Game theory: Bidding strategy during an auction in a card game

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?

• Is the distribution of the points each round functionally random? – David Greydanus Aug 28 '14 at 15:02
• I think your examples don't contain all the relevant data -- you need to know how many points you expect to score both as the high bidder and as the low bidder. It's possible (e.g., if your hand is very balanced) that it might not much matter to you what suit trump is, in which case perhaps you shouldn't bid even if you expect to score some points. – Micah Aug 28 '14 at 15:03
• Is it preferable to maximize your own score, or the difference between your score and your opponent's score? – paw88789 Aug 28 '14 at 15:07
• @paw88689: That's a good question. The match ends when one player accumulates a fixed number of points (usually 11). For the purposes of this question, maximizing the net difference makes sense, but towards the end of the march, the absolute scores become important. One can lose the match while successfully bidding and taking three points, but I consider that to be outside the scope of this particular model. – brianberns Aug 28 '14 at 19:58
• @Micah: That's a good point, but in practice, there's very little the low bidder can know about the probable outcome. For the purpose if this question, we can assume a low bid has a constant probable value. Empirically, the low bid can expect to take about one point or less. – brianberns Aug 28 '14 at 20:04