Happiness and the Art of Sizing Bets

6 January 2004

By Alan Krigman

Bet size relative to a player's bankroll influences the chances of surviving a session of desired duration, and also of reaching a win goal, as opposed to busting out. Even more, bet size affects the happiness to be expected from a gamble. The idea is to pick an amount that balances the pleasure of adding a win to a purse against the pain of subtracting a loss from it. Of course, "happiness" is a vague notion and consultants can charge clients extra by talking about "utility," but they mean the same thing.

Were expected happiness directly proportional to the money forecast having after a round, it would be tantamount to the "expected value" of a stake. Here's an example. Say you play double-zero roulette with $100. You're considering bets of $10, $25, or $50 on a single spot. Hits pay 35-to-1, so wins or losses will leave you $450 or $90, $975 or $75, or $1,850 or $50, respectively. Probability of success is 1/38. Expected value of your bankroll, and expected happiness, for these options are:

bet	expected value and/or happiness
$10	1/38 x ($450) + 37/38 x ($90) = $99.47
$25	1/38 x ($975) + 37/38 x ($75) = $98.68
$50	1/38 x ($1850) + 37/38 x ($50) = $97.37

Not betting would leave you with a sure $100 rather than the chance of either more or less. This would be the "certainty equivalent" of your happiness. Bets of $10, $25, and $50 yield 99.5, 98.7, and 97.4 percent of the certainty equivalent, respectively. With the "linear" correlation, house edge drives expected happiness down as wagers go up.

Most gamblers get more elated as their bankrolls swell. But the link is usually not a straight line. Depending on the person, happiness may grow more slowly or rapidly than the cash involved.

For instance, solid citizens who hit the slots for $50,000 don't normally scream half as loudly or long as those who win $100,000. The $100,000 is more pleasing, but not necessarily twice as much. This shows happiness rising with amount, but at a declining rate. Precise relationships not only defy definition, but vary among individuals and over time. Economists find "square roots" useful for making calculations in cases of this type. Illustrating with the single-spot roulette bet, and using the square root of bankroll to gauge happiness or utility, not betting at all has a value of 10. The arithmetic for the $10, $25, and $50 bets gives expected happiness values of 98.0, 92.5, and 80.2 percent of the certainty equivalent, respectively. The slide here is steeper than that perceived by players with a linear utility function.

Punters may alternately think of $100 as chump change they can always scrape together, and fantasize about leaving the casino with lucre that proves illusive, if not an impossible dream. That is, $1,000 is worth more to them than merely 10 times $100, $5,000 more than 50 times $100, and so forth. The satisfaction these folks derive from winning escalates faster than the actual amounts. Again, a precise function is neither possible nor necessary. A plausible assumption might be that happiness climbs in step with the square of the money -- the amount multiplied by itself. In this situation, the expected happiness associated with not betting is 100 percent of 100 x 100 or 10,000. Doing the math for the three roulette bets, expected happiness is 132, 305, and 925 percent of the certainty equivalent at bets of $10, $25, and $50, respectively. Here, higher bets anticipate soaring joy.

When the house has an edge, the expected value of players' fortunes drops as bets mount. But few people gamble in casinos because of expected value. They do it in the hopes of a win under the cloud of a possible loss. The pleasure they perceive in increasing sums of money can serve as a factor in deciding how big a bet represents the best trade-off for themselves.

Betting is a blend of science and art. Science in the relation between sizes and progressions of wagers and the probabilities associated with session characteristics. Art in the impact of choices on players' success in meeting personal goals and preferences. It's as the beloved bard, Sumner A Ingmark, noted: