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...

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:

Voting is simultaneous. Write down the (three-dimensional) payoff table, and show that in the Nash equilibrium the raise fails unanimously. Examine how a repeated relationship among the members can secure them salary increases every year if

(i) every member serves a 3-year term,

(ii) every year in rotation one of them is up for reelection, and

(iii) the townspeople have short memories, remembering only the votes on the salary increase motion of the current year and not those of past years.