Game Theory. Monika Köppl-Turyna. Winter 2017/2018. Institute for Analytical Economics Vienna University of Economics and Business

Transcription

1 Monika Köppl-Turyna Institute for Analytical Economics Vienna University of Economics and Business Winter 2017/2018

2 Static Games of Incomplete Information

3 Introduction So far we assumed that payoff functions are common knowledge. In games of incomplete information: at least one player is uncertain about the payoff of another player. E.g. Sealed bid auction: each bidder knows their valuation but does not know any other bidder s valuation.

4 Example: Cournot Competition under Asymmetric Information I Consider a Cournot duopoly with P(Q) = a Q where Q = q 1 + q 2 Firm 1 s cost function is C 1 (q 1 ) = cq 1 Firm 2 s cost is C 2 (q 2 ) = c H q 2 with probability θ and C 2 (q 2 ) = c L q 2 with probability 1 θ, where c L < c H. Firm 2 knows their own as well as opponent s cost, whereas firm 1 only knows their own cost and probabilities θ and 1 θ. All of this is common knowledge.

5 Example: Cournot Competition under Asymmetric Information II Let q 2 (c H) and q 2 (c L) denote firm 2 s quantity choices and q 1 a choice of firm 1. Firm 2 solves when its cost is high and when it is low. max q 2 [(a q 1 q 2 ) c H ]q 2 max q 2 [(a q 1 q 2 ) c L ]q 2

6 Example: Cournot Competition under Asymmetric Information II Let q 2 (c H) and q 2 (c L) denote firm 2 s quantity choices and q 1 a choice of firm 1. Firm 2 solves when its cost is high and when it is low. Firm 1 solves: max q 2 [(a q 1 q 2 ) c H ]q 2 max q 2 [(a q 1 q 2 ) c L ]q 2 max q 1 {θ[(a q 1 q (c H )) c]q 1 + (1 θ)[(a q 1 q 2 (c L )) c]q 1 }

7 Example: Cournot Competition under Asymmetric Information III Solutions to the three first order conditions are: q 2 (c H ) = a 2c H + c 3 q 2 (c L ) = a 2c L + c θ 6 (c H c L ) θ 6 (c H c L ) q 1 = a 2c + θc H + (1 θ)c L 3

8 Normal Form Representation of Static Bayesian Games I Player s i possible payoff is given by u i (a 1,..., a n ; t i ) where t i is called player s i type t i T i. Each t i corresponds to a different payoff that player i might have.

9 Normal Form Representation of Static Bayesian Games I Player s i possible payoff is given by u i (a 1,..., a n ; t i ) where t i is called player s i type t i T i. Each t i corresponds to a different payoff that player i might have. In Cournot example we had two types of firm 2 with two different payoffs: π 2 (q 1, q 2 ; c L ) = [(a q 1 q 2 ) c L ]q 2 and So we had T 2 = {c L, c H } π 2 (q 1, q 2 ; c H ) = [(a q 1 q 2 ) c H ]q 2

10 Normal Form Representation of Static Bayesian Games I Player s i possible payoff is given by u i (a 1,..., a n ; t i ) where t i is called player s i type t i T i. Each t i corresponds to a different payoff that player i might have. In Cournot example we had two types of firm 2 with two different payoffs: π 2 (q 1, q 2 ; c L ) = [(a q 1 q 2 ) c L ]q 2 and So we had T 2 = {c L, c H } π 2 (q 1, q 2 ; c H ) = [(a q 1 q 2 ) c H ]q 2 Players may know their own type but be uncertain about other players types.

11 Normal Form Representation of Static Bayesian Games II p i (t i t i ) is player i s belief about other players types t i. Usually types are independent so p i (t i t i ) = p i (t i )

12 Normal Form Representation of Static Bayesian Games II p i (t i t i ) is player i s belief about other players types t i. Usually types are independent so p i (t i t i ) = p i (t i ) Definition The normal form representation of an n player static Bayesian game specifics the players action spaces (A 1,..., A n ), their type spaces (T 1,..., T n ), their beliefs p 1,..., p n, and their payoff functions u 1,..., u n. Player i s type t i is privately known by player i and determines player i s payoff function, u i (a 1,..., a n ; t i ). Player i s belief p i (t i t i ) describes i s uncertainty about the n 1 other players possible types.

13 Normal Form Representation of Static Bayesian Games III In the first (fictional) stage of such a game nature draws types from the prior probability distribution p(t) and reveals t i to player i but not to other players. Knowing her own type and the prior distribution, player i can compute their belief about other types using Bayes Rule: p i (t i t i ) = p(t i, t i ) p(t i ) p(t i, t i ) = t i T i p(t i, t i )

14 Normal Form Representation of Static Bayesian Games III Definition In the static Bayesian game G = {A 1,..., A n ; T 1,..., T n ; p 1,..., p n ; u 1,..., u n } a strategy for player i is a function s i (t i ), where for each type in T i, s i specifies the action from the feasible set A i that t i would choose if drawn by nature.

15 Normal Form Representation of Static Bayesian Games III Definition In the static Bayesian game G = {A 1,..., A n ; T 1,..., T n ; p 1,..., p n ; u 1,..., u n } a strategy for player i is a function s i (t i ), where for each type in T i, s i specifies the action from the feasible set A i that t i would choose if drawn by nature. In a pooling strategy all types choose the same action. In a separating strategy each type chooses a different action.

16 Normal Form Representation of Static Bayesian Games III Definition In the static Bayesian game G = {A 1,..., A n ; T 1,..., T n ; p 1,..., p n ; u 1,..., u n } a strategy for player i is a function s i (t i ), where for each type in T i, s i specifies the action from the feasible set A i that t i would choose if drawn by nature. In a pooling strategy all types choose the same action. In a separating strategy each type chooses a different action. In Cournot example we had firm 2 s strategy (q 2 (c H), q 2 (c L)) and firm 1 s strategy q 1.

17 A Bayesian Nash Equilibrium Definition In the static Bayesian game G, the strategies (s1,..., s n ) are a (pure strategy) Bayesian Nash Equilibrium if for each player i and for each type in T i, si solves max u i (s1 (t 1 ),..., si 1(t i 1 ), a i, si+1(t i+1 ),..., sn (t n ); t)p i (t i t i ). a i A i t i T i That is no player wants to change her strategy, even if the change involves only one action by one type.

18 Applications of Static Bayesian Games I First Price Sealed Bid Auction n bidders labelled i = 1, 2,..., n Bidder i has valuation v i, so his payoff is v i p if he gets the good. Valuations are independently and uniformly distributed on [0, 1] Higher bidder wins and pays his bid; In case of a tie a coin is flipped. The bidders are risk neutral. All of this is common knowledge.

19 Applications of Static Bayesian Games II First Price Sealed Bid Auction Formulation as a static Bayesian game: 1 Actions: submit a non negative bid b i if her type is v i 2 Formally: Action space is A i = [0, ) and type is is T = [0, 1]. 3 Beliefs: valuations are independently draw so i believes that v j is U[0, 1] 4 Payoff for player i: v i b i if b i > max(b j ) u i (b i, b j ; v i, v j ) = (v i b i )/m if b i = max(b j ) 0 if b i < max(b j )

20 Applications of Static Bayesian Games III First Price Sealed Bid Auction A bid b i is optimal if for each v i [0, 1], b i (v i ) solves max b i (v i b i )P[b i > max(b j (v j ))] + 1 m (v i b i )P[b i = max(b j (v j ))].

21 Applications of Static Bayesian Games III First Price Sealed Bid Auction A bid b i is optimal if for each v i [0, 1], b i (v i ) solves max b i (v i b i )P[b i > max(b j (v j ))] + 1 m (v i b i )P[b i = max(b j (v j ))]. Let us look only at linear strategies: b i (v i ) = α + βv i.

22 Applications of Static Bayesian Games IV First Price Sealed Bid Auction Player i s best response solves: (P[b i = b j ] = 0!!!) max b i (v i b i )P[b i > α + β max(v j )] [ P[b i > b j ] = P v j < b ] i α = b i α β β [ ] n 1 bi α P[b i > max(b j )] = β

23 Applications of Static Bayesian Games IV First Price Sealed Bid Auction Player i s best response solves: (P[b i = b j ] = 0!!!) max b i (v i b i )P[b i > α + β max(v j )] [ P[b i > b j ] = P v j < b ] i α = b i α β β [ ] n 1 bi α P[b i > max(b j )] = β Maximization problem becomes: [ bi α max(v i b i ) b i β ] n 1

24 Applications of Static Bayesian Games V First Price Sealed Bid Auction F.O.C. : b i = n 1 n v i + α n Linearity requires α = 0, otherwise a function is first flat, and later increasing.

25 Applications of Static Bayesian Games V First Price Sealed Bid Auction F.O.C. : b i = n 1 n v i + α n Linearity requires α = 0, otherwise a function is first flat, and later increasing. Finally: bi = n 1 n v i

26 Applications of Static Bayesian Games VI Second Price Sealed Bid Auction Recall that in a second price sealed bid auction a dominant strategy is to submit your valuation v i This is also a Bayesian Nash Equilibrium!

27 Applications of Static Bayesian Games VII A Double Auction Players are: a seller and a buyer. Seller s cost is c; buyer s valuation is v; Both uniformly distributed over [0, 1] Both submit bids: If b 1 b 2 then they trade at a price b 1 + b 2 2 If b 1 > b 2, no trade and both get 0. Payoffs: Seller: (b 1 + b 2) c if b 1 b 2 otherwise 0 2 Buyer: v (b1 + b2) 2 if b 1 b 2 otherwise 0

28 Applications of Static Bayesian Games VIII A Double Auction In a complete information case we have a continuum of Nash equilibria b 1 = b 2 = b [c, v] All of these are Pareto efficient.

29 Applications of Static Bayesian Games VIII A Double Auction In a complete information case we have a continuum of Nash equilibria b 1 = b 2 = b [c, v] All of these are Pareto efficient. Seller solves: max b 1 Buyer maximizes: max b 2 ( b1 + E(b 2 b 2 > b 1 ) 2 ( v b 2 + E(b 1 b 1 < b 2 ) 2 ) c P (b 2 > b 1 ) ) P (b 1 < b 2 ) Consider linear strategies: b 1 = α 1 + β 1 c and b 2 = α 2 + β 2 v

30 Applications of Static Bayesian Games IX A Double Auction When c U[0, 1] then b 1 U[α 1, α 1 + β 1 ] Thus: E(b 1 b 1 < b 2 ) = α 1 + b 2 2

31 Applications of Static Bayesian Games IX A Double Auction When c U[0, 1] then b 1 U[α 1, α 1 + β 1 ] Thus: E(b 1 b 1 < b 2 ) = α 1 + b 2 2 Buyer s problem becomes: max b 2 ( v b 2 + α1+b2 2 2 ) b 2 α 1 β 1 With a solution: b 2 = 2v + α 1 = v + α 1 3 When seller plays a linear strategy, a buyer s response is also linear.

32 Applications of Static Bayesian Games X A Double Auction When v U[0, 1] then b 2 U[α 2, α 2 + β 2 ]. Thus: E(b 2 b 2 > b 1 ) = b 1 + α 2 + β 2 2 Seller s problem becomes: max b 1 ( b1 + b1+α2+β2 2 2 ) ( c 1 b ) 1 α 2 β 2 Solution: Again linear response. b 1 = 2 3 c + α 2 + β 2 3

33 Applications of Static Bayesian Games XI A Double Auction For the two strategies to be mutual best responses we must have: b 1 = 2 3 c and b 2 = 2 3 v Trade occurs only if v c Compare with full information case (v c): Inefficiency! It can be shown that linear BNE gives highest payoffs, thus there is no other BNE which is efficient.

34 Mechanism Design I We will consider a particular class of Bayesian games in which one player - the principal would like to condition her actions on a private information held by the other player the agent. The agent will only truthfully provide this information if given a correct incentive. The principal should choose a mechanism that gives him a highest expected utility, e.g. auction type which yields highest revenue, price schedule of a monopolist etc. A three step procedure: 1 Principal designs a mechanism (or a contract) 2 Agent(s) accept or reject a contract. 3 Those agents who accept play the game specified by a mechanism.

35 Mechanism Design II The revelation principle assures that the principal can assure a highest expected utility by restricting attention to mechanisms that are accepted in stage 2 and in stage 3 all agents simultaneously and truthfully reveal their types.

36 Mechanism Design II The revelation principle assures that the principal can assure a highest expected utility by restricting attention to mechanisms that are accepted in stage 2 and in stage 3 all agents simultaneously and truthfully reveal their types. Consider an auctioneer that wishes to maximize his revenue. A game: 1 The bidders simultaneously make (possibly dishonest) claims about their types (their valuations) τ i T i (this can be any τ i, irrespective of his real type t i ). 2 Given (τ 1,..., τ n), bidder i pays x i (τ 1,..., τ n) and receives the good with probability q i (τ 1,..., τ n)

37 Mechanism Design II The revelation principle assures that the principal can assure a highest expected utility by restricting attention to mechanisms that are accepted in stage 2 and in stage 3 all agents simultaneously and truthfully reveal their types. Consider an auctioneer that wishes to maximize his revenue. A game: 1 The bidders simultaneously make (possibly dishonest) claims about their types (their valuations) τ i T i (this can be any τ i, irrespective of his real type t i ). 2 Given (τ 1,..., τ n), bidder i pays x i (τ 1,..., τ n) and receives the good with probability q i (τ 1,..., τ n) A games of this type a called a direct mechanisms.

38 Mechanism Design III A direct mechanism in which truth telling (e.g. τ i = t i ) is a Bayesian Nash Equilibrium is called incentive compatible. The Revelation Principle Any Bayesian Nash equilibrium of any Bayesian game can be represented by an incentive compatible direct mechanism.