December 5th, 2008

Shut the box, continued

In my earlier post, I asked you to consider whether "shut the box" was fair. luckylefty got the correct answer quickly: it's not fair -- and (surprisingly) it's the first player who is at a disadvantage. The nice thing about this game is that the state space is not large, so you can actually evaluate all possible games played and formulate optimal strategy. Given optimal play by both players, on average the first player will lose 2.5% of their stake each game. This is despite the fact that the first player can expect to shut the box 9.5% of the time, while the second player can only shut the box 8.2% of the time. The second player wins 43.1% of the times the box is not shut, compared to the 1st player's 38.2%. The second player knows what score they need to beat to win, and that confers a substantial advantage. In fact, the first player can't expect to profit unless they score a 27 or lower on their turn.

The computer can calculate an optimal strategy: but can it be described in terms simple enough for a human to learn and use? I'll describe a reasonable "human playable" strategy in my next post on this theme.

You might also want to ponder these questions: how might the strategy change if the number of players in the game increased, from 2 to, say, 10? Or 100?

And would making the players strictly alternate their turns make the game more fair?