by Edward A. Bender

In a pool, one tries to guess the “winners” in a set of games. For example, one may have ten matches this weekend and one bets on who the “winners” will be. We’ve put winners in quotes because the pool may handicap the matches so that it is expected that each side has an equal chance. For example, if the Jets are playing the Sharks and the Sharksare a weaker team, then there will be point spread—the Jets must score some speciﬁed number of points more than the Jets to be declared the winner in the pool. There a two common types of pools. • One is among a group of friends and the person with the most correct guesses wins all the money that was bet. If several people have the same number of correct guesses, the money is divided evenly amongthem. We’ll call this the oﬃce pool since that’s where it’s often done. • The other type of pool is like the lotteries that are run in many states: How much you win depends on how many correct guesses you have and it is set up so that the organizers expect to pay out less than they take in. We’ll call this the for-proﬁt pool. In both kinds of pools, a player normally does not know what winners theother players have chosen.

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Can You Make Money in a Football Pool?

Suppose you play the football pool many times. Can you expect to come out ahead in the long run or must you certainly lose? 1

Suppose the handicapping is fair; that is, in each match, both teams have a 50% chance of being declared the (handicapped) winner. In that case, everyone in the pool may as well guess randomlyat the outcome of each game and each person has an equal chance of winning. • Since everything that comes in is paid out in the oﬃce pool, you should tend to break even there in the long run. • Since not everything is paid out in the for proﬁt pool, you should tend to lose there in the long run. This seems to be the best you can do. Of course, you can do better if you somehow know that some teamsare better or worse than believed. We’ll explore some consequences of this in the for proﬁt pool. Amazingly, you can do better than break even in the oﬃce pool! How is this possible? The idea is to enter more than one bet each time, say two bets. Since each bet only breaks even in the long run, how can it help to make two bets, each of which seems to break even in the long run? Before we go intothis, which is a little complicated mathematically, let’s look at a simpler situation which has the same strange property. Three people are told that they will play a game as a team against the Mad Hatter. Here’s how it works. • The Mad Hatter will place either a red or a blue hat on each person’s head. He’ll choose the hats randomly. • Each person can see the other two hats, but not his own. Theywill not be allowed to communicate. Each person must write on a slip of paper either a guess (red or blue) of his hat color or “no guess.” • If everyone writes “no guess” or if someone guesses the wrong color, the team loses. Conversely, if there is at least one guess and all guesses are correct, the team wins. The team is told to work out a strategy and then they will play the game. One personsays:

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It’s clear that a guess can only be right half the time since seeing the other two hats is no information at all. Thus, if one person guesses, we have an even chance of winning. If two people guess, there is only one chance in four that both will be right. If all of us guess, there is only one chance in eight that all of us will be right. Therefore, it is obvious that we should simplychoose one of us to guess and the other two should write “no guess.” Everything this person said is correct, but the strategy is not the best possible. In fact, there is a strategy gives a 75% chance of winning! Here’s the winning strategy. All three people do the same thing, namely: If the other two people have the same color hat, write the opposite color; otherwise write “no guess.” Let’s see...