Algorithmic Strategy Complexity

Transcription

1 Algorithmic Strategy Complexity Abraham Neyman Hebrew University of Jerusalem Jerusalem Israel Algorithmic Strategy Complexity, Northwesten 2003 p.1/52

2 General Introduction The classical paradigm of game theory assumes full rationality of the interactive agents. Algorithmic Strategy Complexity, Northwesten 2003 p.2/52

3 General Introduction The classical paradigm of game theory assumes full rationality of the interactive agents. In particular, it often assumes unlimited computational power. Algorithmic Strategy Complexity, Northwesten 2003 p.2/52

4 General Introduction The classical paradigm of game theory assumes full rationality of the interactive agents. In particular, it often assumes unlimited computational power. However, there are many decision problems and games for which it is impossible to assume that the agents (players) can either compute or implement an optimal (or best response or approximate optimal) strategy. Algorithmic Strategy Complexity, Northwesten 2003 p.2/52

5 Design and Implementation Algorithmic Strategy Complexity, Northwesten 2003 p.3/52

6 Design and Implementation It is often argued that evolutionary self selection leaves us with agents that act optimally. Algorithmic Strategy Complexity, Northwesten 2003 p.3/52

7 Design and Implementation It is often argued that evolutionary self selection leaves us with agents that act optimally. Therefore, the complexity of finding an optimal (or approximate optimal) strategy is conceptually less disturbing. Algorithmic Strategy Complexity, Northwesten 2003 p.3/52

8 Design and Implementation It is often argued that evolutionary self selection leaves us with agents that act optimally. Therefore, the complexity of finding an optimal (or approximate optimal) strategy is conceptually less disturbing. However, the computational feasibility and the computational cost of implementing various strategies should be taken into account. Algorithmic Strategy Complexity, Northwesten 2003 p.3/52

9 Design and Implementation One can imagine scenarios where the design and choice of strategies is by rational agents with (essentially) unlimited computation power and the selected strategies need be implemented by players with restricted computational resources. Algorithmic Strategy Complexity, Northwesten 2003 p.4/52

10 Design and Implementation One can imagine scenarios where the design and choice of strategies is by rational agents with (essentially) unlimited computation power and the selected strategies need be implemented by players with restricted computational resources. A corporation The USA Navy A soccer team A chess player A computer network Algorithmic Strategy Complexity, Northwesten 2003 p.4/52

11 Pure Mixed and Behavioral Algorithmic Strategy Complexity, Northwesten 2003 p.5/52

12 Pure Mixed and Behavioral In theory, mixed and behavioral strategies are equivalent (in games of perfect recall). Algorithmic Strategy Complexity, Northwesten 2003 p.5/52

13 Pure Mixed and Behavioral In theory, mixed and behavioral strategies are equivalent (in games of perfect recall). In practice, mixed and behavioral strategies are not equivalent. Algorithmic Strategy Complexity, Northwesten 2003 p.5/52

14 Pure Mixed and Behavioral In theory, mixed and behavioral strategies are equivalent (in games of perfect recall). In practice, mixed and behavioral strategies are not equivalent. Recall that A mixed strategy reflects uncertainty regarding the chosen pure strategy, and A behavioral strategies randomizes actions at the decision nodes. Algorithmic Strategy Complexity, Northwesten 2003 p.5/52

15 Strategies in the Repeated Game The number of pure strategies of the repeated game grows at a double exponential rate in the number of repetitions. Many of the strategies are not implementable by reasonable sized computing agents. Algorithmic Strategy Complexity, Northwesten 2003 p.6/52

16 General Motivation: Soccer Algorithmic Strategy Complexity, Northwesten 2003 p.7/52

17 General Motivation: Soccer What makes a good soccer player? Algorithmic Strategy Complexity, Northwesten 2003 p.7/52

18 General Motivation: Soccer What makes a good soccer player? What makes a good soccer coach? Algorithmic Strategy Complexity, Northwesten 2003 p.7/52

19 General Motivation: Soccer What makes a good soccer player? What makes a good soccer coach? Why Argentina failed in the recent world cup? Algorithmic Strategy Complexity, Northwesten 2003 p.7/52

20 General Motivation: Soccer What makes a good soccer player? What makes a good soccer coach? Why Argentina failed in the recent world cup? Answer: failure to understand the MINMAX THEOREM Algorithmic Strategy Complexity, Northwesten 2003 p.7/52

21 Further Motivation: Chess Algorithmic Strategy Complexity, Northwesten 2003 p.8/52

22 Further Motivation: Chess Why is Chess of any interest? Algorithmic Strategy Complexity, Northwesten 2003 p.8/52

23 Further Motivation: Chess Why is Chess of any interest? Answer: It is not interesting! Algorithmic Strategy Complexity, Northwesten 2003 p.8/52

24 Further Motivation: Chess Why is Chess of any interest? Answer: It is not interesting! It is a trivial game: (Zermelo s theorem.) Algorithmic Strategy Complexity, Northwesten 2003 p.8/52

25 Further Motivation: Chess Why is Chess of any interest? Answer: It is not interesting! It is a trivial game: (Zermelo s theorem.) A life threatening experience in my meeting with Kasparov. Algorithmic Strategy Complexity, Northwesten 2003 p.8/52

26 General Objective The impact on strategic interactions the value and equilibrium payoffs of variations of the game where players are restricted to employ Simple Strategies Algorithmic Strategy Complexity, Northwesten 2003 p.9/52

27 Simple Strategies Computable Strategies Algorithmic Strategy Complexity, Northwesten 2003 p.10/52

28 Simple Strategies Finite Automata Algorithmic Strategy Complexity, Northwesten 2003 p.11/52

29 Sample of References: F.A. Ben-Porath (1993) J. of Econ. Theory Repeated Games with Finite Automata Neyman (1985) Economics Letters Bounded Complexity Justifies Cooperation in the Finitely Repeated Prisoner s Dilemma Neyman (1997) in Cooperation: Game-Theoretic Approaches, Hart and Mas Colell, (eds.). Cooperation, Repetition, and Automata Neyman (1998) Math. of Oper. Res. Finitely Repeated Games with Finite Automata Algorithmic Strategy Complexity, Northwesten 2003 p.12/52

30 References: Finite Automata Neyman and Okada (2000) Int. J. of G. Th. Two-person R. Games with Finite Automata Amitai (1989) M.Sc. Thesis, Hebrew Univ. Stochastic Games with Automata (Hebrew) Aumann (1981) in Essays in Game Theory and Mathematical Economics in Honor of O. Morgenstern Survey of Repeated Games Algorithmic Strategy Complexity, Northwesten 2003 p.13/52

31 References: F. A. Ben-Porath and Peleg (1987) Hebrew Univ. (DP). On the Folk Theorem and Finite Automata Papadimitriou and Yannakakis (1994) On Complexity as Bounded Rationality (extended abstract) STOC - 94 Papadimitriou and Yannakakis (1995, 1996) On Bounded Rationality and Complexity manuscript (1995, revised 1996, revised 1998). Stearns (1997) Memory-bounded game-playing computing devices. Mimeo. Algorithmic Strategy Complexity, Northwesten 2003 p.14/52

32 References: Finite Automata Kalai (1990) in Game Theory and Applications, Ichiishi, Neyman and Tauman (eds.) Bounded Rat. and Strat. Complexity in R. G. Piccione (1989) Journal of Economic Theory Finite Automata Eq. with Discounting and Unessential Modifications of the Stage Game Rubinstein (1986) Journal of Economic Theory Finite Automata Play the R. P. s Dilemma Zemel (1989) Journal of Economic Theory Small Talk and Cooperation: A Note on Bounded Rationality Algorithmic Strategy Complexity, Northwesten 2003 p.15/52

33 Simple Strategies Recall Bounded Recall Algorithmic Strategy Complexity, Northwesten 2003 p.16/52

34 References: Bounded Recall Lehrer (1988) Journal of Economic Theory R.G.s with Bounded Recall Strategies Lehrer (1994) Games and Economic Behavior Many Players with Bounded Recall in Infinite Repeated Games Algorithmic Strategy Complexity, Northwesten 2003 p.17/52

35 References: Bounded Recall Neyman (1997) in Cooperation: Game-Theoretic Approaches, Hart and Mas Colell (eds.) Cooperation, Repetition, and Automata Aumann and Sorin (1990) GEB Cooperation and Bounded Recall Bavly and Neyman (forthcoming) Concealed Correlation by Boundedly Rational Players Algorithmic Strategy Complexity, Northwesten 2003 p.18/52

36 Simple Strategies Bounded Strategic Entropy Algorithmic Strategy Complexity, Northwesten 2003 p.19/52

37 References: Bounded Entropy Neyman and Okada Strategic Entropy and Complexity in Repeated Games Games and Economic Behavior (1999) Repeated Games with Bounded Entropy Games and Economic Behavior (2000) Algorithmic Strategy Complexity, Northwesten 2003 p.20/52

38 Simple Strategies Algorithmic Complexity Algorithmic Strategy Complexity, Northwesten 2003 p.21/52

39 References/Origin Solomonov (1964) A formal theory of inductive inference, Information and Control Kolmogorov (1965) Three approaches to the quantitative definition of information, Problems in Information Transmission Chaitin Stearns (1997) Memory-bounded game-playing computing devices. Mimeo. Algorithmic Strategy Complexity, Northwesten 2003 p.22/52

40 Simple Strategies Computable Strategies Finite Automata Bounded Recall Bounded Strategic Entropy Algorithmic Complexity Algorithmic Strategy Complexity, Northwesten 2003 p.23/52

41 Definition: Algoritmic Strategy Complexity The algorithmic complexity of a strategy is the length of shortest description of the strategy. length of shortest program that outputs the strategy. Algorithmic Strategy Complexity, Northwesten 2003 p.24/52

42 Examples of simple strategies in the repeated matching pennies where each players has two actions H and T: Algorithmic Strategy Complexity, Northwesten 2003 p.25/52

43 Examples of simple strategies in the repeated matching pennies where each players has two actions H and T: 1. Tit for Tat: Start playing H; at each other stage i play the action of the other players in the previous stage. Algorithmic Strategy Complexity, Northwesten 2003 p.25/52

44 Examples of simple strategies in the repeated matching pennies where each players has two actions H and T: 1. Tit for Tat: Start playing H; at each other stage i play the action of the other players in the previous stage. 2. At stage i play H whenever i is not a multiple of n, and play T otherwise. Algorithmic Strategy Complexity, Northwesten 2003 p.25/52

45 Examples of simple strategies in the repeated matching pennies where each players has two actions H and T: 1. Tit for Tat: Start playing H; at each other stage i play the action of the other players in the previous stage. 2. At stage i play H whenever i is not a multiple of n, and play T otherwise. 3. At stages i n play H. At stage i > n, play the action of the other player at stage i n. Algorithmic Strategy Complexity, Northwesten 2003 p.25/52

46 Algoritmic Strategy Complexity of Examples 1, 2, and 3. Algorithmic Strategy Complexity, Northwesten 2003 p.26/52

47 Algoritmic Strategy Complexity of Examples 1, 2, and 3. What are the shortest binary computer programs describing each of these strategies? Algorithmic Strategy Complexity, Northwesten 2003 p.26/52

48 Algoritmic Strategy Complexity of Examples 1, 2, and 3. What are the shortest binary computer programs describing each of these strategies? 1) The first strategy is definitely simple. It requires a short program. Algorithmic Strategy Complexity, Northwesten 2003 p.26/52

49 Algoritmic Strategy Complexity of Examples 1, 2, and 3. What are the shortest binary computer programs describing each of these strategies? 1) The first strategy is definitely simple. It requires a short program. 2) The second strategy needs essentially to count up to n, and its algorithmic complexity is roughly log n. Algorithmic Strategy Complexity, Northwesten 2003 p.26/52

50 Algoritmic Strategy Complexity of Examples 1, 2, and 3. What are the shortest binary computer programs describing each of these strategies? 1) The first strategy is definitely simple. It requires a short program. 2) The second strategy needs essentially to count up to n, and its algorithmic complexity is roughly log n. 3) The algorithmic complexity of strategy (3) is also roughly log n. Algorithmic Strategy Complexity, Northwesten 2003 p.26/52

51 More examples Algorithmic Strategy Complexity, Northwesten 2003 p.27/52

52 More examples 4. Fix a binary sequence y. The strategy σ y plays y i at stage i. Algorithmic Strategy Complexity, Northwesten 2003 p.27/52

53 More examples 4. Fix a binary sequence y. The strategy σ y plays y i at stage i. 5. Fix a positive integer n and a distribution p = ( m n, n m ) on {H, T }. n Algorithmic Strategy Complexity, Northwesten 2003 p.27/52

54 More examples 4. Fix a binary sequence y. The strategy σ y plays y i at stage i. 5. Fix a positive integer n and a distribution p = ( m n, n m ) on {H, T }. n Fix an n-periodic (p-typical) sequence y, i.e., n i=1 I(y i = H) = np(h) = m Consider the strategy σ y. Algorithmic Strategy Complexity, Northwesten 2003 p.27/52

55 An Important Example 6. Fix 0 x 1, x 2 1, Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

56 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

57 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

58 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

59 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: 1 i t 1 n I(y i = H) = [t 1 x 1 n] Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

60 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: 1 i t 1 n I(y i = H) = [t 1 x 1 n] I.e., in the first t 1 n stages, the frequency of Hs is x 1. Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

61 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: 1 i t 1 n I(y i = H) = [t 1 x 1 n] I.e., in the first t 1 n stages, the frequency of Hs is x 1. And, t 1 n<i n I(y i = H) = [t 2 x 2 n] Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

62 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: 1 i t 1 n I(y i = H) = [t 1 x 1 n] I.e., in the first t 1 n stages, the frequency of Hs is x 1. And, t 1 n<i n I(y i = H) = [t 2 x 2 n] I.e. in the last t 2 n stages of the cycle the frequency of Hs is x 2. Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

63 An Important Example 6. Fix 0 x 1, x 2, t 1 1, t 2 = 1 t 1, and n. Fix an n-periodic (x 1, t 1 ; x 2, t 2 )-typical sequence y: 1 i t 1 n I(y i = H) = [t 1 x 1 n] I.e., in the first t 1 n stages, the frequency of Hs is x 1. And, t 1 n<i n I(y i = H) = [t 2 x 2 n] I.e. in the last t 2 n stages of the cycle the frequency of Hs is x 2. Consider σ y. Algorithmic Strategy Complexity, Northwesten 2003 p.28/52

64 Algorithmic Complexity of σ y Algorithmic Strategy Complexity, Northwesten 2003 p.29/52

65 Algorithmic Complexity of σ y 4) The algorithmic complexity of a strategy σ y is the Kolmogorov-Chaitin (K-C) complexity y. Algorithmic Strategy Complexity, Northwesten 2003 p.29/52

66 Algorithmic Complexity of σ y 4) The algorithmic complexity of a strategy σ y is the Kolmogorov-Chaitin (K-C) complexity y. 5) The K-complexity of strategy (5) is roughly nh(x) where H denotes the entropy function H(x) = x log x (1 x) log(1 x) Algorithmic Strategy Complexity, Northwesten 2003 p.29/52

67 Algorithmic Complexity of σ y 4) The algorithmic complexity of a strategy σ y is the Kolmogorov-Chaitin (K-C) complexity y. 5) The K-complexity of strategy (5) is roughly nh(x) where H denotes the entropy function H(x) = x log x (1 x) log(1 x) 6) The K-complexity of strategy (6) is roughly t 1 nh(x 1 ) + t 2 nh(x 2 ) Algorithmic Strategy Complexity, Northwesten 2003 p.29/52

68 A Machine U Algorithmic Strategy Complexity, Northwesten 2003 p.30/52

69 A Machine U is a map from a subset P U of the set of finite length binary strings {0, 1} (the program input) Algorithmic Strategy Complexity, Northwesten 2003 p.30/52

70 A Machine U is a map from a subset P U of the set of finite length binary strings {0, 1} (the program input) to the set of strategies in the repeated game. Algorithmic Strategy Complexity, Northwesten 2003 p.30/52

71 A Machine U is a map from a subset P U of the set of finite length binary strings {0, 1} (the program input) to the set of strategies in the repeated game. Thus U : P U Σ. Algorithmic Strategy Complexity, Northwesten 2003 p.30/52

72 A Machine U is a map from a subset P U of the set of finite length binary strings {0, 1} (the program input) to the set of strategies in the repeated game. Thus U : P U Σ. The set of strategies describable the machine U is U(P U ). Algorithmic Strategy Complexity, Northwesten 2003 p.30/52

73 Computable strategies A = A i action profiles of the stage game. A all finite strings of elements of A and Σ strategies of player (i) in the repeated game. A strategy σ is a map from A to A i, the stage actions of player i. A has a natural identification with the natural numbers N. Algorithmic Strategy Complexity, Northwesten 2003 p.31/52

74 Computable strategies in the Repeated Game A computable strategy in the infinitely repeated game is an A i -valued recursive strategy, i.e., a partial recursive function defined over all histories. Algorithmic Strategy Complexity, Northwesten 2003 p.32/52

75 Computable strategies in the Repeated Game A computable strategy in the infinitely repeated game is an A i -valued recursive strategy, i.e., a partial recursive function defined over all histories. A computable strategy in the n-stage repeated game is an A i -valued partial recursive function that is defined over all histories of length < n. Algorithmic Strategy Complexity, Northwesten 2003 p.32/52

76 Programable strategies Fix a partial recursive function of two variables, called a programable machine, U : {0, 1} A A i For every p {0, 1} the function h U(b, h) is a partial strategy. The strategies programable on U are those strategies σ s.t. p {0, 1} s.t. h we have U(p, h) = σ(h). Algorithmic Strategy Complexity, Northwesten 2003 p.33/52

77 Universal Machines A universal machine is a programable machine U such that for every partial recursive strategy σ there is p {0, 1} such that U(p, h) = σ(h) where the equality holds whenever either side is defined. Algorithmic Strategy Complexity, Northwesten 2003 p.34/52

78 Independence of the Universal Machine The set of strategies computable by a universal machine is independent of the specific universal machine, i.e., for every two universal machines U and A we have U(P U ) = A(P A ). Algorithmic Strategy Complexity, Northwesten 2003 p.35/52

79 Independence of the Universal Machine The set of strategies computable by a universal machine is independent of the specific universal machine, i.e., for every two universal machines U and A we have U(P U ) = A(P A ). Indeed, if U and A are two universal machines then there is a string x {0, 1} such that for every binary string y {0, 1} we have U(y) = A(y, x). Algorithmic Strategy Complexity, Northwesten 2003 p.35/52

80 A Machine: a function of the program and history We can think of a (strategic) universal computer U as a function of two variables, x and y. Algorithmic Strategy Complexity, Northwesten 2003 p.36/52

81 A Machine: a function of the program and history We can think of a (strategic) universal computer U as a function of two variables, x and y. The first binary string x {0, 1} can be thought as the program. Algorithmic Strategy Complexity, Northwesten 2003 p.36/52

82 A Machine: a function of the program and history We can think of a (strategic) universal computer U as a function of two variables, x and y. The first binary string x {0, 1} can be thought as the program. The second finite string y B is viewed as the history input. Algorithmic Strategy Complexity, Northwesten 2003 p.36/52

83 A Machine: a function of the program and history We can think of a (strategic) universal computer U as a function of two variables, x and y. The first binary string x {0, 1} can be thought as the program. The second finite string y B is viewed as the history input. The output U(x, y) is either an action of the player or undefined. Algorithmic Strategy Complexity, Northwesten 2003 p.36/52

84 A Machine: a function of the program and history We can think of a (strategic) universal computer U as a function of two variables, x and y. The first binary string x {0, 1} can be thought as the program. The second finite string y B is viewed as the history input. The output U(x, y) is either an action of the player or undefined. Thus, y U(x, y) is a partial function. Algorithmic Strategy Complexity, Northwesten 2003 p.36/52

85 Independence of U Thus, U, A x {0, 1} s.t. U(x x, y) = A(x, y) Algorithmic Strategy Complexity, Northwesten 2003 p.37/52

86 Definition of K U (σ) Definition 1 The algorithmic strategic complexity K U (σ) of a strategy σ is K U (σ) = min p p:u(p)=σ Algorithmic Strategy Complexity, Northwesten 2003 p.38/52

87 Definition of K U (σ) Definition 2 The algorithmic strategic complexity K U (σ) of a strategy σ w.r.t. the universal computer U is K U (σ) = min p p:u(p)=σ The K complexity of a strategy depends obviously on the universal computer U. Algorithmic Strategy Complexity, Northwesten 2003 p.38/52

88 Definition of K U (σ) Definition 3 The algorithmic strategic complexity K U (σ) of a strategy σ w.r.t. the universal computer U is K U (σ) = min p p:u(p)=σ The K complexity of a strategy depends obviously on the universal computer U. Moreover, for every (partial) recursive strategy σ there is a universal interactive computer U such that K U (σ) = 1. Algorithmic Strategy Complexity, Northwesten 2003 p.38/52

89 Asymptotic Independence of the Universal Machine However, the K strategic complexity is asymptotically computer independent: Proposition 1 Let U and A be two universal computers. There is a constant c = c(u, A) s.t. for every strategy σ K U (σ) K A (σ) c Algorithmic Strategy Complexity, Northwesten 2003 p.39/52

90 A simple counting argument U there are less then 2 m strategies with K U (σ) < m. Algorithmic Strategy Complexity, Northwesten 2003 p.40/52

91 A simple counting argument U there are less then 2 m strategies with K U (σ) < m. Therefore, if Σ is a set of strategies with Σ 2 m, then U σ Σ s.t. K U (σ) > m Algorithmic Strategy Complexity, Northwesten 2003 p.40/52

92 Bounded Recall and K-complexity Algorithmic Strategy Complexity, Northwesten 2003 p.41/52

93 Bounded Recall and K-complexity Most strategies of k-recall have K-complexity o( A k ) Algorithmic Strategy Complexity, Northwesten 2003 p.41/52

94 Bounded Recall and K-complexity Most strategies of k-recall have K-complexity o( A k ) Let σ be a strategy of k-recall. Then K U (σ) A k log A i +2 log k c(u) + 2 log A C A k log A i Algorithmic Strategy Complexity, Northwesten 2003 p.41/52

95 Finite Automata and K-complexity Algorithmic Strategy Complexity, Northwesten 2003 p.42/52

96 Finite Automata and K-complexity Corollary 2 Most strategies implementable by an automaton of size m have K-complexity m log m Algorithmic Strategy Complexity, Northwesten 2003 p.42/52

97 Finite Automata and K-complexity Corollary 3 Most strategies implementable by an automaton of size m have K-complexity m log m Proposition 4 Let σ be a strategy implementable by an automaton of size m. Then K U (σ) A i m log m + m log A i + 4 log m + c(u) + 2 log A i Cm log m Algorithmic Strategy Complexity, Northwesten 2003 p.42/52

98 K-strategy-complexity and Bounded strategy entropy Every mixed strategy σ is a probability distribution over the set of pure strategies. Algorithmic Strategy Complexity, Northwesten 2003 p.43/52

99 K-strategy-complexity and Bounded strategy entropy Every mixed strategy σ is a probability distribution over the set of pure strategies. Proposition 6 Let σ be a mixture of pure strategies of Kolmogorov complexity k. Then the strategic entropy of σ is k. Algorithmic Strategy Complexity, Northwesten 2003 p.43/52

100 Lemma Let 0 t 1, P, Q are distributions on A. Algorithmic Strategy Complexity, Northwesten 2003 p.44/52

101 Lemma Let 0 t 1, P, Q are distributions on A. Let X = (X i ) be an n-sequences of independent A-valued random variables s.t. X i P if i tn and X i Q if i > tn. Algorithmic Strategy Complexity, Northwesten 2003 p.44/52

102 Lemma Let 0 t 1, P, Q are distributions on A. Let X = (X i ) be an n-sequences of independent A-valued random variables s.t. X i P if i tn and X i Q if i > tn. The probability that K(X) tnh(p ) (1 t)nh(q) εn Algorithmic Strategy Complexity, Northwesten 2003 p.44/52

103 Lemma Let 0 t 1, P, Q are distributions on A. Let X = (X i ) be an n-sequences of independent A-valued random variables s.t. X i P if i tn and X i Q if i > tn. The probability that K(X) tnh(p ) (1 t)nh(q) εn goes to zero as n. Algorithmic Strategy Complexity, Northwesten 2003 p.44/52

104 Lemma II Assume that X i sequences of independent A-valued random variables such that the distribution of X i is a recursive function of i. Let σ n be the mixed strategy equivalent to the behavioral strategy in the n stage game where the distribution of X i is σ n,i ( ). Then Pr σn ( K(x) n i=1 H(X i ) εn) n 0 Algorithmic Strategy Complexity, Northwesten 2003 p.45/52

105 The function u and cav u Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

106 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

107 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. Define the functions u : [0, ) IR by: Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

108 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. u : [0, ) IR by: action x of player 1 to Define the functions Map each mixed min j g(x, j) Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

109 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. u : [0, ) IR by: Define the functions Maximize over all mixed actions x with entropy H(x) α max min x:h(x) α j g(x, j) Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

110 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. Define the functions u : [0, ) IR by: Set u(α) = max min x:h(x) α j g(x, j) Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

111 The function u and cav u Fix a 2-person 0-sum game G = I, J, g. Define the functions u : [0, ) IR by: u(α) = max min x:h(x) α j g(x, j) The function cav u is the least concave function that is u. Algorithmic Strategy Complexity, Northwesten 2003 p.46/52

112 The Game G n (K(k), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.47/52

113 The Game G n (K(k), Σ) is the n-stage repeated game where Algorithmic Strategy Complexity, Northwesten 2003 p.47/52

114 The Game G n (K(k), Σ) is the n-stage repeated game where Player 1 is restricted to strategies of algorithmic complexity k, Algorithmic Strategy Complexity, Northwesten 2003 p.47/52

115 The Game G n (K(k), Σ) is the n-stage repeated game where Player 1 is restricted to strategies of algorithmic complexity k, and Player 2 is unrestricted. Algorithmic Strategy Complexity, Northwesten 2003 p.47/52

116 The Game G n (K(αn), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.48/52

117 The Game G n (K(αn), Σ) is the n-stage repeated game where Algorithmic Strategy Complexity, Northwesten 2003 p.48/52

118 The Game G n (K(αn), Σ) is the n-stage repeated game where Player 1 is restricted to strategies of algorithmic complexity αn, Algorithmic Strategy Complexity, Northwesten 2003 p.48/52

119 The Game G n (K(αn), Σ) is the n-stage repeated game where Player 1 is restricted to strategies of algorithmic complexity αn, and Player 2 is unrestricted. Algorithmic Strategy Complexity, Northwesten 2003 p.48/52

120 The Result Val G n (K U (αn), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

121 The Result Val G n (K U (αn), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

122 The Result (cav u)(α) Val G n (K U (αn), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

123 The Result (cav u)(α) Val G n (K U (αn), Σ) n Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

124 The Result (cav u)(α) Val G n (K U (αn), Σ) n Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

125 The Result (cav u)(α) Val G n (K U (αn), Σ) n (cav u)(α) Algorithmic Strategy Complexity, Northwesten 2003 p.49/52

126 Proof: Part 1 Algorithmic Strategy Complexity, Northwesten 2003 p.50/52

127 Proof: Part 1 The number of strategies in K U (αn) is 2 αn. Algorithmic Strategy Complexity, Northwesten 2003 p.50/52

128 Proof: Part 1 The number of strategies in K U (αn) is 2 αn. Therefore the strategic entropy of a mixture of pure strategies in K U (αn) is αn. Algorithmic Strategy Complexity, Northwesten 2003 p.50/52

129 Proof: Part 1 The number of strategies in K U (αn) is 2 αn. Therefore the strategic entropy of a mixture of pure strategies in K U (αn) is αn. Therefore, by a result of N and Okada, (cav u)(α) Val G n (K(αn), Σ) Algorithmic Strategy Complexity, Northwesten 2003 p.50/52

130 Proof: Part 2 Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

131 Proof: Part 2 Let α Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

132 Proof: Part 2 Let α (cav u)(α) Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

133 Proof: Part 2 Let α 1 α α 2 (cav u)(α) Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

134 Proof: Part 2 Let α 1 α α 2 and 0 t 1 = 1 t 2 1 be such that α = t 1 α 1 + t 2 α 2 and (cav u)(α) Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

135 Proof: Part 2 Let α 1 α α 2 and 0 t 1 = 1 t 2 1 be such that α = t 1 α 1 + t 2 α 2 and (cav u)(α) = t 1 u(α 1 ) + t 2 u(α 2 ). Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

136 Proof: Part 2 Let α 1 α α 2 and 0 t 1 = 1 t 2 1 be such that α = t 1 α 1 + t 2 α 2 and (cav u)(α) = t 1 u(α 1 ) + t 2 u(α 2 ). Let x i, i = 1, 2, be mixed strategies of the stage game with H(x i ) α i and min j g(x i, j) = u(α i ). Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

137 Proof: Part 2 Let α 1 α α 2 and 0 t 1 = 1 t 2 1 be such that α = t 1 α 1 + t 2 α 2 and (cav u)(α) = t 1 u(α 1 ) + t 2 u(α 2 ). Let x i, i = 1, 2, be mixed strategies of the stage game with H(x i ) α i and min j g(x i, j) = u(α i ). The algorithmic complexity of each n-periodic (x 1, t 1, x 2, t 2 )-typical sequence is t 1 nh(x 1 ) + t 2 nh(x 2 ) + o(n) αn. Algorithmic Strategy Complexity, Northwesten 2003 p.51/52

138 Corollaries and Implications Algorithmic Strategy Complexity, Northwesten 2003 p.52/52

139 Corollaries and Implications U (τ n ) s.t. σ K U (αn) g n (σ, τ n ) (cav u)(α) Algorithmic Strategy Complexity, Northwesten 2003 p.52/52

140 Corollaries and Implications U (τ n ) s.t. σ K U (αn) g n (σ, τ n ) (cav u)(α) (τ n ) s.t. U σ K U (αn) g n (σ, τ n ) (cav u)(α + c(u) n ) Algorithmic Strategy Complexity, Northwesten 2003 p.52/52

141 Corollaries and Implications U (τ n ) s.t. σ K U (αn) g n (σ, τ n ) (cav u)(α) (τ n ) s.t. U σ K U (αn) g n (σ, τ n ) (cav u)(α + c(u) n ) U (τ n ) n σ K U (αn) s.t. g n (σ, τ n ) (cav u)(α c(u)/n) Algorithmic Strategy Complexity, Northwesten 2003 p.52/52