Why Maximize Expected Value?

Question

In many instances I've come across (in Game Theory, etc), when trying to choose an optimal strategy it has the criterion that it wants to maximize expected value much of the time. To simplify this question, imagine we are playing a game, $X$, (BlackJack, Slots, Rock-Paper-Scissors) that can be repeated many times if desired. I see that maximizing expected value is a good idea in situations where we can apply Law of Large Numbers (e.g. repeating game many times) and know that value of wins will over $n$ iterations of game will somewhere close to $nE(X)$.

But I have a problem accepting this as a reasonable criteria, when the Law of Large Numbers can't be applied. Lets say we play game $X$ one time or $n$ times where this many times where Law of Large Numbers doesn't reasonable apply. In this situation our actual winnings could vary quite a bit from $nE(X)$. So I don't see how maximizing expected value would still be the best criterion here. Wouldn't other criterions that possible mix variance and expected value or perhaps one that looks at which strategy produces "best" 95% confidence interval of value of winnings be a better criterion?

NOTE: assume I am someone who is risk neutral and utility of winnings is equal to monetary value of winnings

Answer 1

Here's an example of a two-player game which I think bears out your point. Suppose I have a hundred boxes and a hundred coins, and that I am free to distribute the coins as I wish. My partner and I then randomly pick two boxes, and I obtain the difference in coins between the two numbers. What should my strategy be?

If we want to maximize expected value, we may reason as such. Let me start with some distribution of coins and imagine moving one coin from a box with fewer coins to one with more. How does this impact my return?

There are then three outcomes:

1. Neither I or my partner pick one of the two affected boxes. The expected value of these cases is unchanged.
2. Exactly one of us picks one of the affected boxes. Then my return will either increase by one or decrease by one, but these outcomes occur with equal probability. so the expected values are again unchanged.
3. My partner and I both pick the affected boxes. In that case, the difference in coins, and thus the expected value of this outcome, has increased by 2.

So it's always to my advantage to put more coins in fewer boxes. Proceeding this way, we conclude that the best return occurs when I put all 100 coins in one box. In that case, I win all 100 coins whenever one of us chooses the 100 coin box, and win nothing otherwise.

So, from the perspective of my expected value, if I'm playing a long time I expect to make the most profit on this game by relying on a jackpot. But this is hardly a stable rate of return: the probability of winning this way is a mere 1/50 (99 ways to choose one of the empties along with the jackpot out of $_{100} C_2=99*50$ outcomes total). On most runs I win nothing.

So if I was only playing for a few games (or if there was, say, a 1-coin buy-in per round) then it might be very reasonable for me to pick a distribution which has less variance in value. (If anyone knows which distribution minimizes, I'd be interested to hear about it...) Thus the 'maximize expected value' principle isn't necessarily the most 'rational' course in every scenario.

Answer 2

Suppose you bet on a sequence of coin tosses, each tossed independently and with bias $p$. On the $k$th toss you bet $B_k$ dollars and win $B_k$ with probability $p > 1/2$ or lose $B_k$ with probability $1-p < 1/2$. After $n$ trials your expected wealth is

$$E(W_n) = W_0 + \sum_{k=1}^{n} (2p-1)B_k.$$

Since the game has a positive expectation you would maximize the expected value by betting all your available wealth on each trial. Unfortunately this bold strategy has probability of ruin $1-p^n \rightarrow 1$, so ruin is almost sure.

Rather than maximize the expected value, a better approach might be to bet a fraction of your wealth to maximize rate of return, minimize probability of ruin, or some other criterion.

Answer 3

This is a difficult and in many ways a nonmathematical question. Mathematicians tend to think of each dollar as of equal value. If you believe that, a one in a million chance of a billion dollars is more valuable than a certainty of 999 dollars. The economists talk about a utility function, which tends to reflect the declining value of money. In that case you should maximize the utility function instead of the expected value function. You may get different answers. In that case, the law of large numbers works fine where the utility function is roughly linear over (most of) the distribution.