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A Free and Fair Economy: A Game of Justice and Inclusion∗
Ghislain H. Demeze-Jouatsa@Roland Pongou‡Jean-Baptiste Tondji§
July 31, 2021
Abstract
Frequent violations of fair principles in real-life settings raise the fundamental question
of whether such principles can guarantee the existence of a self-enforcing equilibrium in a
free economy. We show that elementary principles of distributive justice guarantee that a
pure-strategy Nash equilibrium exists in a finite economy where agents freely (and non-
cooperatively) choose their inputs and derive utility from their pay. Chief among these
principles is that: 1) your pay should not depend on your name; and 2) a more productive
agent should not earn less. When these principles are violated, an equilibrium may not exist.
Moreover, we uncover an intuitive condition—technological monotonicity—that guarantees
equilibrium uniqueness and efficiency. We generalize our findings to economies with social
justice and inclusion, implemented in the form of progressive taxation and redistribution,
and guaranteeing a basic income to unproductive agents. Our analysis uncovers a new
class of strategic form games by incorporating normative principles into non-cooperative
game theory. Our results rely on no particular assumptions, and our setup is entirely non-
parametric. Illustrations of the theory include applications to exchange economies, surplus
distribution in a firm, contagion and self-enforcing lockdown in a networked economy, and
bias in the academic peer-review system.
Keywords: Market justice; Social justice; Inclusion; Ethics; Discrimination; Self-enforcing
contracts; Fairness in non-cooperative games; Pure strategy Nash equilibrium; Efficiency.
JEL Codes: C72, D30, D63, J71, J38
∗The authors thank Frank Riedel, Sarah Auster, Jan-Henrik Steg, Niels Boissonet, Andr´e Casajus, Hulya
Eraslan, Roberto Serrano, and seminar participants at The University of Texas Rio Grande Valley, Bielefeld
University, and the Texas Economic Theory Camp for their valuable and insightful comments and suggestions.
Demeze-Jouatsa gratefully acknowledges financial support from the DFG (Deutsche Forschungsgemeinschaft /
German Research Foundation) via grant Ri 1128-9-1 (Open Research Area in the Social Sciences, Ambiguity in
Dynamic Environments), Bielefeld Young Researchers’ Fund, the BGTS Mobility Grants, and the University of
Ottawa.
@Center for Mathematical Economics, University of Bielefeld; Email: demeze jouatsa@uni-bielefeld.de.
‡Department of Economics, University of Ottawa; Email: rpongou@uottawa.ca.
§Department of Economics and Finance, The University of Texas Rio Grande Valley; Email: jeanbap-
tiste.tondji@utrgv.edu.
1
“For Aristotle, justice means giving people
what they deserve, giving each person his or
her due.”
Sandel [2010, P. 187]
1 Introduction
It is generally acknowledged that justice is the foundation of a stable, cohesive, and productive
society.1However, violations of fair principles are highly prevalent in real-life settings. For
example, discriminations based on race, gender, culture and several other factors have been widely
documented (see, for instance, Reimers [1983], Wright and Ermisch [1991], Sen [1992], Bertrand
and Mullainathan [2004], Anderson and Ray [2010], Pongou and Serrano [2013], Goldin et al.
[2017], Bapuji et al. [2020], Hyland et al. [2020], Card et al. [2020], and Koffi and Wantchekon
[Forthcoming]). These realities raise the fundamental question of how basic principles of justice
affect individual incentives, and whether such principles can guarantee the stability and efficiency
of contracts among private agents in a free and competitive economy. That the literature has
remained silent on this question is a bit surprising, given the long tradition of ethical and normative
principles in economic theory and the relevance of these principles to the real world [Sen, 2009,
Thomson, 2016]. The main goal of this paper is to address this problem. In our treatment
of this question, we incorporate elementary principles of justice and ethics into non-cooperative
game theory. In doing so, we uncover a new class of strategic form games with a wide range of
applications to classical and more recent economic problems.
We precisely address the following questions:
A: How do fair principles affect the stability of social interactions in a free economy?
B: Under which conditions do fair principles lead to equilibrium efficiency?
To formalize these questions, we introduce a model of a free and fair economy, where agents
freely (and non-cooperatively) choose their inputs, and the surplus resulting from these input
choices is shared following four elementary principles of distributive justice, which are:
1The Merriam-Webster dictionary defines justice as“the maintenance or administration of what is just especially
by the impartial adjustment of conflicting claims or the assignment of merited rewards or punishments.”
2
1. Anonymity: Your pay should not depend on your name.2
2. Local efficiency: No portion of the surplus generated at any profile of input choices should
be wasted.
3. Unproductivity: An unproductive agent earns nothing.
4. Marginality: A more productive agent should not earn less.
It is generally agreed that these ideals form the core principles of market (or meritocratic)justice,
and are of long tradition in economic theory. They have inspired eighteenth centuries writers
like Rousseau [1762] and Aristotle [1946], and contemporary authors like Rawls [1971], Shapley
[1953], Young [1985], Roemer [1998], De Clippel and Serrano [2008], Sen [2009], Sandel [2010],
Thomson [2016], and Posner and Weyl [2018], among several others. However, a number of
empirical observations have suggested that the real world does not always conform to these
elementary principles of justice. Studies have shown that anonymity is violated in job hiring
[Kraus et al., 2019, Bertrand and Mullainathan, 2004], in wages [Charles and Guryan, 2008, Lang
and Manove, 2011], in scholarly publishing [Laband and Piette, 1994, Ellison, 2002, Heckman
et al., 2017, Serrano, Akerlof, 2020, Card et al., 2020], in school admission [Francis and Tannuri-
Pianto, 2012, Grbic et al., 2015], in sexual norm enforcement [Pongou and Serrano, 2013], in
health care [Balsa and McGuire, 2001, Thornicroft et al., 2007], in household resource allocations
[Sen, 1992, Anderson and Ray, 2010], in scholarly citations [Card et al., 2020, Koffi, 2021], and
in organizations [Small and Pager, 2020, Koffi and Wantchekon, Forthcoming]. These studies
generally show that discrimination based on name, race, gender, culture, religion, and academic
affiliation is prevalent in these different contexts. Violations of basic principles of justice therefore
raise the fundamental question of how these principles affect individual incentives, the stability
of social interactions, and economic efficiency.
We examine these questions through the lens of a model of a free and fair economy. This model
is a list E= (N, ×j∈NXj, o, f , φ, (uj)j∈N), where Nis a finite set of agents, Xja finite set of
actions (or inputs) available to agent j,o= (oj)j∈Na reference profile of actions, fa production
(or surplus) function (also called technology) that maps each action profile x∈ ×j∈NXjto a
measurable output f(x)∈R,φan allocation scheme that distributes any realized surplus f(x)
2Here, name designates any unproductive individual characteristic such as first and last names, skin color,
gender, religious or political affiliation, cultural background, etcetera. Anonymity means that a person’s pay
should not depend on their identity; in other words, given my input choice and that of others, my pay should not
vary depending on whether I am called “Emily/Greg” or “Lakisha/Jamal”[Bertrand and Mullainathan, 2004], or
depending on whether my skin color is black, white or green, or depending on whether I am a man or a woman.
3
to agents, and ujthe utility function of agent j. The reference point ocan be interpreted as
an unproduced endowment of goods (or resources) that can be either consumed as such, or may
be used in the production process when production opportunities are specified. Agent j’s action
set Xjcan be interpreted broadly, as we do not impose any particular structure on it other than
it being finite. It may be viewed as a capability set [Sen, 2009], or may represent the set of
different occupations (or functions) available to agent jbased on agent j’s skills, or the set of
effort levels that agent jmay supply in a production environment. The nature of the set of
actions can also be different for each agent. For each input profile x, the allocation scheme
φdistributes the generated surplus f(x)following the aforementioned principles of anonymity,
local efficiency,unproductivity, and marginality, and each agent jderives utility from her payoff
uj(x) = φj(f, x).3
To define an equilibrium concept that captures individuals’ incentives in a free and fair
economy, we first observe that any economy Einduces a corresponding strategic form game
GE= (N, ×j∈NXj,(uj)j∈N).4Then, a profile of actions x∗∈ ×j∈NXjis said to be an equilib-
rium in the free and fair economy Eif and only if it is a pure strategy Nash equilibrium of the
game GE.
Our first main result shows that the principles of market justice stated above guarantee the
existence of an equilibrium (Theorem 1). Moreover, when an economy violates these principles,
an equilibrium may not exist. These findings have profound implications. One implication is
that fair rules guarantee the existence of self-enforcing contracts between private agents in a free
economy. A second implication is that fair rules prevent output (and income) volatility, given that
action choices at equilibrium are pure strategies. Moreover, from a purely theoretical viewpoint,
the incorporation of normative principles into non-cooperative game theory has led us to identify
an interesting class of strategic form games that always have a pure strategy Nash equilibrium in
spite of the fact that each player has a finite action set.5
3The formalization of these principles differ depending on the context. Ours is a generalization of the classical
formalization of Shapley [1953] and Young [1985] to our economic environment. Indeed, we show that these four
principles uniquely characterize a pay scheme that generalizes the classical Shapley value (Proposition 1). This
pay scheme is a multivariate function defined at each input profile x; see also Pongou and Tondji [2018] and
Aguiar et al. [2018, 2020]. Also, note that uj(x)can be any increasing function of the payoff φj(f, x), and the
functional form might be different for each agent.
4The class of free and fair economies therefore defines a large class of games that can be characterized as fair.
Any strategic form game is either fair or unfair, and some unfair games are simply a monotonic transformation of
fair games.
5As is well known, a pure strategy Nash equilibrium does not exist in a finite strategic form game in general
[Nash, 1951]. A growing literature seeks to identify conditions under which a pure strategy Nash equilibrium exists
4
Although a pure strategy equilibrium always exists in any free and fair economy, this equilib-
rium may be inefficient. We uncover a simple structural condition that guarantees equilibrium
efficiency. More precisely, we show that if the technology is strictly monotonic, there exists a
unique equilibrium, and this equilibrium is Pareto-efficient (Theorem 2). Quite interestingly, we
find that when a monotonic economy fails to satisfy the principles of market justice, even if an
equilibrium exists, it may be inefficient.6A clear implication of this finding is that in the class
of monotonic economies, any allocation scheme that violates the principles of market justice is
welfare-inferior to the unique scheme that respects these principles.
Next, we extend our analysis to economies with social justice. The principles of market justice
imply that unproductive agents (for example, agents with severe disabilities) should earn nothing.
In most societies, however, social security benefits ensure that a basic income is allocated to
agents who, for certain reasons, cannot produce as much as they would like to (see, for example,
among others, David and Duggan [2006], and Hanna and Olken [2018]). To account for this
reality, we extend our model to incorporate social justice or inclusion. Generally, social justice
includes solidarity and moral principles that individuals have equal access to social rights and
opportunities, and it requires consideration beyond talents and skills since some agents have
natural limitations, not allowing them to be productive.
Social justice is incorporated into our model in the form of progressive taxation and redis-
tribution. At any production choice, a positive fraction of output is taxed and shared equally
among all agents, and the remaining fraction is allocated according to the principles of market
justice. This allocation scheme satisfies the principles of anonymity and local efficiency, but vi-
olates marginality and unproductivity. Income is redistributed from the high skilled and talented
(or more productive agents) to the least well-off. However, the income rank of a free and fair
economy (without social justice) is maintained, provided that the entire surplus is not taxed. We
generalize each of our results. In particular, a pure strategy equilibrium always exists regardless of
the tax rate (Theorem 3). Consistent with Theorem 2, we also find that if the production technol-
ogy is strictly monotonic, there exists a unique equilibrium, and this equilibrium is Pareto-efficient
(Corollary 1).
We uncover additional results on the efficiency of economies with social justice. In particular,
we find that there exists a tax rate threshold above which there exists a pure strategy Nash
in a finite game (see, for example, Rosenthal [1973], Monderer and Shapley [1996], Mallick [2011], Carmona and
Podczeck [2020], and the references therein). But unlike our paper, this literature has not approached this problem
from a normative perspective. We therefore view our analysis as a contribution.
6A clear example is the prisoner’s dilemma game. Economies that are modeled by such games are monotonic,
although their unique equilibrium is Pareto-inefficient.
5
equilibrium that is Pareto-efficient, even if the economy is not monotonic (Theorem 4). Moreover,
we show that one can always change the reference point of any non-monotonic free economy
with social justice to guarantee the existence of an equilibrium that is Pareto-efficient (Theorem
5). This latter finding implies that if a free economy is able to choose its reference point, then it
can always do so to induce a Pareto-efficient outcome that is self-enforcing.
We develop various applications of our model to classical and more recent economic problems.
In particular, we develop applications to exchange economies [Walras, 1954, Arrow and Debreu,
1954, Shapley and Shubik, 1977, Osborne and Rubinstein, 1994], surplus distribution in a firm,
self-enforcing lockdown in a networked economy with contagion, and bias in the academic peer-
review system [Akerlof, 2020]. This variety of applications is possible because we impose no
particular assumptions on the structure of action sets, and the action set of each agent may be
of a different nature. We start with applying our theory to a production environment where an
owner of the firm (or team leader) uses bonuses as a device to incentivize costly labor supply
from rational workers. Our analysis shows that in addition to guaranteeing equilibrium existence,
the owner can also achieve production efficiency, provided that the costs of labor supply are not
too high. Next, we provide an application to contagion in a networked economy in which rational
agents freely form and sever bilateral relationships. Rationality is captured by the concept of
pairwise-Nash equilibrium, which refines the Nash equilibrium. Using a contagion index [Pongou
and Serrano, 2013], we show how the costs of a pandemic can induce self-enforcing lockdown.
Our application to academic peer-review in the knowledge economy shows that discrimination in
the allocation of rewards results in a Pareto-inferior outcome, which indicates that bias reduces
the incentive to study “soft”, “important”, and relevant topics in equilibrium.7Finally, we recast
the model of an exchange economy in our framework, and show that our equilibrium is generally
different from the Walrasian equilibrium. This difference is in part explained by the fact that the
Walrasian model assumes linear pricing, whereas our model is fully non-parametric.
The rest of this paper is organized as follows. Section 2 introduces the model of a free and
fair economy. In Section 3, we prove the existence of a pure strategy Nash equilibrium in a free
and fair economy. Section 4 is devoted to the analysis of efficiency. In Section 5, we extend our
model to incorporate social justice and inclusion, and we generalize our results. In Section 6,
we present some applications of our analysis. Section 7 situates our paper in the closely related
literature, and Section 8 concludes. Some proofs are collected in an appendix.
7See, for example, a recent study by Akerlof [2020] on the consequences of mostly rewarding“hard” research
topics in the field of economics.
6
2 A free and fair economy: definition, existence and unique-
ness
In this section, we introduce preliminary definitions and the key concepts of the paper. We
then show that there exists a unique economy that is free and fair.
2.1 A free economy
A free economy is an economy where agents freely choose their actions and derive utility from
their pay. It is modeled as a list E= (N, ×j∈NXj,(oj)j∈N, f, φ, (uj)j∈N).N={1,2, ..., n}
is a finite set of agents. Each agent jhas a finite set of feasible actions Xj. We refer to an
action profile x= (xj)j∈Nas an outcome, and denote the set ×j∈NXjof outcomes by X. The
reference outcome (also called reference point) is o= (oj)j∈N; it can be interpreted as the
inaction point, where agents do nothing or do not engage in any sort of transactions with other
agents. A production (or surplus) function (also called technology) ftransforms any choice xto
a real number f(x)∈R, with f(o)=0.8We denote by P(X) = {g:X→R,with g(o)=0}
the set of production functions on X.φ:P(X)×X→Rnis a distribution scheme that assigns
to each pair (f, x)a payoff vector φ(f, x). At each input profile x, each agent jderives utility
uj(x) = φj(f, x).9
2.2 A free and fair economy
A free and fair economy is a free economy E= (N, ×j∈NXj,(oj)j∈N, f, φ, (uj)j∈N)in which the
surplus distribution scheme φsatisfies elementary principles of market justice. These principles,
of long tradition in economic theory, are those of anonymity,local efficiency,unproductivity,
and marginality stated in the Introduction. These principles are naturally interpreted, but their
formalization varies depending on the context. A few preliminary definitions and notations will
be needed for their formalization in our setting.
Definition 1. Let x∈Xa profile of actions. An outcome x0∈Xis a sub-profile of xif either
x0=xor [x0
i6=xi=⇒x0
i=oi], for i∈N.
8We normalize the surplus at the reference point to 0for expositional purposes. It is possible that the surplus
realized at ois not zero, and in this case, f(x)should be interpreted as net surplus at x, that is, the realized
surplus at xminus the realized surplus at o. We assume the reference oto be exogenously determined.
9As noted in the Introduction, uj(x)can be any increasing function of φj(f , x), where the functional form
may be different for each agent.
7
For each x∈X, we denote by ∆(x)the set of sub-profiles of x. Given a production function
f∈P(X), and an outcome x∈X, we define the function fxas the restriction of fto ∆(x):
fx: ∆(x)→R,such that fx(y) = f(y),for each y∈∆(x).
Definition 2. Let i∈N. We define the relation ∆i
oon Xby:
[x0∆i
ox]if and only if [x0∈∆(x)and x0
i=oi].
Let x∈Xbe an outcome. We denote ∆i
o(x) = {x0∈X:x0∆i
ox}, and by Nx={i∈N:
xi6=oi}the set of agents whose actions in xare different from their reference points. We also
denote |x|=|Nx|the cardinality of Nx.
Definition 3. Let f∈P(X),x∈X, and x0∈∆i
o(x). The marginal contribution of agent iat
a pair (x0, x)is:
mci(f, x0, x) = f(x0
−i, xi)−f(x0),
where (x0
−i, xi)∈Xis the outcome in which agent ichooses xi, and every other agent jchooses
x0
j.
Definition 4. Let f∈P(X). Agent iis said to be unproductive if for each x∈Xand all
x0∈∆i
0(x),mci(f, x0, x) = 0.
A permutation πof Nis a bijection of Ninto itself. We denote by Snthe set of permutations
of N. Let x∈Xbe a profile of inputs, and let πx∈ Snbe a permutation of Nwhose restriction
to N\Nxis the identity function, that is πx(i) = ifor each i∈N\Nx. Remark that πx
permutes only agents that are active in the profile x, and is therefore equivalent to a permutation
πx:Nx→Nxover Nx; we denote by Sx
nthe set of such permutations.
Let x∈X,πx∈ Sx
n, and y∈∆(x). We define the profile πx(y) = (πx
j(y))j∈N, where
πx
j(y) = (xjif yk6=ok, j =πx(k)
ojif yk=ok, j =πx(k).
We now formalize the principles of market justice below.
Anonymity. An allocation φsatisfies x−Anonymity if for each i∈Nand πx∈ Sx
n,
φi(πxfx, x) = φπx(i)(fx, x),where πxfx(y) = fx(πx(y)),for y∈∆(x).
The value φsatisfies Anonymity if φsatisfies x−Anonymity for all x∈X.
8
Local Efficiency.P
j∈N
φj(f, x) = f(x)for any f∈P(X)and x∈X.
Unproductivity. If agent iis unproductive, then φi(f, x)=0for each f∈P(X)and x∈X.
Marginality. Let f, g ∈P(X), and xan outcome. If
mci(f, x0, x)≥mci(g, x0, x)for each x0∈∆i
o(x)
for an agent i, then φi(f, x)≥φi(g, x).
These axioms are interpreted naturally. Anonymity means that an agent’s pay does not
depend on their name. It states that every agent is treated the same way by the allocation
rule: if two agents exchange their identities, their payoffs will remain unchanged. An important
property that is implied by anonymity is symmetry (or non-favoritism), which means that equally
productive agents should receive the same pay. Local efficiency simply requires that the surplus
resulting from any input choice be fully shared among productive agents participating in the
economy. Unproductivity means that an agent whose marginal contribution is zero at an input
profile should get nothing at that profile. Marginality means that, if the adoption of a new
technology increases the marginal contribution of an agent, that agent’s pay should not be lower
under this new technology relative to the old technology. In other words, more productive agents
should not earn less compared to less productive agents. Throughout the paper, we abbreviate
the four principles as ALUM.
Definition 5. Afree and fair economy is a free economy (N, X, o, f, φ, u)such that the
distribution scheme φsatisfies ALUM.
We have the following result.
Proposition 1. There exists a unique distribution scheme, denoted Sh, that satisfies ALUM.
For any production function f∈P(X), and any given outcome x∈Xand agent i∈N:
Shi(f , x) = X
x0∈∆i
o(x)
(|x0|)!(|x|−|x0| − 1)!
(|x|)! mci(f, x0, x).(1)
Proof of Proposition 1. See Appendix.
Remark that for each agent i, the value Shi(f, x)is interpreted as agent i’s average con-
tribution to output f(x). It can be easily shown that the allocation rule Sh generalizes the
classical Shapley value [Shapley, 1953]. In fact, to obtain the classical Shapley value, one only
has to assume that each agent’s action set is the pair {0,1}; the classical Shapley value is simply
9
Shi(f , x)where x= (1,1, ..., 1), which effectively corresponds to the assumption that the grand
coalition is formed. Our setting generalizes the classical environment in three ways. First, it is
not necessary to assume that all players have the same action set. Second, the action set of
a player may have more than two elements. Third, the value can be computed for any input
profile x, which effectively means that Shi(f, x)as a multivariate function of x. Our model also
generalizes that in Pongou and Tondji [2018] (when the environment is certain), Aguiar et al.
[2018], and Aguiar et al. [2020]. Following these latter studies, we will call Sh the Shapley pay
scheme.
Below, we illustrate the notion of a free and fair economy, and provide an example of a free
economy that is unfair.
Example 1. Consider a small economy E= (N, X, o, f, φ, u), where N={1,2},X1={a1, a2},
X2={b1, b2, b3},o= (a1, b1),X=X1×X2,fis given by f(a1, b1) = 0, f(a1, b2) = 5 =
f(a1, b3), f(a2, b1)=2,and f(a2, b2)=4=f(a2, b3), and for each x∈X,φ(f, x) = u(f, x)
is given in Table 1 below:
Agent 1
Agent 2
b1b2b3
a1(0,0) (0,5) (0,5)
a2(2,0) (0.5,3.5) (0.5,3.5)
Table 1: A 2-agent free and fair economy
For each of the six payoff vectors presented in Table 1, the first component represents agent 1’s
payoff (for example, u1(f, (a2, b1)) = 2) and the second component represents agent 2’s payoff
(for instance, u2(f, (a2, b1)) = 0). We can check that for each x∈X,u(f, x) = φ(f, x) =
Sh(f , x). Therefore, Eis a free and fair economy.
Agent 1
Agent 2
b1b2b3
a1(0,0) (2,3) (3,2)
a2(1,1) (3,1) (2,2)
Table 2: A 2-agent free and unfair economy
Now, we consider another economy E0with the same characteristics as in Eexcept for the
distribution scheme φthat is replaced by a new scheme ψdescribed in Table 2. In addition to the
10
fact that ψ6=Sh, it is straightforward to show that the distribution ψviolates the marginality
axiom. Therefore, E0is not a free and fair economy.
One of our goals in this paper is to answer the question of whether fair principles guarantee
the existence of a pure strategy Nash equilibrium. We can observe that in the free and fair
economy described by Table 1, there are two pure strategy Nash equilibria, which are (a2, b2)and
(a2, b3). However, the modified economy E0represented by Table 2 admits no equilibrium in pure
strategies. In the next section, we will show that fair principles guarantee the existence of a pure
strategy Nash equilibrium in a free economy, and when an economy violates these principles, a
pure strategy Nash equilibrium may not exist.
3 Equilibrium existence in a free and fair economy
In a free and fair economy, agents make decisions that affect their payoff and the payoffs of
other agents. One natural question that therefore arises is whether an equilibrium exists. In this
section, we first show that a free economy can be modeled as a strategic form game and use the
notion of pure strategy Nash equilibrium [Nash, 1951] to capture incentives and rationality. Our
main result is that a free and fair economy always has a pure strategy Nash equilibrium.
3.1 A free and fair economy as a strategic form game
Astrategic form game is a 3-tuple (N, X, v), where Nis the set of players, X=×j∈NXjis
the strategy space, and v:X→Rnis the payoff function. For each x∈X,vi(x)is agent i’s
payoff at strategy profile x, for each i∈N. A strategic form game is said to be finite if the set
of agents Nis finite, and for each agent i, the set Xiof actions is also finite.
A strategy profile x∗∈Xis a pure strategy Nash equilibrium in the game (N, X, v)if and
only if for all i∈N,vi(x∗)≥vi(x∗
−i, yi), for all yi∈Xi, where (x∗
−i, yi)is the strategy profile in
which agent ichooses yiand every other agent jchooses x∗
j.
A free economy E= (N, X, o, f, φ, u)generates a strategic form game GE= (N, X, uE),
where for each x∈Xand each i∈N,uE
i(x) = ui(f, x) = φi(f , x). In the case Eis a free
and fair economy, then for each outcome x,P
j∈N
uE
j(x) = f(x)since the distribution scheme φ
satisfies local efficiency. For this reason, when Eis a free and fair economy, we may refer to the
production function fas the total utility function of the strategic form game GE.
Definition 6. Let E= (N, X, o, f, φ, u)be a free economy. A profile x∗∈Xis an equilibrium
if and only if x∗is a pure-strategy Nash equilibrium in the strategic form game GE.
11
3.2 Existence of an equilibrium
In this section, we state and prove our main result.
Theorem 1. Any free and fair economy E= (N, X, o, f, φ, u)admits an equilibrium.
The proof of Theorem 1 uses the concept of a cycle of deviations that we introduce below.
Definition 7. Let G= (N, X, v)be a strategic form game and Lk= (x1, x2, ..., xk)be a list
of outcomes, where each xl∈X(l= 1, ..., k) is a pure strategy. The k-tuple Lkis a cycle of
deviations if there exist agents j1, ..., jk∈Nsuch that
xl+1 = (xl
−jl, xl+1
jl)and vjl(xl+1)> vjl(xl)
for each l= 1, ..., k, and xk+1 =x1.
Example 2. In the strategic form game represented in Table 3, consider the list L4= (x1, x2, x3, x4),
where x1= (c, a),x2= (d, a),x3= (d, b), and x4= (c, b).
Agent 1
Agent 2
a b
c(0,4) (3,0)
d(1,0) (0,2)
Table 3: A 2-agent game that admits a cycle of deviations
L4forms a cycle of deviations. Indeed, agent 1 has an incentive to deviate from x1to x2. By
doing so, agent 1 receives an excess payoff of 1. Similarly, agent 2 receives an excess payoff of 2
by deviating from x2to x3. Agent 1 receives an excess payoff of 3 by deviating from x3to x4;
and agent 2 receives an excess payoff of 4 by deviating from x4to x1. The sum of excess payoffs
in the cycle L4is therefore equal to 10.
Agent 1
Agent 2
b1b2b3b4
a1(0,0) (0,0) (0,12) (0,6)
a2(13,0) (13
2,−13
2) (3
2,1
2) (4,−3)
a3(3,0) (8,5) (−1,8) (−1,2)
Table 4: A 2-agent game with Shapley payoffs
12
In the strategic form game in Table 4, the sum of excess payoffs in any cycle of outcomes
equals 0. Therefore, the game does not admit a cycle of deviations. The profile x∗= (a2, b3)is
the only pure strategy Nash equilibrium of the game.
Note that the game in Table 4 is generated from a free and fair economy. From Definition 7,
a sufficient condition for a finite strategic form game to admit a pure strategy Nash equilibrium
is the absence of a cycle of deviations. The sum of excess payoffs in any cycle of deviations
has to be strictly positive, as illustrated in Table 3 in Example 2. Such an example of a cycle
of deviations can not be constructed in a strategic form game generated from a free and fair
economy (see Table 4 in Example 2). We prove that in a strategic form game generated by a
free and fair economy, the sum of excess payoffs in any cycle of deviations equals 0.
Lemma 1. Let E= (N, X, o, f, φ, u)be a free and fair economy, and GE= (N, X, uE)the
strategic form game generated by E. Then, the sum of excess payoffs in any cycle of deviations
in GEequals 0.
Proof of Lemma 1. In this proof, we simply denote the payoff function uEby u. Let Lk=
(x1, x2, ..., xk)be a cycle of deviations in the game GE, and let agents j1, ..., jk∈Nbe the
associated sequences of defeaters. We denote by S(Lk, u)the sum of excess payoffs in the cycle
Lk:
S(Lk, u) = ujk(x1)−ujk(xk) +
k−1
X
l=1
[ujl(xl+1)−ujl(xl)].
We show that in the game GE
S(Lk, u) = 0.
For each agent i∈N, let Ribe a total order on the set Xisuch that oiRixifor all xi∈Xi.
For each outcome x∈X, define
fx(T, y) = (|Nx|if Nx⊆Tand xiRiyifor all i∈Nx
0otherwise
for all T⊆Nand y∈X.
We also define the following production function:
fx(z) = fx(Nz, z)for all z∈X.
Note that the family {fx, x ∈X\{o}} forms a basis of the set of production functions on the
the same set of players N, same set of outcomes X, and same reference outcome o. Therefore,
there exists (αx)x∈X\{o}such that
f(z) = X
x∈X
αxfx(z)for all z∈X. (2)
13
Furthermore, each fx,x∈X, is the total utility function of a strategic form game with
Shapley utilities Gx= (N, X, vx), where for each i∈N,vx
iis given by
vx
i(z) = (1if i∈Nx, Nx⊆Nz, xjRjzjfor all j∈Nx
0otherwise. for all z∈X.
Step 1. We show that the sum of excess payoffs of the cycle Lkequals 0in each strategic
form game Gx. First observe that vx
i≡0for all i /∈Nx, and vx
i≡vx
jfor all i, j ∈Nx. This
means that the sum of excess payoffs in any cycle of the game Gx, and in particular in the cycle
Lk, equals the sum of excess payoffs of any i∈Nx, which is obviously 0.
Step 2. We show that S(Lk, u)=0.
Using equation (2), f=P
x∈X
αxfx, we have that u=P
x∈X
αxvx. Given that S(Lk, vx)=0for
each outcome x, we can deduce that S(Lk, u)=0.
Now, we derive the proof of Theorem 1.
Proof of Theorem 1. From Lemma 1, the game GEadmits no cycle of deviations. As GEis finite,
we conclude that GEadmits a pure strategy Nash equilibrium.
The principles of market justice that define a free and fair economy are only sufficient condi-
tions for the existence of a pure strategy Nash equilibrium. However, an economy that violates
the fair principles may not have a pure strategy Nash equilibrium.
4 Equilibrium efficiency in a free and fair economy
In Section 3.2, we prove the existence of a pure strategy equilibrium (Theorem 1) in a free
and fair economy. However, there is no guarantee that each equilibrium is Pareto-efficient. For
instance, consider the strategic form game described in Table 4 in Example 2. The game admits
a unique pure strategy Nash equilibrium x∗= (a2, b3)with Sh(f, x∗) = (3
2,1
2). However,
the equilibrium x∗is Pareto-dominated by the strategy x= (a3, b2)with Sh(f, x) = (8,5).
Below, we provide two conditions on the production function that address this issue. The first
condition—weak monotonicity—guarantees the existence of a Pareto-efficient equilibrium in a
free and fair economy, and the second condition—strict monotonicity—guarantees that there is
a unique equilibrium and that this equilibrium is Pareto-efficient. Importantly, we also find that in
a free economy that is not fair, these monotonicity conditions do not guarantee the existence of
an equilibrium that is Pareto-efficient. Before presenting these results, we need some definitions.
Let E= (N, X, o, f, φ, u)be a free economy, and for i∈N, we denote X−i=
n
Q
j=1, j6=i
Xj.
14
Definition 8. An order Rdefined on Xis semi-complete if for all i∈Nand x−i∈X−i, the
restriction of Rto Aiis complete, where Ai={x−i} × Xi.
Definition 9. f∈P(X)is:
1. weakly monotonic if there exists a semi-complete order Ron Xsuch that for any x, y ∈X,
if xRy, then f(x)≤f(y).
2. strictly monotonic if there exists a semi-complete order Ron Xsuch that for any x, y ∈X,
[xRyand x6=y]implies f(x)< f(y).
Definition 10. A free and fair economy E= (N, X, o, f ,φ, u)is weakly (resp. strictly) monotonic
if fis weakly (resp. strictly) monotonic.
We have the following result.
Theorem 2. A weakly monotonic free and fair economy E= (N, X, o, f, φ, u)admits an equi-
librium that is Pareto-efficient. If Eis strictly monotonic, then, the equilibrium is unique and
Pareto-efficient.
Proof of Theorem 2. The result in Theorem 2 follows from the fact that each agent i’s payoff
Shi(f , x)at xdepends only on the marginal contributions {f(y−i, xi)−f(y), y ∈∆i
o(x)}of that
agent at x. Since fis weakly monotonic, the underlying semi-complete relation, say R, satisfies
the following condition: there exists x∈Xsuch that freaches its maximum at x, and for all
i∈Nand x−i∈X−i, we have x R (x−i, xi). Therefore, each marginal contribution of agent
iat a given outcome xis less than or equal to his or her corresponding marginal contribution
at the outcome (x−i, xi). Given that the Shapley distribution scheme, Sh(f, .), is increasing in
marginal contributions, agent i’s choice xiis a weakly dominant strategy of agent iin the game
GE. Therefore, xis a Nash equilibrium. The profile xis also Pareto-efficient as it maximizes f.
If fis strictly monotonic, then each xiis strictly dominant and xis the unique Nash equilibrium
of the game GE.
Theorem 2 ensures the uniqueness and Pareto-efficiency of the equilibrium in a strictly mono-
tonic free and fair economy. The strategic form game described in Table 4 admits the profile
x∗= (a2, b3)as the only pure strategy Nash equilibrium. However, x∗is Pareto-dominated by the
profile x= (a3, b2), which is not an equilibrium. Such a result can not arise in a strictly mono-
tonic free and fair economy. In addition to providing a condition that guarantees the existence of
a Pareto-efficient equilibrium, Theorem 2 also provide a condition that rules out multiplicity of
equilibria in the domain of free and fair economies.
15
In Theorem 2, we show that each weakly monotonic free and fair economy admits an equi-
librium that is Pareto-efficient. Consider the strategic form game described in Table 5 below.
The latter is derived from a free and fair economy with the profile o= (c, a)as the reference
point. The economy admits two equilibria, namely, outcomes (c, a)and (d, b). The profile (d, b)
is Pareto-efficient and it dominates the outcome (c, a).
Agent 1
Agent 2
a b
c(0,0) (0,0)
d(0,0) (1,1)
Table 5: A 2-agent free and fair economy with a Pareto-dominated equilibrium
Agent 1
Agent 2
b1b2
a1(0,0) (2,−1)
a2(2,0) (1,2)
Table 6: A 2-agent strictly monotonic free and unfair economy
We relate the existence of an equilibrium that is Pareto-dominated in the free and fair economy
described in Table 5 to the fact that the production function is weakly monotonic. However, it is
essential to emphasize that the existence of an equilibrium is due to the fact that the economy
is fair and not to the monotonicity property of the technology. For instance, consider a free
economy Ef, where agents 1 and 2 have strategies, X1={a1, a2}, and X2={b1, b2}, and the
production function fis given by: f(a1, b1) = 0,f(a1, b2)=1,f(a2, b1)=2, and f(a2, b2) = 3.
Agents’ payoffs are described in Table 6. The environment Efdescribes a strictly monotonic
economy, but it is unfair. Similarly, by replacing the production function fby another function g
defined by: g(a1, b1)=0,g(a1, b2) = g(a2, b1)=1, and g(a2, b2)=3, we obtain a weakly free
monotonic and unfair economy Egwith agents’ payoffs described in Table 7.
Note also that neither strategic form game GEfdescribed in Table 6, nor GEgdescribed in
Table 7 admit a pure strategy Nash equilibrium. This shows that the monotonicity conditions
do not guarantee the existence of a pure strategy Nash in a free economy that is unfair; and
even when an equilibrium exists in such an economy, it may be Pareto-inefficient. This latter
situation occurs, for example, in the prisoner’s dilemma game. An economy that is represented
by a prisoner’s dilemma game is monotonic, but its unique equilibrium is Pareto-inefficient (see,
16
Agent 1
Agent 2
b1b2
a1(0,0) (2,−1)
a2(2,−1) (1,2)
Table 7: A 2-agent weakly monotonic free and unfair economy
for instance, the game described in Table 8; the unique pure strategy Nash equilibrium (Defect,
Defect) is Pareto-inefficient).
Agent 1
Agent 2
Cooperate Defect
Cooperate (0,0) (−2,1)
Defect (1,−2) (−1,−1)
Table 8: A prisoner’s dilemma game
5 A free economy with social justice and inclusion
Our conception of a free economy with social justice embodies both the ideals of market justice
and social inclusion. Members of a society do not generally have the same abilities. Consequently,
distribution schemes that are based on market justice alone will penalize individuals with less
opportunities or those who are unable to develop a positive productivity to the economy.
One of the goals of social justice is to remedy this social disadvantage that results mainly
from arbitrary factors in the sense of moral thought. Social justice requires caring for the least
well-off and those who have natural limitations not allowing them to achieve as much as they
would like to. This requirement goes beyond the considerations of a free and fair economy in
which agents have equal access to civic rights, wealth, opportunities, and privileges. The ideal
of social justice could be implemented in a fair society through specific redistribution rules, and
that is the main message that we intend to provide in this section.
Market justice as defined in the previous sections requires that the collective outcome must
be distributed based on individual marginal contributions. Thus, a citizen who is not able to
contribute a positive value to the economy shouldn’t receive a positive payoff.
Social justice differs to market justice in the sense that everyone should receive a basic worth
for living. This principle is consistent with the results found by De Clippel and Rozen [2013]
17
in a recent experimental study in which neutral agents (called “Decision Makers”) are called
upon to distribute collective rewards among other agents (called“Recipients”). They show that
even if collective rewards depend on complementarity and substitutability between recipients,
some decision markers still allocate positive rewards to those who bring nothing to the economy.
Moreover, a linear convex combination of the Shapley value [Shapley, 1953] and the equal split
scheme arises as a one-parameter allocation estimate of data. This convex allocation is also
known as an egalitarian Shapley value [Joosten, 1996]. Intuitively, this pay scheme can be viewed
as implementing a progressive redistribution policy where a positive amount of the total surplus
in an economy is taxed and redistributed equally among all the agents. We use this distribution
scheme to showcase our purpose. We will see that some properties of an economy that embeds
the idea of social justice depends on the tax rate. Below, we define the equal-split, and an
egalitarian Shapley value schemes.
Definition 11. Let E= (N, X, o, f ,φ, u)be a free economy.
1. φis the equal split distribution scheme, if
φi(f, x) = f(x)
n,for all f∈P(X), x ∈X, and i∈N.
2. φis an egalitarian Shapley value if there exists α∈[0,1] such that for all f∈P(X), and
i∈N,
φi(f, x) = α·Shi(f, x) + (1 −α)·f(x)
n,for all x∈X.
We denote by ESαthe egalitarian Shapley value associated to a given α∈[0,1]. The
mixing equal split and Shapley value satisfies the principles of anonymity and local efficiency,
but violates marginality and unproductivity when α∈[0,1). The allocation scheme ESαhas a
very natural interpretation. Given an outcome x, the technology fproduces the output f(x). A
share (α) of the latter is shared among agents according to their marginal contributions, while
the remaining (1−α) is shared equally among the entire population; the fraction 1−αis the tax
rate. Immediately, those who are more talented will still receive more under a given egalitarian
Shapley value scheme, but less compared to what they receive in a free and fair economy (when
α= 1). Additionally, those who do not have the opportunity to contribute to their optimum
scale will still be rewarded. We have the following definition.
Definition 12. E= (N, X, o, f , φ, u)is a free economy with social justice if there exists α∈[0,1[
such that φ=ESα. We call Eα= (N, X, o, f, ESα, u)an α-free economy with social justice.
18
In Section 5.1, we analyze equilibrium existence and Pareto-efficiency in free economies with
social justice. Our methodology is similar to the one followed in Sections 3 and 4. In Section 5.2,
we prove that an economy can always choose its reference point to induce equilibrium efficiency,
even when the economy is not monotonic.
5.1 Equilibrium existence and efficiency in a free economy with social
justice
In what follows, we study the existence of equilibrium in an α-free economy with social justice.
As defined in Section 3.1, a free economy with social justice admits an equilibrium if the strategic
form game derived from that economy possesses a pure strategy Nash equilibrium. A meritocratic
planner will choose a higher αwhen allocating resources since talents and merits have more value
in such a society. An egalitarian planner will put a higher weight on equal distribution. It follows
that a choice of αreveals a trade-off between market justice and egalitarianism. The good news
is that there exists a self-enforcing social contract irrespective of the size of α. We have the
result hereunder.
Theorem 3. Any α-free economy with social justice Eα= (N, X, o, f, ESα, u)admits an equi-
librium.
Proof of Theorem 3. Consider α∈[0,1] such that φ=ESα. In the proof of Theorem 1, we show
that the sum of excess payoffs in any cycle of deviations from any strategic form game derived
from a fair economy equals 0. The same result holds for any strategic form game derived from
an α-free economy with social justice, since an egalitarian Shapley value is a linear combination
of the Shapley value and equal division. Thus, we conclude the proof.
We also provide a condition under which a free economy with social justice has a Pareto-
efficient economy. We have the following definition.
Definition 13. Let Eα= (N, X, o, f, ESα, u)be an α-free economy with social justice. An
optimal outcome is any outcome x∈arg max
y∈Xf(y)at which fis maximized.
The following result is deduced from Theorem 2.
Corollary 1. A weakly monotonic α-free economy with social justice Eα= (N, X, o, f, ESα, u)
admits an equilibrium that is Pareto-efficient. If fis strictly monotonic, then, the equilibrium is
unique and Pareto-efficient.
19
The proof of Corollary 1 is similar to that of Theorem 2. Next, we provide an additional result
about Pareto-efficiency of equilibria in a free economy with social justice.
Theorem 4. There exists α0∈(0,1) such that for all α∈[0, α0], the α-free economy with
social justice Eα= (N, X, o, f, ESα, u)admits an equilibrium that is Pareto-efficient.
Proof of Theorem 4. Assume that αis sufficiently small. If fadmits a unique optimal outcome
x, then xis a pure strategy Nash equilibrium of the game generated by any α-free economy
with social justice Eα. In the case fadmits two or more optimal outcomes, then, for strictly
positive but sufficiently small α, no agent has any incentive to deviate from an optimal outcome
to a non-optimal outcome. As games generated by α-free economies with social justice do not
admit cycles of deviations, it is not possible to construct any cycle of deviations within the set of
optimal outcomes. It follows that at least one optimal outcome is a Nash equilibrium. The latter
profile is also Pareto-efficient as it maximizes the sum of agents’ payoffs.
Example 3 (Taxation and Social Justice).Consider a small economy involving three agents,
N={1,2,3}, who live in three different states or regions in a given country. One can assume
that each agent is the “typical” representative of each state. Agents face different occupational
choices. Agent 1 can decide to stay unemployed (strategy “a”), work in a middle class job
(strategy “b”) that provides an annual salary of $188,000, or accumulate experience to land
a higher skilled job (strategy “c”) that pays an annual salary of $200,000. Agent 2 can only
choose between strategies “a” and “b”. For many reasons including health concerns, natural
disasters such as hurricane, pandemics or wildfire, or civil war violence, agent 3 does not have
the opportunities available to other agents; he or she can not work, and is therefore considered
as unemployed. The government uses marginal tax rates to determine the amount of income
tax that each agent must pay to the tax collector. The aggregate annual fiscal revenue function
ffor the economy depends on agents’ strategies and it is described as follows: f(a, a, a) = 0,
f(a, b, a) = $41,175.5,f(b, a, a) = $41,175.5,f(b, b, a) = $82,351,f(c, a, a) = $45,015.5,
and f(c, b, a) = $86,191. Numerous countries over the world use marginal tax brackets to
collect income taxes (see, for example, a report by Bunn et al. [2019] for the Organisation for
Economic Co-operation (OECD) and the Development and European Union (EU) countries).
The function fis a simplified version of such fiscal revenue rules. With the tax revenue collected,
the government provides public goods. However, the type of public investment received by an
agent’s state depends on the agent’s marginal contribution to the aggregate annual fiscal revenue.
Using the Shapley scheme φ=Sh in the distribution of public investments yields the outcome
x∗= (c, b, a)as the unique pure strategy Nash equilibrium in this free and fair economy. At
20
this equilibrium, the state of agent 1 receives a public good that is worth $45,015.5, agent 2’s
state receives a public investment of $41.175.5, and agent 3’s state receives nothing. However,
if the egalitarian Shapley scheme φ=ES4/5is used instead to redistribute the fiscal revenue,
then x∗= (c, b, a)is still the unique pure strategy Nash equilibrium in the free economy with
social justice. In that case, the outcome x∗is still Pareto-efficient and the ranking of the size
of investment across states does not change. Agent 3’s state receives a public investment of
$5,746, agent 2’s state receives $38,686.5, and agent 1’s state receives $41,758.5. Although the
allocation ES4/5(f, x∗) = ($41,758.5,$38,686.5,$5,746) might not be the “best” decision for
some people living in that society, it is a significant improvement (at least for agent 3’s state)
from the market allocation Sh(f, x∗) = ($45,015.5,$41,175.5,0).
Using Theorem 4, we deduce the following corollary.
Corollary 2. Let Eα= (N, X, o, f, ESα, u)be an α-free economy with social justice. Assume
that fonly takes non-negative values. Then, each agent receives a non-negative payoff at any
equilibrium.
The intuition behind Corollary 2 is straightforward. Assuming that at a given outcome x∈X,
f(x)is non-negative, then for all i∈N, agent i’s payoff is non-negative if instead of choosing
xi, the agent chooses the reference point oi.
5.2 Choosing a reference point to achieve equilibrium efficiency
So far, we have assumed that the reference point ois exogenously determined and that in a free
economy, the surplus function fis such that f(o1, o2, ..., on) = 0. As noted earlier, this latter
point is just a simplifying normalization. We have also shown that in a free and fair economy,
all the equilibria may be Pareto-inefficient, especially in the absence of monotonicity. Similarly,
in a free economy with social justice, if the tax rate (1−α) is too small, a Pareto-efficient
equilibrium may not exist either. This section shows that we can achieve equilibrium efficiency
simply by changing the reference point of any free and fair economy or any free economy with
social justice.
Without loss of generality, we assume that f(o)is strictly positive and modify the Shapley
distribution scheme such that for i∈N, and x∈X, agent i’s payoff at (f, x), denoted S h(f, x),
is given by Sh(f, x) = Shi(f−f(o), x)+ f(o)
n. Let us denote P(X) = {g:X→R,with g(o)>
0}. Our next result says that any optimal outcome can be achieved via an equilibrium profile
in any α-free economy with social justice endowed with the distribution scheme ESα, where
ESα(f, x) = α·Shi(f, x) + (1 −α)·f(x)
n,for all x∈Xand f∈P(X).
21
Theorem 5. For all free economy Eα(o) = (N, X, o, f, ESα, u), there exists another reference
outcome o0such that the α-free economy Eα(o0) = (N, X, o0, f, ESα, u)admits an optimal
equilibrium x∗.
Proof of Theorem 5. Assume α= 1. Let o0be a profile of inputs such that f(o0) = max
x∈Xf(x).
No agent has any strict incentive to deviate from o0. Indeed if agent ideviates and chooses
xi, then agent iis the only active agent at the new outcome (o0
−i, xi). As each inactive agent
receives f(o0)
nat (o0
−i, xi), and f(o0)maximizes the production, it follows from the local efficiency
axiom of the Shapley distribution scheme that the deviation xiis not strictly profitable. A
similar argument holds for any other α∈[0,1). Indeed, at the profile (o0
−i, xi), agent ireceives
αf(o0
−i, xi)−f(o0) + f(o0)
n+ (1 −α)f(o0
−i,xi)
n, which is less than f(o0)
n.
Remark that this result holds for any value of α, including for α= 1, which corresponds to a
situation where the tax rate is zero. In that case, the entire surplus of the economy is distributed
following the Shapley value. The analysis implies that if an economy can choose its reference
point, it can always do so to lead to equilibrium efficiency.
6 Some applications
There a wide variety of applications of our theory. In this section, we provide applications to the
distribution of surplus in a firm, exchange economies, self-enforcing lockdowns in a networked
economy facing a pandemic, and publication bias in the academic peer-review system.
6.1 Teamwork: surplus distribution in a firm
In this first application, we use our theory to show how bonuses can be distributed among
workers in a way that incentivizes them to work efficiently.
Consider a firm which consists of a finite set of workers N={1,2, ..., n}. Each worker i∈N
privately and freely chooses an effort level xj
i∈Xi, and bears a corresponding non-negative cost
cj
i=c(xj
i), where c(.)denotes the cost function. The cost of labor supply includes any private
resources or extra working time that worker iputs into the project (for example, transportation
costs, time, etcetera). Workers’ labor supply choices are made simultaneously and independently.
The owner of the firm (or the team leader) knows the cost associated to each effort level. At each
effort profile x= (x1,··· , xn), a corresponding monetary output F(x)is produced. A fraction of
the monetary output, f=γ·F, with γ∈(0,1), is redistributed to workers in terms of bonuses.
22
The existence of a pure strategy Nash equilibrium in this teamwork game follows from Lemma
1. To see this, observe that the payoff function of a worker can be decomposed in two parts:
the bonus that is determined by the Shapley payoff and the cost function. Lemma 1 shows that
the sum of excess payoffs in any cycle of deviations equals 0 in any free and fair economy (or
any strategic game with Shapley payoffs). The reader can check that the sum of excess costs
in any cycle of strategy profiles is zero as well in the game. The latter implies that the sum of
excess payoffs in any cycle of strategy profiles of the teamwork game is equal to 0. Therefore, the
teamwork game admits no cycle of deviations. As the game is finite, we conclude that it admits
at least a Nash equilibrium profile in pure strategies. (Recall that the total output of the firm,
F, and the total bonus, f, are perfectly correlated.) We should point out that a pure strategy
Nash equilibrium always exists in the teamwork game, even if costs are high. In the latter case,
some workers, if not all, might find it optimal to remain inactive at the equilibrium. In such a
situation, the owner might want to raise the total bonus to be redistributed to workers.
Illustration. We now provide a numerical example with two workers called Bettina and Di-
ana. Bettina has four possible effort levels: b1, b2, b3and b4; and Diana has four possible
effort levels as well: d1, d2, d3and d4. The cost functions of the two workers are given by:
c(b1) = c(d1)=0, c(b2) = c(b3) = c(d2) = c(d3)=4,c(b4)=3, and c(d4)=5. The fraction
fof the output to be redistributed as bonus is described in Table 9. The number f(b, d)is the
bonus to be distributed at the profile of efforts (b, d); for instance, f(b1, d1)=0.
Bettina
Diana
d1d2d3d4
b10 5 1 13
b22 8 10 2
b35 13 1 13
b43 9 13 2
Table 9: Total bonus function in a teamwork game
The corresponding Shapley payoffs are described in Table 10 and the net payoffs of Bettina
and Diana in the teamwork game are described in Table 11.
The profile (b4, d3)is a pure strategy Nash equilibrium. Therefore, the owner of the firm can
implement the profile (b4, d3)without any need of monitoring the actions of Bettina and Diana,
as (b4, d3)is self-enforcing. The owner can implement the profile (b1, d4)as well. Note that the
set of equilibrium effort profiles depend on the cost functions, and that no worker receives a non
23
Bettina
Diana
d1d2d3d4
b1(0,0) (0,5) (0,1) (0,13)
b2(2,0) (5
2,11
2) (11
2,9
2) (−9
2,13
2)
b3(5,0) (13
2,13
2) (5
2,−3
2) (5
2,21
2)
b4(3,0) (7
2,11
2) (15
2,11
2) (−4,6)
Table 10: Shapley payoffs: redistribution of total bonus in a teamwork game
Bettina
Diana
d1d2d3d4
b1(0,0) (0,1) (0,−3) (0,8)
b2(−2,0) (−3
2,3
2) (3
2,1
2) (−17
2,3
2)
b3(1,0) (5
2,5
2) (−3
2,−11
2) (−3
2,11
2)
b4(0,0) (1
2,3
2) (9
2,3
2) (−7,1)
Table 11: Bettina and Diana’s net payoffs in a teamwork game
positive bonus at the equilibrium. The reason is that each worker ialways has the option to remain
inactive, which is equivalent to Bettina choosing b1or Diana choosing d1in this illustration. The
two equilibria in this teamwork game are Pareto-efficient.
6.2 Contagion and self-enforcing lockdown in a networked economy
In this section, we provide an application of a free and fair economy to contagion and self-
enforcing lockdown in a networked economy. We show how the costs of a pandemic from a virus
outbreak can affect agents’ decisions to form and sever bilateral relationships in the economy.
Specifically, we illustrate this application by using the contagion potential of a network [Pongou,
2010, Pongou and Serrano, 2013, 2016, Pongou and Tondji, 2018].
Consider an economy Minvolving agents who freely form and sever bilateral links according
to their preferences. Agents’ choices lead to a network, defined as a set of bilateral links. Assume
that rational behavior is captured by a certain equilibrium notion (for example, Nash equilibrium,
pairwise-Nash equilibrium, etc.). Such an economy may have multiple equilibria. Denote by
E(M)the set of its equilibria. Our main goal is to assess agent’s decisions in response to the
spread of a random infection (for example, COVID-19) that might hit the economy. As the
pandemic evolves in the economy, will some agents decide to sever existing links and self-isolate
24
themselves? How does network structure depend on the infection cost?
To illustrate these concepts, consider an economy involving a finite set of agents N=
{1, ..., n}. All agents simultaneously announce the direct links they wish to form. For every agent
i, the set of strategies is an n-tuple of 0 and 1, Xi={0,1}n. Let xi= (xi1, ..., xii−1,1, xii+1 , ..., xin)
be an element in Xi. Let xij denote the jth coordinate of xi. Then, xij = 1 if and only if i
chooses a direct link with j(j6=i), or j=i(and thus xij = 0, otherwise). We assume that
the formation of a link requires mutual consent, that is, a link ij is formed in a network if and
only if xijxji = 1. We denote X=×j∈NXj. An outcome x∈Xyields a unique network
g(x). However, a network can be formed from multiple outcomes. We denote o= (0, .., 0) the
reference outcome, and g(o)the empty network. It follows that the networked economy Mcan
be represented by a free economy (N, X, o, f, φ, u), where fis the production function and u=φ
the payoff function (see below).
Assume that rationality is captured by the notion of pairwise-Nash equilibrium as defined by,
among others, Calv´o-Armengol [2004], Goyal and Joshi [2006], and Bloch and Jackson [2007].
The concept of pairwise-Nash equilibrium refines Nash equilibrium building upon the pairwise
stability concept in Jackson and Wolinsky [1996]. Pairwise-equilibrium networks are such that no
agent gains by reshaping the current configuration of links, neither by adding a new link nor by
severing any subset of the existing links. Let gbe a network and ij ∈ga link. We let g+ij
denote the network found by adding the link ij to g, and g−ij denote the network obtained
by deleting the link ij from g. Formally, gis a pairwise-Nash equilibrium network if and only if
there exists a Nash equilibrium outcome x∗that supports g, that is g=g(x∗), and for all ij /∈g,
φi(f, g +ij)> φi(f, g)implies φj(f, g +ij)< φj(f, g).
The contagion function is the contagion potential of a network [Pongou, 2010, Pongou and
Serrano, 2013, 2016, Pongou and Tondji, 2018]. To define this function, we consider a network
gthat has kcomponents, where a component is a maximal set of agents who are directly or
indirectly connected in g; and njthe number of individuals in the jth component (1 ≤j≤k).
Pongou [2010] shows that if a random agent is infected with a virus, and if that agent infects his
or her partners who also infect their other partners and so on, the fraction of infected agents is
given by the contagion potential of g, which is:
P(g) = 1
n2
k
X
j=1
n2
j.
However, in a network g, each agent is exogenously infected with probability 1
n, and given that
agents are not responsible for exogenous infections, the part of contagion for which agents are
25
collectively responsible in gis:
˜c(g) = P(g)−1
n.
We assume that the infection by a communicable virus leads to a disease outbreak in the economy.
Measures that are implemented to fight the pandemic bring economic costs to society. To assess
those costs, we assume that the collective contagion function ˜cgenerates a pandemic cost function
Cso that, for each network g:
C(g) = F(˜c(g)), F being a well-defined function.
The pandemic and network formation affect economic activities. The formation of a network g
brings an economic value v(g)∈Rto the economy. Given the cost function C, the economic
surplus of a network gis:
f(g) = v(g)− C(g).
Our main goal is to examine each agent’s behavior in forming or severing bilateral links as the
pandemic spreads in the economy. Let gbe a network and Sbe a set of agents. We denote by gS
the restriction of the network gto S. This restriction is obtained by severing all the links involving
agents in N\S. Also, let ibe an agent. We denote by gS+ithe network gS∪{i}obtained from gS
by connecting ito all the agents in Sto whom iis connected in the network g. The structure of
the networked economy provides a natural setting for the use of the Shapley distribution scheme.
In a competitive environment where marginal contributions are the only inputs that matter in the
economy, we can expect that an agent who adds no value to any network configuration receives
no payoff, and a more productive agent in a network structure receives a payoff that is greater
relative to that of less productive agents. Assuming that the output from individual contributions
are entirely shared among agents, it becomes natural to consider that agent i’s payoff in a network
gis given by the Shapley distribution scheme (1):
φi(f, g)≡Shi(f, g) = X
S⊆N, i6=S
s!(n−s−1)!
n!f(gS+i)−f(gS), s =|S|.
The networked economy M= (N, X, o, f, Sh, u)describes a free and fair economy. We
have the following result.
Proposition 2. Pairwise-Nash equilibrium networks always exist: E(M)6=∅.
This result partly follows from Theorem 1, but is stronger because the notion of pairwise-Nash
equilibrium refines the Nash equilibrium. The proof is left to the reader. We illustrate it below.
26
Illustration. Let N={1,2,3}. Assume the set of an agent i’s direct links in a network gis
Li(g) = {jk ∈g:j=ior k=i, and j6=k}, of size li(g). The size of gis l(g) = P
i∈N
li(g)/2.
Note that l(g) = 0 if and only if gis the empty network. For illustration, we assume that for
each network g:
v(g)=[l(g)]1/2
C(g) = λ˜c(g) = λ[P(g)−1
n], λ > 0
f(g)=[l(g)]1/2−λ[P(g)−1
n], λ > 0.
We can rewrite fas follows (note that P(∅) = 1
n):
f(g) =
0if l(g) = 0
1−2λ
9if l(g) = 1
√2−2λ
3if l(g) = 2
√3−2λ
3if l(g) = 3
Given that there is only three agents, we can fully represent the set of networks in M. The agents
are labeled as described in Figure 1. In Figure 2, we display the different network configurations
Agent 2 Agent 3
Agent 1
Figure 1: Disposition of agents in a network
in M. In each network, the payoff of each agent is given next to the corresponding node. The
pairwise stability concept facilitates the search of equilibrium networks. We have the following
result. We denote by gNthe complete network.
Proposition 3. Let gbe a network. If:
1. λ < 1.8√2−0.9, then E(M) = {gN}.
2. 1.8√2−0.9< λ < 3√3
2, then g∈ E(M)if and only if l(g)∈ {1,3}.
3. 3√3
2< λ < 4.5, then g∈ E(M)if and only if l(g) = 1.
4. λ > 4.5, then E(M) = {g(o)}.
27
0 0
0
g1
01
2−λ
9
1
2−λ
9
g2
1
2−λ
9
0
1
2−λ
9
g3
1
2−λ
9
1
2−λ
9
0
g4
1
3(√2+1−8λ
9)1
3(√2−1
2−5λ
9)
1
3(√2−1
2−5λ
9)
g5
1
3(√2−1
2−5λ
9)1
3(√2+1−8λ
9)
1
3(√2−1
2−5λ
9)
g6
1
3(√2−1
2−5λ
9)1
3(√2−1
2−5λ
9)
1
3(√2+1−8λ
9)
g7
1
3(√3−2λ
3)1
3(√3−2λ
3)
1
3(√3−2λ
3)
g8
Figure 2: Possible network formation in M
The proof of Proposition 3 is straightforward and left to the reader. Clearly, Proposition 3
shows that pandemic costs affect agents’ decisions in the networked economy. The parameter
λsummarizes the negative effects of the contagion in the economy. When there is no disease
outbreak, or the pandemic costs are very low (lower values of λ), each agent gains by keeping
bilateral relationships with others. In that situation, the complete network is likely to sustain
as the equilibrium social structure in the economy. No agent has an incentive to self-isolate.
However, as the pandemic costs rise, agents respond by severing some bilateral connections. For
intermediate values of λ(3√3
2<λ<4.5), only networks with one link will be sustained in the
equilibrium. This means that some agents find it rational to partially or fully self-isolate in order
to reduce the spread of the virus. In the extreme case where the contagion costs are very high
(λ > 4.5), a complete lockdown arises, and the empty network is the only equilibrium.
Interestingly, the value of λdepends on the nature of the virus. Viruses induce different severity
levels. For example, COVID-19 and the flu virus have different values, inducing different network
configurations in equilibrium. The different network configurations in Figure 2 can therefore be
interpreted as the networks that will arise in different scenarios regarding the nature of the virus.
28
6.3 Bias in academic publishing
In this section, we apply the model of a free and fair economy to academic publishing in a
knowledge environment. Generally, academic researchers have freedom to choose research topics
that are likely to be published either in peer-reviewed or non-peer-reviewed outlets. However,
studies show that the peer-review process is not generally anonymous, and it involves some
biases (see, for example, Ellison [2002], Heckman et al. [2017], Serrano, Akerlof [2020], and
the references therein). Following Ellison [2002], we consider a model of producing scientific
knowledge in which researchers differentiate topics along two quality dimensions: importance
(or q-quality) and hardness (or r-quality).10 In the hypothetical and straightforward knowledge
economy that we analyze, we assume that both importance and hardness levels are discrete,
ordered, and are homogeneous among researchers. Formally, Q={q0, q1, ..., qm}denotes the
set of importance levels, with q0< q1< ... < qm, and R={r0, r1, ..., rm}denotes the set of
different degrees of hardness, with r0< r1< ... < rm.
We consider a knowledge environment involving a finite set of researchers N={1, ..., n}.
Each researcher selects a topic of a given importance level and degree of hardness. For every
researcher i, a strategy xi= (qi, ri)∈Xi⊆Q×R, where qi∈Q, and ri∈R. We denote
X=×j∈NXj. We consider oi= (q0, r0)as the reference choice for researcher i, and o= (oj)j∈N
the reference outcome. A knowledge function (or technology) ftransforms any outcome x∈X
to the number of published articles f(x)∈R, with f(o)=0. An allocation φdistributes f(x)
to active researchers so that the utility of researcher i,ui, at the profile xgiven the knowledge
function f, is ui(x) = φi(f, x).
The knowledge economy Eφ= (N, X, o, f, φ, u)defines a free economy. Thanks to Theorem
1, the free and fair knowledge economy ESh = (N, X, o, f, Sh, u)admits a pure strategy Nash
equilibrium. The allocation φ(6=Sh) in the free knowledge economy can be viewed as the current
academic publishing system. As mentioned above, the latter could lead to an equilibrium outcome
that shows a bias towards“hardness”and against“importance”. To illustrate our point, we consider
a simple knowledge economy involving two active researchers N={1,2}with the same“abilities”
of producing scholarly articles. Each researcher i’s criteria for a choice of topic belongs to the set
Xi={0,1,2,3}, where each number represents a pair in Q×R: “0”: (soft, less important),“1”:
10Tough the trade-off between the two quality dimensions can be viewed as a rational decision, the consequences
can be detrimental to economics, as a discipline and profession. For instance, some general interest journals suffer
from the “incest factor” [Heckman et al., 2017], and Akerlof [2020] shows that the tendency of rewarding “hard”
topics versus“soft”topics in economics results in“sins of omissions”where issues that are relevant to the literature
and can not be approached in a “hard” way are ignored.
29
(soft, important),“2”: (hard, less important), and“3”: (hard, important). The knowledge function
fmatches any profile of decisions x= (x1, x2)made by the researchers to f(x), the number
of academic articles produced in the economy: f(0,0) = 0,f(1,0) = f(0,1) = f(2,2) = 10,
f(1,1) = 20,f(2,0) = f(0,2) = f(3,1) = f(1,3) = f(2,3) = f(3,2) = 8,f(3,0) = f(0,3) =
4,f(1,2) = f(2,1) = 14, and f(3,3) = 6. We assume that the current academic publishing
system allocates articles in the knowledge economy Eφaccording to the allocation scheme φ
described in Table 12 below.11 The free economy Eφadmits a unique equilibrium x∗= (3,2)
Researcher 1
Researcher 2
0 1 2 3
0 (0,0) (0,10) (0,8) (0,4)
1 (5,5) (5,15) (4,10) (2,6)
2 (3,5) (6,8) (1,9) (3,5)
3 (3,1) (4,4) (4,4) (2,4)
Table 12: Academic Knowledge under φ
where both researchers display favor for hardness relative to importance: Researcher 1 favors
hard and important, and Researcher 2 favors hard and less important. At that equilibrium x∗, the
economy produces 8 scientific papers. The profile x∗is Pareto-dominated by the outcome (1,1)
that produces 20 articles in the economy.
Note that there is another distortion in Table 12. Researcher 1 does not receive the same
treatment as Researcher 2. For instance, when Researcher 1 moves from the reference point
to the strategy ”1”, he or she receives the same reward of 5 as Researcher 2. However, when
Researcher 2 does the same move, he or she keeps all the benefits, and Researcher 1 receives 0
even if the knowledge function produces the same output at both profiles (0,1) and (1,0). What
would happen in this knowledge economy Eφif the Shapley distribution scheme Sh replaces φ?
Well, it is straightforward to show that the researchers are symmetric under the knowledge
function f. Using Anonymity and the other principles of merit-based justice, Table 13 below
describes the allocation of academic articles under the allocation Sh.
From Table 13, we can easily conclude that the free economy Eφis unfair. The identity-
bias that we observe under the academic publishing system φdoes not arise in the free and
fair knowledge environment because the distribution scheme Sh allocates rewards based on
11Although we do not have a clear evidence to support the allocation φ, studies such as Heckman and Moktan
[2020], Colussi [2018], Sarsons [2017], and Card and DellaVigna [2013] document that there exists a preferential
treatment for some group of authors in the academic publishing process.
30
Researcher 1
Researcher 2
0123
0 (0,0) (0,10) (0,8) (0,4)
1 (10,5) (10,10) (7,7) (4,4)
2 (8,0) (7,7) (5,5) (4,4)
3 (4,0) (4,4) (4,4) (3,3)
Table 13: Academic Knowledge under Sh
marginal contributions. The free and fair knowledge economy ESh admits the unique profile
x∗∗ = (1,1) as equilibrium in which both researchers exhibit preferences for soft and important
topics. The outcome x∗∗ is Pareto-optimal and it maximizes the quantity of articles produced
in the economy. Importantly, researchers produce the same number of articles at the equilibrium
given their“abilities” and the fact that they choose the same strategy. The profile x∗∗ in the free
and fair economy ESh strictly dominates the equilibrium outcome in the free knowledge economy
Eφwith the academic publishing system φ.
6.4 Exchange economies
In this section, we apply our theory to pure exchange economies (Section 6.4.1) and markets with
transferable payoff (Section 6.4.2).
6.4.1 Pure exchange economies
There are no production opportunities in a pure exchange economy (or, simply, an exchange
economy), and agents trade initial stocks, or endowments, of goods (or commodities) that they
possess according to a specific rule and attempt to maximize their preferences or utilities. Gen-
erally, an exchange economy consists of a list Ω = (N, l, (wi),(ui)), where:
(a) Nis a finite set of agents (|N|=n < ∞);
(b) lis a positive integer (the number of goods or commodities);
(c) the vector wiis agent i’s endowment vector (wi∈Xi⊆Rl
+), with R+being the set of
non-negative real numbers, and Xithe agent i’s consumption set; and
(d) ui:Xi−→ Ris agent i’s utility function.
31
The amount of good kthat agent idemands in the market is denoted xik, so that agent i’s
consumption bundle is denoted xi= (x11, x12, ..., x1l)∈Xi. An allocation is a distribution of
the total endowment among agents: that is, an outcome x= (xj)j∈N, with xj∈Xjfor all
j∈Nand P
j∈N
xj≤P
j∈N
wj. A competitive equilibrium of an exchange economy is a pair (p∗, z∗)
consisting of a vector p∗∈Rl
+, with p∗6= 0 (the price vector), and an allocation x∗= (x∗
j)j∈N
such that, for each agent i, we have:
p∗x∗
i≤p∗wi,and ui(x∗
i)≥ui(xi)for which p∗xi≤p∗wi, xi∈Xi.
We say that x∗= (x∗
j)j∈Nis a competitive allocation.
In an exchange economy, we can assimilate an agent’s consumption bundle to that agent’s
action in the market. In that respect, we can formulate an exchange economy under mild assump-
tions as a free and fair economy. Consider an exchange economy Ω = (N , l, (wi),(ui)) in which
the number of goods is finite (l < ∞), and each agent i’s consumption set Xiis finite (|Xi|<∞).
For instance, one can assume that agents can only purchase or sell indivisible units of goods in
the market. We can model Ωas a free and fair economy EΩ= (N, X =×j∈NXj, o, F, Sh, u)
where:
(i) each agent i’s action xi∈Xi;
(ii) the reference outcome ois the vector of endowments w;
(iii) F:X−→ Ris the net aggregate utility function, i.e., for x= (xj)j∈N∈X,
F(x) = X
j∈N
[uj(xj)−uj(wj)],with F(w) = 0; and
(iv) the Shapley allocation scheme Sh =udistributes the net aggregate utility F(x)between
agents at each profile x∈X:ui(x) = Shi(F, x)for each i∈N.
Only allocations in the free and fair economy can be selected in the equilibrium. This means that
an outcome x= (xj)j∈N∈Xis an equilibrium in the free and fair economy if
(1) P
j∈N
xj≤P
j∈N
wj, and
(2) xis a pure strategy Nash equilibrium of the strategic game (N, X, Sh).
Our model differs from the exchange economy in at least two important respects. First, the
incentive mechanism is different. Second, the equilibrium prediction from free exchanges between
32
agents in both economies is different in general. A competitive equilibrium exists in an exchange
economy when some assumptions exist on agents’ utilities and endowments. For instance, when
utilities are continuous, strictly increasing, and quasi-concave and each agent initially owns a
positive amount of each good in the market, a competitive equilibrium exists, and many equilibria
might arise. However, under such assumptions on agents’ utilities, the net aggregate utility
function Fis strictly increasing, and thanks to Theorem 2, the free and fair economy admits a
unique equilibrium. Additionally, it is not necessary to impose any assumptions on utilities and
endowments to guarantee the existence of an equilibrium in a free and fair economy. We illustrate
these points in the following examples.
Example 4. Consider an exchange economy with two goods (1 and 2) and two agents (A and
B) in which agent A initially owns a positive amount of good 1, wA= (1,0), while agent B owns
a positive amount of both goods, wB= (2,1). We assume that agent A’s consumption set is
XA={(1,0),(0,0)}and utility is uA(xA) = uA(xA1, xA2) = xA1+xA2. Agent B’s consumption
set is XB={(2,1),(1,1),(0,1),(2,0),(1,0),(0,0)}and utility is uB(xB) = uB(xB1, xB2) =
min{xB1, xB2}. An allocation x= (xA, xB)∈XA×XBis such that xA1+xB1≤3and
xA2+xB2≤1. We can show that there is no competitive equilibrium in this exchange economy
(one reason is the fact that agent A owns zero units of good 2), while the free and fair economy
admits two equilibria xSh
1= (wA, wB)and xSh
2= (wA,(1,1)). Each equilibrium maximizes the
net aggregate utility, F(xS h
1) = F(xSh
2) = 0, with ShA(F, xS h
1) = ShB(F, xSh
1) = 0, and
ShA(F, xSh
2) = ShB(F, xSh
2)=0. This example shows that a free and fair exchange economy
has an equilibrium while a competitive equilibrium does not exist. The next example will show
that the equilibrium of a free and fair exchange economy can coincide with the competitive
equilibrium.
Example 5. Consider a Shapley-Shubik economy [Shapley and Shubik, 1977] in which there
are two agents and two goods. Agent A is endowed with 2 units of good 1, wA= (2,0),
and agent B is endowed with 2 units of good 2, wB= (0,2). We assume that agent A’s
consumption set is XA={(0,0),(0,1),(0,2),(1,0),(1,1),(1,2),(2,0),(2,1),(2,2)}and his
or her utility function is uA(xA1, xA2) = xA1+ 3xA2−1
2(xA2)2; agent B’s consumption set is
XB={(0,0),(1,0),(2,0),(0,1),(1,1),(2,1),(0,2),(1,2),(2,2)}and his or her utility function
is uB(zB1, xB2) = xB2+ 3xB1−1
2(xB1)2. Assume that good 1 is the numeraire (p1= 1),
and let p=p2and X=XA×XB. It is straightforward to note that not all pairs of actions
in Xare feasible in the economy. We can show that the pair E∗= (p∗, x∗), where p∗= 1,
and x∗= (x∗
A= (0,2), x∗
B= (2,0)), is the unique competitive equilibrium of the market. At
the equilibrium allocation (p∗, x∗), agents exchange endowments, and that transaction results in
33
utilities: uA(x∗
1) = uB(x∗
2) = 4. Similarly, strategic interactions among agents in the free and
fair market yield the same outcome x∗. To show that result, we use an approach that allows us
to simplify calculations in the free and fair economy.
Let us denote by Xthe subset of allocations (X⊂X), and consider the following decisions:
a“keep the full endowment”, b“sell 1 unit of good”, and c“sell the full endowment.” Consider
XA=XB={a, b, c}as each agent’s set of decisions. Each vector of decisions in XA×XB
yields a unique outcome (xA, xB)∈X. Precisely, the vector (a, a)entails the unique profile
x= (wA, wB) = ((2,0),(0,2));(a, b)corresponds to x= ((2,1),(0,1));(a, c)corresponds
to x= ((2,2),(0,0));(b, a)corresponds to x= ((1,0),(1,2));(b, b)corresponds to x=
((1,1),(1,1));(b, c)corresponds to x= ((1,2),(1,0));(c, a)corresponds to x= ((0,0),(2,2));
(c, b)corresponds to x= ((0,1),(2,1)); and (c, c)corresponds to x= ((0,2),(2,0)). The
net aggregate utility function Fis defined as: F(x) = F(xA, xB) = uA(xA) + uB(xB)−4.
Using the strategy profile (a, a)as the reference point, Table 14 describes agents’ utilities in
the free and fair economy. For each agent, decision cstrictly dominates decisions aand b. It
follows that the vector (c, c)which corresponds to the outcome xSh = ((0,2),(2,0)) = x∗is the
unique equilibrium in the free and fair economy. In this case, the equilibrium coincides with the
competitive allocation.
Agent A
Agent B
a b c
a(0,0) (0,1.5) (0,2)
b(1.5,0) (1.5,1.5) (1.5,2)
c(2,0) (2,1.5) (2,2)
Table 14: Utilities in the free and fair economy
6.4.2 Markets with transferable payoff
A market with transferable payoff is a variant of a pure exchange economy in which each agent
in the economy is endowed with a bundle of goods that can be used as inputs in a production
system that the agent operates. All production systems transform inputs into the same kind of
output (i.e., money), and this output can be transferred between the agents. In a market, the
payoff can be directly transferred between agents, while in a pure exchange economy only goods
can be directly transferred. Following Osborne and Rubinstein [1994], a market with transferable
payoff consists of a list Π=(N, l, (wi),(fi),(ui)), where:
34
(a) Nis a finite set of agents (|N|=n < ∞);
(b) lis a positive integer (the number of input goods);
(c) the vector wiis agent i’s endowment vector (wi∈Xi⊆Rl
+), with Xibeing the agent i’s
input set;
(d) fi:Xi−→ Ris agent i’s continuous, non-decreasing, and concave production function;
and
(e) uiis agent i’s utility function: ui(fi, p, xi) = fi(xi)−p(xi−wi), with p∈Rl
+(the vector
of positive input prices), and xi∈Xi.
In the market, an input vector is a member of Xi, and a profile (xj)j∈Nof input vectors for which
P
j∈N
xj≤P
j∈N
wjis an allocation. We denote w= (wj)j∈N. Agents can exchange inputs at fixed
prices p∈Rl
+, which are expressed in terms of units of output. At the end of the trade, if agent i
holds the bundle xi, then his or her net expenditure, in units of output, is p(xi−wi). Agent ican
produce fi(xi)units of output, so that his or her net utility is ui(fi, p, xi). A price vector p∗∈Rl
+
generates a competitive equilibrium if, when agent ichooses his or her trade to maximize his or
her utility, the resulting profile (x∗
i)i∈Nof input vectors is an allocation. Formally, a competitive
equilibrium of a market is a pair (p∗,(x∗
i)i∈N)consisting of a vector p∗∈Rl
+and an allocation
(x∗
i)i∈Nsuch that, for each agent i, the vector x∗
imaximizes his or her utility ui(fi, p∗, xi), for
each xi∈Xi. The list (N, l, w, (fi),(ui)) defines a competitive market with transferable payoff.
In a market with transferable payoff, we can view an agent’s input vector as an agent’s action
in the market. Therefore, as in section 6.4.1, we can write a market with transferable payoff
under mild assumptions as a free and fair economy. Consider a market with transferable payoff
Π=(N, l, w, (fi),(ui)) in which the number of input goods is finite (l < ∞), and each agent i’s
input set Xiis finite (|Xi|<∞). As in Section 6.4.1, we can model Πas a free and fair market
EΠ= (N, X =×j∈NXj, o, F, Sh, u), with the difference that for x= (xj)j∈N∈X,
F(x) = X
j∈N
[fj(xj)−fj(wj)].
As in the analysis in section 6.4.1 below, we provide examples that show similarities (Example
6) and differences (Example 7) between the predictions of free and fair markets and markets with
transferable payoff.
Example 6. We consider a single-input market with transferable payoff in which there are two
homogeneous agents who have the same production, w1=w2= 1,fi(xi) = √xi,i∈ {1,2},
35
and X1=X2={0,1,2}. The pair E∗= (p∗=1
2, x∗= (w1, w2)) is the unique competitive
equilibrium of the market, and u1(p∗, x∗
1) = u2(p∗, x∗
2) = 1. Similarly, strategic interactions
among agents in the free and fair market yield the same outcome x∗.
Example 7. As mentioned in Section 6.4.1, generally, the equilibrium predictions of a free and
fair economy and a market with transferable payoff do not coincide. To showcase this point,
we consider a market in which agents’ production functions are not concave. Consider a single-
input market with transferable payoff in which there are two heterogeneous agents in production:
w1= 1,X1={0,1,2,3}, and f1(x1) = 1
2x2
1; and w2= 2,X2={0,1,2,3}, and f2(x2) = x2
2. In
the competitive market, the utility functions are convex and given by: u1(p, x1) = 1
2x2
1−p(x1−1)
and u2(p, x2) = z2
2−p(x2−2). There is no exchange in this market, while strategic interactions
among agents in the free and fair market yield a different outcome: xS h = (0,3).
7 Contributions to the closely related literature
In this paper, we propose a model of a free and fair economy, defining a new class of non-
cooperative games, and we apply it to a variety of economic environments. We prove that four
elementary principles of distributive justice, of long tradition in economic theory, guarantee the
existence of a pure strategy Nash equilibrium in finite games. In addition, we show that when an
economy violates these principles, a pure strategy equilibrium may not exist, resulting in instability
in agents’ actions and in income volatility. We extend this model to incorporate social justice and
inclusion. In this more general model, we also prove several results on equilibrium existence and
efficiency.
Our work contributes to several literatures. It is related to studies of group incentives in
multi-agent problems under certainty. Holmstrom [1982] explores the effects of moral hazard in
individual incentives and efficiency in organizations with and without uncertainty. Like Holmstrom
[1982], we consider that in a free economy, any agent has the freedom to choose any action (or
input) from his or her set of strategies, and the combination of actions from agents generates a
measurable output. However, unlike Holmstrom [1982], there is no uncertainty in the supply of
inputs, and we assume that our allocation scheme follows basic principles of distributive justice.
It follows that our scope, analysis and applications are very different. Moreover, Holmstrom
[1982] finds an impossibility result in his setup (see, Holmstrom [1982, Theorem 1, p. 326]),
but our analysis implies that this result does not extend under fair principles in a framework with
finite action sets. Moreover, we show that any free and fair economy which is strictly monotonic
admits a unique equilibrium, and this equilibrium is optimal and Pareto-efficient (Theorem 2).
36
Our findings therefore underscore the role of justice in shaping individual incentives, stabilizing
contracts among private agents, and enhancing welfare.
By incorporating normative principles into non-cooperative game theory, we have introduced a
new class of finite strategic form games that always admit a Nash equilibrium in pure strategies.
We view this paper as contributing to the small but growing literature that seeks to uncover
conditions under which a pure strategy Nash equilibrium exists in a non-cooperative game with
simultaneous moves. Nash [1951] shows a very prolific result on the existence of equilibrium
points in a finite non-cooperative games. Nash [1951] also shows that there always exists at
least one pure strategy equilibrium in finite symmetric games. However, Nash [1951] was silent
about the existence of pure strategy equilibrium in either finite or infinite non-symmetric strategic
form games. Subsequent research has searched for sufficient and necessary conditions for the
existence of pure strategy Nash equilibrium in different structure of strategic form games. Early
contributions in this respect include, among others, Debreu [1952], Glicksberg [1952], Gale [1953],
Schmeidler [1973], Mas-Colell [1984], Khan and Sun [1995], Athey [2001] in continuous games;
Dasgupta and Maskin [1986a], Dasgupta and Maskin [1986b], Reny [1999], Carbonell-Nicolau
[2011], Reny [2016], Nessah and Tian [2016] in discontinuous economic games; Monderer and
Shapley [1996] in potential games; and Ziad [1999] in fixed-sum games. In these studies, scholars
use different concepts of continuity, convexity and appropriate fixed point results along with
some restrictions on utility functions to prove the existence of a pure strategy Nash equilibrium.
Other contributions that guarantee the existence of equilibrium in pure strategies for finite games
include, among others, Rosenthal [1973], Mallick [2011], Carmona and Podczeck [2020], and
the references listed therein. We follow a different approach from this literature. Unlike our
paper, this literature has not approached the issue of equilibrium existence in a non-cooperative
game from a normative angle. We also apply our theory to different economic environments,
including applications surplus distribution in a firm, exchange economies, self-enforcing lockdown
in networked economies facing contagion, and bias in academic publishing.
Finally, in addition to the previous point, our work can also be viewed as contributing to
the Nash Program [Nash, 1953], which bridges non-cooperative and cooperative game theory.
However, we significantly depart from the main approach taken in this literature so far. This
approach has generally sought to define a non-cooperative game whose solution coincides with
the outcomes of a cooperative solution concept; see Serrano [2021] for a recent survey on this
literature. Our approach, on the contrary, follows the opposite direction. It asks if equilibrium can
be found in a strategic form game in which payoffs obey natural axioms inspired by cooperative
game theory.
37
8 Conclusion
In this paper, we examine how elementary principles of justice and ethics, of long tradition
in economic theory, affect individual incentives in a competitive environment and determine the
existence and efficiency of self-enforcing social contracts. To formalize this problem, we introduce
a model of a free and fair economy, in which each agent freely and non-cooperatively chooses
their input from a finite set, and the surplus generated by these choices is distributed following
four ideals of market justice, which are anonymity, local efficiency, unproductivity, and marginality.
We show that these ideals guarantee the existence of a pure strategy Nash equilibrium. However,
an equilibrium need not be unique or Pareto-efficient. We uncover an intuitive condition—strict
technological monotonicity—, which guarantees equilibrium uniqueness and efficiency. Interest-
ingly, this condition does not guarantee equilibrium efficiency (or even existence) when ideals of
justice are violated in an economy. These ideals therefore lead to positive incentives, given their
desirable equilibrium and efficiency properties.
We extend our analysis to incorporate social justice and inclusion, implemented in the form of
progressive taxation and redistribution and guaranteeing a basic income to unproductive agents.
In this more general setting, we generalize all of our findings. In addition, we examine how the tax
policy affects efficiency, showing that there is a tax rate threshold above which an equilibrium that
is Pareto-efficient always exists in the economy, even in the absence of technological monotonicity.
Moreover, we show that if a free economy is able to choose its reference point, it can always do
so to induce an efficient outcome that is self-enforcing, even if this economy is not monotonic.
By incorporating normative principles into non-cooperative game theory, we have defined a
new class of finite strategic form games that always admit a pure strategy Nash equilibrium. We
develop applications to some classical and recent economic problems, including the allocation
of goods in an exchange economy, surplus distribution in a firm, self-enforcing lockdown in a
networked economy facing contagion, and publication bias in academic publishing. This variety
of applications is possible because we impose no particular assumptions on the structure of agents’
action sets, and our setting is fully non-parametric.
38
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44
Appendix
Proof of Proposition 1.
Sufficiency. We show that the allocation scheme Sh satisfies ALUM.
Anonymity. Let f∈P(X),x∈X,πx∈ Sx
n, and ibe an agent. We show that
Shi(πxfx, x) = Shπx(i)(fx, x).
1. If i /∈Nx, then xi=oi, and πx(i) = i.
Shi(fx, x) = X
a∈∆i
0(x)
ϕ(a, x){fx(a+xiei)−fx(a)}
=X
a∈∆i
0(x)
ϕ(a, x){fx(a)−fx(a)}
= 0.
Similarly,
Shi(πxfx, x) = X
a∈∆i
0(x)
ϕ(a, x){πxfx(a+xiei)−πxfx(a)}
=X
a∈∆i
0(x)
ϕ(a, x){fx(πx(a+xiei)) −fx(πx(a))}.
For a∈∆i
0(x)and xi=oi, we have πx(a+xiei) = πx(a), and Shi(πxfx, x)=0.
Therefore, for each i /∈Nx, we can conclude that Shi(πxfx, x) = Shπx(i)(fx, x).
2. If i∈Nx, then xi6=oi. Assume that πx(i) = j. Then, j∈Nxand xj6=oj.
Shj(fx, x) = X
a∈∆j
0(x)
ϕ(a, x){fx(a+xjej)−fx(a)}
=X
a∈∆j
0(x)
ϕ(a, x){f(a+xjej)−f(a)}).
Similarly,
Shi(πxfx, x) = X
a∈∆i
0(x)
ϕ(a, x){πxfx(a+xiei)−πxfx(a)}
=X
a∈∆i
0(x)
ϕ(a, x){fx(πx(a+xiei)) −fx(πx(a))}
=X
a∈∆i
0(x)
ϕ(a, x){f(πx(a+xiei)) −f(πx(a))}
45
a∈∆i
0(x)implies a= (a1, ..., oi
|{z}
ith component
, ..., an). The vector πx(a) = (πx
1(a), ..., πx
j(a)
|{z}
jth component
, ..., πx
n(a)).
Given that j=πx(i)and ai=oi, it follows that πx
j(a) = ojand πx(a)∈∆j
0(x). We also
have a+xiei= (a1, ..., xi
|{z}
ith component
, ..., an). Given that j=πx(i)and (a+xiei)i=xi6=oi,
it follows that πx
j(a+xiei) = xj. Note that we can write πx(a+xiei) = πx(a) + xjej.
Therefore,
Shi(πxfx, x) = X
a∈∆i
0(x)
ϕ(a, x){f(πx(a) + xjej)−f(πx(a))}
=X
b∈∆j
0(x)
ϕ(b, x){f(b+xjej)−f(b)},where b=πx(a)
=Shj(fx, x).
It follows that the allocation Sh satisfies x-Anonymity for each x∈X. Hence, Sh satisfies
Anonymity.
Local Efficiency. For any f∈P(X)and x∈X, it is immediate that P
i∈N
Shi(f , x) = f(x).
Unproductivity. If agent iis unproductive, then for any f∈P(X)and x∈X, it is
immediate that Shi(f, x)=0, since mc(i, f, a, x)=0for each a∈∆i
0(x).
Marginality. Let f, g ∈P(X)such that mc(i, f, x0, x)≥mc(i, g, x0, x)for all i∈N,x∈X
and x0∈∆i
o(x). By the definition of the value Sh, it is immediate that Shi(f, x)≥Shi(g, x).
Necessity. In this part of the proof, we prove the uniqueness of the Shapley value. Consider
another allocation procedure φwhich satisfies ALUM.
Define the following production function fx∈P(X)for each x∈Xby:
fx(y) = (1if x ∈∆(y)
0if x /∈∆(y)
where x∈∆(y)if and only if [xi6=yi⇒xi=oi].
Lemma 2 (Pongou and Tondji [2018]).Any production function is a linear combination of the
production functions fx:
f=X
x∈X
cx(f)fx,where cx(f) = X
x0∈∆(x)
(−1)|x|−|x0|f(x0).
Let f∈P(X). Define the index Iof the production function fto be the number of non-zero
terms in some expression for fin (2). The theorem is proved by induction on I.
46
a) If I= 0, then f≡0. Let x∈Xand i∈N. Then, mc(i, f, a, x)=0for all a∈Xsuch
that a∈∆i
o(x). Therefore, by Unproductivity, Shi(f, x) = φi(f, x) = 0.
b) If I= 1, then f=cx(f)fxfor some x∈X. Consider Nx={l∈N:xl6=ol}.
Step 1. Let i /∈Nx, i.e., xi=oi.
For any a∈Xsuch that a∈∆i
0(x), we have f(a+xiei)−f(a) = 0, i.e., mc(i, f, a, x) = 0.
It follows that Shi(f, x) = 0. Let y∈Xwith y6=x. Then, x∈∆(y)or x /∈∆(y).
•If x∈∆(y), then xl=ylfor each l∈Nx. If yi=oi, then φi(f, y) = 0 = Shi(f, y).
Assume yi6=oi. Then, for any a∈∆i
0(y), we have mc(i, f, a, y) = f(a+yiei)−f(a).
If x∈∆(a), we also have x∈∆(a+yiei)because xi=oiand yi6=oi. Similarly if
x /∈∆(a), then x /∈∆(a+yiei). Therefore, mc(i, f, a, y)=0for each a∈∆i
0(y),
and Shi(f , y) = 0.
•x /∈∆(y), then f(y)=0. If yi=oi, then Shi(f, y) = 0. Assume yi6=oi. Then, for
any a∈∆i
0(y), we have mc(i, f, a, y) = f(a+yiei)−f(a). If x∈∆(a), then for
each l∈Nx,xl=al6=ol. Or a∈∆(y)implies that for each l∈Nx, we will have
al=yl, because al6=ol. Therefore, for each l∈Nx,al=yl=xl, and given that
yi6=oiand xi=oi, we have x∈∆(y), a contradiction. In fact x∈∆(a)if and only
if x∈∆(a+yiei). Thus, mc(i, f, a, y)=0for each a∈∆i
0(y), and Shi(f , y)=0.
Given that agent iis unproductive, it follows that φi(f, y) = Shi(f, y) = 0 for each y∈X.
Step 2. Let i, j ∈Nsuch that i, j ∈Nxand y∈X. Let πy= (ij)a permutation. Given
that φsatisfies Anonymity, it follows that φsatisfies y-Anonymity, and φi(πxfy, y) =
φj(fy, y). For each z∈∆(y), we have πyfy(z) = fy(z). Thus, πyfy=fy, and
φi(fy, y) = φj(fy, y). By Local efficiency, P
k∈Nx
φk(fy, y) = fy(y) = f(y). Therefore,
P
k∈Nx
φk(fy, y) = |Nx|φk(fy, y), and for each k∈Nx,φk(fy, y) = fy(y)
|Nx|=f(y)
|Nx|. If
x∈∆(y), then f(y) = cx(f). Otherwise, f(y) = 0, and for each k∈Nx,φk(f, y) =
φk(fy, y) = Shk(f, y).
c) Assume now that φis the value Sh whenever the index of fis at most Iand let fhave
index I+ 1, with:
f=
I+1
X
k=1
cxk(f)fxk,all cxk6= 0,and xk∈X.
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For k∈ {1,2, ..., I + 1}, consider:
Nxk=l∈N:xk
l6=ok, N =
I+1
\
k=1
Nxk,and assume i /∈N .
Define the following production function:
g=X
k:i∈Nxk
cxk(f)fxk.
The index of gis at most I. Let x, a ∈Xsuch that a∈∆i
0(x). Then f(a+xiei)−f(a) =
g(a+xiei)−g(a). Consequently, using Marginality, φi(f, x) = φi(g, x). By induction, we
have:
φi(f, x) = X
k:i∈Nxk
cxk(f)fxk(x)
|xk|=Shi(f , x),for x∈X.
It remains to show that for each x∈X,φi(f, x) = Shi(f, x)when i∈N. Let x∈X.
By Anonymity, φi(f, x)is a constant ϕfor all members of N; likewise the value Shi(f, x)
is some constant ϕ0for all members of N(with N > 0). By Local efficiency,
|N|φi(f, x) = |N|ϕ=f(x),
so that,
ϕ=f(x)
|N|.
Similarly,
|N|Shi(f , x) = |N|ϕ0=f(x),
so that,
ϕ0=f(x)
|N|.
It follows that ϕ=ϕ0, and concludes the proof.
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