Notes for Section: Week 7

Economics 160 Professor Steven Tadelis Stanford University Spring Quarter, 004 Notes for Section: Week 7 Notes prepared by Paul Riskind (pnr@stanford.edu). spot errors or have questions about these notes. Let me know if you 1 More Information Can Be Bad Here s a simple example of a Bayesian game that illustrates the idea that having more information can actually harm a player by changing the way others act toward her. I ve taken this example from Watson, p. 58. Two players, 1 and, play a game. Let player 1 s type be θ. Player 1 can have two possible types, θ A and θ B. Assume P [θ = θ A ]=P[θ = θ B ]=1/. When θ = θ A,thepayoffs for the game are as follows: L R T 6, 0 4, 1 1 M 0, 0 0, 1 B 5, 1 3, 0 When θ = θ B, the payoffs for the game are: L R T 0, 0 0, 1 1 M 6, 0 4, 1 B 5, 1 3, 0 Notice that changing 1 s type from θ A to θ B just switches 1 s payoffs for strategies T and M. Suppose contrary to our usual assumptions that player 1 doesn t know which type she is. Assume that also doesn t know 1 s type. Since neither player knows 1 s type, neither can condition her behavior on this information. Therefore, the game described above is equivalent to a game in which both players have payoffs equal to the expected value of their payoffs in the game above. Thus, the payoffs inthisgameare: 1

L R T 3, 0, 1 1 M 3, 0, 1 B 5, 1 3, 0 We can solve this game by iterated elimination of strictly dominated strategies. Strategy B is strictly dominant for player 1, sointhefirst round of eliminations, we eliminate T and M for player 1. In the second round, we eliminate R for player since L gives astrictlygreaterpayoff. We conclude that the only strategy profile surviving IESDS, (B,L), is the unique NE for this game. Now, suppose that 1 knows her type, but doesn t know 1 s type. We now have a game of incomplete information one player doesn t know the other s payoffs. Let s find the Bayesian NE. Recall that in a Bayesian NE, each type of each player plays a best response to the other players strategies. Also recall that in a Bayesian game, a strategy assigns an action to each type each player can have. As usual, we solve for the NE by finding the players best responses. We could solve for the NE by writing out the matrix representation of the Bayesian game, but writing out the matrix representation turns out to be unnecessary for this game. Start with player. Note that 1 s type doesn t affect s payoffs. Player s best responses are as follows. BR (T ) = BR (M) =R BR (B) = L Now, suppose that θ = θ A. Then we use the first payoff matrix listed above, and T is a strictly dominant strategy for 1. If instead, θ = θ B,thenweusethe second payoff matrix above, and M is a strictly dominant strategy for 1. So in any BNE, 1 must play T if she is type θ A,andM if she is type θ B. Thus, 1 s strategy in any BNE must be s 1 (θ A )=T, s 1 (θ B )=M. Since 1 never plays B in any BNE, s best response to 1 s strategy must be R. We conclude that the BNE of the game is s 1 (θ A )=T, s 1 (θ B )=M, s = R. The expected payoffs are(4, 1). When player 1 did not know her type, the expected payoffs wereinstead(5, 1), soplayer1 is worse off when she knows her type than when she doesn t. Player 1 is better off not knowing her type because knowing her type changes the way treats her. 1 wants to play L. If knows that 1 knows her type, then knows 1 will never play B. Accordingly, plays R, hurting 1. Bayesian Games With Correlated Types Remember what it means for two random variables to be correlated: If x and y are correlated random variables, then the probability distribution for y changes

when x takes on different values. Thus, when random variables are correlated, knowing the value of one of them gives us information about the probability distribution of the other. In most of the examples we ve seen, the players types are uncorrelated, so knowing your type gives you no information about other players types. In some games we ll see later on, however, types will be correlated. To see how to attack these games, let s do an example. Thekeypointisthatwhentypes are correlated, you can no longer get the probability of a particular vector of types by multiplying the probabilities of individual types, as you could when types were independent. When types x and y are correlated, we cannot say P [x = bx, y = by] =P [x = bx]p [y = by]. We can still solve for the Bayesian Nash equilibria using the matrix representation of the game (if the strategy sets are finite), but we have to make sure we use the right probabilities when we calculate the expected payoffs for each strategy profile. Warning: The algebra in this example is tedious, so I m going to skip most of it. Just try to understand the main idea how to solve for the BNE of a Bayesian game with correlated types. The game we ll analyze is a version of the battle of the sexes with incomplete information. The game has two players, 1 and, who are friends. The two friends must decide which movie to attend, A or B. Each player can have type α or type β. Both types of each player would rather choose the same movie than choose different movies, but different types prefer different movies. Type α prefers film A, while type β prefers film B. The two friends have similar tastes in movies, so their types are correlated. If 1 is type α, sosheprefersfilm A, then she thinks her friend is likely (but not certain) to be type α and prefer A as well. Similarly, if 1 is type β, shethinks is likely (but not certain) to be type β. If the friends attend different movies, they both receive payoffs of0. If both attend movie A, thenplayersoftypeα (who prefer A to B) getapayoff of and players of type β (who prefer B to A) get1. IfbothattendmovieB, then players of type α get a payoff of 1 and players of type β get. Let player i s type be θ i, and let the vector of types be θ =(θ 1,θ ). It will help to write out the payoff matrix for each type of player 1 first. When 1 has type α, she prefers movie A, soherpayoffs are: A B 1 A 0 B 0 1 Similarly, when 1 has type β, she prefers movie B, and her payoffs are: A B 1 A 1 0 B 0 The probability distribution of the two types is as follows: 3

P [θ = (α, α)] = P [θ =(β,β)] = 0.4 P [θ = (α, β)] = P [θ =(β,α)] = 0.1 Notice that the types of the two players are correlated: The types are more likely to be the same than to differ, so knowing one player s type gives you information about the other player s type. Also, notice that both types are equally likely for both players. As usual in Bayesian games, the probability distribution of types is common knowledge but each player knows only her own type. Before we solve for the Bayesian NE of this game, let s digress to calculate the conditional probability distribution of one player s type given the other player s type. We don t need to calculate the conditional probability distributions to solve this game, but you ll need to be able to calculate conditional probability distributions later in the class, so we ll give you some practice now. To calculate the conditional probability distribution of player j s type given player i s type, we use Bayes rule, which says that P [Y X] = P [X Y ] P [X]. Using Bayes rule we get the following probability distribution for s type given 1 s type: P [θ =(α, α)] P [θ = α θ 1 = α] = P [θ 1 = α] P [θ = P [θ =(β,β)] β θ 1 = β] = P [θ 1 = β] P [θ = P [θ =(α, β)] β θ 1 = α] = P [θ 1 = α] P [θ = P [θ =(β,α)] α θ 1 = β] = P [θ 1 = β] = 0.4 0.5 =0.8 = 0.4 0.5 =0.8 = 0.1 0.5 =0. = 0.1 0.5 =0. And with similar calculations, we obtain the following distribution for 1 s type given s: P [θ =(α, α)] P [θ 1 = α θ = α] = P [θ = α] P [θ 1 = P [θ =(β,β)] β θ = β] = P [θ = β] P [θ 1 = P [θ =(β,α)] β θ = α] = P [θ = α] P [θ 1 = P [θ =(α, β)] α θ = β] = P [θ = β] = 0.4 0.5 =0.8 = 0.4 0.5 =0.8 = 0.1 0.5 =0. = 0.1 0.5 =0. So to sum up all the algebra so far, each friend knows that the probability her friend has the same type is 0.8, and the probability her friend has the other type is 0.. 4

We ll now solve for the BNE of this game by writing out the matrix representation. A pure strategy for a player specifies her action when she has type α and her action when she has type β. We ll write a strategy in the form (a α,a β ),wherea α is the player s action when she has type α and a β is her action when she has type β. Each player has four pure strategies: AA, AB, BA, andbb. To write out the matrix representation, we need to calculate the expected payoffs foreachprofile of pure strategies. Calculating the expected payoffs isnotdifficult, but it s complicated, so you may want to try a few of these calculations on your own. We ll go through a couple of the calculations in detail here. Suppose both players play AA. With probability 0.4, the types are θ = (α, α), and since both players play A, the resulting payoffs are(, ). With probability 0.4, the types are θ =(β,β), and since both players again play A, the resulting payoffs are(1, 1). With probability 0.1, the types are θ = (α, β), and since both players choose A, the resulting payoffs are(, 1). With probability 0.1, the types are θ =(β,α), and since both players choose A, the resulting payoffs are(1, ). Accordingly, for this strategy profile, u 1 (AA, AA) = u (AA, AA) =(0.4) + (0.4)1 + (0.1) + (0.1)1 = 1.5. Next, suppose 1 plays AB and plays BA. With probability 0.4, the types are θ = (α, α). Player 1 chooses A and player chooses B, so the resulting payoffs are(0, 0). With probability 0.4, the types are θ =(β,β), and since the players again choose different movies, the resulting payoffs are (0, 0). With probability 0.1, the types are θ =(α, β). Both players now choose A, sotheresultingpayoffs are(, 1). With probability 0.1, the types are θ =(β,α), and since both players choose B, the resulting payoffs are again (, 1). Accordingly, u 1 (AB, BA) =(0.4)0 + (0.4)0 + (0.1) + (0.1) = 0.4, and u (AB, BA) =(0.4)0 + (0.4)0 + (0.1)1 + (0.1)1 = 0.. How do the calculations above differ from those we would do if types were uncorrelated? If types were uncorrelated and each type were equally likely, then to get the probability of a particular vector of types, all we d have to do is multiply the probabilities of the individual types. For example, we d have P [θ =(α, β)] = P [θ 1 = α]p [θ = β] =0.5 0.5 =0.5. All four possible vectors of types would have probability 0.5. As you can see, calculating the expected payoffs for each strategy profile is easy but tedious, so instead of leading you through the rest of the calculations, I ll just write down the results. The matrix representation for this game is: AA AB BA BB AA 1.5, 1.5 0.9, 1 0.3, 0.5 0, 0 1 AB 1, 0.9 1.6, 1.6 0.4, 0. 1, 0.9 BA 0.5, 0.3 0., 0.4 0.8, 0.8 0.5, 0.6 BB 0, 0 0.9, 1 0.6, 0.5 1.5, 1.5 Using our usual method of drawing a line over payoffs when is playing a best response and drawing a line under payoffs when1 is playing a best response, we can easily find all the BNE for this game. There are four BNE 5

in pure strategies: (AA, AA), (AB, AB), (BA, BA), and(bb, BB). It makes sense that this game has many symmetric equilibria. In this game, you want to choose the same action as your friend. If you choose the same strategy as your friend, then because types are correlated, your actions will also be correlated, and you ll both be happy playing the strategy. (Note though, that some of the symmetric equilibria will vanish if the types are less highly correlated than they are in this example.) If you want to give yourself more practice, change this game so that the two friends have different tastes in movies. Assume that given one friend s type, the probability that the other friend has the same type is 0. and the probability that she has a different type is 0.8. The bottom line: You can use Bayes rule to calculate the conditional probability distribution of one player s type given another player s type. When types are correlated, you can still find all the BNE of the game by writing out the matrix representation of the game. When types x and y are correlated, we cannot say P [x = bx, y = by] =P [x = bx]p [y = by]. When you calculate the expected payoffs to strategy profiles in a Bayesian game, be careful not to assume that types are independently distributed unless you can verify that they actually are. 3 Bayesian Games With Continuous Types You need to be able to solve problems involving Bayesian games with continuous sets of types, so here are a couple of examples. (The first is adapted from pp. 11-3 of Fudenberg and Tirole. I obtained the second from Osborne, pp. 94-5.) 3.1 Example 1: Lazy Roommates Two roommates (players 1 and ) own a dog. They must feed the dog, but they re lazy, so each wants the other to do the job. If someone feeds the dog, the dog stops chewing the furniture, giving each roommate a benefit of1. Whichever roommate i feeds the dog incurs a cost of c i,however,becausefeeding the dog requires getting up and walking across the room. Each roommate knows her own cost of feeding the dog, but not the other s. Each believes that the other s cost of feeding the dog is a random variable uniformly distributed over the interval [0, ]. Assume that the roommates costs of feeding the dog are 6

independent. The roommates choose simultaneously whether to feed the dog (F )ornot(d). Here s the payoff matrix for this game: Roommate F D Roommate 1 F 1 c 1, 1 c 1 c 1, 1 D 1, 1 c 0, 0 For player i, a strategy in this game is a function assigns an action, a i A i = {F, D}, to each possible type. Each type of player i has a different cost, c i [0, ]. So a strategy for player i is a function s i :[0, ] {F, D}. We want to find the Bayesian Nash equilibria of this game. Recall that in a Bayesian NE, each type of each player plays its best response to the other players strategies, given the player s beliefs about the distribution of the other players types. So we need to find the players best response functions. As we try to find the best response functions, keep in mind that neither roommate knows the other s type. Suppose player j plays s j (c j ). Player i s best response depends upon the probability with which she thinks player j will choose each action, which depends upon j s strategy and the distribution of costs. Suppose that j s strategy and the distribution of costs imply that j will choose F with probability p j. Then player i s expected payoffs from her two actions are: u i (F, s j ; c i ) = 1 c i u i (D, s j ; c i ) = p j +0=p j Clearly i s best response is to play F if 1 c i p j and to play D if 1 c i p j. Rewriting these inequalities tells us that i should play F if c i 1 p j bc i,and D if c i 1 p j bc i. We can derive similar inequalities to describe j s best response. It follows that each player plays F if her cost c i is at or below some threshold, bc i,andplaysd if her cost is at or above the threshold. So to find the players equilibrium strategies, all we need to do is find the values bc 1 and bc at which both players are playing best responses. Now that we know the form of each roommate s strategy, we can easily calculate the probability that each roommate chooses F. Let the probability that i has a cost less than or equal to c i be G i (c i ). Our assumption that c i is uniformly distributed on [0, ] implies that G 1 (c) =G (c) G(c) =c/. (Ifyou can t prove the preceding statement, see me.) The probability that roommate plays F is thus p = G(bc )=(bc )/, andsimilarly,p = G(bc 1 )=(bc 1 )/. Now, recall that when roommate i is playing a best response, bc i =1 p j. Thus, if both players are playing best responses as they must be in a BNE then bc 1 =1 p =1 (bc )/ and bc =1 p 1 =1 (bc 1 )/. We can solve the preceding equations to obtain bc 1 = bc =/3. We conclude that the only BNE in this game is for roommate i {1, } to choose F when c i [0, /3), choosed when c i (/3, ], andchooseeitherf or D when c i =. 7

If you want to review probability, calculate the probability in equilibrium that at least one of the two roommates feeds the dog. The bottom line: When the set of types is continuous, strategies can potentially be very complicated. Before trying to solve for players best responses, try to narrow the set of possible equilibrium strategies. For example, show that a player s best response is one action if her type is above some threshold and a different action if her type is below the threshold, or show that a player s best response is an increasing (or decreasing) function of her type. 3. Example : The First-Price Sealed Bid Auction We ll now analyze another example of a Bayesian game in which the players have continuous sets of types the first-price sealed bid auction. In a first-price sealed bid auction, each player submits a bid, b i R +, indicating how much she s willing to pay for the asset being sold. Because all bids are sealed, no bidder sees any other bidder s bid before submitting her own bid. The winning bidder receives the asset and pays b i ; the losers pay nothing. We ll analyze a first-price sealed bid auction with two bidders. We ll assume that if b 1 = b, then the seller randomly selects a winner from the two bidders, choosing each with probability 0.5. Assume that bidder i assigns a value of v i to the asset being sold, and assume that for i {1, }, v i is randomly distributed on the interval [0, 1]. Assume that the bidders values for the asset are independent. As usual, each player knows only her own type (value). Bidder i has the following payoff function: u i (b i,b j ; v i )= v i b i if b i >b j (0.5)(v i b i ) if b i = b j 0 if b i <b j A strategy in this game is a bid function, b i (v i ), specifying bidder i s bid for any possible valuation she might have. Bayesian games with both continuous sets of types and continuous actions are complicated enough that there s no simple, general method for solving them. Accordingly, if you get a problem of this sort on a problem set or exam, Steve will likely ask you to verify that a particular strategy profile is a BNE or that the game has a BNE with a particular form, rather than asking you to find a BNE on your own. In this example, we ll just verify that the strategy profile b 1 (v 1 )=0.5v 1, b (v )=0.5v is a BNE. To verify that this strategy profile is a BNE, we just need to show that when the bidders use these bid functions, each is playing a best response to the other. So assume bidder bids b (v )=0.5v. The bidders values are independent, so bidder 1 s value tells her nothing about bidder s value. Bidder 1 knows, however, that since bidder s value is distributed uniformly on the interval [0, 1], bidder s bid is uniformly distributed on the interval [0, 0.5]. Therefore, if bidder 1 bids b 1 0.5, her probability of winning the asset is b 1,soher 8

expected payoff is b 1 (v 1 b 1 ). (Note that the probability of a tie is zero because the probability that bidder makes any particular bid is zero.) If instead, bidder 1 bids b 1 > 0.5, she wins for certain, so her expected payoff is v i b i. Clearly bidder only reduces her payoff by increasing her bid above 0.5, so in equilibrium she can never choose b 1 > 0.5. Assume then that she chooses b 1 0.5, andtrytofind the value of b 1 (given v 1 ) that maximizes her expected payoff. If we maximize the function b 1 (v 1 b 1 ) with respect to b 1,weobtain the first order condition v 1 4b 1 =0, so the solution to the first order condition is b 1 =0.5v 1. The second order condition is satisfied, so b 1 =0.5v 1 does indeed maximize 1 s expected payoff under the assumption that b 1 0.5. We conclude that bidder 1 isplayingabestresponsetobidder s strategy when they use the bid functions b 1 (v 1 )=0.5v 1, b (v )=0.5v. Since the players and strategies are identical, we could have performed the same calculations to show that bidder is also playing a best response. We conclude that the strategies b 1 (v 1 )=0.5v 1, b (v )=0.5v are a BNE of this first-price sealed bid auction. 9