Consider the following two-player game.
Both players roll a fair die. They can see their own roll, but not their opponent's roll.
Then, both players simultaneously choose a (possibly fractional) amount to bid.
Whoever bids higher wins the sum of the two die rolls minus their bid, while the lower bidder gains nothing. When they bid the same amount, both players get nothing.
What is the best strategy for this game? What are the Nash equilibria?
Let n be the number you roll, then the random variable for the sum is uniform between $[n+1, n+6]$. The expected value for the opponent’s roll is 3.5 and so the expected sum would be $n + 3.5$, I will bid as close to the expected sum as possible (so the first whole number that is below the expected sum $n + 3.5$), which would give me an expected payoff of $0.5$. Is this solution correct? I don't know if adding the expected value part is the best strategy, but I assume I want to lower the chance of the opponent getting around the expected value.
Edit: The question was not clear about whether the bid is announced to the other player, but since if it is announced we have to consider the sequence (whether I am first or second), I assume the bids are secret. Though I want to know what the strategies would be for the two different cases (secret and non-secret), if I can choose to bid first/second? You can bid any amount and the opponent is a rational player.
To give an example of the run, you got 3 and your opponent got 6, and if you bid 7 and the opponent bids 8, whoever wins the bid will get 9 points minus the points they bid, so the opponent wins and get 1 point in this case. If there is a tie in the bids, then there is no prize. If you bid a 10 instead of a 7, then you would win the bid, and get -1 point.