Analyzing floor configurations for casino slot machines

OMEGA Int. J. of Mgmt Sci.. Vol. 13, No. 6, pp, 561-567, 1985

0305-0483/85 S3.00+0.00 Copyright ~ 1985 Pergamon Press Ltd

Printed in Great Britain. All rights reserved

Analyzing Floor Configurations for Casino Slot Machines BARRY

L BAYUS 1

SHIV K GUPTA University of Pennsylvania, USA (Received February 1985; in revisedforrn June 1985) Casino management desires to make effective use of the scarce floor space available for slot machines. Tradeoffs among the key variables were found by combining managerial judgement and data analysis; these were then used to develop an interactive micro-computer model which analyzes slot machine configurations. Although we present a single application, our approach may be employed over a wider range of problems, e.g. salesroom configurations, facility layout, etc.

INTRODUCTION AND BACKGROUND GAMBLINGactivities in Atlantic City have been tremendously successful over the last few years and have begun to rival the activity of Las Vegas. Still, there is room for improvement in current planning. At the time of this project, nine casinos were in full operation. Table 1 displays some industry statistics for 1983. The Sands Hotel and Casino offers the widest variety of slot machines in Atlantic City [3]. With the different profit potentials of the various slot machines, it is not surprising that management is interested in making the most effective use of the available casino floor space. Earlier efforts by the authors resulted in the development of a deterministic simulation model of slot activity on the casino floor. This simulation model incorporated a wide range of player behaviors. Resulting patterns of slot play for various floor arrangements lead to suggestions for specific machine placements. In addition, an empirical study of the slot player was conducted through the use of surveys and statistical analyses, This research empirically demonstrated that slot machine revenue was a t Currently a member of the Corporate OR Group at RCA.

function of three components: machine location on the casino floor, denomination of machine, and type of machine. Details of this project are discussed in [2]. Since this earlier effort, Sands has expanded the casino floor area and increased the number and types of slot machines. The possible locations for slot machines has been fixed to aid in the overall casino planning. Approximately 1500 locations on the casino floor have been designated for slot machines. The problem considered in this paper is to determine the 'best' configuration of available slot machines in these locations. Decisions regarding these one-armed bandits have largely been limited to an intuitive approach. Policies were based upon previous observations of player behavior from various slot configurations tried. Staff involved in such decisions have done quite well when considering single dimensions (e.g. locating machines according to performance within denomination o r within machine type). The purpose of this project was to incorporate the simultaneous effects of several variables into decision making. Important issues which were identified in our earlier research and which have guided this effort include the following: 561

562

Bayus, Gupta--Floor Configurationsfor Casino Slot Machines

• How should individual machines be arranged within the available casino floor space? • What will consumer reaction be to changes in the floor configuration? • How can Sands' management evaluate alternative policies? This paper discusses our efforts to develop a microcomputer model which can evaluate and provide guidelines regarding the configuration of slot machines on the casino floor. Several obstacles confronted us, however. (I) At the time of this project there was no systematic data collection procedure in place. Data on the various 'type' characteristics of slot machines (e.g. size, game features) was not readily available, making it difficult to determine the consequences of previous floor changes.

Through a series of meetings and discussions with key Sands personnel, we were able to formulate a model structure. We have implemented this model on a microcomputer and tested it using historical data. This core model structure can be extended to include other complexities such as seasonality and multiple machine characteristics.

THE MODEL STRUCTURE The focus of our study is concerned with t w o aspects: (I) devising a way by which we can predict the profit for different arrangements of slot machines on the casino floor, and (2) choosing the 'best' alternative out of all the possible arrangements. The method should be simple enough so that Sands management can: • Thoroughly understand the methodology and the underlying assumptions. • Know the advantages and limitations of the model.

(2) Historical slot data is of no real use in developing a model structure, since the new floor configuration is larger and different than in the past.

• Give proper inputs to the model. • Interpret and implement the results.

(3) The unit of analysis (i.e. individual slot machine or 'grouped' data) is an issue, since ideally homogeneous group definitions with equal numbers of machines is desired.

• Use it confidently as an aid in decision making.

(4) Interactive effects between location, type, and denomination exist. It is very difficult to separate out individual effects, especially since empirical data is unavailable.

The method should incorporate data readily available and which is generally collected and used by the casino. Finally, any results should be intuitively appealing to management.

Table I. Atlantic City casino statistics for 1983 (Source: [3]) Slot denomination

Casino

Resorts Caesars Bally's Sands Harrah's Golden Nugget Playboy Claridge Tropicana

Floor size (ft z)

No. of machines

60,000 49,06 I

94 75 86 56 66 63 68 55 76

60,000 32,496 44,698 40,717 51,085 34.408 50,873

S1.00

S0.05

$0.25

Handle~ Return ~ No. of (SMM) ('~,~) machines

Handle (SMM)

Return (~/)

No. of machines

Handle (SMM)

Return (,','.~)

38.6 30.6 43.5 24.9 36.2 35.3 30.5 21.2 38.8

85.1 84.9 85.3 85.3 86.8 86.2 85.5 86.6 86.0

323 179 162 68 228 179 95 94 105

25.4 20.4 16.4 4.7 21.9 26.7 4.1 6.5 9.8

89.4 92.9 85.8 86.0 89.1 87.9 86.7 88.9 87.8

2.52 1.64 2.51 1.02 1.66 1.53 1.31 I. 12 1.62

86. I 85.8 86.1 84.8 84.8 86.5 85.1 84.8 84.1

"Handle is the total amount of money wagered. "Return is the difference between the handle and the casino gross win.

1051 886 I101 625 689 727 890 635 981

Omega, Vol. 13, No. 6

they are reasonable. Changes can be made to the tradeoff and profit values if desired. Also managerial judgement can be used for any 'missing values' at this stage. Floor optimizer. The output from the slot micro model is input to the 'optimizing" phase of the model. Management can introduce constraints at this stage (e.g. limitations on the floor configuration, grouping of machines, etc.). The suggested configuration can then be examined and sensitivity analyses on the proposed changes can be conducted to see in greater detail the impact of proposed floor changes. Actual changes in the slot floor configuration can then be made; generating further data which would update our database.

Oc e rt"ie w

An overview of the approach that we have adopted for analyzing slot floor configurations is represented in Fig. 1. Each of the major components is described below. Database. The database contains data on current and past slot floor configurations. This information includes the profit associated with individual slots, the tradeoff values for various combinations of machine denominations and types, and floor arrangements at each point in time. From this data we can calculate the effects of changes on the slot floor performance over time. This database will be continuously updated as new data becomes available, and old data will be removed from it. Data analyzer. The purpose of the data analyzer is to analyse the slot data and calculate the tradeoff parameters. Any effects due to seasonality should be removed at this stage. Other features could include graphical and statistical analyses. Slot micro model. This model is the center of our approach. Here we 'predict' profit values for all possible locations of available machines. The tradeoff parameters, the current configuration and the profit values are analyzed to ensure that

new

data

I ANALYZER DATA I

563

Underlying theory

The problem of predicting profit for all possible configurations of slot machines is simply too big and complicated at an individual machine level. There are about 1,500 machines; if each one is treated individually, there would be a total of 1,500 x 1,499 x 1,498 x ... x 2 x 1 possible configurations. However, not all these 1,500 machines are unique. There are groups of machines which are similar to each other on the

DATA BASE

~ I

~ I old ~, data tl

so0jectve

tradeoff parameters

profit coefficients

Floor Optimizer

=

constraints

1

~g

este d ~ - , - - - - - . ~

change slot floor

fications

~

nfig~ urrent

J j

sensitivity analysis

Fig. 1. Overview of the model.

/

564

Bayus. Gupta--Floor Configurations for Casino Slot Machines

important characteristics of denomination and We will assume that f is a multiplicative functype (i.e. the game on the machine, the size of tion, i.e. the effects of location, denomination, the machine, and manufacturer). In our earlier and machine type can be multiplied together to study, for example, we used a total of 50 groups. obtain the profit for that machine in that loWe make the assumption that the machines cation. In other words, are independent of each other and of the other Ca = u (Li) r ( D i ) w(T,) (1) games in the casino. What this means is that the profit for a machine is assumed to depend only where u, v and w are functions of single varion the machine and its location on the floor, and ables. The assumption that slot denomination and not on what machines or other games are next machine type are independent variables enables to it. Thus, we are removing interaction effects us to factor out effects due to a particular from the analysis, since they are difficult to location. We accomplish this as follows. L e t j be estimate and analyze. In fact, interactive effects some other slot machine in location Lj with are not as significant as one might expect. current profit C~, denomination Dj, and type Tj. Tables 2 and 3 show some examples of interUsing our previous mathematical assumptions, active tradeoff values. Values not equal to one the profit of this machine in location i is indicate the existence of interactive effects. The benefit of this simplification is that we can treat Cj, =f(L,, D,, r,) groups of machines independently, allowing us = u(L,)v(D,), w I L ) to estimate the profit for all possible locations. = C,i v(O)) w(TI) The total profit for the casino floor is simply the v(D,) w(T,) sum across individual groups. = C, d~ to (2) The problem now reduces to 'predicting' the profit for a particular group of machines in a Here, d,~ is the denomination 'tradeoff' in changspecific location. To do this, we assume that ing from denomination i to j (the factor by profit for a certain location is a function of the which the profit of a machine of denomination location, the denomination of the machine in j has to be multiplied by if it is replaced by a that location, and the 'type' characteristics of machine of denomination i, given no differences the machine in that location. by type of machine), and t,j is the type 'tradeoff' Defining in changing from type i to type j. We assume these tradeoff values are symmetric in that C,i = profit for ith m a c h i n e in location i L, = ith location effect d~j = I/d/~ and tij = 1/t/~. D, = d e n o m i n a t i o n of ith m a c h i n e We already know Cii; to get Cji, we need to 7", = type o f ith machine, have values for d u and tij. Two methods can be we can mathematically represent this as used to obtain these values: managerial judgement and data analysis. Our computer model c,, = f(L,, D,, L). combines both approaches. Based on experience built up from familiarity with slot machine performances, Sands staff can provide subjecTable 2. Tradeoff values for quarter slots by manufacturer tive input for the specific tradeoff values by (Ju~¢ 1983) denomination and type of slot machine. It is To Bally Jennings Sircoma Other also possible that estimates of d,j and t,j can be From Bally 1.00 0.75 1.05 1.18 obtained from observed slot data. Our comJennings 1.33 1.00 1.40 1.57 puter model allows management to interactively Sircoma 0.95 0.71 1.00 1.12 Other 0.85 0.64 0.89 1.00 modify the tradeoff values calculated based on past data. For denomination tradeoffs, we can classify all the groups into five denominations Table 3. Tradeoff values for Bally slots and calculate the average profit per group, for by denomination (June 1983) each denomination category. Letting Ai be the To $0.25 0.50 1.00 average profit for machines of denomination v~, From $0.25 1.00 t.80 2.09 we can e s t i m a t e d U. by 0.50 1.00

0.56 0.48

1.00 0.86

1.17 1.00

d,j =

AriA.

Omega, gol. 13, No. 6 Table 4. Denomination tradeoff matrix

From

50.05 0.1 0.25 0.5 I

50.05

0.10

1.00 0.83 0.54 0.33 0.3l

1.21 1.00 0.65 0.40 0.37

To 0.25

0.50

1.00

1.86 1.54 1.00 0.62

3.01 2.48 1.62 1.00

3.26 2.69 1.75 1.08

0.57

0.92

1.00

Using 1983 slot data, tradeoff values for the various denominations are shown in Table 4. We note that these values are calculated using individual slot machine data. Similarly, for estimating the type tradeoff parameters, we can classify the groups into various type categories. Data on slot machine manufacturers was used to calculate the type tradeoff values used in all our model testing, since this data is readily available. Table 5 shows tradeoff values by manufacturer for 1983 using individual slot machine data. The denomination and type tradeoff matrices are the same for all locations because of the independence of the location effect. The 'predicted' profit is then given by (2). To find the 'best' floor configuration based on actual and 'predicted' values, an assignment algorithm based on the linear programming methodology can be used. In practice, a 'greedy' heuristic is a good approximation and can be Table 5. Manufacturer tradeoff matrix To Bally From

Bally 1.00 Jennings 1.85 Sircoma 1.62 Others 1.02

Jennings

Sircoma

Others

0.54 1.00 0.88 0.55

0.62 1.14 1.00 0.63

0.98 1.82 1.60 1.00

Table 6. Initial slot floor configuration Location

Profit (S)

Denomination

Manufacturer

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

274 224 137 456 256 368 211 178 219 210 305 120 128 163 132 179 313 322 492 499

0.25 0.25 0.1 1.00 1.00 0.25 0.25 0.25 0.25 0.25 0.5 0,05 0.05 0. I 0.25 0.25 0.25 0,5 0.5 1,00

Other Sircoma Sircoma Bally Bally Other Bally Jennings Jennings Bally Bally Sircoma Sircoma Sircoma Jennings Bally Sircoma Sircoma Bally Bally

565

easily programmed (e.g. [4, 5]). We note that this heuristic gave optimal solutions for the cases tested in our research. This heuristic can be easily described. The highest profit value from the matrix C is chosen first. The corresponding machine is assigned to that location, and the next highest profit value is picked, etc. IMPLEMENTATION In order to integrate this approach into decision making, we have developed the mathematical model described above on a microcomputer. A computer program was written in BASIC which reads data from a file and calculates the tradeoff matrices for denomination and type (only the manufacturer was used in the initial model variables). These matrices can be examined and modified interactively if desired. The final tradeoff matrices, together with current profit information, is used to calculate the 'predicted' profit values for all possible locations. A 'greedy" heuristic is used instead of an optimization routine to obtain a suggested slot floor configuration. ROBUSTNESS AND SENSITIVITY OF THE M O D E L Before using the model with confidence, questions regarding the validity of the underlying structure and key assumptions made must be addressed. In particular, several test runs of the computer model were conducted to examine the following issues:

• Is the model consistent in the answers it provides? • Is the model robust in the face of deviations from assumptions; in this case, the 'predicted' values? In other words, how does the model perform over time when actual slot play is not as predicted by the model? • Are the results intuitive, and do the suggested floor modifications fall within the range of usual management planning? Details of these analyses are in [1]. In general, the model gave consistent results for a variety of initial floor plans and parameter values; model

566

Bayus. Gupta--Floor Configurationsfor Casino Slot Machines

(,,~Bac k door

results converged over time in the face of deviations from assumptions: and floor configuration suggestions were within the range of managerial action.

NN

USING THE SLOT MICRO MODEL In order to see how the model can be used to guide in arranging slot machines on the casino floor, we discuss an example of 20 groups below. Tradeoff data for this example is in Tables 4 and 5, and data by specific location is in Table 6. Consider the hypothetical casino floor layout shown in Fig. 2. Twenty groups of slot machines have been arranged on the floor. This scenario uses the following policies:

@ k_Y Front door

Fig. 2. Casino floor configurations for twenty groups of

slots.

(1) Locate nickel and dime slots in the rear corner (groups 12-14).

/ f ' - ~ Back door

(2) Locate dollar machines near an entrance (groups 4, 5, 20). (3) Group quarter machines together, including an 'exclusive' section (groups 6--10). In general, this policy also seems to locate groups of slots homogeneously, i.e. locates high (50¢ and $1.00), low (5¢ and 10¢) and medium (25¢) denomination slots close together. The modified floor configuration suggested' by the model is shown in Table 7. This arrangement is graphically presented in Fig. 3. Aside

Fq½ N@N

@ I~"~F ro nt door

Fig. 3. Suggested casino floor for twenty groups of slots.

Table 7. Model results Current profit

Predicted profit

From location

To location

(S)

(S)

4 5 20

17 14 9 13 6 2 12 3 8 18 19 20 I 4 15 7

456 256 499 305 492 210 I79 322 274 368 224 313 178 219 132 137 163 120 128

888 712 707 624 604 363 362 342 329 318 300 176 172 141 132 114 84 76 59 48

5186

6551

11 19 7 10 16 18 1 6 2 17 8 9 15 3 14 12 13

211

10 II 16 5

Total

Omega, Vol. 13, .Vo. 6

from the specific location changes of individual machines, the following general policies emerge: (I) Group the dime machines together and locate in the front corner. (2) Spread the nickel and dollar machines around the floor. (3) Locate a group of quarter machines in the middle of the floor. (4) Locate quarter machines near the front entrance. Thus, a very different slot policy is suggested by the model. This high and low payoff slots (dollar and nickel) are spread over the floor in the suggested configuration. Both policies can be argued for based on intuitive notions. It is only by trying the various arrangements and tracking the results can the 'best' policy be found. Further evaluation and sensitivity analysis is warranted before actually implementing any suggested modifications in the floor plan.

567

(2) Appropriate data is being collected systematically, and can be used to validate the model. Data tracking and performance evaluations of the model are being conducted using actual slot data. (3) The 'type' characteristics of slot machines can be expanded to incorporate other important aspects such as game, size of machine, and marketing factors (e.g. availability of stools and/or prize tickets). The computer model can be modified to include a broader definition of machine type. (4) A scheme for including constraints in the model is under development. Limitations with respect to floor restrictions and placements of specific slots are part of management planning and can be included.

(5) A data analysis 'package' needs to be developed. Effects due to seasonality, day of week, etc. can then be included in the model. In addition, graphical analysis can aid in the planning and evaluation of slot performance. ACKNOWLEDGEMENTS

CONCLUSIONS Over the last few years our research efforts have led to a greater understanding of the casino slot player. In our initial effort, we focused on the individual consumer; the result was a dynamic simulation model of slot players on the casino floor. This paper has discussed our efforts of modelling slot play as a function of aggregate variables; the result is an interactive microcomputer model which can perform a static evaluation of casino slot floor configurations. This model has added another dimension of information to the decisions regarding slot placements. Several steps are planned for the future: (1) The data flow between the mainframe and microcomputer systems still needs to be designed and coordinated. Storage of data at each point in time, definition of variables used in the analysis, and size of problem are all issues which can be addressed.

We would like to thank Brad Stone, the President of Sands for initiating this project; Len DeAngelo and Colleen Hogan of Sands for their key role in the development of the model; Pradeep Bansal for his assistance in the computer programming and data analysis; Wenda Foster for her excellent typing support; and Sam Eilon and an anonymous referee for their encouragement and suggestions.

REFERENCES 1. Bayus BL, Gupta SK and Bansal P (1984) A microcomputer model for analyzing slot machine configurations. Technical Report, Wharton Applied Research Center, University of Pennsylvania. 2. Bayus BL, Banker RL, Gupta SK and Stone BH (1985) Evaluating slot machine placement on the casino floor. Interfaces 15(2), 22-32. 3. Headliner Publications (1984) The Complete Pocket Guide to Atlantic City Casinos. Reading. Pennsylvania. 4. Tien BN (1977) Error bounds and the applicability of the greedy solution to the coin-changing problem. Ops Res. 25(3). 404-418. 5. Zanakis SH and Evans JR (1981) Heuristic "optimization': why, when, and how to use it. Interfaces ! 1(5), 84--90. FOR CORRESPONDENCE: Professor S Gupta, Department of Marketing, The Wharton School of the University of Pennsylvania, Philadelphia 19104, USA.

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