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Bayesian Nash Equilibrium Unless otherwise noted, the content of this course material is licensed under a Creative Commons Attribution – Non-Commercial 3.0 License. http://creativecommons.org/licenses/by-nc/3.0/ Copyright 2008, Yan Chen The following information is intended to inform and educate and is not a tool for self-diagnosis or a replacement for medical evaluation, advice, diagnosis or treatment by a healthcare professional. You should speak to your physician or make an appointment to be seen if you have questions or concerns about this information or your medical condition. You assume all responsibility for use and potential liability associated with any use of the material. Material contains copyrighted content, used in accordance with U.S. law. Copyright holders of content included in this material should contact [email protected] with any questions, corrections, or clarifications regarding the use of content. 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Viewer discretion advised: Material may contain medical images that may be disturbing to some viewers. 1 SI 563 Lecture 6 Normal Form Games of Incomplete Information Professor Yan Chen Fall 2008 Some material in this lecture drawn Maggie Levenstein, Ross School of Business 2 – Games of incomplete information – Random events and incomplete information – Risk and incentives in contracting – Bayesian Nash equilibrium – Lemons and Auctions 3 • To say that a game is of incomplete information is to say something about what is known about the circumstances under which the game is played • Games having moves of nature that generate asymmetric information between the players • Type: different moves of nature that a single player privately observes 4 • Online auctions – unrealistic to assume that bidders know the other bidders’ valuations or risk attitudes • Russian roulette – Both players have an incentive to pretend to be more reckless than they actually are • Oligopoly – Unrealistic to assume that one firm knows the cost structure of the other firm 5 • Harsanyi’s theory of incomplete information offers a means to get a handle on such matters: a technique for completing a structure in which information is incomplete • Main technique: expected utility calculation 6 Normal Form Games Extensive Form Games Subgame Nash Perfect Nash Complete Information Equilibrium Equilibrium (Perfect Incomplete Information Bayesian Nash Bayesian Equilibrium Equilibrium) 7 (Watson Chapter 24) 8 1,1 F A F 1 F 2 N G 0,0 Friend (p) R -1,0 A 1,-1 Enemy (1 – p) 0,0 E E E N 1 G R -1,0 Chance node: nature’s decision node; Nature determines player 1’s type: Friend (with probability p) or Enemy (1-p); Player 1 observes Nature’s move, so he knows his own type; Player 2 does not observe player 1’s type. 9 2 1 A R F E G G 1,2p-1 -1,0 Strategy: If F, G; if E, G F E G N p,p -p,0 F E N G 1-p,p-1 p-1,0 F E N N 0,0 0,0 In games of incomplete info, rational play require a player who knows his own type to think about what he would have done had he been another type. 10 2 1 C D 12 with probability 2/3 A x,9 3,6 x = B 6,0 6,9 0 with probability 1/3 Player 1’s payoff number x is private information; Player 2 knows only that x=12 with probability 2/3 and x=0 with probability 1/3. This matrix is not the true normal form of the game. 11 C 12,9 2 12 A 1 D 3,6 C 6,0 x = 12 12 B (2/3) D 6,9 C 0,9 (1/3) 0 A x = 0 1 D 3,6 C 6,0 0 B D 6,9 12 2 1 C D 12 0 A A 8,9 3,6 12 0 A B 10,6 4,7 12 0 B A 4,3 5,8 12 0 6,0 6,9 B B Player 1’s decision: (1) whether to select A or B after observing x=0; (2) whether to select A or B after observing x=12. 13 Nature moves at the end of the game … - the simplest case (Watson Chapter 25) 14 • More than one possible outcome can occur. • Probability refers to the likelihood that an outcome will occur. • Objective probabilities • Subjective probabilities • The expected value (average, mean) of a random variable is a weighted average of the values of all possible outcomes, with the probabilities of each outcome used as the weights . • Variance and standard deviation are measures of dispersion of individual outcomes from the mean. 15 Expectation: E(X) = Σ pi(Xi) = p1(X1) + p2(X2) + ... + pn(Xn), where pi = probability of outcome i, Xi = value of random variable associated with outcome i, and p1 + p2 + ... + pn = 1. Variance: n 2 = 2 σ Σ pi[Xi - E(Xi)] i=1 Standard Deviation: σ 16 • Risk neutral individuals maximize expected value • Everyone maximizes expected utility • St Petersburg paradox: – A gamble of consecutive tossing of a fair coin – Payoff doubles for every consecutive heads that appears 17 • The Bet: – 50% chance of winning $100 – 50% chance of winning $400 • Risk Neutral Player: – EV = (.5)(100) + (.5)(400) = 250 – Bet is equivalent to having $250 for sure – Player would be willing to pay up to $250 for a lottery ticket with these odds – Player would be willing to pay up to $250 for insurance rather than assume the risk of any bet that is worse than this bet 18 • The Bet: – 50% chance of winning $100 – 50% chance of winning $400 • Risk Averse Player might have U = (I)½ – EU = (.5)(100)½ + (.5)(400)½ = 15 » What income will give him U = 15 for sure? » 15 = (I) ½ ⇒ I = 225 – Bet is equivalent to having $225 for sure – Player would be willing to pay up to $225 for a lottery ticket with these odds – Player would be willing to pay up to $225 for insurance rather than assume the risk of any bet that is worse than this bet 19 Utility Utility Curve Utility from $20,000 Utility from $10,000 Income 10 20 30 40 50 $1,000s • Risk aversion implies a “concave” utility function, or “diminishing marginal utility of money.” Source: Maggie Levenstein, Ross School of Business 20 Utility Utility Curve Expected utility from a bet with 50-50 odds of 10 and 20 Income 10 20 30 40 50 $1,000s Source: Maggie Levenstein, Ross School of Business 21 Utility Curve Expected utility of 15 for sure Utility (15) > .5 U(10) + .5 U (20) Expected utility from a bet with 50-50 odds of 10 and 20 Income 10 20 30 40 50 $1,000s Source: Maggie Levenstein, Ross School of Business 22 Receive $950 A You H Receive $2000 (1/2) B (1/2) T Receive $0 A: sure thing; B: lottery 23 v v(2000) v(x) v(1000) Concave utility v(0)/2+v(2000)/2 function v(0) x 0 1000 2000 v(x): utility of receiving x dollars 24 v v(2000) v(x) Linear utility v(1000) = v(0)/2+v(2000)/2 function v(0) x 0 1000 2000 25 v(x) Risk neutral Slight risk aversion Greater risk aversion x 26 • Set of players: {Pat, Allen} – Pat: Principal, risk neutral, v(x) = x – Allen: Agent, risk averse, v(x) = x^a • Pat: write a contract, (wage, bonus) • Allen: exert high or low effort • Success depends on Allen’s high effort as well as a random factor 27 2 – w, a U (w -1) (1/2) H (1/2) A 6 – w - b, S a w,b Y (w + b -1) P L a A 2 – w, w N 0, 1 w: wage b: bonus, paid only if successful High effort cost: 1 28 2 –w + (4 - b)/2, w,b H a a A (w + b -1) /2 + (w - 1) /2 P Y A L a 2 – w, w N 0, 1 Pat would like Allen to exert high effort. Can she write a contract that induces it? 29 • If Pat sets b=0 – Allen has no incentive to exert high effort – The best Pat can do is to offer w = 1 – Allen is willing to accept • Best no-bonus contract: w=1, b=0 – Payoff vector (1, 1) 30 • Last step: Incentive-compatibility constraint – Allen’s expected payoff from H has to be at least as great as his payoff from L • Second to last step: Individual rationality (or voluntary participation) constraint – Allen’s expected payoff from H has to be at least as great as his outside option • The best contract has to satisfy the IC and IR constraints with equality • Solution: w = 1, b = 2^(1/a) 31 (Watson Chapter 26) 32 • Method 1 – Write down Bayesian normal form – Solve for Nash equilibrium of the normal form: Bayesian Nash equilibrium – Or, solve for the set of strategies which survive iterated elimination of dominated strategies: Bayesian rationalizability • Method 2 – Treat types as separate players (omit) 33 2 1 C D 12 with probability 2/3 A x,9 3,6 x = B 6,0 6,9 0 with probability 1/3 Player 1’s payoff number x is private information; Player 2 knows only that x=12 with probability 2/3 and x=0 with probability 1/3.
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