I'm trying to create a mathematical model for the auction process in a card game called Pitch. The specific question I'm interested in solving is:
Let $p_i$ represent the probability of a specific player taking exactly $i$ points during a game. Given values for $p_i$ for $i$ = 0 to 4 as input, what is the optimal bid that player should make? Note that $\sum_{i=0}^4 p_i = 1$.
There are many variations of this game, so I've simplified the specific rules I'm interested in modeling below:
There are two players, A and B.
During the auction, each player makes a single bid. Player A bids first, then B bids.
The possible bids are Pass, Two, Three, and Four. (Note that there is no One bid, even though it is possible to take only one point during the game.)
Player B must either Pass or exceed player A's bid. For example, if A bids Three, then B must either Pass or bid Four.
After the auction completes, the players will be competing to take four distinct points. For example, if A takes three points, then B takes the one remaining point.
A player's bid represents the minimum number of points that player commits to taking. For example, a Two bid means that the player must take a least two points. A Four bid requires that player to take all four points. A Pass bid means that the player is not committing to take any points.
The player who makes the highest bid wins the auction and receives an advantage during the rest of the game. (Specifically, she gets to choose trump and lead first, but I don't think that matters here.) We call this player the "high bidder".
If both players Pass, then there is no high bidder and the game immediately ends. Neither player scores any points in this case.
If the high bidder takes fewer points than she agreed to, then she suffers a loss equal to the size of her bid. For example, if she bids Three but only takes two points, then her score goes down by three points. (She does not receive credit for any of the points she took.)
If the high bidder takes at least as many points as she agreed to, then her score goes up by the number of points that she took. For example, if she bids Two and actually takes three points, then her score goes up by three points.
The low bidder is not committed to anything and gets to keep any points she takes. For example, assume that player A bids Two but is over-bid by player B's Three bid. If player A then takes one point, her score increases by one point.
Examples:
A bids 2 and B bids 4. A then takes 1 point and B takes 3 points. A scores +1 point, and B scores -4 points.
A passes and B bids 2. Both players take 2 points. A scores +2 points and B scores +2 points.
A bids 3 and B passes. A takes all 4 points. A scores +4 points and B scores 0 points.
A bids 2 and B passes. B takes all 4 points. A scores -2 points and B scores 4 points.
Examples of the actual problem I'm trying to solve:
A assesses that she has a 50% chance of taking 2 points and a 50% chance of taking 3 points. Should she bid Two or Three? Bidding Three is riskier, but has a better chance of being the high bid.
A assesses that $p_0$ = 0, $p_1$ = .1, $p_2$ = .2, $p_3$ = .3, and $p_4$ = .4. What is her optimal bid in this case?