After a couple of recent conversations about the dice game puzzle proposed a few months ago, I spent some time experimenting with the following game that has a vaguely similar “feel,” and that also has the usual desirable feature of being approachable via either computer simulation or pencil and paper.
Suppose you and n other players are playing the following game. Each player in turn spins a wheel as many times as desired, with each spin adding to the player’s score a random value uniformly distributed between 0 and 1. However, if a player’s cumulative score ever exceeds 1, she immediately loses her turn (and the game).
After all players have taken a turn, the player with the highest score (not exceeding 1) wins the game. You get to go first– what should your strategy be, and what are your chances of winning?
The obvious similarity to the dice game Pig is in the “jeopardy”-type challenge of balancing the risk of losing everything– in this case, by “busting,” or exceeding a total score of 1– with the benefit of further increasing your score, and thus decreasing the other players’ chances of beating that score.
I like this “continuous” version of the problem, for a couple of reasons. First, it’s trickier to attack with a computer, resisting a straightforward dynamic programming approach. But at the same time, I think we still need the computer, despite some nice pencil-and-paper mathematics involved in the solution.
We can construct an equally interesting discrete version of the game, though, as well: instead of each spin of the wheel yielding a random real value between 0 and 1, suppose that each spin yields a random integer between 1 and m (say, 20), inclusive, where each player’s total score must not exceed m. The first player who reaches the maximum score not exceeding m wins the game.
This version of the game with n=3 and m=20 is very similar to the “Showcase Showdown” on the television game show The Price Is Right, where three players each get up to two spins of a wheel partitioned into dollar amounts from $.05 to $1.00, in steps of $.05. The television game has been analyzed before (see here, for example), but as a computational problem I like this version better, since it eliminates both the limit on the number of spins, as well as the potential for ties.