Suppose two dice are rolled, and Alice and Bob are playing an auction-style game on the sum of the two dice. The first dice roll is shown to Alice, and the second one is shown to Bob. Then, Alice proposes a bid. Bob, after hearing Alice’s bid, proposes his bid. The winner of the auction (whoever bids higher) must pay the loser the value of the winning bid, and the loser pays the winner the sum of the two dice.

Example Game: Suppose Alice is shown a die roll of 4, and bids $$10$. Bob sees a $5$, and in response, bids $$11$. Then, Bob, the “winner” of the auction, pays Alice $$11$ and Alice pays Bob $$4 + $5 = $9$. The net gain for Alice is $2, and Bob loses the same amount in a zero-sum fashion.

I’m looking to find the Nash Equilibrium of this game. I know that if Alice proposes a bid of $a$, Bob has two strategies: bidding $a + 1$ to win the auction optimally, or bidding anything less than $a$ to purposefully lose the auction.