Chapter 10 Reputation Formation In a comp lete information game, it is assum ed that the play e rs kno w exactly what other pla yers’ pa yoffs are. In real life this assum ptio n almost nev e r holds. W ha t w ou ld happen in equilibrium if a play er has a small amount of doubt about the other play er’s pay offs? It turns out that in dynamic games such small changes may ha ve profound effects on the equilibrium behav ior. In particular, when the game is long and pla yers are patien t, the pla yers’ concern regarding forming a reputation for having an advantageous pa y off function overwh elm s all the other concerns, altering equilibrium behavior dramatically. Kreps and Wilson (1982) and Milgrom and Roberts (19 82) ha ve illustrated this on ex-amples, such as centipede game and c hain-store paradox. The analysis is extended later to more general repeated games, most notably by Fudenberg and Levine. In this lecture, I will illustrate the basic idea on the centipede game. I will start with a simp le examp le, which also illustrates ho w one compu tes a mixed-stra teg y sequential equilibrium. 10 .1 A Sim p l e Exa mp le Consider the gam e in Figure 10.1. In this game, Player 2 does not kno w the pa yoffsof Pla yer 1. She thinks at the beginning that his pa yoffsare as in theupper branch with high probability 0.9, b ut she also a ssign s the small probability of 0.1 to the possibility that he is a verse to pla y dow n, exiting the game. The first ty pe of pla yer 1 is called "norm a l" t ype, and the second t y pe of pla yer 1 is called the "crazy" ty pe. If it w e re common knowled ge that play e r 1 is "norma l", then backwards inductio n w o u ld yield the 103104 CHAPTER 10. REPUTATION FORMATION 12 1112121(1,-5)(1,-5)(1,-5).9.9(4,4) (5,2) (3,3)(4(4,4) (5,2)(3,3),4) (5,2)(3,3).1.112112 1(0,-5)(0,-5)(-1,4) (0,2) (-1,3)(-1,4) (0,2)(-1,3)Figure 10.1: following: pla yer 1 goes do w n in the last decision node; player 2 goes across, and player 1 goes downinthe first node. W hat happens in the incomp lete information game, in which the a bove common knowledge assump tion is relaxed? By sequen tial rationalit y, the "crazy" t ype (in the lo wer branch) alw ays goes acr oss. In the last decision n ode, the normal type a gain goes down. Can it be the case that the normal type goes do wn in his first decision node, as in the com plet e information case? It turns out tha t the an swer is No. If in a sequential equilib rium " no rmal" t ype goes down in the first decision node, in her informat ion set, player 2 must assign proba bility 1 to the crazy type. (B y B ayes rule, Pr (crazy|across)=0.1/ (0.1+(.9) (0)) = 1. This is required for consistency.) G iv en this belief and the actions that we have already determ ined , she gets -5 fro m going across and 2 from going dow n, and she must go down for sequen tial rationality. But then "norm al" type sh ou ld go across as a best rep ly, w hich con trad icts the assump tion that he goes down. Similar ly, one can also show that there is no sequential equilibrium in which the normal ty pe goes across w ith probabilit y 1. If that w ere the case, then by consistency, pla yer 2 wou ld assign 0.9 to normal type in her information set. Her best response w o uld be to go across for sure, and in that case the normal type w ould prefer to go dow n in the first node. In any sequential equilibrium, norm al type must mix in his first decision node. Write105 10.1. A SIMPLE EXAMPLE α =Pr(across|normal) and β for the proba bility of going across for pla yer 2. Write also μ for the probabilit y player 2 assigns to the upper node (the normal type) in her informat ion set. Since normal ty pe mixes (i.e. 0 <α< 1), he is indifferent. Across yields 3β +5(1− β) while down yields 4. Therefore, it must be that 3β +5(1− β)=4, i.e. β =1/2. Since 0 <β< 1, pla y er 2 m ust be indifferent betw e en going down, which yields 2 for sure, and going across, whic h yields the expected payoff of 3μ +(−5) (1 − μ)=8μ − 5. That is, 8μ − 5=2,and μ =7/8. But this belief must be consistent: 70.9α = μ = . 8 0.9α + .1 Therefor e, α =7/9. This completes the comp uta tion of the uniqu e seq uential equilibrium, wh ich is depicted in Figure 10.2. Exercise 15 Check that the p a ir of mixe d strate gy profile and the belief assessment is indeed a sequential equilibrium. Notice that in sequ ential equilibrium , after observing that p lay er 1 g oes across, pla yer 2 increases her probabilit y for play er 1 being a crazy t y pe wh o will go across all the way, from 0.1to0.125. If she assigned 0probabilitytothat typeat the beginning, shewould not chan ge her beliefs after she observes that h e goes across. In the latter case, play er 1 could nev er convince her that he will go across (no matter ho w m any times he goes across), and he would not try. When that p robab ility is positive (no matter ho w small it is), she increases her probabilit y of him being crazy after she sees him going across, and player 1 w o uld try going across with some probability even he is not crazy.106 CHAPTER 10. REPUTATION F ORMATION1 α=7/9 2 β=1/2 1 (1,-5) .9 μ=7/8 (4,4) (5,2) (3,3) .1 12 1 (0,-5) (-1,4) (0,2) (-1,3) Figure 10.2: … 100 100 98 101 99 99 97 100 98 98 1 1 0 3 2 2 1 2 1 1 2 1 2 …1001009810199999710098981103221 2112 12Figure 10.3: Cen tipede Game Exercise 16 In the above game,compu te the sequential equilibrium for any initial prob-ability π ∈ (0, 1) of crazy type (in the figu re π =0.1). 10.2 Re pu tation in Centipede G am e Consider the perfect-information game depicted in Figure 10.3. In this game, there are t wo play ers in a relationship. The players alte rnatin gly get a n o pportunity to end the relationship and get extra payoff in the expense of th e other pla yer. Sta ying in the relationsh ip is beneficial for each player, so that a pla yer wou ld like to rem ain in the107 10.2. REPUTATION IN CENTIPEDE GAM E relationship if she kno w s that th e other pla yer will stay in the relatio nship in the following period. If the other player breaks up in the n ext period, ho wev er, sh …