# Theoretically Optimal Bridge System (In the mathematical sense)

I'm wondering from a mathematical perspective, what the best theoretically possible bridge system is. I know this is a bit of a muddy question, such as what systems are allowed, and what criteria best is, so I'll try to rigorously define it. In human terms, "Bidding must be deterministic, but may be defensive. Try to maximize IMPs." (To keep things simple, assume hands are scored using doubly dummy analysis, but this could be extended to include information given during the bidding in gameplay.)

To be mathematically precise: Find a system S*, so that for all systems S, and summing over all possible bridge hands, the total sum of IMP(Table(S*, S, S*, S), Table(S, S*, S, S*)) >= 0.

Wherein a system is defined as a function that takes as input:

• The current game state. (Specifically, the vulnerability, the bid, the declarer, and how many passes until end of bidding.)
• A bridge hand.
• For each of the four players, the set of bridge hands that player is representing.

And gives as output:

• A legal bid, including pass.

For instance, a player playing Standard American is first to bid, and they pick up AQJ6c 9d K632h Q984s. Then, we apply the function SA(No sides vulnerable, no bid, 4 more passes to end auction, hand = AQJ6c 9d K632h Q984s, {all possible hands}, {all possible hands}, {all possible hands}, {all possible hands}) This evaluates to 1c.

Then, the next player playing the optimal system S*, and picking up hand H, would apply to get their bid: S*(No sides vulnerable, 1c by E, 3 more passes to end auction, hand = H, {all possible hands SA would bid 1c with}, {all possible hands}, {all possible hands}, {all possible hands})

Note that if the first person was instead playing precision, their bid would instead be: S*(No sides vulnerable, 1c by E, 3 more passes to end auction, hand = H, {all possible hands Precision would bid 1c with}, {all possible hands}, {all possible hands}, {all possible hands}) which may or may not be the same bid.

This problem is difficult: there are 52C13 possible bridge hands. That means there are 2^(52C13) sets of bridge hands. The number of systems, which takes sets of bridge hands as input, is much larger than that. It's likely that even if S* were found, it would be too difficult to use in practice.

EDIT: I would like to clarify: I'm asking about the optimal system, from a game theoretic perspective. "Depends on what system your opponent is using." makes sense when you talk about systems from a human standpoint, but is a non-answer in the game theoretic sense.

To use an analogy from chess. From a human perspective, a strategy is something like "castle early, protect your king", which is great against "go for a quick checkmate", but less great against "prioritize development". From a game theory perspective, a strategy is something like: "If white, open d4. If black, and white's first move is a3, open e5. If white's first move is a4, ..." (due to combinatorial explosion, a complete strategy wouldn't even fit in a library) There is an optimal strategy, which does not depend on the opponent's strategy, because the definition of "strategy" includes responses to all possible opponent moves. (Note that if you were to score a strategy by time to mate, the optimal strategy would not be maximally exploitative; but it would be unexploitable.)

Take the example of a maximally constructive and artificial system, A. Now, modify A so that it plays more destructively, and call it A+. It should be uncontroversial that A+ is +EV against A. One may point out that A++, which is more destructive than A+, is +EV against +A, and from there, it's a race to the bottom. But of course it has to stop somewhere: a maximally destructive system would always bid 7NT, and that's clearly not a winning strategy. Where does it stop? One may claim that it depends on the opponent's system, but again, see the definition of system used above: a system function is allowed to output different values for the same bids, if the bid represents a different subset of the 52C13 possible bridge hands.

(There is some real life basis in this as well: players will overcall more aggressively against artificial systems than natural systems.)

• Barring significant advances in general purpose AI, I don't think this is a problem with a meaningful solution.
– Joe
Nov 30, 2023 at 21:11