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1 Advanced Microeconomics Static Bayesian games Harald Wiese University of Leipzig Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

2 Part E. Bayesian games and mechanism design Static Bayesian games The revelation principle and mechanism design Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

3 Static Bayesian games Conditional probability Introduction and an example 3 De nitions 4 The Cournot model with one sided cost uncertainty 5 Revisiting mixed-strategy equilibria 6 The rst-price auction Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 7

4 Conditional probability De nition Given: nonempty set M, probability distribution prob on M; events A and B. If prob (B) 6= 0, the conditional probability of A given B is de ned by prob (A jb ) = prob (A \ B). prob (B) If prob (A jb ) = prob (A), A and B are independent. Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 7

5 Conditional probability Exercises Problem Throw a dice. What is the conditional probability of, or 3 pips (spots) if the number of pips is odd. Problem If events A and B are independent what about the probability of A \ B? Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 7

6 Static Bayesian games First-price auction Simultaneous bids where the highest bidder obtains the object and pays his bid. Two bidders and who know their own willingness to pay (their type ). Sequence: Nature decides the players types. Every player learns his own type and can condition his bid on his own type. Static Bayesian games = all players act simultaneously after learning their own types. Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 7

7 De nitions Static Bayesian game example for extensive form N = f, g, T = t, t and T = t, t, A = fa, bg and A = fc, dg. At the initial node, nature chooses a type combination t = (t, t ) T T. jt j informations sets for player 0 ( t,t ) ( t,t ) ( t,t ) ( t,t ) a b a b a b c d c d c d c d c d c d a b d c d Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 7

8 De nitions Static Bayesian game De nition (Static Bayesian game) A static Bayesian game is a quintuple where Γ = N, (A i ) in, (T i ) in, τ, (u i ) in = (N, A, T, τ, u), N = f,..., ng is the player set, A i is the action set for player i N with Cartesian product A = in A i and elements a i and a, respectively, T = (T i ) in is the tuple of type sets T i for players i N, τ is the probability distribution on T, and u i : A T i! R is player i s payo function. Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 7

9 De nitions Beliefs Ex ante, before the players learn their own types, their beliefs are summarized by τ. The (a priori) probability for type t i is given by τ (t i ) := t i T i τ (t i, t i ). De nition (Belief) Let Γ be a static Bayesian game with probability distribution τ on T. Player i s ex-post (posterior) belief τ i is the probability distribution on T i given by the conditional probability τ i (t i ) := τ (t i jt i ) = τ (t i, t i ) τ (t i ) = τ (t i, t i ) t i T i τ (t i, t i ). () Harald Wiese (University of Leipzig) Advanced Microeconomics 9 / 7

10 De nitions Beliefs Problem Two bidders and with T = T = fhigh, lowg (willingness to pay) and Find τ (high, high) = 3, τ (high, low) = 3, τ (low, high) = 9, τ (low, low) = 9. τ (t = high) (ex ante) and τ (t = high) if player has learned that his own willingness to pay is high (ex post). Harald Wiese (University of Leipzig) Advanced Microeconomics 0 / 7

11 De nitions Actions, strategies, and equilibria De nition Let Γ be a static Bayesian game. A strategy for player i N is a function s i : T i! A i. We sometimes write s (t) instead of (s (t ),..., s n (t n )) A. A strategy combination s = (s, s,..., s n ) is a Bayesian equilibrium (ex post) if si (t i ) arg max a i A i τ i (t i ) u i (a i, s i (t i ), t i ) t i T i holds for all i N and all t i T i obeying τ (t i ) > 0. The quali cation τ (t i ) > 0 is necessary because τ i (t i ) is ill-de ned otherwise. τ (t i ) = 0 implies anything goes. Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

12 The Cournot model with one sided cost uncertainty The model Static Bayesian game Γ = (N, (A, A ), (T, T ), τ, (u, u )) with the set of two rms N = f, g, the action sets A = A = [0, ), the type sets T = c I, ch = f5, 5g, T = f0g, (c = 0 known to both, c known to, only) probability distribution τ on T given by τ (5, 0) = τ (5, 0) = inverse demand function given by p (X ) = 80 X and hence the payo functions u (x, x, t ) = (p (X ) c ) x = (80 (x + x ) 0) x and u (x, x, t ) = (80 (x + x ) 5) x, t = c l (80 (x + x ) 5) x, t = c h Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

13 The Cournot model with one sided cost uncertainty The model The types are independent (trivial). Ex-ante probabilities: τ (0) : = τ (5, 0) + τ (5, 0) = + =, τ (5) : = τ (5, 0) =, and τ (5) : = τ (5, 0) =. Ex-post probability for c = 0 (player s belief): τ (0) = τ (t, 0) τ (t ) = = = τ (0). Ex-post probability for c = 5 (player s belief): τ (5, 0) τ (5) = = τ (0) = = τ (5). Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 7

14 The Cournot model with one sided cost uncertainty The static Bayesian equilibrium Strategy sets: player : S = fs n : fc n g! o[0, )g to be o identi ed with A player : S = s : c I, ch! [0, ). Firm s choice depends on its type: s R (t ) = = arg maxx [0, ) 80 (x + x ) c l x, t = c l arg max x [0, ) 80 (x + x ) c h x, t = c h x, t = c l x, t = c h Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 7

15 The Cournot model with one sided cost uncertainty The static Bayesian equilibrium Firm s pro t is the expected value i τ (5) 80 hx l + x i +τ (5) 80 hx h + x h i = 60 x l + x h x x which leads to the reaction function x R = s R (0) = 30 0 x 0 h i x l + x h. 4 Three equations in the three unknowns x, x l, and x h. They lead to the Nash equilibrium x = 0, s c l = 45, and s c h = 35. Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 7 x

16 Revisiting mixed-strategy equilibria Continuous types We assume two players i =, with types t i from T i = [0, x], x > 0. Types are independent. Density function: τ x (a) = x, a [0, x] 0, a / [0, x] /x Hence, for 0 a b x, τ x ([a, b]) = Z b a τ x (t) dt = b a. x 0 a b x t i The probability for a speci c type a is zero: τ x ([a, a]) = a a x = 0. Careful: Distinguish τ x ([a, a]) from τ x (a). Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 7

17 Revisiting mixed-strategy equilibria Introducing uncertainty Static Bayesian games allow a fresh look on mixed equilibria. For a given matrix game with a mixed-strategy equilibrium, we construct a sequence of static Bayesian games that converges towards that game. In every Bayesian game, no player i N randomizes. However, from the point of view of the other players from Nn fig who do not know the type t i, it may well seem as if player i is a randomizer. Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 7

18 Revisiting mixed-strategy equilibria Introducing uncertainty theatre Peter football Cathy theatre + t C, 0, 0 football 0, 0, + t P Three equilibria. t C = t P = 0. 3, 3, 3, 3 is the properly mixed one for Result (as shown in the manuscript): Cathy s probabilities in the static Bayesian games (which depend on x) converge towards her equilibrium mixed strategy as x goes towards 0. Harald Wiese (University of Leipzig) Advanced Microeconomics 8 / 7

19 The rst-price auction The model Two bidders and with types t, t [0, ]. Player s distribution is given by 0 a b t Harald Wiese (University of Leipzig) Advanced Microeconomics 9 / 7

20 The rst-price auction The model Formally, the rst-price auction is the static Bayesian game where Γ = (N, (A, A ), (T, T ), τ, (u, u )) N = f, g is the set of the two bidders, A = A = [0, ) are the sets of bids chosen by the bidders, T = T = [0, ] are the type sets, τ is the a probability distribution on T given by τ ([a, b], [c, d]) = (b a) (d c) where 0 a b and 0 c d hold, and Harald Wiese (University of Leipzig) Advanced Microeconomics 0 / 7

21 The rst-price auction The model the payo functions are de ned by 8 < u (a, t ) = u (a, t ) = : 8 < : t a, a > a, t a, a = a, 0, a < a, 0 a > a, t a a = a, t a a < a. and Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

22 The rst-price auction Solution In order to solve the rst-price auction, we use the ex-post equilibrium de nition. For example, if player is of type t [0, ], his condition for the equilibrium strategy combination (s, s ) is 0 s (t ) arg max a A (t a ) τ (ft [0, ] : a > s (t )g) {z } probability that player s bid is higher than player s bid + (t a ) τ (ft [0, ] : a = s C (t )g) A. {z } probability for equal bids =0 Harald Wiese (University of Leipzig) Advanced Microeconomics / 7

23 The rst-price auction Solution Following Gibbons (99, pp. 5), we restrict our search for equilibrium strategies to linear strategies of the forms By... (next slide) s (t ) = c + d t (d > 0), s (t ) = c + d t (d > 0). τ (ft [0, ] : a > c + d t g) = τ t [0, ] : t < a c d = τ 0, a c = a c, d d Harald Wiese (University of Leipzig) Advanced Microeconomics 3 / 7

24 The rst-price auction Solution player s maximization problem is solved by s R (t ) = arg max (t a ) a c = c + t. a A d Player s best response to c (and hence to player s strategy) is a linear strategy with c = c and d =. Analogously, bidder s best response is s R (t ) = c + t with c = c and d =. Now, c = c = c = c 4 implies c = 0. Equilibrium candidate: s : [0, ]! R +, t 7! s (t ) = t and s : [0, ]! R +, t 7! s (t ) = t Harald Wiese (University of Leipzig) Advanced Microeconomics 4 / 7

25 The rst-price auction Solution These strategies form an equilibrium because the strategies are best reponses to each other. If player uses the (linear) strategy s, s (t ) = t is a best response as shown above. Thus, s turns out to be a linear strategy. Therefore, we have found an equilibrium in linear strategies but cannot exclude the possibility of an equilibrium in non-linear strategies. Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 7

26 The rst-price auction First-price or second-price auction? We now take the auctioneer s perspective and ask the question whether the rst-price auction is preferable to the second-price auction. The auctioneer compares the prices min (t, t ) for the second-price auction and max t, t for the rst-price auction. We assume a risk-neutral auctioneer who maximizes the expected payo. It can be shown (see manuscript) that the auctioneer is indi erent between the rst-price and the second-price auction! Harald Wiese (University of Leipzig) Advanced Microeconomics 6 / 7

27 (a) Assume that both players are informed which game they play before they choose their actions. Formulate this game as a static Bayesian game! (b*) Assume that player learns whether they play G A or G B while player does not. Formulate this game as a static Bayesian game and determine all of its equilibria! Harald Wiese (University of Leipzig) Advanced Microeconomics 7 / 7 Further exercise Game G A with probability p >, game G B with probability L > M >. p. Assume G A left right up M, M, L G B left right up 0, 0, L down L, 0, 0 down L, M, M

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