Fifty men are trying to win a date with one woman. Each man has been classified in various ways: tall or short, dark hair or light, muscular or wimpy, etc. The woman chooses a category and then her preference -- for example, in the category of "Intelligence" she may have the choice of "Einstein" or "Beer stein". Whichever men do not match her preference have to leave the stage.
After a few choices (and many eliminations), another round is played with the remaining men. In this round the men are forced to do silly things (e.g., serenade the woman while wearing a rooster mask) and she again makes some choices. Eventually the crowd of men is reduced to 3 suitors. Here's where it gets fun.
The woman has been asked a series of questions beforehand, and has made her choice between 2 possible answers. Each man, starting from the left each time, is asked what he thinks her answer will be. Once each man has answered (the second and third men can hear the previous answers) the woman reveals her choice. Whoever guessed correctly takes a step forward toward the woman and the process repeats. The first man to reach the woman (about 5 steps away) wins a date.
As I was sitting there, lamenting my dateless condition, I started thinking that there was no way that I could win such a game -- I would have no idea what the woman would say. For me, I would have to make essentially a random guess each time.
Hmm...could anyone else do any better? Probably not -- it seems like all the men would be guessing (since women are so difficult to predict).
So who would have the advantage? The man on the left who always announces his choice first? The middle man who hears what the first one said? Or the last man who has heard both of the others before making his choice?
And therein lies the game.
The Game: Two players, A and B, each announce what they think the results of a coin flip will be. A always announces his selection first, and B can hear the choice A made. Whoever makes the correct guess gets a point. The first player to 5 points wins the game.
Clearly there is no winning strategy -- a sequence of choices that either A or B could make to guarantee a win -- but is there an optimal strategy for either player? Is there a way for one of the players to maximize his chance of winnin g? The answer is, yes, there is -- Player B has an optimal strategy: on the first flip of the coin, make the opposite choice as Player A. If Player B wins the first round, then from then on he should always pick the same choice as Player A . Players A and B will either both gain a point or both not gain a point -- there is no chance that A can catch up to B. Eventually B will win. If Player B loses the first round, then he will have to continue to pick the opposite as Player A until he g ets ahead (then pick the same as A). But there is a positive probability that this will happen, so Player B will have a greater chance to win than A. Of course, following this strategy does not guarantee a win for B. It is possible (though unlikely) that Player A could correctly guess the flip of the coin 5 times in a row, and so win the game (or tie if B were to guess the same as A each 5 times -- not the optimum strategy for B). So Player A could win the game, but how likely is that? In other words, what is the probability that A will win, provided B follows his optimum strategy? To answer that question, we will need to know how to count the ways that A could win. The Problem(s):Please mail comments or questions about this page to Kennan Shelton