Suppose: 1) you have an advantage in a sequence of games, 2) you bet an optimal proportion of your bankroll on each individual game, and 3) you win approximately the expected proportion of the games. When you compare your net win to your total action (total bets) you will discover that you have won at about half the expected arithmetic rate.

Mathematical Development If you have played blackjack to support yourself, you have probably tried proportional betting -- that is betting a proportion of your bankroll depending on your advantage. If you have been reading my publications, you have probably tried betting a proportion equal to your advantage divided by the variance of possible outcomes. This is a smart thing to do for two reasons. First you won't go broke with proportional betting -- on a bad streak your bankroll will shrink, but it will not disappear. Second, your bankroll will grow at the maximum rate when you use optimal proportional betting.

Definition of optimal proportional betting. Maximization of the logarithm of wealth, sometimes called the Kelly criterion (Kelly, J. L. "A New Interpretation of Information Rate," Bell System Technical Journal, Vol. 35, pp 917-926, 1956), is generally recognized as the best betting system.

[Equation 1]

where

G = exponential rate of growth of bankroll,

B₀ = initial bankroll,

B_n = bankroll after n plays.

	[Equation 2]

where

f = fraction of bankroll bet on each game.

m = the number of different payoffs

x_i = a particular payoff, expressed as a multiple of bet size. (Example: one particular x_i has the value 1.5 -- when you get an untied natural.)

w_i = actual number of wins of payoff x_i.

[Equation 3]

[Equation 4]

where p_i = probability of payoff x_i.

Optimal bet size. To find the optimal bet size, maximize G with respect to f. Optimal bet size turns out to be the expected arithmetic win rate divided by the sum of the squares. For small expected win rates, such as you have in blackjack, the denominator is approximately equal to the variance. This result first appeared in Professional Blackjack, 1975 edition, pages 78-79.

[Equation 5]

where f^* = optimal bet size as a fraction of bankroll.

For games in which the only possible payoffs are +1 and -1, the denominator of equation 5 equals unity, and optimal bet size equals bankroll times expected win rate.

Break-even point on overbetting. Set G equal to zero and solve for f. There are two solutions, of which the trivial one is f = 0. The other solution is two times optimal bet size, as was first noted in Professional Blackjack, 1975 edition, page 83.

f0 = 2f*

[Equation 6]

where f₀ = bet size at which exponential rate of growth of bankroll is zero.

Rate of return with optimal bets. To find your rate of return when you bet the optimal proportion of your bankroll on each game, first find the antilogarithm of the exponential growth rate.

r = eG - 1

[Equation 7]

r* = eG* - 1	[Equation 8]

where

r = rate of return on bankroll under proportional betting.

e = base for natural logs; base 10 gets the same results.

G^* = maximum exponential rate of growth of bankroll.

r^* = maximum rate of return on bankroll under proportional betting.

[Equation 9]

[Equation 10]

Equation 10, r^*/f^*, is the rate of return on action under proportional betting. The derivation of Equation 10 is messy, but can be done with the aid of Taylor's formula. Equation 10 says that when you bet the optimal proportion of your bankroll, your expected win divided by your bet size is half of your expected arithmetic win rate.

This disadvantage of proportional betting comes as no surprise to anyone who has ever used the method. Under proportional betting, when you lose you have to scale down your bet sizes; you then have to win more bets to get even than what you lost in going down. You can break even or win in terms of number of hands won, yet lose in terms of dollars. If a win causes you to raise your bets and a loss causes you to lower your bets, any win-lose or lose-win sequence costs you money.


Examples
Example 1. Suppose you start with $10,000, and are going to bet 1% of your bankroll each game. On the first game you bet $100. If you win the first game, you bet $101 on the second game. If you lose the first game, you bet $99 on the second game. A win followed by a loss or a loss followed by a win will leave you $1 poorer after the two-game set. You broke even in games, but you lost money. That happens with proportional betting, but not with flat betting.

Example 2. Suppose you play a game with payoffs +1 and -1. Suppose you think you have a 1% edge, and thus bet 1% of your bankroll every game. Suppose you play 200 games. Your expectation is to win 101 games and lose 99, for a net win of 2 games. If you actually win exactly 101 games out of the 200, your ending bankroll will be only one bet larger than your beginning bankroll.

B₂₀₀ = (1.01)¹⁰¹ * (.99)⁹⁹ * B₀

B₂₀₀ = 1.01 * B₀

Table 1 shows the ratio of final bankroll to beginning bankroll for various numbers of wins out of the 200 games. Note that you are better off with proportional bets than with flat bets (all bets equal to your first bet, and your first bet is 1% of your beginning bankroll) at 108 or more wins, and also with 93 or fewer wins. For 94 to 107 wins, flat betting woud outperform proportional betting. The difference is as much as half of the arithmetic expectation.

The exact amount of your action will depend on your sequence of wins and losses, but it will likely be around 200% of your initial bankroll. Thus, if you exactly hit your expectation by winning 101 out of 200 games, your net win divided by your action will be 0.5%.

Equation 6 predicts that if your actual number of games won is half of the expected number, then you should break even with optimal proportional betting. Winning at half the expectation for this example would mean a net win of one game out of 200. Doing the math for 100.5 wins and 99.5 losses yields a final to initial bankroll ratio of 1.0000.

As you can see from Table 1, when your actual number of wins is in the vicinity of your expected number of wins, proportional betting costs you about half of your arithmetic expectation. You can think of this as being the premium you have to pay for the insurance against going broke that you get with proportional betting.

TABLE 1
FINAL TO INITIAL BANKROLLS -- FOR 200 GAMES
wins-losses	B200/B0 flat bets	B200/B0 proportional bets
110- 90	1.20	1.2093
109- 91	1.18	1.1853
108- 92	1.16	1.1618
107- 93	1.14	1.1388
106- 94	1.12	1.1163
105- 95	1.10	1.0942
104- 96	1.08	1.0725
103- 97	1.06	1.0513
102- 98	1.04	1.0305
101- 99	1.02	1.0101
100-100	1.00	0.9900
99-101	0.98	0.9704
98-102	0.96	0.9512
97-103	0.94	0.9324
96-104	0.92	0.9139
95-105	0.90	0.8958
94-106	0.88	0.8781
93-107	0.86	0.8607
92-108	0.84	0.8437
91-109	0.82	0.8270
90-110	0.80	0.8106

Example 3. This example shows the relationship between expected arithmetic rate of return and rate of return on action for a proportional bettor. Suppose you play a sequence of five games, each of which has payoffs of +1 or -1. Suppose you expect to win three out of five, which is a 20% edge, so you bet 20% of your bankroll on each game.

There are 32 possible sets of outcomes, ranging from five consecutive wins to five consecutive losses. Table 2 lists all 32 possible sets of outcomes, the amount you would win for each set, your action for each set, and your win divided by your action for each set. It also gives the probability of each set occurring, given that the probability of win is 0.6.

Here is a sample calculation. Suppose you win the first two games, lose the next two, and win the final game.

 

Game	Beginning Bankroll	Bet	Outcome	Final Bankroll
1	1.0000	0.20000	win	1.20000
2	1.2000	0.24000	win	1.44000
3	1.4400	0.28800	lose	1.15200
4	1.1520	0.23040	lose	0.92160
5	0.9216	0.18432	win	1.10592
		_______
Total		1.14272

In the sample calculation, your final bankroll is 10.592% higher than your initial bankroll, and your wagers total 114.272% of your initial bankroll. You thus made about 9.3% on your wagers (10.592/114.272), which is about half of your expected arithmetic win rate of 20%.

Note that all the overall totals in Table 2 behave the way you expect them to. The overall totals have been calculated by multiplying each outcome by its probability and summing. The overall net win is exactly 20% of the overall action, as you expect. A disproportionate amount of the total win comes from the set with five consecutive wins.

Now look at those ten sets with three wins out of five games. These sets contain the expected number of wins, yet their rates of return on action are about half of the expected arithmetic win rate. A 20% edge and a "normal" outcome of three wins out of five means about a 10% return on action to someone who bets the optimal proportional amount of the bankroll on each game.

This is the general rule and not just the special case. It occurs with all schedules of payoffs -- not just +1, -1 games. It occurs at blackjack. You might be sufficiently skillful to be playing with say a 1% expected arithmetic win rate; but if you are betting the optimal proportion of your bankroll, after a "normal" number of wins in relation to losses you should be ahead by about 0.5% of your action.

TABLE 2
ALL POSSIBLE SETS OF 5 GAMES
Outcome	Prob.	Win	Action	Win/Action
WWWWW	0.07776	1.48832	1.48832	1.00
WWWWL	0.05184	0.65888	1.48832	0.44
WWWLW	0.05184	0.65888	1.35008	0.49
WWLWW	0.05184	0.65888	1.23488	0.53
WLWWW	0.05184	0.65888	1.13888	0.58
LWWWW	0.05184	0.65888	1.05888	0.62
WWWLL	0.03456	0.10592	1.35008	0.08
WWLWL	0.03456	0.10592	1.23488	0.09
WWLLW	0.03456	0.10592	1.14272	0.09
WLWWL	0.03456	0.10592	1.13888	0.09
WLWLW	0.03456	0.10592	1.04672	0.10
WLLWW	0.03456	0.10592	0.96992	0.11
LWWWL	0.03456	0.10592	1.05888	0.10
LWWLW	0.03456	0.10592	0.96672	0.11
LWLWW	0.03456	0.10592	0.88992	0.12
LLWWW	0.03456	0.10592	0.82592	0.13
WWLLL	0.02304	-0.26272	1.14272	-0.23
WLWLL	0.02304	-0.26272	1.04672	-0.25
WLLWL	0.02304	-0.26272	0.96992	-0.27
WLLLW	0.02304	-0.26272	0.90848	-0.29
LWWLL	0.02304	-0.26272	0.96672	-0.27
LWLWL	0.02304	-0.26272	0.88992	-0.30
LWLLW	0.02304	-0.26272	0.82848	-0.32
LLWWL	0.02304	-0.26272	0.82592	-0.32
LLWLW	0.02304	-0.26272	0.76448	-0.34
LLLWW	0.02304	-0.26272	0.71328	-0.37
WLLLL	0.01536	-0.50848	0.90848	-0.56
LWLLL	0.01536	-0.50848	0.82848	-0.61
LLWLL	0.01536	-0.50848	0.76448	-0.67
LLLWL	0.01536	-0.50848	0.71328	-0.71
LLLLW	0.01536	-0.50848	0.67232	-0.76
LLLLL	0.01024	-0.67232	0.67232	-1.00
Total	1.00000	0.216653	1.083265	0.20