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Similarity As an Extension of Symmetry and Its Application to Superrationality

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Similarity As an Extension of Symmetry and Its Application to Superrationality Similarity as an Extension of Symmetry and its Application to Superrationality 1 Carlos Maximiliano Senci http://orcid.org/0000-0001-9131-3843 1IIESS-Conicet and Departamento de Humanidades-UNS Bahía Blanca, Buenos Aires Argentina [email protected] 2 Fernando Abel Tohmé https://orcid.org/0000-0003-2988-4519 2IMABB-Conicet and Dept. of Economics-UNS Bahía Blanca , Buenos Aires Argentina [email protected] Article Info CDD: 501 Received:05.11.2020; Revised: 01.02.2021; Accepted: 05.03.2021 https://doi.org/10.1590/0100-6045.2021.V44N2.CF Keywords: Game theory Symmetry Similarity Superrationality Abstract: In this paper we present a concept of similarity in games, on which to ground alternative solution concepts, some of which differ from the classical notions in the field. Manuscrito – Rev. Int. Fil. Campinas, v. 44, n. 2, pp. 128-156, Apr.-Jun. 2021. Similarity as an Extension of Symmetry and its... 129 In order to do this we impose a constraint on players’ be- liefs that amounts to a variant of the well-known symmetry principle in classical bargaining theory. We show how this similarity relation helps to identify different Nash equilibria in games, and how these “similar Nash equilibria” can be extended to non-symmetric games. While the notion is nor- mative, it is nonetheless inspired by phenomena in which similarities between players lead to outcomes detected in behavioral studies. We study the strategic properties of the concept of similarity and discuss its relationships with Hofstadter’ notion of superrationality. 1 Introduction As we know, Game Theory studies situations of inter- action between intentional agents. By ‘interactive sit- uations’ we understand those in which more than one player intervenes, and in which the potential results are the product of the joint actions (simultaneous or not) of several players. Game theory proposes solution con- cepts, that is, rules indicating the choices to be made by the players in order to obtain results that satisfy some theoretical criteria of appropriateness. Solution concepts yield a set of strategy profiles (eventually the empty set) for each finite strategic game. In this sense, Game Theory is normative, since it prescribes solu- tions that disregard the actual decision-making pro- cesses carried out by flesh and blood humans. It rather provides solutions to be chosen by idealized agents. Nash Equilibrium (from now on denoted NE; Nash, 1950; Myerson, 1991, p. 105) is the most important so- lution concept in Non-Cooperative Game Theory. In- formally, a NE is a profile of strategies such that none of the players has incentives to change her strategy, given the strategies chosen by the other players. It is well known that the epistemic conditions required by the NE of the players are quite restrictive. In par- Manuscrito – Rev. Int. Fil. Campinas, v. 44, n. 2, pp. 128-156, Apr.-Jun. 2021. Carlos Maximiliano Senci 130 ticular, in the simplest case of two players, the NE requires rationality and mutual knowledge of rational- ity, i.e. both players must be rational and know that they are rational (Aumann and Brandenburger, 1995). These players are clearly idealized agents. In many ubiquitous situations the NE does not pro- vide satisfactory solutions. The typical case is the Prisoner’s Dilemma, but there are other well-studied games, such as the Stag Hunt or Hi-Lo (Colman and Gold, 2018), in which there are payoff-dominant results that the standard theory cannot justify. An outcome is dominant if all players get larger payoffs than with any other results. In many games in which there are multiple NE, nothing indicates which one should be chosen. Then, it becomes necessary to arbitrate solutions that refine the concept of NE. The NE refinement program has gen- erated a large literature (see, in particular, Harsanyi and Selten, 1988). The attempt to refine concepts of strategic rationality does not seem to have found a general solution like the one sought by philosophers, able to accommodate all the contingencies that arise in relation to the justification of any solution. In the same way, it does not seem that they have been able to accommodate in the vast array of anomalies that arise from the experimental results. Behavioral Game Theory can be seen as a body of solution concepts cor- responding to those results, going beyond the limits of rational choice (Ross, 2014). But this proliferation of solution concepts, while an indication of the impetuous development of Behavioral Game Theory, also testifies to a certain failure, partial at least, to provide a uni- fied framework (Tohmé et. al., 2017). In this search of alternative solutions, sometimes it is convenient to assume that players are somewhat similar to each other. A similarity relation could refer Manuscrito – Rev. Int. Fil. Campinas, v. 44, n. 2, pp. 128-156, Apr.-Jun. 2021. Similarity as an Extension of Symmetry and its... 131 to normative aspects, such as rationality. But empir- ical aspects without normative content may be also relevant. Such is the case if we think about the abili- ties or qualities of the players. This leads to solutions based on the strategic depth of players (k-level rea- soning; Camerer et al., 2004), their level of conformity to norms (Bernheim, 1994), or their levels of inequity aversion (Fehr and Schmidt, 1999), among others. The similarity between the players reduces the uncertainty experienced by each of them when facing players she considers similar to her. This behavioral phenomenon has been documented in a series of recent studies, in which different similarity primings had effects on co- ordination or cooperation (eg, Fischer, 2009; Di Guida and Devetag, 2013; Mussweiler and Ockenfels, 2013 ; Chierchia and Coricelli, 2015; Rubinstein and Salant, 2016). There are different attempts to capture similarity relationships between players. In this paper we pro- vide a formal representation of the concept of similar- ity, showing how it helps to identify different NE in games. We also show how these ‘similar Nash equi- libria’ can be extended to non-symmetric games, and under additional restrictions we will show how to ex- tend it to superrationality. Superrationality was proposed by Hofstadter (1983, 1985) to account for the cooperative solution in the Prisoner’s Dilemma (PD) (Flood and Dresher 1952). It is one of the alternative solutions that have been ad- vanced according to which players could do better than the mutual defection in the PD. Proponents of the co- operative outcome usually enrich the description of the PD with an additional assumption according to which the players are somehow similar. Most notably, one of those lines of thought focuses on symmetry as the basic argument for cooperation, rooted in the alleged ability of the players to recognize that they are facing Manuscrito – Rev. Int. Fil. Campinas, v. 44, n. 2, pp. 128-156, Apr.-Jun. 2021. Carlos Maximiliano Senci 132 the same situation and that their strategies are the same (Davis, 1977; Campbell, 1989; Rapoport, 1960). Hofstadter devised the idea of superrational thinkers as a solution for symmetric dilemmatic situations, that is, one in which the incentives are the same for all the players. Superrational thinkers reason their way to the solution by applying a meta-rationality norm implying that every thinker knows what the rational answer is, and knows that every other player knows this fact. So, superrational thinking is not a simple property of an individual but a relational property: “You need to depend not just on their being rational, but on their depending on every- one else to be rational, and on their de- pending on everyone to depend on every- one to be rational-and so on. A group of reasoners in this relationship to each other I call superrational. Superrational thinkers, by recursive definition, include in their cal- culations the fact that they are in a group of superrational thinkers.” (Hofstadter, 1985, p. 748). Since superrational thinking applies to symmetric situations, the (unique) rational solution identified at which each player arrives through her/his own rational reasoning is mirrored by the one found by any other player. Then, superrationality indicates that the solu- tion to each symmetric game is a profile in the diagonal of the game matrix. In Hofstadter’s words: “Any number of ideal rational thinkers faced with the same situation and undergoing similar throes of reasoning agony will nec- essarily come up with the identical answer eventually, so long as reasoning alone is the Manuscrito – Rev. Int. Fil. Campinas, v. 44, n. 2, pp. 128-156, Apr.-Jun. 2021. Similarity as an Extension of Symmetry and its... 133 ultimate justification for their conclusion. Otherwise reasoning would be subjective, not objective as arithmetic is. A conclu- sion reached by reasoning would be a mat- ter of preference, not of necessity. Now some people may believe this of reason- ing, but rational thinkers understand that a valid argument must be universally com- pelling, otherwise it is simply not a valid argument.” (Hofstadter, 1985, p. 746). Following the lead of Hofstadter, who circumscribed superrationality to symmetric games, we start by con- sidering normal form symmetric games to propose a definition of similarity that extends to non-symmetric games. As a first step we need to find a way to ‘sym- metrize’ non-symmetric games, since in these games it is not clear what could mean ‘to make the same choice’. In order to do this we impose a constraint on players’ beliefs that amounts to a variant of the well-known symmetry principle in classical bargaining theory. While we do not allow players to perceive oth- ers as similar with regards to arbitrary features (such as beauty, trustworthiness, race, etc.) we assume them to reason on the basis of their strategic features.
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