Randomness, Predictability, and Complexity in Repeated Interactions v0.02 Preliminary version, please do not cite. Shuige Liua, Zsombor Z. Meder´ b,∗ aFaculty of Political Science and Economics, Waseda University, 1-6-1, Nishi-Waseda, Shinjuku-Ku, 169-8050, Tokyo, Japan. bHumanities, Arts and Social Sciences, Singapore University of Technology and Design, 8 Sompah Road, 487372 Singapore. Abstract Nash equilibrium often requires players to adopt a mixed strategy, i.e., a randomized choice between pure strategies. Typically, the player is asked to use some randomizing device, and the story usually ends here. In this paper, we will argue that: (1) Game theory needs to give an account of what counts as a random sequence (of behavior); (2) from a game-theoretic perspective, a plausible account of randnomness is given by algorithmic complexity theory, and, in particular, the complexity measure proposed by Kolmogorov; (3) in certain contexts, strategic reasoning amounts to modelling the opponent’s mind as a Turing machine; (4) this account of random behavior also highlights some interesting aspects on the nature of strategic thinking. Namely, it indicates that it is an art, in the sense that it cannot be reduced to following an algorithm. Keywords: repeated games, mixed strategy, Kolmogorov complexity, Turing machine. JEL: C72, C73. IThe authors gratefully acknowledge the support of Grant-in-Aids for Young Scientists (B) of JSPS No. 17K13707, Grant for Special Research Project No. 2017K-016 of Waseda University, and Start-up Research Grant SRHA 15-110 of the Singapore University of Technology and Design. ∗Corresponding author Email addresses: [email protected] (Shuige Liu), [email protected] (Zsombor Z. Meder)´ Submitted to Games 2017 April 16, 2017 Contents 1 Introduction 3 2 Repeated games, Nash equilibrium, and repeated Situations 3 2.1 The problem of learning in repeated games . .3 2.2 Interpretations of probability for mixed strategies . .4 2.3 Act rationality and process rationality . .4 2.4 The limits of ‘strategy’, and modeling repeated interaction . .5 3 Categories of Stage Situations 6 3.1 Predictable and unpredictable situations . .6 3.2 Mixed-strategy equilibria . .8 3.3 Combined situations . .8 4 Understanding repeated interactions through Kolmogorov complexity 9 4.1 Prelimiaries . .9 4.1.1 Binary strings . .9 4.1.2 The stage game . 10 4.1.3 Encoding games . 10 4.2 Turing machines . 11 4.2.1 The halting problem . 12 4.3 Kolmogorov complexity . 12 References 12 2 1. Introduction Repeated interactions are a core topic for the social sciences. Two principal reasons might be cited to account for this fact. First, compared to one-shot interactions, repeated ones account for the dominant share of social behavior. Our family members, friends, sexual partners, work colleagues, or business partners are usually stable over time. Indeed, even on the competitive market, a large share of interactions take place not between anonymous buyers and sellers, but between partners who engage in a series of trades and who are acquinted with each other. Second, stable behavioral and epistemic patterns, such as customs, conventions, and culture are possible and meaningful only through repeated interactions. Conversely, if our lives consisted of a succession of one-shot interactions, discovering regularities in social behavior would likely be prohibitively difficult. The goal of many a social scientist is thus to acquire an understanding of repeated interactions. This involves dealing with three, interconnected problems: 1. Modeling the rules of repeated interaction. 2. Developing a descriptive and normative account of individual decision-making in such situations. 3. Discovering and describing the stable behavioral and epistemic patterns which emerge from these interactions. In this paper, we discuss these three problem from the perspective of game theory. Game theory’s standard solution for problem (1) is to model a repeated interactions as a stage game which is recurring over time. To problems (2) and (3), game theory provides several answers, including the notion of a mixed-strategy Nash-equilibrium, refinements of the Nash equilibrium concepts applied to extensive-form games (subgame-perfect equilibrium etc.), the Folk Theorem, etc. We will argue that the idea of mixed-strategy equilibrium fails to capture essential elements of strategic decision-making in repeated situation. Further, we will argue that there is conceptual inconsistency between the theory of repeated games and mixed- strategy equilibria. Based on these arguments, in Sections 3 and 4, we will give an outline of our approach and show that it can overcome those difficulties and provide insights for our understanding of individual’s behavior in repeated situations. 2. Repeated games, Nash equilibrium, and repeated Situations 2.1. The problem of learning in repeated games The theory of repeated games was developed to model repeated situations, and individuals’ behavior in them (Mertens and Sorin, 2015). Here, a repeated game is formulated as a super-game constituted by a sequence of stage games. An individual’s behavior in a repeated situation is then formulated as a strategy in this super game, that is, a pre-determined plan recommending an action for a player at every possible node. Therefore, this approach gives little space for the study of learning and adjustment which are important and essential in an individual’s behavior in repeated situations; after all, what does it mean by learning and adjustment if one’s behavior is nothing but following a pre-determined plan of actions? This problem seems have long been noticed among game theorists, and, correspondingly, substantial work has been devoted to it. For example, learning theory (Fudenberg and Levine, 1998) and evolutionary game theory (Weibull, 1997) use various statistical methods to formulate players’ learning process in a repeated game. However, the core of this statistical method is set-theoretical model (Aumann, 1976), which itself has some conceptual problems (Kaneko, 2002; Liu, 2016). In addition, the statistical method cannot capture an individual’s initiative learning, which is thought by many psychologists to be an essential part of learning process (Levine, 1975). It can hardly be said that the problem of learning in repeated games has been fully solved.1 The notion of Nash equilibrium was originally developed to describe some necessary condition for an outcome in a one- shot non-cooperative game (Nash, 1950, 1951). Like the notion of the market equilibrium, the Nash equilibrium concept can 1Recently, the statistical approach has been adopted and extended by computer scientists on researches of artificial intelligence, such as pattern recognition and machine learning and has acheived some significant breakthroughs (Bishop, 2010; Murphy, 2012). However, the problems mentioned above persist. 3 be interpreted as a stable pattern of behavior in a repeated situation (Kaneko, 2004, Chapter 2). Further, as Nash equilibrium is a vector of strategies in which no one can improve his payoff by a unilateral deviation, it is conceptually more convenient to apply the Nash equilibrium concept in repeated, rather than one-shot situations, since in the latter achieving a Nash equilibrium is more demanding on each player’s epistemic abilities (Aumann and Brandenburger, 1995; Kaneko and Hu, 2013). There is a literature using pure-strategy Nash equilibrium to study social behavior patterns, (e.g. Kaneko and Matsui, 1999). On the other hand, there are some critical problems with the notion of mixed-strategy Nash equilibrium. 2.2. Interpretations of probability for mixed strategies Let us start from the notion of a mixed strategy. Mathematically, a mixed strategy is a probability distribution over the set of pure strategies of a player. The first question concerns the meaning of “probability” in this context. Among various interpretations of probability (Hajek,´ 2012), only the subjective and frequentist interpretation seem applicable to the game- theoretic setting. The subjective interpretation is a standard one for mixed strategies in one-shot games. According to this view, a mixed strategy represents the uncertainty about the opposites’ choices (cf. Aumann and Brandenburger, 1995; Binmore, 1994). One difficulty that appears here is that this interpretation is incompatible with the original meaning behind the concept of strategy, that is, a strategy of a player is, first of all, some plan of behavior carried out by that player, rather than some belief in other players’ mind (von Neumann and Morgenstern, 1947; Luce and Raiffa, 1957). Second, a player is meant to choose a strategy from a strategy set, that is, he is assumed to have causal control over the strategy he would follow. However, it is not immediately clear that a player can have such a direct influence about his opponent’s beliefs. These difficulties disappear if we adopt the frequentist interpretation of probability. According to this, in a repeated game, the probability vector which represents a mixed strategy indicates the relative frequency of the use of each pure strategy. However, besides the problems inherent to the frequentist interpretation in general (cf. ?)), a basic difficulty here is that setting merely the frequencies of strategies to a particular value is neither necessary, nor sufficient as a proper guide to a player’s behavior. To see that it is not necessary, consider a game with a Nash equilibrium for the stage game which prescribes a particular pure strategy to be used with probability 50%. If the game is played an odd number of times, it is impossible for the strategy to be played exactly with this relative frequency, even assuming the player has access to a proper randomizing device. To see that having the proper frequency is not a sufficiently good guide for behavior, consider a player who follows a simple pattern in his behavior, while following the desired frequency. If this behavioral pattern is detected by his partner, who adjusts his behavior accordingly, the game might be out of equilibrium soon.