Lecture Note II-3 Static Games of Incomplete Information. Games of incomplete information. Cournot Competition under Asymmetric Information (cont )

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1 Lecture Note II- Static Games of Incomplete Information Static Bayesian Game Bayesian Nash Equilibrium Applications: Auctions The Revelation Principle Games of incomplete information Also called Bayesian Games At least one player in uncertain about another player s playoff function Example of static game of incomplete information: A sealed-bid auction Each bidder knows his or her own valuation for the good being sold but does not know any other bidder s valuation Bids are submitted in sealed envelopes (players moves can be thought of as simultaneous Example: Cournot Competition under Asymmetric Information Cournot duopoly model with inverse demand given by P(Qa-Q, where Qq +q Firm s cost function is C (q cq Firm s cost function is C (q c H q with probability θ and C (q c L q with probability - θ Information is asymmetric: firm knows its cost function and firm s, but firm know its cost function and only that firm s marginal cost is C H with probability θ and C L with probability - θ All of this is common knowledge: firm knows that firm has superior information, firm knows that firm know this, and so on Cournot Competition under Asymmetric Information Let q (c H and q (c L denote firm s quantity choice as a function of its cost, and let q * denote firm s single quantity choice If firm s cost is high, it will choose q (c H to solve max[( a q * q c ] q q H If firm s cost is low, it will choose q (c L to solve max[( a q * q c ] q q L Cournot Competition under Asymmetric Information Firm chooses q * to solve * * max θ[( a q q ( ch cq ] + ( θ[( a q q ( cl cq ] q The first-order conditions for these three optimization problem are a q * ch q * ( ch a q * cl q *( cl * * θ[( a q ( ch c] + ( θ [( a q ( cl c] q* Cournot Competition under Asymmetric Information The solutions to the three first-order conditions are a c *( H + c θ q ch ( ch cl; 6 a c *( L+ c θ q cl ( ch cl 6 + a c+ θc + ( θ q* If information is symmetric (player know player s cost (c H or c L, say c a c+ c q **( c Implication a c+ c q **( c H q *( ch > q **( ch ; q *( cl < q **( cl q **( cl < q* < q **( ch cl

2 Static Bayesian Games Definition: The normal-form representation of an n-player static Bayesian game specifies the players action spaces A,,A n, their type spaces T, T n, their beliefs p,,p n, and their playoff functions u,,u n. Player i stype t i, is privately known by player i, determines player i s payoff function, u i (a,,a n ;t i, and t i is a member of the set of possible types, T i. Player i s belief p i (t -i t i describes i s uncertainty about the n- other players possible types, t -i, given i s own type, t i. We denote this game by G{A,,A n ;T,,T n ;p,,p n ;u,,u n } Static Bayesian Games Example: Cournot game The firms actions are their quantity choices, q and q Firm has two possible profit or payoff functions π( q, q; c L ( a q q c L q π ( q, q; c H ( a q q c H q Firm has only one possible payoff function (, ; ( π q q c a q q c q Firm s type space is T {c L,c H } and that firm s type space is T {c} Static Bayesian Games When player i s payoff depend not only on the actions (a,,a n but also on all the types (t,,t n, we write this payoff as u i (a,,a n ;t,,t n Assume nature draws a type vector t(t,,t n according to the prior probability distribution p(t player i s belief probability p i (t -i t i can be computed using Bayes rule pt ( i, ti pt ( i, ti pi ( t i ti pt ( pt (, t i t i T i i i Bayesian Nash Equilibrium Definition: In the static Bayesian game G{A,..,A n ;T,,T n ;p,,p n ;u,,u n }, a strategy for player i is a function s i (t i where for each type t i in T i, s i (t i specifies the action from the feasible set A i that type t i would choose if drawn by nature i.e. A strategy is a function from types to actions Definition: In the static Bayesian game G{A,..,A n ;T,,T n ;p,,p n ;u,,u n }, the strategies s*(s *,...,s n * are a (pure strategy Bayesian Nash equilibrium if for each player i and for each of i s types t i in T i, s i *(t i solves * max ui( s * ( t,..., si ( ti, ai, si+ *( ti+,..., sn * ( tn; t pi ( t i ti ai Ai t i T i Example: Battle of Sexes Game Player Fight Player Fight +t c,,+t p Player s payoff if both attend is +t c, where t c is privately known by player Player s payoff if both attend Fight is +t p, where t p is privately known by player t c and t p are independent draws from a uniform distribution on [0,x] Example: Battle of Sexes Game Static Bayesian game G{A c,a p ;T c,t p ;p c,p p ;u c,u p } The action spaces are A c A p {, Fight} The type spaces are T c T p [0,x] The beliefs are p c (t p p p (t c /x for all t c and t p Strategies Player plays if t c exceeds a critical value c, and plays Fight otherwise With probability (x-c/x to play Player plays Fight if t p exceeds a critical value p, and plays otherwise With probability (x-p/x to play Fight

3 Example : Battle of Sexes Game Player s optimal strategy is play if p p p p ( + tc + [ ] [ ] x x x x p p ( + tc x x x tc c p Player s optimal strategy is play Fight if Example : Battle of Sexes Game p c; p + p x 0 x c x p x x x x When x is small x c x p x lim lim lim( x 0 x x 0 x x 0 x c c c c [ ] 0 + ( + tp [ ] + 0 x x x x c c x ( + tp tp p x x c Fight +t c, Fight,+t p Implication: player play with probability / player play Fight with probability / Mixed strategy Nash Equilibrium VS. Pure Strategy Bayesian Nash Equilibrium A mixed- strategy Nash equilibrium is a game of complete information can be (almost always be interpreted as a pure-strategy Bayesian Nash equilibrium in a closely related game with a little bit of incomplete information Example: Battle of Sexes q -q Fight r -r Fight,, Two pure Nash equilibrium (, and (Fight, Fight A mixed strategy Nash Equilibrium (r,-r(/,/ and (q,-q(/, / Application : An auction First-price, sealed-bid auction The bidders simultaneously submit their bids. The higher bidders wins the good and pays the price she bided In case of a tie, the winner is determined by a flip of a coin Two bidders, labeled i, Bidder i has value v i for the good. The two bidders valuations are independently and uniformly distributed on [0,] If bidder i gets the good and plays the price p, then i s payoff is v-p Application : An auction Bayesian game G{A,A ;T,T ;p,p ;u,u } Action spaces A I [0, ] Type space is T i [0,] Player i s action is to submit a bid b i and her type is her valuation v i Player i s payoff function vi bi if bi > bj ui( b, b ; v, v ( vi bi / if bi bj 0 f bi < bj Application : An auction Player s strategy b (v is a best response to player s strategy b (v and vice versa The pair of (b (v,b (v is a Bayesian Nash equilibrium if for each v i in [0,], b i (v i solves max( vi bi Pr obb { i > bj ( vj } + ( vi bi probb { i bj ( vj } b i

4 Application : An auction Linear equilibrium :b i (v i a i +c i v i e.g. b (v a +c v and b (v a +c v bi aj max ui ( vi bi Pr obb { i > bj ( vj } ( vi bi bi c st.. a j j bi aj + cj Note: bi aj bi aj Pr obb { i > bj( vj } Pr obv { j < } cj cj ( vi + aj / if vi aj bi ( vi aj if vi < aj a i a j 0; c i c j / bi ( vi vi / Application : An auction Symmetric equilibrium: suppose player j adopts the strategy b(., and assume b(. is strictly increasing and differential max ui ( vi bi Pr obb { i > bj ( vj } b i Pr obb { i > bv ( j } Pr obb { ( bi > vj} b ( bi v i First order condition d b ( bi + ( vi bi b ( bi 0 dbi bv ( i vi b ( bj vj if bj bv ( j d b ( bi + ( vi bi b ( bi 0 dbi dvi vi + ( vi bv ( i 0 dbv ( i vi + ( vi b( vi 0 b ( vi b ( vi vi + bv ( i vi bv ( i vi vi + k; b(0 0 Application : A Double Auction The buyer s valuation for the seller s good is v b, the seller s is v s. These valuations are private information and are drawn from independent uniform distributions on [0,] The seller names an asking price, p s, and the buyer simultaneously names an offer price, p b If p b > p s, then trade occurs at price p(p b + p s /; if p b < p s then no trade occurs Application : A Double Auction v b x v b >x>v s Trade v b >v s x One price equilibrium The buyer offers x if v b >x The seller demands x if v s <x v s Application : A Double Auction A pair of strategies {p b (v b, p s (v s } is a Bayesian Nash equilibrium if the following two conditions hold Expected price the seller will demand pb+ E[ ps ( vs pb ps ( vs ] max vb prob{ pb ps( vs } pb ps+ E[ pb ( vb pb ( vb ps] max vs prob{ pb( vs ps} ps Expected price the buyer will offer Application : A Double Auction Assume linear BNE p b and p s can be derived by solving as+ pb pb as max vb { pb + } pb cs ps ab cb ab cb ps max { p + + s+ } v + s ps cb Solve FOC simultaneously pb vb + as pb vb + ps vb + ( ab + cb p s vs+ 4 ps( vs as+ cv s s; pb ( vb ab + cv b b as+ cs 0 ps( vs pb as + pb E ps ( vs pb ps ( v s ps pb ( vb ab + cb ps + ab + cb E[ pb ( vb pb ( vb p s] pb as prob{ pb ps( vs } prob{ vs } cs ps ab prob{ pb ( vs ps} prob{ vs } cb ab + cb ps cb 4

5 Application : A Double Auction Application : A Double Auction pb ( vb vb+ ps ( vs vs+ 4 p s,p b /4 p s (v s p b (v b / /4 pb ps vb vs + 4 v b v b v s +/4 v b v s Trade / /4 /4 v s,v b Linear equilibrium v s Direct Mechanisms Direct Mechanisms Static Bayesian games in which each player s only action is to submit a claim about his or her type A new static Bayesian game with the same types spaces and beliefs as original static Bayesian game but with new action spaces and new payoff functions The playoff functions are chosen so as to confront each player with a choice of exactly of its kind The Revelation Principle (Cont Direct Mechanisms Design Consider the Bayesian Nash equilibrium s*(s *,,s n * in static Bayesian game G{A,,A n ;T,,T n ;p,,p n ; u,,u n } A direct mechanism is a new static Bayesian game is G{T,,T n ;T,,T n ;p,,p n ; v,,v n } The Revelation Principle (Myerson 979 Incentive-compatible direct mechanisms Truth-telling is a Bayesian Nash equilibrium Theorem (The revelation Principle Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive-compatible direct mechanism Direct Mechanism Implementation of a direct mechanism ( Each player signs a contract that allows the neutral outsider to dictate the action you will take when we later play the game G ( Each of the players write down a claim about your type τ i, and submit to the neutral outsider ( The neutral outsider use each player s type report in new game τ i, together with the player s equilibrium strategy from the old game s i *, to compute the action the player would have taken in the equilibrium s* if the player s type s type really were (4 The neutral outsider will dictate that each of the players to take the action the neutral outsider have computed for the players, and the players will receive the resulting playoff 5

6 The Revelation Principle (Cont The Revelation Principle (truth-telling is a Bayesian Nash equilibrium (Proof Player i s action τ i and the players type report τ(τ,.., τ n Player i s playoff v i (τ, tu i [s*(τ,t], where t(t,,t n For each of i s types t i in T i, s i *(t i is the best action for i to choose from A i, given that other players strategies are (s *,, s i- *, s i+ *,,s n * If other players tell the truth, then when i s type is t i the best type to claim to be is t i.the new Bayesian Nash equilibrium of is for each player i to play the truth-telling strategyτ i (t i t i for every t i in T i Homework # Problem set.,.,,6,.7,.8 (from Gibbons Due date two weeks from current class meeting Bonus credit Propose new applications in the context of IT/IS or potential extensions from examples discussed 6