Is Game Theory Applicable Here?

[AaronBrown]

It's trivially easy if you make decisions by expected value, but most people don't. Almost everyone accepts both deals. This has been measured by poll and by real choices, in some cases with significant amounts of money at stake. It has been verified in many cultures and situations.

In A, you give up a 10% chance to make $2 million (converting $500,000 to $2,500,000) in order to eliminate a 1% chance of losing $500,000 (going from $500,000 to zero).

In B, you give up a 1% chance of getting $500,000 instead of zero, in order to get a 10% chance of getting $2,500,000 instead of $500,000.

Here is a good link to the paradox (without my poker gloss).

My point is that if you don't use expected value to make decisions, it's easy to find yourself making inconsistent decisions.

I think the poker example is not so good for this board because people here tend to think in EV terms at least wrt poker. "Who Wants to be a Millionaire" provides a good example of Allais' Paradox. Suppose you are at the $32000 question and use your last lifeline the 50/50 giving you 2 possible choices. You have absolutely no clue which one is the correct answer. You can walk away with a guaranteed $16000 or can take a guess where 50% of the time you win 32000 and 50% of the time you win $1000. Guessing has a higher expected value especially if you consider that a correct guess leads to more money potential. Yet, how often do people walk away with the $16,000. And how often would you walk away instead of guess.

The Allais "paradox" isn't an example of people behaving incorrectly, it's an example of a problem with the standard economic way of analyzing decisions under uncertainty (ie assume a concave utility function to account for risk aversion). Realisticly, people aren't bankrolled for life. You're going to have to deviate from maximizing EV once you get outside of the realm of poker, and it's not entirely clear what the best way to do that is. I'm not comfortable with calling something that occurs so frequently as the preference reversal problem (Allais paradox) a reasoning error.

[ebroil]

No, this is not an example of Allais paradox. This is an example of utility theory. People frequently makes choices that have a smaller EV because their marginal utility for money increases at a decreasing rate. That is, having 16,000 for sure has a greater utility than having a possibility of 32,000, even though that option has a (slightly) higher EV.

Allais paradox occurs when people make choices that are inconsistent with expected utility theory. The example you provided is simply an example of utility theory.

This is correct. The first example appeared to be a correct version of the "paradox" though I didn't read it very carefully. The original question is appropriately answered by utility theory, as is the millionair question. I'd certainly take the 240 straight up rather than the 1/4 chance at 1000 mostly because 240 is a lot of money in graduate school. My main point is that it's not entirely clear that people who exhibit preference reversals are making inconsistent decisions, just decisions inconsistent with expected utility theory (specifically the independence axiom).

[AaronBrown]

Now we have gone full circle, as happens so often in message boards. The original post asked if game theory could justify sacrificing expected value. phzon gave an excellent reply explaining it had nothing to do with game theory, it was utility theory that sometimes argued for accepting less than the maximum expected value.

I seconded that post, but added you have to be careful deviating from maximum expected value as a criterion. It is quite easy to construct examples in which people make decisions that conflict with utility theory, Allais' Paradox is one simple way. No one said the original question was Allais' Paradox, or any paradox at all.

So I think we all agree. People often sacrifice expected value. While there are some simple explanations of why they might do so, the explanations do not correlate well with real human behavior. In fact, it's easy to find behavior that does not correlate with any imaginable rational calculation. So if you're tempted to sacrifice expected value, it's prudent to examine the decision closely before you make it.

Right. I agree with what you're saying. And I agree that people ought to closely examine why they are deviating from maximizing EV. I just find it moderately distressing when some economists point to the Allais paradox as an example of people behaving incorrectly simply because it deviates from our theories of what they ought to do. I think it was a good thing for you to bring up the Allais paradox so that people are careful when they think about these issues, I just wanted to provide a different perspective that preference reversals aren't necessarily wrong, just incosistent with our theories.