Question

Elections between Hillary and Trump. There are only three voters and all of them strictly prefer Hilary to Trump. We assume that voters' utility only depends on the candidate that wins the election. In the first round, all voters vote simultaneously. If a candidate is voted by all the three voters, this candidate is elected. If none of the two candidates receives all the votes, then a second round takes place. All voters vote again simultaneously, but the winner is the candidate that wins the majority of the votes (i.e. 2/3 votes). The game ends after the second round. Consider the following statements:

S1: There is a Sub-game Perfect Nash equilibrium where Trump is elected.
S2: Hillary is the winner in all Sub-game Perfect Nash equilibria of this game
S3: To compute the Subgame Perfect Nash equilibrium, we need to know the utility that each voter receives if a candidate wins.
S4: Each voter has four strategies:

1. (vote for Hillary at the first round, vote for Hillary at the second round)
2. (vote for Hillary at the first round, vote for Trump at the second round)
3. (vote for Trump at the first round, vote for Hillary at the second round)
4. (vote for Trump at the first round, vote for Trump at the second round)

S5: Voting for Hillary is a strictly dominant strategy for all voters.

Select one:

a. The only correct statements are S1 and S4

b. The only correct statements are S2, S4 and S5.

c. S3 is the only correct statement.

d. S1 is the only correct statement.

e. The only correct statement is S2

Answer 1

The answer is a) The only correct statements are S1 and S4.

S4 is correct because that is how we define strategies in extensive form game. A strategy is defined by the action a player takes at all decision nodes in the game. Since a player decided between 2 actions at 2 decision nodes, he has 4 different strategies as specified above.

S1 is correct because the strategy profile in which each player's strategy is to always vote for Trump is a subgame perfect Nash equilibrium. This is because given that the other 2 players are always voting for Trump, a player cannot really change the outcome by voting for Hillary at any of the decision nodes. Thus, no player has a profitable deviation at any decision node. Clearly, in this subgame perfect equilibrium, the outcome is that Trump is elected after the 1st round.

Thus, S1 and S4 are correct. Then, S2 is clearly incorrect simply because S1 is correct. S3 is incorrect because we have found SPE without using the utility values. S5 is incorrect because, as we have shown above, a player is indifferent between voting for Trump and voting for Hillary when other players always vote for Trump. Therefore, voting for Hillary is not a strictly dominant strategy.