology, which inspired us for

the constraction of the present model, were introduced by Carf`ı and coauthors, for instance:

•Carf`ı and Perrone 2011a,b,c,d,e, 2012b,a, 2013;

•Carf`ı and Schilir`o 2010, 2011b,a,c,d,f,e, 2012d,b,c,a,f,e, 2013a, 2014a,b,c

•Carf`ı and Ricciardello 2012a,c, 2013b,a;

•Carf`ı et al. 2011a,b; Carf`ı and Trunﬁo 2011a,b; Agreste et al. 2012; Baglieri et al. 2012a,b,

2015; Carf`ı 2012; Carf`ı and Fici 2012a,b; Carf`ı and Pintaudi 2012a,b; Carf`ı et al. 2013; Carf`ı

and Lanzafame 2013; Okura and Carf`ı 2014; Arthanari et al. 2015; Carf`ı and Romeo 2015.

2.5 Possible future developments in view of Schwartz Linear Algebra

General ideas on the possible future applications of the methodologies introduced in the previ-

ous and present works could be devised under the view of the following researches: Carf`ı 2004a,e,c,b,

2006c,a, 2007c,a, 2008c,b, 2009d,e, 2011b,c,g,a; Carf`ı and Caristi 2008; Carf`ı and Cvetko-Vah 2011,

as well as in Carf`ı 2015b, 2010d,c, 2007b,d, 2005a,b, 2004d, 2003a,b, 2002a,b, 2001a,b, 2000, 1998,

1997, 1996; Carf`ı and German`a 2003, 2000b,c, 1999a,b, 2000a; Carf`ı and Magaudda 2007, for what

concerns a more speciﬁcally econophysical perspective, by adopting tools from Diﬀerential Geometry

Volume I, Issue 1(1), Winter 2015

and Functional Analysis, with a wide application of Laurent Schwartz distributions.

3. Potential Game Model and Bounded-Rational Equilibria

We consider a game with a ﬁnite number Nof "Bank" (investment) players, and all of these

players belong to the set

Λ := {i ∈N :i≤ N} .

At any moment in time, a Bank player i∈ Λ can select an action or strategy (y (i)

1, y (i)

2)∈F, where

Frepresents a convenient subset of the Cartesian square [− 1,1]2 and the y (i)

1and y (i)

2play the role

of the so called strategy variables of the i -th player.

The strategy y (i)

1represents the proportion of its resources that Bank i spends on the oil spot

market, and y (i)

2represents the proportion of its resources spent on the (US) dollar futures market

from its total resources M > 0.

Aconﬁguration ~y of the system is any possible state of the system:

~y = ( y (1)

1, y (1)

2),(y (2)

1, y (2)

2),...,(y (N)

1, y (N)

2) ,

where each pair (y (i)

1, y (i)

2) belongs to F. The set of all possible conﬁgurations of the game is

ΦΛ : =Y

i∈ Λ

F(i) ,

which is called (pure) state space. The F (i) := F here is the diamond set

F:= n ( y(i)

1, y (i)

2)∈[ −1, 1] 2 :k(y (i)

1, y (i)

2)k 1 =  y(i)

1  +  y(i)

2  ≤1o .(1)

Now we will deﬁne the payoﬀ functions as in Carf`ı and Musolino 2014b. The real economic

subject ("Air") is a player in the game, and is assigned zero as its player number. Its strategy variable

x∈[0 ,1] represents the proportion of its resources M(0) spent on purchasing oil futures as a hedge.

The remaining proportion of Air's resources, 1 − x, is spent purchasing jet fuel on the spot market.

The oil payoﬀ function for Air is then what it spent: the amount of jet fuel it bought on the spot

market,

(1 − x )M (0) ,

multiplied by it's savings on the price of jet fuel (hedge prices minus the actual prices, which depends

on the actions of Bank and contains a negated 7 component related to what Bank spent on US dollar

futures). In the case of only a single Bank (player 1), this reduces to (see Carf`ı and Musolino 2014b)

f(0)

O(x, ~y) = M (0) (1 − x)( u ( un − ν) y (1)

1+uky (1)

2),(2)

where u is the capitalization factor (u = 1 + r , for risk-free interest rate r ) resulting from the

transaction occurring in the previous time step. The parameter n > 0 represents represents the eﬀect

of Bank's strategy y (1)

1on the oil spot market price at time 1, and k > 0 is the negative inﬂuence of

Bank's strategy y (1)

2on the price of oil futures. Both n and k depend on Bank's ability to inﬂuence

the oil spot market and the behavior of other ﬁnancial agents. A tax parameter ν , 0 ≤ν ≤ nu , can

be set within the range of no taxation (ν = 0) to full taxation (ν = un ). For the single Bank player,

the Bank payoﬀ function for dollar futures is the product of the amount purchased, and returns per

unit of dollar futures:

f(1)

$(x, ~y) = y (1)

2M(−u 2 ny (1)

1+u( k− κ) y (1)

2−umx) , (3)

7It is pointed out in Carf`ı and Musolino 2014b, and references therein, that rises in oil prices are associated

with the depreciation of the US dollar. Hence, we see a leading negative sign in front of y (2)

1, in each of

equations (2) and (3).

Journal of Mathematical Economics and Finance

where k− κ = 0 in Carf`ı and Musolino 2014b as a result of a tax, and there Bank gains nothing

from its actions on the oil spot market. Here, we can vary κ within 0 ≤κ ≤k to represent the range

from no taxation (κ = 0) to full taxation (κ =k ). The parameter m > 0 measures the inﬂuence of

Air's strategy xon the price of oil futures and the ability of Air to inﬂuence the oil market and the

behavior of other ﬁnancial agents. Similarly, Bank's payoﬀ function from the oil spot market is the

product of the amount purchased and returns per unit:

f(1)

O(x, ~y) = y (1)

1M((un − ν )y (1)

1+uδy (1)

2),(4)

where as above, the tax parameter νis set so that un − ν = 0 in Carf`ı and Musolino 2014b as a

result of a tax (we can take 0 ≤ν≤ un to represent the range from full taxation to no taxation),

and m > 0 measures the inﬂuence of Air's strategy xon the price of oil futures and the ability of Air

to inﬂuence the oil market and the behavior of other ﬁnancial agents.

The model can be generalized to a single large-scale economic subject (Air player zero) and

many investors (Bank players each labeled i , 1 ≤i ≤N ). For simplicity, we assume the Bank agents

are interaction homogeneous 8 , so that they all have the same resources and are identically aﬀected

by each other and by Air, within markets. The gains of each Bank ifrom the dollar futures

market would then be aﬀected by Air and all Bank players:

p$ ( x, ~y) =

j=1 −u 2 n

Ny (j)

1+u(k− κ )

Ny (j)

2−umx, (5)

where the "interaction" terms are

−u 2 n

N, u(k− κ )

and the "ﬁeld" term umx . As mentioned in Campbell 2005, the interaction terms are divided by N

so that demand is based on the average production. Thus demand stays nonnegative for large N.

For example, if each Bank used all resources for oil spot market purchases (all y (j)

1= 1), then

p$ ( x, −−−→

(1, 0)) = −u2 n − umx

is well-behaved and non-trivial (i.e., doesn't go to negative inﬁnity or zero). In a similar manner, the

gain from the oil spot market for each Bank agent would be

pO ( x, ~y) =

j=1  un −ν

Ny (j)

1−uδ

Ny (j)

2.(6)

From these gains, Bank i (1 ≤i ≤N ) has:

•oil spot market payoﬀ function

f(i)

O=y ( i)

1M p O ,

•dollar futures market payoﬀ

f(i)

$=y (i)

2M p $ .

Adding these yields the payoﬀ function for Bank i below:

f(i) ( x, ~y) = f (i)

O+f ( i)

=y(i)

j=1  un −ν

Ny (j)

1−uδ

Ny (j)

2+

+y(i)

j=1 −u 2 n

Ny (j)

1+u(k− κ )

Ny (j)

2−uMmxy (i)

(7)

For brevity, we relabel the interaction and ﬁeld terms:

E:= M( un − ν)≥ 0 , D := M uδ ≥0 , K := M u2 n≥0 ,

8Bank agents are heterogeneous agents, since they can play diﬀerent strategies.

Volume I, Issue 1(1), Winter 2015

J:= M u( k− κ)≥ 0 , hx : =−uM mx ≤0.

A potential game (Monderer and Shapley 1996) with potential V (~q) and payoﬀ functions fi (~q),

~q = ( q1 , ..., qN ), for each agent i∈ Λ satisﬁes, by deﬁnition,

∂

∂qi

fi ( ~q) = ∂

∂qi

V( ~q) . (8)

The salient point is that, for each i, the gradient of the potential with respect to the variables

of agent iis the same as the gradient of the ith agent's payoﬀ (with respect the ith agent's variables).

Agents would follow the gradient of their payoﬀ function for "myopic decisions" (agents look at

the best local choice), and for potentials with an interior maximum, this would lead to the Nash

equilibrium (Monderer and Shapley 1996). In the model presented here, each bank agent has two

variables (y (i)

1, y (i)

2). The conditions (8) above for a potential require the "externality symmetry"

condition D =K , i.e., uδ =u2 n which is to say the negative correlation of the US dollar and oil

markets must have the same eﬀect on each other (accounting for u) for there to be a potential. If

this is the case, then the potential for the payoﬀ functions (7) is:

V( hx , ~y) =

i,j=1

2Ny (i)

1y (j)

i,j=1

2Ny (i)

2y (j)

2−K

i,j=1

y(i)

1y (j)

2−K

i=1

y(i)

1y (i)

i=1 hy ( i)

1i 2 +J

i=1 hy ( i )

2i 2 +h x

i=1

y(i)

(9)

For computations and dynamics, it will be easier to change variables from ~y to ~v ∈ΩΛ , where

for the agents i (1 ≤i ≤N ) in the set Λ,

v(i)

1= y (i)

2+y (i)

1/2, v (i)

2= y (i)

2−y (i)

1/2, (10)

with v(i)

α∈[− 1 / 2, 1/ 2], for α = 1, 2, the cartesian product

F(i) = [− 1 /2 , 1 / 2] × [ − 1/ 2 ,1 /2]

is a square, and

ΩΛ : =Y

i∈ Λ

F(i) .

The potential is then

V( hx , ~v ) =

i,j=1

I−

Nv (i)

1v (j)

i,j=1

I+

Nv (i)

2v (j)

2−2I

i,j=1

v(i)

1v (j)

2−2I

i=1

v(i)

1v (i)

+I−

i=1 hv ( i)

1i 2 +I +

i=1 hv ( i)

2i 2 +h x

i=1 v ( i)

1+v (i)

2

(11)

where

I:=( J− E) /2 , I+ :=( J+ E) /2 + K, I− :=( J+ E) /2− K. (12)

The stochastic dynamics are then given by the It¯o diﬀusion (Langevin) equations:

for 1 ≤i ≤ N, we see

dv (i)

1(t), dv (i)

2(t) = ∂

∂v (i)

f(i) ( ~v, t) dt + ν dw (i)

1(t) ,∂

∂v (i)

f(i) ( ~v, t) dt + ν dw (i)

2(t)! , (13)

Journal of Mathematical Economics and Finance

where w (i)

1(t) and w (i)

2(t) are zero-mean, unit-variance normal random variables from a Wiener pro-

cess, and ν is a variance parameter. Using the deﬁnition of a potential (8), we can rewrite relation

(13) above compactly as

d~v = ~

∇V dt + νd ~w( t) , (14)

with ~v = v (1)

1, v (1)

2,...,v (N)

1, v (N)

2,d ~w = dw (1)

1, dw (1)

2,...,dw (N)

1, dw (N)

2, and ~

∇V having compo-

nents consistent with (13).

Below, we develop the stochastic dynamical system that yields the Gibbs measure as the

equilibrium measure.

Proposition 1 Let ρ ( ~v ) be the joint density function over decision space ΩΛ for a potential game

with a ﬁnite number of agents N and potential V. Consider the dynamics9

d~v = ~

∇V dt + νd ~w ( t) , (15)

where ~v ∈ ΩΛ ,

∇V= ∂V /∂ v(1)

1, ∂V /∂v (1)

2, . . . , ∂V /∂v (N)

2

and the ~w a vector of 2 N standard Wiener processes which are identical and independent across

agents and time. Furthermore, the w (i)

α(α= 1 ,2), have mean zero and variance one and reﬂecting

boundary conditions10 are used.

If the process ~v ( t) satisﬁes the dynamics of (13), then the joint density satisﬁes the Fokker-

Planck equation

∂ρ( ~v, t)

∂t =− ~

∇ · [ ~

∇V( ~v( t )) ρ ( ~v, t)] + ν 2

2∇ 2 ρ (~v, t) (16)

and the corresponding equilibrium measure for variance ν2 is the Gibbs state

ρ( ~v, t) = ρ ( ~v) = exp  2

ν2 V(~v) 

RΩ Λ exp 2

ν2 V(~v0 ) d~v0 . (17)

In statistical mechanics, the term in the exponent of (17) is

−E( ~v)/( k T ),

where k is Boltzmann's constant, Tis temperature, and E (~v) is the energy of conﬁguration ~v.

Hence the analogy of a potential game to statistical mechanics is that ν2 (deviation from rationality;

inﬂuence of the noise in dynamics (13) ) is proportional to 'temperature' and the potential Vis the

negative 'energy' of the system (c.f. axiom 1 in Campbell 2005).

The end goal is to ﬁnd maximums of the potential (9), which is the appropriate reﬁnement

of the Nash equilibrium for a potential game (c.f., Carbonell-Nicolau and McLean 2014). We will

determine a form of the potential that will later facilitate this. The potential is quadratic in the

v(i)

α,1≤i ≤N, α = 1 ,2, and by continuity it will have a maximum on the domain ΩΛ, which may

occur in the interior or on the boundary depending on the parameters I , I+ , and I− . To this end,

the second-degree part of the potential is an 2N× 2N quadratic form Q equal to half of the

second-derivative matrix D2 V (1 ≤ i, j ≤ N ; 1 ≤ α, ¯ α≤2),

9Note that agents are only playing pure strategies in these dynamics.

10This requires zero time derivatives on the boundary, speciﬁcally that the last equation for stationary

states in Appendix A of Campbell 2005 be satisﬁed for boundary points ~v ∈ ∂ ΩΛ.

Volume I, Issue 1(1), Winter 2015

Q=1

2" ∂ 2 V

∂v (i)

α∂v ( j)

¯ α#=1

2I− I−I−I− 2 I I I I

I− 2I−I−I− I2 I I I

I−I− I I

I−I

I−I−I− 2I− I I I 2I

2I I I I 2I+I+I+I+

I2 I I I I+ 2 I+I+I+

I I I+ I+

I I+

I I I 2 I I+ I+I+ 2I+













(18)

where the upper-left quadrant of the matrix contains the α = ¯ α= 1 terms, the upper-right contains

the α = 1, ¯ α= 2 terms, the lower-left has α= 2 ,¯ α= 1 terms, and the lower-right has α = ¯ α= 2

terms. We will use the LDL∗ decomposition (Lis an invertible, lower-triangular matrix with ones

on the diagonal, L∗ is the transpose of L ,D is a diagonal matrix) of the symmetric quadratic form

Qof the potential Vin (18) to facilitate ﬁnding the maximum of the potential V. The matrix L

is determined from Qusing elementary row operations (EROs) as outlined in Beightler and Wilde

1966. The ﬁrst step is to reduce Qto upper triangular form using EROs, and then to determine D

and L , which is done in Appendix A.

Now that the quadratic form corresponding to the potential Vhas been diagonalized as

Q= LDL∗ , the potential can be written as inner products

V( ~v) = h ~v, Q~vi+ hx D ~

1, ~v E

=h~v, LD L∗ ~vi+ hx D ~

1, ( L∗ ) −1 L∗~v E

=hL∗ ~v , DL∗ ~v i+ hx D L−1 ~

1, L∗ ~v E,

(19)

where the column vector

1 = [1 1 ·· · 1]∗

has 2N rows. Using

v:= L∗~v,

the matrix D can be written as a direct sum of its N× N negative deﬁnite and positive deﬁnite parts

D= D− ⊕D+,

and the vector ~

vcan be decomposed into a direct sum over the subspaces corresponding to the

N-dimensional negative deﬁnite and positive deﬁnite parts of Das

v=~

v− ⊕ ~

v+ .

The same can be done for

M:= L−1 ~

to get

M= ~

M− ⊕ ~

M+.

This direct sum decomposition will split the inner products as

Journal of Mathematical Economics and Finance

V(~

v) =  (D− ⊕D+ )  ~

v− ⊕ ~

v+  ,~

v− ⊕ ~

v+  +hx D ~

M− ⊕ ~

M+ ,~

v− ⊕ ~

v+ E =

= D− ~

v− ,~

v−  +hx D ~

M− , ~

v− E + D+ ~

v+ ,~

v+  +hx D ~

M+ ,~

v+ E =

=−



|D− |1/2~

v− −h x

2|D− |−1/ 2 ~

M− 



+



D1/2

v+ +h x

2D −1/2

M+ 



+h 2

2



|D− | −1/ 2 ~

M− 



2−h 2

2



D−1/2

M+ 



(20)

where the last line is a result of completing the square,

|D− |:= −D− ≥ 0

as a matrix, and the square root of a diagonal matrix with non-negative entries dii is the matrix

having diagonal entries √ d ii.

4. Conclusions

A potential game for the speculative/hedging model (Carf`ı and Musolino 2014b) arises when

the externality symmetry condition is assumed; i.e., that currency and oil markets aﬀect each other

in a symmetric way. We have seen that the use of a potential is a powerful tool, insofar as it allowed

a systematic and tractable generalization to any number of banks, as well as to a bounded-rational

model. A bounded-rational model is a realistic assumption in the case when there is a large number

of bank players. This is because of the overwhelmingly large amounts of information to characterize

the state of all players at various times. It is reasonable to assume a single agent is making deci-

sions based on partial information, "intuition", etc., because of the impracticality of acquiring and

processing such large amounts of information. The next goal for this extended model is to ﬁnd the

Nash equilibrium.

Acknowledgments. I, Michael Campbell, am grateful to Stephen Goode for his introduction and

reference to David Carﬁ, as well as his mentorship in my early years, which pointed me to where I am

today. I also wish to acknowledge the inﬂuential mentorship, patience, and commitment of Harriet

Edwards, Greg Pierce, Vyron Klassen, and William Gearhart in my early years.

I, David Carf`ı, wish to thank Francesco Musolino and Emanuele Perrone for their helpful comments

and remarks.

Appendix A: Calculation of L∗ ,D , and ( L∗ ) −1

The matrices L∗ ,D , and (L∗ ) −1 from the LDL∗ decomposition of the quadratic form from the

potential V is needed to ﬁnd the maximum of V on its domain. To ﬁnd L∗ , we start by (1) reducing

the upper-left quadrant of Qand, (2) keeping track of the eﬀects on the upper-right quadrant. Then

we will (3) reduce the lower-left quadrant to a zero submatrix, and (4) keep track of the eﬀects on

the lower-right quadrant. Finally, (5) we will reduce the lower-right quadrant to upper-triangular

form. These ﬁve steps will be done with recursion relations that track the EROs (elementary row

operations).

For part (1) , the upper-left quadrant only consists of two distinct entries, diagonal entries

d(1) := 2 I− and non-diagonal entries e(1) : = I− . The entries below the diagonal in the upper-left

quadrant are all eventually reduced to zero, so we only need keep track of the diagonal and entries to

the right-of-diagonal. First we divide the ﬁrst row by d (1) . We do not alter the ﬁrst row any more,

and it will only be used to zero out the entries in the ﬁrst column in the upper-left quadrant, below

the diagonal. Then the entries for the ﬁrst row in L∗ are L∗

1, 1= 1 and

L∗

1,j =e (1) /d (1) = 1/2

and the ﬁrst diagonal entry of Dis

D1,1 = d (1) = 2I−.

Volume I, Issue 1(1), Winter 2015

The EROs to zero out the ﬁrst column will alter all the the entries below. Continuing this process

results in the recursion relations for the subsequent diagonal and right-of-diagonal entries for the

upper-left quadrant of Q :

d(k +1) =d(k) − e(k) e (k)

d(k) ,(21)

e(k +1) = e (k) −e(k) e (k)

d(k) ,(22)

where the ﬁnal entries for row k are d(k) on the diagonal and e (k) to the right of the diagonal. We

then divide row k by d (k) to get a diagonal entry L∗

k,k = 1, entries to the right of the diagonal

L∗

k,j =e ( k) , and zeros to the left of the diagonal. The diagonal entry for D is D k,k =d ( k) . These

recursion relations can be solved, by noting that

d(k) −e(k) = d (1) −e(1) = I −

and by using the forms

e(k +1) = e(k)  d (k) − e (k)

d(k)  = I−

e(k)

d(k) = I −

e(k)

e(k) + I−

=I−

1 + I −

e(k)

.(23)

The relation above can be scaled to

e(k) := e(k) /I−

and solved by iteration. This solution can be substituted into (21) to yield the solutions

d(k) = k + 1

kI − ,(24)

e(k) =1

kI − .(25)

For part (2) , we will track what part (1) did to the upper-right quadrant of Q , with

the recurrence relations for the diagonal and right-of-diagonal entries

δ(k +1) = δ(k) − e(k)  (k)

d(k) ,(26)

(k +1) =(k) − e(k)  (k)

d(k) ,(27)

respectively, where the initial values are δ (1) : = 2I and (1) : =I . Since the form of the upper-right

quadrant is symmetric to that of the upper-left quadrant, the upper-right quadrant is also reduced to

an upper-triangular form. This can be seen from the last (i.e., k+1) ERO on row kfor left-of-diagonal

elements in the upper-right quadrant:

0 = (k) −e(k) [δ (k) /d(k) ] ,

using the solutions (28), (29) below. The relations (26) and (27) can be solved using (24) and (25)

and then iterating:

δ(k) = k+ 1

kI, (28)

(k) =1

kI . (29)

For part (3) , the lower-left quadrant again consists of only two distinct entries, diagonal

entries ˜

δ(1) := 2 Iand non-diagonal entries ˜ (1) := I. All entries in the lower-left quadrant are all

eventually reduced to zero. However, we need keep track of the eﬀects on the lower-right quadrant,

which we will do in part (4) below. First we multiply row 1 by ˜

δ(1) and subtract that from row

N+ 1, the ﬁrst row of the lower-left quadrant. This results in row N+ 1 entries of the lower-left

Journal of Mathematical Economics and Finance

quadrant being zero (i.e., columns 1 to N). We do not alter row N+ 1 any more. We then go on to

use row 1 to get zeros in the remaining entries of column 1 (for rows N+ 2 to 2N). Continuing this

process results in the recursion relations for the subsequent diagonal and right-of-diagonal entries for

the lower-left quadrant of Q:

δ(k) =˜

δ(k−1) −˜ (k−1) e(k−1)

d(k−1) ,(30)

˜ (k) =  (k−1) − ˜ (k−1) e(k−1)

d(k−1) ,(31)

which again has solutions

δ(k) = k+ 1

kI, (32)

˜ (k) =1

kI. (33)

The last ERO on row N +k results in all zero entries for the lower-left quadrant of that row

(columns 1 to N ), and is indicated by

δ(k) =˜

δ(k) −˜

δ(k) d (k)

d(k) = 0 ,(34)

˜ (k) = ˜ (k) −˜

δ(k) e (k)

d(k) =1

kI− k+ 1

kI I − /k

(k + 1)I− /k = 0.(35)

Part (4), the eﬀects of what part (3) did to the lower-right quadrant of Q , is somewhat

more complicated since the entries in the upper-right quadrant do not have ones on the diagonal. As

such, we have to track the eﬀects of the zeroing out of the columns in step (3) on the left-of-diagonal

elements in the lower-right quadrant, ˜ c(k) . The lower-right quadrant entries

QN+j,N+(k −1) , k −1 < j ≤ N,

below the diagonal position N + (k− 1), N + (k− 1), all start oﬀ as ˜ e(k) . But after zeroing out column

k+ 1 in step (3) they become, say, ˜ c(k) . The resulting recurrence relations for the left-of-diagonal,

diagonal, and right-of-diagonal entries are then

˜ c(k) = ˜ e(k) − ˜ (k) δ (k)

d(k) ,(36)

d(k) =˜

d(k−1) −˜ (k−1)  (k−1)

d(k−1) ,(37)

˜ e(k) = ˜ e(k−1) − ˜ (k−1)  (k−1)

d(k−1) ,(38)

d(k) =˜

d(k) −˜

δ(k) δ (k)

d(k) , (39)

˜ e(k) = ˜ (k) − ˜

δ(k)  (k)

d(k) ,(40)

where (39) and (40) is the right-half of the last ERO done on row N +k (c.f., (34) and (35)). With

the initial values ˜

d(1) := 2 I+ and ˜ e(1) := I+ , the solutions to the above relations (36), (39) and (40)

are

˜ c(k) = I − I + − I 2

I−

,(41)

δ(k) = 2 I − I + − I 2

I−

,(42)

˜ (k) = I − I + − I 2

I−

.(43)

Volume I, Issue 1(1), Winter 2015

This is identical to the original form of the upper-left quadrant of Q.

Finally, for part (5) , we reduce the lower-right quadrant to upper-diagonal form in

the same manner that we did with part (1), since the lower-left quadrant has all zero entries at this

point. This results in the ﬁnal forms for L∗ and D in the LDL∗ decomposition

L∗ =

1 1/2 1/ 2 1/ 2 I/I−I / (2I− ) I /(2I− ) I/(2I− )

0 1 1/ 3 1/ 3 0 I/I−I/ (3I− ) I/(3I− )

0 0 1 1/ 4 0 0 I/I−I / (4I− )

0 0 0 1 0 0 0 I/I−

0 0 1 1/ 2 1/ 2 1/2

0 1 1/ 3 1/3

0 0 1 1/4

0 0 0 0 0 1













(44)

and the diagonal matrix

D= diag  2 I−

N,..., ( k+ 1)I−

kN ,..., ( N + 1) I −

NN , 2∆

N,..., ( k+ 1)∆

kN ,..., ( N + 1)∆

NN  , (45)

∆: = I− I+ − I2  /I− , (46)

where we will presently only consider the case ∆ >0 since the assumptions in Carf`ı and Musolino

2014b that J =E = 0 (as a result of the tax parameters κ =k and νtax =ν ) along with the

deﬁnitions in (12), result in ∆ = K > 0. Note, however, that we are not making the assumption that

I= 0.

Now we will ﬁnd (L∗ ) −1 using (44) and reducing the left side of [L∗ |I2N ] to the 2N× 2 N

identity matrix I2N . As before, this results in ERO recursion relations to reduce the left side. These

EROs are also applied to the right side, and result in (L−1 )∗ . With 1/ (k + 1) the right-of-diagonal

entries of row k, N + 1 ≤k≤ 2N of L∗ in (44), it is clear that the recursion relations to reduce row

kof the upper-left of L∗ are

R(j− k )

k=R ( j−k−1)

k−1

k+ 1 R #

j, k < j ≤ N , (47)

where we iterate (47) upward: k =N− 1, N − 2,..., 1 (i.e., start with row N− 1 and work up to

row 1), and R #

jindicates the ﬁnal form of row jafter all iterations reduce it (columns 1 to N) to a

row of the N× N identity IN .

Iterating (47) will reduce the upper-left quadrant of L∗ to IN , and it will reduce the upper-

right quadrant of L∗ to a multiple of the identity matrix: (I/I− ) IN . To reduce the lower-right

quadrant of L∗ to IN , we do the same procedure as with the upper-left quadrant:

R(j− k)

N+ k=R ( j−k−1)

N+ k−1

k+ 1 R #

N+ j, k < j ≤ N , (48)

for k =N− 1 , N − 2,..., 1. Finally, we zero out the diagonal upper-right quadrant of L∗ with

R##

k=R #

k−I

I−

N+ k,1≤k≤N, (49)

where R ##

kis the ﬁnal form of row kafter the upper-right quadrant of L ∗ is reduced to all zero

entries. Now we will iterate these recursion relations on the identity matrix; i.e., on the right side of

[L∗ |I2N ].

Journal of Mathematical Economics and Finance

Iterating (47), we see that

k=R ( N−k )

=R (N− k−1)

k−1

k+ 1 R #

=R(N− k−2)

k−1

k+ 1 R #

N−1−1

k+ 1 R #

N=·· · =

=R(1)

k−

j= k+1

k+ 1 R #

(50)

Using (50) for R #

k+1, we can eliminate all but one term in the summation with the diﬀerence

k−k+ 2

k+ 1 R #

k+1 =R (1)

k−k+ 2

k+ 1 R (1)

k+1 − 1

k+ 1 R #

k+1,(51)

which can be simpliﬁed to

k=R (1)

k−k+ 2

k+ 1 R (1)

k+1 +R #

k+1.(52)

Note that row N of L∗ in (44) is initially in reduced form, therefore R #

N=R (1)

N, and iterating

(52) to row Nresults in

k=R (1)

k−1

k+ 1 R (1)

k+1 − 1

k+ 2 R (1)

k+2 − · ·· − 1

NR (1)

N.(53)

Since the original form of row k ,R (1)

k, was that of row kof the identity matrix IN , we see

that (47) reduces the upper-left and lower-right quadrants of ( L∗ ) −1 to

k,j =R #

N+ k,N + j= 









0 1 ≤ j < k ≤ N,

1 1 ≤j =k ≤ N,

−1

k+ 1 1≤ k < j ≤ N.

(54)

Note that during the iteration of (47) on the upper-left quadrant of I2N , the upper-right

quadrant of I2N initially had all zero entries, thus the upper-right quadrant has all zero entries after

iterating EROs (47) and (48). The ﬁnal state of the upper-right quadrant of I2N is due to iterating

the N EROs in (49). The end result is

(L∗ )−1 = (L−1 )∗=

1−1

2− 1

3− 1

N− I

I−

2I−

3I−

NI−

0 1 − 1

3− 1

N0− I

I−

3I−

NI−

0 0 1 − 1

N0 0 − I

I−

NI−

0 0 0 1 0 0 0 − I

I−

0 0 1 − 1

2− 1

3− 1

0 1 − 1

3− 1

001 −1

0 0 0 0 0 1













(55)

Volume I, Issue 1(1), Winter 2015

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Carf`ı, D. and F. Lanzafame (2013). A Quantitative Model of Speculative Attack:

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Carf`ı, D. and F. Musolino (2011a). Fair Redistribution in Financial Markets: a Game

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Carf`ı, D. and F. Musolino (2011b). Game complete analysis for ﬁnancial markets stabilization. In

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Carf`ı, D. and F. Musolino (2012c). Game theory and speculation on government bonds. Economic

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Carf`ı, D. and F. Musolino (2012d). Game theory model for European government bonds market sta-

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Carf`ı, D. and F. Musolino (2012e). Game Theory Models for Derivative Contracts: Financial

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Carf`ı, D. and F. Musolino (2013b). Game theory application of Monti's proposal for European

government bonds stabilization. Applied Sciences 15, 43–70. http://www.mathem.pub.ro/apps/

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Carf`ı, D. and F. Musolino (2013c). Model of Possible Cooperation in Financial Markets in presence of

tax on Speculative Transactions. AAPP — Physical, Mathematical, and Natural Sciences 91 (1),

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Carf`ı, D. and F. Musolino (2015b). Tax Evasion: A Game Countermeasure. AAPP — Physical,

Mathematical, and Natural Sciences 93 (1), 1–17. https://dx.doi.org/10.1478/AAPP.931C2.

Carf`ı, D., F. Musolino, A. Ricciardello, and D. Schilir`o (2012). Preface: Introducing pisrs. AAPP —

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Carf`ı, D., G. Patan`e, and S. Pellegrino (2011a). Coopetitive games and sustainability in Project

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Carf`ı, D. and E. Perrone (2011a). Asymmetric Bertrand Duopoly: Game Complete Analysis by

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Volume I, Issue 1(1), Winter 2015

DOI: https://doi.org/10.14505/jmef.v1.1(1).02

The Easterlin threshold:

a ﬁrst glimpse

Francesco Strati

Department of Economics and Statistics,

University of Siena, Italy

francesco.strati@unisi.it

Abstract:

This work is supposed to introduce and set up a theoretical model which depicts the time dy-

namics of the relations between income and happiness. By using two distortions: the materialism

and the run eﬀect, the model conceives a happiness-income ratio as dependent on the locus of the

graph in which they are placed. Their positive or negative relations depend on the level of income:

for a level beyond the Easterlin threshold the happiness decreases, for a low level of income happiness

increases, while in the midst of these loci the level of happiness increases in average but marginal ly

decreases as income increases.

Keywords: Well-Being, Consumption, Dynamic Analysis, Life Satisfaction, Happiness.

JEL Classiﬁcation: I131, E21, C61.

1. Introduction

Is there any threshold beyond which the happiness decreases as income increases? Easterlin

1974 shows that an increase in income per-capita is not followed by an increase in happiness arising

in the so called Easterlin paradox. The present work is intended to be a very ﬁrst introduction to a

theoretical model about the relationship between income and happiness, in particular it is devoted to

study what I called: the Easterlin threshold. In what follows, I shall not take into account an absent

or negative relation between income and happiness paths at once, but I shall let it depend on the

level of income. Moreover, the problem faced here is about wealth which stems only from working

eﬀorts and can be modiﬁed only through them.

By exploiting a dynamic Ramsey model, I shall develop a theory for which income and hap-

piness move either towards the same direction or not, depending on the loci in which they will be

placed. In particular, for low level of income, its increase brings about a sudden increase in happi-

ness. Along this path the happiness over income ratio is marginally decreasing, but on the average

it increases until that path reaches the Easterlin threshold beyond which any increase in income

triggers a decrease in happiness. Of course this may be not true for a rich heiress who only enjoys

her richness. What the model is going to depict is fairly simple: if the eﬀort of obtaining a higher

amount of income tears down leisure and social connections (i.e. she works for too many hours) then

happiness comes to be lower. This means that the heiress above may nurture high level of social

connections and be very happy just because she is rich and she has to do nothing for producing it.

What the work is about is to model the path through which a human being has to work for a hour

in order to earn one euro (for the sake of clarity). The theory implies that if a worker wants to

become wealthier she has to work harder because the model takes into account horizontal income

comparisons, that is: workers belonging to the same level. It is then plain that the paper does not

concern a clerk who becomes manager, but a manager (or a clerk as well) who works harder so as to

earn more than before.

The present work can be seen as studying a horizontal happiness/income ratio for which the

comparisons are among the same working income levels (see Duesenberry 1949, for instance).

Journal of Mathematical Economics and Finance

2. The model

First of all it is of the utmost importance to deﬁne the relation between income and the

working eﬀort. It is assumed that 1eeuro of income is related to 1hworked hour, while 0eeuro of

income is related to 1lhour of leisure. This means that a worker may earn maximum 24ein a working

day, and a minimum of 0ein a leisure one which is much devoted to nurture social connections that

are supposed to increase happiness. The magnitude of this increase is depicted by the model and

hinges on the locus that the happiness and income ratio are placed on.

Let us think about happiness in an analytical fashion, and let us denote happiness by H , then:

H= f(( R, T , ˜

I)α , I1−α ) (1)

or in a more compact form

H= Sα I1−α .(2)

In Eq.(1) Rare the social connections, I means income, Ttrust in institutions and ˜

Ithe

horizontal (or perhaps vertical) social comparisons among income levels, or better: the income in

terms of goods bought by neighborhood. In the compact form of Eq.(2), Ssums up each term but

I, here Scan be thought of as a general social capital (albeit not exactly)1. The theory underpins

the relations among income-happiness-consumption-social capital, can be thought of as a well known

Ramsey model. By dividing Eq.(2) by I

I=  S

Iα

that is to say h = iα (3)

Moreover, i , from Eq.(3), can be seen as the social capital-income ratio (SCIR). Thus ican be

thought of as the intensity of the social capital with respect to the cold income measure. Its meaning

is plain if it is shown in motion by the ﬁrst derivative

f0 ( i) = αiα−1 = α

i1−α

that means: if igoes down, then the income motives are greater than non -income ones and vice

versa. By the famous Inada conditions

lim

i→0f 0 (i) = +∞ and lim

i→∞ f 0 (i) = 0

it can be said that the importance of income decreases if i goes up, that is Sgrows faster than I.

Moreover it is assumed, as usual, that the optimization process will use

dt = i α − c− δi and U ( c ) = Z ∞

eρi  c 1−σ

1−σ dt

It is interesting that δi is the part of the SCIR which is not exploited. Of course cis the

consumption and U ( c) the well known CRRA utility function.

Consumption is very important in the model, because the more cgoes up the more the

environment becomes consumerist and thus the SCIR goes down as can be seen later.

In order to ﬁnd an optimal and eﬃcient relationship between happiness and income motives,

it is interesting to employ a simple dynamic optimization model by using the hamiltonian (M)

M= c 1−σ

1−σ +λ[ iα − c− δi] (4)

so that the ﬁrst order conditions (FOC) with respect to consumption and SCIR are

Mc = 0 with λ = c−σ (5)

1Here and throughout this work, social capital is intended in a more general view. It is clear that social

connection may fairly diﬀer from the common notion of social capital. It is thus used as a general social box

so as of being separated from those growth-leading variables.

Volume I, Issue 1(1), Winter 2015

with the second derivative (where the circle means derived with respect to time)

λ=− σc−σ−1 ˚ c(6)

then, with respect to SCIR, by using the below general formula of dynamical optimization

Mi = ρλ − ˚

we obtain

λ[αiα−1 − δ ] = ρλ − ˚

˚ c

c=1

σ[ αi α−1 − ρ−δ](7)

where 1/σ is the elasticity of intertemporal substitution. Thus the two dynamics are depicted by

( ˚ c/c = 1 /σ [αiα−1 − ρ− δ ]

i= iα − c− δi (8)

Figure 1: Optimal relation between consumption and SCIR

and can be seen in the plain Fig.1.

For ˚ c= 0 it is obtained (by Eq.(7)) that

σαi α−1 =1

σδ+ρ

and then at ˚ c= 0 we have

i∗ = α

δ+ ρ 1

1+α

(9)

Furthermore, by the second equation of the dynamics and posing ˚

i= 0

c∗ = iα − δi (10)

The points are very straightforward by looking at Fig.1.

Journal of Mathematical Economics and Finance

The very interesting thing is the derivative of Eq.(10) with respect to i, which gives the relation

in motion

di = αi α−1 − δ= α

i1−α − δ (11)

Eq.(11) is really important for our aim: it states that if igoes up then dc/di goes down

and vice versa. Thus, for example, by improving social relations, ceteris paribus, the consumption

motives will be lowered (I assume that δis a constant). The eﬀect of an increase in ccaused by a

decrease in imay be called materialism eﬀect2 .

From an empirical point of view it is known now that His in a relationship with social

connections, conﬁdence in institutions, and reference income ( ˜

I). From 1974 to 2004 all indicators of

social connections and conﬁdence in institutions seem to have declined in U.S (Bartolini et al. 2013).

This means that the numerator of Eq.(3) has grown slower than the denominator I. In Bartolini

et al. 2013 U.S. people have faced up to an increase in income (I) to the detriment of mostly R , ˜

and T which in turn oﬀset the increase in I itself, hence: i is in the downward sloping locus. This

result is not well-forecasted by the only changes in household income, reference income, work status,

and demographic characteristic.

The dangerous decrease of social connections stems from what is summarized by Eq.(11),

rather it makes clear that a higher cmeans a decrease in S. By looking at Fig.1, after the (c∗ , i∗ )-

point (or Easterlin threshold), to reach higher happiness, cmust decrease so that it triggers an

increase in i which means an increase3 in S , faster than a possible increase in I.

In order to recap, the paradoxical tendency of the happiness trend in U.S., as far as Bartolini

et al. 2013 is concerned, is caused by

+ an increase in income;

– a decrease in social connections;

– a decline of trust in institutions;

– a strong reference income attitude.

3. The run eﬀect

Another eﬀect comes to be important and can be seen as how fast a man becomes materialist

from a neutral position (of course in the Eq.(11) meaning). I call this velocity: run eﬀect and can

be formalized (from Eq.(11)) as

L= ` α 2 − α

i2−α  (12)

Now, ` is generally greater than zero, until it approaches to 1 there is not run eﬀect, instead

there is a much more awareness of the time wasted and the more `→ 0 the more people want to

maximize their scarce time to enrich social connections. But, as ` > 1 there is a run eﬀect and

the numb attitude starts to increase by accelerating the materialism eﬀect (which is now given by

2In a more complete reasoning, ". . . Materialism consists in ascribing great importance in life to extrinsic

motivations and low priority to intrinsic motivations. The distinction between extrinsic and intrinsic moti-

vations refers, respectively, to the instrumentality or lack thereof of the motivations for doing something. In

fact, the term extrinsic refers to motivations that are external to an activity, such as money, while intrinsic

refers to internal motivations, such as friendship, solidarity, civic sense and the like. In short, individuals

who adopt materialistic values attribute a higher priority to goals such as money, consumption and success,

whereas they ascribe a limited priority to aﬀections, to relations in general and to pro-social behavior. . . "

(Bartolini 2013)

3In this case I am concerned with a ceteris paribus example given that only R moves

dS = dRα ( Tγ ˜

I1−α−γ )

of course an increase in Imust be slower than an increase in R.

Volume I, Issue 1(1), Winter 2015

the second derivative of Eq.(11) because we are interested in the acceleration c 00 and its velocity

`). Plain examples of run eﬀect stem from smartphone apps, social networking and bad acquisition

of information. Their eﬀects depend of course on the way they are used. For example, if social

networking isolates a man around a laptop, then his choices will become very elastic with respect to

the digital world's happenings. On the contrary a man who uses digital supplies as a mere device

for the real life, then the elasticity of his choices with respect to the digital world becomes almost

inelastic. The more the isolation around a device increases, the more a man becomes a lonely man

and thus the more social connections go down. In this case the materialism comes to be a perfect

substitute of social connections in order to reduce the feeling of loneliness. Moreover, so as to become

materialist one has to work harder and so forth.

It is clear that the above example is not the general one, but it could be seen as a benchmark

instance for lot of people out there.

4. Conclusions

The present work is intended to be an introduction to the issues discussed above. Surprisingly

enough, it isolates two important problems for the subjective well being (SWB): the materialism and

the run eﬀect. It is now clear how they shoot up the demand for higher incomes to the detriment

of SWB. Looking at a horizontal comparisons, same-level workers try to earn more in order to

compensate the lack of S-determinants by intensifying materialism which in turn eases the lack of S.

This can turn out to be a vicious cycle for SWB. Materialism eﬀect erodes S -determinants and the

run eﬀect makes this process faster, in this case materialism is a disguise of a lack of happiness.

Further works are still in process in order to enrich the model sketched throughout this paper.

References

Bartolini, S. (2013). Manifesto per la felicit´a: Come passare dalla societ´a del ben-avere a quella del

ben-essere. Feltrinelli.

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Americans' happiness? Social Indicators Research 110 (3), 1033–1059.

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Economics 79, 123–144.

Brickman, P. and D. T. Campbell (1971). Hedonic relativism and planning the good society. In H. A.

Mortimer (Ed.), Adaptation level theory: A symposium. New York Academic Press.

Duesenberry, J. S. (1949). Income, savings and the theory of consumer behaviour. Cambridge and

London: Harvard University Press.

Easterlin, R. A. (1974). Does economic growth improve the human lot? Some empirical evidence. In

R. David and M. Reder (Eds.), Nations and households in economic growth: Essays in honor of

Moses Abramovitz, pp. 89–125. New York Academic Press.

Easterlin, R. A. (1995). Will raising the incomes of all increase the happiness of all? Journal of

Economic Behavior and Organization 27, 35–47.

Easterlin, R. A. (2005). Feeding the illusion of growth and happiness: A reply to Hagerty and

Veenhoven. Social Indicators Research 74, 429–443.

Ferrer-i-Carbonell, A. (2005). Income and well-being: An empirical analysis of the comparision

income eﬀect. Journal of Public Economics 89 (5-6), 997–1019.

Proto, E. and A. Rustichini (2013). A reassessment of the relationship between GDP and life satis-

faction. PLoS ONE 8 (11).

Proto, E. and A. Rustichini (2015). Life satisfaction, income and personality. Journal of Economic

Psychology 48.

Journal of Mathematical Economics and Finance

DOI: https://doi.org/10.14505/jmef.v1.1(1).03

Effectiveness and Efﬁciency Trade-Off in the Demerit Goods Taxation: a

Non-Standard Approach

Francesco Musolino

Viale Regina Elena 313, 98121 Messina, Italy

Abstract:

The aim of this paper is to study a ﬁrst approach to the trade-off relation between effectiveness and efﬁ-

ciency about the intervention of the State to protect merit goods, in presence of a constant consumer's marginal

utility. Moreover, we focus on how efﬁciency and effectiveness vary according to proportional and progressive

taxes.

Keywords: Merit Goods; Demerit Goods; Tax; Efﬁciency; Effectiveness; Political Economics.

JEL Classiﬁcation: C0, H2, H3.

1. Introduction

With our paper, we analyze the effects of the introduction of a tax by the State on behaviors adverse to

merit goods. Precisely, we compare effectiveness and efﬁciency of proportional and progressive taxes.

What are merit goods?. They are those goods or services at which the community gives a particular

importance for moral and social development of the community itself: (education, health, theater performances

etc). The state often satisﬁes these needs, not based on a speciﬁc demand, but as result of the evaluation of the

advantages that the entire community obtains. Other times, the State gets involved in prohibiting certain behav-

iors, such as smoking in public places.

What is the effectiveness? A tax that tends to limit a certain behavior adverse to merit goods is effective

if it avoids this behavior. In this regard, the tax effectiveness is a value 0 ≤E ≤ 1, which is

•1 when, because of the tax, the consumer do not adopt the adverse behavior;

•0 if the consumer, despite the tax, continues to adopt the adverse behavior in the same way as he would

do without tax.

Formally, the effectiveness E (q), at a certain behavior level q (with q∈ [ 0, q max ] ), is the difference be-

tween maximum possible adverse behavior qmax and the behavior qadopted by the consumer, divided by maxi-

mum possible adverse behavior qmax . We have

E( q) = q max − q

qmax

=1−q

qmax

,(1)

for every q∈ [ 0, qmax ] .

What is the efﬁciency? A tax is efﬁcient if, with the introduction of the tax, there is no loss of global

wealth than global wealth without tax. In this regard, tax efﬁciency is a value 0 ≤I≤1, which is

•0 if there is a total loss of global wealth;

•1 when the global wealth does not change.

Volume I, Issue 1(1), Winter 2015

Formally, the efﬁciency at q is the ratio between the global utility Ut (q ) with tax and the maximum global

utility

max

q∈[ 0, qmax ] U

without tax. We have

I( q) = U t ( q)

maxq∈[0 , qmax ] U (2)

for every q∈ [ 0, qmax ] .

2. Literature review

In this paper, we shall refer to various economic literature. First of all, we refer to literature about

Public Economics and its insights of welfare policy (see Besanko and Braeutigam 2005; Bosi 2010; Musgrave

1959; Head 1974; Pigou 1920). Moreover, our study follows a general streak of mathematical and Game Theory

methodologies whose roots can be understood and considered reading some papers and articles about Game

Theory and Decision Theory, written since 2008 by Carfì (see Carfì 2008a, 2009c,f,b,a,e, 2010a,b,c, 2012) and

by Carfì et al. (see Baglieri et al. 2010, 2012; Carfì et al. 2010; Carfì and Ricciardello 2009, 2010, 2011,

2012a,c,e,g,d,h,b,i,f,j, 2013b,a; Agreste et al. 2012; Carfì and Fici 2012; Carfì and Perrone 2012b,a, 2013; Carfì

and Pintaudi 2012; Carfì and Schilirò 2012c,b,a,d, 2013; Carfì et al. 2013; Carfì and Lanzafame 2013). Moreover,

interesting perspectives on possible future developments of this lane of study could be devised by the integration

among:

•the applications of methodologies illustrated by Carfì and coauthors (see Carfì et al. 2011; Carfì and

Perrone 2011a,b,c, 2012b,a, 2013; Carfì and Schilirò 2011a,c,b, 2012c,b,a,d, 2013, 2014a,b; Carfì and

Trunﬁo 2011; Agreste et al. 2012; Baglieri et al. 2012, 2016; Carfì 2012; Carfì and Fici 2012; Carfì and

Pintaudi 2012; Carfì and Ricciardello 2012a,c, 2013b,a; Carfì et al. 2013; Carfì and Lanzafame 2013;

Okura and Carfì 2014; Arthanari et al. 2015; Carfì and Romeo 2015), by Carfì and Musolino (seeCarfì

and Musolino 2011b,a, 2012a,b,c, 2013a,b,c, 2014b,a, 2015a,b) and by Musolino (see Musolino 2012);

•the mathematical-ﬁnancial researches developed in Carfì 2004a,d,c,b, 2006b,a, 2007b,a, 2008c,b, 2009d,

2011; Carfì and Caristi 2008; Carfì and Cvetko-Vah 2011;

•the results we are going to show in this paper.

In fact, by considering a generic economic trouble (ﬁnancial speculation or tax evasion, for example) as

demerit goods, the idea explained in the present paper could be implemented by adopting the above literature, for

further supports and developments in the research ﬁeld of economic policy - about efﬁciency and effectiveness

of the countermeasures adopted by the State (or by any subject we are interested) in order to obtain its economic

policy goals.

3. Model description

We suppose that the consumer adopts a behavior that endangers merit goods (buying cigarettes or making

speculation) and he has a linear and increasing utility function Uc , equal to

Uc ( q) = mq,

where:

•0≤ q≤ qmax represents the quantity of adverse behavior that the consumer adopts. Assuming qmax = 1,

we can consider qas a percentage of the maximum quantity;

•m>0 is a coefﬁcient that measures the increase in utility due to the increase of one unit of q.

Remark. We assume that the consumer always acts to maximize his utility function.

The State, in order to protect merit goods, decides to introduce a tax T (q ) on the adoption of adverse

behavior. The consumer's utility function Uc (q ) becomes

Uc ( q) = mq − T ( q). (3)

Journal of Mathematical Economics and Finance

The utility function Us (q ) of the State is

Us ( q) = T (q). (4)

The global utility function Ut is equal to the sum of Uc and Us . We have

Ut ( q) = Uc (q) + Us (q) = mq − T ( q) + T (q) = mq. (5)

4. Trade-off between effectiveness and efﬁciency

Intuitively, we immediately note an inverse trend between effectiveness and efﬁciency.

Proposition. Let I( q) and E ( q) be respectively the efﬁciency and the effectiveness of a tax on demerit

goods. Then,

E( q) + I (q) = 1.

Proof. We substitute Eq.(5) into Eq.(2) and we obtain

I( q) = mq

mqmax

qmax

,(6)

where q is the quantity of adopted adverse behavior.

Recalling Eq.(6) and (1), we have

E( q) = 1− I (q) or I (q) = 1− E( q). (7)

This completes the proof. 

In the following sections we introduce new variables inﬂuencing the tax T (q), by depending on the type

of tax we study. So, we pass from a generic one variable tax T (q ) to a particular two variable tax T (q,a).

5. Proportional tax

We assume that a function tax

T:[ 0,1]× [ 0,1]→R ,

introduced by State is proportional to utility obtained by the adverse behavior. We have

T( q,a) = amq, (8)

where 0 <a ≤ 1 is the tax rate (if a> 1, it would be expropriation).

And substituting Eq.(8) in equations (3) and (4):

Uc ( q,a) = m (1− a) q and Us ( q,a) = amq.

We ﬁnd the value q∗ maximizing the consumer's utility Uc (q,a ) (for ﬁxed a):

∂Uc

∂q( q,a) = m (1− a),

that is positive if a< 1, and therefore the function is increasing for every q∈ [ 0,1] . So, given a proportional tax

with 0 <a < 1, the consumer adopts the quantity of adverse behavior q∗ =qmax = 1.

Proportional tax effectiveness. The proportional tax is totally ineffective. Indeed, recalling Eq.(1), we

have

E( q∗ ) = 1− q ∗

qmax

=1− 1 =0,

Volume I, Issue 1(1), Winter 2015

where q∗ is the quantity of adverse behavior adopted by consumer.

Proportional tax efﬁciency. Recalling Eq.(7), the proportional tax is maximally efﬁcient:

I( q∗ ) = 1− E ( q∗ ) = 1 .

Revenue from the proportional tax. Since q∗ = 1, the tax revenue for the State is

Us ( q∗ ,a) = amq∗ =am. (9)

Remark. If a= 1, consumer chooses independently any quantity q∗ , because his utility is constant

(Uc ( q, 1) = 0 for every q).

6. Progressive tax

We assume that tax

T:[ 0,1]×]1, +∞[→R ,

introduced by State is progressive and its marginal rate increases for increasing levels of q, up to 1 in correspon-

dence of qmax . Therefore

T( q,y) = mqy , (10)

where y> 1 (if y< 1 , it would be expropriation by the State).

Substituting Eq.(10) in equations (3) and (4), we have

Uc ( q, y) = mq(1− qy−1 )

and

Us ( q, y) = mqy . (11)

We ﬁnd the value q∗ maximizing the consumer's utility Uc (q,y):

∂Uc

∂q( q,y) = m(1− y)qy−1 ,

that is positive if

q< y1/( 1−y ).

The value y 1/( 1−y) is lower than 1 for every y> 1 and therefore the function is increasing up to y 1/( 1−y )

and after decreases. The maximum is

q∗ ( y) = y1/( 1−y) . (12)

Remark. If a= 1, the consumer chooses independently any quantity q∗ , because his utility is constant

(Uc ( q, 1 ) = 0 for every q).

Progressive tax effectiveness. Recalling Eq.(1), we have

E( y) = 1− q ∗ ( y)

qmax

=1−y 1/( 1−y) (13)

Progressive tax efﬁciency. Recalling Eq.(7), the progressive tax efﬁciency is

I( y) = 1− E (y) = y1/( 1−y) (14)

Analysis of progressive tax effectiveness and efﬁciency. Progressive tax effectiveness and efﬁciency

depend on y (see Eq.(13) and (14)).

Revenue from progressive tax. Recalling Eq.(11) and (12), the revenue for the State is

Us ( y) = m (q∗ (y)) y =myy/( 1−y) (15)

Journal of Mathematical Economics and Finance

Multiplying Eq.(15) by

1=q ∗ (y)qmax

q∗ (y)qmax

we obtain

Us ( y) = m ( q∗ (y)) y−1 qmax

q∗ (y)

qmax

Recalling Eq.(12) and that

I( y) = q ∗ ( y)

qmax

we have

Us ( y) = mq max

yI(y). (16)

Analysis of progressive tax revenue. The value y, inversely proportional to efﬁciency, is also inversely

proportional to revenue of the State (see Eq.(16)). If the State wants to obtain with progressive tax the same

revenue of proportional tax, it has to choose the variable yto satisfy the relation putting Eq.(9) equal to Eq.(15).

We obtain

a= yy/( 1−y) .

Progressive tax maximum effectiveness. We maximize the progressive tax effectiveness

E( y) = 1− ( y1/(1− y ) ).

We know that

∇(fg ) = fg  g0 ln f+ gf 0

f ,

and so, putting

M( y) = y1/( 1−y) ,

we have

∇M (y ) = y1/( 1−y)  ln y

(1− y)2 + 1

y(1− y) . (17)

This derivative is positive if

ln y

(1− y)2 + 1

y(1− y)>0,

and multiplying by ( 1− y) , that is always negative because y> 1, we obtain

ln y

1−y+ 1

y<0,

that is equivalently to

ln y>− 1−y

that is

ln y> 1− 1

As we see in the following Fig.1, the function ln is greater than the function

y7→ 1 − 1

after the point y 0 =1, therefore the derivative ∇M (y ) is always positive (for y> 1) and the function

E=1− M

should reveal decreasing in ] 1, +∞[.

Volume I, Issue 1(1), Winter 2015

Figure 1: Graphical representation of ln y and 1 − ( 1 /y) with y> 1.

Concluding:

•sup E< 1< +∞

•sup E= limy→1+ E ( y) .

Putting ε> 0 a small value ad lib and substituting y= 1+ε in Eq.(13) we have:

sup E≈ E ( 1+ε ) = 1− 1

(1 +ε )1/ε .

By choosing ε= 10−10 , we obtain

sup E≈ 63.2%.

Consequently (see Eq.(7)), the minimum efﬁciency of the progressive tax is

inf I= 1− sup E≈ 1− 63. 2% = 36.8%.

Consequently (see Eq.(7)), the minimum efﬁciency of the progressive tax is

inf I= lim

ε→0

(1 +ε )1/ε = 1

e≈36.8%.

7. Conclusions

In this paper we address the introduction of a tax to project merit goods. We showed that:

1. there is a trade-off relation between effectiveness and efﬁciency, that is

Effectiveness = 1− ( Efﬁciency);

2. a proportional tax

T( a,q) = aq,

with tax rate a< 1 chosen by State, is maximally efﬁcient (there are not losses of global utility), but is

totally ineffective;

Journal of Mathematical Economics and Finance

3. a progressive tax of type

T( q, y) = qy ,

with y chosen by State has:

(a) effectiveness and revenue inversely proportional to y;

(b) efﬁciency directly proportional to y;

4. the progressive tax has

(a) 0% ≤ Effectiveness ≤ 63.2%;

(b) 36. 8% ≤ Efﬁciency ≤100%;

5. by introducing a progressive tax, the State can:

(a) adopt value y to achieve its objectives of effectiveness and efﬁciency;

(b) calculate in advance the tax revenue, according to relation

Us ( y) = mq max

yI(y ) = myy/( 1−y) .

Acknowledgments. The author wishes to thank Dr. Prof. David Carfì for his help in the formulation

of the model and its resolution. Moreover, the author wishes to thank two anonymous referees for their useful

suggestions and comments.

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Journal of Mathematical Economics and Finance

DOI: https://doi.org/10.14505/jmef.v1.1(1).04

A model for coopetitive games

David Carf`ı

Department of Mathematics, University of California Riverside, USA

Department of Economics, California State University Fullerton, USA

david.carfi@ucr.edu

davidcarfi@gmail.com

Abstract:

In the present introductory work we propose a survey of an original analytical model of coopet-

itive game, conceived and introduced in 2009 by the author himself. Much of the material presented

here has been already published in their complex, at diﬀerent stages of development, in numerous pa-

pers, during the last 5 years. Here we explain the present state of the theory, in a virtually complete,

organized and self-contained version. We also suggest here - after the presentation of the model -

general types of feasible solutions - again in a coopetitive perspective - in the form of sophisticated

bargaining solutions (in a rational decision theory context) viewed as reasonable mediations among

the partially diverging interests driving the players of the coopetitive games themselves.

Keywords: Games and Economics, competition, cooperation, coopetition, normal form games,

games in Management.

JEL Classiﬁcation: C7,C61,C70,C72,C78,C79.

1. Introduction

In this paper, we develop and exemplify the mathematical model of a coopetitive game intro-

duced by David Carf`ı in Carf`ı and Schilir`o 2011f; Carf`ı 2010a and already applied to economics and

Finance by Carf`ı et al. (see Baglieri et al. 2010, 2012a,b; Carf`ı et al. 2010a,b; Carf`ı and Ricciardello

2010, 2011, 2012a,c,e,g,d,h,b,i,f,j, 2013b,a; Agreste et al. 2012; Carf`ı and Fici 2012a,b; Carf`ı and

Perrone 2012b,a, 2013; Carf`ı and Pintaudi 2012a,b; Carf`ı and Schilir`o 2012d,b,c,a,e,f, 2013a,b; Carf`ı

et al. 2011, 2013; Carf`ı and Lanzafame 2013). The idea of coopetitive game is already presented

and used, in a mostly intuitive and non-formalized way, in Strategic Management Studies (see for

example Brandenburger and Nalebuﬀ 1995, Stiles 2001, Bouncken et al. 2015). Here, we propose a

survey of the new original analytical model of coopetitive game, conceived by Carf`ı and employed

by him and coauthors in 2009 and later. The need of a precise and quantitative mathematical def-

inition of a coopetitive game appears very strong in the applications to economics and ﬁnance and

management, especially because, after the preliminary qualitative analysis, economics needs a quan-

titative, previsional or checkable mathematical analysis. Much of the material presented here has

been already published in their complex, at diﬀerent stages of development, in numerous papers,

during the last 5 years, by Carf`ı and Musolino (Carf`ı and Musolino 2011b,a, 2012c,a,e,b,d, 2013a,b,c,

2014b,a, 2015a,b) and by Carf`ı and coauthors (Carf`ı et al. 2011a,b; Carf`ı and Perrone 2011a,b,c,d,e,

2012b,a, 2013; Carf`ı and Schilir`o 2010, 2011b,a,c,d,f,e, 2012d,b,c,a,e,f, 2013a,b, 2014a,b,c; Carf`ı and

Trunﬁo 2011a,b; Agreste et al. 2012; Baglieri et al. 2012a, 2015; Carf`ı 2012; Carf`ı and Fici 2012a,b;

Carf`ı and Pintaudi 2012a,b; Carf`ı and Ricciardello 2012a,c, 2013b,a; Carf`ı et al. 2011, 2013; Carf`ı

and Lanzafame 2013; Okura and Carf`ı 2014; Arthanari et al. 2015; Carf`ı and Romeo 2015; Carf`ı and

Gambarelli 2015). Here we lay out the present state of the theory, in a virtually complete, organized

and self-contained version together with new possible future developments and considerations. We

also suggest here - after the presentation of the deﬁnitions regarding the basics of coopetitive games -

general types of feasible solutions for the model itself - again in a coopetitive perspective - in the form

of very sophisticated bargaining solutions (in a rational decision theory context) viewed as reasonable

Volume I, Issue 1(1), Winter 2015

mediations among the partially diverging interests driving the game players.

2. Organization of the paper

The work is organized as follows:

•section 3 presents the literature review;

•section 4 presents the general idea of the paper;

•section 5 presents the original model of coopetitive game introduced in the literature by D.

Carf`ı;

•section 6 proposes possible solutions concepts for the original model of coopetitive game;

•section 7 presents a dynamical interpretation of the coopetitive game model;

•section 8 provides a ﬁrst sample of coopetitive game in an intentionally simpliﬁed fashion

(without direct strategic interactions among players) to emphasize the new role and procedures

of coopetition;

•section 9 provides a second sample of coopetitive game, showing possible coopetitive solutions;

we propose a linear model, with a direct strategic interactions among players;

•conclusions end up the paper.

The concept of coopetition was essentially devised at micro-economic level for strategic man-

agement solutions (see Brandenburger and Nalebuﬀ 1995), who suggest, given the competitive

paradigm (see Porter 1985), to consider also a cooperative behavior to achieve a win-win outcome

for both players.

3. Literature review

In this paper, we shall refer to a wide variety of literature. First of all, we shall consider some

papers on the complete study of diﬀerentiable games and related mathematical backgrounds, intro-

duced and applied to economic theories since 2008 by Carf`ı (see Carf`ı 2008a, 2009b,c,g,a,f, 2010a,b,c,

2012) and by Carf`ı et al. (see Baglieri et al. 2010, 2012a; Carf`ı and Magaudda 2009; Carf`ı et al.

2010a,b; Carf`ı and Ricciardello 2009, 2010, 2011, 2012a,c,e,g,d,h,b,i,f,j, 2013b,a; Agreste et al. 2012;

Carf`ı and Fici 2012a,b; Carf`ı and Perrone 2012b,a, 2013; Carf`ı and Pintaudi 2012a,b; Carf`ı and

Schilir`o 2012d,b,c,f,a,e, 2013a,b; Carf`ı et al. 2011, 2013; Carf`ı and Lanzafame 2013). Speciﬁc applica-

tions of the previous methodologies, also strictly related to the present model, have been illustrated

by Carf`ı and Musolino (see Carf`ı and Musolino 2011b,a, 2012a,c,e,b,d, 2013a,b,c, 2014b,a, 2015a,b).

Other important applications of the complete examination methodology were introduced by Carf`ı

and coauthors (see Carf`ı et al. 2011a,b; Carf`ı and Perrone 2011a,b,c,d,e, 2013, 2012b,a; Carf`ı and

Schilir`o 2010, 2011b,a,c,d,f,e, 2012d,b,c,a,e,f, 2013a,b, 2014a,b,c; Carf`ı and Trunﬁo 2011a,b; Agreste

et al. 2012; Baglieri et al. 2012a,b, 2015; Carf`ı 2012; Carf`ı and Fici 2012a,b; Carf`ı and Pintaudi

2012a,b; Carf`ı and Ricciardello 2012a,c, 2013b,a; Carf`ı et al. 2013; Carf`ı and Lanzafame 2013; Okura

and Carf`ı 2014; Arthanari et al. 2015; Carf`ı and Romeo 2015). General ideas on the possible future

applications of the methodologies introduced in the previous works could be devised under the view

of the following researches (see Carf`ı (2004a,d,c,b, 2006b,a, 2007b,a, 2008c,b, 2009d,e, 2011d,b,c,e,a);

Carf`ı and Caristi (2008); Carf`ı and Cvetko-Vah (2011)).

4. The idea

A coopetitive game is a game in which two or more players (participants) can interact co-

operatively and non-cooperatively at the same time. Even Brandenburger and Nalebuﬀ, creators of

coopetition, did not deﬁne, precisely, a quantitative way to implement coopetition in the Game Theory

context.

The problem to implement the notion of coopetition in Game Theory is summarized in the

following question:

•how do, in normal form games, cooperative and non-cooperative interactions coexist simulta-

neously, in a Brandenburger-Nalebuﬀ sense?

Journal of Mathematical Economics and Finance

In order to explain the above question, consider a classic two-player normal-form gain game

G= ( f, >) - such a game is a pair in which f is a vector valued function deﬁned on a Cartesian

product E× F with values in the Euclidean plane R2 and > is the natural strict sup-order of the

Euclidean plane itself (the sup-order is indicating that the game, with payoﬀ function f, is a gain

game and not a loss game). Let E and F be the strategy sets of the two players in the game G . The

two players can choose the respective strategies x∈ E and y∈ F

•cooperatively (exchanging information and making binding agreements);

•not-cooperatively (not exchanging information or exchanging information but without possi-

bility to make binding agreements).

The above two behavioral ways are mutually exclusive, at least in normal-form games:

•the two ways cannot be adopted simultaneously in the model of normal-form game (without

using convex probability mixtures, but this is not the way suggested by Brandenburger and

Nalebuﬀ in their approach);

•there is no room, in the classic normal form game model, for a simultaneous (non-probabilistic)

employment of the two behavioral extremes cooperation and non-cooperation.

4.1 Towards a possible solution

David Carf`ı (Carf`ı and Schilir`o 2011f and Carf`ı 2010a) has proposed a manner to overcome

this impasse , according to the idea of coopetition in the sense of Brandenburger and Nalebuﬀ. In a

Carf`ı's coopetitive game model,

•the players of the game have their respective strategy-sets (in which they can choose coopera-

tively or not cooperatively);

•there is a common strategy set C containing other strategies (possibly of diﬀerent type with

respect to those in the respective classic strategy sets) that must be chosen cooperatively;

•the strategy set C can also be structured as a Cartesian product (similarly to the proﬁle

strategy space of normal form games), but in any case the strategies belonging to this new set

Cmust be chosen cooperatively.

5. Coopetitive games

5.1 The model for n -players

We give in the following the deﬁnition of coopetitive game proposed by Carf`ı (in Carf`ı and

Schilir`o 2011f and Carf`ı 2010a).

Deﬁnition (of n-player coopetitive game). Let E = (Ei ) n

i=1 be a ﬁnite n-family of non-

empty sets and let Cbe another non-empty set. We deﬁne n -player coopetitive gain game over

the strategy support (E, C ) any pair G = (f, > ) , where f is a vector function from the Cartesian

product × E× C (here × E denotes the classic strategy-proﬁle space of n-player normal form games,

i.e. the Cartesian product of the family E ) into the n -dimensional Euclidean space Rn and > is the

natural sup-order of this last Euclidean space. The element of the set C will be called cooperative

strategies of the game.

A particular aspect of our coopetitive game model is that any coopetitive game Gdetermines

univocally a family of classic normal-form games and vice versa; so that any coopetitive game could

be deﬁned as a family of normal-form games. In what follows we make precise this very important

aspect of the model.

Deﬁnition (the family of normal-form games associated with a coopetitive game).

Let G = (f, > ) be a coopetitive game over a strategic support ( E , C) . And let

g= ( gz )z∈C

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be the family of classic normal-form games whose member gz is, for any cooperative strategy z in C,

the normal-form game

Gz := ( f ( ., z), >),

where the payoﬀ function f( ., z ) is the section

f( ., z) : × E→R n

of the function f , deﬁned (as usual) by

f( ., z)( x) = f( x, z ),

for every point xin the strategy proﬁle space × E. We call the family g(so deﬁned) family of

normal-form games associated with (or determined by) the game G and we call normal

section of the game G any member of the family g.

We can prove this (obvious) theorem.

Theorem. The family gof normal-form games associated with a coopetitive game Guniquely

determines the game. In more rigorous and complete terms, the correspondence G 7→ g is a bijection

of the space of all coopetitive games - over the strategy support ( E, C ) - onto the space of all families

of normal form games - over the strategy support E- indexed by the set C.

Proof. This depends totally on the fact that we have the following natural bijection between

function spaces:

F(× E ×C, Rn ) → F ( C, F(× E , Rn )) : f 7→ (f( ., z))z∈C ,

which is a classic result of theory of sets. 

Thus, the examination of a coopetitive game should be equivalent to the examination of a

whole family of normal-form games (in some sense we shall specify).

In this paper we suggest how this latter examination can be conducted, as well as the solu-

tions corresponding to the main concepts of solution, which are known in the literature as the classic

normal-form games, in the case of two-player coopetitive games.

5.2 Two players coopetitive games

In this section we specify the deﬁnition and related concepts of two-player coopetitive games;

sometimes (for completeness) we shall repeat some deﬁnitions of the preceding section.

Deﬁnition (of coopetitive game). Let E ,F and C be three nonempty sets. We deﬁne

two player coopetitive gain game carried by the strategic triple (E, F , C ) any pair of the form

G= ( f, >) , where f is a function from the Cartesian product E× F× C into the real Euclidean plane

R2 and the binary relation >is the usual sup-order of the Cartesian plane (deﬁned component-wise,

for every couple of points p and q , by p > q iﬀ pi > qi , for each index i).

Remark (coopetitive games and normal form games). The diﬀerence between a two-

player normal-form (gain) game and a two player coopetitive (gain) game is the fundamental presence

of the third strategy Cartesian-factor C. The presence of this third set Cdetermines a total change

of perspective with respect to the usual exam of two-player normal form games, since we now have

to consider a normal form game G (z ), for every element z of the set C ; we have, then, to study an

entire ordered family of normal form games in its own totality, and we have to deﬁne a new manner

to study these kinds of game families.

5.3 Terminology and notation

Deﬁnitions. Let G = (f, > ) be a two player coopetitive gain game carried by the strategic

triple ( E, F, C) . We will use the following terminologies:

Journal of Mathematical Economics and Finance

•the function fis called the payoﬀ function of the game G ;

•the ﬁrst component f1 of the payoﬀ function f is called payoﬀ function of the ﬁrst player

and analogously the second component f2 is called payoﬀ function of the second player;

•the set E is said strategy set of the ﬁrst player and the set F the strategy set of the

second player;

•the set C is said the cooperative (or common) strategy set of the two players;

•the Cartesian product E× F× C is called the (coopetitive) strategy space of the game G.

Memento. The ﬁrst component f1 of the payoﬀ function fof a coopetitive game Gis the

function of the strategy space E× F× C of the game G into the real line Rdeﬁned by the ﬁrst

projection

f1 ( x, y, z) := pr1 ( f( x, y , z)),

for every strategic triple (x, y, z ) in E× F× C ; in a similar fashion we proceed for the second com-

ponent f2 of the function f.

Interpretation. We have:

•two players, or better an ordered pair (1, 2) of players;

•any one of the two players has a strategy set in which to choose freely his own strategy;

•the two players can/should cooperatively choose strategies zin a third common strategy set C;

•the two players will choose (after the exam of the entire game G) their cooperative strategy z

in order to maximize (in some sense we shall deﬁne) the vector gain function f.

5.4 Normal form games of a coopetitive game

Let G be a coopetitive game in the sense of above deﬁnitions. For any cooperative strategy z

selected in the cooperative strategy space C , there is a corresponding normal form gain game

Gz = ( p( z ) , >),

upon the strategy pair (E , F ), where the payoﬀ function p (z ) is the section

f( ., z) : E× F→ R2 ,

of the payoﬀ function fof the coopetitive game - the section is deﬁned, as usual, on the competitive

strategy space E× F , by

f( ., z)(x, y) = f( x, y , z),

for every bi-strategy (x, y ) in the bi-strategy space E× F .

Let us formalize the concept of game-family associated with a coopetitive game.

Deﬁnition (the family associated with a coopetitive game). Let G = (f , > ) be a two

player coopetitive gain game carried by the strategic triple (E , F, C ) . We naturally can associate with

the game Ga family g = (gz )z∈C of normal-form games deﬁned by

gz := Gz = ( f ( ., z), >),

for every z in C , which we shall call the family of normal-form games associated with the

coopetitive game G.

Remark. It is clear that with any above family of normal form games

g= ( gz )z∈C ,

with gz = (f ( ., z ) , > ), we can associate:

Volume I, Issue 1(1), Winter 2015

•a family of payoﬀ spaces

(imf ( ., z))z∈C ,

with members in the payoﬀ universe R2 ;

•a family of Pareto maximal boundary

(∂∗ Gz )z∈C ,

with members contained in the payoﬀ universe R2 ;

•a family of suprema

(supGz )z∈C ,

with members belonging to the payoﬀ universe R2 ;

•a family of Nash zones

(N (Gz ))z∈C ;

with members contained in the strategy space E× F ;

•a family of conservative bi-values

v# = (v#

z) z∈ C;

in the payoﬀ universe R2 .

And so on, for every meaningful known feature of a normal form game.

Moreover, we can interpret any of the above families as set-valued paths in the strategy space

E× For in the payoﬀ universe R2 .

It is just the study of these induced families which becomes of great interest in the exami-

nation of a coopetitive game G and which will enable us to deﬁne (or suggest) the various possible

solutions of a coopetitive game.

6. Solutions of a coopetitive game

6.1 Introduction

The two players of a coopetitive game G - according to the general economic principles of

monotonicity of preferences and of non-satiation - should choose the cooperative strategy z in Cin

order that:

•the reasonable Nash equilibria of the game Gz are f -preferable than the reasonable Nash

equilibria in each other game Gz 0 ;

•the supremum of Gz is greater (in the sense of the usual order of the Cartesian plane) than

the supremum of any other game Gz 0 ;

•the Pareto maximal boundary of Gz is higher than that of any other game Gz 0 ;

•the Nash bargaining solutions in Gz are f -preferable than those in Gz 0 ;

•in general, ﬁxed a common kind of solution for any game Gz , say S( z) the set of these kind

of solutions for the game Gz , we can consider the problem to ﬁnd all the optimal solutions (in

the sense of Pareto) of the set valued path S , deﬁned on the cooperative strategy set C . Then,

we should face the problem of selection of reasonable Pareto strategies in the set-valued

path S via proper selection methods (Nash-bargaining, Kalai-Smorodinsky and so on).

Moreover, we shall consider the maximal Pareto boundary of the payoﬀ space im(f) as an

appropriate zone for the bargaining solutions.

The payoﬀ function of a two person coopetitive game is (as in the case of normal-form game)

a vector valued function with values belonging to the Cartesian plane R2 . We note that in general

the above criteria are multi-criteria and so they will generate multi-criteria optimization problems.

In this section we shall deﬁne rigorously some kind of solution, for two player coopetitive

games, based on a bargaining method, namely a Kalai-Smorodinsky bargaining type. Thus initially,

we need to specify what kind of bargaining method we are going to use.

Journal of Mathematical Economics and Finance

6.2 Bargaining problems

In this paper, we shall propose and use the following original extended (and quite general)

deﬁnition of bargaining problem and, consequently, a natural generalization of Kalai-Smorodinsky

solution. In the economic literature, several examples of extended bargaining problems and extended

Kalai-Smorodinski solutions are already presented. The essential root of these various extended

versions of bargaining problems is the presence of utopia points not-directly constructed by the dis-

agreement points and the strategy constraints. Moreover, the Kalai-type solution, of such extended

bargaining problems, is always deﬁned as a Pareto maximal point belonging to the segment joining

the disagreement point with the utopia point (if any such Pareto point does exist): we shall follow the

same technique. In order to ﬁnd suitable new win-win solutions of our realistic coopetitive economic

problems, we need such new types of versatile extensions. For what concerns the existence of our

new extended Kalai solutions, for the economic problems we are facing, we remark that conditions

of compactness and strict convexity will naturally hold; we remark, otherwise, that, in this paper,

we are not interested in proving general or deep mathematical results, but rather to ﬁnd reasonable

solutions for new economic coopetitive context.

Deﬁnition (of bargaining problem). Let S be a subset of the Cartesian plane R2 and let

aand bbe two points of the plane with the following properties:

•they belong to the small interval containing S, if this interval is deﬁned (indeed, it is well

deﬁned if and only if Sis bounded and it is precisely the interval [inf S, sup S ]≤ );

•they are such that a < b;

•the intersection

[a, b]≤ ∩∂∗ S,

among the interval [ a, b]≤ with end points a and b (it is the set of points greater than aand

less than b , it is not the segment [a, b ] ) and the maximal boundary of Sis non-empty.

In these conditions, we call a bargaining problem on Scorresponding to the pair of

extreme points ( a, b) , the pair

P= ( S, ( a, b)).

Every point in the intersection among the interval [ a, b]≤ and the Pareto maximal boundary of S

is called possible solution of the problem P . Some time the ﬁrst extreme point of a bargaining

problem is called the initial point of the problem (or disagreement point or threat point) and

the second extreme point of a bargaining problem is called utopia point of the problem.

In the above conditions, when Sis convex, the problem Pis said to be convex and for this case

we can ﬁnd in the literature many existence results for solutions of P enjoying prescribed properties

(Kalai-Smorodinsky solutions, Nash bargaining solutions and so on ...).

Remark. Let S be a subset of the Cartesian plane R2 and let a and b two points of the plane

belonging to the smallest interval containing Sand such that a≤ b . Assume the Pareto maximal

boundary of S be non-empty. If a and b are a lower bound and an upper bound of the maximal

Pareto boundary, respectively, then the intersection

[a, b]≤ ∩∂∗ S

is obviously not empty. In particular, if a and b are the extrema of S(or the extrema of the Pareto

boundary S∗ =∂∗ S ) we can consider the following bargaining problem

P= ( S, ( a, b)) ,( or P = ( S∗ ,( a, b)))

and we call this particular problem a standard bargaining problem on S (or standard bargaining prob-

lem on the Pareto maximal boundary S∗ ).

Volume I, Issue 1(1), Winter 2015

6.3 Kalai solution for bargaining problems

Note the following property.

Property. If (S, (a, b )) is a bargaining problem with a<b , then there is at most one point

in the intersection

[a, b ]∩∂∗ S,

where [ a, b] is the segment joining the two points a and b.

Proof. Since if a point pof the segment [a, b] belongs to the Pareto boundary ∂∗ S , no other

point of the segment itself can belong to Pareto boundary, since the segment is a totally ordered

subset of the plane (remember that a < b ). 

Deﬁnition (Kalai-Smorodinsky). We call Kalai-Smorodinsky solution (or best com-

promise solution) of the bargaining problem ( S, ( a, b)) the unique point of the intersection

[a, b ]∩∂∗ S,

if this intersection is non empty.

So, in the above conditions, the Kalai-Smorodinsky solution k(if it exists) enjoys the following

property: there is a real r in [0,1] such that

k= a+ r( b− a),

k− a= r( b− a),

hence k 2 − a 2

k1 − a1

=b 2 −a2

b1 − a1

if the above ratios are deﬁned; this last equality is the characteristic property of Kalai-Smorodinsky

solutions.

We end the subsection with the following deﬁnition.

Deﬁnition (of Pareto boundary). A Pareto boundary consists of every subset Mof an

ordered space which has only pairwise incomparable elements.

6.4 Nash (proper) solution of a coopetitive game

Let N := N ( G ) be the union of the Nash-zone family of a coopetitive game G, that is the

union of the family (N ( Gz ))z∈C of all Nash-zones of the game family g = (gz )z∈C associated to the

coopetitive game G . We call Nash path of the game G the multi-valued path

z7→ N ( Gz )

and Nash zone of Gthe trajectory N of the above multi-path. Let N∗ be the Pareto maximal

boundary of the Nash zone N. We can consider the bargaining problem

PN = ( N∗ , inf( N∗ ) , sup( N∗ )).

Deﬁnition. If the above bargaining problem PN has a Kalai-Smorodinsky solution k, we say

that k is the properly coopetitive solution of the coopetitive game G.

The term "properly coopetitive" is clear:

•this solution kis determined by cooperation on the common strategy set Cand to be selﬁsh

(competitive in the Nash sense) on the bi-strategy space E× F .

Journal of Mathematical Economics and Finance

6.5 Bargaining solutions of a coopetitive game

It is possible, for coopetitive games, to deﬁne other kind of solutions, which are not properly

coopetitive, but realistic and sometime aﬀordable. We call these types of solutions super-cooperative.

Let us show some of these types of solutions.

Consider a coopetitive game Gand

•its Pareto maximal boundary Mand the corresponding pair of extrema (aM , bM );

•the Nash zone N(G ) of the game in the payoﬀ space and its extrema (aN , bN );

•the conservative set-value G# (the set of all conservative values of the family gassociated with

the coopetitive game G) and its extrema (a# , b# ).

We call:

•Pareto compromise solution of the game G the best compromise solution (K-S solution)

of the problem

(M, (aM , bM )),

if this solution exists;

•Nash-Pareto compromise solution of the game G the best compromise solution of the

problem

(M, (bN , bM ))

if this solution exists;

•conservative-Pareto compromise solution of the game G the best compromise of the

problem

(M, (b# , bM ))

if this solution exists.

6.6 Transferable utility solutions

Other possible compromises we suggest are the following.

Consider the transferable utility Pareto boundary M of the coopetitive game G, that is the set

of all points pin the Euclidean plane (universe of payoﬀs), between the extrema of G, such that their

sum

+(p ) := p1 +p 2

is equal to the maximum value of the addition + of the real line Rover the payoﬀ space f (E× F× C )

of the game G.

Deﬁnition (TU Pareto solution). We call transferable utility compromise solution

of the coopetitive game G the solution of any bargaining problem ( M, ( a, b)) , where

•aand bare points of the smallest interval containing the payoﬀ space of G

•bis a point strongly greater than a;

•Mis the transferable utility Pareto boundary of the game G;

•the points a and b belong to diﬀerent half-planes determined by M.

Note that the above fourth axiom is equivalent to require that the segment joining the points

aand bintersect M.

Volume I, Issue 1(1), Winter 2015

6.7 Win-win solutions

In the applications, if the game G has a member G0 of its family which can be considered as

an "initial game" - in the sense that the pre-coopetitive situation is represented by this normal form

game G0 - the aims of our study (following the standard ideas on coopetitive interactions) are

•to "enlarge the pie";

•to obtain a win-win solution with respect to the initial situation.

So that we will choose as a threat point ain TU problem (M, (a, b )) the supremum of the

initial game G0 .

Deﬁnition (of win-win solution). Let (G, z0 ) be a coopetitive game with an initial

point, that is a coopetitive game G with a ﬁxed common strategy z0 (of its common strategy set C).

We call the game Gz 0 as the initial game of (G, z0 ) . We call win-win solution of the game

(G, z0 )any strategy proﬁle s = (x, y, z ) such that the payoﬀ of G at s is strictly greater than the

supremum L of the payoﬀ core of the initial game G( z0 ) .

Remark 1. The payoﬀ core of a normal form gain game Gis the portion of the Pareto

maximal boundary G∗ of the game which is greater than the conservative bi-value of G.

Remark 2. From an applicative point of view, the above requirement (to be strictly greater

than L ) is very strong. More realistically, we can consider as win-win solutions those strategy proﬁles

which are strictly greater than any reasonable solution of the initial game Gz 0 .

Remark 3. Strictly speaking, a win-win solution could be not Pareto eﬃcient: it is a situa-

tion in which the players both gain with respect to an initial condition (and this is exactly the idea

we follow in the rigorous deﬁnition given above).

Remark 4. In particular, observe that, if the collective payoﬀ function

+(f) = f 1 +f 2

has a maximum (on the strategy proﬁle space S ) strictly greater than the collective payoﬀ L1 + L2

at the supremum L of the payoﬀ core of the game Gz 0 , the portion M ( > L ) of Transferable Utility

Pareto boundary M which is greater than Lis non-void and it is a segment. So that we can choose

as a threat point ain our problem (M, (a, b )) the supremum Lof the payoﬀ core of the initial game

G0 to obtain some compromise solution.

6.7.1 Standard win-win solution. A natural choice for the utopia point bis the supremum of the

portion M≥a of the transferable utility Pareto boundary Mwhich is upon (greater than) this point

M≥a ={ m∈ M: m≥ a}.

6.7.2 Non standard win-win solution. Another kind of solution can be obtained by choosing b

as the supremum of the portion of Mthat is bounded between the minimum and maximum value of

that player i that gains more in the coopetitive interaction, in the sense that

max(pri(imf )) − max(pri(imf0 )) > max(pr3−i (imf )) − max(pr3−i (imf0 )).

6.7.3 Final general remarks. In the development of a coopetitive game, we consider:

•a ﬁrst virtual phase, in which the two players make a binding agreement on what coopera-

tive strategy z should be selected from the cooperative set C, in order to respect their own

rationality.

•then, a second virtual phase, in which the two players choose their strategies forming the proﬁle

(x, y ) to implement in the game G (z ).

Journal of Mathematical Economics and Finance

Now, in the second phase of our coopetitive game Gwe consider the following 4 possibilities:

1. the two players are non-cooperative in the second phase and they do or do not exchange info,

but the players choose (in any case) Nash equilibrium strategies for the game G (z ); in this

case, for some rational reason, the two players have devised that the chosen equilibrium is the

better equilibrium choice in the entire game G; we have here only one binding agreement

in the entire development of the game;

2. the two players are cooperative also in the second phase and they make a binding agreement

in order to choose a Pareto payoﬀ on the coopetitive Pareto boundary; in this case we need

two binding agreements in the entire development of the game;

3. the two players are cooperative also in the second phase and they make two binding agreements,

in order to reach the Pareto payoﬀ (on the coopetitive Pareto boundary) with maximum col-

lective gain (ﬁrst agreement) and to share the collective gain according to a certain subdivision

(second agreement); in this case we need three binding agreements in the entire development

of the game;

4. the two players are non-cooperative in the second phase (and they do or do not exchange

information), the player choose (in any case) Nash equilibrium strategies; the two players have

devised that the chosen equilibrium is the equilibrium with maximum collective gain and they

make only one binding agreement to share the collective gain according to a certain subdivision;

in this case we need two binding agreements in the entire development of the game.

7. Dynamics

Consider a coopetitive game f :S× C→ Rn .

The function

Φ : C→ Diﬀ(M ) : z 7→ Φ(z ) ,

where

Φ(z ) : f (., z0 )(S )→f ( ., z)(S ) : f ( x, z0 ) 7→ f ( x, z ),

is well deﬁned if, for every point P in f ( ., z0 ), we have

f( x, z) = f ( x0 , z ),

for any z∈ C and any x, x0 ∈ S such that f (x, z0 ) = f ( x0 , z0 ) = P.

Moreover, if C is the real interval [a, b ], note that, for every x0 in the strategy space S , the

curve

f( x0 , .) : C→R2 : z 7→ f( x0 , z)

is smooth and well deﬁned, we call it the payoﬀ evolution of the initial strategy x0 . In general, we

cannot consider this evolution as an orbit of the initial payoﬀ f (x0 , a ), but, if we deﬁne the z-state

spaces

Mz ={ ( x, X)∈ S× f( ., z )( S ) : X= f( x, z)}

and more general the ﬁber-space

F= ( E, C, ρ),

where E is the disjoint union of the family M, that is

E= {(z, x, X ) ∈C ×S ×f( S) : X= f( x, z) },

and the projection is the map

ρ:E → C: ( z, x, X ) 7→ z,

we have the evolution of the element (z0 , x0, X ) into the ﬁbration E, deﬁned by

γ( x0 , X) : C → E : z 7→ ( z, x0 , f ( x0 , z)).

Volume I, Issue 1(1), Winter 2015

8. First example

8.1 Payoﬀ function of the game

We consider a coopetitive gain game with payoﬀ function given by

f( x, y, z )=( x+ 1 /( x+ 1) − z, (1 + m ) y+ (1 + n ) z) =

= (x + 1/ (x + 1),(1 + m )y ) + z (− 1, 1 + n)

for every x, y, z in [0,1].

Figure 1: 3D representation of the initial game (f ( ., 0), <).

8.2 Study of the game G = (f, >)

Note that, for a ﬁxed cooperative strategy z in U , the section game G (z )=( p( z) , >) with

payoﬀ function p (z ), deﬁned on the square U× U by

p( z)( x, y) = f( x, y, z ),

is the translation of the game G (0) by the "cooperative" vector

v( z) = z(− 1 ,1 + n),

so that we can study the initial game G(0) and then we can translate the various informations of the

game G (0) by the vector v (z ).

So, let us consider the initial game G(0). The strategy square S =U2 of G (0) has vertices 02,

e1 , 12 and e2 , where 02is the origin, e1 is the ﬁrst canonical vector (1 , 0), 12 is the sum of the two

canonical vectors (1, 1) and e2 is the second canonical vector (0,1).

8.3 Topological Boundary of the payoﬀ space of G0

In order to determine the Pareto boundary of the payoﬀ space, we shall use the techniques

introduced by D. Carf`ı in Carf`ı (2009g). We have

p0 ( x, y) = ( x + 1 /( x + 1) , (1 + m) y) ,

Journal of Mathematical Economics and Finance

for every x, y in [0,1]. The transformation of the side [0, e1 ] is the trace of the (parametric) curve

c: U→R2 deﬁned by

c( x ) = f( x, 0 , 0) = ( x + 1 / ( x + 1) , 0),

that is the segment

[f (0), f (e1 )] = [(1, 0),(3/2,0)].

The transformation of the segment [0, e2 ] is the trace of the curve c :U→ R2 deﬁned by

c( y) = f(0 , y, 0) = (1 , (1 + m) y) ,

that is the segment

[f (0), f ( e2 )] = [(1, 0),(1, 1 + m )].

The transformation of the segment [e1 , 12 ] is the trace of the curve c :U→R2 deﬁned by

c( y) = f(1 , y, 0) = (1 + 1 / 2 , (1 + m) y) ,

that is the segment

[f (e1 ), f (12 )] = [(3/2 , 0),(3/2, 1 + m )].

Critical zone of G(0) . The Critical zone of the game G(0) is empty. Indeed the Jacobian

matrix is

Jf ( x, y) =  1 + (1 + x) −2 0

0 1 + m ,

which is invertible for every x, y in U.

Payoﬀ space of the game G(0) . So, the payoﬀ space of the game G(0) is the transforma-

tion of topological boundary of the strategic square, that is the rectangle with vertices f (0, 0), f (e1 ),

f(1 ,1) and f( e2 ).

Figure 2: Initial payoﬀ space of the game (f, <).

Nash equilibria. The unique Nash equilibrium is the bistrategy (1, 1). Indeed,

1 + (1 + x ) −2 >0

so the function f1 is increasing with respect to the ﬁrst argument and analogously

1 + m > 0

so that the Nash equilibrium is (1,1).

Volume I, Issue 1(1), Winter 2015

8.4 The payoﬀ space of the coopetitive game G

The image of the payoﬀ function f, is the union of the family of payoﬀ spaces

(impz )z∈C ,

that is the convex envelope of the union of the image p0 (S ) (Sis the square U× U ) and of its

translation by the vector v(1), namely the payoﬀ space p1 (S ): the image of fis an hexagon with

vertices f (0, 0), f ( e1 ), f (1, 1) and their translations by v(1).

Figure 3: Payoﬀ space of the game (f , <).

8.5 Pareto maximal boundary of payoﬀ space of G

The Pareto sup-boundary of the coopetitive payoﬀspace f (S ) is the segment [P0 , Q0 ], where

P0 = f(1 ,1) and

Q0 = P0 + v (1).

Possibility of global growth. It is important to note that the absolute slope of the Pareto

(coopetitive) boundary is 1 + n. Thus the collective payoﬀ f1 +f2 of the game is not constant on

the Pareto boundary and, therefore, the game implies the possibility of a global growth.

Trivial bargaining solutions. The Nash bargaining solution on the segment [P0 , Q0 ] with

respect to the inﬁmum of the Pareto boundary and the Kalai-Smorodinsky bargaining solution on

the segment [P0 , Q0 ], with respect to the inﬁmum and the supremum of the Pareto boundary, coin-

cide with the medium point of the segment [P0 , Q0 ]. This solution is not acceptable from the ﬁrst

player point of view, it is collectively better than the supremum of G0 but it is disadvantageous for

ﬁrst player (it suﬀers a loss!): this solution can be thought as a rebalancing solution but it is not

realistically implementable.

8.6 Transferable utility solution

In this coopetitive context it is more convenient to adopt a transferable utility solution, indeed:

Journal of Mathematical Economics and Finance

•the point of maximum collective gain on the whole of the coopetitive payoﬀ space is the point

Q0 = (1 /2 , 2 + m+ n).

8.6.1 Rebalancing win-win best compromise solution. Thus we propose a rebalancing win-win

kind of coopetitive solution, as it follows (in the case m = 0):

1. we consider the portion sof transferable utility Pareto boundary

M:= (0 ,5 /2 + n) + R (1 ,−1),

obtained by intersecting Mitself with the strip determined (spanned by convexifying) by the

straight lines e2 +R e1 and

(2 + n ) e2 + Re1 ,

these are the straight lines of maximum gain for the second player in games G(0) and G

respectively.

2. we consider the Kalai-Smorodinsky segment s0 of vertices (3/2 , 1) - supremum of the game

G(0) - and the supremum of the segment s.

3. our best payoﬀ coopetitive compromise is the unique point Kin the intersection of segments

sand s0 , that is the best compromise solution of the bargaining problem

(s, (sup G0 , sup s )).

Figure 4: Two Kalai win-win solutions of the game (f, < ), represented with n = 1/2.

8.7 Win-win solution

This best payoﬀ coopetitive compromise Krepresents a win-win solution with respect to the

initial supremum (3/2 , 1). So that, as we repeatedly said, the ﬁrst player can also increase his initial

proﬁt from coopetition.

Win-win strategy procedure. The win-win payoﬀ K can be obtained (by chance) in a

properly coopetitive fashion in the following way:

Volume I, Issue 1(1), Winter 2015

•the two players agree on the cooperative strategy 1 of the common set C;

•the two players implement their respective Nash strategies of game G(1); the unique Nash

equilibrium of G (1) is the bistrategy (1,1);

•ﬁnally, they share the "social pie"

5/ 2 + n = (f1 +f2 )(1,1,1),

in a cooperative fashion (by contract) according to the decomposition K.

9. The second example

9.0.1 Main Strategic assumptions. We assume that:

•any real number x, belonging to the interval E := [0, 3], represents a possible strategy of ﬁrst

player;

•any real number y, in the same interval F := E , represents a possible strategy of the second

player;

•any real number z, again in the interval C = [0, 2], can be a possible cooperative strategy of

the two players.

9.1 Payoﬀ function of the game

We consider a coopetitive gain game with payoﬀ function f :S→R2 , given by

f( x, y, z) = (2 + x− y/3− z , 2− 2 x/3 + (1 + m) y + (1 + n) z ) =

= (2, 2) + (x− y/ 3, − 2 x/3 + (1 + m )y ) + z ( − 1, 1 + n ),

for every (x, y, z ) in S := [0, 3]2 ×[0,2].

Figure 5: 3D representation of (f, <).

9.2 Study of the second game G = (f, >)

Note that, ﬁxed a cooperative strategy zin 2U , the section game G (z ) = (p (z ), > ) with payoﬀ

function p (z ), deﬁned on the square E2 by

p( z)( x, y) := f( x, y, z ),

Journal of Mathematical Economics and Finance

is the translation of the game G (0) by the "cooperative" vector

v( z) = z(− 1 ,1 + n),

so that, we can study the initial game G(0) and then we can translate the various information of the

game G (0) by the vectors v (z ), to obtain the corresponding information for the game G (z ).

So, let us consider the initial game G (0). The strategy square E2 of G (0) has vertices 02 , 3e1 ,

32 and 3e2 , where 02is the origin of the plane R2 , e1 is the ﬁrst canonical vector (1, 0), 32 is the

vectors (3, 3) and e2 is the second canonical vector.

9.3 Topological Boundary of the payoﬀ space of G0

In order to determine the the payoﬀ space of the linear game it is suﬃcient to transform the

four vertices of the strategy square (the game is an aﬃne invertible game), the critical zone is empty.

9.3.1 Payoﬀ space of the game G(0) . So, the payoﬀ space of the game G(0) is the transformation

of the topological boundary of the strategy square, that is the parallelogram with vertices f (0,0),

f(3 e1 ), f (3 , 3) and f(3 e2 ). As we show in the below Figure 6.

B' = (4,3)

C' = (1,5)

D' = (2,2)

A' = (5,0)

Figure 6: Initial payoﬀ space of the game (f, <).

9.3.2 Nash equilibria. The unique Nash equilibrium is the bistrategy (3,3). Indeed, the function

f1 is linear increasing with respect to the ﬁrst argument and analogously the function f2 is linear

and increasing with respect to the second argument.

9.4 The payoﬀ space of the coopetitive game G

The image of the payoﬀ function f, is the union of the family of payoﬀ spaces

(impz )z∈C ,

that is the convex envelope of the union of the image p0 (E2 ) and of its translation by the vector v(2),

namely the payoﬀ space p2 (E2 ): the image of fis an hexagon with vertices f (0, 0), f (3e1 ), f (3,3)

and their translations by v (2). As we show below in Figure 7.

Volume I, Issue 1(1), Winter 2015

B' = P' = (4,3)

D' = (2,2)

A' = (5,0)

Q' = B'' = (2,6)

C'' = (-1,8)

D'' = (0,5)

Figure 7: Payoﬀ space of the game (f , <).

9.5 Pareto maximal boundary of the payoﬀ space of G

The Pareto sup-boundary of the coopetitive payoﬀ space f (S ) is the union of the segments

[A0 , B0 ], [P0 , Q0 ] and [Q0 , C 00 ], where P0 =f (3,3, 0) and

Q0 = P0 + v (2).

9.5.1 Possibility of global growth. It is important to note that the absolute slopes of the seg-

ments [A0 , B0 ], [P0 , Q0 ] of the Pareto (coopetitive) boundary are strictly greater than 1. Thus the

collective payoﬀ f1 +f2 of the game is not constant on the Pareto boundary and, therefore, the game

implies the possibility of a transferable utility global growth.

9.5.2 Trivial bargaining solutions. The Nash bargaining solution on the entire payoﬀ space, with

respect to the inﬁmum of the Pareto boundary and the Kalai-Smorodinsky bargaining solution, with

respect to the inﬁmum and the supremum of the Pareto boundary, are not acceptable for ﬁrst player:

they are collectively (TU) better than the Nash payoﬀ of G0 but they are disadvantageous for the

ﬁrst player (it suﬀers a loss!): these solutions could be thought as rebalancing solutions, but they are

not realistically implementable.

9.6 Transferable utility solutions

In this coopetitive context it is more convenient to adopt a transferable utility solution, indeed:

•the point of maximum collective gain on the whole of the coopetitive payoﬀ space is the point

Q0 = (2 , 6).

9.6.1 Rebalancing win-win solution relative to maximum gain for the second player in

G. Thus we propose a rebalancing win-win coopetitive solution relative to maximum gain for the

second player in G , as it follows (in the case m = 0):

1. we consider the portion sof transferable utility Pareto boundary

M:= Q0 +R (1 ,−1),

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obtained by intersecting Mitself with the strip determined (spanned by convexifying) by the

straight lines P0 +R e1 and C 00 +R e1 , these are the straight lines of Nash gain for the second

player in the initial game G(0) and of maximum gain for the second player in G , respectively.

2. we consider the Kalai-Smorodinsky segment s0 of vertices B0 - Nash payoﬀ of the game G(0) -

and the supremum of the segment s.

3. our best payoﬀ rebalancing coopetitive compromise is the unique point Kin the intersec-

tion of segments s and s0 , that is the best compromise solution of the bargaining problem

(s, (B0 , sup s)).

Figure 8 below shows the above extended Kalai-Smorodinsky solution Kand the Kalai-

Smorodinsky solution K0 of the classic bargaining problem (M, B 0 ). It is evident that the distribution

Kis a rebalancing solution in favor of the second player with respect to the classic solution K0 .

D' = (2,2)

A' = (5,0)

Q' = B'' = (2,6)

C'' = (-1,8)

D'' = (0,5)

B' = P' = (4,3)

KK'

Figure 8: Two Kalai win-win solutions of the game (f, < ), represented with n = 1/2.

9.6.2 Rebalancing win-win solution relative to maximum Nash gain for the second

player. We propose here a more realistic rebalancing win-win coopetitive solution relative to maxi-

mum Nash gain for the second player in G, as it follows (again in the case m = 0):

1. we consider the portion sof transferable utility Pareto boundary

M:= Q0 +R (1 ,−1),

obtained by intersecting Mitself with the strip determined (spanned by convexifying) by

the straight lines P0 +R e1 and Q0 +R e1 , these are the straight lines of Nash gain for the

second player in the initial game G(0) and of maximum Nash gain for the second player in G,

respectively.

2. we consider the Kalai-Smorodinsky segment s0 of vertices B0 - Nash payoﬀ of the game G(0) -

and the supremum of the segment s.

3. our best payoﬀ rebalancing coopetitive compromise is the unique point Kin the intersec-

tion of segments s and s0 , that is the best compromise solution of the bargaining problem

(s, (B0 , sup s)).

Figure 9 below shows the above extended Kalai-Smorodinsky solution Kand the Kalai-

Smorodinsky solution K0 of the classic bargaining problem (M, B 0 ). The new distribution Kis

Volume I, Issue 1(1), Winter 2015

a rebalancing solution in favor of the second player, more realistic than the previous rebalancing

solution.

D' = (2,2)

A' = (5,0)

Q' = B'' = (2,6)

C'' = (-1,8)

D'' = (0,5)

B' = P' = (4,3)

KK'

Figure 9: Two Kalai win-win solutions of the game (f, < ), represented with n = 1/2.

9.7 Win-win solution

The payoﬀ extended Kalai-Smorodinsky solutions K represent win-win solutions, with respect

to the initial Nash gain B0 . So that, as we repeatedly said, also ﬁrst player can increase his initial

proﬁt from coopetition.

9.7.1 Win-win strategy procedure. The win-win payoﬀ Kcan be obtained in a properly

transferable utility coopetitive fashion, as it follows:

•the two players agree on the cooperative strategy 2 of the common set C;

•the two players implement their respective Nash strategies in the game G (2), so competing `a

la Nash; the unique Nash equilibrium of the game G (2) is the bistrategy (3,3);

•ﬁnally, they share the "social pie"

(f1 +f2 )(3,3,2),

in a transferable utility cooperative fashion (by binding contract) according to the de-

composition K.

10. Conclusions

Our new mathematical model of coopetitive game is a game theoretic system in which two or

more players (participants) can interact cooperatively and non-cooperatively at the same time, i.e.

simultaneously. Even Brandenburger and Nalebuﬀ, the authors who introduced coopetition in scien-

tiﬁc literature, did not propose a clear quantitative model to implement and represent coopetition in

a formalized Game Theory context: we, in this work, specify clearly and univocally such a possible

mathematical model in a quite versatile fashion.

Journal of Mathematical Economics and Finance

The problem to implement the notion of coopetition in Game Theory is summarized by the

following question:

•how do, in normal form games, cooperative and non-cooperative interactions coexist simulta-

neously, in a Brandenburger-Nalebuﬀ sense?

In order to understand the above question, consider a classic two-player normal-form gain

game G = (f, > ) - such a game is an ordered system (pair) in which fis a vector valued function,

deﬁned on a Cartesian product (proﬁle strategy space) with values in the Euclidean plane (payoﬀ

proﬁle universe), and the mathematical object >is the natural strict sup-order of the Euclidean

plane itself (the sup-order is indicating that the game G, with payoﬀ function f, is a gain game and

not a loss game).

Let E and F be the strategy sets of the two players in the game G. The two players can

choose the respective strategies (x in E and y in F ) in two distinct and mutually exclusive way:

•cooperatively (exchanging information and making binding agreements);

•not-cooperatively (not exchanging information or exchanging information but without possi-

bility to make binding agreements).

The above two behavioral ways are mutually exclusive, at least in normal-form games:

•the two ways cannot be adopted simultaneously in the model of normal-form game (without

using convex probability mixtures, but this is not the way suggested by Brandenburger and

Nalebuﬀ in their approach);

•there exists no room, in the classic normal form game model, for a simultaneous (non-probabilistic)

employment of the two behavioral extremes represented by cooperation and non-cooperation.

Here we propose a manner to overcome that impasse, according to the idea of coopetition in

the sense of Brandenburger and Nalebuﬀ. In our coopetitive game model:

•the players of the game dispose of their respective strategy-sets (in which they can choose

cooperatively or not cooperatively);

•there exists a common strategy set Ccontaining other strategies (possibly, of diﬀerent type

with respect to those in the classic strategy sets) which must be chosen cooperatively;

•the strategy set C can also be structured as a Cartesian product (similarly to the proﬁle

strategy space of normal form games), but in any case the strategies belonging to this new set

Cmust be chosen cooperatively.

In our paper we oﬀer:

•an original model of coopetitive game, introduced in the literature by D. Carf`ı;

•several ways to construct and deﬁne possible solutions concepts for the new original model of

coopetitive game;

•a dynamical interpretation of the coopetitive game model;

•a basic analysis of a sample of coopetitive game (in an intentionally simpliﬁed fashion - without

direct strategic interactions among players) to emphasize the new mathematical exam proce-

dures needed for coopetitive games and to show how to build up new concept of game solutions

in such a coopetitive context;

•the complete examination of a second sample of coopetitive game, showing other possible

coopetitive solutions; we propose in this second case a linear model, with a direct strategic

interactions among players.

Volume I, Issue 1(1), Winter 2015

Finally, we desire to emphasize that the model and solutions provided by our coopetitive game

theory approach:

•aim, ﬁrstly, at enlarging the pie (payoﬀ space) of the classic game theory models - in a newly

conceived dynamical way - and, secondly, they succeed in sharing that enlarged pie fairly, by

using sophisticated Kalai-Smorodinsky and Nash bargaining solutions;

•can show several win-win and rebalancing strategy proﬁles and outcomes, for all the game

participants, within a coopetitive non constant-sum game dynamic path.

Acknowledgments. The authors wish to thank Dr. Eng. Alessia Donato for her valuable

help in the preparation of the ﬁgures. Moreover, the author wish to express his gratitude to Prof. Dr.

Daniele Schilir`o, Dr. Francesco Musolino and Dr. Emanuele Perrone for their helpful comments