Here is my question for my problem set for game theory.
Can anybody solve the problem?
1. 11. (Exam) Terrence and Philip are debating over how to split a dollar. Both Terence
and Philip simultaneously yell out their shares which we will denote by sT and sP .
The rules that apply are simple.
• If the sum of sT and sP is greater than 1, both players get zero.
• If the sum of sT and sP is less than or equal to 1, they both get the share they
call.
(a) Define the set of players (N), the set of strategies (S), and the pay-off function
(u), of this game.
(b) Find all (!) pure strategy Nash equilibria of the game.
2. A town council consists of three members who vote every year on their own salary increases. Two YES votes are needed to pass the increase. Each member would like a higher salary but would like to vote against it herself because that looks good to the voters. Specifically, the payoffs of each are as follows :
Raise passes, own vote is NO : 10
Raise fails, own vote is NO:5
Raise passes, own vote is YES:4
Raise fails, own vote is YES:0
Voting is simultaneous.
(a) Write down the (three-dimensional) payoff table, and show that in the Nash equilibrium the raise fails unanimously.
(b) Examine how a repeated relationship among the members can secure them salary increases every year if (1) every member serves a 3-year term, and (2)every year in rotation one of them is up for reelection, and (3) the townspeople haver short memories, remembering only votes on the salary-increase motion of the current year and not those of past year.
Can anybody solve the problem?
1. 11. (Exam) Terrence and Philip are debating over how to split a dollar. Both Terence
and Philip simultaneously yell out their shares which we will denote by sT and sP .
The rules that apply are simple.
• If the sum of sT and sP is greater than 1, both players get zero.
• If the sum of sT and sP is less than or equal to 1, they both get the share they
call.
(a) Define the set of players (N), the set of strategies (S), and the pay-off function
(u), of this game.
(b) Find all (!) pure strategy Nash equilibria of the game.
2. A town council consists of three members who vote every year on their own salary increases. Two YES votes are needed to pass the increase. Each member would like a higher salary but would like to vote against it herself because that looks good to the voters. Specifically, the payoffs of each are as follows :
Raise passes, own vote is NO : 10
Raise fails, own vote is NO:5
Raise passes, own vote is YES:4
Raise fails, own vote is YES:0
Voting is simultaneous.
(a) Write down the (three-dimensional) payoff table, and show that in the Nash equilibrium the raise fails unanimously.
(b) Examine how a repeated relationship among the members can secure them salary increases every year if (1) every member serves a 3-year term, and (2)every year in rotation one of them is up for reelection, and (3) the townspeople haver short memories, remembering only votes on the salary-increase motion of the current year and not those of past year.