There are 7 participants in this vote, who each get a ballot and are allowed to cast one vote for every contestant including themselves on their ballot. (So each ballot can be used to assign 7 votes and each vote is a point) The topic of the vote is negative and so the aim is to not get the highest number of votes. You do not want to win the vote. However, A and B have dared each other to try and get the highest vote. Everybody else is voting honestly. In other words, A and B are both competing with one another for the negative outcome and will use strategy, while everyone else will cast genuine ballots. For A and B winning is the most desired outcome. But both would rather draw than lose (So the expected outcomes are Win= 1 Draw= 0 and Losing= -1 for A and B). For the other 5 voters not winning is the most desired outcome. (Loss and draw= 1 and win= -1)

So to make a Nash Equilibrium between A and B possible we will assume that the only contestants able to vote for themselves are A and B, and nobody else will vote for themselves, as they do not want to give themselves a point. However, everyone else can be assumed to cast a random ballot other than that. So, another contestant can make any possible combination of votes as long as it does not include their own name.

So... How would A and B distribute their votes to achieve Nash Equilibrium?

I am sorry if this was not explained clearly. If not I will clear anything up. The immediate common sense answer I had was that A casts a vote for A and B casts a vote for B, leaving the rest up to chance. But when I tried to work this out and prove it I realized that this problem filtered me hard. So I would love to see how you guys approach this.