Advanced Microeconomics


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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 mixedstrategy equilibria 6 The rstprice 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 Firstprice 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 expost (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 illde 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). Exante probabilities: τ (0) : = τ (5, 0) + τ (5, 0) = + =, τ (5) : = τ (5, 0) =, and τ (5) : = τ (5, 0) =. Expost probability for c = 0 (player s belief): τ (0) = τ (t, 0) τ (t ) = = = τ (0). Expost 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 mixedstrategy 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 mixedstrategy equilibria Introducing uncertainty Static Bayesian games allow a fresh look on mixed equilibria. For a given matrix game with a mixedstrategy 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 mixedstrategy 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 rstprice 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 rstprice auction The model Formally, the rstprice 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 rstprice 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 rstprice auction Solution In order to solve the rstprice auction, we use the expost 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 rstprice 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 rstprice 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 rstprice 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 nonlinear strategies. Harald Wiese (University of Leipzig) Advanced Microeconomics 5 / 7
26 The rstprice auction Firstprice or secondprice auction? We now take the auctioneer s perspective and ask the question whether the rstprice auction is preferable to the secondprice auction. The auctioneer compares the prices min (t, t ) for the secondprice auction and max t, t for the rstprice auction. We assume a riskneutral auctioneer who maximizes the expected payo. It can be shown (see manuscript) that the auctioneer is indi erent between the rstprice and the secondprice 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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