The rules of the competition say that, if they disagree, they should vote (secret ballot, one member one vote) and that, if and only if the vote is tied, the winner will be the candidate for whom member 1 vote.

Suppose that each committee member gets a payoff of 1 if her most preferred candidate is elected, 0 if her second most preferred candidate is elected, and -1 if her least preferred candidate is elected.

Suppose each committee member votes for her most preferred candidate, the payoff for member 1 is _____________________

From what i've found, there are 3 member of the comitee. The member 1 prefers candidate A and hates candidate C. The member 2 prefers candidate B and hates candidate A, and the member 3 prefers candidate C and hates candidate B.

If everyone votes for their favourite candidate (member 1 for A, member 2 for B and member 3 for C), then every candidate will be tied and A will win since it has member 1's vote, therefore the payoff for member 1 is 1.

Now, what would happen if member 2, noticing that in this situation he will get a payoff of -1, decides to vote for candidate C instead? This means that candidate C will get 2 votes (from members 2 and 3 respectively) and therefore, it will be the clear winner, thus, the payoff for member A is -1 and he cant do anything to change this. Also, note that member 3 doesnt have a motive to change his vote, if he votes goes to candidate A, then

If member 1 voted also candidate A, then member 3 could have voted candidate C instead for an equal or greater payoff (candidate B woudnt win in any case)
If member 1 voted for candidate B, then candidate A would only get 1 vote and it woudnt win, thus, he could have voted for candidate C instead.

Therefore, we have that

Member 3 always votes for candidate C, the one member 1 hates
Member 2 gets a payoff of 0 by voting also for candidate C.
In this case, member 1 gets a payoff of -1, but in this case he can go for candidate B instead because:
If member 2 votes for candidate B thinking member 1 will switch its vote, then both members 1 and 2 will get a better payoff than before, however member 1 can take advantage of this situation and vote for candidate A for an even greater payoff, and the worst possible for B.

However, if everyone goes for the simplest approach and vote for their favourite candidate, then Member 1 gets a payoff of 1.