Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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of U i (a i ,a j ) rather than u i (a i ,a j ), where the material payoffs are given byColumnL RRow U 1,1 −1,2D 2,−1 0,0Do the equilibriumpredictions change? Explain the role that cplays in changing theagents’motives.(b). For what values of c are (F,B) or (B,F) pure-strategy Nash equilibria in the Battleof the Sexes game, and for what values of c are (F,F) and (B,B) pure-strategy Nash equilibriain the Battle of the Sexes game? Explain why different values of c lead to differentequilibrium outcomes.(c). Behavioral economists have noted that, “the same people who are altruistic to otheraltruistic people are also motivated to hurt those who hurt them” (Advances in BehavioralEconomics, by Camerer, Loewenstein and Rabin, p. 296). We’ll call this idea reciprocalaltruism. Is the above model a good model of reciprocal altruism? Explain why or why not.(d). Explain whether the following statement is true or false: “The contribution of behavioraleconomics is to explain why the payoffs take the values they do, because once we fixthe players’ payoffs we can then use standard game theory to analyze the strategic form.Therefore, it is without loss of generality to focus on self-regarding players.”(a). The strategic form for battle of the sexes becomesL RU 7/2,−1/2 0,0D 0,0 −1/2,7/2The profiles (U,L) and (D,R) are no longer Nash equilibria. In fact, this becomeslike a matching pennies game, where there is no pure strategy Nash equilibrium. Byincreasing c, the players have become “vindictive”, and prefer to be alone rather thango to the other agent’s prefered event. So players prefer to be unkind.The strategic form for prisoners’ dilemma becomesL RU 1,1 1/2,1/2D 1/2,1/2 0,0Now there is a unique equilibrium at (U,L) rather than (D,R). By decreasing c, playersworry about “hurting” their opponent, so neither player wants to move from Silent toConfess and impose a burden on the other player. So player dislike being unkind.(b). The strategic form isL RU 2+c/2,1−c/2 0,0D 0,0 1−c/2,2+c/210
The usual profiles (U,L) and (D,R) are PSNE ifor 2 ≥ c, and1−c/2 ≥ 02+c/2 ≥ 0so that c > −4. So if 2 ≥ c ≥ −4, then (U,L) and (D,R) are PSNE.The other profiles are PSNE ifor 2 ≤ c, and1−c/2 ≤ 02+c/2 ≤ 0or c ≤ −4. So if c ≤ −4, (U,R) and (D,L) are PSNE.What happens when c > 2? Try 10: The game becomes like matching pennies, withno pure strategy equilibria.As c becomes more positive, the players respond more to the unkindness of theiropponent, and want to hurtpeoplewhohurt them. As c becomes more negative, playersdon’t want to be unkind themselves, and go out of their way to give their opponenttheir preferred outcome.(c). This model can capture one half of reciprocal altruism; it is like a “one-sided”altruism model. Typically, altruism assumes that I just want to make other peoplebetter off per se, even when their actions harm me. In this one-sided altruism model,I either want to hurt players who are unkind to me (c > 0)), or I don’t want to hurtplayers who are kind to me (c < 0). So it captures some of the observations behavioraleconomists have made, but doesn’t incorporate both motivations into the same model.(d). In this version of behavioral economics, we used a hypothesis about humanbehavior to transform the payoffs, but then we used standard game theory to analyzethe transformed payoffs. In this sense, game theory and behavioral economics arecomplementary, and there is no reason you can’t freely mix both together. This exerciseshows that you can build in any behavioral aspects you want simply by transformingthe payoffs the right way. This suggests you can study game theory and behavioraleconomics separately without loss of generality, but then bring them together wheneverit’s appropriate.11
Page 1 and 2: Problem Set 11. Games in Strategic
Page 3 and 4: The Nash equilibrium is (A,A), sinc
Page 5 and 6: (1) CL RU 2,2,3 1,1,2B D 1,3,4 1,4,
Page 7 and 8: ight-side up and you drop the ball
Page 9: the strategy d is a best response t

Problem Set 1 1. Games in Strategic Form: Do questions 1-4 and ...

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