My grandfather was a fantastic bridge player, and working on CUBE Backgammon gives me a reason to look at his game from another angle. To compare backgammon moves, a program can play out many possible futures and average what each move earns. What happens when we try that with cards the player cannot see?
At backgammon, both players can inspect the whole position. In bridge, much of the information needed to choose a plan sits in the opponents’ concealed hands. A calculation that ignores that difference can recommend a plan no player at the table could follow, however accurately it solves each exposed deal.
The bidding and the cards already played offer clues, so a bridge program can generate possible deals consistent with what it has seen, play them out and compare the results.
A tempting shortcut is to give the simulation all four hands. With the cards exposed, a solver can work out what each side could achieve. Repeat the calculation over many plausible deals and average the results, and it seems that the uncertainty has been handled. The trouble is that the simulated player may now choose a different continuation in each deal using knowledge the real player does not have.
The cards visible at the table
Bridge is played by two partnerships of two players. After the bidding establishes a contract and the opening lead is made, the declarer plays both their own hand and their partner’s exposed hand, called dummy. The defenders’ hands remain concealed from the declarer, while each defender knows their own cards and sees dummy.
A player can therefore be uncertain whether a missing queen is held by the opponent on the left or the opponent on the right. If those two possibilities call for different choices, a solver that sees the queen has an advantage that cannot be reproduced by simply thinking longer at the table.
A winning choice in every deal is not always a usable plan
The following small game strips away bridge’s rules so we can see exactly where the error enters. It gives the simulated player the same extra information that can distort a bridge calculation.
The order of the operations matters. With hidden cards, you choose an action and average what that action earns across the possible arrangements. Revealing the cards first lets the solver choose the best action separately in each arrangement, then average those results. That second calculation answers how well an informed player could do.
In the example, a faulty analysis could rate “play the card-guessing game” at 100 and prefer it to the guaranteed 60. The implementable guessing strategies are worth only 50. Adding more arrangements to the sample makes the wrong comparison more stable.
Why researchers call it strategy fusion
The term strategy fusion describes this combining of choices that require different private knowledge into one apparent strategy. Ian Frank and David Basin studied the failure in search methods for hidden-information games, including bridge. The central restriction is that a player must make the same decision in situations they cannot distinguish, unless their strategy deliberately randomises that decision.
Randomising does not rescue our queen example: choosing either side with any fixed mix still finds an equally likely queen half the time. Information arriving later could change the calculation, but the example explicitly supplies none. In real bridge, the challenge is to model which future cards and plays will actually reveal something.
The problem was already visible in GIB
In his 2001 paper on GIB, Matthew Ginsberg examined both solving exposed-card positions and choosing under concealed cards. GIB’s work on achievable sets addressed weaknesses in ordinary sampling.
In 2024, Jérôme Arjonilla, Abdallah Saffidine and Tristan Cazenave investigated delaying the point at which a search hands the position to a perfect-information evaluator. Keeping uncertainty in the search for longer reduced strategy-fusion problems in their experiments. That result concerns the tested algorithms and games; it is not a claim that bridge as a whole has been solved.
Why future dice are a different uncertainty
At backgammon, you and your opponent see the same board and the dice already rolled. Nobody knows next turn’s dice, but the player will see them before choosing how to use them. A simulated player is entitled to choose a different play after each different future roll.
The timing is crucial. A bridge continuation may wrongly use the queen’s location before it is revealed; a backgammon continuation may correctly use a roll after it becomes public. Both calculations involve uncertainty, but only one is granting the player unavailable information.
Backgammon simulations have other ways to mislead, including a small sample or weak continuation play. The bridge example adds a separate question to ask of any simulation: when it chooses a move, does its player know only what the real player would know at that moment?
Sources: Ian Frank and David Basin, A Theoretical and Empirical Investigation of Search in Imperfect Information Games (2001); Matthew Ginsberg, GIB: Imperfect Information in a Computationally Challenging Game (2001); and Arjonilla, Saffidine and Cazenave, Perfect Information Monte Carlo with Postponing Reasoning (2024). The queen-and-blank game, its payoffs and its calculations are original teaching material, not a historical bridge deal or a reported benchmark.