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Best of Alan Krigman
Happiness and the Art of Sizing Bets6 January 2004
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:
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.
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:
Of gamblers wise there is a dearth,
Who recognize both price and worth.
Best of Alan Krigman