ECONS STRATEGY AND GAME THEORY HOMEWORK #7 ANSWER KEY Exercise 5-Chapter 8-Watson (Signaling between a judge and a defendant) a. This game has a unique PBE. Find and report it. After EE, the judge chooses yy such that: MMMMMM (yy ) yy Taking FOCs with respect to yy, we obtain: (yy ) = yy = Similarly, after EE, the judge chooses yy such that: MMMMMM yy yy Taking FOCs with respect to yy, we obtain: Hence, the game becomes: yy yy = (-, ) E x= / N q y ( -y, - (y-) ) (-, ) E x= / N (-q) y ( -y, - y ) Let us first check for the existence of a separating PBE where EE and NN :. Belief: q= since N only comes from x=. Judge (second mover): After observing N, the judge selects y assigning full probability to being in the open node of his information set (see figure below) Hence, MMMMMM (yy ) yy Taking FOCs with respect to yy, we obtain: (yy ) =, which implies yy =. Defendant (first mover): o If xx =, the defendant compares: - if he chooses EE - if he chooses NN So, NN is better.
o If xx =, the defendant compares: - if he chooses EE - if he chooses NN So, EE is better. (-, ) E x= / N q y ( -y, - (y-) ) (-, ) E x= / N (-q) y ( -y, - y ) Hence, this separating PBE can be supported. Let us now check the separating NN EE. Beliefs: qq = since N only comes from xx =. Judge: After observing N, the judge assigns full probability to lower node of his information set. Then, he selects y such that: MMMMMM yy yy Taking FOCs with respect to yy, we obtain: yy, which implies yy =. Defendant: o o If xx =, the defendant compares: - if he chooses EE if he chooses NN So, NN is better Deviation from the prescribed separating NN EE. If xx =, the defendant compares: - if he chooses EE if he chooses NN So, NN is better. Hence, the separating NN EE cannot be supported as PBE. Let us now check if a pooling PBE where NN NN can be sustained. Beliefs: qq = + =. Judge: After observing N, given his beliefs q=/, he must choose y in order to maximize his expected utility: MMMMMM yy [ (yy ) ] + [ yy ]
Taking FOCs with respect to yy, we obtain: (yy ) yy =, which implies yy =. Defendant: o If xx =, the defendant compares: if he chooses EE = 5 if he chooses NN So, NN is better o If xx =, the defendant compares: if he chooses EE = 5 if he chooses NN So, NN is better Deviation from the prescribed pooling. Hence, the pooling NN NN cannot be sustained. Let us now check if a pooling PBE where EE EE can be sustained.. Beliefs: qq [, ] since N is only observed off-the-equilibrium. Judge: From his beliefs, he chooses y in order to maximize his expected utility: MMMMMM yy qq[ (yy ) ] + ( qq)[ yy ] Taking FOCs with respect to yy, we obtain: qq (yy ) ( qq) yy =, which implies yy = qq. Defendant: o o If xx =, the defendant compares: if he chooses EE qq if he chooses NN So, NN is better for any qq < Deviation from the prescribed pooling. If xx =, the defendant compares: if he chooses EE qq if he chooses NN So, EE is better for any qq > / the pooling EE EE cannot be supported as PBE either. b. Explain why the result of part (a) is interesting from an economic standpoint? The only equilibrium that we can support in this game is the separating equilibrium in which the innocent defendant provides evidence of his innocence, whereas the guilty defendant does not provide such evidence. This is something desirable, since the judge can perfectly infer the true innocence of a defendant by simply observing whether he/she presented evidences. c. When xx [, KK] with each value equally likely, compute the PBE. We are going to test the equilibrium where all types of x={,,k-} present evidence (E), but the last type x=k presents no evidence (N). ) Beliefs After observing the evidence presented by the defendant, the judge can perfectly observe his type,,,,k-. In these cases we don t need to specify beliefs. When no evidence(n) is presented, the judge s beliefs are:
μμ tt jj NN = jj = {,, KK } μμ(tt KK NN) = Which implies that after receiving no evidence, the judge assigns full probability to the K- type, and therefore no probability to any of the,,,,k- types. ) Judge s Best Response: Given N: max (yy yy KK) yy NN = KK Given E(where there is no information set and the judge knows what type has played E): max yy [ (yy xx) ] where x is the specific type of the defendant that presented evidence (a type that is observed by the judge thanks to the presentation of evidence). Taking FOCs with respect to y, we obtain yy + xx = yy EE = xx ) Defendant s Best Response: For types,,k- : if he provides evidence, E, then they get yy EE from the judge, providing: yy EE which must exceed his payoff from not presenting evidence: yy NN = KK (in this case the judge interprets that the defendant is a K-type and chooses a sentence yy NN = KK) o e, type x= prefers the payoff he obtains by presenting evidence, yy EE =, than his payoff from not presenting evidence, KK (since K > given that there are more than two types of defendants). o Similarly for type x=, where yy EE = > KK; and for all other types x=,, o The defendant who obtains the lowest equilibrium payoff from providing evidence is x=k-, who obtains yy EE = (KK ). Let us check if his equilibrium payoff from providing evidence is larger than from deviating, that is: KK + > KK KK + > KK >
This obviously holds, so the defendant behaves as prescribed when his type is x=,,k- For type K: if he doesn t provide evidence, N, (as initially prescribed) then he gets a sentence yy NN = KK from the judge, providing a payoff of: KK This must exceed his alternative payoff from providing evidence (E): KK The condition reduces to: KK > KK > This as well holds, showing the initially stated strategy, where types x=,,,,k- present evidence but type x=k does not, can be sustained as a PBE. 5
Exercise -Chapter -Harrington The extensive form of the WMD game: Nature Probability = w WMD No WMD Probability = (-w) (h) (-h) Allow Allow Allow Allow Bush Bush Bush Bush (b) (-b) (b) (-b) Bush 5 6 8. Nature moves first determining a presence of WMD: with probability w has WMD with probability ( ww) he does not, where < ww < /. After observing his own type, s strategies are the following: when he has WMD then he does not allow inspections with probability. when he does not have WMD then he can choose either to allow inspection with probability h or do not allow with probability ( h).. Assumptions: If has WMD, then Bush found out it and then Bush wants to invade; If does not have WMD, then Bush does not found it and he prefers not to invade. After observing s decision about inspection, Bush strategies are: 6
if allows inspections and WMD are found, then invade with probability. if allows inspections and WMD are not found, then do not invade with probability. if does not allow inspections, then Bush can guess that with some probability has WMD, so that Bush invade with probability b. See graph below: Nature Probability = w WMD No WMD Probability = (-w) (h) (-h) Allow Allow Allow Allow Bush Bush Bush Bush (b) (-b) (b) (-b) Bush 5 6 8 Steps: Step : Bush s beliefs If does not allow inspections, then the probability of s having WMD is given by Bayes s rule: PP(WWWWWW NNNNNN AAAAAAAAAA) = PP(WWWWWW) PP(NNNNNN AAAAAAAAAA, WWWWWW) PP(NNNNNN AAAAAAAAAA) = ww ww + ( ww) ( h) where Sadam has WMD with probability w, and in that event, he does not allow inspections with probability ; and while with probability ( ww), Sadam has WMD and, in that event, he does not allow inspections with probability ( h). Step : Bush s optimal strategy given his beliefs 7
Its optimality is clear when there are inspections, whether WMD are found or not. o When inspections are not allowed, Bush is content to randomize (that is, < bb < ) if and only if: EE BBBBBBh [IIIIII WWWWWW oooo NNNN WWWWWW] = EE BBBBBBh [NNNN IIIIII WWWWWW oooo NNNN WWWWDD aaaaaa NNNN] ww ww + ( ww) ( h) + 6 ww ( h) ww + ( ww) ( h) ww = ww + ( ww) ( h) + ww ( h) ww + ( ww) ( h) The left-hand expression is the expected payoff from invading, and the right-hand expression is the expected payoff from not invading. Solving this equation for h yields: e that: h = 5ww ( ww) < 5ww ( ww) < wwheeee < ww < 5 The latter condition was assumed. See graph below. o When Sadam has WMD, it is clearly optimal for him to not allow inspections. When he does not have WMD, it is optimal to randomize if and only if: bb + 8( bb) = where he earns a payoff of by allowing inspections in which case there is no invasion-- and gets an expected payoff of bb + 8( bb) = from not allowing inspections (where there is an invasion with probability b). Solving this equation, we can get bb = /. 8
Nature Probability = w WMD No WMD Probability = (-w) (h) (-h) Allow Allow Allow Allow Bush Bush Bush Bush (b) (-b) (b) (-b) Bush 5 6 8 EE BBBBBBh [IIIIII WWWWWW oooo NNNN WWWWWW] = PP(NNNN WWWWWW) UU(IIIIII) + PP(NNNN NNNN WWWWWW) UU(IIIIII) = PP(NNNN WWWWWW) + 6 PP(NNNN NNNN WWWWWW) EE BBBBBBh [NNNNNNNN WWWWWW oooo NNNN WWWWWW aaaaaa NNNN] = PP(NNNN WWWWWW) + PP(NNNN NNNN WWWWWW)
Exercise 5-Chapter -Harrington Consider the cheap talk game: Nature t t Probability =.6 Probability =. Sender Sender m m m m Receiver Receiver a b c a b c a b c a b c Sender Receiver 5 5 5 a. Find a separating PBNE. With a separating equilibrium, the sender chooses distinct messages, so let us presume that the sender chooses m when his type is t and chooses m when his type is t. (We could instead have supposed that the sender s strategy is to choose m when his type is t, and m when his type is t.) Receiver s beliefs After observing message mm, And after observing message mm, μμ(tt mm ) = μμ(tt mm ) = μμ(tt mm ) = μμ(tt mm ) =
Receiver s optimal response After observing mm, the receiver believes that such a message can only originate from a tt - type of sender. Hence, his optimal response is bb given that it yields a payoff of (higher than what he gets from aa,, and cc,.) After observing message mm, the receiver believes that such a message can only originate from a tt -type of sender. Hence, his optimal response is either bb or cc, since both yield a payoff of 5, rather than aa, which only provides a payoff of. For simplicity, we choose cc. Sender s optimal messages If his type is tt, by sending mm he obtains a payoff of (since mm is responded with bb), but a lower payoff of if he deviates towards message mm (since such message is responded with cc). Hence, the sender doesn t want to deviate from mm. [e that if mm were responded with bb, then the sender would be indifferent between mm and mm (both would yield a payoff of ). Strictly speaking, he wouldn t have incentives to deviate from message mm ]. If his type is tt, he obtains a payoff of by sending message mm (which is responded with cc) and a payoff of if he deviates to message mm (which is responded with bb). Hence, he doesn t have incentives to deviate from mm. [Similarly as above, note that if message mm were responded with bb the sender would obtain the same payoff sending mm and mm,. Nonetheless, tt -sender wouldn t have incentives to deviate from his initially prescribed message of mm ]. Hence, the initially prescribed separating strategy profile can be supported as a PBE. Also note that there is another separating equilibrium in which m and m are exchanged. b. Find a pooling PBNE. With a pooling PBNE, the sender chooses the same message regardless of his type. Let this message be m. Receiver s beliefs After observing message mm (in problem),.6 μμ(tt mm ) =.6 +. =.6. μμ(tt mm ) =.6 +. =.
After receiving message mm (off-the-equilibrium), μμ(tt mm ) =.6.6 +. = and hence beliefs must be arbitrarily specified, i.e. μμ [,]. Receiver s optimal response After receiving a message mm, the receiver s expected utility from responding with actions aa, bb, and cc are Action a:.6 +. =. Action b:.6 +. =.8 Action c:.6 +. 5 =. Hence, the receiver s optimal strategy is to choose action a in response to message mm. After receiving message mm (off-the-equilibrium), the receiver s EUs from each of his three possible responses are EEUU RRRRRRRRRRRRRRrr (aa mm ) = μμ + ( μμ) = μμ EEUU RRRRRRRRRRRRRRRR (bb mm ) = μμ + ( μμ) 5 = 5 μμ EEUU RRRRRRRRRRRRRRRR (cc mm ) = μμ + ( μμ) 5 = 5 5μμ Where it s clear that the EU from responding with bb, 5 μμ, is the highest EU payoff the responder can obtain given that μμ [,]. Sender s optimal message If his type is tt, the sender obtains a payoff of from sending mm (since it is responded with aa), but a payoff of only when deviating towards mm (since it is responded with bb). Hence, he doesn t have incentives to deviate from mm. If his type is tt, the sender obtains a payoff of by sending mm (since it is responded with aa), but a payoff of only by deviating towards mm (since it is responded with bb). Hence, he doesn t have incentives to deviate from mm. Therefore, the initially prescribed pooling strategy profile where both types of sender select mm can be sustained as a PBE of the game. There are other babbling equilibria that differ in terms of the message sent by the sender and the receiver s beliefs in response to a message that the sender never sends (according to his strategy). For any babbling equilibrium, it must be the case that the receiver ends up choosing action a.
c. Suppose the probability that the sender is type t is p and the probability that the sender is type t is ( pp). Find the values for p such that there is a pooling PBNE in which the receiver chooses action b. For any pooling equilibrium, the sender s strategy has him choose the same message let it be m for any type and, in response to observing that message; the receiver s beliefs are her prior beliefs. Receiver s beliefs After observing message mm, μμ(tt mm ) = μμ(tt mm ) = After receiving message mm (off-the-equilibrium), μμ(tt mm ) = pp pp + ( pp) = pp ( pp) pp + ( pp) = pp pp pp + ( pp) = And hence beliefs must be arbitrarily specified, i.e. μμ [,]. Receiver s optimal response After receiving a message mm (in equilibrium), the receiver s expected utility from responding with actions aa, bb, and cc are Action a: pp + ( pp) = pp Action b: pp + ( pp) = + pp Action c: pp + ( pp) 5 = 5 5pp For it to be optimal to choose action b, it must be the case that + pp pp pp aaaaaa + pp 5 5pp pp Thus if pp < /, then the receiver does not choose action b at a pooling equilibrium, as she would prefer action a. If pp, then it is the receiver s optimal strategy to choose bb in response to message mm. After receiving message mm (off-the-equilibrium), the receiver s EUs from each of his three possible responses are EEUU RRRRRRRRRRRRRRRR (aa mm ) = μμ + ( μμ) = μμ EEUU RRRRRReeiiiiiiii (bb mm ) = μμ + ( μμ) 5 = 5 μμ
EEUU RRRRRRRRRRRRRRRR (cc mm ) = μμ + ( μμ) 5 = 5 5μμ Where it s clear that the EU from responding with bb, 5 μμ, is the highest EU payoff the responder can obtain given that μμ [,]. Sender s optimal message If his type is tt, the sender obtains a payoff of from sending mm (since it is responded with bb), but a payoff of only when deviating towards mm (since it is responded with bb). Hence, he doesn t have incentives to deviate from mm. If his type is tt, the sender obtains a payoff of by sending mm (since it is responded with bb), but a payoff of only by deviating towards mm (since it is responded with bb). Hence, he doesn t have incentives to deviate from mm. Therefore, the initially prescribed pooling strategy profile where both types of sender select mm can be sustained as a PBE of the game when pp.
Exercise 7-Chapter -Harrington Consider the Courtship Game with Cheap Talk Nature Jack loves Rose Probability = p Jack does not love Rose Probability = (-p) Nature Nature Rose loves Jack Rose does not love Jack Rose loves Jack Rose does not love Jack Probability = p Probability = (-p) Probability = p Probability = (-p) Jack Jack Jack Jack Suggests Does not suggest Suggests Does not suggest Suggests Does not suggest Suggests Does not suggest Rose Rose Accept Decline Rose Accept Decline Rose Jack Rose m + s m + s m m m m Accept Decline m + s m + s m m m m Accept Decline s u s u Show that there is no PBNE in which premarital sex occurs. Consider a strategy profile in which Rose accepts Jack s proposal when it is made. To begin, it is clear that she would never accept having sex with someone she doesn t love. Doing so results in a payoff of uu < (as she knows she isn t going to marry Jack), while not having sex results in a payoff of zero. Thus, if there is an equilibrium with premarital sex, it would only involve Rose having sex with Jack if she loves him. Suppose Rose does act in that manner, accepting if she loves Jack but declining if she does not. Jack. What is an optimal response for Jack? Regardless whether or not he loves Rose, his payoff is higher by having sex. Thus, he ll ask for sex; he has nothing to lose. If Rose doesn t love him, then she ll decline and his payoff is zero. If she does love him, then his payoff is higher by s. More specifically, if he loves Rose, then his expected payoff from asking for sex is: 5
EEUU JJJJJJJJ (ssssss llllllllll RRRRRRRR) = pp (mm + ss) + ( pp) = pp(mm + ss) and from not asking is EEUU JJJJJJJJ (nnnnnn ssssss llllllllll RRRRRRRR) = pp mm + ( pp) = pp mm. Thus, Jack asks for sex. Rose. Is Rose s strategy of accepting if she loves Jack optimal given Jack asks regardless whether he loves her? Her payoff from accepting his proposition is: EEUU RRRRRRRR (aaaaaaaaaaaa llllllllll JJJJJJJJ) = pp (mm + ss) + ( pp) uu Since both Jack types ask, Rose doesn t learn anything about whether he wants to marry her from the fact that he wants to have sex with her. Recall that we are evaluating this in the case when she loves Jack. If she doesn t have sex, then her expected payoff is: EEUU RRRRRRRR (rrrrrrrrrrrr llllllllll JJJJJJJJ) = pp mm + ( pp) = pppp Thus, it is indeed optimal for Rose to accept if and only if: which is equivalent to pppp + ( pp)uu >. pp (mm + ss) + ( pp) uu > pp mm If this condition does not hold, then Rose would prefer to decline even if she loves Jack. In that case, there is no PBE in which premarital sex occurs. 6
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