CHAPTER 4 4.1 LEARNING OUTCOMES By the end of this section, students will be able to: Understand what is meant by a Bayesian Nash Equilibrium (BNE) Calculate the BNE in a Cournot game with incomplete information about costs Understand how to interpret the BNE, and how it compares with the Cournot outcome in the case where information is complete Understand how to calculate the BNE when players do not know other players preferences Understand how to interpret the BNE (when players are not fully informed about the other players preferences) Understand what is meant by Perfect Bayesian Equilibrium (PBE) Understand how a PBE relates to the other equilibrium concepts studied so far Understand how to find a PBE 4.2 INTRODUCTION So far we have assumed that the players have complete information. This means that the players payoff functions are common knowledge. In this lecture, the assumption of complete information will be relaxed. We will consider the situation in which at least one player is uncertain about another player s payoff function. In other words, we will look at games characterized by incomplete information (see Gibbons, 1992). 4.3 EXAMPLE: STATIC GAMES OF INCOMPLETE INFORMATION Consider 2 firms who compete in quantities (Cournot competition). Assume that firm 2 knows its own cost function as well as firm l s cost function. Firm 1 has incomplete information about firm 2 s costs: it knows that firm 2 s marginal cost is with probability and with probability. 80
Note that the source of uncertainty is that firm 2 may be a new entrant, or it could have just invented a new technology. We assume that the probabilities and are common knowledge. The inverse demand function and the firms cost functions are as follows: with probability with probability We note that firm 2 will tailor its quantity to its cost. That is, if firm 2 is a high (low) cost firm, then it will produce a low (high) amount of output. Firm 1 anticipates this. Definition of the Bayesian Nash Equilibrium (BNE) The BNE is a Nash equilibrium of the Bayesian game. A BNE of this game is a triple ( ) such that: i. ii. iii. That is, each firm s quantity is optimal in the sense that it is a best response to the other firm s quantity. Notice also that the definition above treats the two types of firm 2 as separate players. Solving for the BNE If firm 2 has high cost (i.e. ), then it will choose to solve: [ ] If firm 2 has low cost (i.e. ), then it will choose to solve: [ ] 81
Finally, firm 1 chooses to solve: [( ) ] [( ) ] The corresponding first order conditions are as follows. From (1), we obtain: (4) Similarly, from (2), we obtain: 2( L ) = L 1 2 (5) From (3), we obtain: ( ) ( ) [ ] [ ] [ ] [ ] (6) Substituting (4) and (5) into (6) we obtain: 82
Thus, depends positively on the linear combination of. This means that if firm 2 s costs increase, then firm 1 will produce more output (because firm 2 produces less when it operates at a higher cost). Substituting into (4) we get: Call this Case 1. Similarly, Call this Case 2. Note that the Cournot quantity when costs differ between firms is given by: A special case of the expression above is the one in which, which implies that each firm produces a level of output equal to, as we have seen in previous lectures. The difference can be thought of as the extent of uncertainly faced by firm 1 (regarding firm 2 s costs). If the uncertainty is resolved, then the present game of incomplete information becomes a game with complete information. 83
Note further that in Case 1 uncertainty is good for firm 2: it produces more output than it would produce under complete information (and thus makes relatively more profit). In contrast, in Case 2 uncertainty is bad for firm 2, as it produces less output than it would produce under complete information (and thus it makes relatively less profit). What is the intuition behind this result? Suppose that information is complete. If firm 2 is a high-cost firm, then firm 1 who knows this will produce a high level of output (since firm 2 will produce a low level of output). Suppose now that information is incomplete. This means that firm 1 does not know firm 2 s costs. Thus instead of producing a high level of output (recall that firm 2 has high costs), firm 1 takes into account the possible marginal costs of firm 2, and, and thus produces a moderate level of output; which is lower than if firm1 knew firm 2 s true marginal cost ( in this case). Therefore, uncertainty is beneficial for firm 2 when its own costs are high. The logic is analogous in the case where firm 2 is a low-cost firm uncertainty hurts firm 2. 4.4 STATIC BAYESIAN GAMES NORMAL FORM REPRESENTATION To state the formal definition of a game with incomplete information, it is useful to start from a game with complete information. A simultaneous-moves game of complete information can be represented as: { } The timing in such a game is as follows: 1) Players simultaneously choose actions (player i chooses action from a feasible set ) 2) Payoffs are received, Note that the game is static (simultaneous-move), so everything occurs in an instant! We now turn to the case of a static Bayesian game (i.e. a game where information is incomplete). 84
In this context, we need to represent idea that a player knows his/her own payoff function but may be uncertain about the payoff functions of the other players. Accordingly, we introduce the concept of the type of a player. Each type of player has a different payoff function. Returning to our previous example of the Cournot game, we note the following: Firm 2 has two different types: { }. Firm 1 has one type: { }. Each type of firm 2 has a different payoff function. Also, firm 1 is uncertain about firm 2 s payoff function (or type). More generally, let { } denote the types of all players other than player i. Player i may not be informed about the other players types. So she forms some beliefs about the possible types of the other players. is the probability distribution which denotes player i s belief about the other players types. (In most of the literature, types are independent so we can write.) Thus, a static Bayesian game consists of the following key elements (Harsanyi, 1967): 1. Players action spaces : 2. Players types : 3. Players beliefs : 4. Players payoff functions : We are now ready to describe the timing in a static Bayesian game: 1. Nature draws a type vector. 2. Nature reveals to player i but not to any other player. 3. Players simultaneously choose actions, i.e. player i chooses from the set. 4. Payoffs are received. 85
Notice that in the timing above there is a new player, i.e. Nature. In stage 2, Nature reveals to player i only. Thus at stage 2 incomplete information is introduced. Notice that, in fact, we have introduced incomplete information in a game of imperfect information. In the game above, we have incomplete information because at stage 3 when each player chooses her own action she does not know the types of the other players. 4.5 STATIC BAYESIAN GAMES: INCOMPLETE INFORMATION ABOUT PREFERENCES So far we have seen that a player may not be informed about another player s payoff function (or type). Thus we relaxed the assumption that the players payoff functions are common knowledge. Likewise, we have assumed that in a Nash equilibrium players hold correct beliefs about the other players actions. This means that players know the other players preferences. However, it is easy to imagine situations in which players may not be fully informed about the other players preferences (this is in fact a more natural assumption to make). For example, bargainers in an auction may not know each other s valuation of the object. Accordingly, our objective is to solve for the Bayesian Nash equilibrium (BNE) of such a game with incomplete information about preferences. 4.6 EXAMPLE: BACH OR STRAVINSKY Two people would rather be together than separate, but they have different preferences. This is the same example that we have seen in previous lectures but with a key difference: so far we have assumed that players know each other preferences (i.e. that they both want to go out with each other); we will now consider the case where player 2 knows player 1 s preferences but player 1 does not know player 2 s preferences (see Osborne, 2004). More specifically: Player 2 knows player 1 s preferences (i.e. that player 1 wants to go out with player 2). Player 1 thinks that with probability ½, player 2 wants to go out with her. Player 1 thinks that with probability ½, player 2 wants to avoid her. 86
The payoffs of player 2 are: 1 2 B S B 2, 1 0, 0 S 0, 0 1, 2 Player 2 wants to go out with player 1 (probability ½) 1 st type 1 2 B S B 2, 0 0, 2 S 0, 1 1, 0 Player 2 wants to avoid player 1 (probability ½) 2 nd type From player 1 s viewpoint, there are 2 types of player 2 so player 1 maximizes expected payoffs. The payoffs of player 1 are: 2 B 2 1 1 0 S 0 ½ ½ 1 For instance, (B, B) is a pair of actions, one for each type of player 2. Given (B, (B, B)), the expected payoff of player 1 is: Similarly one case obtain the rest of the payoffs in the above payoff matrix. 87
4.6.1 DEFINITION OF BAYESIAN NASH EQUILIBRIUM (BNE) A Bayesian Nash equilibrium is a triple of actions with the property that: the action of player 1 is optimal, given the actions of the 2 types of player 2. the action of each type of player 2 is optimal, given the action of player 1. Note: the definition above treats the 2 types of player 2 are treated as separate players. Thus, it is as if there a game among 3 players, with corresponding payoffs given by the matrices above. 4.6.2 SOLVING FOR THE BNE To find the BNE, we will propose a candidate equilibrium and check whether it satisfies the definition of a BNE. Claim: (B, (B, S)) is a Bayesian Nash equilibrium. For (B, (B, S)) to be a BNE, it must be the case that B is a best response to (B, S); and (B, S) is a best response to B. We proceed to show that this claim is true (it may not be so the proposed equilibrium is not a BNE). Given (B, S), it is optimal for player1 to choose B; this follows from the matrix of player 1 s payoffs. Given B by player 1, it is optimal for player 2 to choose B (if she is type 1 ) also to choose S (if she is type 2 ). It follows that (B, (B, S)) is a Bayesian Nash equilibrium. What is the interpretation of the BNE? The BNE says that player 2 believes that player 1 will choose B. Also, player 1 believes that player 2 will choose B if she wishes to meet player 1; but player 1 believes that player 2 will choose S if player 2 wishes to avoid player 1. 88
4.7 DYNAMIC GAMES OF INCOMPLETE INFORMATION: PERFECT BAYESIAN EQUILIBRIUM (PBE) So far we have studied three equilibrium concepts: Nash equilibrium in static games of complete information. SPNE in dynamic games of complete information. Bayesian Nash Equilibrium in static games of incomplete information. In this lecture, we will introduce a new equilibrium concept: Perfect Bayesian Equilibrium (PBE) in dynamic games of incomplete information. Why do we require a new equilibrium concept for each class of games? The equilibrium concepts above are closely related they are not new per se. We also need to strengthen our equilibrium concept, as we consider progressively richer games. For example, a SPNE eliminates Nash equilibria that involve non-credible threats. Similarly, a Perfect Bayesian Equilibrium can be thought of as a refinement of a Bayesian Nash Equilibrium Also, a Perfect Bayesian Equilibrium strengthens the requirements of SPNE (see example below) by considering explicitly the players beliefs. (Recall that beliefs are important in the context of games characterized by incomplete information.) 89
4.8 MOTIVATION FOR THE USE OF A PBE Consider the following dynamic game (characterized by complete but imperfect information). 1 R L M 1,3 [p] 2 [1-p] L R L R 2,1 0,0 0,2 0,1 Normal form Representation: 1 2 L R L 2, 1 0, 0 M 0, 2 0, 1 R 1, 3 1, 3 It is easy to see that, in the normal form game above, there are 2 (pure strategy) Nash equilibria: (L, L ) and (R, R ). Are these Nash equilibria subgame-perfect? The answer is yes. The reason is that the only subgame is the entire game. (Recall that a SPNE is a Nash equilibrium in every subgame.) However, there is a problem with the equilibrium (R, R ), as it involves a non-credible threat: For player 2, L dominates R (so player 2 would not play R ). Thus, we need to strengthen our equilibrium concept to eliminate the SPNE (R, R ). This is the reason why we need to consider a PBE not just a SPNE. 90
4.9 REFINING OUR EQUILIBRIUM PREDICTIONS A PBE imposes 3 requirements on our equilibrium predictions, 2 of which a presented below: Requirement 1 ( Beliefs ) At each information set, the player with the move must have a belief about which node in the information set has been reached (see Kreps and Wilson, 1982). Non-singleton information set: belief = probability distribution. Singleton information set: probability 1 is assigned to the single decision node. Based on our example above, requirement 1 is represented by the probabilities p and l-p (see Figure). Requirement 2 ( Sequential Rationality ) Players act optimally given their beliefs and the other players strategy. Based on our example above (L ) Thus, Requirement 2 prevents player 2 from choosing R because (L ) In consequence, requiring that each player has a belief and acts optimally given this belief suffices to eliminate (R, R ). That is, player 2 won't play R, so player 1 won t be induced to play R. Thus, we are left with (L, L ) as our unique SPNE outcome. 91
4.10 ANOTHER EXAMPLE TO ILLUSTRATE SEQUENTIAL RATIONALITY 1 C [2/3] 2 D [1/3] E J 1 F K G F G 0, 1 1 1 2, 0 2, 0 3, 0 0, 3 Suppose that player 2 assigns probability 2/3 to history C Suppose that player 2 assigns probability 1/3 to history D Sequential rationality requires that player 2 s strategy be optimal, given the subsequent behavior specified by player 1's strategy: i.e. Thus, Sequential Rationality requires that player 2 chooses G. Sequential rationality also requires that player 1 s strategy is optimal at her two information sets, given player 2 s strategy: i.e. after history (C, F) J optimal at the beginning of the game D, E optimal, given G Thus, there are 2 optimal strategies for player 1: DJ, EJ; given G. (Recall that a strategy is complete plan of action, specifying what the player is going to do at each decision node she may be called upon to decide so we need to specify an action at each of player 1 s decision nodes.) 92
4.11 REQUIREMENT 3 Requirement 3 says: Each player s belief is consistent with the equilibrium strategy profile ( consistency of beliefs with equilibrium strategies ). Based on our initial example, requirement 3 simply says that, in the SPNE (L, L ), player 2's belief is p=1. This completes our analysis of the 3 requirements related to a PBE. 4.12 CALCULATING BELIEFS A GENERAL APPROACH Example: Entry Game Consider the following entry game: [p] Challenger Out Ready P 0 Unready P R P u 2,4 Incumbent [1-p] Accom Fight Accom Fight 3, 2 1, 1 4, 2 0, 3 Suppose that the Challenger attaches probability P R, P U and P O to Ready, Unready and Out, respectively. The Incumbent s probabilities that Ready and Unready will occur are p and 1-p, respectively. We have the following possibilities: If, then Requirement 3 does not restrict the Incumbent s belief. If, then Requirement 3 says that the Incumbent assigns probability to Ready and probability to Unready. 93
Both of the last 2 probabilities are consistent with Bayes rule. Thus we have arrived at the following definition of a PBE: DEFINITION: A Perfect Bayesian Equilibrium consists of strategies and beliefs satisfying Requirements 1-3. 94
4.13 ADDITIONAL EXERCISES 4.13.1 EXERCISE Two people are involved in a dispute. Person 1 does not know whether person 2 is strong or weak. She assigns probability α to person 2 s, being strong. Each person s preferences are represented by the expected value of a (Bernoulli) payoff function that assigns payoff 0 if she yields (regardless of the other person s action) & payoff of 1 if she fights & the opponent yields, if both people fight, then their payoffs are (-1,1) if person 2 is strong & (1,-1) if person 2 is weak. Find the Nash equilibrium of this Bayesian game if α < ½ and α > ½. 4.13.2 EXERCISE Consider the game tree below: 1 A D 2, 0, 0 2 L R subgame [p] 3 [1-p] L R L R 1, 2, 1 3, 3, 3 0, 1, 2 0, 1, 1 (i) (ii) Find the Perfect Bayesian Equilibrium (PBE). Consider the strategies (A, L, L ) and the belief p=0. Explain whether these strategies and the belief constitute a PBE. 95
4.14 CHAPTER SUMMARY In this chapter, we relaxed our previous assumption that information is complete. Instead, we assumed that a player may be uncertain about another player s payoff function or type. Thus, we studied incomplete information and we did so in the context of the Cournot game. We considered the case where one of the firms (firm 1, say) is uncertain about the other firm s costs. We defined the Bayesian Nash equilibrium of this game, and we solved for it. We obtained an interesting interpretation of the Bayesian Nash equilibrium in this context: when firm 2 has high (low) costs, then uncertainty is beneficial (harmful) for itself. Finally, we saw how incomplete information can be formalized and we did so in the context of a game characterized by imperfect information (see the timing above in a static Bayesian game). We noted that a key assumption behind a Nash equilibrium is that players hold correct beliefs about the other players actions. This means that players know the other players preferences. We relaxed this assumption given that in many situations in which players may not be fully informed about the other players preferences (e.g. bargainers in an auction may not know each other s valuation of the object). We solved for the Bayesian Nash equilibrium of such a game with incomplete information about preferences, and interpreted the outcome. Next, we saw that all equilibrium concepts studied so far are closely related. We noted that, as we progressively consider richer game, we need to strengthen our equilibrium concept. For example, a Perfect Bayesian Equilibrium can be thought of as a refinement of a Bayesian Nash Equilibrium in the sense that it considers explicitly the players beliefs. We examined 3 requirements implied by a Perfect Bayesian Equilibrium: in brief, (i) beliefs, (ii) sequential rationality and (iii) consistency of beliefs with equilibrium strategies. We defined a PBE as strategies and beliefs satisfying requirements 1-3. Through a series of examples, we studied how the 3 requirements can be applied so as to obtain a PBE. 96
4.15 FURTHER READING [1] Fudenberg, D. and Tirole, J. (1991) Perfect Bayesian Equilibrium and Sequential Equilibrium, Journal of Economic Theory, 53, 236-260. [2] Mas-Colell, A., Whinston, M., and Green, G. (1995) Microeconomic Theory, Oxford University Press. 4.16 REFERENCES [1] Gibbons, R. (1992) A Primer in Game Theory, Prentice Hall. [2] Harsanyi, J. (1967) Games with Incomplete Information Played by Bayesian Players, Parts I, II and III, Management Science, 14, 159-82, 320-334, 486-502. [3] Kreps, D. and Wilson, R. (1982) Sequential Equilibrium, Econometrica, 50, 863-894. [4] Osborne, M.J. (2004) An Introduction to Game Theory, Oxford University Press. 97
BIBLIOGRAPHY [1] Binmore, K. (1992) Fun and games, a text on game theory, D.C. Heath and Company. [2] d Aspremont, C. and Jacquemin, A. (1988) Cooperative and Noncooperative R&D in Duopoly with Spillovers, American Economic Review, 78, 1133-37. [3] Fudenberg, D. and Tirole, J. (1991) Game Theory, Cambridge, MA: MIT Press. [4] Gibbons, R. (1992) Game theory for applied economists, Princeton University Press, 1992. [5] Matsumura, T. and Ogawa, A. (2012) Price versus quantity in a mixed duopoly, Economics Letters, 116, 174-177. [6] Myerson, R.B. (2013) Game theory: analysis of conflict, Harvard university press. [7] Rasmusen, E. (2007) Games and Information: An Introduction to Game Theory, Fourth Edition, Blackwell Publishing. [8] Varian, H. (1992): Microeconomic Analysis, 3rd edition, Norton: New York. 98