Diane Da-Hyun Lee POLSCI30 Homework 2 Professor Bawn (TA : Tamar) April 23rd, 2019 [Scenario A-1] [probability = 50%] expected value = [(0.5)(2)] + [(0.5)(-5)] = 1 - 2.5 = -1.5 optimal decision : Candidate A [probability = 75%]expected value = [(0.75)(2)] + [(0.25)(-5)] = 1.5 - 1.25 = 0.25 optimal decision : Candidate B [probability = p] expected value = [2p] + [(1-p)(-5)] = 2p - 5 + 5p = 7p - 5 7p - 5 < 0 , p < 5/7 The probability of candidate B being a star has to be less than 5/7 (about 0.71) for the party leader to support candidate A. [Scenario A-2] If the probability that candidate B is a star is 50%, the party leader’s ex post mistake would indicate that after he made his decision to support candidate A, given the uncertainty, candidate B turned out to be a star, which would have given the leader a higher payoff than choosing candidate A. In the case that the probability of candidate B being a star is 50%, the party leader’s probability of making an ex post mistake would be 50%, or 0.5. When the probability is 75%, the ex post mistake would be that the party supports candidate B but he turns out to be a dud, not a star. If the probability of candidate B being a star is 75%, then the likelihood of making an ex post mistake would be 25%, or 0.25. In the first scenario, since the probability of an ex post mistake is 50%, I think that the party leader would not have changed his decision because the risk of candidate B being a dud is too high for the leader to risk. However, if the probability of an ex post mistake is 25%, there is a possibility that the leader may change his decision and choose to support candidate A for his party. On the other hand, 25% may seem relatively low for the alternative decision and the party leader may continue supporting candidate B.[Scenario B-1] Rollback Equilibrium : (A ; A if A, B if B) Given this game tree, I expect both the party leader an the activists to support candidate A because if party leader supports candidate B, the activists are likely to also support candidate B, which would result in a lower payoff for the party leader. Since the leader and the activists are able to unite behind candidate A, their party will beat the competing party and win the upcoming election. [Scenario B-2] The point of this game is to balance the party leader and the activists’ preferences regarding the candidate that they want as the representative of their party in the upcoming election. Given that the only way for their party to win is to unite under one candidate (A or B), we can see that in both nodes the activists end up supporting the same candidate (A if A, B if B). Regarding the internal dynamics of the party’s nomination process, we can see that winning is more important for both players than supporting the candidate of their choice, best represented by the fact that if the party fails to unite, regardless of the decision that both the leaders and activists choose independently, their payoffs are far lower than if the party unites and wins the election.[Scenario C-1] If the activists move first, the rollback equilibrium changes to (B ; A if A, B if B). In this scenario, we expect the activists to support candidate B and the party leader to support candidate B, uniting the party once again, but behind a different candidate. [Scenario C-2] Compared to the predicted outcome in B-2, we can see that the outcomes that the second player chooses remains the same, but when the party leader acts first, the party chooses candidate A, and if the activists move first, the party unites behind candidate B. This is a reflection of the players’ preferences because the activists prefer candidate B and the party leader prefers candidate A. Thus the first player determines the candidate that the party unites behind. [Scenario D] The outcome in scenario B is pareto efficient because the resulting outcome gives the players a payoff of (5,3) - 5 for the party leader and 3 for the activists - and when the party fails to unite, the payoffs are worse off for both players. Additionally in the outcome where both players unite behind candidate B, the payoff of the activists increases to 6, but that of the party leaders decreases to 3. Since there is no outcome that results in a higher payoff for at least one player and no worse off payoff for either player, the outcome of scenario B is pareto efficient. Similarly in scenario C, the predicted outcome reveals a payoff of 6 for the activists and 3 for the party leader. Both players receive a lower payoff if the party fails to unite (B if A, A if B), and while the payoff for the leader may be higher in (A if A), the activists receive a worse off payoff, making (B if B) a pareto efficient outcome.[Scenario E-1] Case 1 : [if (5 -5p) > (2p +1) ; 4/7 > p] Rollback Equilibrium : (A; B if A and star, A if A and dud, B if B and star, B if B and dud) Case 2 : [if (5- 5p) < (2p + 1) ; 4/7 < p] Rollback Equilibrium : (B; B if A and star, A if A and dud, B if B and star, B if B and dud) [Scenario E-2] In this game tree, the uncertainty determines whether the party leader supports candidate A or B. The uncertainty does not have an effect on the decisions that the activists make, which is why the nature node is located above the activists’ decision nodes. Unlike scenario B, where the decision tree only affected the payoffs of the party leader, the uncertainty regarding the credibility of candidate B influences the payoffs of both players, which is why a nature node needs to be included in all possible outcomes. When p is large (greater than 4/7), the payoff for the party leader is greater if he supports candidate B, resulting in a rollback equilibrium of (B; B if A and star, A if A and dud, B if B and star, B if B and dud). If p is small (less than 4/7), then the leader is better off supporting candidate A, resulting in (A; B if A and star, A if A and dud, B if B and star, B if B and dud). We can see that in each rollback equilibrium, the decisions made by the activists remain constant, further indicating that the activists’ decisions are not influenced by the nature node.[Scenario F] If p = 0.25, the probability that candidate B is a star is less than 4/7 (0.57), which leads to case 1, resulting in the party leader supporting candidate A and the activists supporting candidate B if B is a star and candidate A if B is a dud. If p = 0.75, the probability that candidate …