It's easy to think about the limit case and work backwards. If there are an infinite number of players, eventually

*someone*will "shut the box" — it happens in about 1 in 10 games, after all. Since the round will eventually end this way, then your final score doesn't matter: all that's important is whether you manage to shut the box. Every player's strategy is identical. (With an infinite number of players, there is no "last" player — in fact, even if there were one, there would be an infinitesimal chance of their touching the dice: they game would almost certainly be ended by a previous player's shutting the box before the "last turn" occurred.)

A strategy which maximizes your chances of shutting the box, ignoring final score, is easy to implement and evaluate, and differences from the "largest result" strategy described previously can again be tabulated on a single side of a sheet of paper. This "most zeros" strategy manages to shut the box in 9.8% of games played, but still loses in a two player game — even though the optimal 1st player strategy only manages the shut the box in 9.5% of its games. As first player against an optimal 2nd player, "most zeros" loses 3.3% of the stake in each game, compared to only 2.5% lost by an optimal 1st player (game results: 9.5% 1st player shut box, 8.2% 2nd player shut box, 37.3% 1st player by points, 43.8% 2nd player by points, 0.9% ties). As second player against an optimal 1st player, "most zeros"

*loses*2.8% of the stake; an optimal second player would

*win*2.5% of the stake (game results: 9.5% 1st player shut box, 8.8% 2nd player shut box, 41.1% 1st player by points, 39.7% 2nd player by points, 0.9% ties).

Since the box is shut with a fairly high probability, we converge to the "most zeros" strategy fairly quickly as the number of players increases. Intermediate strategies are parameterized by the "score to beat" (the minimum score achieved by previous players) and the probability distribution of the minimum score to be achieved by future players. The complexity of computing exact solutions increases quickly.

In my next post on the topic, I'll discuss an "alternating turns" variant of shut-the-box which might be a more enjoyable real-time-collaborative computer game.