two persons roll dice and bid game: optimal strategy

Question

Two persons $A, B$ roll a fair $n$-face dice separately and get $1 \le x,y \le n$ points. Then the third party will put $x + y$ dollars in a black box. $A$ and $B$ only know the point they roll and don't know the other's.

Then they bid on the black box. $A$ bids a integer price $p_1$, then $B$ can only bid at least $1$-dollar higher integer price $p_2$ or give up. If $B$ gives up, then $A$ must buy the black box with price $p_1$. If $B$ bids $p_2$, then $A$ can bid at least $1$-dollar higher integer price $p_3$ or give up, etc. Until one gives up, the other one should buy the black box with the latest price.

Question:

1. Assuming that $A$ and $B$ are rational, what's the optimal strategy of first player $A$ and second player $B$? How can, from the other one's bidding price, infer the range of points the other one has?

2. What if the other one should have to bid at least $k$ dollars higher ($k$ is integer)?

3. What if when there are $m$ players? (same question as Q1 and Q2).

Is there any terminology of this problem? Is there any reference or literature which thoroughly discusses this problem?

Answer 1

Assume strategies are known by the other player.

n=1. Always bid 1. You win 1 dollar or the other player bids 2 and no one wins.

n=2. When $x$ is rolled, the pot is $x+1$ or $x+2$, so bidding $x+1$ nets 0 or 1. Bidding 2 each time gives no information to player 2, so if player 2 rolls a 1, then player 2 is risking a loss by bidding 3, so player 2 will only bid 3 if they roll a 2. Thus bidding 2 each time will net a profit of 1 half the time (and 0 otherwise). The second strategy (bidding 2 every time) has the advantage that your opponent cannot cut you out of your profits by responding to a bid of 3 with 4. Though a bid of 4 may be considered irrational as it will never net a profit, as it is possible that your opponent's strategy may change. If you don't want your opponent to win money, bidding $x+1$ each time ensures he cannot profit. If you only care about your own winnings, then either strategy is suitable. Player 2 should always bid $y+1$. As you can see from this example, the problem is not specific enough to have a single answer, even in this very simple case. So the problem needs more specification around preferences, though more could be said in the instance where $k > 1$ .