Class Note for ECON 702 at UMass(3)
Class Note for ECON 702 at UMass(3)
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IV Sequential Games under Complete Information In dynamic games players move sequentially Games under complete information can be classified into two cases Perfect versus imperfect information In a game under perfect information each player knows where she is on the game tree when it is her turn to move In a game under imperfect information some players have to move without knowing the previous moves of other players We can de ne perfect and imperfect information more precisely using an inform ation s et De nition An information set ofplayer i is the set ofnodes among which playeri cannot distinguish If each information set is a singleton then the game has perfect information If some information sets are not singletons the game is one of imperfect information Imperfect information Perfect information We have to impose some information set rules if we do not want to have a strange structure of games Rule 1 All the nodes in any information set must belong to the same player Rule 2 If a node D1 is a predecessor ofnode D2 then D1 and D2 cannot be contained in the same information set Rule 3 Exactly the same set ofmoves can be taken at each of the decision nodes in an information set This Violates Rule 1 15 32 24 03 1 This Violates Rule 2 This Violates Rule 3 Actions versus strategies As in the case ofsimultaneous move games under incomplete information actions are different from strategies in dynamic games under complete information An action or a move is a choice a player can make at each decision node A strategy on the other hand is a detailed set ofplans for playing the game that speci es the player s move for every contingency Thus a strategy maps out a plan of actions under all eventualities Because a Nash equilibiium is de ned in terms of strategies not of actions we must translate an extensive form represented in terms of actions into its normal form representation in terms of strategies Example 1 L R U 10 01 D 21 13 11 L R U U 11 11 D 21 13 21 L D R 13 10 L L L L R KL KR U 1 0 10 01 0 1 D 2 1 21 13 1 3 Subgames and proper subgames De nition A proper subgame in an extensive form game is a set ofnodes that 1 begins at a noninitial decision node that is a singleton 2 includes all the decision and teiminal nodes following that decision node and 3 does not cut any information sets The entire game is also a subgame Subgames that are not the entire game are called proper subgames Example Three subgames and two proper subgames No proper subgame Backward Induction and Incredible threats Consider the following game of perfect information 1 Player 1 chooses an action al from the feasible set AH1 2 After observing a1 player 2 chooses an action 612 from AH2 3 Payoffs are given by r1ala2 and r2ala2 One way to approach the problem of prediction in this game is to simply derive its normal form representation and then apply the Nash equilibrium concept to it In this type of games however the Nash equilibrium may include incredible threats Selten believed that any Nash equilibrium incorporating incredible threats is a poor predictor of human behavior and thus should be eliminated Nash equilibria a er eliminating those incorporating incredible threats are called subgame perfect Nash equilibria In games of perfect information subgame perfect Nash equilibria can be found by using the method of backward induction Example Entrant versus Incumbent Consider a model in which an incumbent faces the possibility of entry by an entrant The entrant may be a rm considering entry into an industry currently occupied by a monopolist a politician competing for the leadership of a party or a nation or an animal competing for the right to mate with a congener of the opposite sex 38 25 Enter 43 0 40 10 80 0 Enter Enter Enter 80 80 Enter 80 80 Expand 38 25 38 25 43 0 43 0 Don t Expand 40 10 80 0 40 10 80 0 There are two pure strategy Nash equilibria Don t Expand Enter Enter and Expand Stay Out Enter Are both of them reasonable Consider Don t Expand Enter Enter The strategy of Enter Enter can be interpreted as an announcement or a commitment of Mr Entrant that he would play Enter regardless of the choice made by Ms Incumbent If she were persuaded by this threat she would play Don t Expand But is this threat really credible If Ms Incumbent actually adopts the strategy of Expand while ignoring this threat there is nothing to stop Mr Entrant from backing away from its threat and not entering at all Thus it might be an incredible threat for clever Ms Incumbent Another way oflooking at the issue is the following Note rst that Enter 80 is strictly dominated by 80 Enter and 80 80 is weakly dominated by 80 Enter So Ms Incumbent can delete them After that she can also delete Enter Enter because it is weakly dominated by 80 Enter Example Ulysses and the Sirens You must bind me hard and fast so that I cannot stir from the spot where you will stand me and ifI beg you to release me you must tighten and add to my bonds The Odyssey The problem of an incredible threat is a problem of dynamic inconsistency To avoid dynamic inconsistency involved in an incredible threat we need to impose the principle of sequential rationality equilibrium strategies should specify optimal behavior from any point in the game onward This principle is intimately related to the procedure of backward induction Now we will describe the method of backward induction in a general way When player 2 gets the move at the second stage of the game heshe will choose a2 Eargmaxr2ala2 Given the action al the solution to this problem is thus given by a function a a2 a Since player 1 can solve player 2 s problem as well as player 2 can player 1 should anticipate player 2 s reaction to each al so player 1 s problem at stage 1 amounts to a E argmaxr1a1aza1 The pair aocza is called a backward induction outcome of this game whereas the pair aoczal is called a backward induction equilibrium of this game Note that all af is a numbers while a2 a is a function of al Example Entrant versus Incumbent continued Stay Out if afExpand Enter if al Don t In this example a Expand and ocal Thus the backward induction equilibrium of this game is Expand Stay Out Enter while the backward induction outcome of the game is Expand Stay Out 7 Example Consider the following game 170 50 Enter Share 150 50 S E 400 0 500 0 Don Expand 80 Share 250 150 Fi g 230 90 Incumbent has eight possible strategies Expand Fight Fight Expand Fight Share Expand Share Fight Expand Share Share Don t Expand Fight Fight Don t Expand Fight Share Don t Expand Share Fight Don t Expand Share Share Entrant has four possible strategies Enter Enter Enter 80 80 Enter so so E E E so so E so so Ex FF 170 50 170 50 400 0 400 0 Ex FS 170 50 170 50 400 0 400 0 Ex SF 15050 15050 400 0 400 0 Ex SS 15050 15050 400 0 400 0 Don t FF 230 90 500 0 230 90 500 0 Don t FS 230 90 500 0 230 90 500 0 Don t SF 250150 500 0 250150 500 0 Don t SS 250150 500 0 250150 500 0 There are four pure strategy NE But the only backward induction equilibrium is Ex FS 80 E Example Stackelberg duopoly There are two companies in the bottled industry Sparkling Water Co and Clear Water Inc They sell the exact same product in the same market and set how much water to put up for sale to maximize pro ts The price of water is set by the market then Sparkling Water is the industry leader so that it sets the quantity rst afterwards Clear Water chooses its quantity The total cost of each rm is TC 3Q where i S C and 10 Q if Q S 10 where Thus the ro t 0 ifQ gt10 Q QC Q3 p inverse demand function is P 7QQ ifQSIO of each rm is 71 ence an out ut above 7 is strictl 3Q1 ifQgt10 H y p y dominated by zero output If this game were played simultaneously then the equilibrium would be the Coumot equilibrium which is Q Q Now we will compare this with the equilibrium obtained by the backward induction method A strategy for Sparkling Water is a real numberQS while a strategy for Clear Water is a function RQS which determines QC for each level of Q3 Let us solve Clear Water s problem rst The rst order condition for Clear Water is 671C an 7 QS 2QC 0 thus RQS 35 S This is the strategy of Clear Water Now Sparkling Water solves maX7 RQsQsQS which gives Q 35 Thus the backward induction equilibrium is 35 RQS 35 QS while the backward induction outcome is 35 RQS 175 In this game the rst mover has an advantage because 713 6125 gt 7 3063 Note There are many pure strategy Nash equilibria of the normal form representation of this sequential move game Clearly QS35 and RQS35 i constitutes a Nash equilibrium But 2 S QS and RQS ie a constant function constitutes another Nash equilibrium because Clear Water s setting output of g is optimal for QS 5 But a Nash equilibrium like QS and RQS is not a backward induction equilibrium because it implies an incredible threat If Sparkling Water were to produce QS 3 instead of then Clear Water s only credible response is to choose QC 2 not g Example Dynamic Bertrand game We have seen that the rst mover has an advantage in a dynamic Coumot game ie the Stackelberg game This is not always true The second mover has an advantage in a dynamic Bertrand game We will study it through a homework exercise TheoremZermelo s Theorem Every finite game of perfect information has a pure strategy Nash equilibrium that can be derived through backward induction Moreover if no player has the same payoffs at any two terminal nodes there is a unique Nash equilibrium that can be derived in this manner 10 Randomization in Dynamic games There are at least two ways of formalizing randomization in the context of dynamic games Recall that actions are different from strategies in dynamic games A mixed strategy is a probability distribution de ned over a strategy space A behavior strategy on the other hand allows agents to randomize at the level of action choices De nition Given player i s pure strategy set Si a mixed strategy of player i is a probability distribution oiSi gt0 1 such that 6iSikEASi for every SikESi A list of mixed strategies 661 62 61 is called a mixed strategy combination Ex 1 clcls11clsfm 1p l 2 3 4 02 0252 5 0282 6282 6282 111 112 113 1111112113 Choosing a mixed strategy means that a player independently1 chooses a random device for selecting the pure strategy to be played The type of random device chosen determines the probabilities with which the different pure strategies will be selected Therefore such a random device represents a probability distribution on the set of pure strategies The expected payoff of player i from a mixed strategy combination 6 116 is the expected value of i s payoff generated by the mixed strategy combination 6 1110 2 0101 0232 3939Gr 31 553 De nition A behavior strategy of player i is a probability distribution biUi gt 0 1 such that biHaEAAH for every aEAH A list ofbehavior strategies bb1 b1 is called a behavior strategy combination Ex 3 Consider the following game 1 If players jointly choose a random device and condition their strategy choices on the outcome of this joint randomization then they would play a correlated strategy 11 10 Sl 1L 1R MaL M R B L B R 61 p1 p2 p3 p4 p5 lp1p2p3p4p5 mixed strategy ofplayer l Aa T M 13 Ade R L b1 q1 q2 lqlqz v lv behavior strategy ofplayer l A behavior strategy combination will induce a probability distribution on the set ofterminal nodes TN Let qbn be the probability that terminal node n is reached ifthe behavior strategy combination b is played This probability can be calculated by multiplying all the probabilities the behavior strategy b assigns to the actions along the path to the terminal node n The expected payoff of player i from a behavior strategy combination b Rib is the expected value of the payoffs given the probability distribution on terminal nodes induced by b that is R b 2 VI 70 VIETN A behavior strategy combination b induces a probability distribution on the set of terminal nodes TN starting not only from the initial node but from any decision node XE NTN Let qbnlx be the probability that terminal node n is reached if the behavior strategy combination b is played and if the game has already reached the decision node x The expected payoff ofplayer i then is given by R blx 2 VI 70 nix HETN 12 Since both mixed strategies and behavior strategies are are obtained by allowing for some kind of randomization the question arises whether the two concepts are essentially the same ie whether the set of all mixed strategy combinations will create exactly the same probability distributions over outcomes as the set of all behavior strategy combinations Unfortunately it is not true in general that there is a behavior strategy combination for every mixed strategy combination that yields the same probability distribution on the terminal nodes But there is a class of games for which two strategy concepts are equivalent De nition An extensive form game is said to have perfect recall if every player always remembers his previous choices Formally a game has perfect recall when for every playeri and for every two information sets H and H ofplayer i if one node x in H comes a er a choice Z at H then every node y in H comes a er a choice in H In other words ifthere is a play that crosses rst H and then H then all the plays that cross H must rst cross H Example Two games not satisfying perfect recall xg I Playe2ltRltq39 Va PlayefltR I 4 XPI XT39 39 X Player 1 yeg playlt1 PIayer1 Pla r 1 V lt lt lt2 Q I k Player Playegt2lt I 13 TheoremKuhn In games with perfect recall for any mixed strategy combination 6 there is a behavior strategy combination b that induces the same probability distribution on the set of terminal nodes TN Therefore mixed strategies and behavior strategies are equivalent for games with perfect recall l4 Subgame Perfect Nash Equilibrium The concept of subgame perfection generalizes the idea of backward induction A SPNE is de ned as a behavior strategy combination such that the behavior strategy of each agent is a best response strategy against the behavior strategies of the other player in every subgame F De nition Suppose n is an arbitrary node A behavior strategy combination bb1 bIquot is a subgame perfect Nash equilibrium if for all players iEI and for all subgames F starting from 11 70771 n 2 7103217 n for all b2 Every SPNE is a NE because the equilibrium behavior strategy combination must be a NE of the subgame To which is identical to the game itself The idea behind the identi cation of the set of SPNE is the generalized backward induction procedure Subgame perfect Nash equilibrium versus subgame perfect outcome in a simple setting Consider the following structure ofa game Let s focus on pure strategies 1 Playerl chooses al 6A1 in stage 1 2 Player 2 choose a2 6 A2 after observing al in stage 2 3 Payoffs are r1ala2and r2ala2 This is the situation we studied under the title of backward induction When player 2 gets the move at the second stage she will face the following problem choose all to maxr2ala2 while taking all as given If we assume that player 2 s optimization problem has a unique solution for each all then we can denote this solution by 052011 This is player 2 s best response to player 1 s action Since player 1 can solve player 2 s problem as well as player 2 can player 1 should anticipate player 2 s reaction to each al that player 1 might take Thus player 1 solves choose al to max r1aloc2 511 Let a be a solution of this optimization problem 15 Then 61052a is a subgame perfect outcome of this game while 6111052011 is a SPNE of this game These are the same as the backward induction outcome and the backward induction equilibrium Subgame perfect Nash equilibrium versus subgame perfect outcome in a more general setting Consider the following structure ofa game 1 Players 1 and 2 simultaneously choose all 6 A1 and a2 6 A2 in stage 1 2 Players 3 and 4 after observing 611612 simultaneously choose a3 6A3 and 614 GA in stage 2 3 Payoffs are rlal 512613 a4 for i l234 In stage 2 players 3 and 4 will play the game while taking 611612 as given Supppose a3czla2 and a4 01020011Stlt11t6 a NE of the second stage subgame for each al a2 In other words 3a15a2 E argmaxr3a15a25a35a4a15a2a and 054611 a2 6 argmaX r4ala2oc3ala2a4 By anticipating 053611 a2 and 054611 a2 correctly players 1 and 2 choose a E argmax Napalaxal aia4a1a and a E argmax r2al5a25 3allal5a4a5a2 39 Thus a SP outcome is afa oc3afaoc4afa while a SPNE is afaa3aazaaaz 16 Some problems of backward induction and subgame perfection I To apply the logic of backward induction we need the assumption that rationality is common knowledge among players This is OK in many straightforward situations but is less compelling ifthere are many rounds in the game Consider the following game played between Coco and Woojin Coco is Woojin s lovely dog C W C W C W 6 0 Leave Leave Leave Leave Leave Leave GT b ab ab ab b ab 10 02 30 04 50 06 According to the logic of backward induction we should predict that Coco will immediately grab the dollar in the rst round to nish the game Why is this outcome predicted Because Coco believes that were the second round reached rational Woojin would nish the game leaving nothing to her Why does she believe so Because she believes that Woojin would believe in the second round that were the third round reached rational Coco would nish the game leaving nothing to him The chain of this way of reasoning should go all the way up to the nal round Now consider the fourth round A key premise here is that rational Woojin would grab four dollars because he would believe that rational Coco would nish the game in the fth round But were rationality common knowledge how would the game get all the way up to the fourth round The assumption that rationality is common knowledge implies that it should not get past the rst round but Coco already passed twice Isn t it then reasonable for rational Woojin to conclude that something has gone wrong If so why is it reasonable for both players to suppose that rational Woojin in the fourth round would believe that Coco would grab the money in the fth round given that she already passed twice 1 If Woojin in the fourth round believes that Coco s passing twice is a simple mistake that would not affect her future actions like a typing error then he can continue to assume that Coco is rational That is if Woojin believes that Coco intended to act rationally but made simple mistakes in the rst and the third rounds and therefore further mistakes are unlikely then he can continue to assume that Coco is rational In this case the logic of backward induction continues to work But the fact that Coco passed twice may mean that Coco may not have made simple mistakes There are goods reasons for rational Woojin to think that Coco might be a different kind ofa player she may be irrational she is a dog in any way or an altruistic dog who always leaves the money to bene t Woojin Or Woojin might believe that she is reciprocal employing a titfor tat strategy which starts from Leave and copies what Woojin does If Woojin believes that way in the fourth round the argument of backward induction collapses The problem is not only this Knowing this rational Coco might pretend to be irrational altruistic or reciprocal by passing twice at the very beginning of the game and snatches the money away later to Woojin s great surprise Many game theorists have argued that any reasonable theories of game should not rule out any belief once an event to which the theory assigns probability zero has occurred because in that case the theory provides no guidance for players to form their predictions conditional on these events One lesson that we can derive from this example is that ifthe number ofrounds is very long the chain reasoning that supports the backward induction outcome is unlikely to hold The failure of backward induction does not imply that players are socially motivated 2 Even if the number of rounds is very small the logic of backward induction may be problematic sometimes Consider the following example 18 V1D65 N 8 3 A 29 103 The backward induction outcome of this model is that player 1 say John chooses A in the rst round and nishes the game immediately This is because player 2 say Jane would expect that player 1 would choose D if she chose X and he would choose C if she chose Y Note however that player 1 earns 10 by choosing A while his maximum payoff from choosing B is 8 and only 6 if the ensuing play is optimal This means that any strategy that assigns A to player l s initial decision node strictly dominates any strategy that assigns B Thus were the game reached at player 2 s decision node she would know that player I chose a strictly dominated strategy In that case should player 2 anticipate that player 1 would respond in an optimal manner to what player 2 would do Probably not The logic of backward induction collapses again 3 Subgame perfection is an extension of backward induction so is vulnerable to the problems just discussed Furthermore subgame perfection requires that all players agree on the play in a subgame even if that play cannot be predicted from backward induction arguments Consider the following threeplayer game suggested by Mattew Rabin 1988 Note that the subgame between players 1 and 3 at the third stage is a coordination game which has two pure strategy NE FF and GG F G F 7107 000 G 000 7107 prlayer 2 expects that players 1 and 3 successfully coordinate their actions in the third stage he would choose L hoping to get the payoff of 10 instead of 6 prlayer 2 expects that players 1 and 3 may fail to coordinate successfully leaving him only the expected payoff of 5 player 2 would choose R yielding the payoff of6 19 According to the logic of backward induction player 1 would choose R in both cases In the rst case the case of successful coordination player receives 7 instead of 6 In the second case the case of coordination failure she receives 8 instead of 6 Nevertheless as Rabin argues player 1 may have an incentive to play L in the rst stage She would do so if she saw no way to coordinate in the third stage in which case she gets 0 and feared that player 2 would think that the third stage would result in successful coordination The point is that subgame perfection supposes not only that the players expect Nash equilibria in all subgames but also that all players expect the same equilibria 1107 l 000 SF 000 710 7 8 6 8 6 0 6 4 A pure strategy SPNE may not exist even in nite games Selten proves however that if we consider mixed strategies or behavior strategies then every nite game has a SPNE Example 20 The normal form representation of this game is u d U L 15 15 U R 15 15 D L 32 03 D R 24 11 Thus there are two pure strategy Nash equilibiia U R d and ULd and one mixed strategy Nash equilibiium 0 0 34 l4 12 12 Player Iwould not put any positive probability on UL or U R because they are weakly dominated by D R None of the pure strategy Nash equilibiia are subgame perfect Thus only a subgame perfect Nash equilibrium of this game is a mixed strategy Nash equilibiium 0 0 34 14 12 12 5 The notion of subgame perfection is intended to capture the p1inciple of sequential rationality But sometimes especially in games under imperfect information it may still fail to eliminate the equilibiia that incorporate incredible threats Consider the following example 21 1 1 pig Fight Accommodate i Im Accom3moda9e Inl 1 1 3 0 Incqlmb 1 1 Inz 1 1 2 1 Fight iEntrant out 0 2 02 2 1 0 2 There are two pure strategy NE 111 Accommodate Out Fight Both of them are also SPNE since the game has no proper subgames Nonetheless Out Fight depends on an incumbent s incredible threat since Accommodate strictly dominates Fight for incumbent the entrant would not be induced to play Out by incumbent s threat to play Fight if given his move Thus the criterion of subgame perfection is of no use here to eliminate the equilibrium incorporating an incredible threat One way to strengthen the equilibrium concept so as to rule out the case as Out Fight is to impose more requirements We will study this issue under the title ofperfect Bayesian equilibrium and sequential equilibrium 22 Perfect Bayesian Equilibrium and Sequential Equilibrium Requirement 1 At each information set the player with the move must have a belief ie a speci cation of a probability about which node in the information set has been reached by the play of the game Requirement 2 Given their beliefs the players strategies must be sequentially rational That is at each information set the action taken by the player with the move must be optimal given the player s belief at that information set and the other players subsequent strategies Requirement 3 At nonsingleton information sets on the equilibrium path beliefs are determined by Bayes rule and must be consistent with the players equilibrium strategies This condition is trivially satis ed for an information set never reached by the equilibrium path Requirement 4 At nonsingleton information sets off the equilibrium path beliefs are determined by Bayes rule and must be consistent with the players equilibrium strategies where possible This condition is trivially satis ed if there is no nonsingleton information set off the equilibrium path Definition A perfect Bayesian equilibrium consists of a pair of behavior strategies bf and a pair ofbeliefs pquot satisfying requirements 1 to 4 for all players The notion of PBE extends the principle of sequential rationality by formally introducing the notion of beliefs An equilibrium no longer consists ofjust a strategy for each player but now also includes a belief for each player at each information set at which the player has the move A PBE is a SPNE but not vice versa A concept slightly stronger than a PBE has been proposed A sequential equilibrium is de ned by replacing requirement 4 with the following limit consistency Definition A behavior strategy combination bf and a belief system pquot form a sequential equilibrium if 1 b pf satis es requirements 13 and 23 2 there are a sequence of behavior strategy combinations bv such that V gt 0 for all nonsingleton information sets and lim b V b and a sequence ofbeliefs uv vgtoo which is derived from bv using Bayes rule such that limyv if vgtoo Note that each behavior strategy combination b in the sequence bv needs not be an equilibrium The only requirement is that this sequence should converge to the equilibrium strategy combination b In many cases a PBE and a sequential equilibrium coincide but in other cases a sequential equilibrium eliminates an unreasonable PBE Example 1 Continued If the player of the game reaches incumbent s nonsingleton information set then incumbent must have a belief about which node has been reached or equivalently about whether entrant has played In or 1112 Let this belief be represented by the probabilities p and 1 H Requirement 2 prevents incumbent from choosing Fight since for any u the payoff from Accommodate is higher than that from Fight Thus requirements 1 and 2 suffice to eliminate the implausible equilibrium Out Fight Requirement 3 makes the beliefs reasonable In the equilibrium In Accommodate incumbent s belief must be pl given entrant s equilibrium strategy Inl incumbent knows which node in the information set has been reached Requirement 4 is trivially satis ed since there is no nonsingleton information set of the equilibrium path Therefore the unique pure strategy PBE is In Accommodate with ul Recall that a PBE is a pro le of strategies together with beliefs Writing down a strategy pro le only is not a complete description of a PBE Showing that In Accommodate with ul is a sequential equilibrium is straightforward Suppose the behavior strategy of Entrant is a probability distribution bVb20111bln2b0utl ll0 This converges to 100 which v v corresponds to playing Inl for sure Now let s construct uvl uv from V using Bayes rule Then because 24 POE b20171 1l Phistory up to the info set b1nl b 1112 v Px1 1 history up to the info set and Px2 b20112 1 Px2 1 history up to the info set V V Ph1story up to the info set bEInl b5 1112 v uVl luv l ll converges to l 0 which is the belief associated with Inl v V Accommodate Example Consider the following three player game between Entrant Monopolist and Govemment 030 0520 0500 lll lll ll0 l00 F A F A NE 030 030 NE 030 030 EH 05 20 0500 EH 0520 1 11 EL 100 111 EL 100 110 G chooses C G chooses NC There are four pure strategy NE NE F C NE F NC NE A NC EL A C Because there are no proper subgames they are all SPNE But NE F C is not 25 reasonable because if Government chooses C then A strictly dominates F This unreasonable equilibrium is eliminated if we use the concept ofPBE There are two information sets Let Ll l be the belief of Monopolist while ul u be the belief ofGovernment Given the belief ul LL Govemment s expected payoff is 1 u from playing C and u from playing NC Case 1 Suppose us that is l yZ u Then Government chooses C Now Monopolist chooses A whatever his beliefbecause A strictly dominates F for all A 6 01 Finally Entrant chooses EL Now these choices assign A 0 but has no further restriction on u Thus EL A C with beliefs 1 0u S is a PBE 0 3 0 05 2 0 05 0 0 111 111 110 1 0 0 Case 2 Suppose u gt that is l LLlt LL Then Government chooses NC Now Monopolist does not have a strictly dominant choice If 2ampZ l ie A S then Monopolist choose F if 2amp lt l ie Z gt he chooses A Case 2 1 Suppose A S Then Monopolist chooses F which implies that Entrant chooses NE Thus NE F NC together with beliefs 1 lt i and u gt constitute a PBE 2 26 030 05 2 0 05 00 lll lll ll0 l00 Case 2 2Suppose Z gt Then Monopolist chooses A which implies that Entrant chooses NE Thus NE A NC together with beliefs A gt and u gt constitute a PBE 030 05 2 0 05 00 l ll l l l ll0 l00 Out of these three sets of PBEs some associated with the second type however may be unreasonable Consider for example a pair of the beliefs 1 0 and LL 1 This pair of belief together with the choice described in Case 21 constitute a PBE but this is unreasonable because the outofequilibrium beliefs of Monopolist and Government are inconsistent Monopolist believes that Entrant has chose EL while Government believes EH is the choice of Entrant 27 It is not dif cult to see that it does not constitute a sequential equilibrium Let bEH bELl b EH b EH be a sequence of the behavior strategy of Entrant while b1 Al b1 A a sequence of the behavior strategy of Monopolist Now let s construct WA iv and uvl uv from b EH b ELl b EH b EH and b1 A 1 by A using Bayes rule Because W V bEEHV and luv V bMEAbEEHV 133 EH 133 EL bMAbEEH 13021 we must have iv uv for all v Thus any seuqnece ofbehavior strategies thatjusti es the belief 1 1 will also justify LL 1 as well But the choices described in Case 21 do no longer constitute a PBE if A u l Summarizing sequential equilibria are 1 EL A C with beliefs 1 u 0 and 2 NE A NC together with beliefs 1 u gt In the literature a pro le of the behavior strategies and beliefs satisfying only requirements 1 to 3 is called a weak perfect Bayesian equilibrium In many cases both concepts coincide for instance requirement 4 provides no extra restrictions on a weak PBE if there is one information set in a game but not always Also a weak PBE is neither necessary nor suf cient for being a SPNE SPN A eak 28 Example Consider the following game In this game player I moves twice Player Ihas four possible strategies and player II has two Thus the normal form representation of this game is as follows and there are four pure strategy Nash equilibria TL u TL d TR u and TR d u d T L 2 0 20 11 d T R 20 20 L 1 0 0 1 B L 10 01 R 01 10 13 R 01 1 0 There is one proper subgame starting from the node ofPlayer I s second move But this proper subgame has no Nash equilibrium which means that none of the above four Nash equilibria are subgame perfect Now we will show that there are weak PBEs although there are no SPNEs and so no PBEs Let s rst assign beliefs u and l u to the two nodes in the information set Given this belief player II obtains the expected payoff of u0l LL1 l u by playing u and ull u0 u by playing d M Suppose u 2 ie l u S u Then playing d is rational for player II Knowing this player I would choose R in his second move which means he would choose T in the rst move Because the information set is not on the equilibrium path requirement 3 is vacuous Thus TRd with the belief u 2 satis es the requirements 29 13 it is a weak PBE But u 2 Violates the requirement 4 because it must be zero according to requirement 4 so it is not a PBE Case 2 Suppose u lt ie l u gt LL Then playing u is rational for player 11 Knowing this player I would choose L in his second move which means he would choose T in the rstmove Again the information set is not on the equilibrium path Thus TLu with the belief u lt satis es the requirements 13 it is a weak PBE But again u lt Violates the requirement 4 because it must be one according to requirement 4 so it is not a PBE 20 10 01 01 10 30 Sometimes we may need further re nements because PBE and sequential equilibria may include unreasonable equilibria In particular both PBE and a sequential equilibrium may attach a positive probability on nodes off the equilibrium path that can be reached only when players play a strictly dominated strategy We may wish to impose a further requirement Requirement 5 prossible each player s beliefs offthe equilibrium path should place zero probability on nodes that are reached only if another player plays a strategy that is strictly dominated beginning at some information Example Consider the following two games 151 00 10 01 u d u d L 31 00 L 151 00 M 10 01 M 10 01 R 22 22 R 22 22 Let s consider the left side problem rst Ifthe belief on the information set is ul LL then player H s expected utility from playing u at this belief is ull u0 u while that from playing d is l u Thus we have two cases Case 1 If u 2 then player 11 will paly u and thus player I will play L which implies that the equilibrium beliefmust be u l Requirement 4 is vacuous for this case because there is no information set off the equilibrium path Thus Lu together with ul is aPBE 31 Case 2 If u lt then player II will paly d and thus player I will play R Requirement 3 is vacuous for this case because the information set is off the equilibrium path Requirement 4 also cannot be applied because there is no further information Thus R 51 together with u lt is another PBE Thus there are two classes ofPBEs But the second PBE seems unreasonable because the belief that u lt is not reasonable Indeed M is strictly dominated by R for player I thus if we assume that player II knows that player I will not take such an action then u must be equal to 1 Requirement 5 eliminates the second PBE Note that there is a quali er ifpossible in Requirement 5 To see the meaning of this quali es let s consider the right side game which is exactly identical to the left side game except the top payoff for player I This game has two classes of PBEs R u with u 2 and R d with u lt Note that both L and M are strictly dominated by R for player 1 But even if player II knows that player I will not take such actions it is not possible to place zero probability on both nodes following M and L thus R5 cannot apply In other words R5 is useless in this case Both PBE and sequential equilibrium eliminate strictly dominated choices at information sets on the equilibrium path but does not eliminate weakly dominated strategies Example 32 The normal form representation of this game is the following and there are two pure strategy Nash equilibria In A and Out F Because there is no proper subgame they are also subgame perfect However Out F seems not reasonable because Fight is weakly dominated by Accommodate for player II So one might wish to apply the notions of PBE and sequential equilibrium to eliminate this unfortunately Out F is not eliminated from applying any of these stronger notions Fight Accommodate In 1 l 2 2 Out 0 5 0 5 To see that both of them are PBE let ul u be the belief attached by player II to nodes X1 and X2 Then player II s expected payoff is u l l u5 5 6y from playing F and u2 l u5 5 3u from playing A Thus player II would play F if LL0 andAif 0lt LLSl Thus Out Fwith LL0 is aPBE Also In Awith ul is a PBE Finally we show that Out F with u 0 is also a sequential equilibrium Let a V l l behav1or strategy of player I be a probability distribution b l attached to In v v and Out respectively This behavior strategy is strictly positive and converges to 01 which corresponds to playing Out for sure Now let s construct uV1 luv from H using Bayes rule Then because PJcl l history up to the info set L i and Phistory up to the info set v P l Px2 l history up to the info set 1 Phistory up to the info set v uV1 luv 11 1 converges to 01 which is the belief associated with Out F v v In like manner we can easily show In A with u l is also a sequential equilibrium 33 V Sequential Games under Incomplete Information Signaling game 1 Nature randomly draws a type for the Sender from a set 8 616n according to a probability distribution pQ where p gt 0 and ipwl 11 2 Sender observes his own type Q and then chooses a message m from a set M 3 Receiver observes m but not Q and then chooses an action a from a setA 4 Payoffs are given by USm ma for Sender and UR m ma for Receiver Note that Sender s pure strategy is a ntuple m61m6n and Receiver s pure strategy is a messagecontingent function am 91 621 In the case of a signaling game the set of subgame perfect Nash equilibria is identical to the set of Bayesian Nash equilibria because there is no proper subgame So if 34 you want to apply any re nement to the signaling game it must be no weaker than the concept of the perfect Bayesian equilibrium We state corresponding requirements ofPBE in the context of a signalling game Requirement 1 A er observing any message m EM the receiver must have a belief about which types could have sent m which we denote by uQ l mj where uQ lm ZO foreach Q and Zylel 5 Requirement 2 R For each message m EM the receiver s action amj must maximize the Receiver s expected utility where the expected utility is formed by using the belief uQ l mj That is a mjeargrgg 01 lmjURmjak01 Requirement 2 S For each Q the Sender s message mQ must maximize the Sender s utility given the Receiver s strategy amj That is mae argmaxUsmamja mJEM Note In the literature Requirement 28 is often called the incentive compatibility condition of the Sender Requirement 3 For each message m EM if there exists Q such that mQ m then the Receiver s belief at the information set corresponding to m must follow from Bayes rule and the Sender s strategy MQ 21991 9 lej Requirement 4 vacuous in signaling games Requirement 5 If the information set following m is off the equilibrium path and m is strictly dominated for type Q that is there is m39eM such that 35 min USmquotaQ gt max USm39a01 then Receiver s belief 01 l m39 should be zero if BEA BEA possible Example Consider the following version ofthe signalling game elm t We rst nd the pure strategy Bayesian Nash equilibria of this game by transforming it into a normal form representation Note that Sender s strategy is a pair rrt61rrt02 thus there are four possible strategies and Receiver s strategy is a pair am1 am2 thus there are four possible strategies u u u d d u d d L L 12 34 12 34 40 01 40 01 L R 11 30 11 32 41 00 40 02 R L 22 14 02 04 20 11 00 01 R R 21 10 01 02 21 10 01 02 Thus there are two pure strategy BNE LL ud and RL ud The former is a pooling equilibrium while the latter is a separating equilibrium 36 Next we solve this game by the concept of PBE To do this we rst compute Receiver s payoff Requirement 1 says that Receiver must attach a belief for each information set let 61 l L be Receiver s belief after observing message L and 61 lR be Receiver s belief after observing message R Requirement 2 says that Receiver must be sequentially rational given these beliefs If Receiver observes message L then he obtains EHUR Lu6 3u01lL 4l 61 lL 4 61 l L by playing u39 and EHURL d0 001lL11 01 l L 1 01 l L by playing d Because 4 u61 lL gt1 u61 lL for all 1161 lL 6 01 playing u is rational for Receiver whatever belief he attaches to the information set passing through message L If Receiver observes message R on the other hand then he obtains EHUR Ru0 lu61 lR 01 61 l R 61 l R by playing u39 and EHURRd0 0u01lR 21 u01lR 2 2u61 l R by playing d Thus when Receiver observes message R playing u is rational for him if 61 l R 2 while playing d is rational if 61 l R lt M Suppose 61 l R 2 while 61 lL can take any value between 0 and 1 Then given these beliefs Receiver s optimal strategy is aL aR u u Hence Sender would choose m61m62 RL But this implies 61 lL 0 and 61 l R1 These beliefs are consistent with the initial assumption thus a pair of strategies m01 m62 aL aR RL u u together with a belief system u61 l L 61 l R 01 constitutes a FEB 37 M Suppose u61 lR lt while u61 lL can take any value between 0 and 1 Then given these beliefs Receiver s optimal strategy is aLaRud Hence Sender would choose m61m62LL But this implies 61 lL 05 while imposing no further restriction on 61 l R Thus a pair of strategies m01 m62 aL aR LL u 1 together with a belief system u61 l L 61 l R 05 r Where r lt constitutes another FEB 38 Note that the concept of PBE is of no help in re ning equilibria we still have two equilibria which are identical to the Bayesian Nash equilibria Requirement 5 is also vacuous because there are no strictly dominated actions off the equilibrium path Example This example shows that some ofPBEs are not reasonable elm 39 We rst compute the PBEs ofthis game Requirement 1 says that Receiver must attach a belief for each information set let 61 l L be Receiver s belief after observing message L and 61 l R be Receiver s belief after observing message R Requirement 2 says that Receiver must be sequentially rational given these beliefs If Receiver observes message L then he obtains EHUR Lu6 21461 lL 0l 61 lL 21461 l L by playing u39 and EHURL d0 001lL11 01 l L 1 01 l L by playing d Thus if 1161 lL Z playing u is rational for Receiver otherwise playing d is rational If Receiver observes message R on the other hand then he obtains EHUR R m0 0u01lRll 61 l R l 61 l R by playing u39 and EHURRd01t01 l R 01 01 l R 01 l R by playing d Thus when Receiver observes message R playing u is rational for him if 1161 l R S while playing d is rational if 61 l R gt Case 1 1161 Lgtl and 01 R S 3 Then Receiver plays u u and Sender plays L R This implies that 61 lL 1 and 61 l R 0 This is one PBE Case 2 61 lLZ and 61 lRgt Then Receiver plays u d and Sender plays L L This implies that 61 lL There is no restriction on 61 l R This is another PBE Case 3 61 lLlt and 61 lRS Then Receiver plays d u and Sender plays L R This implies that 1161 lL 1 and 61 l R 0 which is a contradiction Case 4 61 lLlt and 1161 lRgt Then Receiver plays d d and Sender plays L L This implies that 61 lL which is a contradiction Thus there are two PBEs one separating and the other pooling But the belief 61 l R gt in case 2 is not reasonable because 61 node in the Receiver s information set following R can be reached only when the Sender plays a strategy that is strictly dominated beginning at the information set Requirement 5 dictates that 61 l R039 thus the second equilibrium is eliminated if we impose Requirement 5 40 u u u d d u d d L L 31 20 31 20 21 01 21 01 L R 32 21 30 20 22 01 20 00 R L 11 00 01 10 11 01 01 11 R R 12 01 00 10 12 01 00 10 In some games there are perfect Bayesian equilibria that seem unreasonable but nonetheless satisfy requirement 5 Cho and Kreps add the following requirement to eliminate such cases Requirement 5 Intuitive criterion of Cho and Kreps If the information set following m is off the equilibrium path and m is equilibriumdominated for type Q that is if type Q s equilibrium payoff is greater than Q s highest payoff from m then the Receiver s belief uQ l mj should place zero probability on type Q if possible Note Requirement 5 is stronger than Requirement 5 because if m is dominated for type Q then m must be equilibriumdominated for type Q Example 41 Let 1161 lLbe Receiver s belief after observing message L and 61 l R be Receiver s belief after observing message R If Receiver observes message L then he obtains EHUR Lu0 101 l L 11 6 L 201lL1 by playing u and EHUR L 10 0u61lL 0l 61 lL 0 by playing d Thus if 61 lL Z playing u is rational for Receiver otherwise playing d is rational If Receiver observes message R on the other hand then he obtains EHUR Ru0 lu61 l R ll 61 lR 21401 l R 1 by playing u39 and EHUR R 10 0u01lR 0l u01lR 0 by playing d Thus when Receiver observes message R playing u is rational for him if 1161 l R 2 while playing d is rational if 61 l R lt Case 1 61 lLZi and u01lRZ Then Receiver plays u u and Sender plays L R This implies that 61 lL 1 and 61 l R 0 which is a contradiction Case 2 y61lLZ and y61lRlt Then Receiver plays u d and Sender plays R R This implies that 61 l R 01 and no restriction on 61 lL 2 This is one PBE Case 3 y61lLlt and y61lRZ Then Receiver plays d u and Sender L 2 plays L L This implies that 1161 lL 01 and no restriction on 61 l RZ This is another PBE Case 4 61 lLlt and y61lRlt Then Receiver plays d d and Sender plays L R This implies that 61 lL 1 and 61 l R 0 which is a contradiction Thus there are two pooling PBEs Out of the two PBEs LL du with 61 lL 01 and 61 l R gt i is not reasonable although it satis es Requirement 5 2 u u u d d u d d L L 10 151 32 00 32 00 32 00 L R 1 1 151 13 10 31 051 33 00 R L 00 151 20 051 02 10 22 00 R R 01 151 23 00 01 151 23 5 00 42 This is not reasonable because if Receiver unexpectedly observe R then the Receiver concludes that the Sender is at least as likely to be type 01 as type 02 ie 61 l R Z even though a type 61 cannot possibly improve on the equilibrium payoff of 3 by choosing R rather than L while b type 02 could improve on the equilibrium payoff of2 by receiving the payoff of3 that would follow if the Receiver held a belief 61 l R lt i For instance type 62 would make the following speech Seeing me choose R should convince you that I am type 02 because choosing L could not possibly have improved the lot oftype 61 by a And if choosing R convinces you that I am type 62 then doing so will improve my lot by b Ifsuch a speech is believed it dictates that 61 l R 0 43 Example Spence s signaling model In Spence s signaling model wages increase with education more than can be explained by the effect of education on productivity The crucial assumption in this model is that low ability workers find signaling more costly than do high ability workers If low ability workers find it more dif cult to acquire the extra education a larger increase in wages to compensate high ability workers for education is sustained at the equilibrium There are two types of workers with unobservable abilities 6H and BL where OH gt 0L gt 0 and Pr6 OH q 6 01 We take an extreme assumption that education does nothing for a worker s output productivity thus the output produced by a type 6 worker who expended e years of education is y86 6 In other words education plays only the role of signaling in this extreme version of Spence s model The main conclusion is not changed even if the productivity is affected by education We assume that the cost ofeducation is given by a function c86 where d 8c86L gt 8c86H 2 Mme 8 88 88 c06 0 The last property is called a single 2 crossing property As a speci c example consider c86 6 Then c06 0 Z 8 8820 lgt0 and 8080L igt 8080H 8 88 6 88 6 88 OH L Let uw 86 w C8 0 be the utility ofa type 6 worker who chooses e and receives w We assume that the reservation wages are equal to zero for both types The rm offering wages is competitive and its profit is given by 7rw 0 y8 0 w 6 w After observing education level e chosen by a worker the rm attaches a belief system 116H 8 to its information sets De nition A PBE of Spence s signaling game is a pair of strategies 86L 80H W8 and a beliefsystem 116H 8 such that 1 wquote e argmax 816 e0H 1 6H 26 w 2 80H e arg max w8 C80H and 80L e arg max w8 080L and 44 3 y6H l e is determined by Bayes rule Because the rm is competitive the zero pro t condition implies that the equilibrium wage must be equal to we wa l e0H l y6H l 86L Thus 0L S we S 6H for all e Knowing this fact we now turn to the issue of the worker s equilibrium strategy her choice of an education level contingent on her type We study two possibilities separately separating and pooling equilibria Case 1 Separating eguilibria Suppose 80L e0H 8182 62 gt 61 is the worker s equilibrium education choice at a separating PBE In any PBE beliefs on the equilibrium path must be derived from the equilibrium strategies using Bayes rule Upon seeing 86L the rm assigns probability 1 to the worker being type 6L which implies that we 86L 6L Likewise upon seeing 86H the rm assigns probability 1 to the worker being type 6H which implies that we 86H OH For other levels of education the PBE does not restrict on we 0 ifee0L 9L ifee0L Thus HHle l ifee0H and we 6H ifee0H In 01 otherwise ODOH otherwise we 6 particular 116H l e L is an appropriate belief system 6H 6L We rst prove that 8012 0 in any separating PBE Suppose 80L e gt 0 By doing so the worker receives the wage of BL but this level of wage is guaranteed even if e 0 Since choosing e 0 would have saved her the cost of education she would be strictly better off by doing so which is a contradiction On the other hand for 80H to be a separating equilibrium requirement ZS dictates that we must have quotwile 8 6L 8611 6L 2 wiF 8 8 8 9 6L 45 G 0L 0039L 2 9H 089H9L and VtE 89H 89H 9 9H 2 MYE 89L 89L agH G 0H 089H9H Z 9L 009H The rst condition requires that Ltype should not envy Htype s deal and the second condition requires that Htype should not envy Ltype s deal For instance if 2 2 0020 23 6 the rst condition becomes 6L 2 6H Q 80H Z l0L0H 6L g L a 2 and the second condition is 6H 86 2 0L Q 86H S JOH 0H 6L e H Thus if we let g be the level ofeducation for which ce6L OH 6L holds and E the level of education for which 080H OH 6L holds then the above two inequalities imply that we must have 86H 6 5 at a separating PBE Summarizing 80L 80H 0 where a e 35 BL ife 60L we OH ife 86H and y6H l e W constitute a separating PBE 9 ODOH otherwise H L Because we have so much freedom to choose the rm s belief offthe equilibrium path many wage schedules can arise that support the equilibrium education choices of workers We illustrate two possibilities Type 6L w W T e 6 yp L Type 6H ype 6H 6H 9H I I I T 39 I 39 r 9L 9L I elteHgt e elteHgt e 46 Note that these various separating equilibria can be Pareto ranked In all of them the rm makes zero pro ts and a low ability worker s utility is 6L But a high ability worker does strictly better in equilibria if she gets a lower level of education Hence all equilibria in which 86H gt g are strictly dominated by an equilibrium in which 86H g These equilibria are eliminated ifwe impose requirement 5 The welfare effects ofsignaling activities in this case are generally ambiguous By revealing information about worker types signaling can lead to a more efficient allocation ofworkers labor and in some instances to a Pareto improvement At the same time because signaling activity is costly workers welfare may be reduced if they are compelled to engage in a high level of signaling activity to distinguish themselves As the fraction of high ability workers grows it becomes more likely that the highability workers are made worse off by the possibility of signaling In fact as this fraction gets close to I nearly everyone is getting costly education just to avoid being thought to be one of the hand ll of bad workers Market competition is devastating Case 2 Pooling equilibria Suppose 86H e0L 818 is the worker s equilibrium education choice at a pooling PBE Then the rm s beliefmust be y6H l e q so that we qHH l q0L E0 Such wage offer is however possible only when it gives the low ability workers as much as if she chose not to study at all and to get wage 6L in other words uE6 8101 Z u0L 00L Hence ifwe let 3 be the level of education for which uE0e0L u0L00L then e S g Summarizing 86L86H 818 where 8 6 03 we E6 and 116H l e q constitute a pooling PBE Note that a pooling equilibrium in which e 0 Pareto dominates any pooling equilibrium with e gt 0 Thus Requirement 5 again eliminates all equilibria in which e gt 0 Nobody takes education at an equilibrium 47 Cheap talk games Cheap talk games are identical to signalling games except that payoffs are given by USaQ for Sender and URaQ for Receiver In other words messages do not affect payoffs of the players in cheap talk games thus cheap talks are often called costless signaling 1 Nature randomly draws a type for the Sender from a set 8 616n according to a probability distribution p where pQ gt 0 and ipQ 11 2 Sender observes his own type Q and then chooses a message m from a set M 0102 3 Receiver observes m but not Q and then chooses an action a from a setA 4 Payoffs are given by USaQ for Sender and UR a39 for Receiver 921 eJ39eir 39 Saw 48 There are a few distinct characteristics of cheap talk games 1 In signalling games all the Sender s types have the same preferences over the Receiver s possible actions For instance all workers prefer higher wages independent of ability In such situations cheap talk cannot be informative and there cannot be equilibrium in which cheap talk affect Receiver s action Thus for cheap talks to be informative different Sender types must have different preferences over the Receiver s actions Proof Suppose there were a pure strategy equilibrium in which 61 sends m1 and 02 sends m2 In equilibrium the Receiver will believe m1 as coming from type Q and so will take the optimal action a given this belief Since the Sender types have the same preferences over actions if one type prefers al to all then all types have this preference thus sned m1 to m2 thereby destroying the putative equilibrium 2 The Receiver s preferences over actions must not be completely opposed to the Senders In general more communication can occur through cheap talks when the player s preferences are more closely aligned Proof Suppose Receiver prefers a when Sender type is If 61 prefers a1 and 02 prefers a2 then communication between the two players can occur But if 01 prefers a2 and 62 prefers al then communication cannot occur because the Sender would like to mislead the Receiver 3 In cheap talk games a pooling equilibrium always exists The interesting question in a cheap talk game is therefore whether nonpooling equilibria exist Proof Because messages have no direct effect on the Sender s payoff pooling is a best response for the Sender if the Receiver ignores all messages Because messages have no direct effect on Receiver s payoff a best response for the Receiver is to ignore all messages ifthe Sender is pooling 49 Example Consider the following game with w 2 y and x Z Z alien939quot 91 921 Let 1161 61be Receiver s belief after observing message 61 and 1161 62 be Receiver s belief after observing message 62 If Receiver observes message 01 then he obtains 61 i 01 by playing u and 1 u6161 by playing d Thus if y61612 playing u is rational for Receiver otherwise playing d is rational If Receiver observes message 62 he obtains OJOD by playing u and 1 u0102 by playing d Thus when Receiver observes message 02 playing u is rational for him if 61 62 2 while playing d is rational if 61 62 lt Case 1 u01012 and 01 02Z Then Receiver plays u u and both types of Sender are indifferent between 01 and 02 The only strategies compatible with Requirement 3 are 6161 with 61 i 61 p 2 and 6262 with 61 02 p 2 Case 2 61 602 and u0102lt Then Receiver plays u d and Sender plays 6162 This implies that 1161 61 1 and 61 i 62 0 50 Case 3 61 61lt and y61l622 Then Receiver plays d u and Sender plays 0201 This implies that 61 lL 0 and and 61 l 02l Case 4 y61l61lt and 61 62lt Then Receiver plays d d and both types of Sender are indifferent between 61 and 62 The only strategies compatible with Requirement 3 are 6161 with 61 l 61 p lt and 6262 with 61 02 p lt Example Suppose 6 is unifome distributed on 01 Receiver s action is ae0l and the message space is M G Let UScz0 a 0b2 be Sender s payoff and UR 516 a 62 Receiver s payoff As we said earlier a pooling equilibrium always exists in cheap talk games The more interesting question is what are the nonpooling equilibria Claim There is a maximum equilibrium number nb of intervals into which the type space is divided ie 01021UClx2UUxnb71l such that at the PBE all the types in a given interval send the same message while types in different intervals send different messages We rst observe that given the Sender s type the Sender s optimal action 0b exceeds the Receiver s optimal action 6 by b Thus if two adjacent steps say xk71xk and xkxk1 were of equal length the boundary type between the steps xk would strictly prefer to send the message associated with the upper step The only way to make the boundary type indifferent between the two steps is therefore to make the upper step appropriately longer than the lower step Suppose the lower step xk1xk is oflength c ie xk 0 x1671 The Receiver s x x optimal action associated With this step is xk and it is gb below Receiver s optimal action for xk namely xkb Thus to make the boundary type xk indifferent between the two steps the Receiver s action associated with the upper step 0 xk xk c xkxk1 must be Eb above the optimal action for xk39 T xk bE b or xk xk c 4b Thus each step must be 4b longer than the last If the rst step us of 51 length d the second must be of length d4b and the nth step which is equal to 1 must be 61 d 4bd n lb Thus 7161 nn 12b 1 Given any n such that nn l2b lt l there exists a value of d that solves this inequality Since the length of the rst step must be positive the largest number of steps in such an equilibrium is the largest integer value of n namely nb less than 1 l2b If 12 nb 1 there is no communication if the players preferences are too similar Thus for partially pooling equilibria to exist b lt Also nb decreases in b and approaches infinity as b approaches zero more communication can occur through cheap talk when the players preferences are more closely aligned Other examples Stein 1989 shows thjat policy announcements by the Fed can be informative but cannot be precise Mattews 1989 studies how a veto threat by the president can in uence which bill gets through Congress AustenSmith 199 shows that in some settings debate among legislators improve the social value of the eventual legislation Farrell and Gibbons 1991 show that in some settings unionization by fascilitating communication from the work force to management improves social welfare 52