Bayesian Games and Auctions Mihai Manea MIT
Games of Incomplete Information ◮ Incomplete information: players are uncertain about the payoffs or types of others ◮ Often a player’s type defined by his payoff function. ◮ More generally, types embody any private information relevant to players’ decision making. . . may include a player’s beliefs about other players’ payoffs, his beliefs about what other players believe his beliefs are, and so on. ◮ Modeling incomplete information about higher order beliefs is intractable. Assume that each player’s uncertainty is solely about payoffs. Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 2 / 49
Bayesian Game A Bayesian game is a list B = ( N , S , Θ , u , p ) where ◮ N = { 1 , 2 , . . . , n } : finite set of players ◮ S i : set of pure strategies of player i ; S = S 1 × . . . × S n ◮ Θ i : set of types of player i ; Θ = Θ 1 × . . . × Θ n ◮ u i : Θ × S → R is the payoff function of player i ; u = ( u 1 , . . . , u n ) ◮ p ∈ ∆(Θ) : common prior Often assume Θ is finite and marginal p ( θ i ) is positive for each type θ i . Strategies of player i in B are mappings s i : Θ i → S i (measurable when Θ i is uncountable). Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 3 / 49
First Price Auction ◮ One object is up for sale. ◮ Value θ i of player i ∈ N for the object is uniformly distributed in Θ i = [ 0 , 1 ] , independently across players, i.e., � ˜ ˜ i , ∀ θ i p ( θ i , ∀ i ∈ N ) = ∈ [ 0 , 1 ] i i ≤ θ θ , ∈ N . i ∈ N ◮ Each player i submits a bid s i ∈ S i = [ 0 , ∞ ) . ◮ The player with the highest bid wins the object (ties broken randomly) and pays his bid. Payoffs: θ i − s i if s i ≥ s j , ∀ j ∈ N |{ j ∈ N | s i = s j }| u i ( θ, s ) = 0 otherwise. Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 4 / 49
An Exchange Game ◮ Player i = 1 , 2 receives a ticket on which there is a number from a finite set Θ i ⊂ [ 0 , 1 ] . . . prize player i may receive. ◮ The two prizes are independently distributed, with the value on i ’s ticket distributed according to F i . ◮ Each player is asked independently and simultaneously whether he wants to exchange his prize for the other player’s prize: S i = { agree , disagree } . ◮ If both players agree then the prizes are exchanged; otherwise each player receives his own prize. Payoffs: θ if s 1 = s 2 = agree 3 − i u i ( θ, s ) = otherwise. θ i Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 5 / 49
Ex-Ante Representation In the ex ante representation G ( B ) of the Bayesian game B player i has Θ i strategies ( s i ( θ i )) θ i ∈ Θ i ∈ S i —his strategies are functions from types to strategies in B —and utility function U i given by ��� � � � s i ( θ i ) = E p ( u i ( θ, s 1 ( θ 1 ) , . . . , s n ( θ n ))) . U i θ i ∈ Θ i i ∈ N Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 6 / 49
Interim Representation The interim representation IG ( B ) of the Bayesian game B has player set ∪ i Θ i . The strategy space of player θ i is S i . A strategy profile ( s θ i ) i ∈ N ,θ i ∈ Θ i yields utility U θ i (( s θ i ) i ∈ N ,θ i ∈ Θ i ) = E p ( u i ( θ, s θ 1 , . . . , s θ n ) | θ i ) for player θ i . Need p ( θ i ) > 0. . . Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 7 / 49
Bayesian Nash Equilibrium Definition 1 In a Bayesian game B = ( N , S , Θ , u , p ) , a strategy profile s : Θ → S is a Bayesian Nash equilibrium (BNE) if it corresponds to a Nash equilibrium of IG ( B ) , i.e., for every i ∈ N , θ i ∈ Θ i , ∀ s ′ ∈ � � �� θ, s ′ , s E p ( ·| θ i ) [ u i ( θ, s i ( θ i ) , s − i ( θ − i ))] ≥ E p ( ·| θ i ) − i ( θ − i ) u i S i . i i Interim rather than ex ante definition preferred since in models with a continuum of types the ex ante game has many spurious equilibria that differ on probability zero sets of types. Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 8 / 49
Connections to the Complete Information Games When i plays a best-response type by type, he also optimizes ex-ante payoffs (for any probability distribution over Θ i ). Therefore, a BNE of B is also a Nash equilibrium of the ex-ante game G ( B ) . BNE ( B ) : Bayesian Nash equilibria of bayesian game B NE ( G ) : Nash equilibria of normal-form game G Proposition 1 For any Bayesian game B with a common prior p, BNE ( B ) ⊆ NE ( G ( B )) . If p ( θ i ) > 0 for all θ i ∈ Θ i and i ∈ N, then BNE ( B ) = NE ( G ( B )) . Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 9 / 49
Business Partnership Two business partners work on a joint project. ◮ Each businessman i = 1 , 2 can either exert effort ( e i = 1) or shirk ( e i = 0). ◮ Each face the same fixed (commonly known) cost for effort c < 1. ◮ Project succeeds if at least one partner puts in effort, fails otherwise. ◮ Players differ in how much they care about the fate of the project: i has a private, independently distributed type θ i ∼ U [ 0 , 1 ] and receives 2 payoff θ i from success. 2 2 Hence player i gets θ i − c from working, θ i from shirking if opponent j works, and 0 if both shirk. e 2 = 1 e 2 = 0 θ 2 1 − c , θ 2 θ 2 1 − c , θ 2 e 1 = 1 2 − c 2 2 2 e 1 = 0 θ , θ 2 − c 0 , 0 1 Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 10 / 49
Equilibrium e 2 = 1 e 2 = 0 θ 2 1 − c , θ 2 θ 2 1 − c , θ 2 e 1 = 1 2 − c 2 2 2 e 1 = 0 θ , θ 2 − c 0 , 0 1 p j : probability that j works—sufficient statistic for strategic situation faced by player i 2 2 2 Working is rational for i if θ i − c ≥ p i ⇐⇒ 1 − p j ) θ i ≥ c . Thus i must ( j θ play a threshold strategy: work for � c ∗ := θ i ≥ θ i . 1 − p j ∗ ∗ , we get Since p j = Prob ( θ j ≥ θ ) = 1 − θ j j � c � c ∗ = � � 4 c θ ∗ = = θ i i , θ ∗ � c j θ ∗ i √ c . In equilibrium, i = 1 , 2 works if θ i √ 3 c and shirks otherwise. so θ ∗ 3 i = ≥ Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 11 / 49
Auctions ◮ single good up for sale ◮ n buyers bidding for the good ◮ buyer i has value X i , i.i.d. with distribution F and continuous density f = F ′ ; supp ( F ) = [ 0 , ω ] ◮ i knows only the realization x i of X i Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 12 / 49
Auction Formats ◮ First-price sealed-bid auction: each buyer submits a single bid (in a sealed envelope) and the highest bidder obtains the good and pays his bid. Equivalent to descending-price (Dutch) auctions. ◮ Second-price sealed-bid auction: each buyer submits a bid and the highest bidder obtains the good and pays the second highest bid. Equivalent to open ascending-price (English) auctions. Bidding strategies: β i : [ 0 , ω ] → [ 0 , ∞ ) ◮ What are the BNEs in the two auctions? ◮ Which auction generates higher revenue? Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 13 / 49
Second-Price Auction Each bidder i submits a bid b i , payoffs given by x i − max j i b j if b i > max j i b j � � u i = 0 if b i < max j � i b j Ties broken randomly. Proposition 2 In a second-price auction, it is a weakly dominant strategy for every player II β ( i x i ) = x i . i to bid according to Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 14 / 49
Second-Price Auction Expected Payments Y 1 = max i � 1 X i : highest value of player 1’s opponents, distributed according to G with G ( y ) = F ( y n ) − 1 Expected payment by a bidder with value x is m II ( x ) = Prob[Win] × E [ 2nd highest bid | x is the highest bid] = Prob[Win] × E [ 2nd highest value | x is the highest value] = G ( x ) × E [ Y 1 | Y 1 < x ] Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 15 / 49
First-Price Auction Each bidder i submits a bid b i , payoffs given by x i − b i if b i > max j � i b j u i = 0 if b i < max j � i b j Ties broken randomly. Clearly, not optimal/equilibrium to bid own value. Trade-off: higher bids increase the probability of winning but decrease the gains. Symmetric equilibrium: β i = β for all buyers i . Assume β strictly increasing, differentiable. Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 16 / 49
Optimal Bidding Suppose bidder 1 has value X 1 = x and considers bidding b . Clearly, b ≤ β ( ω ) and β ( 0 ) = 0. Bidder 1 wins the auction if max i � 1 β ( X i ) < b . Since β is s. increasing, i ) = β ( max i � 1 X i ) = β ( Y 1 ) , so 1 wins if Y 1 < β − ( b ) . His 1 max i � 1 β ( X expected payoff is G ( β − 1 ( b )) × ( x − b ) . G ′ ( β − 1 ( b )) b ) − G ( β − ( b )) = 0 1 FOC : ( x − β ′ ( β − 1 ( b )) b = β ( x ) ⇒ G ( x ) β ′ ( x ) + G ′ ( x ) β ( x ) = xg ( x ) ⇐⇒ ( G ( x ) β ( x )) ′ = xg ( x ) x 1 � β ( 0 ) = 0 ⇒ β ( x ) = yg ( y ) dy G ( x ) 0 = E [ Y 1 | Y 1 < x ] . Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 17 / 49
Equilibrium Proposition 3 The strategies I β ( x ) = E [ Y 1 | Y 1 < x ] constitute a symmetric BNE in the first-price auction. Mihai Manea (MIT) Bayesian Games and Auctions February 22, 2016 18 / 49
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