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### lec23

Course: ECON 134A 62360, Spring 2012
School: UC Irvine
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Word Count: 898

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Economic Standard Model: Game Theory Game theory is a tool for studying strategic behavior of several players. The behavior of each player in a game should take into account the behavior of others and the mutual recognition of this interdependence. Applications: Industrial organization (Cournot equilibrium), auctions, bargaining, mechanism design, real games (tennis, baseball, poker) Dominant Strategy...

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Economic Standard Model: Game Theory Game theory is a tool for studying strategic behavior of several players. The behavior of each player in a game should take into account the behavior of others and the mutual recognition of this interdependence. Applications: Industrial organization (Cournot equilibrium), auctions, bargaining, mechanism design, real games (tennis, baseball, poker) Dominant Strategy Equilibrium ALICE: Confess ALICE: Deny BOB: Confess Alice: -3 Bob: -3 Alice: -10 Bob: -1 BOB: Deny Alice: -1 Bob: -10 Alice: -2 Bob: -2 A dominant strategy for agent I is a strategy that is better than any other choice regardless of the choices made by other agents. If each agent has a dominant strategy, then this is an equilibrium in dominant strategies. Iterated Dominant Strategy Equilibrium ALICE: Confess ALICE: Deny BOB: Confess Alice: -3 Bob: -3 Alice: -10 Bob: -5 BOB: Deny Alice: -1 Bob: -10 Alice: -2 Bob: -2 If agents can repeatedly eliminate dominated strategies (for themselves and other agents), then they may arrive at a single choice. For example, Bob can predict that Alice should confess. Then he should confess as well. This outcome is called iterated dominant strategy equilibrium. Nash Equilibrium Nash equilibrium is an outcome where each player uses a strategy that is best against the equilibrium strategies of other players. There can be several NE. ALICE: Confess BOB: Confess ALICE: Deny Alice: -6 Bob: -6 Alice: -11 Bob: -11 (Confess, Confess) and (Deny, Deny) are both NE. BOB: Deny Alice: -11 Alice: -4 Bob: Bob: -4 -11 Nash equilibrium in Mixed Strategies Bob Call the bet Nash equilibrium in pure strategies need not exist. Suppose that Bob Fold to bet The bank is \$10 The possible bet is \$20 Alice has great hand or no hand with equal probabilities. Bob has medium hand that beats no hand but loses to great hand. Alice Always bet Alice: 5 Bob: 5 Alice Alice: 15 Bet only Bob: -5 with great hand Alice: 10 Bob: 0 Alice: 5 Bob: 5 Nash equilibrium in Mixed Strategies Bob Call the bet Key idea: if an agent randomizes, then he has the same expected utility from both strategies. Let p be probability of always betting In order to randomize, Bob must have equal expected payoffs Alice Always bet Alice: 5 Bob: 5 Bob Fold to bet Alice: 10 Bob: 0 5p -5(1-p) = 5(1-p) p = 10/15 = 2/3 Alice Alice: 15 Bet only Bob: -5 with great hand Alice: 5 Bob: 5 Nash equilibrium in Mixed Strategies Key idea: if an agent randomizes, then he has the same expected utility from both strategies. Bob Call the bet Bob Fold to bet Let q be probability of calling any bet In order to randomize, Alice must have equal expected payoffs 5q + 10(1-q) = 15q + 5(1-q) q = 5/15 = 1/3 Alice Always bet Alice: 5 Bob: 5 Alice Alice: 15 So Alice always bets with probability 2/3 and Bob calls with Bet only Bob: great probability -5 with 1/3. hand Alice: 10 Bob: 0 Alice: 5 Bob: 5 Randomization in experiments Subjects do not randomize very well. they alternate strategies too often (Rapoport and Budescu, 92) they generate samples that are too balanced What is the probability that exactly 5 out of 10 coin flips are tails? < 25% People tend to generate balanced sequences far too often. Walker and Wooders (01) observe that players alternate too often between strategies in tennis matches. Behavioral Models of Randomization Feature Matching Heuristic: Rapoport and Budescu (97) Subjects have memory of length m and they are more likely to pick a strategy at step (m+1) that balances the previous sequence of length m. For example, if CCCFFCCF have been previously chosen, and m= 5, then the next choice is more likely to be C If m =7, then the next choice is more likely to be F. Empirical data: m = 7 on average Equilibrium in beliefs Subjects do not exploit the patterns in randomization of other agents, but focus mostly on the average frequency of various strategies. This explains why tennis players who exhibit patterns in their randomization are rarely exploited. Bloomfield (94), Shachat (02) some players do not randomize at all, and still do fairly well in a population of agents who do randomize. Quantal Response Equilibrium Players do not choose the best response with certainty, but they choose better responses with higher probabilities. This behavior can be motivated by bounded rationality, uncertainty, framing etc. Iterated Dominance in Experiments Nagel (95) Each player picks a number from 0 to 100. The guess close half of the mean pick wins the prize. Then x>50 never wins and is dominated. If nobody plays x> 50, then x>25 never wins and is dominated. If nobody plays x>25 then x>12.5 never wins and is dominated . So the only iterated dominant strategy equilibrium is x = 0. Yet people seem to iterate only a few times (1 to 3) Some popular answers are 25 (one iteration), 12.5 (two iterations), 6.25 (three iterations) K-Level Thinking People operate on different levels: Level 0: picks a random strategy, or pick some status quo Level 1: picks the best response against Level 0 agents Level 2: picks the best response against Level 1 agents Level 3: picks the best response against Level 2 agents Etc. Stahl and Wilson (95) identify other types as well Nave Nash: assuming others play Nash Real-life Nash: assuming some others play Nash, but some others are Level k K-Level Thinking Camerer, Ho, and Chong (04) propose a cognitive hierarchy theory where the distribution of each level of thinking has a Poisson distribution. The single parameter in this distribution can be estimated from empirical data. This behavioral model does better than plain NE, and is capable of explaining spikes in observed strategies.
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