MS&E 246: Lecture 12 Static games of incomplete information. Ramesh Johari

Transcription

1 MS&E 246: Lecture 12 Static games of incomplete information Ramesh Johari

2 Incomplete information Complete information means the entire structure of the game is common knowledge Incomplete information means anything else

3 Cournot revisited Consider the Cournot game again: Two firms (N = 2) Cost of producing s n : c n s n Demand curve: Price = P(s 1 + s 2 ) = a b (s 1 + s 2 ) Payoffs: Profit = Π n (s 1, s 2 ) = P(s 1 + s 2 ) s n c n s n

4 Cournot revisited Suppose c i is known only to firm i. Incomplete information: Payoffs of firms are not common knowledge. How should firm i reason about what it expects the competitor to produce?

5 Beliefs A first approach: Describe firm 1 s beliefs about firm 2. Then need firm 2 s beliefs about firm 1 s beliefs And firm 1 s beliefs about firm 2 s beliefs about firm 1 s beliefs Rapid growth of complexity!

6 Harsanyi s approach Harsanyi made a key breakthrough in the analysis of incomplete information games: He represented a static game of incomplete information as a dynamic game of complete but imperfect information.

7 Harsanyi s approach Nature chooses (c 1, c 2 ) according to some probability distribution. Firm i now knows c i, but opponent only knows the distribution of c i. Firms maximize expected payoff.

8 Harsanyi s approach All firms share the same beliefs about incomplete information, determined by the common prior. Intuitively: Nature determines who we are at time 0, according to a distribution that is common knowledge.

9 Bayesian games In a Bayesian game, player i s preferences are determined by his type θ i T i ; T i is the type space for player i. When player i has type θ i, his payoff function is: Π i (s ; θ i )

10 Bayesian games A Bayesian game with N players is a dynamic game of N + 1 stages. Stage 0: Nature moves first, and chooses a type θ i for each player i, according to some joint distribution P(θ 1,, θ N ) on T 1 x L x T N.

11 Bayesian games Player i learns his own type θ i, but not the types of other players. Stage i: Player i chooses an action from S i. After stage N: payoffs are realized.

12 Bayesian games An alternate, equivalent interpretation: 1) Nature chooses players types. 2) All players simultaneously choose actions. 3) Payoffs are realized.

13 Bayesian games How many subgames are there? How many information sets does player i have? What is a strategy for player i?

14 Bayesian games How many subgames are there? ONE the entire game. How many information sets does player i have? ONE per type θ i. What is a strategy for player i? A function from T i S i.

15 Bayesian games One subgame SPNE and NE are identical concepts. A Bayesian equilibrium (or Bayes-Nash equilibrium) is a NE of this dynamic game.

16 Expected payoffs How do players reason about uncertainty regarding other players types? They use expected payoffs. Given s -i ( ), player i chooses a i = s i (θ i ) to maximize E[ Π i ( a i, s -i (θ -i ) ; θ i ) θ i ]

17 Bayesian equilibrium Thus s 1 ( ),, s N ( ) is a Bayesian equilibrium if and only if: s i (θ i ) arg max ai S E[ Π i i ( a i, s -i (θ -i ) ; θ i ) θ i ] for all θ i, and for all players i. [Notation: s -i (θ -i ) = (s 1 (θ 1 ),, s i-1 (θ i-1 ), s i+1 (θ i+1 ),, s N (θ N )) ]

18 Bayesian equilibrium The conditional distribution P(θ -i θ i ) is called player i s belief. Thus in a Bayesian equilibrium, players maximize expected payoffs given their beliefs. [ Note: Beliefs are found using Bayes rule: P(θ -i θ i ) = P(θ 1,, θ N )/P(θ i ) ]

19 Bayesian equilibrium Since Bayesian equilibrium is a NE of a certain dynamic game of imperfect information, it is guaranteed to exist (as long as type spaces T i are finite and action spaces S i are finite).

20 Cournot revisited Back to the Cournot example: Let s assume c 1 is common knowledge, but firm 1 does not know c 2. Nature chooses c 2 according to: c 2 = c H, w/prob. p = c L, w/prob. 1 - p (where c H > c L )

21 Cournot revisited These are also informally called games of asymmetric information: Firm 2 has information that firm 1 does not have.

22 Cournot revisited The type of each firm is their marginal cost of production, c i. Note that c 1, c 2 are independent. Firm 1 s belief: P(c 2 = c H c 1 ) = 1 - P(c 2 = c L c 1 ) = p Firm 2 s belief: Knows c 1 exactly

23 Cournot revisited The strategy of firm 1 is the quantity s 1. The strategy of firm 2 is a function: s 2 (c H ) : quantity produced if c 2 = c H s 2 (c L ) : quantity produced if c 2 = c L

24 Cournot revisited We want a NE of the dynamic game. Given s 1 and c 2, Firm 2 plays a best response. Thus given s 1 and c 2, Firm 2 produces:

25 Cournot revisited Given s 2 ( ) and c 1, Firm 1 maximizes expected payoff. Thus Firm 1 maximizes: E[ Π 1 (s 1, s 2 (c 2 ) ; c 1 ) c 1 ] = [p P(s 1 + s 2 (c H )) + (1 - p)p(s 1 + s 2 (c L ))] s 1 c 1 s 1

26 Cournot revisited Recall demand is linear: P(Q) = a - bq So expected payoff to firm 1 is: [ a - b(s 1 + ps 2 (c H ) + (1 - p)s 2 (c L )) ]s 1 - c 1 s 1 Thus firm 1 plays best response to expected production of firm 2.

27 Cournot revisited: equilibrium A Bayesian equilibrium has 3 unknowns: s 1, s 2 (c H ), s 2 (c L ) There are 3 equations: Best response of firm 1 given s 2 (c H ), s 2 (c L ) Best response of firm 2 given s 1, when type is c H Best response of firm 2 given s 1, when type is c L

28 Cournot revisited: equilibrium Assume all quantities are positive at BNE (can show this must be the case) Solution: s 1 = [ a -2c 1 + pc H + (1 - p)c L ]/3 s 2 (c H ) = [a -2c H + c 1 ]/3 + (1 - p)(c H - c L )/6 s 2 (c L ) = [a -2c L + c 1 ]/3 - p(c H - c L )/6

29 Cournot revisited: equilibrium When p = 0, complete information: Both firms know c 2 = c L NE (s (0) 1, s (0) 2 ) When p = 1, complete information: Both firms know c 2 = c H NE (s (1) 1, s (1) 2 ) For 0 < p < 1, note that: s 2 (c L ) < s (0) 2, s 2 (c H ) > s (1) 2, s (0) 1 < s 1 < s (1) 1 Why?

30 Coordination game Consider coordination game with incomplete information: Player 2 L R Player 1 l r (2 + t 1,1) (0,0) (0,0) (1,2 + t 2 )

31 Coordination game t 1, t 2 are independent uniform r.v. s on [0, x]. Player i learns t i, but not t -i. Player 2 L R Player 1 l r (2 + t 1,1) (0,0) (0,0) (1,2 + t 2 )

32 Coordination game Types: t 1, t 2 Beliefs: Types are independent, so player i believes t -i is uniform[0, x] Strategies: Strategy of player i is a function s i (t i )

33 Coordination game We ll search for a specific form of Bayesian equilibrium: Assume each player i has a threshold c i, such that the strategy of player 1 is: s 1 (t 1 ) = l if t 1 > c 1 ; = r if t 1 c 1. and the strategy of player 2 is: s 2 (t 2 ) = R if t 2 > c 2 ; = L if t 2 c 2.

34 Coordination game Given s 2 ( ) and t 1, player 1 maximizes expected payoff: If player 1 plays l: E [ Π 1 (l, s 2 (t 2 ); t 1 ) t 1 ] = (2 + t 1 ) P(t 2 c 2 ) + 0 P(t 2 > c 2 ) If player 1 plays r: E [ Π 1 (r, s 2 (t 2 ) ; t 1 ) t 1 ] = 0 P(t 2 c 2 ) + 1 P(t 2 > c 2 )

35 Coordination game Given s 2 ( ) and t 1, player 1 maximizes expected payoff: If player 1 plays l: E [ Π 1 (l, s 2 (t 2 ) ; t 1 ) t 1 ] = (2 + t 1 )(c 2 /x) If player 1 plays r: E [ Π 1 (r, s 2 (t 2 ) ; t 1 ) t 1 ] = 1 - c 2 /x

36 Coordination game So player 1 should play l if and only if: t 1 > x/c 2-3 Similarly: Player 2 should play R if and only if: t 2 > x/c 1-3

37 Coordination game So player 1 should play l if and only if: t 1 > x/c 2-3 Similarly: Player 2 should play R if and only if: t 2 > x/c 1-3 The right hand sides must be the thresholds!

38 Coordination game: equilibrium Solve: c 1 = x/c 2-3, c 2 = x/c 1-3 Solution: 9+4x 3 c i = 2 Thus one Bayesian equilibrium is to play strategies s 1 ( ), s 2 ( ), with thresholds c 1, c 2 (respectively).

39 Coordination game: equilibrium What is the unconditional probability that player 1 plays r? = c 1 /x 1/3 as x 0 Thus as x 0, the unconditional distribution of play matches the mixed strategy NE of the complete information game.

40 Purification This phenomenon is one example of purification: recovering a mixed strategy NE via Bayesian equilibrium of a perturbed game. Harsanyi showed mixed strategy NE can almost always be purified in this way.

41 Summary To find Bayesian equilibria, provide strategies s 1 ( ),, s N ( ) where: For each type θ i, player i chooses s i (θ i ) to maximize expected payoff given his belief P( θ -i θ i ).