It’s a simple game. You put down a dollar, I put down a dollar, and we flip a coin. If it comes up heads, you keep both dollars; if it comes up tails, I keep them both. We stop when one of us is broke.
And yes, it’s a fair coin. Cripes.
Now, obviously, if it’s a fair coin and we both start with a thousand dollar bills, over a long time this game is going to come out pretty even: sometimes I’ll win, sometimes you will, but in the end it’ll come out nearly even; it will take a long long time before one of us is busted. What’s more, we’ll always have exactly $2000 in the game. (By the way, this is what games theorists call a “zero sum game”, a term primarily of interest in economics because it’s what a real economic system never is.)
So let’s change the game a little and say I start with $1990, and you start with $10. Some rules otherwise, and obviously there’s still $2000 in the game. Now things are rather different: most of the time, the game is going to end relatively quickly, and it’s going to end up with me having $2000 and you having nothing, because you only have to lose ten more flips than you win to go bust, while I’ve got to love 1990 more times than I win. Still, if we were to play this game out thousands and millions of times, at least some times I’ll end up broke, and you’ll end up with $2000.
This is what probability theorists call Gambler’s Ruin: even in a fair game — and there’s no game fairer that this one — a player with a small stake usually eventually loses all their money. (In fact, it turns out that the probability I’ll win all the dollars in this game is 0.995 and that you will is 0.005. The Gambler’s Ruin article I liked will explain why.)
So, let’s expand the game a little more. Instead of just the two of us playing, we’ll get another 99 players with $10 each, and I’ll put in $1000, so we have 100 $10 players and me with my $1000. Our total money involved is still $2000, and we’ll play it out round-robin, one of the $10 players successively playing against me for the coin flip. If we play it out until all but one of the players is broke — well, the game’s not so good for me: it’s only 50/50 that I’ll be the one. But that means it’s now 50/50 that one of the $10 players will end up with the $2000.
At which point, the Coin Flipping Game Journal will run a front page story on the winner and his “genius strategy” at the coin flipping game, one that made him 200 times his initial investment. He’ll probably write a book, go on lots of talk shows, and make a lot more money talking about his brilliant strategy; if he’s smart, he won’t put it back into the coin-flipping game.
We could carry this one a good bit further, if we have the time and the coffee necessary to stay awake, but this is enough for my purposes right now. The point is that even in a perfectly balanced, perfectly fair, perfectly random game, a big player can still lose big, and if there are a lot of small players, at least one of the small players will very likely win big.
Having gone to Duke and the University of North Carolina, this time of year I’m somewhat vulnerable to the seasonal disease “March Madness”, which is oddly similar to the coin-flipping game. If you look at lots and lots of games, the difference in number of points scored and games won and lost is actually pretty small; among the best teams, in the usual year, no team is really dominant. So, the outcome of the tournament is going to be at least similar to simply laying out the 64 teams in the tournament tree, and flipping a coin to decide the games. It’s no big surprise when UNC, U Conn, or Duke make it fairly far into the tournament, the coin isn’t completely fair that way, but the randomness of the situation is such that often one of the less-favored teams will go much farther than expected in the tournament, becoming a “Cinderella” team and getting the cover of Sports Illustrated.
Does this mean the “Cinderella team” was unexpectedly good? Well, maybe. In statistics, we talk about how to distinguish from the null hypothesis, which simply means we want to tell the difference between a result that “means something”, and a result that could have happened simply by chance. The Cinderella team may be better than we thought, but it’s awfully hard to tell that from their having just, by chance, gotten the coin flip their way. After all, at least one in four teams has to make it to the Sweet Sixteen, and one team in sixteen has to make it to the Final Four, every year, no question, and in a game where the winner is often decided by a few points scored in the last 30 seconds, there’s a lot of room for random chance.
So now — at last — we come to another bit of March Madness, the recent collapse of Bear Stearns. The investment market in mortgage backed securities isn’t a simple coin flip, and it’s not a zero sum game — over time there will be more money around in the game than the players started with. It’s still essentially random — you don’t know who will default on a mortgage, you don’t know what else will happen, and as happened with Bear Sterns, you don’t know when the coin you’re flipping will start being unfair. But being essentially random, investment in mortgage backed securities will always have a certain small probability of a Gambler’s Ruin: no matter how big you are and how effective your risk management strategies might be, sometimes you will lose big.
In Bear Stearns’ case, the turning point came when they went from being the big player, taking the other side of thousands or tens of thousands of small transactions, to being the small player: suddenly, as their assets began to fall, all the little players started pulling their money out of the game. This is a classic “run on the bank”, and it can kill a bank, any bank, because the bank’s investments are bets where the money is off the table for a relatively long time. Bear Stearns owned a lot of mortgages, which owned a lot of houses, that people suddenly didn’t know how to value.
The ones that could started taking their money away, and the fair coin started leaning way against Bear. Then the really big player came in. (There just isn’t a bigger player in the financial markets than the Federal Reserve Bank.) The next thing you know, Bear doesn’t have the money to play against a run of “bad luck.”
When you listen to financial pundits talk about mismanagement by the Fed, or call for more regulation, or demand criminal prosecutions, keep this in mind: before you can say something went wrong, you need to consider whether it was really just a random event.
And remember, neither being the winner in the coin-flipping game, nor being the loser on Wall Street, may mean anything at all.
Charlie Martin is a Colorado computer scientist and nearly-successful screenwriter who contributes to the Flares Into Darkness political blog as ‘Seneca the Younger,’ and blogs under his own name at the aggressively non-political Explorations blog.
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