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# Playing it smart - Expected utility and the law of diminishing returns

10 September 2007

"Expected value" is the textbook measure of the quality of a gamble. For each possible outcome, multiply the probability by what you'd have at the end, and sum the products. The result is the "expected value" of the bet. How far this is above or below the bet tells you whether it's good, bad, or indifferent.

But, even when chances and returns are known precisely, expected value isn't a reliable decision standard. It's based on the false premise that a dollar is always worth the same thing to everyone.

Here are some examples of why this is false. Poor folks might covet a chance to earn \$5 while wealthy people might not. Or, compare a \$10 bet with chances of 60 percent to grow to \$50 and 40 percent to tank, versus a \$10 million business deal with the respective likelihoods of becoming \$50 million and of going sour. Expected values are \$30 and \$30 million in the one and the other, both triple the amount at risk. Many individuals would jump at the first, but even the very rich might balk at the second.

The explanation is that a person doesn't necessarily gauge money purely by magnitude, but also by the perception of its usefulness or "utility." Were quantity and efficacy related through a mathematical formula or function, the decision criterion could shift from expected value to a more meaningful expected utility.

One type of relationship which seems appropriate would embody the law of diminishing returns. The idea is that money has decreasing marginal utility. For instance, going from \$1 to \$2 is different than going from \$100 to \$101. And \$2 billion is twice \$1 billion, but those who have it won't buy twice as many cars or shoes.

Diminishing returns could be expressed by equating utility to the square root of value. To see how this would work, picture an investment as one unit and its possible final values as multiples of that unit. Say chance is 20 percent of losing (value = 0, utility = 0), 30 percent of falling to a quarter unit (value = 0.25, utility = 0.5), 40 percent of growing to two and a quarter units (value = 2.25, utility = 1.5), and 10 percent of becoming nine units (value = 9, utility = 3). The accompanying table shows expected value is 1.875 times the outlay moderately favorable; the data indicate that expected utility is 1.05 times the outlay also favorable, but arguably not enough to justify the risk.

Expected value and expected utility for hypothetical investment with utility equal to the square root of value

```probability	value	expected	utility	expected
value		utility
0.2	0.00	0.000	0.0	0.00
0.3	0.25	0.075	0.5	0.15
0.4	2.25	0.900	1.5	0.60
0.1	9.00	0.900	3.0	0.30
totals		1.875		1.05
```

The weakness of utility for decision support is the lack of a "universal" function or class of functions relating it to value under all conditions. Diminishing marginal utility serves in many cases, and even covers gamblers who'd be unhappy losing \$10 but dissatisfied winning \$10. However, it fails to account for instances like those in which marginal utility is variously increasing, flat, and decreasing depending on monetary level.

As an illustration, someone with a \$100 bankroll might not gamble merely to earn another \$100, while viewing a \$1,000 as far more desirable than 10 wins of \$100 each. This same bettor might feed \$0.50 slots with \$100,000 jackpots but not \$1 machines with \$1 million potential payouts. An suitable function would show a flat, then an increasing, and last a decreasing marginal utility.

With no recipe to convert value to utility that holds over a wide spectrum of cases, is the concept practical for anything? Those \$10 million investments the big mahafs make now and then warrant developing utility function to fit particular circumstances. For other situations, like casino visits, solid citizens may find they're able to make more sophisticated choices of games, bet sizes, and exit strategies if they think about how their own marginal utility changes at various levels of loss and gain. The punter's poet, Sumner A Ingmark, waxed about it this way:

A bettor who's wise is rarely one who spurns,
A principle like diminishing returns.

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Best of Alan Krigman
Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.