A central question in game theory, learning, and other fields is how a rational intelligent agent should behave in a complex environment, given that it cannot perform unbounded computations. We study strategic aspects of this question by formulating a simple model of a game with additional costs (computational or otherwise) for each strategy. While a zero-sum game with strategy costs is no longer zerosum, we show that its Nash equilibria have an interesting structure and the game has a new type of “value. ” We also show that potential games with strategy costs remain potential games. Both zero-sum and potential games with strategy costs maintain a very appealing property: simple learning dynamics converge to Nash equilibrium. 1 The Approach and Basic Model How should an intelligent agent play a complicated game like chess, given that it does not have unlimited time to think? This question reflects one fundamental aspect of “bounded rationality, ” a term coined by Herbert Simon [1]. However, bounded rationality has proven to be a slippery concept to formalize (prior work has focused largely on finite automata playing simple repeated games such

This talk studies the implications of bounding the complexity of players' strategies in long term interactions. The complexity of a strategy is measured by the size of the minimal automaton that can implement it. A finite automaton has a finite number of states and an initial state. It prescribes the action to be taken as a function of the current state and its next state is a function of its current state and the actions of the other players. The size of an automaton is its number of states. The results study the equilibrium payoffs per stage of the repeated games when players' strategies are restricted to those implementable by automata of bounded size. The first talk will concentrate maily on the 0-sum case and address the following topics/questions. 1 What is the relation between the bounds of the automata sizes and the quantitative advantage of the player with the larger bound. (Theorems 1 and 3 of the enclosed paper) 2 What is the duration (number of repetition) needed for an unrestricted player to exploit fully his advantage over a player with bound automata (Conjecture 2 of the enclosed paper including a positive solution of its second part). 3 The existence of a deterministic periodic sequence (with period n) which is asymptotically random for every automata of size o(n= log n) (Proposition 2 of the enclosed paper). 1

We model differences among agents in their ability to recognize temporal patterns of prices. Using the concept of DeBruijn sequences in two dynamic models of markets, we demonstrate the existence of equilibria in which prices fluctuate in a pattern that is independent of the fundamentals and that can be recognized only by the more competent agents.

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We study nitely repeated two-person zero-sum games in which Player 1 is restricted to mixing over a xed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 along with an optimal mixed strategy. We show that the entropy of this mixed strategy appears as a factor in an exact formula for the value of the game and thus is seen to have a direct numerical eect on the game's value. We develop upper and lower bounds on the value of these games that are within an additive constant and discuss how our results are related to the work of Neyman and Okada on strategic entropy (Neyman and Okada, 1999, Games Econ. Behavior 29, 191-223). Finally, we use these results to bound the value of repeated games in which one of the players uses a computer with a bounded memory and is further restricted to using a constant amount of time at each stage. Journal of Economic Literature Classi cation Number: C72 Key Words: Bounded rationality, entropy, repeated games, nite automata This is a preprint of an article that appears in Games and Economic Behavior 43 (2003) 107-136. Page numbering and gure placement may dier.

In Part I, we study bounded rationality in repeated two-person zero-sum games. First we investigate infinitely repeated games in which both players are restricted to pure strategies that can be executed on a finite automaton. In particular, we provide an upper bound on the number of states that Player 2 needs to defeat Player 1 when Player 1 is restricted to simple cycles of length m. Next we argue that the finite automaton approach to bounded rationality is not satisfactory. As an alternative, we propose limiting the number of strategies available to the players. We provide a thorough study of finitely repeatedly zero sum games in which Player 1 is restricted to mixing over a fixed number of pure strategies while Player 2 is unrestricted. We describe an optimal set of pure strategies for Player 1 and a method for describing these strategies such that any strategy from this set can be efficiently executed given its description. We develop upper and lower bounds on the value of these games and discuss how the value is related to the strategic entropy function defined by Neyman and Okada (1999). Finally, we show that an approximately optimal set can be produced in time which is linear in the size of the set. This set achieves a total expected payoff that is within an additive constant of the optimal.

Unpredictable behavior is central for optimal play in many strategic situations because a predictable pattern leaves a player vulnerable to exploitation. A theory of unpredictable behavior is presented in the context of repeated two-person zero-sum games in which the stage games have no pure strategy equilibrium. Computational complexity considerations are introduced to restrict players ’ strategy sets. The use of Kolmogorov complexity allows us to obtain a sufficient condition for equilibrium existence. The resulting theory has implications for the empirical literature that tests the equilibrium hypothesis in a similar context. In particular, the failure of some tests for randomness does not justify rejection of equilibrium play.

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Let G = 〈I, J, g 〉 be a two-person zero-sum game. We examine the two-person zero-sum repeated game G(k, m) in which player 1 and 2 place down finite state automata with k, m states respectively and the payoff is the average per stage payoff when the two automata face off. We are interested in the cases in which player 1 is “smart ” in the sense that k is large but player 2 is “much smarter ” in the sense that m ≫ k. Let S(g) be the value of G were the second player is clairvoyant, i.e., would know the player 1’s move in advance. The threshold for clairvoyance is shown to occur for m near min(|I|, |J|) k. For m of roughly that size, in the exponential scale, the value is close to S(g). For m significantly smaller (for some stage payoffs g) the value does not approach S(g).

We study learning effectiveness as a function of memory size. We quantify the maximal level that a bounded memory machine (or agent) can match (or reproduce) a long string of inputs as a function of the input length k and the memory size n. The input string is an element of I k and the output string is an element of J k and the loss of the agent when matching an input coordinate i ∈ I with an output coordinate j ∈ J is g(i, j). This level is expressed by a function v(p, θ) of two variables: a probability p on I and a nonnegative θ ≥ 0. The function v(p, θ) is defined as a function of the triple G = 〈I, J, g〉. It equals the minimum of EQg(i, j), where the minimization is over all distributions Q on action pairs with marginal p on I, denoted QI, and the mutual information IQ(i; j) = H(QI) + H(QJ) − H(Q) ≤ θ, where H is the entropy function. If i1,..., ik are iid I-valued random variables with distribution p, then for T ⊂ J k we have E min (j1,...,jk)∈T 1 ∑k k I ∗ of I-elements to J we have E minτ∈T 1 ∑k k log |T | v(p, k log |T |