The unbearable lightness of expected utility

In chapter 8 of my book, Risk Intelligence, I discuss the concept of expected utility.  To calculate the expected utility of a course of action, the first step is to estimate – separately – the probability of success, and the potential gains and losses that success and failure would entail.  Next, one does a little math, multiplying the probability of success by the potential gains, and multiplying the probability of failure by the potential losses.  Finally, we adding these two figures together to end up with the expected utility of that course of action.  After doing this for each possible action, the rational choice is to pick the action with the highest expected utility.

Expected utility is an abstract concept. It doesn’t refer to any actual win or loss: it’s the average amount you would win or lose per bet if you placed the same bet an infinite number of times. But this ethereal figure takes on an almost physical nature for the expert gambler, looming even larger in his consciousness than the actual profit or loss that hypnotises the rest of us. Expert gamblers see something different when they look at a poker table or a roulette wheel. Most people see a range of prizes; they see a single abstract figure.

This was brought home to me one night when a highly successful gambler invited me to accompany him to a casino. This particular gambler made all his money betting on horses. He shunned casinos, and only ventured in on this occasion because he wanted to give me and his other companions the thrill of seeing someone playing for high stakes. We followed him over to the craps table.

Craps is a dice game in which players place wagers on the outcome of the roll of two dice. My gambler friend proceeded to show off by placing bets of 1000 US dollars on the pass line, one after the other. A pass line bet is won immediately if the first roll is a 7 or 11. If the first roll is a 2, 3 or 12, the bet loses (this is known as “crapping out”). If the roll is any other value, it establishes a point; if that point is rolled again before a seven, the bet wins. If, with a point established, a seven is rolled before the point is re-rolled, the bet loses (“seven out”). A pass line win pays even money; in other words, my friend stood to win or lose a thousand dollars on each bet.

However, as he told me later over cocktails in the casino bar, it wasn’t this figure that was at the forefront of his mind. Instead, he just treated each bet as a bit of fun that cost him fourteen bucks. Although not a casino gambler, he was familiar enough with craps to know that the expected value of a pass line bet is -0.014. And this meant that, on average, each bet of a thousand dollars would leave him fourteen dollars less well off. The actual profit or loss at the end of an hour on the craps table could be anything from plus fifty thousand dollars to minus fifty thousand, and my friend would be aware of this figure too. But the figure that mattered most for him was the expected value of minus fourteen bucks per bet. That’s not a good-value bet, of course. Which is why, in any other circumstance, my gambler friend would never play craps.

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