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Thinking and Deciding 10: Normative Theory of Choice Under Uncertainty

09/13/2013

Expected utility theory

The probability of an exhaustive list of possibilities will add up to one. Each possibility has a finite probability of one or less. Multiply the probability of an event by the utility that it would bring, and you’ve got the expected utility of your option. Expected utility theory, as a normative standard of decision making, states that the best choice to make is the one with the highest expected utility.

The theory is normative instead of prescriptive because no one can do the math for every decision they make, and function in the world. Instead the theory is an idealized standard by which we can judge how good the choices are.

Why expected utility theory is normative

This seems like a weird question to even ask, but I guess I’ve got my own unquestioned assumptions here. Why wouldn’t people try to maximize their expected goal fulfillment? There are a few approaches. The long run argument says pretty much the above. If we want to achieve our goals to the maximum, we ought to follow expected utility theory. The argument takes basic, unobjectionable (presumably) principles, and uses those to justify the theory. These are the weak-ordering principle and the sure thing principle. Not gonna describe them.

The utility of money

This section takes a thought experiment proposed a gamble that lead to an expected value of infinity. Flip a coin the first time. If heads, you get a dollar and can flip again. If tails, you lose. If you flip again, heads gets you 2 dollars and can flip again. Tails, you are done. Each consecutive flip leads to a doubling of the prize. The expected value is infinity, but even a reasonable person would probably sell their chance to play this game for $20. Where’s the disconnect? Well, the more money you get, the less value each dollar has. The utility doesn’t double with each doubled dollar. It may be a logarithmic relationship.

But what if the gamble was different? What if you found how many more dollars led to a doubling of the utility? Then wouldn’t the value be infinite? Or what if we change it from dollars to an extension in life, or maybe some other good that doesn’t stop being valuable? Some type of wager like this would seem to have infinite value. But I suppose that’s the whole point. You can’t just double dollars or years of life extension to double utility. The utility provided is pretty tough to nail down.

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