Question

Consider three players (1, 2, 3) and three alternatives (A, B, C). Players vote simultaneously for an alternative and abstention is not allowed.The alternative with more votes wins. If no alternative receives a majority, alternative A is chosen.
U1 (A) = U2 (B) = U3 (C) = 2
U1 (B) = U2 (C) = U3 (A) = 1
U1 (C) = U2 (A) = U3 (B) = 0
Obtain all Nash equilibria in
pure strategies.

Answer 1

Let us understand how the subgame pure strategies will be obtained :-

1) It is very evident from the question itself that if All the players vote for an alternative then :-

A ) In case of majority - The player with highest number of votes will win .

B ) In case of no majority the first choice will win .

2) Thus there are chances of winning for all three players .

However if Choice is player 1 then gain is 2

Choice is player 2, then equilibrium gain is 1 and

If choice is player 3 equilibrium gain is 0 for all .

3) However in case of majority only the player who backs majority wins , else all loose .

In the given figure first choice only Player 1 wins with majority ,

Second choice player 2 wins with majority

While the other two choices show all equilibrium strategies with majority of player 3 or win individual choices and no majority.