Friday, January 2, 2009

time-ordering and information-ordering

There is a famous puzzle, which some googling suggests is known as Newcomb's paradox, involving an expert on human nature (or something) who presents each player of a game with two envelopes, one of which the player knows to contain $1000. The player is permitted to receive either just the other envelope, or both envelopes; if this expert believes both envelopes will be taken, the second envelope is empty, while if the expert believes that only that second envelope will be taken, then it contains $1,000,000. After observing several other players, for each of whom the expert's prediction was correct, do you choose to accept both envelopes, or do you decline the $1,000 to take just the second?

My answer is that I take only the second envelope. I don't know what's going on in precise detail, but it appears to me that, one way or another, my decision is available to the expert when the envelopes are sealed. I apparently take my action after the expert acts first, but, the way the game appears to me, the information I have available when I act is circumscribed — I don't know what's in the second envelope — but the expert's decision is made knowing what I will do. The game, in information order, is that I make my decision, and then the expert places the checks, even though that is not the time-ordering of events.

There are a lot of situation in which uncertainty is of importance in economics, and it is very rarely the case that it matters whether the uncertainty is due to a lack of knowledge about the present or a lack of knowledge about the future. If you and I are stuck together for six hours, and we know that a football game has taken place during that time but that neither of us knows how it has gone, it is just as reasonable for us to bet on it at the end of six hours as at the beginning; in the former case we are betting on an event that has, in a time sense, already happened, but for which we are just as uninformed as if it hadn't taken place yet. Similarly, if I am about to have a test done to determine whether I have a genetic predisposition to some disease, it seems reasonable to ask an insurance company to provide me insurance against an adverse result, provided I don't initially know any more than the insurance company does, even though the genes are already there and the information in some sense already exists.

Studying actions of policy-makers or financial markets is invariably complicated by causal relationships running both directions in time; the stock market may rise because the economy is likely to improve in six months, but the economy may improve because the stock market rose. In that case, the effects likely reintensify their own causes — a "positive feedback loop" — but there are negative (i.e. stabilizing) feedback loops as well. Monetary economists speak of a "price puzzle" when one does a naive analysis of the effect of monetary policy on the economy, where tighter monetary policy seems to be followed by an increase in inflation for a short period of time; this is what one would expect if monetary policy is being done competently — the monetary authority should tighten policy when an increase in inflation is coming. Because the earlier event is being taken on the basis of anticipation of the later event, the causal relationship runs backward in time (though, in these cases, it runs forward as well).

I think the real-world solutions to a lot of game theory conundrums — incidentally, I've done less reading on this than I should — involve effects of this nature. People will work out that a repeated Prisoners' dilemma can yield cooperation, at least for a while, so long as future results are discounted relative to current ones, or some such, but, while time-preferences can be screwy and extreme, it usually seems to require too big a discount to generate the results you see in experiments (or real life), and almost certainly isn't in accord with how the agents themselves would describe their rationales. They might talk in moral terms, but it seems likely to me that a certain amount of what is going on is that people know that other people are somewhat cooperative, and — especially in real life — they believe they can tell "what kind of person" some counterparty to some arrangement is. Insofar as one can be read ahead of time, one is at least partially precommitted before the game formally begins.

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