Risk Ajustment for Poker Players William Chin DePaul University, Chicago, Illinois Marc Ingenoso Conger Asset Management LLC, Chicago, Illinois September, 2006 Introuction In this article we consier risk aversion for avantage poker players. We iscuss a sense in which poker players can proportionally Kelly-bet. This results in a metho of prescribing bankroll requirements epening on risk aversion. It also provies a companion to the results of [CI] (in this volume) whose formulæ will apply to poker players exercising bankroll management an stake selection as prescribe. Employing a continuous iffusion moel as in [CI], we show that if stakes are chosen in accorance with bankroll as prescribe, then the bankroll will follow the same stochastic process as for a fractional Kelly bettor with unit variance. One s risk-aversion can be parameterize by an instantaneous risk of ruin, or equivalently by a Kelly fraction k, which in turn correspons to a utility function. The Kelly fractions we recommen are ones commonly use by blackjack professionals. In the secon section we use the concept of Certainty Equivalent (CE) to quantify how big a pot must be in orer to call a bet with a rawing han, epening on risk aversion. This makes precise a common assertion that one shoul forgo borerline positive expectation wagers that have high variance. As one might expect, except for longshot raws an low bankroll situations, the effect of risk-aversion is small but not negligible. We solve for the CE break-even points exactly for a range of Kelly fractions an provie numerical computations. Benefitting from the brevity of the one-half Kelly CE break-even formula, we obtain a simplifie formula to approximate CE break-even points, in the last subsection. Our investigation is base on limit holem. With sufficient ata the same consierations shoul apply to other wagering, such as no-limit holem 1
an tournaments. The excess pot os values are analogous to risk-averse playing inices in blackjack. 1 Proportional Betting for Poker Players A common rough benchmark for limit holem players (at least up to mistakes) is an hourly expectation of 1 big bet an a variance of 100 big bets square. Thus the unitless ratio of hourly expectation to hourly stanar eviation is 10. While your mileage may vary, we shall see that for our analysis, the ratios of expectation to variance an expectation to stanar eviation are what matter. Some games may have higher expectation, but often their variance is higher, an the reverse may occur. We moel the benchmark situation as Brownian motion with linear rift r = 1 (big bets/hour) an hourly stanar eviation s = 10 big bets. Poker players o not resize their bets as they o in blackjack. But they essentially choose their betting levels by which game they play. Let us assume a fixe ratio of expectation to stanar eviation equal to a positive constant r/s. We moel this game as Brownian motion with linear rift as earlier in this section. The resulting risk of ruin is given in [CI, 3.3 Corollary 3] as exp( 2xr/s 2 ) where x is the bankroll. Now let us assume that the player s risk tolerance is specifie by her Kelly fraction k, an their attenant risk of ruin exp( 2/k). Setting the two risks of ruin equal, we get i.e. xr/s 2 = 1/k x = s 2 /kr This says that the k times Kelly player ieally has a bankroll always equal to s 2 /kr. We note for our benchmark (s 2 /r = 100) game: An optimal (full Kelly) player ieally always choose a game where the bankroll is 100 big bets, an this gives the optimal geometric growth rate. 2
As we note in [CI, section 1], betting with k > 1 is always suboptimal. Therefore is always wrong to have less than s 2 /r (=100 big bets in 1 benchmark) as a bankroll. A 3 Kelly player will have 300 big bets. The more conservative quarter-kelly players nee 00 big bets, etc. This gives an answer to bankroll requirements for holem players in terms of fractional Kelly betting. The range of answers seem be higher than bankroll recommenations commonly given (sometimes by erroneous reasoning) by poker experts, e.g. [Mal], www.twoplustwo.com. Overall, more conservative money management is recommene. Important reasons for scaling back k for poker versus blackjack inclue the relative lack of certainty about r an s. Using a value a k = 1/6 an a bank of 600 big bets woul not seem unreasonable. As we pointe out above, even smaller values of k may be appropriate. For an iniviual, the bankroll is one s total net worth minus expenses (incluing the present value of future earnings). However, many efy this efinition an play with a gambling bankroll or what they can affor to lose. In such artificial cases, playing closer to the optimal k = 1 may be recommene. As a practical matter, suppose one settles at a risk tolerance of say aroun k = 1/6 or k = 1/5 an is playing a game with 600 big bets. If a losing streak of 100 big bets occurs, then it is time to consier scaling back to a smaller game. Although the resizing is not perfect, one can operate within certain risk tolerance bouns. One s bankroll B with constant rate of return r an stanar eviation s (for a session of fixe length) is moele by the stochastic equation B = rt + sw, which results in Brownian motion with linear rift at rate r. Mathematically, the player that chooses stakes accoring to his or her bankroll as we have just prescribe is equivalent to a fractional Kelly bettor. Precisely, the equivalence is given by the following observation, whose proof is a simple algebraic simplification. Conclusion 1 Consier the equation B = rt + sw where W is a stanar Wiener process. If µ = r/s is a positive constant an stakes are always chosen so that B = s 2 /(kr), then the equation is the same as B = kµ 2 Bt+kµBW. The bankroll will thus follow the same process as the fractional Kelly bettor. The latter equation is our iffusion for k times Kelly, with unit, as iscusse in section 1 in [CI]. Proof. Set µ = r s 3
an assume that this ratio is constant. A player who continuously ajusts stakes so that B = s2 kr satisfies B = r B B t + sb B W = r kr Bt + s kr BW s 2 s 2 ( r ) 2 r = k Bt + k s s BW = kµ 2 Bt + kµbw This is proportional betting paraigm for fractional Kelly betting, with growth rate µ = r s an σ = 1. 2 Micro Risk Ajustment A basic wager is to win a pot of p bets, risking a call of 1 unit where the rawing os are 1 :, i.e., the wager will be won with probability 1 +1. The expecte value of the wager is p + 1 + 1 = p + 1. Let x = p the excess pot os. It is ruimentary that p = (i.e. x = 0) is the break-even point for expectation. The Certainty Equivalent (CE) of a wager X is the risk-free value that has utility equal to the expecte utility of X. CE is use as a risk-ajuste way of comparing bets, epening on the choice of utility function u(t). We use utility functions u(t) = t1 1 k 1 1 k which correspon to the Kelly fraction k with 0 < k < 1. Full Kelly betting at k 1 correspons to u(t) = ln t. We CE equal to zero an solve for x using various values of, k an B. It is important to keep in min that the values of B an x are measure in units of bets. So if when the (limit holem) pot is raise your bankroll is numerically halve. The values of x give the excess pot size require to break even in CE, in excess of what woul be require by pure expectation. Thus, to break even in CE the pot size nees to be p = x +.
2.1 k = 1 The CE break-even point for logarithmic utility is expresse by v(u(b 1) + 1 + u(b + x + )( 1 + 1 )) = B where u(t) = ln t an v(t) = exp t,which has exact solution for x: 2.2 k =.5 f(, B) = x = B exp ( (ln(b 1) + (ln B) ) B We set u(t) = t 1 = v(t) an obtain the relatively simple CE break-even point function ( + 1) f(x, B) = B 1. 2.3 k = 1/3 Let u(t) = 1/t 2 an v(t) = ( t) 1 2. quaratic break-even equation is The meaningful solution to the x = 1 2 (1 + 2B + B 2 2B) (2B 2 2B 2 2 + 2B 2 + B 2 + B 2 2B 3 2 (B 2 + B 2 B 3 + 5B B 3 + 6B 2B 5 B 5 + B 6 )). 2. k = 1/ Let u(t) = 1/t 3 an v(t) = ( t) 1 3. The meaningful solution to the cubic break-even equation is B x = + 1 + 3B 2 3B B 3 + 3B 2 3B ( ( B 3 + 3B 2 3B + 1 ) ( + 1 + 3B 2 3B B 3 + 3B 2 3B ) ) 2 1 3 B 5
2.5 Break-even tables with fixe k Using the break-even formulae above we tabulate values of x. The values of are chosen to reflect some holem rawing os. E.g. = 6 is a longshot raw to a single out, whereas = is approximately for a straight or flush raw. k = 1 6 B = 25. 35 1. 2 3. 1. 0 1. 92. 5 B = 50. 208. 596 1. 3. 08 5. 98 30.6 B = 100. 102. 289. 690 1. 5 2. 5 12. 8 B = 200.051. 12. 338. 01 1. 32 5. 8 k = 1/2 6 B = 50. 1. 3 3. 8. 2 18. 21 B = 100. 2. 609 1. 5 3. 28 6. 5 0. 8 B = 200. 292. 02 1. 9 2. 89 1. 1 B = 00.051. 13. 30. 12 1. 3 6. 13 k = 1/3 6 B = 5. 8 1. 36 3. 60 8. 81 21. B = 150. 2. 613 1. 52 3. 35 6. 81. 2 B = 300. 293. 0 1. 50 2.90 1. B = 600.0506. 13. 31. 13 1. 35 6. 6
k = 1/ 6 B = 100. 50 1. 3 3. 6 9. 1 23. 2 B = 200. 212. 6 1. 53 3. 39 6. 9 51.8 B = 00. 292. 09 1. 51 2. 91 15.0 B = 800.0 51. 13. 32. 15 1. 36 6. 2 The missing values o not exist as real numbers, so unless the pot has imaginary chips, you shoul not call in these situations. The paraoxical lack of real solutions is an is explaine by the fact that the utilty function is boune above. Notice that the values for fixe kb (e.g. kb = 100 in bolface) are very close for a range of values of B an. This is perhaps unsurprising given the popular continuous CE approximation E s2 p 2kB with expectation E = +1, though it is an approximation that we have iscare as inaccurate for the current computations. 2.6 Approximations We use k =.5 as a moel an ajust for other k s proportionally x = ( + 1) 2k(B 1) or as a slightly more crue unerestimate: ( + 1) x = 2kB For the sake of comparison, we show the approximate values versus the more cruely unerestimate ones for two values of kb. The result is that the approximate values are quite goo except for the longshot raws. kb = 50 6 k = 1. 208. 595 1. 3. 08 5. 98 30. 6 k =.5. 210. 609 1. 50 3. 28 6. 5 0. 8 k = 1/3. 2. 613 1. 52 3. 35 6. 81. 2 k = 1/. 212. 6 1. 53 3. 39 6. 9 51.8 (+1) 2kB. 20. 56 1. 3 2. 2 5. 06 21. 6
kb = 100 6 k = 1. 102. 289. 689 1. 5 2. 5 12. 8 k =.5. 292. 02 1. 9 2. 86 1. 1 k = 1 3. 293. 06 1. 50 2. 90 1. k = 1. 293. 09 1. 51 2. 92 15.0 (+1) 200. 10. 28.66 1. 2.5 References [CI] W. Chin an M. Ingenoso, Risk Formulæ for Proportional Betting. [Mal] M. Malmuth, Gambling Theory an Other Topics, Two Plus Two Publishing Las Vegas 1999. Poker Essays, Two Plus Two Publishing 1996. 8