An Introduction to Decision Theory

Imagine you have just won your first professional golf tournament. The chairman of the United States Golf Association offers you to either collect the $1,000,000 prize check right away or toss a fair coin. If the coin lands heads he will double the prize check, but if it lands tails you lose the money and have to walk away with $0. As a relatively poor newcomer to the world of professional golf you strictly prefer $1,000,000 to the actuarially fair gamble in which you either double or lose your prize money. This is hardly controversial. It may take years before you win your next tournament. Few would question that decision makers are sometimes rationally permitted to be risk averse, especially if the stakes are high. That said, it remains to explain how risk aversion should be defined and analyzed, and to determine whether it is always rationally permissible to be risk averse.

The term “risk aversion” has several different but interconnected meanings in decision theory. Some of these are compatible with the principle of maximizing expected utility, while others are not. In what follows we shall discuss three of the most influential notions of risk aversion discussed in the literature:

Each of these notions of risk aversion can be characterized formally. As we will see shortly, this makes it easier to assess their applicability and limitations.

Actuarial Risk Aversion

Kenneth Arrow, winner of the Nobel Prize in Economics, characterizes what we may call actuarial risk aversion as follows: “[a] risk averter is defined as one who, starting from a position of certainty, is unwilling to take a bet which is actuarially fair.” A golfer who is unwilling to accept a bet worth $1,000,000 on the toss of a fair coin is risk averse in this sense. For another example, imagine that you are offered a choice between getting three apples for sure and a lottery that gives you a fifty-fifty chance of winning either six apples or nothing. Unless you desperately need six apples for making an apple pie it seems rationally permissible to prefer three apples for sure over the actuarially fair lottery.