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1 Ealy Round Bluffing in Poke Autho(s): Califonia Jack Cassidy Souce: The Ameican Mathematical Monthly, Vol. 22, No. 8 (Octobe 25), pp Published by: Mathematical Association of Ameica Stable URL: Accessed: :2 UTC You use of the JSTOR achive indicates you acceptance of the Tems & Conditions of Use, available at info/about/policies/tems.jsp JSTOR is a not-fo-pofit sevice that helps scholas, eseaches, and students discove, use, and build upon a wide ange of content in a tusted digital achive. We use infomation technology and tools to incease poductivity and facilitate new foms of scholaship. Fo moe infomation about JSTOR, please contact suppot@jsto.og. Mathematical Association of Ameica is collaboating with JSTOR to digitize, peseve and extend access to The Ameican Mathematical Monthly.

2 Ealy Round Bluffing in Poke Califonia Jack Cassidy Abstact. Using a simplified fom of the Von Neumann and Mogensten poke calculations, the autho exploes the effect of hand volatility on bluffing stategy, and shows that one should neve bluff in the fist ound of Texas Hold Em.. INTRODUCTION. The phase the mathematics of bluffing often bings a puzzled esponse fom nonmathematicians. Isn t that an oxymoon? Bluffing is psychological, they might say, o, Bluffing doesn t wok in online poke. You can t see people s faces. Thee is a definite psychological aspect to bluffing in poke, as thee is in any competitive game. But fist and foemost it is a mathematical technique. The basic mathematics of bluffing in the final betting ound has been known fo yeas. We ll ecap some of that in this pape. What has not been known peviously is how to apply mathematics to bluffing in the ealy ounds. This pape pesents techniques to calculate bluffing stategy fo any ound of betting in poke. We use these techniques to demonstate a useful fact that it is not pofitable to bluff in the fist ound of Texas Hold Em poke. 2. VON NEUMANN AND MORGENSTERN. The mathematical study of bluffing has its oots in papes published in the ealy 2th centuy by Emile Boel and John Von Neumann. The biggest influence came fom Von Neumann s aticle, which he oiginally published in Geman [6]. He tanslated the aticle to English fo his book with Oska Mogensten, Theoy of Games and Economic Behavio [7], which we will efe to as VNM. In ode to talk about poke mathematically, the authos made a numbe of simplifications. Thei game can be descibed by the extensive fom diagam in Figue. At the stat of the game, thee is one unit in the pot. The fist move is a move of Natue, assigning a andom numbe fom [, ] to each playe. In ou desciption, the lowe hand is bette, that is, zeo is the best possible hand. Playe (we ll efe to Playe as him and Playe 2 as he ) then makes a decision to Bet o to Check. If he checks, thee is an immediate showdown, which is to say that the hands ae exposed and compaed. Whoeve has the lowest hand wins the one-unit pot. If Playe bets, putting one unit into the pot, then thee is a move fo Playe 2. She decides to call o dop. If she dops, then Playe gets the pot, + fo him. If she calls, adding one unit to the pot, thee is a showdown with the lowest hand winning the thee unit pot. This makes a win of 2 units o a loss of unit, depending on the values of the hands. VNM showed that this game has a pue stategy solution. (They also show a mixed stategy solution, and multiple pue stategy solutions. We will use only the simplest pue stategy.) The stategy is that Playe 2 selects a theshold c, such that she will call wheneve he hand is bette (lowe) than c, and dop if he hand is wose. The Playe stategy that we will analyze has two paametes, b (bet) and d (deceive). He will bet if his hand is bette than b. He will also bet (that is to say bluff ) if his hand is wose than d. MSC: Pimay S5, Seconday 7A2; C c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

3 show + 2/ N BET CALL 2 DROP + CHECK show + / Figue. Extensive fom of the VNM poke game c 2 b d Figue 2. Payoff squae fo Playe in the VNM game A complete stategy fo the game is given by s = (b, d) and s 2 = (c). We want a Nash Equilibium, which you will ecall is a stategy in which neithe playe can benefit fom unilateally changing thei own stategy. When we say equilibium, we mean Nash Equilibium. Following Zhang s example [], in Figue (2) we pesent a payoff squae giving the payoffs of the game. VNM calculated the equilibium stategy fo this game to be s = ( 2, 8 ), s 2 = ( 4 ). Playe always bets with a hand bette than 2, and always bluffs with a hand wose than 8. Playe 2 always calls with a hand bette than 4. (Note that fo these calculations, and all the othes we discuss, the playes behave hype-ationally. They ae awae of all the odds and assume that thei opponent is, too.) The value of the game, which is the amount Playe can expect to win on aveage using the equilibium stategy, is 5. Note that in this solution < b < c < d <. Octobe 25] EARLY ROUND BLUFFING IN POKER 727

4 Definition 2.. A bluff is a bet made with a hand in the ange [d, ]. Ou definition of bluffing involves betting with you wost hands. This does not include aggessive betting with hands almost good enough fo a nomal bet (see discussion of a semibluff in Section ). No does it include betting with evey hand (b = d). We say thee is no bluffing without the stict inequality b < d <. The VNM game may seem vey simple, but at the time it was published, it was a majo beakthough. Fo one thing, it established the bluff as a mathematical stategy. VNM shows that the bluff is a pofitable stategy, even if you opponent knows exactly which hands you use fo bluffing. In the equilibium solution, each playe s stategy makes thei opponent indiffeent to which stategy they choose fo thei maginal hands. That is to say that the values b, c, and d will satisfy equations () below. The function u is the utility, o expected etun fo a playe. It has two aguments, the hand that the playe holds, and the action that they take. The equations say that at the magin between betting and checking b, Playe gets an equal etun whethe he bets o checks. The same is tue fo the magin between checking and bluffing, and the magin between calling and dopping. Conside the equations u (b, bet) = u (b, check), u (d, bet) = u (d, check), u 2 (c, call) = u 2 (c, dop). () A published emak by poke theoist Noman Zadeh [8] led us to use such indiffeence equations to calculate poke stategies in a pevious pape [2]. This technique was efined by Feguson et al. [4], who intoduced the tem indiffeence equation. 3. INDIFFERENCE EQUATIONS. We d like to justify the use of indiffeence equations and demonstate thei usefulness in poke calculations. We begin with a gaphical, visual pesentation of Playe s expectation in the VNM game using two simple stategies. In Figue 3 (left), he uses the stategy b = (bet with any hand less than, which is to say always), and d = (neve bluff). Playe 2 is using the equilibium stategy discussed in Section 2, which is to call with hands lowe than 4, and fold othewise. When Playe uses this all-bet stategy, his expectation fo the best possible hand (zeo) is two units when Playe 2 calls, and one unit when she dops, fo an expectation of = 3. His expectation deceases linealy fo highe hand values, until it eaches Playe 2 s calling theshold of 4. Fo hands wose than that, he loses one unit when she calls, and wins one unit when she dops, fo a constant expectation of. In Figue 3 (ight), Playe is using the stategy b = and d =. He neve bets. His expectation is just what he gets fom a showdown fo the unit in the pot. Figue 4 is a supeposition of the othe two. It illustates that fo hands between and 2, Playe has a highe expectation fom betting. Fo hands between 2 and 8, he has a highe expectation fom checking. And fo hands between 8 and, he has a highe expectation fom betting. These values ae in fact the ones found by Von Neumann, as discussed in Section 2. Feguson et al. [4] povide a poof of the validity of indiffeence equations fo some games with no volatility. We cannot pove the geneal case, but do povide some discussion in Section 7, and make the following Claim. 728 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

5 Claim 3.. A solution to indiffeence equations is a Nash equilibium. Once we accept indiffeence equations as a tool, poke calculations become much simple. We can easily calculate the VNM esult in just a few lines S = (,), bet S = (,), check Figue 3. Playe pue stategy payoffs vs value of hand Figue 4. Supeposition of simple stategies Octobe 25] EARLY ROUND BLUFFING IN POKER 72

6 The final line of () comes into play only if Playe has a hand in [, b] o[d, ], which causes him to make some kind of bet. The left side of the equation is Playe 2 s cost ( ) plus he possible win (3) times the conditional pobability that Playe was bluffing, given that he bet. The ight side of the equation is what Playe 2 gets by dopping, which is zeo. This gives us + 3( d) =. (2) (b + d) This simplifies to b + 2d = 2. Switching to Playe s indiffeence equations, the top two lines of (), we note that if he checks any abitay hand h, his expectation is one (the pot size) times the pobability that the opponent has a wose hand. Since his opponent s hand is unifomly distibuted in [, ], that means u (h, check) = h, whee h is Playe s hand. His betting expectation is the cost of the bet ( ), plus 2 units if Playe 2 dops, plus 3 units if she calls with a wose hand, u (h, bet) = { + 2( c) + 3(c h) if h c, + 2( c) if h > c. (3) Using ou knowledge that b < c < d, and letting h = b and h = d in the above, we obtain two moe equations in b, c, and d, + 2( c) + 3(c b) = b, + 2( c) = d. (4) Equations (2) and (4) ae thee equations with thee unknowns. Solving these equations gives us the VNM solution. With a small adjustment, these equations can give us the equilibium solution fo diffeent bet sizes. They also let us solve the poblem of diffeent hand distibutions fo the two playes, fo example if Playe s hand is dealt fom U(.5,.), and Playe 2 s hand is dealt fom U(,.85). With moe elaboation, they can be used to solve a two-peson game whee both playes have the option of betting, as in nomal poke [2]. In this pape we will use the simplicity of indiffeence equations to allow us to examine betting stategies fo othe foms of poke. 4. ADDING VOLATILITY. Ou main topic in this pape is the effect of hand volatility on betting stategy. In the VNM game, each playe aleady has thei final hand it s not going to change. But in eal poke, most of the betting takes place in the ealy ounds, when thee ae still moe cads to be dealt. Thee ae many ways that hand values can change. Moe cads can be added to the hand (stud poke); you can exchange some cads fo othes (daw poke); community cads can be dealt (Texas Hold em, which we will call Holdem, as well as Omaha); o cads can suddenly become wild (Follow the Queen). The net effect is that elative hand value changes ove the couse of the game. An elegant pape by Benasconi et al. [] addesses volatility by adding a step in which the hand values ae evesed some of the time. We ll discuss the Benasconi pape futhe in Section 8. We model these changes by eniching the VNM model to make a game we call VNMplus. In VNMplus, befoe any showdown, thee is an additional move of Natue. Each playe gets a andom numbe between and. This is added to thei oiginal 73 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

7 U(, ) numbe. The playe with the lowest total wins the pot. Neithe playe knows thei final hand duing the betting ound. If we set =, we get the oiginal VNM game. Thee is moe volatility when is lage, so we call the elative volatility numbe. Definition 4.. VNMplus is the VNM game, with the addition of a andom U(, ) deal befoe the showdown. We call the elative volatility. To illustate how this can affect bluffing, we stat with a case of vey lage. The exteme case of = is simila to =, but can moe easily be dealt with numeically. We switch teminology now, efeing to Playe X and Playe Y. It s not impotant fo this example, but late we ll use Playe X fo the playe whose exact hand x is known, and Playe Y fo the playe whose hand is known to occupy a ange [, y]. Fo this example, Playe X begins with known hand x and at showdown time has a hand epesented by the pobability density function X (t) coesponding to the unifom distibution U(x, x + ). Playe Y begins with known hand y and at showdown time has a hand epesented by the pobability density function Y (s) fo the unifom distibution U(y, y + ). These functions fo the case x < y ae illustated in Figue 5. The pobability of Playe X getting a hand bette than o equal to a paticula hand h is h X (t) dt. The pobability of Playe Y getting a hand bette than Playe X is given by the double integal t X (t) Y (s) ds dt. (5) X (t). x x + Y (s). y y + Figue 5. X(t) and Y(s) fo the vey high volatility game Fo the Y (s) we e looking at, assuming t y +, the inne integal is t Y (s) ds = t. ds =.(t y). The oute integal, when x < y, becomes y which is x+ y (.)(.)(t y) dt 2 +.(x y) + 2 (.)(.)(x y)2. The wost case fo Playe Y is when y = and x =, which gives Playe Y winning chances of.45. Theefoe, each playe, X o Y, has a pobability of winning a Octobe 25] EARLY ROUND BLUFFING IN POKER 73

8 showdown that is slightly moe o slightly less than, depending on who has the 2 bette stating hand. Theoem 4.2. Fo the equilibium solution of VNMplus with =, thee is no bluffing. Poof. Fo Playe 2 s stategy, we obseve that calling is always bette than dopping, u 2 (h 2, call) = + 3P[h 2 wins showdown].475, u 2 (h 2, dop) =. Fo Playe s stategy, we see that betting is bette than calling when his chance of winning the showdown is geate than 5%, i.e., when his hand is less than.5, The stategy when = is u (h, check) = P[h wins showdown], u (h, bet) = + 3P[h wins showdown]. s = () s 2 = (.5, ). In othe wods, Playe has no motivation to bluff with a bad hand, because thee is no chance that Playe 2 will dop. 5. REALISTIC VOLATILITY. With no volatility, =, bluffing is a key pat of Playe s stategy. With extemely high volatility, =, bluffing should be avoided altogethe. Thee must be some point, < <, whee the stategy of bluffing disappeas. We call this special value R the volatility theshold. This value is useful in pactical tems. Fo a game situation with volatility geate than R, we know not to bluff with bad hands. Fo situations with volatility less than R, bluffing adds to ou etun. Definition 5.. R theshold. Fo VNMplus, thee must be some value between = and = whee bluffing disappeas fom Playe s stategy. We call this value R. We can use indiffeence equations to calculate R. We intoduce the notation p (h, h 2 ) to epesent the pobability that Playe wins a showdown when he has stating hand h and Playe 2 has stating hand h 2. When Playe bets with hand b, and Playe 2 calls, we know that Playe 2 has a hand in the ange [, c]. We use the notation p (b, [, c]) to epesent the pobability of Playe winning, conditional on h = b and h 2 [, c]. With this notation, equations () become + + 2( c) + 3p (b, [, c]) = p (b, [, ]), + 2( c) + 3p (d, [, c]) = p (d, [, ]), 3 b + d (b( p ([, b], c)) + ( d)( p ([d, ], c))) =. (6) 732 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

9 We ll come back to these equations, o thei equivalent using the p y notation intoduced below, thoughout the pape. In Sections 5 and 6 we find the volatility theshold R. In Section 7 we move on to othe values of, and in Sections 8 though we discuss vaious applications to eal life poke games. Fo ou fist task, finding the theshold R, we e looking fo the point whee d, Playe s maginal bluff, equals exactly. So d is a constant. The vaiables ae b, c, and. The indiffeence equations ae + 2( c) + 3p (b, [, c]) = p (b, [, ]), + 2( c) + 3p (, [, c]) = p (, [, ]), + 3( p ([, b], c)) =. (7) Note that does not appea in these equations. It aises fom the calculation of p. At this point we need to bing back ou Playe X, Playe Y teminology. In each of the p functions, one of the playes has a fixed hand x, while the othe playe has a hand in the ange [, y]. In the fist two equations, Playe is Playe X. In the thid equation, he is Playe Y. This leads us to intoduce a new notation, p y (x, [, y], ). This function epesents the pobability that Playe Y will win a VNMplus showdown when Playe X has hand x, Playe Y has a hand in the ange [, y], and the elative volatility is. Using p y, the indiffeence equations ae + 2( c) + 3( p y (b, [, c], )) = p y (b, [, ], ), + 2( c) + 3( p y (, [, c], )) = p y (, [, ], ), + 3( p y (c, [, b], )) =. (8) To calculate p y, we begin with the double integal (5) we used fo the high volatility case. Resticting the integals to thei nonzeo anges gives us the integal x+ t p y (x, [, y], ) = X (t) Y (s) ds dt. () x As in the high volatility case, X (t) is the pobability distibution of U(x, x + ). Y (s) is moe inteesting. It is the pobability distibution of a convolution of U(, y) and U(, ). These distibution functions ae illustated in Figues 6 and 7. X (t) x x + Figue 6. X(t) pobability density function Octobe 25] EARLY ROUND BLUFFING IN POKER 733

10 Y (s) y y + Figue 7. Y(s) pobability density function Unfotunately, the evaluation of this double integal is dependent on the elative values of x, y, and. Diffeent situations give ise to fouteen diffeent integations. We enumeate these cases and show how to evaluate two of them in the Appendix. The two cases we need fo discoveing the R theshold ae y < x <, which we efe to as p ylo, and x < y <, which we efe to as p yhi. Once again estating the indiffeence equations, we get + 2( c) + 3( p ylo (b, [, c], )) = p yhi (b, [, ], ), + 2( c) + 3( p ylo (, [, c], )) = p ylo (, [, ], ), + 3( p yhi (c, [, b], )) =. () The evaluation of p ylo and p yhi in the Appendix gives p ylo (x, [, y], ) = 2 2 ( 2 x 2 + xy + 2x y) y2 6 2, p yhi (x, [, y], ) = (2x + y) 2 + (y x)3 x 3 6y 2. () A simple sanity check fo these equations is to look at the limit as appoaches infinity, when each playe should have an equal chance of winning a showdown. In this limit, we indeed find p ylo (x, [, y], ) = p yhi (x, [, y], ) = 2. Anothe sanity check is when x = y. Fo that case, p ylo (x, [, y], ) = p yhi (x, [, y], ) = ( x x 2 )/6 2. To veify that this makes sense, we can set x = y =. Both playes stat with a fixed hand of, and so both should have an equal chance of winning. The equation evaluates to SOLVING THE EQUATIONS. Plugging () into () gives us thee equations with thee unknowns, b, c, and. Unfotunately the equations ae too cumbesome fo an analytical solution, as fa as we can see. The altenative is to find a numeical solution. Speadsheets such as Excel and Open Office Calc (and packages like Matlab) have algoithmic solves that can be used to solve such poblems. It is impotant to use a nonlinea solve fo these nonlinea equations, and to put appopiate limits on the vaiables. The Excel speadsheet that we used to solve fo the volatility theshold R is available online as Supplemental Mateial. By enteing equations () and () into the nonlinea solve, and setting the limits c b and b we get the solution 734 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

11 b =.462, c =.44, =.84, (2) which is unique. Claim 6.. The R theshold, at which bluffing disappeas (fo a bet size equal to the pot), is at =.84. You should have bluffing as pat of you stategy when is less, and neve bluff when is geate than this value. This is the solution we ve been looking fo. But it s not obvious how to make use of it. We ll discuss in Section 8 what this means fo eal life games such as Holdem. 7. OPTIMAL/EQUILIBRIUM STRATEGIES FOR OTHER VALUES OF R. We have calculated the R theshold. In so doing, we developed techniques that can be used to calculate equilibium stategies fo any value of. We can solve any game of VNMplus by making a constant and using ou speadsheet solve to detemine b, c, and d. Fo example, if you have a pactical poke case whee =.7, you can solve VNMplus to find the stategy s = (.268,.33), and s 2 = (.43). Rathe than following this exact stategy, you would use it to diagnose you opponent s weaknesses, and take advantage. If you opponent was following this equilibium stategy, they would be calling about 43% of the time. If they call less than that, you can take advantage by bluffing moe than you othewise would. If, on the othe hand, you think you opponent is bluffing moe than 6.7% of the time (% 3.3%), you should call moe than the equilibium amount. Fo calculations such as this, whee R, you need vaious p y integations, shown in Table A. of the Appendix. The pocedue is this: choose a value of, and use a solve to combine the appopiate p y expession with the indiffeence equations (6). Let the solve calculate b, c, and d. This is how we calculated the =.7 case, as well as all the plots and othe esults in the emainde of this pape. Fo the plots, we also use a compute pogam (available in Supplemental Mateial) to calculate Playe s expected etun fo any given set of stategies. Expeimenting with a vaiety of cases shows that fo most values of, the equilibium solution has b < c. But fo values nea the R theshold, b > c. It would be inteesting to know why. Since we do not have a geneal poof of 3., it would be nice to have the VNMplus equivalent of Figue 4 to eassue ouselves that ou indiffeence equation solutions ae in fact Nash equilibia. A easonable example is the case =.7, discussed above. (Any value of whee bluffing is useful would seve equally well.) Figue 8 shows Playe s expected etun fo both of his actions (bet o check) fo each stating hand, when Playe 2 is using this stategy. The gaph makes it appaent that he does best by betting with hands bette than.268 and bluffing with hands wose than.33. Convesely, Figue shows that Playe 2 does best by calling with hands bette than.43 and dopping (fo a etun of zeo) with wose hands, when she is confonted by Playe s stategy. Thee is a diffeence hee between VNM and VNMplus. Fo a zeo volatility game, when you opponent is bluffing too much, it is pofitable to call with any hand that can beat a bluff. Fo VNMplus, this is not the case. If, fo example, s = (.268,.), it would not be pofitable to call with h 2 =.8. Octobe 25] EARLY ROUND BLUFFING IN POKER 735

12 P Bet vs Check, =.7, c= bet_value check_value Hand Value Figue 8. P s etun, betting vs checking with vaious hands P2 Call, =.7, b=.268, d= Hand Value Figue. P2 s etun, always calling call_value 8. VOLATILITY AND HOLDEM. It would be useful to find a way to apply this to eal games, like Holdem. We need a way to elate the volatility of community cads to ou abstact VNMplus volatility. Also, unlike VNMplus, Holdem has fou ounds of betting, and the bet size is usually not equal to the pot size. Holdem teminology and play: Two face-down cads dealt to each playe (playes look at thei own cads); Fist ound of betting; Thee face up cads dealt to middle of table, the flop ; Second ound of betting; 736 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

13 Anothe face up cad; Thid ound of betting; Final face up cad; Fouth ound of betting; Playes make best hand fom combination of two face-down cads and five face-up (community) cads. Best hand wins. In a subgame pefect Nash equilibium fo Holdem, thee would be a continuation value of the game fo each playe at the game node just befoe the flop is dealt. That is to say that the game has a theoetical value fo each playe depending on the contents of thei hands and on the amount of money in the pot. The equilibium stategy fo the fist ound of betting should depend on these continuation values, and on the amount of volatility in the est of the game. To study eal life Holdem, we use a simplified vesion that we call FlopShow N. Definition 8.. FlopShow N (2 N 6) is defined as follows: 2 face down cads dealt to each playe; N 2 face up cads dealt to the middle; One ound of betting; 7 N face up cads dealt to the middle, the flop ; Playes make best hand fom combination of two face down cads and five face up cads. Best hand wins. FlopShow N is played with a eal 52 cad deck. It uses the nomal hand valuations fo poke (e.g., thee-of-a-kind is bette than two pai), which we will not enumeate. FlopShow 2 is an appoximation of the fist ound of betting in Holdem. Each playe has two face down cads with five face up cads still to be dealt. FlopShow 5 is an appoximation of the second ound of betting in Holdem (two face down, thee face up, two to be dealt). FlopShow 6 is an appoximation of the thid ound. (FlopShow 3 and FlopShow 4 would appoximate nonexistent states of Holdem, with one o two face up community cads.) If we can figue out a betting stategy fo FlopShow N, it will be a good guide to betting in Holdem. The measuement of volatility in VNMplus is simply the value. Fo FlopShow N and Holdem we need a quantitative measue fo the amount of change that can be intoduced by cads still to be dealt. A geneal measue of volatility that is useful fo all thee games comes fom asking the question, How often do elative hand values emain unchanged by the deal? We call this metic HOM (Hand Ode Maintained). Fo a small value, o with few cads to be dealt, hand values usually hold up, so HOM is a little less than. Fo lage values, o with many cads to be dealt, hand values change a lot, so HOM is slightly geate than.5. We claim, without poof, that if VNMplus and FlopShow N have the same HOM, they have the same equilibium stategy. Definition 8.2. H O M (Hand Ode Maintained) is the pobablity that the best hand befoe cads ae dealt is still the best hand afte cads ae dealt. We calculate HOM fo VNMplus in a staightfowad way. Fo any value of, simulate many ounds of VNMplus, and count the numbe of times that hand ode is maintained. We do this with a shot compute pogam, available as Supplemental Octobe 25] EARLY ROUND BLUFFING IN POKER 737

14 Mateial. Having un this fo many values, we noticed that the elationship between and HOM looks somewhat exponential. We did a cuve fitting execise fo the egion of most inteest, 4, and found HOM = e.834. While this appoximation is numeically close, we doubt any analytical accuacy. Calculation of HOM fo FlopShow N is conceptually simple. The most staightfowad technique is to deal out many hands fom a physical deck, and count the esults. We began by doing this, but got tied afte seveal hunded hands. To get lage amounts of data, one must esot to compute simulation. Any simulation equies you to deal with the question of which hand is best befoe the flop. In this case, best means most likely to win afte the flop. You can find tables fo this online. They tell you, fo example, that in a two playe game, an Ace-King of the same suit is bette than a pai of sevens, but wose than a pai of eights. At this point, having assumed a elationship between VNMplus and FlopShow N, and a elationship between FlopShow N and Holdem, we can ask questions about Holdem stategy. The question that got us stated on this inquiy was, Is it pofitable to bluff in the fist ound of betting in Holdem? This tanslates to the question, Does the equilibium stategy fo FlopShow 2 include bluffing? Ou physical deck dealing and compute simulations of FlopShow 2 evealed that HOM 6%. That coesponds to VNMplus = 2.85, which is fa geate than the volatility theshold R =.84 whee bluffing becomes unpofitable. Claim 8.3. In Holdem it is unpofitable to bluff in the fist ound of betting. If you have an opponent who bluffs in the fist ound of Holdem, you should call moe often than you othewise would. We leave othe Holdem questions fo the eade to exploe. But we ll mention that ou FlopShow 5 simulation yielded an value close to the R theshold, suggesting that bluffing is of limited use in the second ound of betting. Ou FlopShow 6 simulation gave.65, which says that bluffing is necessay in the thid ound of betting in Holdem. Fo the final ound of betting, see ou ealie pape [2].. BERNASCONI S COIN FLIP. Benasconi et al. [] took an appoach that is simila to ous. Like us, they add volatility to the oiginal VNM game. They do this by intoducing a biased coin flip that occus afte the betting and befoe the showdown. With some pobability q, the values of the hands ae evesed. A highe q 2 value coesponds to moe volatility in ou scheme. In Section 8 we intoduced the HOM (hand ode maintained) metic. It is always the case that HOM. 2 The HOM of Benasconi s game equals q. Among othe things, Benasconi exploes the inteesting fact that the value of the game to Playe ises as q ises between zeo and.atq = thee is a maximum, 3 3 then the game value declines as q inceases to. In shot, a small amount of uncetainty, in thei scheme, makes the game bette fo Playe. 2 Supisingly, unde ou scheme, the opposite is tue. Fo small amounts of elative volatility, that is to say when HOM is close to but less than one, the game value to Playe is less. A look at the diffeence between the of VNMplus and the q of Benasconi s game explains this. The basic diffeence is that Benasconi s coin flip q will evese the value of two hands egadless of thei oiginal value. But elative volatility is moe likely to evese the hands if thei oiginal values ae simila. Fo widely diffeent hand values, a small has no effect at all. Fo example, in the game with no volatility, as Von Neumann showed, the equilibium stategy is s = ( 2, 8 ), s 2 = ( 4 ). Fo a small q value, say., the final hand 738 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

15 ( - HOM) vs Game Value P Game Value HOM Figue. VNMplus game value fo inceasing volatility, as measued by HOM values ae swapped % of the time. This is an advantage to Playe, because his bluffing hands, which othewise have no chance of winning, now become winnes % of the time. Using ou scheme, if we set HOM =., it coesponds to an value of.33. This has no effect on Playe s bluff hands. They still lose evey time, because d c is geate than.33. Howeve, this value inceases the etun fo Playe 2 s calling hands that ae geate than b. With no volatility, these hands would always lose to legitimate bets h < 2. With =.33, they win a fai pecentage of the time. These diffeences between the coin flip q method and the elative volatility method means that elative volatility moe closely models standad poke games such as Holdem. Howeve, the coin flip method is pobably a bette model fo games that have occasional sudden evesals, such as Follow the Queen. Benasconi et al. chat the value of thei VNM vaiation fo vaious values of q. We do the same fo VNMplus. Ou calculations let us see the value of the game fo the equilibium stategy at vaious values of, see Figue. When HOM = (no volatility) the oiginal game has a value of 5. With HOM =.5, infinite volatility, the value is.5. In between, thee is a local minimum at HOM.8 ( =.74) and a maximum at HOM.62 ( = 2.5).. DIFFERENT BET SIZES. We d like to eview ou definition of bluffing. It means to bet with you wost hands, in the hope that the opponent who has a hand bette than yous will dop, and that the opponent s dops will bing you moe money than not betting with these hands. (Compae this with the semibluff, discussed below.) In Holdem tems, this might mean betting o aising with a 7 and 2 of diffeent suits (a vey bad hand) in the fist ound. You hope would be that eithe the opponents will dop immediately, o they will dop as you continue to bluff in futue ounds, o that the boad will fall lucky fo you. A lage pat of you expected value must come fom winning immediately in the fist ound. What Claim 8.3 says is that hand values in the fist ound ae so likely to change, that you opponents pobably have a pofitable call. But those calculations wee made assuming a pot-size bluff. One might say, Sue, they won t be scaed off by a pot-size bluff. But I m playing no limit. I can epesent a bette hand with a huge bet, and that will scae them off. O one might sumise that smalle bet sizes will make a bluff pofitable. Octobe 25] EARLY ROUND BLUFFING IN POKER 73

16 The same speadsheet solve we used befoe can be used to facto in the effect of diffeent size bets. The initial pot size is still but the bet size becomes β. The new equations ae β + ( + β)( c) + ( + 2β)( p ylo (b, [, c], )) = p yhi (b, [, ], ), β + ( + β)( c) + ( + 2β)( p ylo (, [, c], )) = p ylo (, [, c], ), β + ( + 2β)( p yhi (c, [, b], )) =. (3) Putting these equations though the solve gives the inteesting esult that the nobluff theshold value R actually deceases fo lage bets. When β = 5, epesenting a bet five times the size of the pot, R goes down to.. One case of lage β coesponds exactly to a eal-wold situation that comes up often in head-to-head (i.e., two playe) tounament play. This situation is analyzed by Chen and Ankenman, in thei excellent book The Mathematics of Poke [3]. They discuss a specific case of tounament poke 2 fo which the best stategy fo both playes is what they call jam o fold. This means you should eithe bet all you money (jam) o dop out of the hand (fold). The situation they analyze (two playe tounament play, whee one playe is close to elimination), is diffeent fom ou situation (two playes with infinite money). Even so, thei equilibium solution says that Playe should bet 58% of the time, with no bluffing. Ou equilibium solution says that with thei bet size, and a volatility of 2.85 (ou calculated fo FlopShow 2 ), Playe should bet 55% of the time, with no bluffing. Significantly smalle bets also lowe the R theshold. Fo β =.3, R =.3. So fo lage o small bets, it takes less volatility to make bluffing unpofitable. The highest value of R is not when β = howeve. The highest theshold value is when β =.784 and R =.87. With that bet size, the stategy is nicely symmetical, s = (.5, ), s 2 = (.5). It would be inteesting to know why this is.. POKER CONCEPTS. The uselessness of bluffing in the fist ound of Holdem (and aguably in the second ound as well) tells us a lot about how the game should be played. A successful playe has to have a paadoxical mindset: in the ealy ounds neve bet with bad cads; in the late ounds fequently bet with bad cads. This is the meaning of the phase tight-aggessive, which is often used by advice books to teach the pope appoach to the game. Fotunately, emoving bluffing fom you ealy ound stategy doesn t mean giving up deception. Two othe techniques, slow play and semibluff, become moe impotant when tue bluffing is ineffective. By its definition, slow play is a mio of tue bluffing. A slow play is to undebet you vey best hands. A tue bluff is to ovebet you vey wost hands. The idea of slow play is to penalize aggession by the othe playes, letting them aise the pot with infeio hands. It can be used effectively at any point in the hand, ealy o late. But in stategic tems the opposite of a tue bluff is a semibluff. A semibluff (which we fist encounteed in Sklansky [5]), is betting with a hand that is faily good, but pobably not the best hand in the pot. If you nomal stategy would be to bet with all hands bette than b, the semibluff has you betting with the best of you nomally nonbetting hands. (Contast this to a tue bluff, which has you betting with the wost of you nomally nonbetting hands.) A semibluff is helpful when a hand has good potential, such as a stong staight o flush daw, especially when facing a pobable 2 One playe has a stack that s equal to only times the big blind, o counting the small blind, 6.67 times the total ante. 74 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

17 pai. It can pay off in vaious ways: win the pot outight, build the pot fo when you make you hand, buy a fee cad fo anothe chance to make you hand, o set the stage fo a tue bluff in a late ound. The semibluff is most useful in the ealy ounds of betting, when thee is still a lot of volatility. This is pecisely when a tue bluff is useless. The tue bluff is most useful in the late ounds of betting, when thee is little o no volatility. This is pecisely when a semibluff is useless. The technique of using VNMplus to calculate a volatility theshold can be used with othe poke games. Sometimes these analyses can have supising esults, which will be the subject of othe papes than this one. But sometimes they just econfim something that expeienced playes ae aleady awae of. Fo example, when playing Holdem with moe than two playes, you need a bette hand to bet o call as the numbe of playes goes up. Unsupisingly, VNMplus simulations show that the volatility theshold is lowe when thee ae moe playes. This is a coollay to the idea that when thee ae moe playes to be foced out of the pot, a smalle pecentage of bluffs ae successful. 2. CONCLUSIONS. VNM povided ealy insight into poke stategy by establishing that the bluff is a mathematical technique. We discuss a simplified way to epoduce thei calculations, which enables us to add some vaiations. We changed the game by adding a andom numbe daw afte the betting ound. When we incease the size of the andom numbe to a cetain point, which we call the elative volatility theshold R, the technique of bluffing becomes unpofitable. We show how to calculate R fo a modified VNM game. By using a simplified vaiation of Holdem, we ae able to come to some conclusions about the eal-wold game. In paticula, it s unpofitable to bluff in the fist betting ound of Holdem. Appendix. The p y integal has a diffeent solution depending on the elative values of x, y, and. Thee ae fouteen total cases. We give details of the integation fo the cases x < y <, which we call p yhi, and y < x <, which we call p ylo. Fo p yhi, the oute integal can be boken into thee integals x+ x = y x x y Fo each one, we ll calculate the aea unde the Y (t) cuve using geomety. Fo t between x and y the gaph of Y (s) is the tiangle shown in Figue with a width of t and a height of t t2. Its aea is. y 2y Fo t between y and the aea unde the Y (t) gaph is the aea shown in Figue 2. It is a tiangle with width y and height / plus the ectangle with width t y and height /. The esult is 2t y. 2 Fo t geate than the aea is one minus that of the tiangle at the ight of Figue 3. That has width + y t and height ( ) +y t y Fo all values of t between x and x +, X (t) = evaluates to 6y 2 (y 3 x 3 ). The second integal y which gives us (+y t)2 (2t y)dt evaluates to ( y) y. so ou fist integal y x t 2 dt 2y Octobe 25] EARLY ROUND BLUFFING IN POKER 74

18 Y (s) t y y + Figue. Integal of Y(s) when t < y Y (s ) y t y + Figue 2. Integal of Y(s) when t is between y and The thid integal +x (+y t)2 ( )dt can be boken into two pieces 2y +x +x dt ( + y t) 2 dt. 2y 2 The second pat benefits fom a change of vaiable u = + y t. We end up with x (y 3 (y x) 3 ). 6y 2 Adding the thee integals, we get p yhi (x, [, y], ) = (2x + y) 2 + (y x)3 x 3 6y 2. (4) Similaly, fo p ylo, we beak the integal into thee pats: x+ x = x y+ x y+ Since x > y, the fist Y (s) integal hee is the same as the second integal above, that is, (2t y). Then, 2 x 2 (2y t)dt = 2 2 ( 2 x 2 + yx y). 742 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22

19 Figue 3. When t is geate than, subtact the tiangle fom Table A.. p y integals fo vaious cases. Case p y x < y < (2x + y) + (y x)3 x 3 2 6y 2 x < < y < x + 2x + y + (y x)3 x 3 2 6y 2 x < < x + < y 3 x 3 + 3x x 6y 2 x + y (x + y )(y x) < x < y < x (y x)3 3 2y 6y x 2 < x < x + < y y y < x < 2 ( 2 x 2 + xy + 2x y) y y < < x < y + (y + x)3 6y 2 y < < y + < x < y < x < y + (y + x)3 6y 2 < y < y + < x left as execise x < y < y < 2y 3 + y 3 3y2 y + (x + y)(x + y ) 2 2 (y y ) 2 2 x < < y < y < x + y 3 y 3 + 3(x + )(y y )(x + y y ) 6 2 (y y ) x < < y < x + < y (x + ) 3 y 3 + 3y2 (x + ) 3y (x + ) (y y ) x + < y The second Y (s) integal fo the y < x case matches the thid integal in the x < y case, (+y t)2. Then, 2y y+ ( + y t)2 ( )dt (using the same change of vaiables) = y 2y y Fo the thid integal, when x > y +, the Y (s) integal is the total aea unde the cuve, i.e.,. Then, Octobe 25] EARLY ROUND BLUFFING IN POKER 743

20 Adding the thee integals gives x+ y+ dt = (x y). p ylo (x, [, y], ) = 2 ( 2 x 2 + xy + 2x y) y (5) 2 Integals fo the othe cases ae evaluated similaly. Thee ae five simple cases fo which x < y: one in which is geate than both x and y, two in which is between them, and two in which is less than eithe. Thee ae five simila cases fo which y < x. One of these is a low volatility case that is left as an execise fo the eade. Thee ae fou cases fo which the bottom of the y ange is not zeo. These cases come up when thee is bluffing, and we want to evaluate p y (c, [d, ], )) in the thid line of equations (6). In these cases, the bottom of the y ange, y, appeas in the case specification and in the p y esult. Please go to supplements to find an Excel speadsheet that has examples and diections fo the calculation of equilibium stategies in games with volatility, and small utility pogams witten in Ruby, to calculate some esults fo the pape. ACKNOWLEDGMENT. Thank you to Pofesso Jeff Rabin of UCSD and novelist Janice Steinbeg, autho of The Tin Hose, fo thei help witing this pape. The MONTHLY eviewes wee wondefully helpful. Thanks also to the elatives, fiends, and stanges who gave me input on the manuscipt ove the last five yeas. REFERENCES. N. Benasconi, J. Loenz, R. Spohel, Von Neumann and Newman poke with a flip of hand values, Discete Math. 3 no. 2 (2) J. Cassidy, The last ound of betting in poke, Ame. Math. Monthly 5 (8) B. Chen, J. Ankenman, The Mathematics of Poke. ConJelCo, Pittsbugh, PA, C. Feguson, T. Feguson, C. Gawagy, U(, ) two peson poke models, in Game Theoy and Applications. Vol. 2. Nova Science Publishes, Hauppauge, NY, 27, D. Sklansky, The Theoy of Poke. Two Plus Two Publishing, Las Vegas, NV, J. Von Neumann, Zu theoie de gesellschaftsspiele, Math. Annu. (28) J. Von Neumann, O. Mogensten, Theoy of Games and Economic Behavio. Pinceton Univ. Pess, Pinceton, NJ, N. Zadeh, Winning Poke Systems. Pentice-Hall, Englewood Cliffs, NJ, 74.. H. Zhang, Two-playe zeo-sum poke models with one and two ounds of betting, Penn Sci. no. (2) CALIFORNIA JACK CASSIDY eceived his B.A. in Mathematics fom Conell Univesity. He woked at Hewlett Packad and elsewhee as a softwae enginee. He is also a playwight and the autho of a shot stoy collection, Winning At Poke and Games of Chance th Ave., San Diego, CA 23 jackcassidy@me.com 744 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 22