paper a day : games people don’t play

Games People Don’t Play
Peter Winkler
Puzzlers’ Tribute, David Wolfe and Tom Rodgers, eds., A K Peters Ltd. (2001)

This is a short note on 4 little games that people don’t really play. Some are unfair, some are violent, but the interesting question arises for each — what is the best strategy for each player? I’ll describe two of them (with variants) — the other two are equally interesting, but there’s no point in repeating everything!

In “Larger Or Smaller”, Paula writes two numbers on different slips of paper. Victor picks one and has to guess if its the larger or smaller, winning $1 if he’s right. In one version the numbers are just different integers and in the other the numbers are drawn from a uniform distribution on [0,1] and Paula picks which one Victor sees.

Victor can do better than even (if only by a tiny amount) by adopting a clever threshholding strategy proposed by Cover. He guesses “larger” or “smaller” based on a comparison to his threshhold and will win slightly more than half the time (think about it). In the random version, the game is completely fair, with a simple strategy for Paula to choose which number to reveal.

In “Colored Hats” a dictator gives blue and red hats to a roomful of dissidents. They cannot communicate. Each can see the hats of the others and the simultaneously guess their own hat color. If they are wrong they are executed. How do we maximize the number of survivors? As a variant, the dissidents are lined up and guess from the back of the line to the front, so they can see all the hats ahead of them and hear the guesses from those behind them.

It turns out you can save floor(n/2) of the dissidents by having them pair up and guess their partner’s hat color. In the sequential version you can save all but 1 by having the first person guess based on the parity of the red hats in front of them. This provides enough bias for everyone to guess their own hat color.

This hat problem is particularly interesting because of its relation to some of the work in my Master’s thesis. So this paper is actually relevant (albeit tangentially) to my research. Plus it’s a fun and entertaining read. Recreational math and little games like this was what really got me interested in mathematics when I was younger. That and Raymond Smullyan, of course.