Representing Subjective Value

Bayesian decision theory, like all models of reasoning and systems of formal logic considered throughout this book, faces some problems and limitations. An examination of all its shortcomings is beyond the scope of this chapter, but it will be worthwhile to consider one of its most obvious challenges: the difficulty of assigning precise values, or utilities, to the various possible outcomes in a decision matrix. In the example discussed on the previous page, I mentioned that it doesn’t matter exactly what numbers we use in a decision matrix. I also said that the units don’t matter, because Bayesian decision theory is just an idealized theoretical model to help us understand how rational deliberation works. However, I’ll admit, that was a slight oversimplification.

It’s true that Bayesian decision theory is an idealized theoretical model to help us understand how rational deliberation works. It’s also true that the exact numbers are irrelevant. What matters is their proportions. For example, if you value outcome x twice as much as outcome y, then its utility in the decision matrix should be represented with a number twice as large as the number you assigned to y.

On the other hand, it’s not quite true that the units don’t matter. We might be tempted to “measure” how much someone values something by how money much he or she would be willing to pay for it. Money does provide a quantitative measure of value, but monetary units (e.g. dollars) don’t always correspond to the degree to which a person values something. Depending on circumstances, someone may be willing to pay a lot of money for something he values very little, or vice versa.  In fact, the degree to which we value (or care about) money itself isn’t constant: it varies depending on how much money we have, and it may also vary for other reasons. This spells disaster for Bayesian decision theory if we try to define utility in terms of monetary value. To illustrate the problem, here’s an interesting real-life example involving the Powerball lottery:

In January 2016, the grand prize jackpot of the Powerball lottery was $1.5 billion. The probability of winning the grand prize was 1 in 292,201,338 (approximately 0.0000000034). The cost of a ticket was $2. Let’s ignore taxes for the purpose of this example, and suppose the winner gets to keep the whole 1.5 billion dollars. Then, in terms of money (not utility), the expected value of buying a ticket was:

expected value in $ = (prize minus cost of ticket × probability of win)  –  (cost of ticket × probability of loss)
= ($1,499,999,998 × 0.0000000034)  –  ($2 × 0.9999999966)
= $3.10

So, ignoring taxes, the expected monetary value of playing the Powerball lottery in January 2016 was $3.10. (With taxes, the expected value was significantly less.Moreover, the payout was dramatically less for anyone who selected an “up-front lump sum” payment option rather than a 30-year distribution of smaller payments. Those factors, together with a 37% federal tax rate on the winnings, made the true expected value negative, as it almost always is for lotteries.) Of course, the expected value of not buying a ticket was just zero, since you neither gain nor lose money by not playing. So, in this case, the expected monetary value of playing was higher than the expected value of not playing!Well, not really. See previous footnote. Does this mean that it was rational to play the Powerball lottery in January 2016?

No. The reason is that monetary values can’t be translated directly into utilities, due to a phenomenon called the “diminishing marginal utility of money.” Consider: gaining or losing a few dollars could make a noticeable difference to the quality of life for a homeless beggar, whereas a billionaire won’t notice any difference at all in his quality of life if he gains or loses only a few dollars. Thus, each dollar is worth more to a beggar than it is to a billionaire.

Even for a person of average wealth, $1 is worth much more (in terms of how it can affect your quality of life) than each dollar of the jackpot. In other words, if you win the jackpot, each of those 1.5 billion dollars will be worth far less to you than the value $1 has for you now. So, unless you’re already a billionaire, playing the lottery isn’t rational. (Oddly, the January 2016 Powerball seems to be a case in which it might have been rational for a billionaire to play the lottery.Then again, probably not. A billionaire would likely make much more money by investing his or her wealth in some other venture, rather than wasting precious time buying lottery tickets with a paltry expected return of $3.10.)

Here’s another way of explaining the point. Those who played the Powerball lottery in January 2016 became $3.10 richer on average. Of course, everyone except the winnerIn point of fact, the winnings of the record-high January 2016 jackpot were split three ways, as several players chose the same winning numbers. lost money, but the huge win pulled up the average. In terms of quality of life, however, the average outcome was a loss. Each dollar spent by the many losers made a more significant difference to their quality of life than each dollar gained by the winner. Of course, all of those dollars together made a very dramatic difference, but only for onetechnically, three people—see previous footnote person; and the difference made to that person—in terms of utility—was not as significant as the collective difference to the many losers.

The upshot of the foregoing example is that money is not a good measure of utility. In fact, there just isn’t any precise way of measuring the degree to which a person values something. Fortunately, this doesn’t diminish the value of Bayesian decision theory as a model: it can still help us to understand the considerations involved in making rational decisions. In real life, obviously, we don’t make decisions by assigning numbers to our values and degrees of belief, then perform mathematical calculations. However, we do deliberate by weighing the pros and cons of various possible outcomes together with our degrees of belief, in a way that is closely analogous to the expected utility calculations described in Bayesian decision theory. For this reason, Bayesian decision theory can be a helpful guide to deepen our understanding of what it means to be reasonable, or rational, in the choices we make.