Betting Pools and Expected Values

How a betting pool is structured will affect how people bet [i.e., to belabour the obvious, people respond to incentives]. Here is an example from our local curling club, but it applies to many office pools that are organized quickly, with low organizational costs.

Last weekend at our curling club, Ms. Eclectic and I entered the pool to predict who would win the 2007 Tim Hortons Brier [the Canadian men's championship in curling, in which it is determined who will represent Canada at the world championship tournament in April]. It is a pretty simple pool: everybody puts in $2 and the name of the team they think will win the tournament. The pot is divided between all those who correctly picked the winner.

Ms. Eclectic thinks it will be a toss-up between Glenn Howard (Ontario) and Kevin Martin (Alberta). Deep in my heart, I predicted Glenn Howard will win.

Who did we pick in the club pool? Not Glenn Howard; and our choices were reasonable because of how the pool works.

Given that we curl in Ontario, and (as a result) given that we expected most of our fellow club members would pick Howard to win the Brier, we realized that if we also picked Howard, and Howard won, then the size of the prize (split among everyone who picked Howard to win) would be pretty small.

So, even though I think there's a somewhat higher chance Howard will win, I picked Brad Gushue (Newfoundland, and gold medalist at the Olympics) to win in the pool, and Ms. Eclectic very reasonably picked Kevin Martin.

Algebraically, for each bettor, E(W) = p(S/N), where

  • E(W) is each bettor's probablistic, expected dollar winnings,
  • p represents that bettor's subjectives guess as to the probability that his/her selection will win the Brier;
  • S is the size of the pot that is to be divided between all those who correctly pick the winner; and
  • N is the number of people each bettor thinks will have selected the same winner s/he did.

For Howard, p might be high, but N is expected to be high, too, offsetting the high value of p. For Martin and Gushue, we expect that N will be much lower so that if we do pick the winner, we will win more money.

Note: this strategy assumes risk neutrality on our part. It also assumes that nobody else (or not many others) in the club have the same betting strategy.