game theory exercise

3. (8 points total) Consider the following model of political party location with the twist that there is a positive payoff to losing. In many countries, Turkey among them, there are large benefits to being in the parliament even if you do not have a majority of seats, we represent this as saying the benefit of winning is 1, of tying for winning is and of losing (but still being in parliament) is a [0,1]. 1 Other than that this is a standard model of political party location. There are five locations for the political parties, arranged in a line, (L,CL,C, CR, R), and voters are located at one of these five locations. They will vote for the party that is closest to them (CL is closer to C than CL is to CR for example) and split their votes if they are indifferent. There are two political parties that choose a location l; E (L,CL,C, CR, R) for i {1, 2} and l1 = 12 is allowed. They have the same utility function: 1 if the votes for party i are more than the votes for the other party U; (1,12)= if the votes for party i are equal to the votes for the other party a if the votes for party i are less than the votes for the other party There are 21 voters in total and the distribution of voters is: L CL 0 CR R 6 3 2 9 1 For the answers to questions a and b assume that a = 4 (a) (5 points) If a = { write down the best responses of party 1 to every location of party 2 in the following table. Explain your answers in the space below. (You may continue your explanations onto the second page of the quiz.) L CL CR R Location of Party 2 Best Location(s) of Party 1 (b) (2 points) If a = 1 write down all of the Nash Equilibria. (c) (1 point) For what range of a [0, 1] will the Nash Equilibria be like you just found? For what range of a will the Nash Equilibria be like in a standard party location model (where a = 0)? 3. (8 points total) Consider the following model of political party location with the twist that there is a positive payoff to losing. In many countries, Turkey among them, there are large benefits to being in the parliament even if you do not have a majority of seats, we represent this as saying the benefit of winning is 1, of tying for winning is and of losing (but still being in parliament) is a [0,1]. 1 Other than that this is a standard model of political party location. There are five locations for the political parties, arranged in a line, (L,CL,C, CR, R), and voters are located at one of these five locations. They will vote for the party that is closest to them (CL is closer to C than CL is to CR for example) and split their votes if they are indifferent. There are two political parties that choose a location l; E (L,CL,C, CR, R) for i {1, 2} and l1 = 12 is allowed. They have the same utility function: 1 if the votes for party i are more than the votes for the other party U; (1,12)= if the votes for party i are equal to the votes for the other party a if the votes for party i are less than the votes for the other party There are 21 voters in total and the distribution of voters is: L CL 0 CR R 6 3 2 9 1 For the answers to questions a and b assume that a = 4 (a) (5 points) If a = { write down the best responses of party 1 to every location of party 2 in the following table. Explain your answers in the space below. (You may continue your explanations onto the second page of the quiz.) L CL CR R Location of Party 2 Best Location(s) of Party 1 (b) (2 points) If a = 1 write down all of the Nash Equilibria. (c) (1 point) For what range of a [0, 1] will the Nash Equilibria be like you just found? For what range of a will the Nash Equilibria be like in a standard party location model (where a = 0)