EC319 Economic Theory and Its Applications, Part II: Lecture 2

Transcription

1 EC319 Economic Theory and Its Applications, Part II: Lecture 2 Leonardo Felli NAB January 2014

2 Static Bayesian Game Consider the following game of incomplete information: Γ = {N, Ω, A i, T i, µ i, u i } where N is the set of players, Ω is the set of states, A i is the action space for each player i T i is the set of types for every player i µ i are the beliefs of player i u i is player i s payoff for a given state ω Ω. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

3 Bayesian Strategies and Payoffs A strategy for player i is a mapping from T i to A i. Let S i be the set of player i s strategies. It is a contingent plan of actions for each type configuration. Given s = (s 1,..., s n ) and µ i, player i s interim (given he knows t i but not t i ) expected payoff for type t i is: E Ω,T i {u i (s i (t i ), s i (t i ); ω) t i } Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

4 Bayesian Nash Equilibrium A strategy profile s = (s1,..., s n) is a Bayesian Nash Equilibrium if for every i N, given the beliefs µ i, si assigns an optimal action for each t i that maximizes player i s interim expected payoff. Definition (Bayesian Nash Equilibrium) The strategy profile s = (s 1,..., s n) is a Bayesian Nash Equilibrium if and only if E Ω,T i { ui (s i (t i), s i (t i); ω) t i } E Ω,T i { ui (a i, s i (t i); ω) t i } a i A i, t i T i, i N Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

5 First-Price Sealed Bid Auction Two bidders N = {1, 2}; Both bidders can submit any non-negative bid: A 1 = {b 1 0} A 2 = {b 2 0} The highest bidder obtains the good and pays a price equal to his bid. We assume that if both players submit the same bid the object is allocated to each bidder with probability 1/2. We assume that the value of the good to each bidder is known to the bidder but it is not known to the opponent. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

6 First-Price Sealed Bid Auction (cont d) Let v 1 and v 2 be the values of the good to player 1 respectively player 2. We assume that these values can be any number between 0 and 1: 0 v 1 1, 0 v 2 1. We assume that each bidder believes that the value of the opponent is uniformly distributed in [0, 1]. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

7 First-Price Sealed Bid Auction (cont d) In other words from the point of view of each player: Pr{v 2 v} = v, µ 1 (v) = 1, 0 v 1 Pr{v 1 v} = v, µ 2 (v) = 1, 0 v 1 The payoffs to the players are: v 1 b 1 if b 1 > b 2 u 1 (b 1, b 2 ) = (1/2) (v 1 b 1 ) if b 1 = b 2 0 if b 1 < b 2 v 2 b 2 if b 2 > b 1 u 2 (b 1, b 2 ) = (1/2) (v 2 b 2 ) if b 2 = b 1 0 if b 2 b 1 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

8 First-Price Sealed Bid Auction (cont d) Strategies for both players are the functions b(v 1 )and b(v 2 ). Assume that b( ) is strictly increasing and differentiable. The best reply of player i, that knows v i, is: max b i (v i b i ) Pr{b i > b(v i )} (v i b i ) Pr{b i = b(v i )} Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

9 First-Price Sealed Bid Auction (cont d) Recall that since µ i is a density then Pr{b i = b(v i )} = 0. Moreover, let v i = b 1 (b i ) be the valuation that bidder i must have to choose action b i. Then: Pr{b i > b(v i )} = Pr{b 1 (b i ) > v i } = = Pr{v i < b 1 (b i )} = b 1 (b i ). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

10 First-Price Sealed Bid Auction (cont d) Hence player i s best reply is: max b i (v i b i ) b 1 (b i ) with first order conditions: b 1 (b i ) + (v i b i ) d b 1 (b i ) d b i = 0 these are a pair of differential equations defining b(v i ) and b(v i ). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

11 First-Price Sealed Bid Auction (cont d) We restrict ourselves to b i = b(v i ): symmetric Bayesian Nash equilibria. Substituting b i = b(v i ) into the first order conditions of player i s best reply, we get: b 1 (b(v i )) + (v i b(v i )) d b 1 (b(v i )) d b i = 0 Clearly b 1 (b(v i )) = v i and d b 1 (b(v i )) d b i = 1 b (b 1 (b i )) = 1 b (v i ). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

12 First-Price Sealed Bid Auction (cont d) In other words: or v i + (v i b(v i )) 1 b (v i ) = 0 b (v i ) v i + b(v i ) = v i Notice that: d b(v i ) v i d v i = b (v i ) v i + b(v i ) Hence d b(v i ) v i d v i = v i Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

13 First-Price Sealed Bid Auction (cont d) Integrating both sides we get: b(v i ) v i = 1 2 v 2 i + K Notice that a boundary condition on individual strategies is that no player bids more than his/her valuation: b(v i ) v i which for v i = 0 implies: b(0) = 0 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

14 First-Price Sealed Bid Auction (cont d) Substituting in the solution to the difference equation we get: K = 0 Hence the unique Bayesian Nash equilibrium of this auction game is for every i = 1, 2: b(v i ) = 1 2 v i. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

15 Second-Price Sealed Bid Auction As above: Two bidders N = {1, 2}; Action spaces: A 1 = {b 1 0} A 2 = {b 2 0} The value of the good to the bidders:. 0 v 1 1, 0 v 2 1 or the type spaces of the bidders are: T 1 = [0, 1], T 2 = [0, 1] We assume that each bidder believes that the value of the opponent is uniformly distributed in [0, 1]. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

16 Second-Price Sealed Bid Auction (cont d) The beliefs of each bidder are: µ 1 (v) = 1, µ 2 (v) = 1 The key difference is that the bidder that submits the highest bid gets the good but only pays the second highest bid. The payoffs to the players are then: u 1 (b 1, b 2 ) = u 2 (b 1, b 2 ) = v 1 b 2 if b 1 > b 2 (1/2) (v 1 b 2 ) if b 1 = b 2 0 if b 1 < b 2 v 2 b 1 if b 2 > b 1 (1/2) (v 2 b 2 ) if b 2 = b 1 0 if b 2 b 1 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

17 Second-Price Sealed Bid Auction (cont d) Symmetric strategies for both players are the functions b(v 1 ) and b(v 2 ) (strictly increasing and differentiable). Once again, we have that {b i > b(v i )} = {b 1 (b i ) > v i } = {v i < b 1 (b i )} The expected payoff of player i, that knows v i and takes expectations on v i, is then: E v i <b 1 (b i ) {v i b(v i )} = b 1 (b i ) 0 (v i b(v i ))dv i Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

18 Second-Price Sealed Bid Auction (cont d) Hence player i s best reply is: max b i b 1 (b i ) 0 (v i b(v i ))dv 1 Using Leibniz s rule: ( ) β(y) G(x, y)dx y α(y) = = G(β(y), y) β (y) G(α(y), y)α (y) + + β(y) α(y) G(x, y) y dx Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

19 Second-Price Sealed Bid Auction (cont d) the first order conditions of player i s best reply problem are: (v i b(b 1 (b i ))) d b 1 (b i ) d b i = 0 in other words (v i b i ) 1 b (v i ) = 0 these are a pair of differential equations defining b(v i ) and b(v i ). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

20 Second-Price Sealed Bid Auction (cont d) We restrict ourselves to strategies such that b (v i ) > 0. From (v i b i ) 1 b = 0 i {1, 2} (v i ) We then conclude that the unique Bayesian Nash equilibrium of this auction game is for every i = 1, 2: b(v i ) = v i. i {1, 2} In other words the unique Bayesian Nash equilibrium of this auction is for each bidder to bid exactly his/her own valuation. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

21 Purification of Mixed Strategies Consider the following battle of sexes game: M C M 1, 2 0, 0 C 0, 0 2, 1 Recall that this game has three mixed strategy Nash equilibria: two pure strategy Nash equilibria (M, M) and (C, C); one non-degenerate mixed strategy Nash equilibrium ( 1 3, 2 ) 3 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

22 Purification of Mixed Strategies (cont d) We focus on the non-degenerate mixed strategy Nash equilibrium. Recall that this equilibrium is characterized by the fact: each player when choosing his/her strategy is completely indifferent among all choices; at every stage of the game each player is uncertain about which strategy the opponent chooses. The important feature of mixed strategies is the fact that there exist uncertainty about the opponent play. It is less important that each player is indifferent when choosing a strategy. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

23 Purification of Mixed Strategies (cont d) To highlight this we are going to show that similar behaviour is foreseen in an environment in which uncertainty arises from a little incomplete information. In particular, we modify the battle of sexes game introducing a small uncertainty that each player has on the payoff faced by the opponent. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

24 Purification of Mixed Strategies (cont d) In particular assume that the game played is the following: M C M 1, 2 + t 2 0, 0 C 0, t 1, 1 where t 1 and t 2 both take values between 0 and x > 0 where we take x to be small. We assume that t 1 is known to player 1 but unknown to player 2. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

25 Purification of Mixed Strategies (cont d) Player 2 believes that t 1 is uniformly distributed in [0, x]: Pr{t 1 y} = y x, 0 y x. We assume that symmetrically t 2 is known to player 2 but unknown to player 1, Player 1 believes that t 2 is uniformly distributed in [0, x]: Pr{t 2 y} = y x, 0 y x. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

26 Purification of Mixed Strategies (cont d) In the terminology of static games of incomplete information: N = {1, 2}, A i = {M, C} T i = [0, x] µ i = 1 x. Consider now the following pure strategies for this incomplete information game: Player 1 plays C if t1 > c 1, where 0 < c 1 < x; Player 1 plays M if t1 < c 1. Player 2 plays M if t 2 > c 2, where 0 < c 2 < x; Player 2 plays C if t2 < c 2. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

27 Purification of Mixed Strategies (cont d) Consider now player 1 s expected payoff if he decides to play C: since (2 + t 1 ) Pr{t 2 < c 2 } + 0 Pr{t 2 > c 2 } = (2 + t 1 ) c 2 x. Pr{t 2 < c 2 } = c 2 x, Pr{t 2 > c 2 } = 1 c 2 x. Player 1 s expected payoff if he decides to play M: 0 Pr{t 2 < c 2 } + 1 Pr{t 2 > c 2 } = 1 c 2 x. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

28 Purification of Mixed Strategies (cont d) Hence for player 1 playing C is optimal if and only if: t 1 x c 2 3. Consider now player 2 s expected payoff if he decides to play M: (2 + t 2 ) Pr{t 1 < c 1 } + 0 Pr{t 1 > c 1 } = (2 + t 2 ) c 1 x. Player 2 s expected payoff if he decides to play C: 0 Pr{t 1 < c 1 } + 1 Pr{t 1 > c 1 } = 1 c 1 x. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

29 Purification of Mixed Strategies (cont d) Hence for player 2 playing M is optimal if and only if: t 2 x c 1 3. Hence in equilibrium it must be the case that: c 1 = x c 2 3, c 2 = x c 1 3. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

30 Purification of Mixed Strategies (cont d) Solving these two non linear equations we get: c 1 = c 2 = c, c 2 + 3c x = 0. or c = x. 2 Hence player 1 will choose M with probability (c/x); while player 2 chooses M with probability 1 (c/x). Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

31 Purification of Mixed Strategies (cont d) Consider now these probabilities as x converges to zero. Notice first that: c lim x 0 x = lim x x 0 2 x = 0 0 Hence using the Hospital rule we get: lim x x 1 lim 2 = 3 x 0 Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32

32 Purification of Mixed Strategies (cont d) This implies that when the incomplete information converges to zero: player 1 chooses M with probability 1 3, while player 2 chooses M with probability 2 3. A pair of strategies that coincide with the mixed strategies of the perfect information game. This construction is known as purification of mixed strategies. Leonardo Felli (LSE) EC319 Economic Theory and Its Applications 23 January / 32