The answer to the backgammon player’s perennial dilemma, “How much should I bet?” was discovered half a century ago by the Scottish mathematician Jon Kelly. Kelly demonstrated that within a universe of bettors, it was precisely the ones who wagered exactly X% of their bankroll whenever they had an X% advantage who would win the race to riches against those employing alternative betting strategies. Like Goldilocks’ porridge, one’s account balance will taste best if it is cooked just right: not too risky (double-Kelly, for example, would instruct you to bet 20% of your bankroll with a 10% edge) or too tight (half-Kelly, for example, would tell you to bet only 5% of your bankroll with a 10% edge). The too-risky strategies go broke too often, while the too-conservative ones don’t grow fast enough.
In my life, I have known only a few cold fish who could follow “the Kelly criterion” faithfully. They are multimillionaires today. The rest of us are victims, in one way or another, of our emotions.
Some percentage of the populace – it seems like a larger one each year – goes through each day with an abiding sense of entitlement. For whatever reason, this type of person has become convinced that that the universe owes him a living – or is at least obliged to treat him fairly. He experiences each small run of good luck as so much manna, showered down upon him from heaven by the gods. Intoxicated by this confirmation that he is one of the chosen, he almost always overbets, going double or triple Kelly. Or he bets with no advantage at all, secure in the “knowledge” that his luck will continue to run hot. Not that he needs good luck to overbet. When he goes on a losing streak, he reacts to his bad luck with indignation. Sure that things will “break even,” that he is “due” (indeed overdue) for a reversal, he presses his bets. Called “steaming” (to evoke the image of the irate, red-faced gambler with smoke coming out of his ears) this anger-entitlement-motivated form of overbetting is a sure recipe for disaster.
On the opposite side of the coin are the pessimists, paranoids and cynics: the people who believe that every game is fixed, all the dice are loaded that the wise guys control everything, that you can’t fight city hall, that the universe is out to get you. Most of these citizens will never find the courage to bet on anything, no matter good it looks; but some of them do, surprisingly, find their way into the gambling arena. But there they are, of course, chronic underbettors. Ask them to offer a million-to-one odds that the sun will come up tomorrow, and they will recoil in horror. “Nothing is worth a million-to-one,” you will hear them snarl. “I’ll give you ten-to-one for ten dollars.”
The Average Backgammon Money Player
Most of us, to be sure (whether we incline to the sunny or shady side of the street) fall somewhere well away from these two extremes. But even among normal, balanced personalities, anomalies in betting behavior (i.e. deviations from Kelly) are the rule, rather than the exception. The reason for this, I believe, is that the need to understand probabilities is a quite recent development in the course of human history – so recent, in fact, that natural selection has not yet had a chance to favor those with the brain circuitry able to handle these problems. In my next article I shall examine some of the normal irrationalities of the average gambler and suggest a few exercises that he can use to make himself more Kelly-ready.