Can You Size Your Bets for Maximum Happiness?

What if you had some cash stashed in a cookie jar and a chance to gamble with it when you had (or thought you had) an advantage? That is, with a payoff multiple greater than the odds you're fighting to win. How do you decide how much of it to bet?

You have the edge, but could easily lose. So you want to weigh the emotive effects of adding to or subtracting from your purse.

A hypothetical case can show the idea. Say your kid's tiddlywinks team is in the semi-finals against Lowell High. Your season is five wins and five losses. Lowell is seven and three. You therefore figure your opponents as seven-to-five favorites; in probability terms, the chance you'll win is 5/12 or 41.7 percent.

A braggadocios Lowell parent offers to pay 2-to-1 for bets on your school, up to $100. Getting 10-to-5 for a 7-to-5 proposition gives you a 25 percent edge. Casinos, of course, do the opposite. They pay solid citizens less than the odds; for instance, 9-to-5 for 10-to-5 wagers when craps players place the four or 10.

Contemplating only expected value, you might bet the maximum because the higher you go the greater your statistical profit. For the tiddlywinks offer, every $1 you bet has an expected value of $1.25. This is "on the average," however. It ignores your bet being a longshot, as well as how despondent you might feel with what's left after losing or how elated with the total if you win.

Economists would view this as a classic "utility" problem. You start with $100 and can end anywhere from zero to $300, depending on your bet and the likelihood of winning. Given the uncertainty, how do you relate to different amounts between 0 and 300? Considering the risk, how happy do the various sums make you?

Few folks would dispute "more is better." Still, as riches rise, every extra dollar tends to add less and less to most people's satisfaction. If you could specify a "utility function" relating your pleasure to dollar amounts and associated probabilities, you could find the fraction of a bankroll to bet that would maximize expected happiness for this proposition. While no such function will be wholly reliable or exact, square roots work well for many individuals. This would yield a utility index of 0 for no money, 10 for $100, 14.14 for $200, and 17.32 for $300. The first $100 increases utility by 10, the second by 4.14, the third by 3.18.

Grinding through the math would show expected utility for the situation in question to be highest betting $25.76. Picture a greater edge, perhaps because the Lowell lunkhead offered 5-to-1 and you continue to figure odds at 7-to-5. The optimum bet would be $66.21. At a 10-to-1 payoff, you might be tempted to go for broke, but your expected happiness will peak at an $81.97 bet.

You can turn the matter around. For example you might figure you have the better team, maybe 6-to-4 -- 60 percent chance of winning. And you find someone willing to go even money against you. The optimum bet for your $100 stake here is $38.46.

At this point, inquiring minds will want to know whether this stuff holds for wagers with no advantage either way, or where you're fighting an edge. The optimum for a "fair" bet is zero. And, up against an edge, the math dictates amounts below zero. Both ways, your expected happiness will be maximum by keeping your certain $100 rather than going for the gamble.

You're not apt to face conditions precisely like those in the tiddlywinks illustration. You'll rarely encounter gambles when you have an edge. And utility functions are artifices at best. The implications that bet sizing is far more complex than most players imagine are worth pondering nonetheless. And a lesson lurks in the fact that as your advantage grows, the optimum bet creeps closer to the total you have in your fanny pack but never quite gets there. Not dissimilar to the view voiced by the versatile versifier, Sumner A Ingmark, when he wisely warbled:

While losing's worse than winning,
Excessive betting's sinning.