Game theory
Exercise 2. Voting Voters in Suburbia are choosing between two candidates, L and R. L is proposing new infras- tructure funded with new taxes. Candidate R is in favour of preserving the status quo, i.e., no new taxes and no new spending. Consider three voters - Alison, Beth, and Chandra - whose preferences and payoff are as follows. Alison strictly prefers L to R. She gets a payoff of 2 if L wins but otherwise. Beth strictly prefers to L. She gets a payoff of 2 if Rwins but otherwise. Chandra is not interested in electoral politics. Nonetheless voting is compulsory in Suburbia so he always votes. He gets 0 if the candidate he votes for wins. Otherwise, his payoff is -1. All three individuals - Alison, Beth, and Chandra - vote. Candidates receiving majority of the votes wins. (1) Write down the 3-person normal form game where Alison, Beth, and Chandra are three players and each of them has two strategies - vote for L or vote for R. As described above, their payoffs depend on the outcome of the election. 2 (ii) Find out all Nash equilibria in pure strategies. (iii) Suppose we are interested in Nash equilibria where neither strictly nor weakly domi- nated strategies are played. A Nash equilibrium satisfying this property is called ad- missible. Are all Nash equilibria in (ii) admissible? Exercise 2. Voting Voters in Suburbia are choosing between two candidates, L and R. L is proposing new infras- tructure funded with new taxes. Candidate R is in favour of preserving the status quo, i.e., no new taxes and no new spending. Consider three voters - Alison, Beth, and Chandra - whose preferences and payoff are as follows. Alison strictly prefers L to R. She gets a payoff of 2 if L wins but otherwise. Beth strictly prefers to L. She gets a payoff of 2 if Rwins but otherwise. Chandra is not interested in electoral politics. Nonetheless voting is compulsory in Suburbia so he always votes. He gets 0 if the candidate he votes for wins. Otherwise, his payoff is -1. All three individuals - Alison, Beth, and Chandra - vote. Candidates receiving majority of the votes wins. (1) Write down the 3-person normal form game where Alison, Beth, and Chandra are three players and each of them has two strategies - vote for L or vote for R. As described above, their payoffs depend on the outcome of the election. 2 (ii) Find out all Nash equilibria in pure strategies. (iii) Suppose we are interested in Nash equilibria where neither strictly nor weakly domi- nated strategies are played. A Nash equilibrium satisfying this property is called ad- missible. Are all Nash equilibria in (ii) admissible
