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1.(strategic voting)
suppose that there are three candidates,X Y Z.thecandidates have noｎｏno action (i.e. the candidates are not the players)　in this game. Suppose that there are five voters(A B C D E )as the players of this game. Simultaneously and independently,five voters select single candidates.
The candidates who obtains the most votes wins the election,and if two candidates have the same vote share ,the probability of winning is 50% for each .The preferences of five votersare as follow
VoterA:X＞Y＞Z
VoterB:X＞Y＞Z
VoterC:Y＞X＞Z
VoterD:Y＞X＞Z
VoterE:Z ＞Y＞X
specifically,assume that the voter's utility is 2 if her most preferred candidate wins.1 if her second preferredcandidate wins .and0 if her least preferredcandidate wins
(a)show that that the strategy profile(X X Y Y Z )is not a nash equilibrium.the strategy profile (X X Y Y Z )means that each player votes for her most preferred candidate   i.e.voters A and B vote for X,VotersCand D vote for Y ,and VoterE Vote for Z (In this case, Candidate Xwins with probability 50% , and Y wins with remainIng probability 50%)
(b)Show that the strategy profile(Z  Z Z Z X)is aNash equilibrium . The Strategy profiles( Z  Z  Z Z X) means that each player Vote forher least preferred Candidate ,
(c)Find One pure  straitegy profile which is a Nash equilibrium Such that CandidateY wins .prove that your Pure Strategy profile is actually a Nash equilibrium

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关键词：博弈论 SIMULTANEOUS equilibrium Probability Independent 博弈论 game strategic suppose winning

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(a)If E choose Y,his expected payoff will be 1 which is larger than the expected payoff of choosing Z.
(b)No one can change the result of the game assuming other players' strategies are unchanged.
(c)(X X Y Y Y)