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April-15-2008,
The Complete Square's Guide to Sports Wagering: The Kelly Criterion...By Jay Graziani

For most gamblers, the primary hurdle to winning comes in selecting winning bets. If you are playing at a long-term disadvantage, you cannot turn a profit regardless of any form of "money management". But once an edge is found, the question becomes how best to utilize that advantage. In other words, how much should you bet? While most gamblers are best served with a simple, fixed-wager size approach, that is not the most efficient way to utilize an advantage. For more complex money management, many gamblers turn to the "Kelly criterion".

John Kelly, while working for AT&T's Bell Labs, published a paper in 1956 entitled "A New Interpretation of Information Rate". The idea drew from recent advances in information theory and dealt with the hypothetical case of a gambler with a "private wire" through which he received horse racing tips. The ideas in Kelly's paper have been adapted by money-makers in numerous industries, from blackjack card counters to Wall Street wizards. For a thorough background and history of the Kelly criterion, "Fortune's Formula" by William Poundstone is worth a read.

Kelly's work focused on how to maximize an information advantage. Let's say that you know that a particular team is 60% likely to win a game, but a bet on them pays even money. That's a hefty advantage, but you shouldn't bet everything you have on it because there is still a 40% chance that the team will lose and you'll go broke. On the other hand, you shouldn't only bet $1 on it because your information is clearly worth more than that and you are wasting your advantage by not cashing in on it. So how much should you bet?

Kelly showed that the answer depends on two factors: the size of your bankroll (how much you have to invest, or alternatively, the amount you can afford to lose) and the accuracy of your information (the size of your edge). The Kelly criterion maximizes long-term bankroll growth when an information advantage exists. It is essentially a formula that relates information to money.

Kelly bet sizes are easy to calculate, simply your advantage divided by the payout odds. When your advantage is zero (or negative), your optimum Kelly bet is zero - you shouldn't bet. You can determine your advantage by multiplying your expected win% by the odds and then subtracting your losing percentage. Odds need to be expressed as the potential winnings in "decimal style", the amount you would win by risking 1. Odds of -110 convert to 0.91, +300 equals 3, and -200 is expressed as 0.5. For -110 bets, the odds are 0.91, so a 55 % win rate would give you an advantage of (0.55 X 0.91) - 0.45, for an advantage of 0.05. Dividing your advantage (0.05) by the odds (0.91) gives you the percentage of your bankroll to wager under Kelly, 5.5 % in this example. In Excel, if you insert your winning % in cell A1 and your payout odds in cell A2, the Kelly formula is [((A1*A2)-(1-A1))/A2].

Thus, Kelly bet sizing does not only depend only on edge, but also on the chance that your ticket will cash. A 5% edge on a -110 bet directs you to bet 5.5% of bankroll. But having that same 5% edge on a +300 bet should only cause you to bet 1.7% of bankroll - the chance of success is less, so your bet size is reduced. Likewise, a 5% edge on a 10-to-1 favorite would direct you to bet 50% of bankroll, since you have an advantage that will be realized greater than 90% of the time. When two bets have the same advantage betting on the one most likely to win is obviously the better investment.

Kelly is a proportional betting system. This means you are betting a percentage of your bankroll on each wager, so as your bankroll grows, so does your bet size, creating a compounding effect. This makes Kelly a very aggressive money management system. Theoretically a gambler can never "go broke" with Kelly, as you are always only betting a potion of your bankroll. But that is not very reassuring when your bankroll has dropped to 5% of its original size and you are now betting pennies per game after experiencing an extended losing streak.

Following standard Kelly bet sizing leaves you a 50% chance of dropping to half of your bankroll, and a 10% chance of dropping to one-tenth of your bankroll. That is a lot of downside to stomach for most gamblers. For sports betting there is the added complication that the true odds on an outcome are not known. When plugging your win percentage into Kelly, your estimate may differ from the true odds enough to significantly affect the optimal bet sizing. The difficulty in determining completely accurate odds in sportsbetting cannot be understated.

Both under- and overbetting the Kelly optimum provides a lessened rate of return. Underbetting provides a more "steady" growth, but that unfortunately also leads to lower profits. Overbetting can be fatal with Kelly, as betting twice the optimal Kelly bet results in almost no long-term growth. For a 55% handicapper playing into -110 lines, Kelly would call for a bet of 5.5% of bankroll, for a 57% handicapper, 9.7%. Intuition tells any experienced gambler that this is just too much to be risking on something as volatile as sports, where your true edge may differ substantially from your expectations. Following a 57% system with bets worth 10% of your total funds would be gambling suicide. Overbetting is penalized more severely than underbetting under Kelly, so most gamblers prefer to take the conservative path by using fractional Kelly bets.

Half Kelly is generally considered a safe route, and while profits are theoretically lessened, the lower volatility makes it a fair tradeoff. Betting half Kelly returns about 25% less bankroll growth than full Kelly, but with much less volatility. For a 55% handicapper, half Kelly translates to about 2.8% of bankroll, hence the standard gambling wisdom that 2-3% of bankroll should represent a typical wager.

Most bettors are probably best served by using a flat 2% of bankroll per play, since figuring edges in sports is often difficult. For a season-long win rate of 55%, a good target for most bettors, this represents a little more than 1/3 Kelly, a conservative compromise between risk and return. Stepping out to 3%, or more rarely, 4% on an especially good play is reasonable. More experienced handicappers, with a good conception of the downsides of Kelly and above-average ability in estimating betting advantages, may wish to take the more aggressive Kelly approach in maximizing returns.

"The Complete Square's Guide to Sports Wagering" is a recurring series aimed at educating novice sports bettors. The next article will take a look at evaluating win-loss records for significance.

4-15-08
Jay Graziani
MajorWager.com
graziani@majorwager.com

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