Lecture 6 Games with Incomplete Information. November 14, 2008

Transcription

1 Lecture 6 Games with Incomplete Information November 14, 2008

2 Bayesian Games : Osborne, ch 9 Battle of the sexes with incomplete information Player 1 would like to match player 2's action Player 1 is unsure about player 2's preferences: a) may like to match player 1 b) may like to avoid player 2 Player 2 knows that player 1 is unsure

3 B S B 2,1 0,0 S 0,0 1,2 B S B 2,0 0,2 S 0,1 1,0 state! 1 ; Pr(! 1 ) = state! 2 ; Pr(! 2 ) = 1

4 Bayesian Game set up 1. Nature chooses state! 2 f! 1 ;! 2 ); with prob (; 1 ) 2. Player 2 observes the realized state!; player 1 does not. 3. The two player's simultaneously choose actions 4. Payos given by the actions chosen and state, as in tables. Structure of this game (1-4) is commonly known by both players e.g. 2 knows that 1 knows that Pr(! 1 ) =

5 (Bayesian) Nash equilibrium: Pure strategy equilibrium: action for player 1, a 1 & pair of actions for player 2, a 2 (! 1 ); a 2 (! 2 ) : Player 1's action is optimal: u 1 [a 1 ; a 2 (! 1 )]+(1 )u 1 [a 1 ; a 2 (! 2 )] u 1 [a 0 1 ; a 2(! 1 )]+(1 )u 1 [a 0 1 ; a 2(! 2 )], 8a 0 1 (player 1's payo does not depend upon state)

6 Player 2's action is optimal at! 1 : u 2 (a 1 ; a 2 (! 1 );! 1 ) u 2 (a 1 ; a 0 2 ;! 1); 8a 0 2 Player 2's action is optimal at! 2 : u 2 (a 1 ; a 2 (! 2 );! 2 ) u 2 (a 1 ; a 0 2 ;! 2); 8a 0 2 Player 2 knows the state, so the probability is not relevant for his calculation.

7 Strategy for player 1: a 1 2 A 1 Strategy for 2: (a 2 (! 1 ); a 2 (! 2 )) Is [B; (B; S)] an equilibrium? If 1 plays B; optimal for 2 to play B at! 1 and S at! 2 : For 1, Eu 1 [B; (B; S)] = 2 + (1 ) 0 = 2:

8 Eu 1 [S; (B; S)] = 0 + (1 ) 1 = 1 : Optimal to play B as long as 2 1 ) 1 3 : If 1 3 ; [B; (B; S)] a Nash equilibrium. If < 1 3 ; [B; (B; S)] is not a Nash equilibrium.

9 Suppose 1 3 : Is there another pure strategy Nash equilibrium? [S; (S; B)]: What conditions on must be satised for this to be a NE? Similar caclulation: if 2 3 ; [S; (S; B)] is an equilibrium, not if < 2 3 : So if 3 2 ; two pure strategy equilibria What happens if < 1 3? Neither is an equilibrium, so equilibrium must be in mixed strategies Fix = 0:25:

10 Solve for a mixed strategy Nash equilibrium. Player 1 chooses a prob. of B; and each type of player 2 chooses a prob. of B: (one type of player 1 might choose a pure action) [p; (q 1 ; q 2 )]

11 Bayesian Game: n players Each player has action set A i 1. Nature chooses state! 2 = f! 1 ;! 2 ; :;! k g state! has probability p(!); P p(!) = 1 2. Each player i receives a signal regarding the state, i (!) signal function: deterministic function of state!

12 i :! T i ; T i is the set of signals for player i If at states! and! 0 ; i (!) 6= i (! 0 ); then player i can distinguish these states if i (!) = i (! 0 ); then i cannot distinguish! 0 from!: For signal t i ; the set of states that are consistent with t i At signal t i ;player i t i has a belief over the set of states that are consistent with 3. Each player i chooses an action in A i 4. payo of player i is u i (a;!); where a = (a 1 ; a 2 ; ::; a n ) Pure strategy for a player is a function s i : T i! A i

13 Battle of sexes with incomplete information states are f!;! 0 g player 1 receives signal from set ft 1 g 2 receives signal from set ft 2 ; t 0 2 g at!; she gets t 2 ; at! 0 ; she gets t 0 2 ;

14 Auctions Single object up for auction. n bidders Independent private values Each player has a valuation v i 2 [0; 1] My valuation does not depend upon any other bidder's valuation or information valuations are independent

15 distributed with density f(v) and cdf F (v): Uniform Example: f(v) = 1; F (v) = v Each bidder is risk neutral: Payo is v i p if she wins object, 0 otherwise Auction rules: a) First price sealed bid auction. highest bid wins object, and pays bid

16 b) Second price sealed bid auction. highest bid wins object, and pays second highest bid In a second price auction, it is a weakly dominant strategy to bid b i = v i Let ~ b = maxfbj : 1 j n; j 6= ig highest bid of everyone else First price auction Suppose each bidder employs an bidding function (v) that is strictly increasing.

17 u(b i ; v i ) = (v i b i ) Pr [(b i b 1 ) and (b i b 2 ) and...and(b i b n )] Since the bids are independent and the values are independent, u(b i ; v i ) = (v i b i ) Pr [(b i b 1 )] Pr(b i b 2 ) :: Pr(b i b n )] : Now suppose that each bid is proportional to the valuation b j = v j

18 Pr(b j b i ) = Pr(v j b i ) = Pr v j b i = F b! i = b i! (uniform distribution) u i (b i ; v i ) = (v i b i )! n 1 b i

19 b i chosen by i to maximize u(b i ; v i i = (v i b i ) (n 1)! n 2 b i 1! n 1 b i = 0: (v i b i )(n 1) 1! b i = 0: (n 1)v i = nb i

20 ^bi = n 1 n v i: So if i expects everyone else to bid a fraction of their valuation, it is optimal for i to bid a fraction n n 1 of his valuation. So if everyone bids a fraction n 1 n of their valuation, this is an equilibrium.

21 Suppose that there are two bidders: What is the expected value of v 2 conditional on v 2 v 1? R v1 E(v 2 jv 2 v 1 ) = 0 vf(v)dv F (v 1 ) R v1 = 0 vdv = 1 v 2 v 1 v 1 2 = v 1 2 : Since the optimal bid is v 1 2 ; we see that 1 is bidding the expected value v 2, conditional on v 2 v 1 :! v1 0

22 More generally, if there are n bidders, ^b(v) = E(max j6=i v jjv j v) In second price auction, the winner pays p = (max j6=i v jjv j v) So the expected payments of any bidder in the event he wins are the same in both auctions

23 Revenue equivalence theorem. Both rst and second price auctions yield the same revenues provided that a)buyers are risk neutral b) buyers are symmetric and valuations are independently drawn from any distribution

24 Private vs Common values In many situations, i directly cares about j's private information Examples: Auctions: common value auctions (Bidding for oil tracts) Voting in committees or elections private values { e.g. political opinions or likes common values { competence

25 In common value case, your information can cause me to change my mind. Key idea: in common value situations, a player should condition upon being pivotal

26 common value auctions Two bidders Second price sealed bid auction suppose that i has an estimate t i of the value of the oil tract v i (t i ) = t i What value should i use in formulating his bid? If i bids t i ; then if he wins, will be subject to the winner's curse.

27 Both player's estimates are equally reliable. v i (t 1 ; t 2 ) = t 1 + t 2 2 If 1 wins, this means that t 2 < t 1 So the expected value conditional on winning is v 1 (t 1 & win) = 0:5t 1 + 0:5E(t 2 jt 2 < t 1 ) < t 1 :

28 Suppose t 1 and t 2 are uniform on [0; 1] and are independent random variables v 1 (t 1 & win) = 0:5t 1 + 0:5(0:5t 1 ) = 3 4 t 1:

29 General model v i (t i ; t j ) = t i + t j private values = 0 oil tracts(perfect common values) { = t 1 and t 2 are independent and uniformly distributed on [0; 1] First price common value auction Suppose that each bidder bids t i

30 Solve for a symmetric equilibrium where is the same for the two bidders What is my optimal bid as function of t 1 ; given my rival's bidding behavior u 1 (b; t 1 ) = Pr (b 2 < b) [E(v 1 jt 1 ; b 2 < b) b] u 1 (b; t 1 ) = Pr (t 2 < b) [E(v 1 jt 1 ; t 2 : t 2 < b) b] u 1 (b; t 1 ) = Pr t 2 < b! "t 1 + E(t 2 jt 2 < b ) b #

31 u 1 (b; t 1 ) = b Dierentiating with respect to b, " t 1 + b 2 = fb [2t 1 (2 )b]g b # 2t 1 2b(2 ) = 0 b = 2 t 1

32 That is, player 1 bids a constant multiplied by his signal In a symmetric equilbrium, b = t 1 ; so = 2 = + : 2

33 Second price sealed bid auction Claim: Bayes Nash equilibrium where each player bids ( + )t i Suppose that my opponent is bidding ( + )t 2 If I bid ( + )t 1 ; I win when t 2 < t 1 and my payo is (t 1 + t 2 ) ( + )t 2 = (t 1 t 2 ) > 0 If I reduce my bid, then I do not change my payo in the event that I win, but lose for some values of t 2 where (t 1 t 2 ) > 0

34 So not advantageous to reduce my bid If I bid more than ( + )t 1 ; then I win for some values of t 2 such that t 2 > t 1 : But then my payo when I win is (t 1 t 2 ) < 0 So it is a Bayesian Nash equilibrium for both of us to bid ( + )t i