Generalizing the Kelly Betting Strategy to Allow for

Generalizing the ‘Kelly’ Betting Strategy to Allow for Multiple Payouts Tristan Barnett Victoria University Alan Brown Swinburne University

All American Poker With perfect play, the total return is 100. 72% - a winning strategy exists!

Analysis of Casino Games A casino game can be defined as follows: There is an initial cost C to play the game. Each trial results in an outcome Oi, where each outcome occurs with profit xi and probability pi. The condition ∑ pi=1 must be satisfied. The percent house margin (%HM) is then -∑Ei/C. The total return is 1+∑Ei/C.

Classical ‘Kelly’ formula Ref: Finding the Edge (2000) Consider a game with two possible outcomes: win or lose. Suppose the player profits k units for every unit wager. Further, suppose that on each trial the win probability p is constant with p+q=1. If kp-q>0, so the game is advantageous to the player, then the optimal fraction of the current capital to be wagered is given by b=(kp-q)/k. Outcome Profit Probability Expected Profit Win $2 0. 35 $0. 70 Lose -$1 0. 65 -$0. 65 Example:

Analysis of Classical ‘Kelly’ The bank B(1) = B(0) * (1+kb) with probability p B(1)= B(0) * (1 - b) with probability q If player attempts to maximise f(b) = E[ B(1 )/ B(0) ] = 1 + (kp –q) b then f′(b) = kp –q > 0 for 0 ≤ b ≤ 1 so solution is b = 1 Great in the short term, but stupid long term. Feller (1966) shows that we minimize ruin by making a minimum bet on each trial, but this also minimizes the expected gain. We can avoid these solutions if we change the objective.

Analysis of Classical ‘Kelly’ As before the bank is B(1) = B(0) * (1+kb) with probability p B(1)= B(0) * (1 - b) with probability q 0. 1 0 g(b) b* 1 b If player attempts to maximise -0. 1 g(b) = E[ log( B(1 )/ B(0)) ] = p log(1+kb) + q log (1 -b) -0. 2 then for 0 ≤ b < 1 g′(b) = pk/(1+kb) –q/(1 -b) -0. 3 2 2 2 g′′(b) = - pk /(1+kb) –q/(1 -b) Observe that g′(0) =kp-q > 0 and g′′(b) <0 in the interval so g(b) have a unique maximum in 0 ≤ b < 1 given by b* = (pk-q)/ (1+k) -0. 4 -0. 5 Note how the asymptote in the objective function at b=1 forces the solution b* < 1.

All American Poker With perfect play, the total return is 100. 72% - a winning strategy exists!

All American Poker Outcome Profit ($) Probability Expected Profit ($) Royal Flush 799 1 in 43, 450 0. 018 Straight Flush 199 1 in 7, 053 0. 028 Four of a Kind 39 1 in 444 0. 088 Full House 7 1 in 91 0. 077 Flush 7 1 in 64 0. 110 Straight 7 1 in 54 0. 129 Three of Kind 2 0. 069 0. 138 Two Pair 0 0. 120 0. 000 Jacks or 0 0. 183 0. 000 %HM= -0. 0072/1 = -0. 72%, Return = 100. 72% Better Nothing -1 0. 581 -0. 581

‘Kelly’ with Multiple Payouts Assume a constant proportion b of the bank is bet, with m discrete finite mixed outcomes. Let B(1) / B(0) = 1+ ki b probability pi for i = 1 to m. Assume the player wishes to maximize g(b) = E[log(B(1) / B(0)) ]. Without loss of generality let k 1 be the maximum possible loss. In the interval 0< b <-1/k 1 1 + ki b > 0 since ki ≥ k 1 for i = 1 to m, so the logarithm of each term is real.

‘Kelly’ with Multiple Payouts (a) g(0) = 0, (b) g′(0) > 0 follows directly from the requirement for a winning strategy, and (c) g′′(b) < 0 for 0<b<-1/k 1 (where k 1 is the MPL) so the first derivate has at most one zero in this interval. Hence whenever there is a winning strategy, the force of growth has a unique maximum given by the root of 0. 1 0 -0. 1 g(b) -0. 2 -0. 3 -0. 4 -0. 5 b* -1/k 1 b

‘Kelly’ with Multiple Payouts Theorem 1: Consider a game with m possible discrete finite mixed outcomes. Suppose the profit for outcome i is ki with probability pi for 1 ≤ i ≤ m , where at least one outcome is negative and at least one outcome is positive. Then if a winning strategy exists, and the maximum growth of the bank is attained when the proportion of the bank bet at each turn, b, is the smallest positive root of

All American Poker Applying Theorem 1, b = 0. 0307% Example: With a $10, 000 bankroll, Theorem 1 suggests that the player should initially bet $3. 07 On average, the player’s bankroll will grow by g(0. 000307) = 0. 00011%

All American Poker Outcome Profit ($) Probability: $800 Jackpot Probability: $1200 Jackpot Royal Flush Jackpot -1 1 in 43, 450 1 in 35, 848 Straight Flush 199 1 in 7, 053 1 in 6, 999 Four of a Kind 39 0. 00225 Full House 7 0. 01098 0. 01096 Flush 7 0. 01572 0. 01505 Straight 7 0. 01842 0. 01846 Three of Kind 2 0. 06883 0. 06888 Two Pair 0 0. 11960 0. 11954 Jacks or Better 0 0. 18326 0. 18336

All American Poker Suppose a player has a bankroll of $11, 000 and is required to bet $5 hand. What jackpot level is required to maximize long term growth? Jackp ot Return Theorem $11, 00 $17, 00 1 0 0 $250 99. 62% - - $800 100. 72 % 0. 0307% $3. 38 $5. 22 $1, 200 101. 74 % 0. 0468% $5. 15 $7. 96

Blackjack Outcome Profit ($) Probabili ty Expected Profit ($) Win 4 units 4 0. 02 0. 08 Win 3 units 3 0. 09 Win 2 units 2 0. 06 0. 12 Blackjack 1. 5 0. 06 0. 09 Win 1 unit 1 0. 25 Draw 0 0. 1 0 Lose 1 unit -1 0. 40 -0. 40 Lose 2 units -2 0. 05 -0. 1 Lose 3 units -3 0. 02 -0. 06 Lose 4 units -4 0. 01 -0. 04 1. 00 0. 03 Ref: Finding the edge (2000), p 177 “It is easy to verify that when there is a spectrum of favourable situations the same recipe, fi*=pi -qi for the ith situation, holds. Again, in actual blackjack fi * would be adjusted down somewhat for the greater variance. ” Advantage = 3% Applying Theorem 1: b=1. 4%

Spread Betting Ref: Haigh (2000). The Kelly criterion and bet comparisons in spread betting Bets are made on the outcome of a quantity X, such as the number of runs in cricket or the number of games in tennis. The index betting firm offers an interval (c, d), known as the spread. A bettor may choose to buy at unit stake β (β>0), in which case he receives β(X-d), or to sell X at unit stake α (α>0), in which case he receives α(c-X). Stop losses are often used to protect the bettor from bankruptcy, by replacing an unbounded Y, by the variable X = A if Y≤A, X = Y if A≤Y≤B, X=B if Y≥B, where A and B are finite constants For a buy bet g(b) = E[ln{1+(X-d)b/d}] g′(b*)=E[(X-d)(d+u*(X-d)]=0

Spread Betting Outcome Profit ($) Probabili ty Expected Profit ($) 20 - games -6 0. 07 -0. 42 20 - games -1 0. 07 -0. 07 21 games -5 0. 05 -0. 25 21 games -0. 83 0. 05 -0. 04 22 games -4 0. 05 -0. 2 22 games -0. 67 0. 05 -0. 03 23 games -3 0. 08 -0. 24 23 games -0. 50 0. 08 -0. 04 24 games -2 0. 08 -0. 16 24 games -0. 33 0. 08 -0. 03 25 games -1 0. 08 -0. 08 25 games -0. 17 0. 08 -0. 01 26 games 0 0. 08 0 27 games 1 0. 08 27 games 0. 17 0. 08 0. 01 28 games 2 0. 10 0. 2 28 games 0. 33 0. 10 0. 03 29 games 3 0. 10 0. 3 29 games 0. 50 0. 10 0. 05 30+ games 4 0. 23 0. 92 30+ games 0. 67 0. 23 0. 15 1. 00 0. 03 Spread – (24 -26), Stop loss - (20 -30) Return = 115%, Applying Theorem 1, b* = 8. 24%

Decision Theory Source: http: //www. cisiova. com/betsizing. asp In investment we often come to problems like this: Company A is currently researching 3 different new products. In an upcoming convention, we know that A might be going to announce the launch of one of the new products. We can also estimate the impact of different outcomes on the stock price: Launching Product 1: 30% increase in stock price (ROI = 30%). Chance of happening: 20%. Launching Product 2: 10% increase in stock price (ROI = 10%). Chance of happening: 15%. Launching Product 3: 12% increase in stock price (ROI = 12%). Chance of happening: 25%. Failure to launch: 15% decrease in stock price (ROI = -15%). Chance of happening: 40%

Decision Theory F = % of your bankroll that you invest in A Wi = ROI of Product i Pi = Probability of Product i Launching B = Initial Bankroll B' = Future Bankroll after N such investments M = The Geometric Mean of N such investments Using the above information, we can formulate: B' = B * (1 + W 1*F)^(P 1*N) * (1 + W 2*F)^(P 2*N) * (1 + W 3*F)^(P 3*N) * (1 + W 4*F)^(P 4*N) M^N = B'/B = (1 + W 1*F)^(P 1*N) * (1 + W 2*F)^(P 2*N) * (1 + W 3*F)^(P 3*N) * (1 + W 4*F)^(P 4*N) M = [(1 + W 1*F)^(P 1*N) * (1 + W 2 * F)^(P 2*N) * (1 + W 3*F)^(P 3*N) * (1 + W 4*F)^(P 4*N)]^(1/N) M = (1 + W 1*F)^(P 1) * (1 + W 2*F)^(P 2) * (1 + W 3*F)^(P 3) * (1 + W 4*F)^(P 4) Therefore, to maximize the geometric return M, we need to find F such that the

Decision Theory Outcome ROI Profit Probabilit y Expected Profit Product 1 30% 0. 30 0. 20 0. 06 Product 2 10% 0. 10 0. 15 0. 015 Product 3 12% 0. 12 0. 25 0. 03 Failure to launch 20% -0. 20 0. 40 -0. 08 1. 00 0. 025 Return = 104. 5% Applying Theorem 1, b = 65. 4%.

Simultaneous Bets Finding the edge (2000), p 182 Suppose we bet simultaneously on two independent favorable coins with betting fractions b 1 and b 2 and with success probabilities p 1 and p 2. Then the expected growth rate is given by g(b 1, b 2) = p 1 p 2 ln(1+b 2) + p 1 q 2 ln(1+b 1 -b 2) + q 1 p 2 ln(1 -b 1+b 2)+q 1 q 2 ln(1 -b 2) ∂g/ ∂b 1 = 0 and ∂g/ ∂b 2 = 0 b 1+b 2 = (p 1 p 2 -q 1 q 2)/(p 1 p 2+q 1 q 2) ≡ c b 1 -b 2 = (p 1 q 2 -q 1 p 2)/(p 1 q 2+q 1 p 2) ≡ d b 1* = (c+d)/2 b 2* = (c-d)/2

Simultaneous Bets Game 1 Game 2 Profit Pro b Ex. Profit 1 p 1 1 p 2 -1 q 1 -1 q 2 -q 2 1. 00 p 1 -q 1 1. 00 p 2 -q 2 Joint Dist b = (p 1 p 2 -q 1 q 2)/(p 1 p 2+q 1 q 2) Profit Unit Profit Prob Ex. Profit Theorem 1 2 1 p 2 p 1 p 2/(1+b ) 0 0 p 1 q 2+p 2 q 1 0 0

Simultaneous Bets b = (p 1 p 2 -q 1 q 2)/(p 1 p 2+q 1 q 2) (Theorem 1) b 1+b 2 = (p 1 p 2 -q 1 q 2)/(p 1 p 2+q 1 q 2) ≡ c (Thorp, Finding the Edge) Theorem 1 is a generalization of Thorp which is a generalization of Kelly When the distributions are identical for each game (but not necessarily independent) bi* = b / n where n is the number of games played

Simultaneous Bets Game 1 Game 2 Profit Prob Ex. Profit Joint Distribution Profit Prob Ex. Profit Kelly 1 0. 5100 0. 51 2 1 0. 2601 0. 26 0. 25 -1 0. 4900 -0. 49 0 0 0. 4998 0. 00 0 1. 00 0. 02 -2 -1 0. 2401 -0. 24 -0. 25 1. 00 0. 02 9 E-07 b* Unit Profit Prob 2. 00% b* 2. 00% c 0. 03998 d 0. 0000 b* (1) 1. 999% b*(2) 1. 999% Total 3. 998% b*=Total 3. 998%

Simultaneous Bets Game 1 Game 2 Profit Prob Ex. Profit Joint Distribution Profit Prob Ex. Profit Kelly 1 0. 5100 0. 51 1 0. 5500 0. 55 2 1 0. 2805 0. 28 0. 251 -1 0. 4900 -0. 49 -1 0. 4500 -0. 45 0 0 0. 4990 0. 00 0 1. 00 0. 02 1. 00 0. 10 -2 -1 0. 2205 -0. 22 -0. 25 1. 00 0. 06 3 E-08 b* Unit Profit 2. 00% b* 10. 00% c 0. 11976 d -0. 0802 b* (1) 1. 980% b*(1) b*(2) 9. 996% b*(2) Total 11. 976% b*=Total 11. 976%

All American Poker Suppose a player is dealt JS, JH, TH, AH, 3 C. Jackpot = $799 Jackpot = $802 Hold Profit Hold Profit JJ___ 0. 408 1 JJ__ _ 0. 408 1 _JTA_ H 0. 407 _JTA 0. 409 8 TP T 0 N _ J+ 3 K S F FH 4 K SF R JJ___ 1621 5 0 1155 9 259 2 185 4 0 0 165 45 0 0 _JTA _ 1081 78 6 207 21 7 15 44 0 0 0 1 Should a player deviate from maximizing the expected payout in order to maximum the long term growth?

All American Poker Outcome Hold: _JTA_ Hold: JJ___ Profit ($) Probability Ex. Profit ($) Probability Ex. Profit ($) Royal Flush 802 1 in 42, 246 0. 0190 1 in 42, 412 0. 0189 Straight Flush 199 1 in 7, 049 0. 0282 Four of a Kind 39 1 in 444 0. 0877 1 in 444 0. 0878 Full House 7 1 in 91 0. 0768 Flush 7 1 in 64 0. 1102 1 in 64 0. 1101 Straight 7 1 in 54 0. 1290 1 in 54 0. 1290 Three of Kind 2 0. 06877 0. 1375 0. 06878 0. 1376 Two Pair 0 0. 11952 0. 0000 0. 11953 0. 0000 Jacks or 0 0. 18292 0. 0000 0. 18298 0. 0000

All American Poker Hold Ex. Profit b* Ex. Bank Return JJ__ _ 0. 00729 09 0. 00030 47 100. 00021 9% <100. 7221 % _JTA _ 0. 00729 10 0. 00030 40 100. 00021 8% 100. 7221 % HI-OPT 1 STRATEGY: 4 Deck Pair Splitting Dealer Up Card You Hold 4 5 6 10, 10 +6 +4 +4

Game Theory Colin A Colin B Rose A 2/3 -1 Rose B 0 1 Outcome • 1/4 as the value of the game • 3/8 A, 5/8 B as Rose’s optimal strategy • 3/4 A, 1/4 B as Colin’s optimal strategy Profit ($) Probabilit y Ex. Profit ($) Theorem 1 Bank Ex. Bank Rose A / Colin A 2/3 9/32 3/16 0. 1342 1. 3974 0. 3930 Rose A / Colin B -1 3/32 -0. 2321 0. 4040 0. 0379 Rose B / Colin A 0 15/32 0 0. 0000 1. 0000 0. 4688 b* = 0. 5960 Rose B / 1 5/32 0. 0979 1. 5960 0. 2494 Colin B Is it possible for either player to alter strategies to increase/decrease the expected bank 1 1/4 0. 00 114. 90%

Further Work Mathematical properties of Theorem 1 Changing strategies to maximize long term growth Simultaneous bets Calculating blackjack probabilities Applying Theorem 1 to other favorable games Game Theory

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