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246 22. Signaling Games This Is Page 2 Printer: Opaq 23 Building a Reputation 246 22. Signaling Games This is page 2 Printer: Opaq 23 Building a Reputation 23.1 Driving a Tough Bargain It is very common to use language such as “he has a reputation for driving a tough bargain” or “he’s known not to yield and has a reputation of being greedy.” What does it really mean to have a reputation of being greedy and ruthless? What can we say about people putting in an effort to build a reputation of being someone they really are not? Using incomplete information, we can shed some light on these questions. First consider the following perfect information bargaining game, which is ad- mittedly a bit contrived, but captures the main ideas of a bargaining problem: the more I get, the less you get, and we need to reach some agreement. So imagine that a secluded and eccentric rich man dies, and in his will leaves two cars: a shiny new Mercedes sports car (M), and a beat-up Hyundai sedan (H). The will, however, outlines that the final owner of these cars will be determined by a rather unusual bargaining game between his nephew, player 1, and his butler, player 2. Specifi- cally, player 1 first chooses whether to offer player 2 one of the cars, M or H, after which player 2 can choose to accept (A), or reject (R). If player 2 rejects, the cars will be donated to a charity, leaving both players with utility 0. If player 2 accepts, he gets what he was offered, and player 1 is left with the other car. Assuming that 248 23. Building a Reputation a Mercedes is worth 2 and a Hyundai is worth 1, the extensive form is depicted in Figure 5.x. 1 HM 2 2 ARAR 2 0 1 0 1 0 2 0 It is easy to see that this game has a unique Subgame-perfect equilibrium: (H,AA), where following our previous conventions, AA means that player 2 plays A in both his information sets, following either H or M. You should easily be able to convince yourself that this game has Nash equilibria that are not SPE, for example (H, AR), which yields the same outcome as (H, AA) but player 2 is not playing a best response if M were offered. The more interesting one is (M,RA), in which player 2 gets the Mercedes, but this is supported by the incredible threat of rejecting a Hyundai. One can interpret this equilibrium as the one in which player 2 “drives a hard bargain”, but since we believe that sequential rationality is an important feature of rational behavior, this equilibrium is not a very convincing one. Now consider a variation of this game to include some incomplete information. In particular, imagine that player 2 can be “normal” (t2 = N), with payoffs as described above, or he can be a “jerk” (t2 = J), who prefers both players to get nothing over getting the inferior car. Imagine further that nature first chooses the type of player 2, who is a jerk with probability p, and assume that the payoff to the jerk of getting the Hyundai is −1, whereas all other payoffs are the same. Player 1 does not know the type of player 2, but he does know that player 2 is a jerk with probability p. This game is described in figure 5.X below. 23.1 Driving a Tough Bargain 249 Nature JNp1-p 1 1 HM HM 2 2 2 2 ARARARAR 2 0 1 0 2 0 1 0 -1 0 2 0 1 0 2 0 As one might expect, since this is a game of incomplete information, we will focus on sequential rationality using Perfect Bayesian equilibria (PBE). For the simpler signaling games we analyzed, we found PBE by first finding the set of BNE in the Bayesian game, given by a matrix, and then checked to see which profiles of BNE strategies can be supported as part of a PBE with appropriate beliefs. Here, we would need a 2 row by 16 column matrix to do this, which is not too demanding, but there is a much simpler way, using backward induction. To see this, consider player 2 at each of the four nodes that follow an offer from player 1. In any PBE, player 2 must be playing a best response to his beliefs in every information set, which are singletons for these for nodes. thus, for each node player 2 must play a best response at that node, which immediately implies that a normal player 2 will accept any offer, while a jerk will accept a Mercedes and reject a Hyundai. Given this behavior of player 2, and the fact that the uninformed player 1 plays after Nature makes its choices, the beliefs of player 1 are immediately pinned down from natures choices and he must believe that Pr{t2 = J} = p. This implies that player 1 will strictly prefer to offer a Hyundai if and only if the following inequality 250 23. Building a Reputation holds: p · 0+(1− p) · 2 >p· 0+(1− p) · 2 offer H offer M or, 1 p < . 2 and will prefer to to offer a Mercedes otherwise. The intuition for the effect of p on the unique PBE is simple: if there is a good chance that player 2 is normal (p<1 ) 2 then player 1 is better offering the Hyundai and risking a rejection than getting the Hyundai for sure. If, however, player 2 is likely enough to be a jerk (p>1 ) 2 then the safe Hyundai is better than the risk of keeping the Mercedes. Now we take our simple game a step further, to allow for the possibility of reputation building. To do this imagine that the rich uncle’s will is modified as follows: If player 1’s initial offer is accepted, then the game ends as before. If the initial offer is rejected by player 2, then the cars are put in storage for a year, after which player 1 has another chance to make an offer. After this second offer, the game proceeds as above (rejection causes donation) and the payoffs are the same. However, a year’s delay will result in discounted payoffs, with a discount factor of δ<1 as we had in our previous bargaining models. The extensive form of this game is given in figure 5.X 23.1 Driving a Tough Bargain 251 Nature J p 1-p N 1 1 μJ HM HM 2 2 2 2 AR AR AR A R 2 1 2 1 -1 2 1 2 1 1 1 1 μJ|H μJ|M HMHMHMHM ARARARARARARARAR 2δ 0 δ 0 2δ 0 δ 0 2δ 0 δ 0 2δ 0 δ 0 - δ 0 2δ 0 - δ 0 2δ 0 δ 0 2δ 0 - δ 0 2δ 0 where µ is used to denote the beliefs of player 1 in each of his information sets: µJ ∈ [0, 1] is his belief at the beginning of the game that player 2 is a jerk, µJ|H ∈ [0, 1] is his belief that player 2 is a jerk conditional on player 2’s rejection of a Hyundai, and µJ|M ∈ [0, 1] is his belief that player 2 is a jerk conditional on player 2’s rejection of a Mercedes. Once again, an attempt to turn this into the normal-form game will lead to an 8-row by 4096-column matrix! (Player 2 has 12 information sets with two actions each, so 212 =4096pure strategies.) However, since we are looking for PBE, we can employ a similar form of backward induction over all the information sets that are singletons. This implies that at the second stage a normal player 2 will accept any offer, and a jerk will accept a Mercedes and reject a Hyundai. Thus, we can perform one stage of backward induction, and taking into account the best response of player 2 at the final stage after a second offer, the game reduces to the one depicted in figure 5.X, which we call “game 2”: 252 23. Building a Reputation Nature JNp 1-p 1 1 μJ HM HM 2 2 2 2 AR AR AR A R 2 1 2 1 -1 2 1 2 1 1 1 1 μJ|H μJ|M HMHMHMHM 0 δ 0 δ 2δδ2δδ 0 2δ 0 2δ δ 2δ δ 2δ If we try to turn this reduced game into its normal form, we still get a rather sizeable matrix with 8 rows and 16 columns, but it seems that we cannot proceed with backward induction due to the information sets of player 1 after a rejection in the first round of bargaining. However, if we think a bit more carefully we can perform another tricky step of backward induction. To do this consider the nodes in the first round at which player 2 has to move after he is offered to keep the Mercedes. If he accepts, he gets a payoff of 2, while if he rejects then the game moves into the next stage of bargaining. Note, however, than in the next stage the most player 2 can get is 2δ<2, so if he is choosing rationally at these information sets, he must accept the Mercedes no matter what his type is! This allows us to reduce the game further since we know that in any PBE player 2 will accept a Mercedes in the first round regardless of his type.
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