atheorist (atheorist) wrote,

mean reversion and organizations

If you were to graph the numeric value of cards drawn successively from a deck that is reshuffled when it runs out of cards then it would be an irregular line. But of course the line is not completely random. If the line has been unusually high, it will fall - the high cards have been depleted, and only the low cards are remaining. Correspondingly, if the line has been unusually low, it will rise. That phenomenon is called mean reversion, and taking advantage of it in cards is called card counting, and taking advantage of it in finance is called statistical arbitrage.

What might cause mean reversion in the real world? Well, some time series, like inflation rates, are actively "steered" by very powerful people who have stable goals, or a goal of stability, or something. Another mechanism is that two commodities, like oil warehoused in location A and oil warehoused in location B might be almost equivalent, so that purchasers of that commodity are likely to go to the cheaper one, causing the price to rise to near-equality with the other one. Similarly, producers might easily switch to produce whichever one is (temporarily) more expensive.

In games of cards, there's events like shuffles that suddenly destroy your painstakingly gathered information. In order to make a game that illustrates mean reversion that might be a bit more applicable to the real world, you can imagine putting each card as its played into a 'time out' queue, and taking whatever card is at the head of the queue and shuffling it into the deck. Then there aren't any shuffles, but there still is mean reversion. If you've seen the high cards recently, then the high cards are in the queue, and the upcoming cards will be low.

There's a family of psychological tests (or exercises) called n-back. In these, you see or hear a stimulus (something like seeing a card dealt) but you have to act according to your memory of the n-th stimulus ago. Imagine a game with 4 cards total, all different, and a queue of 2 cards, where the idea is that you predict what card is coming up. If you've been playing for a while and can keep the queue in your head and figure out the contents of the deck, then you can predict correctly 50% of the time, but someone who just sat down can only predict correctly 25% of the time. (If you can only remember 1 back then you can still get some advantage by picking anything other than that one.)

There's a problem called 'gambler's ruin' (a few related problems, actually). If you have a finite bankroll and your opponent doesn't and you keep gambling rather than stopping at some point then the only way the game can end is if you lose. If both players have finite bankrolls, and the game ends when either one is "ruined", then the one with a larger bankroll has a substantial advantage in the full game even if the micro game is fair. Considering how to avoid gambler's ruin leads to bet sizing, in particular Kelly bet sizing.

If you modify the previous card game, so that the player not only predicts, but chooses how much to bet on their prediction, then they might reasonably bet a constant fraction of their bankroll (that's Kelly bet sizing, given that the structure of the game isn't changing).

Outside of a gaming context though, the mean-reverting time series that I mentioned are pretty abstract. It's not necessarily clear (it's certainly not clear to me) how you might translate Kelly bet sizing advice into sizing trades on two commodities when you believe they might have a mean-reverting price difference (or some other predictability). One possibility is to assume that there are derivatives available - puts and calls and swaps. However, I'd rather not, firstly because they're not always available and secondly because they have implementation details (even if it's weird things like "short sellers get castigated in the public eye") that complicate things.

Here's a story for how this might work. If a large organization kept a lot of holdings in a standard ratio (e.g. 50% in gold and 50% in pork bellies), then as the two goods drift in price, it would need to re-evaluate its holdings and rebalance. If you were responsible for holding a small piece of it, then you could shift your piece off the standard ratio. When rebalance comes around, the rest of the organization expects you to have gained or lost the same as everyone else, but if the goods drift apart or together then you'll have gained or lost more or less.

Lets suppose that I like being perfectly balanced, with half of my wealth in gold and half of my wealth in pork bellies. Suppose the current market price of gold is 3 pork bellies, so I apparently have N units of gold and N*3 units of pork bellies. Then the relative price of gold and pork bellies changes. Now gold is worth 4 pork bellies. I'm not sure whether I gained or lost, but I'm certainly unbalanced now. So I want to exchange a little gold for pork bellies, so that I have half of my wealth in gold and half in pork bellies again. With a little algebra, I figure out that N/8 gold would become N/2 pork bellies, and then I'd have 7N/8 gold and 3N+N/2=3.5N=4*(7/8)N pork bellies.

Now, if I had known that the price change was coming, and exchanged epsilon pork bellies for epsilon/3 gold, then I would have had a bit more (N+epsilon/3) gold and a bit less (3N-epsilon) pork bellies before the price change, but in the rebalance I could exchange a bit more ((7*epsilon + 3*N)/24) gold than before, and cover the epsilon pork bellies that are missing, and end up with a bit better situation which is nonetheless balanced according to policy (epsilon/24 + (7*N)/8 gold and epsilon/6 + (7*N)/2 pork bellies). That extra epsilon/24 gold and epsilon/6 pork bellies is the profit from a correctly predicted price movement, and epsilon is (somehow) related to the Kelly bet size.

But even if there are some reasons to prefer one ratio over another, there are also costs of continually rebalancing; fees and the bid/ask gap. Maybe the correct way to handle this is to introduce a utility function. Then if you trade epsilon pork bellies for epsilon/3 gold you can keep track of that fact in an account denominated in utility and intended to measure your trading as opposed to your capital gains/losses. Then when prices change, the trading account starts showing positive utility, even if there isn't any rebalance operation. Then I guess a Kelly bet-sizing rule would be something like a cap on the amount of utility that flows into the trading account?

Why obsess about this? Well, I think it might be helpful to be able to intentionally build organizations that behave a certain way because of their structure - Friendly organizations, non-psychopathic organizations, epistemologically grounded organizations. Stross references this idea in Rule 34 - one of his characters, Dorothy Straight, works as an auditor investigating whether organizations have psychopathic incentives. David Brin has a (publicly available) chapter of his latest that describes a smart mob that computes collectively.

That n-back thing where if you can remember what happened n back then you can predict 50% correctly can be perceived as a sort of spam filter, reducing the influence (money) of bettors who can't or won't remember n back. In order to design a smart mob that can say 'sure, anyone can join', you're going to need that kind of moderation / karma system.
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