Table of Links Abstract and 1. Introduction Abstract and 1. Introduction A free and fair economy: definition, existence and uniqueness 2.1 A free economy 2.2 A free and fair economy Equilibrium existence in a free and fair economy 3.1 A free and fair economy as a strategic form game 3.2 Existence of an equilibrium Equilibrium efficiency in a free and fair economy A free economy with social justice and inclusion 5.1 Equilibrium existence and efficiency in a free economy with social justice 5.2 Choosing a reference point to achieve equilibrium efficiency Some applications 6.1 Teamwork: surplus distribution in a firm 6.2 Contagion and self-enforcing lockdown in a networked economy 6.3 Bias in academic publishing 6.4 Exchange economies Contributions to the closely related literature Conclusion and References A free and fair economy: definition, existence and uniqueness 2.1 A free economy 2.2 A free and fair economy A free and fair economy: definition, existence and uniqueness 2.1 A free economy 2.1 A free economy 2.2 A free and fair economy 2.2 A free and fair economy Equilibrium existence in a free and fair economy 3.1 A free and fair economy as a strategic form game 3.2 Existence of an equilibrium Equilibrium existence in a free and fair economy 3.1 A free and fair economy as a strategic form game 3.1 A free and fair economy as a strategic form game 3.2 Existence of an equilibrium 3.2 Existence of an equilibrium Equilibrium efficiency in a free and fair economy Equilibrium efficiency in a free and fair economy Equilibrium efficiency in a free and fair economy A free economy with social justice and inclusion 5.1 Equilibrium existence and efficiency in a free economy with social justice 5.2 Choosing a reference point to achieve equilibrium efficiency A free economy with social justice and inclusion A free economy with social justice and inclusion 5.1 Equilibrium existence and efficiency in a free economy with social justice 5.1 Equilibrium existence and efficiency in a free economy with social justice 5.2 Choosing a reference point to achieve equilibrium efficiency 5.2 Choosing a reference point to achieve equilibrium efficiency Some applications 6.1 Teamwork: surplus distribution in a firm 6.2 Contagion and self-enforcing lockdown in a networked economy 6.3 Bias in academic publishing 6.4 Exchange economies Some applications 6.1 Teamwork: surplus distribution in a firm 6.1 Teamwork: surplus distribution in a firm 6.2 Contagion and self-enforcing lockdown in a networked economy 6.2 Contagion and self-enforcing lockdown in a networked economy 6.3 Bias in academic publishing 6.3 Bias in academic publishing 6.4 Exchange economies 6.4 Exchange economies Contributions to the closely related literature Contributions to the closely related literature Contributions to the closely related literature Conclusion and References Conclusion and References Conclusion and References Appendix Appendix 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 noncooperatively) 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 nonparametric. 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. 1 Introduction It is generally acknowledged that justice is the foundation of a stable, cohesive, and productive society.[1] However, 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? fair principles B: Under which conditions do fair principles lead to equilibrium efficiency? fair principles 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: free and fair economy Anonymity: Your pay should not depend on your name.[2] Local efficiency: No portion of the surplus generated at any profile of input choices should be wasted. Unproductivity: An unproductive agent earns nothing. Marginality: A more productive agent should not earn less. Anonymity: Your pay should not depend on your name.[2] Anonymity: Your pay should not depend on your name.[2] Anonymity name Local efficiency: No portion of the surplus generated at any profile of input choices should be wasted. Local efficiency: No portion of the surplus generated at any profile of input choices should be wasted. Local efficiency: Unproductivity: An unproductive agent earns nothing. Unproductivity: An unproductive agent earns nothing. Unproductivity Marginality: A more productive agent should not earn less. Marginality: A more productive agent should not earn less. Marginality 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, 2018, 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. 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] Although a pure strategy equilibrium always exists in any free and fair economy, this equilibrium 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.[6] A 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 redistribution. 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 violates 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 technology 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 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 peerreview 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.[7] Finally, 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. This paper is available on arxiv under CC BY 4.0 DEED license. This paper is available on arxiv under CC BY 4.0 DEED license. available on arxiv [1] The 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] Here, 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. [4] The 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. [5] As 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 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. [6] A clear example is the prisoner’s dilemma game. Economies that are modeled by such games are monotonic, although their unique equilibrium is Pareto-inefficient. [7] See, for example, a recent study by Akerlof [2020] on the consequences of mostly rewarding “hard” research topics in the field of economics. Authors: (1) Ghislain H. Demeze-Jouatsa, Center for Mathematical Economics, University of Bielefeld (demeze jouatsa@uni-bielefeld.de); (2) Roland Pongou, Department of Economics, University of Ottawa (rpongou@uottawa.ca); (3) Jean-Baptiste Tondji, Department of Economics and Finance, The University of Texas Rio Grande Valley (jeanbaptiste.tondji@utrgv.edu). Authors: (1) Ghislain H. Demeze-Jouatsa, Center for Mathematical Economics, University of Bielefeld (demeze jouatsa@uni-bielefeld.de); (2) Roland Pongou, Department of Economics, University of Ottawa (rpongou@uottawa.ca); (3) Jean-Baptiste Tondji, Department of Economics and Finance, The University of Texas Rio Grande Valley (jeanbaptiste.tondji@utrgv.edu).
