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Making shut-the-box fair
cananian
The standard game of "shut the box", analyzed in previous posts, has a fundamental flaw: it's not fair! On average in a two-player game, the first player will lose 2.5% of their stake in each game. Can we make a more fair two-player game?

The simplest solution is to change how ties are treated: with optimal strategies, 1% of the games end in ties. We can resolve all ties in favor of the first player, but that's not good enough. We actually need to treat ties like box-shutting, and award double payoffs. And it's still not enough, especially since the 2nd player can alter their strategy to avoid ties if they become too painful.

So let's leave ties alone, and alter the box-shutting payoff. Although the 2nd player has an advantage in the points race, because they know what 1st player score they have to beat, the 1st player has an advantage in shutting the box: if there are able to do so, the game ends immediately and the 2nd player doesn't have a turn. The 2nd player doesn't have the opportunity to, for example, achieve a shut box themselves, tying the game.

In the standard game, box-shutting pays out double: if everyone had to put up a $1 stake to play, then they have to pay another $1 to the player who is able to shut the box. We'll call this a "2x" payout. Even odds work out to a 3.8x payout (accounting for the fact that optimal strategies change as the payout rises). It would be a little awkward to ante up $2.80 a person in a $1 ante game, but it turns out that a 4x payout ($3 more from each person on a shut box) is probably okay: this only gives the 1st player a 0.1% advantage. That might be enough for a casino to live on, but it's probably acceptable among friends.

Another way to even the odds is to force the players to alternate turns. This is hard in the physical game, where there's only one physical box with tiles, but it is natural in a computer version of shut-the-box — and it probably improves gameplay by elimininating the long waits between a player's turns. The first player still has a slight advantage, but computation indicates that this advantage is limited to 0.4% of the stake ("acceptable among friends"). It's hard to compute optimal strategies in a 3 player game, due to combinatorial explosion, but it appears the first player's advantage grows as the number of players does.

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