1 Grim Trigger in the Repeated Prisoner s Dilemma (70 points)

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1 Solutios to Problem Set 4 David Jimeez-Gomez, Fall 2014 Due o 11/7. If you are workig with a parter, you ad your parter may tur i a sigle copy of the problem set. Please show your work ad ackowledge ay additioal resources cosulted. Questios marked with a ( ) are iteded for math-ad-game-theory-heads who are iterested i deeper, formal exploratio, perhaps as preparatio for grad school. The questios typically demostrate the robustess of the results from class or other problems, ad the aswers do ot chage the iterpretatio of those results. Moreover, this material will ot play a large role o the exam ad teds to be worth relatively little o the problem sets. Some folks might cosequetly prefer to skip these problems. 1 Grim Trigger i the Repeated Prisoer s Dilemma (70 poits) I oe istace of the prisoer s dilemma, each player chooses whether to pay some cost c > 0 i order to cofer a beefit b > c oto the other player. The payoffs from a sigle iteratio of this prisoer s dilemma are therefore: Cooperate Defect Cooperate (b c, b c) ( c, b) Defect (b, c) (0, 0) The repeated prisoer s dilemma 1 is built out of several stages, each of which is a copy of the above game. At the ed of each stage, the two players repeat the prisoer s dilemma agai with probability δ, where 0 δ 1. A strategy i the repeated prisoer s dilemma is a rule which determies whether a player will cooperate or defect i each give stage. This rule may deped o which roud it is, ad o either player s actios i previous rouds. For example, the grim trigger strategy is described by the followig rule: cooperate if both players have ever defected, ad defect otherwise. The goal of this problem is to show that the strategy pair i which both players play grim trigger is a Nash equilibrium if δ > c b. 1. Suppose that player 1 ad player 2 are both followig the grim trigger strategy. What actios will be played i each stage of the repeated game? What are the payoffs to players 1 ad 2 i each stage? Aswer: I each stage of the repeated game, players 1 ad 2 will cooperate. The payoffs will the be b c to each player i a give stage. 2. Usig your result from part 1, write dow the expected payoff to player 1 from the etire repeated prisoer s dilemma i terms of c, b, ad δ. 1 Please cosult Sectio 5 of the Game Theory hadout o Repeated Games for details. 1

2 Hit: Remember that, if δ < 1: Aswer: The payoff to player 1 is: a + aδ + aδ 2 + aδ = a 1 δ (b c) + (b c)δ + (b c)δ = b c 1 δ 3. Now we will check whether player 1 ca improve his payoff by deviatig from the grim trigger strategy. Argue that we oly eed to check the case where player 1 plays all-d, that is, player 1 defects i every roud. Aswer: If player 1 deviates, the he must defect i some roud. If he has icetive to defect i some roud k, the by symmetry, player 1 has icetive to defect i the first roud. But if player 1 defects i the first roud, the player 2 defects forever, so it could ot possibly be Nash for player 1 to cooperate i ay roud k. Thus, if player 1 has icetive to deviate to ay strategy at all, he also has icetive to deviate to all-d. 4. Suppose that player 2 plays grim trigger ad player 1 deviates from grim trigger ad plays all-d. What is the total payoff to player 1 from the etire repeated prisoer s dilemma? Aswer: Player 1 receives payoff b i the first roud ad 0 i each subsequet roud, so his total payoff is b. 5. For grim trigger to be a Nash equilibrium, we eed that the payoff to player 1 from playig grim trigger is greater tha or equal to the payoff to player 1 from playig all-d, assumig player 2 s strategy is fixed. Usig your results from parts 2 ad 4, write dow a iequality that must be satisfied i order for grim trigger to be a Nash equilibrium. Simplify this iequality to obtai the coditio δ > c b. Aswer: We have: This is the desired result. b < b c 1 δ 1 δ < b c b 1 b c < δ c b b < δ 6. ( ) - 10 poits. Show that the Grim Trigger is a Subgame Perfect equilibrium i additio to beig a Nash equilibrium [Hit: use the oe-stage deviatio priciple] 2

3 Aswer. To show that the Grim Trigger is SPE, we eed to show that players do ot have icetives to deviate eve i stories which are impossible whe both players follow the strategy. Namely, we have to cosider strategies where at least oe player defected. Followig the same logic as i 3, we oly eed to make sure that player 1 does ot have a icetive to play C if somebody ever played D. If player 1 sticks to the Grim Trigger strategy ad plays D, she obtais 0. If she deviates oly this period, ad later coforms to Grim Trigger, she obtais c (because player 2 plays D sice he is usig Grim Trigger ad we are i a history where somebody defected). Therefore, the deviatio is ot profitable i this kid of histories; we already kew deviatios where ot profitable i histories where both players have always cooperated. Therefore, by the oe-stage deviatio priciple, the Grim Trigger is a SPE. So far we have focused o the Grim Trigger because it is a relatively simple strategy to uderstad, but ot ecessarily because we thik it is used i practice. Importatly, may of the isights we have leared from studyig the Grim Trigger geeralize to ay Nash equilibrium. 7. ( ) - 10 poits. Show that i ay Nash equilibrium i which both players play C at each period, player 2 must cooperate less i the future if player 1 were to deviate ad play D at ay period istead of C. Iterpret this result i terms of reciprocity, as discussed i lecture. Aswer. Let s be a Nash equilibrium where both players choose C at each period. That meas that player 1 s payoffs of s are (b c)/(1 δ). Let W be the total payoff that player 1 would get from roud 2 owards if she chose D i the first period; the her payoffs for playig D would be b + δw. I order for s to be a Nash equilibrium it must be the case that b c 1 δ b + δw. Now, if player 2 cooperated at all future rouds, the W would be at least (b c)/(1 δ), ad possibly more. That meas that i that case for s to be a Nash equilibrium it must be the case that b c 1 δ b + δ b c 1 δ b c b, which is a cotradictio! We obtaied this cotradictio from assumig that player 2 cotiued to cooperate at all periods after player 1 defected: therefore it must be the case that player 2 defects at least oe period after player 1 defects, provig our claim. 3

4 2 No Cooperatio for Small δ (50 poits) I lecture, we argued that cooperative equilibria exist i the repeated prisoer s dilemma if ad oly if δ > c b. I problem 1, you showed that we ca have a Nash equilibrium i which both players always cooperate (specifically, the equilibrium i which both players play grim trigger) if δ > c b. I this problem, we will show that if δ < c b, the the oly Nash equilibrium is (all-d, all-d). That is, cooperative equilibria exist oly if δ > c b. Combied, your resposes to these two questios thus provide a complete proof to our claim from lecture. 1. Suppose that the strategy pair (s 1, s 2 ) is a Nash equilibrium, ad let U 1 (s 1, s 2 ) ad U 2 (s 1, s 2 ) be the payoffs to players 1 ad 2, respectively. Show that U 1 (s 1, s 2 ) 0 ad U 2 (s 1, s 2 ) 0. Aswer: Suppose either player received some egative payoff u i (s 1, s 2 ) < 0. The player i could improve his payoff by deviatig to all-d, sice this strategy guaratees a payoff u i (all-d, s i ) 0. Hece player i has a icetive to deviate, cotradictig that (s 1, s 2 ) is Nash. We coclude that each player s payoff is oegative. 2. Notice that, i each roud of the prisoer s dilemma, the sum of the payoffs to players 1 ad 2 is either 2(b c), b c, or 0. Show that, if s 1 ad s 2 are ay two strategy pairs, the U 1 (s 1, s 2 ) + U 2 (s 1, s 2 ) 2(b c). Aswer: The sum of the payoffs i each roud is 2(b c), b c, or 0. I either case, the sums of the payoffs is at most 2(b c). Summig over all rouds, u 1 (s 1, s 2 ) + u 2 (s 1, s 2 ) 2(b c) + 2(b c)δ +... = 2(b c) 1 δ (1) 3. Now assume δ < c b. Usig your results from part 2, show that U 1(s 1, s 2 ) + U 2 (s 1, s 2 ) < 2b for ay strategy pair (s 1, s 2 ). Use this to coclude that, if (s 1, s 2 ) is a Nash equilibrium, at least oe player receives total payoff less tha b. Aswer: If δ < c b, the 1 < b b c by algebra. Pluggig this ito the result of (b), u 1 (s 1, s 2 ) + u 2 (s 1, s 2 ) 2(b c) b = 2b (2) b c The at least oe player receives total payoff less tha b (if both received payoff greater tha b, the sum would exceed 2b, a cotradictio). 4. Suppose that, whe players 1 ad 2 play s 1 ad s 2, both players cooperate i some roud k. Without loss of geerality, we may assume that k = 1 (otherwise we repeat the argumet from parts 1-3 to the subgame startig at roud k, itroducig a factor of δ k 1 ). Usig your result from part 3, show that oe of the players ca improve his payoff by deviatig. Aswer: Suppose player 1 receives payoff less tha b. The if players 1 ad 2 cooperate i roud 1, player 1 ca improve his overall payoff by playig all-d ad gettig a total payoff at least b. Thus (s 1, s 2 ) caot be a Nash equilibrium. 4

5 5. Next we eed to rule out the possibility of a roud i which oe player cooperates ad the other defects. Repeat the argumet of part 2 usig the additioal result that players 1 ad 2 ever simultaeously cooperate (so the sum of their payoffs i a give roud is either b c or 0). Show that U 1 (s 1, s 2 ) + U 2 (s 1, s 2 ) b c. Aswer: Sice players 1 ad 2 ever simultaeously cooperate, we have that the sum of their payoffs i a give roud is at most b c. Thus by the argumet above, u 1 (s 1, s 2 ) + u 2 (s 1, s 2 ) < b c 1 δ (3) 6. Agai assume that δ < c b. Use your results from parts 1 ad 5 to coclude that each player s payoff is less tha b; that is, U 1 (s 1, s 2 ) < b ad U 2 (s 1, s 2 ) < b. Sice we have also show 1 < b b c, we have: u 1 (s 1, s 2 ) + u 2 (s 1, s 2 ) < b c 1 δ < b (4) Sice the payoffs are oegative, both players receive payoff less tha b. 7. Now suppose that, i the first roud, player 1 cooperates ad player 2 defects. By your reasoig from part (f), player 2 receives total payoff less tha b. Show that player 2 ca improve his payoff by deviatig, so that (s 1, s 2 ) is ot a Nash equilibrium. Aswer: I the first roud, player 2 defects ad receives payoff b. But by part (e), player 2 receives total payoff less tha b over the whole game. Thus player 2 has a icetive to deviate ad play all-d, which would ear him payoff at least b. So we coclude that (s 1, s 2 ) is ot a Nash equilibrum. Usig this proof by cotradictio, you have showed that a strategy pair (s 1, s 2 ) which ivolves cooperatio i ay period caot be a Nash equilibrium if δ < c b. It follows that (all-d, all-d) is the oly equilibrium i this case. 3 Icorporatig Altruism s Quirks: Observability ad Iattetio to Efficacy (50 poits) Cosider the followig simple twist to the repeated PD: the first period agets are ot payig full attetio to the game, ad with probability 1 p they do ot observe what actios were played. For the rest of periods, both agets are payig attetio ad kow which actios were played. Note that whe p = 1, we are back to the stadard case of the repeated PD. However, whe p < 1, it might be that actios where ot observed i the first period. 1. For a arbitrary p, we defie Grim Trigger as the strategy that cooperates if the player ever observed D, ad defects otherwise. We kow from lecture that whe p = 1, the strategy 5

6 profile where both players use the Grim Trigger is a Nash equilibrium whe δ c/b. Show that both players usig Grim Trigger is a Nash equilibrium if p δ c b c. [Hit: Thigs become much easier if you defie a ew variable W which is the total payoff player i will receive from period 2 ad owards if both play accordig to the strategy profile. That allows us to write payoffs as U i (s) = b c + δw ]. Aswer. Cosider the deviatio which plays D i the first period: i that case o period 2 owards payoffs will be V. This deviatio will ot be profitable if b c + δw b + δ(1 p)w, which is equivalet to p c δw. Now, takig ito accout that W = (b c)/(1 δ), we get that the deviatio is ot profitable whe p 1 δ δ c b c. 2. ( ) - 5 poits. Show that whe p < c δ b c at least oe player must play D i the first period [Hit: i additio to W, defie a ew variable V as the payoff player i will get o period 2 ad owards if he deviates from the strategy profile. Also, otice there is always a player whose utility is at most (b c)/(1 δ).] Aswer. We kow that U 1 (s) + U 2 (s) b c. Therefore there is at least oe player i such that U i (s) b c. Let V be the payoff player i would get o roud 2 ad owards if player 1 observed him playig D, ad W be the payoff player i would get o period 2 ad owards if player i did ot observe him playig D. The player i will defect if or equivaletly b c + δw < b + δ(1 p)w + δpv, (5) Now, we kow that W b c c/(b c). δpw < c + δpv. (6) ad V 0, ad therefore Equatio 9 holds if p < (1 δ)/δ 3. I light of these results, discuss the coectio betwee altruism ad observability. How does this relate to the observability experimetal results discussed i class, such as the eye spots experimet? 6

7 Aswer. These results show that observability is fudametal for players beig able to cooperate. If player 2 caot observe the actios of player 1, the player 2 use puishmets to icetivize player 1 to cooperate - ad therefore cooperatio breaks dow. Therefore, we would expect players to cooperate more whe they are beig observed. The eye spots experimet shows that eve a cue of observability is eough to obtai this effect. 4. Now suppose that i all periods, i additio to D ad C, there is a extra actio E, which has the same payoffs as playig C, except it costs e to play ad it yields 2e extra to the other player, for e 0. I the first period, we ow suppose that, with probability 1 p players caot tell whether the other player chose C or E. I period 2 ad subsequet periods, players ca always tell which actio their oppoet chose. Suppose that δ c+e b+2e. Cosider the Efficiet Grim Trigger strategy, which plays E if othig differet tha E was observed, ad D otherwise. Show that the strategy profile where both players use the Efficiet Grim Trigger strategy profile is a Nash equilibrium if p (1 δ)e δ(b + e c) Aswer. Cosider first the deviatio where player 1 plays C i the first roud. This deviatio is ot profitable if b + e c + δw b + 2e c + δ(1 p)w, or equivaletly p e δw. Takig ito accout W = c+e b, last expressio becomes p (1 δ)e δ(b + e c). (7) Therefore whe Equatio 7 holds, playig C is ot a profitable deviatio. Now, let s cosider a deviatio to D. This deviatio is always observable, ad it will be profitable wheever ad takig ito accout W = b c b + e c + δw b + 2e, ad solvig for δ we fid that D is ot profitable if 7

8 δ c + e b + 2e, which was assumed i the descriptio of the problem. 5. ( ) - 5 poits. Show that whe p < ()e δ(b+e c), there is o Nash equilibrium i which both players play E i the first roud. Aswer. We will prove this usig a similar method as i 2. We kow that U 1 (s) + U 2 (s) b+e c. Therefore there is at least oe player i such that U i(s) b+e c. Let V be the payoff player i would get o roud 2 ad owards if player i observed him playig C, ad W be the payoff player i would get o period 2 ad owards if player i observed him playig E (or could ot differetiate betwee C ad E). The player i will play C if or equivaletly b + e c + δw < b + 2e c + δ(1 p)w + δpv, (8) δpw < e + δpv. (9) Now, we kow that W b+e c ad V 0, ad therefore Equatio 9 holds if p < ()e δ(b+e c). This proves that C is a profitable deviatio, ad therefore player 1 will ot play E i the first roud. 6. Coect your aswer to 4 with what you leared i lecture about the iteractio betwee observability, efficiecy, ad altruism. I particular: what happes as the efficiecy parameter e ad the observability parameter p chage? Aswer. Our aswer to 4 suggests that players will choose the efficiet altruistic actio E wheever the other player ca observe whether the altruistic actio is efficiet (i.e. ca distiguish C from E) with eough probability. Whe the other player caot differetiate C ad E (i.e. p = 0), the either player will ever play E. We saw i class that people seem oblivious to the effect of their cotributios (whether they are savig 1,000 or 100,000 lives; whether their doatios are matched, etc.) most of the time - we could iterpret that as p = 0, because others do ot kow how efficiet the doatio was. But we also saw a experimet i which people doated more efficietly whe aother perso kew could observe both the doatio ad how efficiet it was - that would correspod with a high p, which i our model correspods to E beig played. 8

9 I particular, as e icreases, ()e δ(b+e c) also icreases, what makes Equatio 7 less likely to hold (ad so less likely that E will be played i the first roud). Recall that players pay a extra cost of e to give a extra beefit of 2e. As e icreases, the iterpretatio is that the efficiet actio, while beig more efficiet, is also more costly for ourselves. For example, as e gets high, people might eed to do extesive research i order to fid out the efficiet NGOs, etc. This makes it less likely that they will doate efficietly. As p icreases, Equatio 7 is more likely to hold (ad so more likely that E will be played i the first roud). We already explaied the ituitio for this: as p icreases, it is easier to observe whether C or E was played - for example if wheever a doatio is matched, it is posted to the perso s Facebook wall. 4 Costly Sigalig as a Extesive Form Game of Icomplete Iformatio (30 poits ( )) We ve see that whe we represet sequetial move (extesive form) games as simultaeous move (matrix form) games, we ca lose meaigful iformatio. So far, we have aalyzed costly sigalig as a matrix form game. We ow cofirm that this did ot somehow yield misleadig results by aalyzig it as a extesive form game. We start by rigorously defiig a simplified versio of the game preseted i problem 1 of problem set 3. For the rest of this problem, assume the followig: There are two types of seders: good ad bad. A fractio p = p are bad). There is oe type of receiver. of seders are good (ad There are two levels of sigal, which we call s 0, s 1. For good types, sedig these sigals costs 0 ad 1 respectively. For bad types, sedig these sigals costs 0 ad 6. Seders receive a payoff of 5 if receivers accept them. Receivers receive a payoff of 10 upo acceptig a good seder, ad a payoff of 10 upo acceptig a bad seder. The game proceeds as follows: (1) To model radom assigmet of seder s type, we assume that Chace moves first i the game. Chace has two possible actios, {Good, Bad}, ad chooses Good with probability p = 1/3. (2) After that, the seder seds a sigal (kowig her type; i.e. kowig what Chace chose). (3) Fially the receiver chooses whether to accept or reject, without kowig what actio Chace took, but kowig the sigal that the seder set. 1. Write this game formally (refer to the appropriate sectio of the Game Theory hadout), ad draw the game tree. [Hit: the game tree will look very similar to the oe for the beer-quiche 9

10 game i the Game Theory hadout] The set of players is N = {Seder, Receiver}. The set of actios is A Seder,h = {s 0, s 1 } at all histories where the Seder moves, ad A Receiver,h = {Accept, Reject} at all histories at which the receiver moves. The player fuctio is P ( ) = Chace, (Good) = P (Bad) = Seder, ad P (h) = Receiver for all other o-termal histories. µ(good ) = 1/3, µ(bad ) = 2/3. The game tree is as follows, where dotted lies idicate that two histories are i the same part of the Receiver s partitio: 2. Prove that each of the followig is a Perfect Bayesia equilibrium of the game: (a) Efficiet separatig: good seders sed sigal s 1, bad seders sed sigal s 0, ad receivers accept sigal s 1. Aswer. First of all, we eed to defie the assessmet, which i this case will be β R (Good s 1 ) = 1, ad β R (Good s 1 ) = 0. This assessmet are cosistet, because they are derived from Bayes rule. Next we check sequetial ratioality. Note that o seder has a icetive to deviate. The high type s payoff is 4, ad would get -1 if she deviated to s 0 ; the low type s payoff is 0 ad would get 1 if she deviated to s 1. Next, the receiver does ot have a profitable deviatio either. Her payoff is 10/3, if she deviated to acceptig all, her payoff would be 10(2/3 1/3), rejectig all, her payoff would be 0 acceptig s 0, rejectig s 1 : her payoff would be 2/3( 10) Because o player has a icetive to deviate, the strategy profile is sequetially ratioal. That, together with the assesmet beigs cosistet shows that they costitute a PBE. (b) Poolig with rejectio: good ad bad seders sed s 0 ad receivers ever accept ay sigal. Aswer. First we defie the assesmet: β R (Good s 0 ) = 1/3. However, o player seds s 1 i equilibrium. Because of that, we eed to cosider a possible tremble: for example the tremble where the bad seder seds s1 with probability ɛ. The, we ca apply Bayes rule to the tremble, ad obtai 10

11 β R (Bad s 1 ) = P (s 1 Bad) P (s 1 Bad) + P (s 1 Good) = ɛ ɛ + 0 = 1. This is a tremble because as ɛ 0, it coverges to our strategy profile (where the bad seder ever seds s 1 ). Because the assessmet is derived from Bayes s rule wheever possible it is cosistet. Next, let s show that the strategy profile is sequetially ratioal. The payoff for the receiver is 0; if she deviated ad accepted s 0 her payoff would be 10(1/3 2/3), therefore it is ot profitable. They payoff for the good ad bad types is 0. If either of them would deviate ad sed s 1, they would icur a cost ad be rejected, so they would have a egative payoff. Therefore either type has a profitable deviatio. Because the assessmet is cosistet ad the strategy profile is sequetially ratioal, they costitute a PBE. 3. Suppose the fractio of good seders icreases to 90%. Is poolig with rejectio still a equilibrium (i.e. if both types of seders sed s 0, would receivers still be better off rejectig all seders)? Show that there is a alterate equilibrium i which receivers accept all seders. Call this equilibrium poolig with acceptace. Aswer. Whe the fractio of good seders icreases to 90%, ay cosistet assessmet will have β R (good s 0 ) =.9. Therefore, the receiver has a profitable deviatio, to accept s 0, which yields payoff 10(.9.1) = 8 > 0. Next, we will show that equilibrium with acceptace is a PBE. Let β R (bad s 1 ) = 1. Agai, o player seds s 1 i equilibrium. Because of that, we eed to cosider a possible tremble: the tremble where the bad seder seds s1 with probability ɛ. The, we ca apply Bayes rule to the tremble, ad obtai β R (Bad s 1 ) = 1. As we have showed above i part 2b, this assessmet is cosistet. We have already showed that the receiver does ot have a icetive to deviate. By a reasoig similar to part 2b, either seder has a icetive to deviate. This shows the strategy profile is sequetially ratioal, ad poolig with acceptace is a PBE. 4. Now suppose that the cost of sedig s 1 for the good type icreases to 4. Are the efficiet 11

12 separatig, poolig with acceptace, ad poolig with rejectio PBE for p = 1/3. How about p =.9? What if the cost had icreased to 6 istead, for p = 1/3 ad p =.9? Aswer. [Complete but somewhat less formal - ay aswer at the same level of formality from studets should get full credit] Whe the cost for the good type icreases to 4, all the reasoigs i the previous parts hold, ad therefore (a) Whe p = 1/3, the efficiet separatig ad poolig with rejectio are PBE; poolig with acceptace is ot a PBE (b) Whe p =.9, the efficiet separatig ad poolig with acceptace are PBE; poolig with rejectio is ot a PBE However, whe the cost for the good type icreases to 6, sedig s 0 becomes a domiat strategy for her. Therefore the efficiet separatig is ever a PBE, ad: (a) Whe p = 1/3, poolig with rejectio is PBE; poolig with acceptace is ot a PBE (b) Whe p =.9, poolig with acceptace is a PBE; poolig with rejectio is ot a PBE 5. Pachaatha ad Boyd First cosider possible deviatios. Startig with the collective game a deviatio takes the form of a defectio. If a player defects here the they have o icetive to cooperate for the rest of the game as they will be shued. As such they will defect i all future rouds as well. We will term this strategy All-D. Give that a player cooperates i the collective game we cosider aother possible deviatio. A defectio i the mutual aid i the first roud where a player is ot eedy is the sole reasoable deviatio. Give that a player iteds to defectio at some poit, if it is profitable for them at a arbitrary stage it will also be profitable i the first possible roud. Ay other deviatio will lead to a payoff less tha these two deviatios. Now calculate the payoffs for these three strategies (Shuer, All D, Mutual Defect (MD): B C + (b c) 1 ω B B C + b 1 ω/ Next we compare the payoff of Shuer to the payoffs from deviatios. First deviatig to All D 12

13 B C + = (b c) 1 ω (b c) 1 ω > B > C This matches the first coditio from our assumptios. Next cosider a deviatio to MD: B C + (b c) > B C + b 1 ω 1 ω/ (b c) > b 1 ω 1 ω/ b c 1 ω > b 1 ω/ This matches our secod coditio. The for it ot to be a beeficial deviatio these coditios must be satisfied. As such all shuer is a NE iff these coditios are met. 6. Itroductio to Iformatio Structures (a) Player oe ca see two states, red ad blue. Player two ca see gree or orage. A state depedet strategy for P1 is A whe blue ad B whe red. For P2 A whe gree, B whe orage. (b) The players payoffs deped upo the proportio of the time a sigle state is realized. Player 1: q b + r d + s c Player 2: q c + r d + s b (c) To show this is a ESDS we will show that there are o better choices i each idividual state. Suppose P1 sees red, the P1 kows with certaity that the state is q ad P2 will see gree ad therefore play A. As such the best choice is A. Suppose P1 sees red blue, the with probability r r+s the state is r, ad P2 will play A. As such P2 s probability of playig A is less tha p by the assumptio ad P1 will play B optimally. Therefore P1 does t have icetive to deviate. Now cosider P2 whe P2 sees orage they kow for certai P1 will play B as such B is their best optio. Suppose they see gree, the the state is either q or r. The probability of it beig state q is q q+r sice this is more tha p P2 s best decisio is A. The either player will deviate ad this is a BNE. 7. Usig Iformatio Structures to Uderstad Higher-Order Beliefs (a) i. Give that P2 has see blue the state is r or s. With probability r r+s play A, therefore P1 beefits from playig A here. > p P1 will 13

14 ii. Now P2 kows P1 will play A to play A as well. s s+t > p of the time, ad therefore is also icetivized iii. Lastly P1 kows that P2 will always play A as such P1 s best decisio here is to play A as well. (b) i. Player 2 thiks P1 sees either blue or yellow. ii. P2 thiks that P1 thiks that P2 sees gree or orage. iii. P2 thiks that P1 thiks that P2 will do A if it is gree. iv. P2 should determie the relative probabilities of r, s ad t ad use it to optimize their payoffs. Playig A always would be a simple way to esure that players always coordiate. 14