Section 7: Contrasting Flat Betting with Kelly Betting
By now, hopefully you have a basic understanding of the basic concepts that define optimal bet sizing. We are aiming for long term growth, a reduction of risk, and an intelligent relationship between sizes. We have discussed the Kelly Criterion as a principle for forming the basis of sizing but have shown the various dangers relying on pure KC creates.
Furthermore, consider that using Kelly growth (versus flat betting) increases your expected value over a given sample, but also decreases your chances of being a net winner. Consider an extremely basic example where you bet on two consecutive games that you have a 55% chance of winning. Flat Bettor has $10,000 and wagers $500 per game, while Kelly Bettor has $10,000 and decides to bet 1/2 Kelly sized wagers, and bets 5% of his current bankroll per game. Here is the outcome distribution:
| Flat | ||
|---|---|---|
| 2-0 | 0.303 | $1,000.00 |
| 1-1 | 0.248 | $0.00 |
| 0-2 | 0.203 | -$1,000.00 |
| Net EV | 100.00 | |
| Win % | 0.303 | |
| Kelly | ||
| 2-0 | 0.303 | $1,025.00 |
| 1-1 | 0.248 | -$25.00 |
| 0-2 | 0.203 | -$975.00 |
| Net EV | 106.44 | |
| Win % | 0.303 | |
Notice that the Kelly Bettor does better on the extremes (wins $25 more when he goes 2-0, loses $25 less when he goes 0-2), and that the flat bettor does better in the middle (breaks even at 1-1, while the Kelly bettor loses $25). The math behind the Kelly Bettor losing despite going 1-1 is simple: $X * 1.05 (win) * 0.95 (loss) = 0.9975 $X.
Also note that although the Flat bettor wins more often, the Kelly Bettor has a slightly higher expected value. The Kelly bettor expects to win $106.44, while the flat bettor expects to win $100 from these bets. Therein, as they say, lies the rub – betting Kelly is more volatile and will cause you to win less frequently, but you will perform much better on both extremes, and have a slightly higher EV.
Consider the same scenario, over 5 bets:
| Flat | ||
|---|---|---|
| 5-0 | 0.050 | $2,500.00 |
| 4-1 | 0.206 | $1,500.00 |
| 3-2 | 0.337 | $500.00 |
| 2-3 | 0.276 | -$500.00 |
| 1-4 | 0.113 | -$1,500.00 |
| 0-5 | 0.018 | -$2,500.00 |
| Net EV | 250.00 | |
| Win % | 0.593 | |
| Kelly | ||
| 5-0 | 0.050 | $2,762.82 |
| 4-1 | 0.206 | $1,547.31 |
| 3-2 | 0.337 | $447.57 |
| 2-3 | 0.276 | -$547.44 |
| 1-4 | 0.113 | -$1,447.68 |
| 0-5 | 0.018 | -$2,262.19 |
| Net EV | 252.51 | |
| Win % | 0.593 | |
Once again, Kelly Betting has a higher EV ($252.51 compared to $250.00). Again, it does better at the extremes, winning $262 more when it goes 5-0 and $47 more when it goes 4-1, and losing $238 less when it goes 0-5 and losing $53 less when it goes 1-4. Flat betting performs better in the middle outcomes, winning $53 more at 3-2 and losing $47 less at 2-3. Thus, the risk/reward of Kelly becomes clearer. Exactly 61.3% of the time, over a five bet sample, flat betting would do better than Kelly betting. However, in the 38.7% of the time that Kelly betting does better, it does so by much larger margins (especially in the 6.8% of 5-0 and 0-5 outcomes).
Once the sample sizes grow larger, the differences become more pronounced. Kelly betting becomes more +EV, but the percentage of the time that it is better overall grows smaller. With a 21-20 record over a 41-game sample at $500 a game with no juice, a flat bettor is +$500, but a Kelly bettor is actually -$12 despite the winning record! But with an unusually good record of 30-11 over that same sample, the Kelly bettor is +$14,583 while the flat bettor is only +$9,500.
In conclusion, bettors who are more concerned with maximizing their win probability in a given season, especially considering that one season’s worth of games is a small enough sample that the extra EV from Kelly betting is quite small, would be well advised to pursue a very flat betting structure. Bettors who are intending to invest for a longer period of eight seasons or more might want to pursue progressive wagers at a fractional Kelly.