## Gambling Wisdom: The Horse Race Paradox

### Gambling (general), Strategy

July 31, 2007

In this series I discuss topics that are well known within the gambling community, but that may be new to players who come to the game of poker from a non-gambling background.

Imagine a race between two horses. Horse A is a remarkably reliable horse, and will always complete the race in essentially the same time. Horse B, on the other hand, is remarkably inconsistent. Sometimes it runs a very fast race, far faster than A, and sometimes it runs a very slow race, far slower than A. Which behavior horse B exhibits is random, and P(fast race) is 0.3 or the odds are 7:3 against horse B running a fast race.

Now, if given the option of betting even money on horse A or horse B, it would of course be sensible to choose horse A since the odds are 7:3 in favor of him winning the race. On average, every time we made that bet, we would end up 0.4 units ahead. This is all very simple and sensible. In some real sense, horse A is superior to horse B.

Now, imagine that we start adding more horses C,D,E etc. with the same properties as horse B to the race for a total of N horses, and continue to offer a flat bet where you win N-1:1 if your horse wins. When N=3 (ie there’s horse A and two inconsistent horses B & C) the odds are almost exactly even as to A beating the field or losing, but the increased payout means that our expectation actually goes up to almost .5 units per bet. There seems to be a pattern – not only is horse A superior to B & clones, but as more of them are added that edge increases even as the raw chance A wins decreases.

However, something rather odd happens when N=4. Betting on A is still profitable, but it’s NOT as profitable as it was when N=3. So our pattern was wrong. More surprisingly, when N is increased to 7, betting on horse A is basically a break-even proposition. A will win about its fair share of races. As N goes higher, horse A actually becomes a losing bet.

So what’s going on here? Simple: P(A wins)= (1-P(B wins))^(number of copies of B). In other words, P(A wins) decays exponentially as horses are added, but the return you get on a bet on A increases only linearly. For a small number of horses, the increase in return dominates the decrease in win rate, but eventually that exponential term catches up with you, and betting on A is no longer profitable.

Now, the exact same situation often arises in poker when a made hand faces many drawing hands. I’ll cover the details of how this works, and the strategic implications, in a future post.