Perfect Bayesian Equilibrium

Transcription

1 Perfect Bayesian Equilibrium Econ 400 University of Notre Dame Econ 400 (ND) Perfect Bayesian Equilibrium 1 / 27

2 Our last equilibrium concept The last equilibrium concept we ll study after Nash eqm, Subgame Perfect Nash eqm, and Bayesian Nash eqm is Perfect Bayesian Equilibrium. Econ 400 (ND) Perfect Bayesian Equilibrium 2 / 27

3 Our last equilibrium concept The last equilibrium concept we ll study after Nash eqm, Subgame Perfect Nash eqm, and Bayesian Nash eqm is Perfect Bayesian Equilibrium. Perfect refers to the fact that the game will be dynamic, like the kind we solved using Subgame Perfect Nash Equilibrium Econ 400 (ND) Perfect Bayesian Equilibrium 2 / 27

4 Our last equilibrium concept The last equilibrium concept we ll study after Nash eqm, Subgame Perfect Nash eqm, and Bayesian Nash eqm is Perfect Bayesian Equilibrium. Perfect refers to the fact that the game will be dynamic, like the kind we solved using Subgame Perfect Nash Equilibrium Bayesian refers to the fact that the game will include incomplete information, so that players are unsure of other players private information or past decisions. Econ 400 (ND) Perfect Bayesian Equilibrium 2 / 27

5 Our last equilibrium concept The last equilibrium concept we ll study after Nash eqm, Subgame Perfect Nash eqm, and Bayesian Nash eqm is Perfect Bayesian Equilibrium. Perfect refers to the fact that the game will be dynamic, like the kind we solved using Subgame Perfect Nash Equilibrium Bayesian refers to the fact that the game will include incomplete information, so that players are unsure of other players private information or past decisions. There are actually other important solution concepts Sequential Equilibrium, Markov Perfect Equilibrium that won t be covered in the course, but are commonly used. PBE was developed in the 1980 s, and is a pretty cutting edge idea, given that Nash Equilibrium was discovered in 1950 by Nash, Subgame Perfect Nash Equilibrium was discovered in 1965 by Reinhard Selten, and Bayesian Nash Equilibrium was discovered in 1967 by John Harsanyi. Econ 400 (ND) Perfect Bayesian Equilibrium 2 / 27

6 Information Sets in the Extensive Form Definition An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. Econ 400 (ND) Perfect Bayesian Equilibrium 3 / 27

7 Information Sets in the Extensive Form Definition An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. Consider this version of the Battle of the Sexes game: The woman W decides whether to Go Out on a date or Stay in. Econ 400 (ND) Perfect Bayesian Equilibrium 3 / 27

8 Information Sets in the Extensive Form Definition An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. Consider this version of the Battle of the Sexes game: The woman W decides whether to Go Out on a date or Stay in. If W goes out, she goes to a football game F or the ballet B. Econ 400 (ND) Perfect Bayesian Equilibrium 3 / 27

9 Information Sets in the Extensive Form Definition An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. Consider this version of the Battle of the Sexes game: The woman W decides whether to Go Out on a date or Stay in. If W goes out, she goes to a football game F or the ballet B. If W goes out, she leaves a message for M saying that she s going to meet him, but it doesn t say where, and her phone goes straight to voice mail. The man M can go to F or B. Econ 400 (ND) Perfect Bayesian Equilibrium 3 / 27

10 Information Sets in the Extensive Form Definition An information set is a set of decision nodes, all belonging to the same player, over which that player cannot distinguish. Consider this version of the Battle of the Sexes game: The woman W decides whether to Go Out on a date or Stay in. If W goes out, she goes to a football game F or the ballet B. If W goes out, she leaves a message for M saying that she s going to meet him, but it doesn t say where, and her phone goes straight to voice mail. The man M can go to F or B. How can we represent this game in an extensive form? Econ 400 (ND) Perfect Bayesian Equilibrium 3 / 27

11 Example The information set is represented by the dashed line. Econ 400 (ND) Perfect Bayesian Equilibrium 4 / 27

12 Subgames Definition A subgame is all of the nodes, branches, and payoffs that follow after a singleton information set. Econ 400 (ND) Perfect Bayesian Equilibrium 5 / 27

13 Subgames Definition A subgame is all of the nodes, branches, and payoffs that follow after a singleton information set. So in this game: there are TWO subgames, not three. We cannot cut information sets up. Econ 400 (ND) Perfect Bayesian Equilibrium 5 / 27

14 Beliefs Definition A set of beliefs are a probability distribution over all the nodes in a given information set. Econ 400 (ND) Perfect Bayesian Equilibrium 6 / 27

15 Beliefs Definition A set of beliefs are a probability distribution over all the nodes in a given information set. Note that beliefs is a technical term, like the word heat or energy in physics. We don t mean that beliefs are feelings (feelings are not real in economics) or anything like that. The word just suggests the right idea about the probability distribution being a subjective assessment of likelihoods. Econ 400 (ND) Perfect Bayesian Equilibrium 6 / 27

16 Beliefs For the example, we have µ and 1 µ as the beliefs: So µ is (the probability that W chose F, given that she went out). Econ 400 (ND) Perfect Bayesian Equilibrium 7 / 27

17 Perfect Bayesian Equilibrium Definition A perfect Bayesian equilibrium is a set of strategies σ = (σ 1,...,σ N ) and beliefs for every player at every information set, so that Bayesian Beliefs: The beliefs are derived from the strategies and common prior beliefs using Bayes rule, wherever possible! Sequential Rationality: The strategies σ are optimal at every point in the game, given the players beliefs. Econ 400 (ND) Perfect Bayesian Equilibrium 8 / 27

18 Perfect Bayesian Equilibrium Definition A perfect Bayesian equilibrium is a set of strategies σ = (σ 1,...,σ N ) and beliefs for every player at every information set, so that Bayesian Beliefs: The beliefs are derived from the strategies and common prior beliefs using Bayes rule, wherever possible! Sequential Rationality: The strategies σ are optimal at every point in the game, given the players beliefs. Note that we use the strategies to derive the beliefs, but the beliefs must be consistent with the strategies. Econ 400 (ND) Perfect Bayesian Equilibrium 8 / 27

19 Perfect Bayesian Equilibrium Then for a Perfect Bayesian Equilibrium is a set of strategies for M and W, and beliefs µ for M, so that no player has a profitable deviation. Econ 400 (ND) Perfect Bayesian Equilibrium 9 / 27

20 Equilibrium 1 Let s think about this strategy profile: W goes out. W goes to B. M goes to B. µ = 0 Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2? Econ 400 (ND) Perfect Bayesian Equilibrium 10 / 27

21 Equilibrium 2 Let s think about this strategy profile: W goes out. W goes to F. M goes to F. µ = 1 Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2? Econ 400 (ND) Perfect Bayesian Equilibrium 11 / 27

22 Equilibrium 3 Let s think about this strategy profile: W goes out. W goes to F with probability 1/3, and B with probability 2/3. M goes to F with probability 2/3, and B with probability 1/3. µ = 1/3 Is this a perfect Bayesian equilibrium if x = 0? x = 1/2? x = 3/2? Econ 400 (ND) Perfect Bayesian Equilibrium 12 / 27

23 Example Let s find some PBE s from scratch: Econ 400 (ND) Perfect Bayesian Equilibrium 13 / 27

24 Signaling Games The most interesting class of games that are solved used the perfect Bayesian Equilibrium concept are signaling games: (i) Nature assigns a privately known type to the Sender (ii) The sender chooses a message to send to the Receiver (iii) The receiver takes an action that determines both their payoffs Econ 400 (ND) Perfect Bayesian Equilibrium 14 / 27

25 Signaling Games The fun part of signaling games is that the message often has nothing to do with the information the sender has per se, but the signals simply vary in cost: Econ 400 (ND) Perfect Bayesian Equilibrium 15 / 27

26 Signaling Games The fun part of signaling games is that the message often has nothing to do with the information the sender has per se, but the signals simply vary in cost: Educational signaling: (i) a student finds out whether he low or high productivity, (ii) the student chooses to get a high school or college degree, and (iii) a firm decides what wage to offer. Econ 400 (ND) Perfect Bayesian Equilibrium 15 / 27

27 Signaling Games The fun part of signaling games is that the message often has nothing to do with the information the sender has per se, but the signals simply vary in cost: Educational signaling: (i) a student finds out whether he low or high productivity, (ii) the student chooses to get a high school or college degree, and (iii) a firm decides what wage to offer. Lawsuits: (i) a client finds out whether he has a genuine or spurious lawsuit against a firm, (ii) the client chooses to get an expensive or cheap lawyer, and (iii) the firm decides whether to settle or go to court. Econ 400 (ND) Perfect Bayesian Equilibrium 15 / 27

28 Signaling Games The fun part of signaling games is that the message often has nothing to do with the information the sender has per se, but the signals simply vary in cost: Educational signaling: (i) a student finds out whether he low or high productivity, (ii) the student chooses to get a high school or college degree, and (iii) a firm decides what wage to offer. Lawsuits: (i) a client finds out whether he has a genuine or spurious lawsuit against a firm, (ii) the client chooses to get an expensive or cheap lawyer, and (iii) the firm decides whether to settle or go to court. Advertising and Introductory Offers: (i) a firm learns whether its new product is good or bad, (ii) the firm chooses to advertise/discount the product or not, and (iii) consumers decide whether to try it out or not. Econ 400 (ND) Perfect Bayesian Equilibrium 15 / 27

29 Signaling Games The fun part of signaling games is that the message often has nothing to do with the information the sender has per se, but the signals simply vary in cost: Educational signaling: (i) a student finds out whether he low or high productivity, (ii) the student chooses to get a high school or college degree, and (iii) a firm decides what wage to offer. Lawsuits: (i) a client finds out whether he has a genuine or spurious lawsuit against a firm, (ii) the client chooses to get an expensive or cheap lawyer, and (iii) the firm decides whether to settle or go to court. Advertising and Introductory Offers: (i) a firm learns whether its new product is good or bad, (ii) the firm chooses to advertise/discount the product or not, and (iii) consumers decide whether to try it out or not. Limit pricing: (i) a firm finds out whether he is low- or high-cost, (ii) the firm chooses whether to charge a low or a high price, and (iii) a potential entrant decides whether to enter the market or not. Econ 400 (ND) Perfect Bayesian Equilibrium 15 / 27

30 Separating, Pooling, and Hybrid Equilibria There are two kinds of perfect Bayesian equilibria we might find in signaling games: Econ 400 (ND) Perfect Bayesian Equilibrium 16 / 27

31 Separating, Pooling, and Hybrid Equilibria There are two kinds of perfect Bayesian equilibria we might find in signaling games: Separating equilibria: All sender types choose different messages to send to the receiver. Econ 400 (ND) Perfect Bayesian Equilibrium 16 / 27

32 Separating, Pooling, and Hybrid Equilibria There are two kinds of perfect Bayesian equilibria we might find in signaling games: Separating equilibria: All sender types choose different messages to send to the receiver. Pooling equilibria: Multiple sender types send the same message to the receiver. Econ 400 (ND) Perfect Bayesian Equilibrium 16 / 27

33 Separating, Pooling, and Hybrid Equilibria There are two kinds of perfect Bayesian equilibria we might find in signaling games: Separating equilibria: All sender types choose different messages to send to the receiver. Pooling equilibria: Multiple sender types send the same message to the receiver. Hybrid equilibria: Some types mix over the signal they send, while others use pure strategies. (We won t cover these) Econ 400 (ND) Perfect Bayesian Equilibrium 16 / 27

34 Separating, Pooling, and Hybrid Equilibria There are two kinds of perfect Bayesian equilibria we might find in signaling games: Separating equilibria: All sender types choose different messages to send to the receiver. Pooling equilibria: Multiple sender types send the same message to the receiver. Hybrid equilibria: Some types mix over the signal they send, while others use pure strategies. (We won t cover these) Separating equilibria are well-behaved, while pooling equilibria can be more complicated to analyze. Econ 400 (ND) Perfect Bayesian Equilibrium 16 / 27

35 Two Examples: Timing Nature gives the sender type S 1 with probability p and type S 2 with probability 1 p Econ 400 (ND) Perfect Bayesian Equilibrium 17 / 27

36 Two Examples: Timing Nature gives the sender type S 1 with probability p and type S 2 with probability 1 p The sender sends message u or message d Econ 400 (ND) Perfect Bayesian Equilibrium 17 / 27

37 Two Examples: Timing Nature gives the sender type S 1 with probability p and type S 2 with probability 1 p The sender sends message u or message d Knowing the message but not the sender s true type the receiver chooses action l or action r Econ 400 (ND) Perfect Bayesian Equilibrium 17 / 27

38 Example 1 Here s a signaling game that has two separating equilibria: Econ 400 (ND) Perfect Bayesian Equilibrium 18 / 27

39 Example 2 Here s a signaling game that potentially has two pooling equilibria: Econ 400 (ND) Perfect Bayesian Equilibrium 19 / 27

40 Example: Educational Signaling Nature assigns a productivity type High or Low to the worker. Econ 400 (ND) Perfect Bayesian Equilibrium 20 / 27

41 Example: Educational Signaling Nature assigns a productivity type High or Low to the worker. The worker chooses to go to High School only, HS, or High School and College, C. Econ 400 (ND) Perfect Bayesian Equilibrium 20 / 27

42 Example: Educational Signaling Nature assigns a productivity type High or Low to the worker. The worker chooses to go to High School only, HS, or High School and College, C. The firm observes the worker s education, but not the worker s type, and offers either a high wage, w h, or a low wage, w l. Econ 400 (ND) Perfect Bayesian Equilibrium 20 / 27

43 Educational Signaling: Extensive Form Econ 400 (ND) Perfect Bayesian Equilibrium 21 / 27

44 Educational Signaling: Separating Equilibria (?) Is there a separating equilibrium of the following type: High types send C, low types send HS. The firm pays the high wage to applicants with degree C, and the low wage to applicants with degree HS. In equilibrium, the firm believes only high types go to college, and only low types go to high school. Econ 400 (ND) Perfect Bayesian Equilibrium 22 / 27

45 Educational Signaling: Pooling Equilibria Is there a separating equilibrium of the following type: High types send C, low types send C. The firm pays the high wage to everyone. In equilibrium, the firm has beliefs that are correct. Econ 400 (ND) Perfect Bayesian Equilibrium 23 / 27

46 Educational Signaling: Beliefs on the equilibrium path To check the firm s beliefs, we need to compute pr[high C] = pr[high C] pr[c] = p(1) p(1)+(1 p)1 = p which uses Bayes rule, and uses the reasoning from the lecture on dartboards and broken machine parts. Econ 400 (ND) Perfect Bayesian Equilibrium 24 / 27

47 Educational Signaling: Beliefs off the equilibrium path Now, we need to compute the firm s beliefs for every information set, including the one that is never reached in equilibrium. In particular, pr[high HS] = pr[high HS] pr[hs] = 0 0 =...? Econ 400 (ND) Perfect Bayesian Equilibrium 25 / 27

48 Educational Signaling: Beliefs off the equilibrium path Off the equilibrium path, we solve for the entire set of beliefs that are consistent with the equilibrium strategies. Econ 400 (ND) Perfect Bayesian Equilibrium 26 / 27

49 Educational Signaling: Beliefs off the equilibrium path Off the equilibrium path, we solve for the entire set of beliefs that are consistent with the equilibrium strategies. In particular, we just need the firm to believe that offering w l in response to HS is better than offering w h : Econ 400 (ND) Perfect Bayesian Equilibrium 26 / 27

50 Educational Signaling: Beliefs off the equilibrium path Off the equilibrium path, we solve for the entire set of beliefs that are consistent with the equilibrium strategies. In particular, we just need the firm to believe that offering w l in response to HS is better than offering w h : pr[high HS]1+pr[Low HS]1 pr[high HS]2+pr[Low HS]0 or 1 2 pr[high HS] Econ 400 (ND) Perfect Bayesian Equilibrium 26 / 27

51 Educational Signaling: Pooling Equilibria The follow strategies and beliefs are a perfect Bayesian equilibrium of the signaling game: All sender types choose C. The receiver pays w h for C and w l for HS. The receiver s beliefs are pr[high C] = p, pr[low C] = 1 p, and 1 2 pr[high HS] Econ 400 (ND) Perfect Bayesian Equilibrium 27 / 27