Here's a fun little puzzle for introducing some issues in decision theory. You want to invest a sum of money that is very large for you (maybe it represents all your present savings, and you are unlikely to save that amount again), but not large enough to perceptibly affect the market. A reliable financial advisor suggests you diversifiedly invest in *n* different stocks, *s*_{1},...,*s*_{n}, putting *x*_{i} dollars in *s*_{i}. You think to yourself: "That's a lot of trouble. Here is a simpler solution that has the same expected monetary value, and is less work. I will choose a random number *j* between 1 and *n*, such that the probability of choosing *j*=*i* is proportional to *x*_{i} (i.e., *P*(*j*=*i*)=*x*_{i}/(*x*_{1}+...+*x*_{n})). Then I will put all my money in *s*_{j}." It's easy to check that this method does have the same expected value as the diversified strategy. But it's obvious that this is a stupid way to invest. The puzzle is: Why is this stupid?

Well, one standard answer is this. This is stupid because utility is not proportional to dollar amount. If the sum of money is large for you, then the disutility of losing everything is greater than the utility of doubling your investment. If that doesn't satisfy, then the second standard answer is that this is an argument for why we ought to be risk averse.

Maybe these answers are good. I don't have an argument that they're not. But there is another thought that from time to time I wonder about. We're talking of what is for you a very large sum of money. Now, the justification for expected-utility maximization is that *in the long run* it pays. But here we are dealing with what is most likely a one-time decision. So maybe the fact that in the long run it pays to use the simpler randomized investment strategy is irrelevant. If you expected to make such investments often, the simpler strategy would, indeed, be the better one—and would eventually result in a diversified portfolio. But for a one-time decision, things may be quite different. If so, this is interesting—it endangers Pascal's Wager, for instance.

## 8 comments:

I don't see the connexion with Pascal's Wager, wasn't the aim of that to make us make a non-diversified investment?

...that is, didn't Pascal's argument claim to give us a reason to make just the one investment. Why would we then diversify? That would be like also investing in things with very poor prospects, which the reliable financial advisor presumably wouldn't advise.

The connection is not so much about diversification, but this: Perhaps expected utility maximization is not the right rule for once or twice in a lifetime choices. Investing a very large sum of money may be such a case, and Pascal's Wager definitely is such a case.

The implicit thought is this. What is the justification for choosing the option that maximizes expected utility? It is something to do with the Law of Large Numbers, presumably. But the LLN is useless for a single case event.

Alex,

There is one obvious reason the latter is a worse way to invest than the former. These two strategies agree with respect the chances P of the total value V. But they differ with respect to the chances of some payoff or other. So, compare two games, G1 and G2. In G1, we have a one-off game, win or lose, with say a .5 chance of winning V and .5 chance of receiving no payoff. EV(G1) = .5V. In G2, I may have chances to win, say, .9(V/2) + .05(V). In the latter, you're very likely to win something, though no more likely to come away with V. If I'm faced with two strategies S and S' with the same expected utility, but where S' is likely to yield some payoff, I should play S'.

I think you're right that this has some very interesting implications for PW. For instance, the believe payoff is much larger, but the non-believe payoff is much surer.

Mike:

I think your main point is exactly right. But the application to PW is a bit more complex

"For instance, the believe payoff is much larger, but the non-believe payoff is much surer."

Remember that Pascal's own formulation said that the probability of theism is 1/2. Now, many people who have pushed Pascal's wager since Pascal's time have taken this "1/2" to stand in for any non-zero probability. But it may well be the case that the 1/2 is important to the argument, or at least to the intuitions. If the probability is 1/2, then it is false that one of the two payoffs is surer.

So, here is a suggestion (I am not endorsing it). Perhaps PW only works where the probabilities are not too close to 0. Perhaps it works as long as P(theism) is at least 1/3 or 1/4.

Remember that Pascal's own formulation said that the probability of theism is 1/2Alex,

Yes, right. But now I'm beginning to think that I've been missing smoe structure in the PW argument. Surely, there is a guaranteed payoff to ~believe that does not depend on God not existing. It is not as though I get the payoff for not believing only if chances fall my way and I don't believe. I get the immediate payoff (such as it is) if living my life without moral constraints. I expect that payoff immediately on ~believing. What am I missing?

If God's non-existence is more likely than His existence, then surely we shouldn't believe that He exists (perhaps we should be agnostic). Would God reward us for believing in Him despite the evidence? He might reward loyalty and love, but surely not belief justified by expected gain. (I seem to recall that Pascal said something similar, about his argument not being a good reason to believe. If so then we can all agree that PW is not a good argument.) Indeed, I suspect that He would punish or at least test such belief until it broke and turned into something else.

A nice analogy may be a lottery. Should you believe that your ticket will win because if it does then you get a big pay-off? Of course not. I imagine that PW is thought to be different because it's about entering into a relationship with your own loving creator. But is that how one enters a loving relationship? You may take a shot in the dark to begin to interact with someone; but we are already created. And surely you don't believe in people just because it would be nice if they were nice. You do it on the basis of the evidence (and loyalty too, but only later).

Diversified portfolios (in non-homogeneous markets) give greater expected returns at each risk level (except the two extremes) than a single stock.

I knew my economics degree would come in handy.

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