Toward Legalization of Poker: The Skill vs. Chance Debate

Transcription

1 Toward Legalization of Poker: The Skill v. Chane Debate obert C. Hannum* Anthony N. Cabot Abtrat Thi paper hed light on the age-old argument a to whether poker i a game in whih kill predominate over hane or vie vera. eent work addreing the iue of kill v. hane i reviewed. Thi urrent tudy onider two different enario to addre the iue: ) a mathematial analyi upported by omputer imulation of one random player and one killed player in Texa Hold'Em, and 2) full-table imulation game of Texa Hold'Em and Seven Card Stud. inding for enario howed the killed player winning 97 perent of the hand. inding for enario 2 further reinfored that highly killed player onviningly beat unkilled player. ollowing thi tudy that how poker a predominantly a kill game, variou gaming juridition might delare poker a uh, thu legalizing and broadening the game for new venue, new market, new demographi, and new media. Internet gaming in partiular ould be expanded and releaed from it urrent illegality in the U.S. with benefit aruing to aino who wih to offer online poker. Keyword: Poker, game of kill, game of hane, mathemati obert C. Hannum* (Contat author) Profeor Dept. of Statiti & Operation Tehnology Daniel College of Buine Univerity of Denver 20 S. Univerity Blvd. Denver. CO Tel: (303) ax: (303) rhannum@du.edu Anthony N. Cabot Partner Lewi and oa 3993 Howard Hughe Pkwy, Suite 600 La Vega, NV 8969 Tel: (702) ax: (702) aabot@lrlaw.om The game of poker ha beome enormouly popular over the lat deade, in large part due to ignifiantly inreaed Internet and televiion expoure. Tournament poker, in partiular, ha beome a viable petator port with the advent of peialized amera that allow the televiion audiene to view the hidden ard of the player. Thi inreaed popularity and the uneven legal tanding of the game in variou juridition ha reulted in oniderable media attention a well a debate over whether ertain form of poker hould be legalized. Central to that debate i whether poker hould be treated a a game of kill or hane. If kill predominate over hane, poker would be onidered lawful under mot legal tandard; if hane predominate over kill, then poker would be deemed illegal under mot interpretation of the law of the federal and tate government (Cabot & Hannum, 2005). In general, prohibited gambling involve any ativity in whih the following element are preent: () the award of a prize, (2) outome determined on the bai of hane, and (3) where onideration i paid (ee, e.g., Cabot & Coka, 2004, and the referene ited therein). If, however, any of thee three element i miing, then the ativity may be allowed. Mot tate exempt kill game from riminal gambling prohibition (Cabot & Coka, 2007, pp ). Three reent legal ae deided in the United State in early 2009 are illutrative of the importane of the kill veru hane debate in poker. In January, a jury in Colorado found the organizer of a poker league not guilty of illegal gambling. The outome of the trial hinged on whether poker i a game of kill or a game of hane; the jury agreed with expert tetimony preented by a tatitiian that poker i a game of kill (Geller, 2009; Jenkin, 2009). Alo in January, a Pennylvania judge agreed with a defene attorney' argument in ruling that kill i the predominant fator in poker and dimied harge againt two people who ran a poker game out of a garage. It may have been UNLV Gaming eearh & eview Journal+ Volume 3 Iue

2 gambling but not unlawfully o, beaue the outome of the game had more to do with kill than hane, Judge Thoma Jame Jr. ruled (ottein, 2009). In a ebruary trial of five defendant arreted in a raid on a home poker game in South Carolina, the judge agreed with the tetimony of two expert in finding that Texa Hold'em i a game of kill, oberving that the evidene and tudie are overwhelming that thi i o. Noting that the dominant fator tet ha been aepted in imilar ruling in other ourt, uh a Pennylvania, California, Miouri, and Nebraka, however, the judge ould not jutify the ue of the predominane tet over the South Carolina law that refer to 'any houe ued a a plae of gaming' and found the defendant guilty (Newell, 2009). In ruling that poker i a game of kill, the trial ourt reognized that thi would be an eay ae for aquittal if the proper tet under South Carolina law i whether a game i predominantly one of kill. The judge onluded, however, that a higher ourt mut deide that iue (Polon, 2009). The five South Carolina defendant are appealing their onvition (Kropf, 2009). Poker differ in ubtantial repet from the variou ativitie often laified a gambling, uh a lotterie and mot aino-tyle game, beaue poker ha many element of kill not preent in traditional game of hane, uh a lotterie and other aino game. Thi paper addree the quetion of kill veru hane in poker and upport the notion that poker i a game of kill. elated Work There are a number of ientifi tudie addreing the iue of kill in poker a well a a plethora of referene to ame in the more general poker literature. Thee two olletion of work are ummarized in the ubetion below. Sientifi Studie Kadane (986) ompared a 'dumb' trategy and a 'mart' trategy to argue that ue at eletroni draw poker depend a great deal on the player' kill. Thi onluion wa baed on the fat that the mart trategy reulted in four time a many win/point a the dumb trategy. Kadane emphaize that in eletroni draw poker, kill mean that there are deiion under the ontrol of the player that greatly influene the ultimate outome, an interpretation that ould well apply to other gambling game. Larkey, Kadane, Autin, & Zamir (997), operating on the premie that kill and kill differene among player are important feature of real game, ue a implified verion of tud poker to better undertand the onept of differential player kill in game. Uing omputerized experiment, the author ompared twelve different trategie for thi implified poker game, ranging from a imple trategy that play randomly (whih the author argue i a player with no kill) through progreively more ophitiated trategie utilizing varying degree of kill. The reult how a wide range of outome aro the different trategie, with better outome generally aoiated with higher degree of kill. The imple random trategy (zero kill) i the wort overall performer in term of winning. Cabot & Hannum (2005) explored the origin and purpoe of poker law and the predominane tet, uggeting that the urrent publi poliy debate over gambling hould onider game of kill, inluding variant of poker, from a different perpetive than game of hane. In addreing the kill element of poker, thee author analyzed the performane of highly killed player againt one or more le killed player in imulated game of limit Texa Hold'Em. The reult howed that the highly killed player beat the low kill player onviningly in a variety of player-mix enario. A Ion (2007) onduted a detailed analyi of everal implified model of poker, whih an be viewed a toy model of Texa Hold'Em. The author note that the implified model allow a preie mathematial analyi and yet apture many of the main propertie of the more ompliated real game. The analyi ugget that kill play 2 UNLV Gaming eearh & eview Journal+ Volume 3 Iue

3 Poker ontain a greater kill element than any other ard game, inluding ontrat bridge, pinohle, gin rummy, and blakjak. Toward Legalization of Poker: The Skill v. Chane Debate an important role in poker. Alan onlude that "kill i far more dominant than luk, and that poker i predominately a game of kill" (p. 7). Croon, ihman, & Pope (2008) preent evidene of kill differential among poker player finihing in one of the final two table in high-take Texa Hold 'Em tournament between 200 and 2005 (data inluded 899 player in 8 eparate tournament). Their main onluion i that "there appear to be a ignifiant kill omponent to poker: Previou finihe in tournament predit urrent finihe" (p. 28). Uing additional data from profeional golf tournament, the author ompare poker to golf, a game univerally held to be predominantly kill-baed. The author onlude that "kill differene among top poker player are imilar to kill differene aro top golfer" (p. 28). Dedonno & Detterman (2008) examine whether poker i a game of luk or kill by omparing player who were taught trategie baed on expert opinion with other player who were taught no trategie. Partiipant who were intruted out-performed thoe who were not intruted. The author onlude: "The unequivoal finding i that poker i a game of kill" (p. 36). They further oberve that the kill involved in poker i omplex and luk (random fator) diguie the fat that poker i a game of kill. iedler and ok (2009) propoe a 'ritial repetition frequeny,' defined a the threhold of repetition at whih a game beome predominantly influened by kill rather than by hane. Their rationale i that hane element anel out in the long run while kill element do not. Analyzing poker data from an empirial urvey of 5,76 poker player, the author onlude that: (a) low kill etablihe itelf muh fater againt the hane element than high kill; (b) poker i in the ontinuum between being a game of hane and being a game of kill; and () for their ample, poker i a game of kill. Uing a game-theoreti approah to tudy the relative role of kill and hane in game, Borm & van der Genugten (200 I) onlude that the level of kill in the three popular variant of poker, Seven Card Stud, Texa Hold 'Em, and Draw Poker, i greater than that in roulette, rap, and blakjak. Game theoreti modeling of poker an alo be found in erguon & erguon (2007) and in the referene ited therein. Kelly, Dhar, & Verbiet (2007) examine the kill veru hane iue and it role in the debate on the legality of poker, oberving that in a erie of hand in a tournament, the individual draw of ard ha a lower impat on the outome than a ingle hand. They argue that in a tournament etting, the outome are ubjet to the law of large number and the differene in kill will be made apparent. Lipton, Lazaru, & Weber (2005) diu game of kill veru game of hane in Canadian law. MCrory (2002) diue legal iue urrounding the kill veru hane debate, partiularly in the ontext of video poker, and note the ditintion between the 'Englih rule,' where any amount of kill i enough to qualify a game a a kill game and the 'Amerian rule,' whih tate a game qualifie a a game of kill if kill i the dominant fator in the outome of the game. In a erie of paper, Heubek (2008a, 2008b, 2008) diue the general iue of meauring kill in game in light of the fat that game offering prize are often exempted from gambling tatute if the game exhibit uffiient kill. Anedotal Evidene Many author of popular book are expert in poker due to their extenive play, informal tudy, and intimate familiarity with the game (e.g., Brunon, 2005; Caron, 200; Krieger, 2000; Sarne, 980; Sklanky, 999). The onenu among thee author i that while the ditribution of ard i random, the kill involved in betting predominate over hane. Sarne (980) devote an entire hapter in hi book to the iene and kill in poker. eferring to the proliferation of ard game in nineteenth-entury Ameria, Sauly (982) oberve that while ome were purely game of hane, poker require kill, judgment, and pyhologial trength. In diuing the iue of hane veru UNLV Gaming eearh & eview Journal+ Volume 3 Iue 3

4 kill in poker, many author admit that hane an have an influene in the hort term, but emphaize that kill ha a more pronouned effet in the long run (Eriman, 999; Levinon, 963; Netor, 2003). Other diu the kill required in poker to that of other game, onluding that poker ontain a greater kill element than any other ard game, inluding ontrat bridge, pinohle, gin rummy, and blakjak (Sarne, 980; Netor, 999). Commenting on the ability to onitently make money playing poker, one expert note that there are a few profeional who earn a living playing blakjak and video poker, and though kill ha a part in thee game, luk and the perentage till hold the greatet way. It' the other way around in poker; bad luk an hurt, but kill alway beat luk over time (Netor, 999). Sue in poker relate to winning the mot money and a mitake often made by novie and unkilled player i to try to win the mot pot. A Krieger (2000) oberve, a quik way to go broke i to play every hand; you'll win more pot along your peronal road to ruin but the objetive i to win the mot money, not the mot pot. Over the long run everybody get the ame proportion of good and bad ard; beginning poker player rely on big hand and luky draw while expert player ue their kill to minimize their loe on their bad hand and maximize their profit on their big hand (Sklanky, 999). A killful poker player an ue poition, pyhology, bluffing, and other method to inreae hi hane to win the pot and inreae the ize of the pot he win (Caron, 200 I). Expert often ite everal omponent to the kill neeary to play poker well. Thee inlude mathemati, pyhology, deeptivene, bluffing, aeing ompetition, reading hand, reognizing tell, exploiting poition, and money management (Brunon, 2005; Sarne, 980; Sklanky, 999). In hi lai book on the theory of gambling, ihard Eptein note that poker game have a large number of trategi alternative, and ertain type are almot purely trategi (Eptein, 995). There are numerou book devoted to the iene and mathemati of poker (e.g., Chen & Ankenman, 2006; Hilger, 2006; Mahmood, 2003); imilarly, everal addre the pyhology of poker (e.g., Hilger & Taylor, 2007; Shoonmaker, 2000; Shoonmaker, 2007). The olletive expert opinion i lear- kill play a large role in determining outome in poker. A Sklanky ( 999) ummarize, poker i not primarily a game of luk; it i a game of kill. The following etion explore the differene between game of hane and game of kill. Sue in poker relate to winning the mot money and a mitake often made by novie and unkilled player i to try to win the mot pot. Game of Chane veru Game of Skill Gambling game an be ategorized a thoe of pure hane and thoe involving an element of kill (Hannum & Cabot, 2005). In game of pure hane, outome i determined by hane alone and no deiion, trategy, or kill an be ued by the player to affet the outome, expetation, or perentage of money won (or lot). Game of pure hane inlude roulette, rap, keno, bingo, (traditional) lot, and lotterie. In game involving kill, typially deiion and trategie an affet the outome and the perentage of money won or lot i a diret refletion of the player' level of kill. Among game involving kill, a further ditintion an be made between houe banked game of kill and player veru player (non-houe banked) game of kill. In the former, a player an exerie kill to alter the houe advantage and the latter the exerie of kill may reult in a true player advantage. Houe banked game of kill inlude blakjak, video poker, and many of the newer poker-baed aino game uh a Caribbean Stud Poker, Let It ide Poker, and Three Card Poker. Thee game are eentially a math between the player' kill and the rule of the game- the houe ue a fixed trategy with no appliation of kill- and proper exerie of kill an redue the houe advantage. In houe banked game of kill, it i often the ae that an optimal trategy i known 4 UNLV Gaming eearh & eview Journal+ Volume 3 Iue

5 Toward Legalization of Poker: The Skill v. Chane Debate and fairly eaily matered. Example of player veru player game of kill are bridge, bakgammon, he, mahjong, and poker. In thee game a player ue hi or her kill againt the kill of the other player() and the more killed player will win over the le killed player over time. In player veru player game of kill, optimal trategy i typially diffiult to mater and the motivated player i alway learning trategie for winning. With thi group of kill game the (motivated) player never top learning and ontinue to learn new trategie for winning. Some game may be ategorized a pure kill, devoid of all probabiliti element. Currently uh game are not typially offered in a aino environment, but would inlude Ti-Ta-Toe, Cheker, Che, and Go, among other (Eptein, p. 337). The following etion more arefully examine the role of kill and hane in poker. In a game of kill the player an alter the expeted outome; not o for a game of pure hane. Poker, Skill, and Chane veru player poker (herein referred to 'multiplayer poker' or imply 'poker') differ in ubtantial repet from the variou ativitie often laified a gambling, uh a lotterie and mot aino-tyle game inluding video poker, beaue poker ha many element of kill not preent in traditional game of hane. Video poker i a houe-banked game with a fixed hold perentage at optimal play wherea multiplayer poker i an interative, non-houe banked game that allow individual to ompete with one another, where ue i baed on the appliation of a player' kill againt other player. Suh kill inlude mathemati (alulating odd), pyhology, aeing ompetition, reading hand, reognizing tell, bluffing, folding, exploiting poition, and money management. Wherea pratie an improve the player' level of kill in player veru player poker, no amount of 'pratie' will remove the houe advantage in roulette, keno, or houe-favorable video poker. In pure game of hane, intentional 'bad play' annot aue a player to loe, or loe fater. In game of kill, on the other hand, it i poible for a player to play badly, whether intentionally or not, and loe more. In partiular, a killed player in a player veru player game of kill ould ue hi kill to intentionally loe fater than an unkilled player. Thi i the ae for poker, but not for game of hane uh a roulette, keno, or lot where it i not poible for a player to purpoely Joe, or loe fater. The ability to intentionally loe i a feature of game involving kill. In a game of kill the player an alter the expeted outome; not o for a game of pure hane. The outome of a bet in roulette, for example, i determined olely by hane. If a player bet on a ingle number in Amerian roulette (where there i a double zero on the wheel), the probability of winning i /38 while the probability of loing 37/38. The payout however, i only 35-to-. Thu, hort of a biaed wheel or heating, there i nothing the player an do to hange thee probabilitie or the expetation, whih i -5.3 perent (i.e., the houe edge i 5.3 perent). Similarly, a player betting at doublezero roulette will loe on average 5.3 perent of the money wagered, regardle of the partiular bet made and the 'pattern' of betting. Beaue the pin of the roulette wheel are independent trial and the probabilitie of winning and loing remain ontant from trial to trial (and the expetation i negative on eah trial), it i mathematially impoible to devie a ytem to hange the expeted win (lo) perentage. A novie roulette player will fare jut a well (or poorly) a an 'experiened' roulette player; a player making random deiion in roulette will fare jut a well or poorly a one who make deiion aording any ort of o-alled trategy. In hort, there i no uh thing a a 'good' (or 'bad') roulette player. The expeted win in poker, on the other hand, depend on the kill or trategy employed by the player. In poker, ertain knowledge an be brought to bear on the deiion made during the game, reulting in a greater expeted win. While there are UNLV Gaming eearh & eview Journal+ Volume 3 Iue 5

6 arguably variou faet of 'kill' in poker, the ombination of thee faet are expreed through one element, betting- the deiion on how muh money (if any) to invet. ting trategy in poker- whether to fold or bet and how muh to bet- i a deiion made of a player' own free will and i omething at whih a player an beome killful. A player' betting method an get better or wore. In poker game, if more than one player remain after the lat round of betting, remaining player expoe and ompare their hand to determine the winner or winner. Thi i alled the howdown. or hand that do not go to howdown, betting trategy, peifially one player' deiion to invet in the hand while all other deide to not invet in the hand, i the ole fator in determining outome; the winner i the one player left who ha not folded. or hand that go to howdown, the betting trategy of folding determine the outome if a player who folded had a better hand than the winner, ine the winner would have been different had thi player not folded. Hope, Mizelle, & MCulloh (2009) examined 03 million hand of Texa Hold'Em played at an online poker ite and etimate that 76 perent of the hand in Texa Hold'Em do not go to howdown. Of the remaining 24 perent, only about half, or 2 perent of all hand, are won by the player at the table who would have had the bet hand. Thi i beaue the player with the lukiet ard that would otherwie have been the bet 5-ard hand folded prior to howdown. Thu, only about 2 perent of the time doe the player with the bet or lukiet ard how hi hand and win. In a game involving kill, the maller the differene in kill level between player, or the horter the duration of the ontet (the 'hort term'), the more unertain i the outome. Even for game in whih kill i a predominant fator, when the gap in kill level i mall the reult tend to reemble hane. Thi doe not mean, however, that kill i not an important fator, but rather that it i more diffiult to detet and ae the impat of kill when the relative kill level of ompeting player are loe. Conider, for example, the game of he (or perhap tenni, or golf), univerally onidered a game of kill. When the gap in kill level between two he player i large, the highly killed player will onitently beat the le killed player. However, when the gap in kill level i mall and the he player are loely mathed in level of kill, the reult may appear to be due to hane. egardle of the partiular kill or ombination of kill, what differentiate a game of kill and a game of hane i the ability of the player to influene the game' expetation, broadly defined a the expeted outome, through knowledge, judgment, deiion, and/or performane. What ontitute kill or hane legally i ubjet to the differing interpretation of the federal government and tate government. Mot tate ue the 'predominane tet'- if the element of kill in a partiular game predominate over hane, then the game i permitted (ee, e.g., Alexander, 2008; Cabot & Coka, 2007; Humphrey, 2007b). Under thi tet, an ativity i onidered illegal gambling if a peron rik omething of value on an ativity predominately determined by hane for the opportunity to win omething of greater value than he or he riked. Mot tate have a ommon definition of the predominane tet. Under the predominane tet, one mut enviion a ontinuum with pure kill on one end and pure hane on the other. The element of hane i met if hane predominate over kill in determining the outome of the ontet, even if the ativity require ome kill. In theory, an ativity roe from kill to hane exatly in the middle of the ontinuum. On the ontinuum, game uh a he would be at the kill end, while traditional lot mahine would be at the hane end. ween thee end, there are many game that ontain both kill and hane. In thi area, a legal rik exit beaue it i a ubjetive aement a to where on the ontinuum a game that i part kill and part hane lie. 6 UNLV Gaming eearh & eview Journal+ Volume I3 Iue I Mot tate ue the 'predominane tet'- if the element of kill in a partiular game predominate over hane, then the game i permitted.

7 Toward Legalization of Poker: The Skill v. Chane Debate Unfortunately, the predominane tet i not the only tet that ourt employ in the variou tate. or example, in ome tate a game i prohibited if hane i a material element in the outome. Thi i a leer tandard than the predominane tet and effetively make offering lawful kill-baed gaming to reident of thoe tate more diffiult if the game in quetion reort to a hane omponent in determining the outome. Yet, in ome tate kill or hane i irrelevant and merely wagering on any event or ontet i uffiient to be illegal gambling (Hannum & Cabot, 2005; Humphrey, 2007a, 2007b). While the type of kill that an be applied and the extent to whih it may affet the outome varie, if kill predominate over hane in determining the outome of a game, it i uually onidered a game of kill. Thi paper argue that in poker game uh a Texa Hold'Em, while there are arguably variou faet of 'kill,' the ombination of thee faet are expreed through one element, namely, betting, and that thi kill i the mot likely fator influening the outome. The following etion examine the kill veru hane iue in poker by analyzing the reult of a one-on-one game of Texa Hold'Em in whih a player making haphazard deiion play againt a (killed) player who employ a winning trategy. Analyzing Skill in Poker (Part A) Senario : andom veru Skilled In poker, player ompete againt eah other and money won or lot i merely tranferred from one player to another (exept for the houe 'rake,' a mall perentage taken from eah pot, if any). Thi i true even in a aino, where ardroom poker differ from other houe-banked game in that player are not playing againt the houe. The aino provide a dealer, who doe not play, and make money by taking a perentage of eah pot, harging an hourly fee, or olleting a flat amount for every hand. The following etion outline the rule and terminology of Texa Hold'Em. Texa Hold'Em The mot popular type of poker played today i Texa Hold'Em. The game typially aommodate between two and ten player. In the initial deal, eah player i dealt two ard fae down (the 'hole ard'). Thee ard are unique to the player to whom they were dealt. A round of betting then take plae. During eah round of betting, a player an either tart the betting (exept the firt round, where the betting ha already been tarted with the fored 'blind' bet prior to any ard being dealt), all the bet oming to him, raie the betting, or fold hi ard. In all betting round exept the firt, a player may hek if no bet ha yet been made on that round. If a player fold that player i removed from that hand. If the number of player i redued to a ingle player, then that player win regardle of hi or her hand. After the firt round of betting, three hared ard (the 'flop') are plaed fae up in the middle of the table. Another round of betting then take plae. Next, a ingle hared ard (the 'turn') i turned up among the ommunity ard in the enter of the table, followed by a third round of betting. inally, the lat hared ard (the 'river') i then added to the hared ommunity ard, and a final round of betting our. At that point, if two or more player are till ative, the peron with the highet hand win (the 'howdown'). The next etion provide the bakground for the 'andom veru Skilled' game to be analyed. andom veru Skilled - Bakground and Game Parameter To hed light on the quetion of whether kill predominate over hane in poker, thi tudy onidered a two-player (head-up) limit game of Texa Hold'Em in whih an unkilled player making betting deiion haphazardly play againt a player employing a non-random betting trategy. The former player i referred to a 'andom,' and the latter a 'Skilled.' The logi behind analyzing thi enario i that if kill play no part in poker, UNLV Gaming eearh & eview Journal+ Volume I 3 Iue 7

8 then a player making random betting deiion hould fare a well a any other player. If kill play a role, there hould be a trategy that an be employed to defeat the player making random deiion; the more onvining the defeat, the larger the role of kill. By omparing the performane of a player with no kill (andom) and a player employing a non-random trategy (Skilled), the impat of kill in thi enario an be aeed in term of perentage of hand and (more importantly) amount of money won. Againt an opponent with no kill whatoever and making deiion randomly a relatively imple trategy an be devied to win onviningly: Alway raie the bet when permitted to do o and to all the bet otherwie. Thi will be the trategy employed by Skilled in thi tudy. (When the gap in kill level i not large, for example when one player i moderately killed and the other very killed, more ompliated trategie would be neeary to defeat the opponent.) Thi analyi onider a $2/$4limit head-up game of Texa Hold 'Em between the unkilled andom player (making betting deiion randomly) and the Skilled player (raie when poible, otherwie all), auming a fourraie per round maximum. There i no houe rake. The following etion provide the mathematial reult of thi andom veru Skilled Texa Hold 'Em game. andom veru Skilled - Mathematial Analyi Exat probabilitie and win rate for thi Skilled veru andom player Texa Hold'Em game an be derived mathematially uing a deiion tree approah. or purpoe of thi mathematial analyi it i aumed that eah player win half the howdown (the iue of tied hand i addreed in the imulation etion below). The analyi require onideration of two ae, when andom i the mall blind (Cae ) and when Skilled i the mall blind (Cae 2). A ummary of the reult of the mathematial analyi follow; further detail howing probability alulation for eah poible deiion path for eah ae and an example tree diagram for the flop deiion in Cae are given in the Appendix. Cae : Small Blind =andom; Big Blind = Skilled. Table how the exat probabilitie of a hand making it to eah tage of the game - flop, turn, river, and howdown -a well a exat win rate for eah player for the ae when andom i the mall blind and Skilled i the big blind. Thu, 35.2 perent of the hand will make it to the flop (the other 64.8 perent of the time andom will fold pre-flop), 7.6 perent of all hand will make it to the turn (50 perent of thoe making it to the flop), 8.8 perent to the river (50 perent of thoe making it to the turn), and 4.4 perent to the howdown (50 perent of thoe making it to the river). Auming howdown hand are divided equally, Skilled win 97.8 perent and andom win 2.2 perent of the hand.!fable. Probabilitie and Win ate -andom i Small Blind I Win ate - Skilled Win ate - andom I ~tage!probability I IPre-lop lop If urn !iver Showdown* PVEALL *Auming eah player win half the howdown. Cae 2: Small Blind= Skilled ; Big Blind= andom. Table 2 how the exat probabilitie of a hand making it to eah tage of the game - flop, turn, river, and howdown -a well a exat win rate for eah player for the ae 8 UNLV Gaming eearh & eview Journal+ Volume 3 Iue

9 Toward Legalization of Poker: The Skill v. Chane Debate when Skilled i the mall blind and andom i the big blind. Note that the overall win rate for Skilled i lightly lower for thi ae (95.9 perent ompared to 97.8 perent when andom i the mall blind, auming howdown hand are divided equally) due to the fat that almot twie a many hand make it to howdown (8.2 perent when Skilled i the mall blind veru 4.4 perent when andom i the mall blind). Put another way, fewer hand are folded by andom when Skilled i the mall blind. Table 2. Probabilitie and Win ate - Skilled i Small Blind Win ate - Skilled Win ate - andom ~tage!probability!!pre-lop !lop !fum jiver Showdown* [oveall *Auming eah player win half the howdown. Overall: Cae I and 2 Combined. Table 3 how the probabilitie of a hand making it to eah tage a well a win rate for eah player when Cae I and 2 are ombined (i.e., half the hand are played under eah ae). The reult how that 45.4 perent of the hand will make it to the flop (the other 54.6 perent will be folded pre-flop by andom), lightly more than half of thee will progre to the tum (o 23.5 perent of all hand will make it to the tum), lightly more than half of the tum hand will move on to the river (2. perent of all hand make it to the river), and lightly more than half the river hand will go to howdown (6.3 perent of all hand will go to howdown). Auming eah player win half the howdown hand the overall win rate are perent for Skilled and 3.4 perent for andom. Table 3. Probabilitie and Win ate - Overall Win ate - Skilled Win ate - andom Stage Probability!! Pre-lop lop Tum iver Showdown* OVEALL *Auming eah player win half the howdown. In the etion below, large ale imulation were ued to obtain an etimate of the perentage of tie in the andom veru Skilled enario and validate the analytial reult above. andom veru Skilled - Simulation Analyi Simulation of one billion hand eah were onduted (implemented uing a C++ program) for the andom veru Skilled Texa Hold'Em game deribed above. In UNLV Gaming eearh & eview Journal+ Volume 3 Iue 9

10 order to ompare the imulation reult with the mathematial analyi preented in the previou etion, the firt imulation were onduted without evaluating and omparing hand at howdown, auming intead (a wa done in the analytial analyi) that half the howdown are won by eah of the two player. In the eond imulation, hand were evaluated and ompared at howdown, allowing for an etimate of the perentage of tie. Table 4 how the reult of one billion imulated hand when the player' hand are not ompared at howdown. The reult reveal that Skilled won perent of the hand and $6.44 billion, or an average of I.6 big blind per hand. andom won 3.I4 perent of the hand and lot $6.44 billion. Sine howdown hand are not evaluated (it i aumed eah player win half the howdown) there are no tie. Thee reult are onitent with and onfirm the analytial analyi and Table 3 final figure preented above.!fable 4. One Billion Hand Played -No Hand Evaluator*! wwon W olded ~Won/Lot!andom 3,4I2,489 r37,76,985-6,439,579,675 Skilled r68,587,5ii p f+-6,439,579,675 lfie p lfotal Hand I,000,000,000 * Auming eah player win half the howdown. Table 5 how the reult of one billion imulated hand when the player' hand are ompared at howdown, allowing for tie. In thi eond imulation, Skilled won perent of the hand outright, andom won 3.04 perent outright, and perent of the hand were tie (plit pot). Note that thee reult are onitent with thoe of the no-hand-evaluation imulation in the ene that had the 62,833,476 (from I,000,000, ,66,524) howdown hand been evenly divided between the two player, the reulting perentage of hand won by Skilled and andom would be perent and 3.I4 perent, repetively, the ame figure a were obtained in Table 4 (a well a thoe from the analytial reult in Table 3). A wa the ae with the previou imulation, the overall win amount for Skilled i approximately $6.44 billion, or an average of.6 big blind per hand. Table 5. One Billion Hand Played - Hand Evaluator WWon Itt olded $Won/Lot andom 30,390,26 r37,66,524-6,439,490,668 Skilled ~67,556,925 p +6,439,490,668 Tie 2,052,84 Total Hand,000,000, andom veru Skilled - Summary When a killed player employing a raie-when-poible trategy i up againt a player making haphazard (random) deiion in a limit Texa Hold'Em game, the killed 0 UNLV Gaming eearh & eview Journal+ Volume 3 Iue I

11 Toward Legalization of Poker: The Skill v. Chane Debate player win full pot 96.8 perent of the time and plit pot 0.2 perent of the time, with a win rate average of.6 big blind per hand (for omparion, three big blind per hour, or 0.0 big blind per hand auming 30 hand per hour, i onidered reaonably good among top poker player). That a player an implement a trategy to defeat another player uh a high perentage of the hand, with uh a large win rate i trong evidene that kill predominate over hane in determining the outome in poker. Sine a player making random deiion would fare jut a well a any other player in a game of pure hane (uh a roulette), thee figure for the Texa Hold'Em game analyzed here peak to the overwhelming dominane of the role of kill, expreed through betting trategy, in poker. The following etion preent (additional) imulation of full-table Texa Hold'Em and Seven Card Stud that further upport the notion that kill i a predominant fator in poker. Analyzing Skill in Poker (Part B) Senario 2: ull-table Simulation Thi etion examine the role of kill in full-table poker game in whih ome player are killed and other are not. Computer imulation of one million hand were run for everal ituation for the two mot popular poker game, Texa Hold'Em and Seven Card Stud, uing Wilon' Turbo Texa Hold' Em and Turbo Seven Card Stud oftware. Beaue the imulation program allow for different player betting profile to be loaded into the game, it i poible to ee the effet, in term of money won or lot, when unkilled player are playing againt killed player. If kill i a predominant fator, the unkilled player will tend to loe a the ard 'even out' and it will beome apparent that the killed player beat the unkilled player a more and more hand are dealt. Suh an approah i viable with poker beaue player profile an be deigned and et to play at different level of kill. (It i impoible, for example, to deign a imulation of roulette in whih the long-run winning of player differed. Thi i beaue there i no kill in roulette; regardle of how a player plae bet, or trie to employ any o-alled 'ytem' or trategy, the long-run expeted winning in double-zero roulette will be a lo of 5.3 perent of the money wagered.) Hourly win rate are omputed auming 30 hand per hour. The next two etion detail the imulation reult for Texa Hold'Em and Seven Card Stud, repetively. Texa Hold'Em: $20/$40 betting truture. our ae were examined for a ten-player $20/$40 limit Texa Hold'Em game: (I) ten killed player, (2) one unkilled player and nine killed player, (3) nine unkilled player and one killed player, and (4) five killed and five unkilled player. The reult are a follow. UNLV Gaming eearh & eview Journal+ Volume 3 Iue

12 igure. Hold'Em- All Skilled $200 $ $ ~~--~~ $50t-----~~~~------~~~~~ $O I ro 00 0oJooo -$ ~~ =~~~--==~~ ~Skilled A -a-- Sklled B ---G-Skilled C --+--SkilledD ~SkilledE Skilled --e--sklled G ;-Skilled H -$ ~ Skilled I ---Skilled J -$50r ~~~ $ r~ $250 Number of Hand igure I how hourly win rate in a game with ten killed player with idential player profile. A expeted due to the idential player profile, differene among final win figure for the ten player were relatively mall The final win figure how that ix player lot and four player won, with net win ranging from -$32,047 to $6,400, or -$3.96 to $3.49 per hour. igure 2. Hold'Em - One Unkilled & Nine Skilled $600 $400 $200 $0 -$200 ~ ~ ~ ~ ~ Skilled A g..._ Skilled B ~SiulledC -r-skilled D -$ Skilled E ---e-skilled -$ tr-skhied G --+--Skilled H -$800 -$,000 -$,200 ~ ~ --Skilled I..._unkilled -$,400 Number of Hand igure 2 how hourly win rate in a game with one unkilled player and nine killed (idential profile) player. The unkilled player lot approximately $38.8 million, ompared to a win of roughly $4.3 million for eah of the killed player (win amount for the killed player ranged from $3.7 million to 4.7 million). Thi onvert to an average hourly win rate of approximately $29 for the nine killed player (range from $ to $42) ompared to an hourly lo of $,65 for the unkilled player. 2 UNLV Gaming eearh & eview Journal+ Volume 3 Iue

13 Toward Legalization of Poker: The Skill v. Chane Debate igure 3. Hold'Em- One Skilled & Nine Unkilled $5,000 l ~.ooor---~~ $3, \ unkilled -----Q-Unk!lled2 -e-unkilled3 ~Unkilled4 ---Unkilled 5 --e-unkilled 6 --tr- Unkilled Unkilled8 -Unkilled9 -killed A ~$3,000 Number of Hand igure 3 how hourly win rate in a game with nine unkilled (idential profile) player and one killed player. Eah of the unkilled player lot money - average lo wa approximately $3.3 million (ranging from $2.4 million to $3.8 million), or almot $00 per hour ($7 to $4)- while the one killed player won almot $30 million, or $880 per hour. $2,000 igure 4. Hold'Em ~ive Skilled & ive Unkilled $,500 t-' $,000 r '~~ '-a-~::;,,,;:::;.,-;-a-, $SOO ----G----Sklled 8 L ~4:':!=~~~~~-~~"'=''=='"""E==:==="L---- -e-sket I ~Skilled D ----e----skilled E -<>- Uookilled $0 r------=rfl---~~ ~ --tr-unkilled Unkilled 3 I -Unki!led4 -$500 ~ =j;::-:==~==~~~~5~~~==~--- l-unkllled5 Number of Hand igure 4 how hourly win rate in a game with five unkilled (idential profile) player and five killed (idential profile) player. The killed player all won and the unkilled player all lot. Win amount for the killed player ranged from $8.2 million to $9.6 million (average $8.8 million), or $547 to $587 per hour (average hourly win $565). Loe for the five unkilled player ranged from $7.2 million to $9.4 million (average $8.8 million), or $57 to $583 per hour (average hourly lo $565). UNLV Gaming eearh & eview Journal+ Volume 3 Iue 3

14 Seven Card Stud: $6/$2 betting truture. our ae were examined for an eight-player $6/$2 limit Seven Card Stud game: () ten killed player, (2) one unkilled player and nine killed player, (3) nine unkilled player and one killed player, and (4) five killed and five unkilled player. The reult are a follow. $60 igure 5. Seven Card Stud -All Skilled ~ ~~ $20 I w I - ~ "" 5 "' -$20 I !z:::,/ --a- Skilled A -a-skilled B --tr- Skilled C ---+-Skilled D killed E :----.-Skilled ---- Sktlled G --e---skilled Hi -~ ~ $60 j Number of Hand igure 5 how hourly win rate in a game with eight killed player (all player profile idential). A wa the ae for the analogou Hold'Em ituation (where all player profile were idential), differene among final win figure were relatively mall. The final won/lo figure how that five player lot money and three player won, with net win ranging from -$0,30 to $8,974, or -$0.30 to $0.57 per hour. igure 6. Seven Card Stud- One Unkilled & Seven Skilled $00 $50~~----~~--~~ $0 ~~~ I $ ~~~ \ -----a-sktlleda -----e--skilledb SkilledC._...SkilledD -killede ~Skilled ---G-Skilled G -tl'----unskilled $00 <6 -$50 Number of Hand igure 6 how hourly win rate in a game with one unkilled player and even killed (idential profile) player. The unkilled player lot $94,74 ompared to an average win of $34,453 for the even killed player (ranging from $03,909 to $72,566). Thi onvert to an average hourly win rate of $4.03 for the even killed player (from $3.2 to $5.8), ompared to an hourly lo of $28.24 for the unkilled player. 4 UNLV Gaming eearh & eview Journal+ Volume 3 Iue

15 Toward Legalization of Poker: The Skill v. Chane Debate igure 7. Seven Card Stud- One Skilled & Seven Unkilled $200 $50 \ $00 $50 $0 -$50 -$00 \ \ ~\,_ ~ ~ -----t--unkllled ---tr--unkilled 2 : -a- Unk~lled 3 -Unkdled ~ ~ ~ unkdled5 'Y~ ~ :>-Unklled6 oonoo e--unkilled Skilled A -$50 Number of Hand igure 7 how hourly win rate in a game with even unkilled (idential profile) player and one killed player. Eah of the unkilled player lot money - average lo wa $30,07 (ranging from $94,242 to $92,203), or $3.90 per hour ($2.83 to $5.77) while the one killed player won nearly $90,7, or $27.30 per hour. igure 8. Seven Card Stud - our Skilled & our Unkilled $0 t-----<>"9'"'=-"~-' ;~-----~ ~.._killed A a-skilled B -$20 t---;;=:====~~2==~~~:::::~=~~~:=::==~~' SkilledD Sklled C -unkllled -,r-unkilled 2 -$ ,L G-Unkllled3 -Q- Unkilled 4 Number of Hand igure 8 how hourly win rate in a game with four unkilled (idential profile) player and four killed (idential profile) player. In thi ae the killed player all won and the unkilled player all lot. Win amount for the killed player ranged from $500,03 to $60,762 (average $556,080), or $5.00 to $8.05 per hour (average hourly win $6.68). Loe for the five unkilled player ranged from $55,633 to $594,786 (average $556,080), or $5.47 to $7.84 per hour (average hourly lo $6.68). UNLV Gaming eearh & eview Journal+ Volume 3 Iue 5

16 Summary of ull-table Simulation The reult of thee full-table imulation of Texa Hold'Em and Seven Card Stud game trongly ugget that the money in poker hift from the weaker player to the more killed player, and that over time a killed player will beat an unkilled player. Conluion and eperuion The analyi and reult preented in thi paper ugget that poker i a game of predominantly kill, the kill element expreed through the player' betting trategy; i.e., the deiion on whether to hek, bet, all, raie, or fold. The importane of thi fatual deiion an not be undertated from a poliy perpetive beaue the very legality of poker hinge on the onluion. Though the tudie here foued primarily on Texa Hold'Em, with ome examination of Seven Card Stud, imilar reult would be expeted for other form of poker. urther reearh examining the quetion of kill veru hane for different variant of poker and game enario i needed, but the evidene preented here ugget trongly that poker i a game of predominantly kill. The ramifiation to the traditional gaming indutry if poker i determined to be a game of kill relate motly to the loe relationhip between the poker indutry and the aino indutry. Many tate that allow aino alo permit poker, inluding New Jerey, Nevada, Miiippi and Mihigan. Likewie, ome tate without traditional land-baed aino alo permit ome form of poker, inluding California and lorida. Thi broader appeal of poker together with it inreaed expoure over televiion and the Internet ha made the aino relevant to a broader bae of patron. or example, the number of player ompeting in the World Serie of Poker (WSOP), held eah year in La Vega at the io Hotel, roe exponentially for the firt 35 year, from ix player in the firt main event to nearly 9,000, eah of whom paid a buy-in of $0,000, in 2006 (PokerCainoGuide.om). The tournament itelf i fed by atellite tournament held aro the world in both aino and non-aino environment. The effet of the expanion of poker on the Internet, televiion and other non-aino loation ha had a diret impat on the aino. A one Lo Angele Time reporter noted in May 2007: Poker ha never been bigger in Vega than right now. It wa almot extint five year ago, but there are now more than 00 poker room in Sin City, and probably two or three new one added eah month. The boom reflet the national poker raze touhed off in 2003 by the Travel hannel' World Poker Tour, whih popularized the ue of the "poket am," the hidden amera that let the audiene ee the hole ard of eah player. Overnight, one of the mot boring ativitie to wath on TV beame high drama. Throw in the no-limit betting rule in Texa Hold 'Em and you have nothing le than weet mui to Vega aino. (Cooper, 2007) The value of the expanion of poker opportunitie doe not imply mean expanion of the exiting poker room. It alo bring a new demographi of player to the traditional aino. Poker player tend to be younger than the average aino patron and repreent the generation that the aino need to remain viable. Moreover, when Harrah' urveyed the player at the 2004 WSOP, it diovered that more than half of World Serie of Poker player enjoy playing other table game and about a quarter alo play lot and plae bet in the port book (Profile of the Amerian Caino Gambler: Harrah' Survey 2004 ). The determination that poker i a kill game will open new market, both potentially land-baed and on the Internet. Creating new growth and demand for the produt will inreae the demand for inreaed poker apaity in the traditional aino, help olidify the popularity of major tournament like the WSOP and the United State Poker Championhip at Trump Taj Mahal in Atlanti City and bring an entire new demographi to the aino where table game and lot mahine an be ro-marketed. 6 UNLV Gaming eearh & eview Journal+ Volume 3 Iue I

17 Toward Legalization of Poker: The Skill v. Chane Debate Appendix Deiion Tree and Probabilitie for Skilled veru andom Cae : andom ( ) = Small Blind; Skilled ( S) = Big Blind IPE-~LOP Outome lop SWin lop SWin lop SWin lop SWin lop S Win I Probability ~~;;;;~! LOP , I'~"' Outome I ;Turn ! Turn S Win C Turn S Win Play_er I Probability TUN Outome I i ----!ive_r i is Wm iver i win C ~iver S Win I Probability IVE Outome I Showdown! SWin I Showdown' SWin ' Showdown I SWin i I I Probabilit ting Legend: C =Call =old = irt aie 2 = Seond aie 3 Third aie 4 = ourth aie = No deiion option Skilled Win Pre-lop: Skilled Win on lop: Skilled Win on Turn: Skilled Win on iver: Skilled Win on Showdown: Pre-lop to lop: lop to Tum: 00 Turn to iver: 00 iver to Showdown: 00 To lop: To Turn: To iver: To Showdown: ANDOM WINS: SKILLED WINS: UNLV Gaming eearh & eview Journal+ Volume 3 Iue 7

18 Cae II: Skilled ( S) = Small Blind; andom ( ) = Big Blind PE-LOP i Outome lop SWin lop SWin lo a 0~ Probabilitv! 0~ I 3 0. I LOP SWin :"~- Tum,Outome Turn is Win I Tum SWin Tum SWn!Turn Probabil~ I ITUN Play_er ioutome ~ ~~t~rn ~iver S Win C iver IS_Win ver SWin \iver 'SWin iver Plaver i : ' Probabilitv: [ ' , IVE 0 ~-~ 'IS Wm Sho~down SWm C Showdown ~ ~~h~;~own x lwin!showdown S Win Showdown Plaver Probabilit , ~:~:~~~~~ ! ] ting Legend: C =Call =old = irt ae 2 = Seond aie 3 = Thrd aie 4 = ourth aie = No deiion option =Chek Skilled Wm Pre-lop: Skilled Wn on lop: Skilled Win on Turn: Skilled Win on iver: Skilled Win on Showdown: Pre-lop to lop: lop to Turn: Turn to ver: ver to Showdown: To lop: To Turn: To iver: To Showdown ANDOM WINS: SKILLED WINS: Example Tree Analyi for Cae, lop Skilled Deiion Jl andom Deiion.ll Skilled Deiion 2 u NA andom Deiion 2 Skilled DeS0n 3 andom Deiion 3 ~ NA I NA I ul I NA I 0 I 0 0 I u< 'L/ I] old-bl NA b NA I NA '==='=~:> () U Call UNLV Gaming eearh & eview Journal+ Volume 3 Iue