Risk Assessment of Progressive Casino Games

Transcription

1 Risk Assessmet of Progressive Casio Games Joh Quigley Uiversity of Strathclyde, Glasgow, UK Matthew Revie* Uiversity of Strathclyde, Glasgow, UK * Correspodig author. Address for correspodece Departmet of Maagemet Sciece, Uiversity of Strathclyde, 4 George St, Glasgow, G1 1QE, UK. Abstract This paper presets a ivestigatio ito the properties of a stochastic process whereby the value of a fud grows arithmetically ad decays geometrically over discrete time periods. While this geeral structure is applicable to may situatios, it is particularly prevalet i may casio games. This ivestigatio was motivated by a request for support by a casio operator. Statistical models were developed to idetify optimal decisios relatig to the casio game cocerig settig the iitial jackpot, the probability of wiig each prize, ad the size of the prizes. It is demostrated that all momets of the process coverge asymptotically ad the limitig distributio is ot Normal. Closed form expressios are provided for the first momet as well as ivestigate the quality of approximatig the distributio with a Edgeworth Expasio. The case that motivated this iitial ivestigatio is preseted ad discussed. Keywords: Stochastic Processes, Gamig, Risk, Optimisatio 1

2 1 Itroductio There were 144 casios operatig i the UK with a workforce of 14,667 i 212 (Gamblig Commissio, 21) with a estimated 5, casios worldwide. I the UK, almost 6 Billio was gambled i 212 (Gamblig Commissio, 21), while worldwide, approximately $1 Billio was gambled i casios. The casio idustry is becomig more substatial with worldwide growth expected to be aroud 9%; fuelled i part by large growth i Asia (PriceWaterhouseCooper, 21). I the largest markets, for example the US, Chia, Sigapore, UK ad Australia, gamblig is heavily regulated by the govermet. Regulatio is used to esure that games are accurately advertised to players ad that miimum payouts are made. This paper focuses o regulatio withi the UK oly; however, similar regulatio exists i all the major markets. A key part of the regulatio is esurig that the risk associated with a game is well uderstood. However, as games become more complex, ituitio is challeged ad there is a greater eed for stochastic models to explicate the relatioship betwee settig the coditios of the games ad outcomes. Decisio support models such as optimizatio, stochastic modellig, data miig, decisio theory ad forecastig, have played a key role i supportig decisio makig withi casios. or example, optimizatio was used by Bayus ad Gupta (1985) to model floor cofiguratios of slot machies, data miig was used by (Hedler ad Hedler, 24) to model reveue maagemet of a casio, prospect theory was used by (Barberis, 212) to model player strategy, while mathematical models have bee used to model casio VIP rebate schemes (Gao et al., 211). (Gaisbury et al., 212) aalysed player data to assess the behaviour of differet player types. More broadly, optimisatio was used by (Mills ad Pato, 1992) to optimizig security officers, decisio theory by (Tsoukias, 28) to model ratioality of players, ad data miig by (aregh ad Leth-Steese, 211)to model game prefereces of players. Statistics has also bee used to support player decisio makig. Werthamer (Werthamer, 25) modelled optimal bettig schemes for players o blackjack, ad the exteded this work i (Werthamer, 28) to model players `hoppig' betwee tables to maximize expected retur. As illustrated, decisio support models have played a key role i supportig decisio makig across a wide rage of casio operatios, icludig the developmet ad desig of casio games. I this paper, we are cocered with supportig a casio set the operatioal parameters of a ew game that is beig implemeted across a chai of casios. This aalysis takes two forms. 2

3 irst, we model the stochastic behaviour of the game. Secod, we optimize the profitability of the game through modificatio of these iitial operatioal parameters ad by capturig the arrival rate of players through a fuctio of the jackpot. The game is structured as follows. The value of the jackpot followig the th had is deoted by. We refer to each play as a had ad a sequece of hads termiatig i the jackpot beig wo is referred to as a game. Such games start with a iitial amout of moey i the jackpot provided by the casio which we deote by. Settig the value of is oe of the key decisios that the casio operators must make for the game. Hads are the played sequetially. Each player pays 1 to play, of which the casio takes h ad the remaiig 1 h is added to the jackpot. The casio ca set the value of h but the regulator bouds it betwee ad.3. A radom mechaism the determies the wiigs to the player, which comprise a) othig, b) a fixed amout, or c) a proportio of the jackpot, which may iclude wiig the etire jackpot. The probabilities associated with the prizes do ot chage throughout the game, but ca be determied by the casio at the begiig. The wiigs are removed from the jackpot ad if the value of the jackpot is greater tha, the the game cotiues. Through derivig a explicit expressio of the value of the jackpot (similar to the work of (Sugde, 21) applied to Keo) we support decisios made by the Director of Operatios for the casio i determiig appropriate probabilities ad associated prizes. The decisios that we support iclude the followig: the size of the iitial jackpot; the size of the cotributio to the jackpot per had; the proportio of the jackpot wo; the size of each prize; ad the probabilities associated with each prize. I additio, our aalysis reveals behaviour of the game that is couterituitive, demostratig the eed for aalytical ivestigatio of the process. or our iitial study, we were iterested i a particular game structure described by the casio. However, this geeral descriptio of progressive jackpots is applicable to various casio games. Progressive side wagers (Nevada Gamig Commissio ad State Gamig Cotrol Board, 25), are games with a progressive jackpot available to players playig a traditioal game, where the radom mechaism that determies the prize is derived from the realizatios o-goig durig the traditioal game. or example, durig a game of blackjack, players will be provided the opportuity to make additioal bets o the outcome of the cards dealt through playig a 3

4 progressive jackpot, where prizes will be allocated depedig o the cards realized. The structure of the game remais idetical so that players ca play both the traditioal game ad the progressive game. The structure of the paper is as follows. I Sectio 2, closed form solutios for the mea of the value of the fud are preseted. The asymptotic behaviour of the first momet is assessed. I Sectio 3, a Edgeworth expasio is derived for approximatig the distributio fuctio of the value of the jackpot. The case that motivated this ivestigatio, based o aalysis commissioed by a UK casio chai, is preseted i Sectio 4. By modellig the arrival rate of customers as a fuctio of the jackpot, we optimize the iitial coditios of the game to maximize profit over caledar time. ially, i Sectio 5 we summaries ad coclude the paper, ad outlie future work. 2 Derivig Momets 2.1 Outlie I this sectio we derive a explicit expressio for the first momet associated with progressive jackpot games. I Lemma 1 we set up a recursive expressio for derivig all momets. This is used to determie a explicit expressio for the first momet described i Theorem 1. This is followed by Corollary 1 which establishes a asymptotic expressio for the first momet of the jackpot, showig it coverges to a fiite mea. Corollary 2 establishes that this covergece is mootoic. Theorem 2 derives coditios for all momets of the series to coverge to a fiite expressio. Theorem 3 presets a expressio for the Momet Geeratig uctio of the series, followed by Corollary 3, which presets a geeral expressio for all asymptotic momets ad ca be used to show that the limitig distributio of the value of the jackpot is ot Normal. All proofs are i the Appedix. The results from this sectio will be used i Sectio 3 for derivig a Edgeworth Expasio to approximate the distributio fuctio of value of the jackpot. The aalysis ad derivatios i Sectio 2 ad Sectio 3 are coditioal o the jackpot ot beig wo. 4

5 Throughout this sectio ad the remaider of the paper, the otatio described i Table 1 will be used. Symbol Descriptio Iitial fud provided by the casio p Probability jackpot is wo i q Radom variable represetig the value of the fud after i hads have bee played Probability of wiig proportio of jackpot Proportio of jackpot wo as proportioal prize I i X J i h Radom variable idicatig whether the proportioal wi has bee made o the i th had Radom variable represetig fixed amout wi give the proportioal prize is ot wo where each X has the same distributio as X N Radom variable idicatig whether or ot the jackpot has bee wo o the i th had Value take by the casio o each had Table 1: List of otatio used throughout the paper 2.2 Closed form solutio of first momet We cosider the progressive jackpot with oe of two types of prizes. A fixed prize is awarded o the th had with a specified probability that will result i a arithmetic chage o the series deoted by X. A proportio of the jackpot is awarded that will result i a geometric chage i the value of the jackpot. We ca express the value of the jackpot followig the th had recursively i the followig maer through a coditioig argumet, where the idicator 5

6 variable I deotes whether the proportioal prize has bee wo. As such the radom variable X is coditioal o the proportioal prize ot havig bee wo: ( (1 h) X )(1 I ) 1 ( (1 h))(1 ) I, 1,2, 1 where: I 1, if proportioal prize awarded;, else. The followig key assumptios are made: 1. The iitial jackpot is. 2. Each X is idepedetly ad idetically distributed for all h is added to the jackpot for each had as it costs 1 uit to play ad the casio take h. 4. A proportio of ca be wo o the th had with probability q. Our first result presets a recursive relatioship, permittig the evaluatio of all momets for the jackpot at a specified had. Lemma 1 j j i wije 1 i E[ ] [ ] Where w ij j i [ [(1 ) j i ](1 ) (1 ) j E h X q h i (1 ) j q] 6

7 The Lemma ca be used to derive closed form expressio for momets. Upo ispectio of the weights, we see that momets ca be expressed as sums of geometric series. Theorem 1 makes use of this relatioship ad provides a closed form expressio for the first momet of the jackpot for ay specified had. Theorem 1 Give that the four assumptios hold, the the expectatio of the jackpot after hads is: 1 (1 q) E[ ] (1 q) ((1 h)(1 ) q ((1 h) E[ X ])(1 q)) q Depedig o the game parameters, the expectatio of the jackpot decreases mootoically, icreases mootoically or remais costat with respect to the umber of hads. This is of iterest i determiig the coditios of the game, such as the probability of wiig, the associated prizes ad the iitial value of the jackpot, as it would be udesirable for a casio to have a expected jackpot that decreases over time. rom this, we ca develop two corollaries. These two corollaries are ot immediately ituitive but they aturally emerge from Theorem 1. Corollary 1 demostrates that the jackpot reaches a limit; which is ot i itself immediately obvious. Corollary 1 The limit of E [ ] as is (1 h)(1 ) q E[1 h X ](1 q) lim E [ ] q 7

8 Upo ispectio of Theorem 1 we ca observe that the series will coverge to a fiite mea uder certai coditios. The explaatio as to why the fud coverges is because the proportioal prize, i.e. $\gamma$, discouts the ifluece of the past geometrically. Corollary 2 provides coditios as to whe it is icreasig, decreasig or statioary. Corollary 2 ((1 h)(1 ) q ((1 h) E[ X ])(1 q)) E [ ] if, q ((1 h)(1 ) q ((1 h) E[ X ])(1 q)) E [ ] if, q ((1 h)(1 ) q ((1 h) E[ X ])(1 q)) E [ ] if. q The impact of Corollary 2 is that the fud mea may be mootoically decreasig. The Director of Operatios stated that this was somethig the casio would wish to avoid. 2.3 Asymptotic Expressios for Momets We ca exted the results from Lemma 1 to derive a recursive asymptotic expressio for all momets of the series. This is stated i Theorem 2. I combiatio with Corollary 2, these are easily derived. Theorem 2 8

9 j1 i wije j i E 1 w i i where: E lime jj ially, we derive a asymptotic expressio for the Momet Geeratig uctio i Theorem 3. We make use of the Momet Geeratig uctio i Corollary 3 to obtai a expressio for the asymptotic cumulats. These are used i Sectio 3 with the Edgeworth Expasio. Theorem 3 The Momet Geeratig uctio of the value of the jackpot after the ot bee wo, i.e. M () z, is: th had assumig it has M z e M z q M z M z q z(1 h)(1 ) ( ) ( (1 )) ( ) (1 h) X ( )(1 ) 1 1 where ( ) [ z(1 h X ) M ] (1 h X) z E e. Corollary 3 We deote the limit of the momet geeratig fuctio as lim M ( Z) M ( Z). rom this, z(1 h)(1 ) e q M ( z) M ( z(1 )) 1 M ( z)(1 q) (1 h) X 9

10 It ca be show through comparig coefficiets from a series expasio i $z$ that the Momet Geeratig uctio of the Normal distributio caot be represeted i the form expressed i Corollary 3. As such, the limitig distributio of is ot Normal. Usig the momet geeratig fuctio, we ca develop the cumulat geeratig fuctio. The relatioship betwee momets ad cumulats ca be expressed recursively through the followig relatioship (Small, 21): i1 i i 1 i j E i,,. j E j1 j 1 rom this, the followig corollary emerges. Corollary 4 j, (1 h)(1 q) (1 q) E[ X ], j 1 q j d l(1 M (1 h) X ( z)(1 q)) j z dz, j 2 j Summary We ca see upo ispectio that the momet geeratig fuctio expressed i Corollary 3 does ot possess the form of the momet geeratig fuctio of a Normal distributio ad as such the limitig distributio of the value of the jackpot does ot coverge to a Normal distributio. However, we have ot established how close a approximatio the Normal distributio will be to the actual asymptotic distributio, which is cosidered i the followig sectio. I Sectio 3, we illustrate that the Edgeworth Expasio will coverge as a limitig series to the asymptotic distributio, ad as such provide a good approximatio through adjustig the Normal 1

11 distributio for skewess ad kurtosis. I Sectio 3, we derive a Edgeworth Expasio to describe the probability desity fuctio (PD) as well as the Cumulative Distributio uctio (CD) of the value of the series. This is evaluated ad bechmarked agaist the Normal distributio, where we see the Edgeworth Expasio substatially outperforms the Normal approximatio. 3 Edgeworth Expasio 3.1 Outlie The Cetral Limit Theorem states that uder certai coditios, a suitably re-scaled average of a sample of radom variables will coverge to the stadard Normal distributio as the sample size icreases. The Edgeworth Expasio was derived to improve upo the Normal approximatio of the distributio of such a average for fiite sample sizes. The adjustmets that are proposed icorporate iformatio pertaiig to the higher momets or cumulats of the distributios. I theory, the expasio ca be exteded to icorporate ay umber of higher momets but most commo are to iclude iformatio about the third ad fourth momet to assess skewess ad kurtosis. (Hall, 1997) credits (Chebyshev, 189), (Edgeworth, 1896) ad (Edgeworth, 197) with the iitial coceptio of the idea. The Edgeworth Expasio is a ifiite series expasio startig after a Normal approximatio to adjust for skewess, kurtosis ad higher momets or cumulats. To esure covergece we require 2 Ee ((Cramer, 1928) as cited i (Hall, 1997). I this case, for each give, ca oly take a fiitely may real values; hece the expectatios are less tha ifiity. We propose usig the Edgeworth Expasio for approximatig the distributio of values of. for large We kow from Sectio 2 that the momets coverge ad the coverget distributio is ot Normally distributed. We seek to obtai a suitable approximatio of the asymptotic value of jackpot to explicate the relatioship betwee the structure of the game ad value of the jackpot. 11

12 I Sectio 3.2 we show a expressio of the CD for from the Edgeworth Expasio. The Edgeworth Expasio was coceived to support iferece o averages, but we propose usig it o a sample of 1 i the sese that it is the realizatio of oe sample path of the stochastic process. As such, we seek to evaluate the error associated with the approximatio. This is cosidered i Sectio Derivatio of the Edgeworth Expasio Iitially we re-scale expressed as such that it has a mea of ad a stadard deviatio of 1. W i the followig: W 1,. 2, This is or the remaider of this paper, we will cosider a Edgeworth approximatio of the distributio of by trucatig the Edgeworth expasio after the fourth momet. This trucated Edgeworth expasio is expressed i (6), see (Small, 21). P W w w w w 6 3, 2 1 where w ad w 3 2 2, 2 4, 3, w w w w w , 72 2, are the CD ad PD of the stadard Normal distributio respectively. Ispectig the expressio of the CD i equatio (\ref{eq_cdf}), we see that the PD of the Normal distributio acts as a weightig fuctio, where the weight will be greatest ear the mea. Moreover, the third cumulat 3, measures the effect of skewess, either icreasig N or decreasig the percetile obtaied from the Normal distributio. 12

13 3.3 Assessig the accuracy of the Edgeworth Expasio I this sectio, we use the above expressio of the Edgeworth expasio to evaluate the accuracy for a rage of differet parameter values. our parameters are varied:, the iitial fud; q, the probability of wiig a proportio of the jackpot;, the proportio of the jackpot wo; ad, the umber of hads played without the jackpot beig wo. We use the asymptotic estimates of the third ad fourth cumulats. Table 2 illustrates the values simulated for each parameter. Parameter Values take, 1, 1, 1 q.1,.2,.1, ,.1,.25,.5,.75 25, 1, 1 Table 2: Values assessed durig simulatio} Usig Matlab R29b, simulatio was carried out to assess the size of the fud for each combiatio above, i.e. 24 differet combiatios. Each combiatio was simulated 1 times. rom this, the empirical CD for the size of the jackpot give the parameter values was calculated. A compariso betwee the empirical distributio, ad the Normal ad Edgeworth expasio was carried out o four differet measures. These measures were the maximum differece betwee the empirical ad the Normal/Edgeworth expasio (deoted by M N ad M E respectively), the differece at the 5th, 5th ad 95th percetiles (deoted by 5 N, 5 N, ad 95 N respectively for the Normal ad similar otatio for the Edgeworth expasio). The results showed that o 96 % of the simulatios, the Edgeworth expasio outperformed the Normal distributio for the maximum error, ad for the error at the 5th, 5th ad 95th percetiles, outperformed the Normal distributio o 95.3 % of the simulatios. igures 1a - 1d 13

14 are scatterplots of the M E agaist M N, 5 E agaist 5 N, 5 E agaist 5 N ad 95 E agaist 95 N. As referece, they have bee grouped by values ad the 45 lie is provided as a referece. Simulatios where the Normal out (uder) performs the Edgeworth expasio lie above (below) this lie. The Edgeworth Expasio outperforms the Normal for almost all iput variables. As ca be see from igures 1a - 1d, the vast majority of the errors fall below the 45 lie, idicatig that the error for the Normal distributio is larger tha for the Edgeworth expasio. 14

15 igure 1: Scatterplot of Normal distributio agaist Edgeworth expasio maximum error idicatig that Edgeworth expasio outperforms Normal distributio (These are labeled i the text as igures 1a (top left), igure 1b (top right), igure 1c (bottom left) ad igure 1d (bottom right) igure 1a illustrates that the maximum error for both the Normal distributio ad the Edgeworth expasio appears to be radomly scattered aroud a cetral poit. However, for the error assessed at percetiles, i.e. igure 1b - 1d, a clear patter emerges. As the error associated with the Normal distributio icreases, for the differet values, the Edgeworth expasio error begis to decrease liearly, reaches zero ad the icreases agai. rom igures 1c ad 1d, we see that the size of the error for the Normal distributio is heavily iflueced by the value. or each of the differet performace measures, chagig the value of has a step chage effect i the error associated with the Normal distributio. Coversely, the Edgeworth expasio, while affected, does ot appear to be affected as largely by chagig. or the 5th percetile ad 95th percetile, see igures 1c ad 1d, it appears that the Edgeworth expasio error is idepedet of. There are 25 observatios (out of 3) where the Normal distributio outperforms the Edgeworth expasio for at least oe of the four performace measures. Table \ref{tb_norm} cotais the parameter values for each of the 25 combiatios. 15

16 Table 3 Parameter values where the Normal distributio outperforms the Edgeworth expasio q q rom Table 3, we see that is the oly variable that has the same value reoccurrig. The oly clear patter that emerges from Table 3 is the effect of. or low values of, it appears that the Normal distributio outperforms the Edgeworth expasio - however, it is worth highlightig that the Normal distributio outperformed the Edgeworth expasio o oly 25 of the 3 differet combiatios ad o oly 6 occasios, did the Normal distributio outperform the Edgeworth expasio o all four performace measures. 4 Applicatio 4.1 Outlie 16

17 I this sectio we illustrate the applicatio of the results derived i Sectio 2 ad 3 through the case which motivated the aalysis. I Sectio 4.2 we describe the iitial coditios of the game as defied by the casio. To date, o statistical techiques have bee used by the casio to set these parameters. I Sectio 4.3 we derive expressios for the momets of the game ad illustrate the implicatios of the coditios set by the casio as well as illustratig the Edgeworth Expasio to approximate the distributio. I Sectio 4.4 we explore the expected value of the jackpot whe it is wo give the iitial parameters. I Sectio 4.5, we review the decisios the casio has made regardig the iitial coditios ad through optimizatio, idetify optimal parameter values over caledar time. 4.2 Iitial Coditios of the Game The progressive jackpot uder cosideratio for this aalysis was a progressive side wager game, where additioal bets would be made available durig a game of blackjack. While playig a game of blackjack a player could choose to compete for the jackpot by payig 1. This is i additio to the miimum wager required by the player to play the stadard blackjack game. The casio take $.3 ad cotribute the remaiig $.7 ito the jackpot. The player is the dealt two cards from a shoe cotaiig six decks. If either card is ot a ace, the player loses their progressive bet. If both cards are aces, the player may cotiue. If the third card is ot a ace, for the purposes of the progressive jackpot, they must stop; however, the player still receives the prize, i.e. oce the chai is broke, they are still rewarded for their previous cards. If the third card is a ace, the player may cotiue. It is assumed that players will always cotiue to draw if they have the chace. As the probability of exceedig 21 ad `bustig' at this stage is zero, this seems a reasoable assumptio. The probability of a player beig dealt their first two cards as aces from the same suit is P(6, 2) P(288,1) The deomiator calculates the umber of ways to deal 3 P(312,3) cards from 312, where order matters (as we must have the first two cards beig aces ad the last oe ot). The umerator calculates the umber of ways the first cards ca come from a particular suit ad the third card ot beig a ace. As there are four particular suits we multiply 17

18 by 4. Similar calculatios are coducted for each sceario. Table 4} is a summary of the differet scearios required to wi, the associated probabilities ad the prize. The casio start the jackpot with $1. Table 4: Probability of player wiig each prize Outcome Probability Prize 2 Aces of the same suit Aces but differet suit Aces of the same suit Aces but differet colour % of jackpot 4 Aces of the same colour % of jackpot 4.3 Momets ad Distributio of the Series We evaluate the coditioal first two momets of the fixed prize awards, give that either the proportioal prize or the jackpot was wo. EX [ ].151 EX 2 [ ] Illustrated i igure 2 is the expectatio of the fud as a fuctio of the umber of games played. We see that the mea approaches its asymptotic value $\uderset{\to \ifty }{\mathop{\lim }}\,E [ {{}_{}} ]= 22,667 assumig that o oe wis the jackpot. At its asymptotic value the expected beefit from eterig the game is derived i the followig way. The expected payout from playig a had whe the jackpot reaches its asymptotic value (3) is derived through combiig the expected fixed payout plus the expected proportioate payout. 18

19 E[ Payout] E[ X I I ] P( I I ) lim E[ ](.1 P( I 1) P( I 1)) 1.26 This exceeds the fee of 1, but does illustrate that the game would require a substatial umber of games to be played before the odds were i favour of the gambler. J J J igure 2: Expected value of the jackpot assumig that o oe wis the jackpot showig the value approachig its asymptotic mea Usig Lemma 1 to determie the secod momet alog with the expressio from Theorem 1, we ca calculate the stadard deviatio for the series. This is illustrated i igure 3 where we see the series approachig its asymptotic stadard deviatio of 5,

20 igure 3: Stadard deviatio of the value of the jackpot assumig ot wo showig it approachig its asymptotic value} ially, usig Corollary 4, we ca derive the asymptotic cumulats. or the Edgeworth 3, Expasio we will require the followig: 1.5 2, 4,.6112 ad 2 2,.6298 igure 4 illustrates the Edgeworth Expasio estimate of the asymptotic distributio of the value of the jackpot assumig it has ot bee wo. Oe of the cosequeces of the Edgeworth Expasio is that it does ot guaratee a proper PD or CD, such that the correctios may take the CD above 1 or the PD below. 2

21 igure 4: Edgeworth expasio for the asymptotic value of the fud igure 5 presets the deviatio betwee the Normal approximatio ad the Edgeworth Expasio, showig the disagreemet to be greatest about 2,, where the Normal approximatio exceeds the Edgeworth Expasio by about.4. The aalysis usig the Edgeworth Expasio shows that the 5th ad 95th percetiles are ( 129,587, 294,644). The Normal approximatio would have provided a rage of ( 137,279, 34,53). 21

22 igure 5: Differece betwee Edgeworth expasio ad Normal Distributio showig greatest differece is almost 4% Uderstadig the distributio of the jackpot whe wo is importat to the casio operatios maager. Due to regulatio i the UK, casios must be able to pay out o ay bet that they accept. As there may be multiple of these progressive games takig place simultaeously, ad the legth of time util a cotributio is made to the jackpot ad the whe the jackpot is fially paid out, it is importat that the casio appreciates the variability i the expected jackpot whe it is wo so they have sufficiet fuds available to pay the jackpot. The aalysis i this sectio has assumed that the jackpot has ot bee wo. I Sectio 4.4 we cosider the size of the jackpot whe it is wo. 4.4 Expected Value of the Jackpot whe it is Wo The aalysis i Sectio 4.2 assumed that the jackpot was ot wo. The radom process for wiig the jackpot follows a geometric distributio. As such, we ca model the had umber 22

23 used i the idexig of the jackpot value to be a radom variable that follows a geometric distributio i order to estimate the value of the jackpot whe it is wo. Theorem 4 Deotig the had whe the jackpot is wo for the first time as N J ad assumig the probability of wiig the jackpot o ay game is p the the expectatio of the value of the jackpot whe it is wo is: p 1 p 1 (1 p )(1 q ) E[ N J ] ((1 h)(1 ) q ((1 h) E[ X ])(1 q)) 1 (1 p)(1 q) q Usig the formula provided i Theorem 4 we ca calculate the expected value of the jackpot whe wo assumig people cotiue to play, which is 113,99. The assumptio that people cotiue to play is critical which we reflect o i Sectio 4.5. Lastly, the casio is iterested i the expected profit ad the probability of losig moey withi ay game as they provide the iitial amout ito the jackpot. The expected umber of hads played util the jackpot is wo is simply the mea of the geometric distributio which for this series is 391,191. As the casio ears.7 per had their icome per had is expected to be 273,834. or the casio to ear back their 1, they require at least 14,286 hads to be played before the jackpot is wo, this has a probability of.96 of beig realized. 5 Optimizig iitial coditios 23

24 Cosiderig the casio as a profit maximizig orgaisatio, it is of iterest to assess the optimal iitial cotributio to the fud, i.e., such that profits are maximized over a period of time. So far i the aalysis the stochastic process is idexed by the umber of hads that have bee played withi ay cycle. However the iter-arrival time of players will be iflueced by the value of the fud ad there is a obvious trade-off betwee havig a high iitial value i the fud to ecourage more players, subsequetly shorteig the caledar legth of a cycle ad cotributig too much so that it is uprofitable. I this sectio, we cosider how optimizatio ca be used to assess the startig coditios of the game based o the arrival rate of players. We assume that the iterval time betwee games decreases expoetially as the value i the jackpot icreases. As such we ca approximate the expected profit, i.e. V, i a give time period i the followig expressio: J E V E N h e, where EN [ ] which is the expected umber of hads per cycle, J h is the profit per had, ad is the arrival rate of players. Here, we model that the arrival rate of the player is a fuctio of the of the iitial fud. The expected profit becomes the product betwee the expected profit per cycle, ad the umber of cycles per uit of time. Optimizig this expressio with respect to, to obtai the optimal value, i.e. E NJ h * followig expressio: h1 E N * results i the 1. This results i a expected optimal profit of J * e E V. The casio's origial settig of was 1, which would be optimal for a value of approximately 1. igure 6 is a illustratio of the expected profits 1 relative to the optimal value for various values of for equal to , 1 1 ad 24

25 igure 6: Illustratio of the profits relative to the optimal profits with respect to for values of 1 1 (dotted lie), 1 1, (dashed lie) ad 1 1, (solid lie) showig that profits are very sesitive for small values of. igure 6 ca be used to set the iitial fud of the jackpot, assumig we kow the iter arrival rate of players. Whe the iter arrival rate is low, we should set to be small. As the iter arrival rate icreases, we should icrease. However, prior to the game beig selected, we are ucertai as to how popular the game will be. Our optimal profit for a give startig coditio is very variable depedig o that startig coditio. or example, we ca see that if we iitially overestimate the umber of players playig, the our expected profit is very low for 5. However, if we set 9, we will achieve approximately 5% of the profit for two of the scearios. 6 Coclusios ad uture Work 25

26 This paper has provided a modelig framework for determiig the key characteristics of a casio game that ca be used to iform decisios cocerig optimal coditios of a game. Give the combiatio of prizes, from fixed amouts to proportios of jackpots with relatively small probabilities, we believe that assessig the effect o profitability is cogitively challegig. The aalysis illustrates that the behavior of the process ca be couterituitive ad as such requires the support of aalysis. The cosequeces i lost opportuity costs for ad hoc decisios ca be substatial with profits beig very sesitive to these choices. I the case explored, we observe that the deviatio betwee the ormal distributio ad the Edgeworth expasio is relatively small (aroud 4% ) ad as such the ormal distributio provided a adequate approximatio. However, i Sectio 3, the simulatios illustrated that the correctios provided by the Edgeworth expasio could be quite substatial. Without this aalysis, it would have bee impossible for the Director of Operatios to assess how good a approximatio the Normal distributio was. Note, however, that while we have discussed the characteristics of oe particular implemetatio of the progressive format i oe game, i.e. Blackjack, the theorems developed are sufficietly geeric that they could be applied to other casio games, or other situatios with a similar structure. ially, casios cosist of games, players ad spectators, ad as such, are a portfolio of iteractig stochastic processes that eed to be maaged. Through this paper, we have explicated the lik betwee reveue ad the game parameters thereby allowig decisio makers to trade-off betwee ivestmet, prizes ad arrival rates of players. Decisios cocerig the parameters of a game will ifluece the rate at which spectators become participats. Each casio has a fiite reserve ad the parameters of each games, such as, must be maaged simultaeously with the aim of icreasig participatio, ot merely movig players from oe game to aother. rom a portfolio perspective, we have ot begu cosiderig simultaeously modellig the collectio of games beig played withi the casio. By doig so, we could optimize the parameters of each game globally agaist profit ad risk subject to costraits such as the availability of fuds to pay out wiigs ad iitial casio reserves. Bibliography 26

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