How Much to Bet on Vdeo Poker Trstan Barnett A queston that arses whenever a gae s favorable to the player s how uch to wager on each event? Whle conservatve play (or nu bet nzes large fluctuatons, t lacks the potental to axze longter growth of the bank. At the other extree, aggressve play (or axu bet runs the rsk of losng an entre bankroll, even though the player has an advantage n each tral of the gae. What s requred s a atheatcal forulaton that nfors the player of how uch to bet wth the objectve of axzng the long-ter growth of the bank. The faous Kelly crteron, as developed by John L. Kelly n A New Interpretaton of Inforaton Rate, acheves ths objectve. The Kelly crteron has been ost recognzed n gaes n whch there are two outcoes: wn $x wth probablty p and lose $y wth probablty -p. When there are ore than two outcoes, a generalzed Kelly forula s requred. Ths artcle wll apply the Kelly crteron when ultple (ore than two outcoes exst through workng exaples n vdeo poker. The ethodology could be used to assst advantage players n the decsonakng process of how uch to bet on each tral n vdeo poker. Kelly Crteron Analyss of casno gaes and percent house argn A casno gae can be defned as follows: There s an ntal cost C to play the gae. Each tral results n an outcoe O, where each outcoe occurs wth proft k and probablty p. A proft of zero eans the oney pad to the player for a partcular outcoe equals the ntal cost. Profts above zero represent a gan for the player; negatve profts represent a loss. The probabltes are all non-negatve and su to one over all the possble outcoes. Gven ths nforaton, the total expected proft SE Sp k. The percent house argn (%HM s then -SE /C, and the total return s +SE /C. Postve percent house argns ndcate the gablng ste, on average, akes oney and the players lose oney. Negatve percent house argns ndcate the gae s favorable to the player and could possbly generate a long-ter proft. Table suarzes ths nforaton when there are possble outcoes. 0 VOL. 4, NO., 0
Classcal Kelly crteron The well-establshed classcal Kelly crteron s gven by the followng result: Consder a gae wth two possble outcoes: wn or lose. Suppose the player profts k unts for every unt wager and the probabltes of a wn and loss are gven by p and q, respectvely. Further, suppose that on each tral, the wn probablty p s constant wth p + q. If -q>0, so the gae s advantageous to the player, then the optal fracton of the current captal to be wagered s gven by b* (-q/k. Consder the followng exaple. A player profts $ wth probablty 0.35 and profts -$ wth probablty 0.65, as represented by Table. Snce the expected proft of x 0.35-0.650.05>0, the gae s advantageous to the player and the optal fracton s gven by b* ( x 0.35-0.65/ 0.05. If a player has a $00 bankroll, then wagerng 00*0.05 $.50 on the next hand wll axze the long-ter growth of the bank. If the player loses $ on that hand, then the next wager should be exactly 99*0.05 $.475 under the classcal Kelly crteron. Snce fractons are often not allowed n gablng gaes, ths fgure should be rounded down to an allowable bettng aount. Kelly crteron for ultple outcoes When there are ultple (ore than two outcoes, as s the stuaton for vdeo poker, a generalzed Kelly forula s requred fro the classcal Kelly forula. Ths generalzed Kelly forula s gven by Theore (see A Proof of Theore. Theore Consder a gae wth possble dscrete fnte xed outcoes. Suppose the proft for a unt wager for outcoe s k wth probablty p for, where at least one outcoe s negatve and at least one outcoe s postve. Then, f > 0 a wnnng strategy exsts, and the axu growth of the bank s attaned when the proporton of the bank bet at each turn, b*, s the sallest postve root of 0 * kb + Let g(b represent the rate of growth of the bank that s the quantty to be A Proof of Theore Assue a constant proporton b of the bank s bet wth dscrete fnte xed outcoes. Let B( / B(0 equal + k b wth probablty p for to, where B(t represents the player s bank at te t. Assue the player wshes to axze g(b E[log(B( / B(0 ]. Wthout loss of generalty, let be the axu possble loss. In the nterval 0< b <-/, +k b > 0 snce k for to, so the logarth of each ter s real. Takng dervatves wth respect to b, dg( b db and d g( b db g'( b + kb ( + kb g''( b Note that (a g(0 0 (b g9(0 > 0 follows drectly fro the requreent for a wnnng strategy (so you should bet soethng (c g99(b < 0 for 0<b<-/ (where s the MPL so the frst dervatve has at ost one zero root n ths nterval. Hence, whenever there s a wnnng strategy, the force of growth has a unque axu gven by the root of * + kb 0 Table Representaton n Ters of Expected Proft of a Casno Gae wth Possble Outcoes Outcoe Proft Probablty Expected Proft O p E p O p E p O 3 k 3 p 3 E 3 p 3 k 3 O k p E p k.0 SE Table A Saple Casno Gae to Deterne the Optal Bettng Fracton Under the Kelly Crteron Outcoe Proft Probablty Expected Proft Wn $ 0.35 $0.70 Lose -$ 0.65 -$0.65.0 0.05 CHANCE
0. 0-0. -0. -0.3-0.4 0. 0-0. -0. -0.3-0.4 Kelly crteron for ultple outcoes: approxatons to the optal Accordng to Stanford Wong n hs 98 Blackjack World artcle, Optal bet sze turns out to be the expected arthetc wn rate dvded by the su of the squares. For sall expected wn rates, such as you have n blackjack, the denonator s approxately equal to the varance. Ths stateent suggests approxatng the optal b* by b' p ( k -0.5 Fgure. Graphcal representaton of the Kelly crteron for the classcal case (left and when ultple outcoes exst (rght, where the optal bettng fracton of b* occurs at a axu turnng pont on g(b. Value s the axu possble loss n the ultple outcoe gae. A Deonstraton That b0 Can Be Larger Than b* Suppose possble outcoes occur n a gae. Suppose + < 0 wth <0 and >0 and the condtons of Theore are satsfed. Then b*< b, whch ples that b*<b also, snce b <b. -0.5 Proof: Let p + p and p + p. Then b - b* / + /( /( ( + /( (p + p +(p +p /( (p + p ( + /( ( + > 0 snce < 0 and > 0 and + <0. Thus b >b*. The forula b s sple to copute gven the probabltes and profts for a sngle play of a gae. Accordng to WzardofOdds.co, Most gablers use advantage/varance as an approxaton, whch s a good estator. Ths second approxaton can be wrtten as b'' Agan, ths forula s sple to copute gven the probabltes and profts. It s obvous that 0<b <b (the denonator of b s larger. It would be useful f we could prove b <b*, because then b would be the superor approxaton to b* always. Unfortunately, t s not so. A proof that t s not so s gven n A Deonstraton That b Can Be Larger Than b*. Despte ths, there are avalable crtera to show when ether approxaton s useful for anagng the rsk of run fro over bettng. Observe n Fgure that g (b > 0 for 0 < b < b*. It s sple to check n practcal exaples f ether (a p( k g (b > 0 and 0 < b < {/ or advantage varance axzed. Fgure shows a graphcal representaton of the Kelly crteron for the classcal case (left and when ultple outcoes exst (rght. Let the value be the axu possble loss n the ultple outcoe gae. The player s bank wll grow as long as g(b>0 and s axzed when g (b0 (represented by g(b* n Fgure. It s portant to note that a player s bank wll not grow (and s lkely to ht run when over bettng the bankroll, even though the gae s stll favorable. Ths s represented on the graph for the values of b such that g(b<0. (b g (b > 0 and 0 < b < {/ If condton (a s satsfed, but condton (b s not, then use b. If condton (b s satsfed, but condton (a s not, then use b. When both these sets of condtons are satsfed, t s preferable to work wth b, snce t s a closer approxaton to the optal value b*. Notce the crtera do not requre pror knowledge of the value of b*. VOL. 4, NO., 0
Vdeo Poker Nonprogressve achnes Vdeo poker s based on the tradtonal card gae of draw poker. Each play of the vdeo poker achne results n fve cards beng dsplayed on the screen fro the nuber of cards n the pack used for that partcular type of gae (usually a standard 5-card pack, or 53 f the joker s ncluded as a wld card. The player decdes whch of these cards to hold by pressng the hold button beneath the correspondng cards. The cards not held are randoly replaced by cards reanng n the pack. The fnal fve cards are pad accordng to the payout table for that partcular type of gae. The pay tables follow the sae order as tradtonal draw poker. For exaple, a full house pays ore than a flush. Wthout a thorough understandng of vdeo poker, t should be clear n the analyss to follow how Theore can be appled to deternng an optal bet sze. A pay table for the outcoes, profts, probabltes, and expected profts for a Jacks or Better achne (known as All Aercan Poker are gven n Table 3. The probabltes were obtaned usng WnPoker (a coercal product avalable at www.zazone.co and assue the player s always axzng the expected proft on deternng the correct playng strateges. Note that $ s bet each gae. It shows that the overall payback for ths achne by playng an optal strategy s 00.7%. The standard devaton s calculated as 5.8. The approxaton forulas above gve b 0.069436% and b 0.069437%. Snce g(b 0.000789 > 0 and 0 < b <, ether approxaton of b and b s useful for anagng the rsk of run fro over bettng. Theore s appled to deterne a bet sze for ths vdeo poker gae by usng the payouts and probabltes gven n Table 3. The solver functon n Excel s used to calculate ths value as b*0.030679%. Exaple: Wth a $0,000 bankroll, Theore suggests the player should ntally bet $3.07 (lkely to be round down to $3. Nuercal Illustraton of Kelly Crteron n Multple (> Outcoe Gae Suppose 3 and the outcoe - occurs wth probablty 0.45, wth 0.45, and k 3 wth probablty 0.0. The expected outcoe s (-(0.45+( (0.45+(0.0 0.0 >0, whch s postve. The approxatons are b 0./[ (- (0.45 + ( *0.45 + ( *0.0 ] 0./.3 0.538 and b 0./[.3 0. ] 0./.6 0.587. s - and both b and b are less than -/. g (b (- (0.45/(-(b + ((0.45/(+(b + (0.0/(+(b 0.0. Slarly, g (b 0.0053. Both condtons (a and (b are satsfed, so t s preferable to work wth b. If soeone has $,000, the bet should be $58.73, whch lkely would be rounded to $58. Table 3 Profts and Probabltes for the All-Aercan Poker Gae Outcoe Return ($ Proft ($ Probablty Expected Proft ($ Royal Flush 800 799 n 43,450 0.08 Straght Flush 00 99 n 7,053 0.08 Four of a Knd 40 39 0.005 0.088 Full House 8 7 0.0098 0.077 Flush 8 7 0.057 0.0 Straght 8 7 0.084 0.9 Three of Knd 3 0.06883 0.38 Two Par 0 0.960 0.000 Jacks or Better 0 0.836 0.000 Nothng 0-0.58076-0.58.00 0.007 Progressve achnes Often, a group of achnes s connected to a coon jacot pool, whch contnues to grow untl soeone gets a royal flush. When ths occurs, the jacot s reset to ts nu value. CHANCE 3
Table 4 Probabltes of Outcoes for Dfferent Jacot Levels for the All-Aercan Poker Gae Outcoe Return ($ Table 5 Kelly Crteron Analyss for Progressve Jacot Machnes Jacot Return Theore $,000 $7,000 $50 99.6% $800 00.7% 0.0307% $3.38 $5. $,00 0.74% 0.0468% $5.5 $7.96 Usually, ths nu value would gve a return of less than 00%, whch creates a wn-wn stuaton for the astute player and the house. The aount bet to obtan the jacot s a fxed aount. Table 4 represents the probabltes of outcoes wth three jacot levels for the All-Aercan Poker gae. The $800 jacot was the gae analyzed earler. The $50 and $,00 jacots gve returns of 99.6% and 0.74%, respectvely. Notce the probablty of obtanng a royal flush ncreases as the jacot ncreases. Ths s logcal, as a player would be ore aggressve toward obtanng a royal flush wth a larger jacot. Suppose a player has a bankroll of $,000 and s requred to bet $5 hands. What jacot level s requred to axze the long-ter growth of the player s bank under the Kelly crteron? Table 5 gves the results and can Prob: $50 Jacot Prob: $800 Jacot conclude that a jacot level of $,00 s requred. A player would need a bankroll of about $7,000 to play the gae at a jacot level of $800. Practcal Dffcultes Prob: $,00 Jacot Royal Flush Jacot n 58,685 n 43,450 n 35,848 Straght Flush 00 n 7,7 n 7,053 n 6,999 Four of a Knd 40 0.006 0.005 0.005 Full House 8 0.00 0.0098 0.0096 Flush 8 0.0588 0.057 0.0505 Straght 8 0.085 0.084 0.0846 Three of Knd 3 0.06899 0.06883 0.06888 Two Par 0.988 0.960 0.954 Jacks or Better 0.8406 0.836 0.8336 Nothng 0 0.5794 0.58076 0.583.00.00.00 Despte the theoretcal advances ade above, t s possble to effectvely pleent the optal Kelly bettng strategy on an All-Aercan Poker achne, or any other vdeo poker gae. There are three an sources of dffculty. The frst s the exstence of a nu bettng unt on a achne. The second s the need to round the bet to avod fractons of a unt. Thrd, to gan an edge n the long run requres httng royal flushes. In the nonprogressve All- Aercan Poker achne, ths occurs on average once every 43,450 trals. Therefore, a player s bankroll would need to wthstand the downward drft between httng jacots to avod over bettng. Conclusons An analyss of casno gaes was gven to dentty when gaes are favorable to the player and could possbly generate a long-ter proft. Analyses were gven for both the classcal Kelly (two outcoes and the Kelly crteron when ultple outcoes exst (ore than two. The Kelly crteron when ultple outcoes exst was appled to favorable vdeo poker achnes. In the case of nonprogressve achnes, an optal bettng fracton was obtaned for axzng the long-ter growth of the player s bankroll. In the case of progressve achnes, the nu jacot sze was obtaned as an entry trgger to avod over bettng, based on the player s bankroll. Approxaton forulas when ultple outcoes exst were appled to vdeo poker and shown to be useful for anagng the rsk. The analyss developed n ths paper could be used by advantage players to assst wth bankroll anageent, whch s recognzed as an portant coponent to longter success. Further Readng Barnett, T., and S. Clarke. 004. Optzng returns n the gang ndustry for players and operators of vdeo poker achnes. Proceedngs of the Internatonal Conference on Advances n Coputer Entertanent Technology. Natonal Unversty of Sngapore, 6. Barnett, T. 009. Gablng your way to New York: A story of expectaton. CHANCE (3. Epsten, R. A. 977. The theory of gablng and statstcal logc. Calforna: Acadec Press. Hagh, J. 999. Takng chances: Wnnng wth probablty. New York: Oxford Unversty Press. Kelly, J. L. 956. A new nterpretaton of nforaton rate. The Bell Syste Techncal Journal July, 97 96. Thorp, E.O. 000. The Kelly crteron n blackjack, sports bettng, and the stock arket. In Fndng the edge, ed. O. Vancura, J. Cornelus, and W. Eadngton, 63 5. Insttute for the Study of Gablng and Coercal Gang. Wong, S. 98. What proportonal bettng does to your wn rate. Blackjack World 3:6 68. 4 VOL. 4, NO., 0