Author: Title of manuscript: Spinning the Big Wheel on The Price is Right : A Spreadsheet Simulation Exercise

Transcription

1 Author: Keith A. Willoughby Department of Finance and Management Science Edwards School of Business University of Saskatchewan Saskatoon, SK Canada S7N 5A7 Phone: (306) Fax: (306) Title of manuscript: Spinning the Big Wheel on The Price is Right : A Spreadsheet Simulation Exercise 1

2 INTRODUCTION The Price is Right is the longest-running game show in broadcast history. The show s format calls for audience involvement. At the beginning of each episode, four audience members are requested to come on down to participate in contestant s row. These four individuals estimate the dollar value of a particular item, with the winner being the player with the closest bid (without exceeding the prize s dollar value). The three winners from each half of a 1-hour episode compete in a Showcase Showdown. This involves the spinning of a large wheel comprised of twenty sections. Each section corresponds to a particular monetary value (ranging in $0.05 increments from $0.05 to $1.00). s can take at most two spins of the wheel. The winner is the player with a total spin amount closest to, but not exceeding, $1.00. Should two (or more) players have the same score after completion of their turn, then a spin-off is held with each player receiving an additional spin of the wheel. The player with the higher (or highest) total on this extra spin wins. Should their initial spin be less than $1.00, would it be wise for the player to take a second spin? The decision to utilize this second spin evokes considerable audience reaction, especially for those initial spin values that fall in the intermediate range of possible amounts. Spinning a second time permits the player to earn a total score closer to $1.00, thus giving the contestant a greater chance of winning the game. Nonetheless, there exists a certain likelihood that the additional spin may catapult the player over the $1.00 threshold, thereby automatically eliminating them from further competition. The probability of exceeding $1.00 would grow as the total of the first spin increases. Given the widespread appeal of game shows, especially among collegiate students (we note that almost two million people have attended studio tapings of the show, with roughly a 2

3 third of these being college students), an instructor may be able to generate considerable classroom interest and participation by effectively utilizing an in-class exercise based on gameshow decision-making. This is the thrust of our teaching brief. We seek to develop an exercise, suitable for introductory undergraduate Decision Sciences or Operations Management courses, to simulate the spinning of the big wheel during the Showcase Showdown. We shall consider the initial contestant s decision during a 3-person game. We will develop our simulation models a powerful spreadsheet add-in. Developing such a model within this context ought to capitalize on our students exposure to spreadsheets as a tool for effectively modeling realworld decisions (see, for example, various textbooks by Albright, Winston and Zappe (2003) and Ragsdale (2004) The Price is Right in particular, and game shows in general, have been the subject of an extensive body of academic journal articles. Specifically, Coe and Butterworth (1995) analytically determined a stopping rule for a player s first spin of the big wheel during the Showcase Showdown. For a 2-person game, they discovered that players should stop at $0.55 (and above). In a 3-player game, the first contestant should stop at $0.70 (and above). SPREADSHEET MODEL DEVELOPMENT We will explore the development of a spreadsheet model, to examine the spinning of the wheel during the Showcase Showdown on The Price is Right. Although Coe and Butterworth (1995) have already analytically determined the optimal stopping points (obviously, the results of our spreadsheet model will support their conclusions), we feel that creating a spreadsheet model of this decision-making situation is a worthwhile objective. It provides certain pedagogical benefits. Students may be more prone to appreciate and understand a specific topic when it is imparted to them within a particularly interesting context. Kohn 3

4 (1997) suggested that in order to promote a deeper understanding of material, students ought to be engaged with what they are doing. A spreadsheet model of the type to be discussed in this paper permits instructors to innovatively introduce (or crystallize) the important subject of simulation. Moreover, we can capitalize on the extensive spreadsheet skills our students are developing in variety of other courses. We shall begin by illustrating the development of a spreadsheet model for the case of a 2- player Showcase Showdown. Our objective is to calculate the probability that the first player ultimately wins this game as a function of the observed value on their initial spin, and whether or not the player takes advantage of their second spin. Table 1 presents a sample spreadsheet for the 2-player game. The value in cell A2 refers to the particular score obtained by 1 after their initial spin. Since we want to model a range of potential first spin values, it is helpful to use a RISKSIMTABLE function in this cell. This function would call up different first spin values. For the case of a player opting to spin again, we would want to explore all values from $0.05 to $0.95 (should a player score $1.00 on their first spin, they would never want to use their second spin they have already guaranteed themselves of at least heading to a spin-off!). If we are examining the decision of a player to stay with their initial spin, we would consider values from $0.05 to $1.00. The random, probabilistic elements of our spreadsheet model are the actual spin values. In cell B2, we need to introduce a function that will generate random spin amounts. We suggest using the function: ROUNDUP(RiskUniform(0,20),0)/20 4

5 This function draws random values from a uniform distribution containing 20 possibilities (0 through 20). By forcing the random element to have 0 decimal places, then using the ROUNDUP function, then subsequently dividing it by 20, we will create a stream of random numbers of the sort 0.05, 0.10, 0.15 and so forth, each with an equal likelihood of occurrence. This is exactly the distribution we need. We envision that students may have some initial questions about this particular function (especially if an instructor is using this example to introduce simulation), but a detailed, thorough discussion of its respective components ought to alleviate any concerns. We note that if we are exploring the case in which the contestant opts not to take their second spin, then the value in cell B2 is, simply, zero. Cell C2 is the sum of the first two spins, in other words, A2 + B2. Cells D2 and E2 are the spin amounts for player 2. The same RiskUniform function described above can be used in these cells. As with player 1, we need to total the score for player 2. This is done in cell F2. Cells G2 and H2 are included in the event that a spin-off is needed in the Showcase Showdown. Recall that this situation is required should the players be tied after their spin(s) of the wheel. The entries in these cells would be our usual RiskUniform function. To provide us with a numerical example, we have incorporated possible spin values and resulting totals in Table 1. This example illustrates a specific simulation run in which the first player earned $0.45 on their first spin and opted to spin again. The result of their second spin provided them with a total of $0.85, which exceeded the subsequent spin total of player 2. 1 won this game. (The spin-off values are listed in the table for sake of completeness). From a student s perspective, one of the more complex parts of this model may involve analytically calculating whether or not the first player wins the Showcase Showdown. We need to create an expression in our spreadsheet to determine this event. Such an expression 5

6 would be entered in cell B4. Borrowing partially from the notation adopted by Coe and Butterworth (1995), let us define the following variables: S ij = spin amount for player i on spin j T i = total spin value for player i (in other words, S i1 + S i2 ) SP i = spin-off amount for player i Discussion with the students ought to evoke four distinct possibilities that lead to victory for 1. These events are: - 1 has a spin total less than or equal to $1.00 and player 2 s first spin is lower than 1 s total. 2 spins again and exceeds the $1.00 threshold. This can be represented as: S 21 < T 1 and S 21 + S 22 > has a spin total less than or equal to $1.00 and player 2 s first spin is lower than 1 s total. 2 spins again and their resulting total is still below 1 s total. This is given by: S 21 < T 1 and S 21 + S 22 < T 1-1 has a spin total less than or equal to $1.00 and player 2 s first spin equals 1 s total. In the spin-off, 1 s spin exceed 2 s spin (note that we are implicitly assuming that 2 will not opt to take advantage of their second spin in the event that they tie 1 after their first spin. They proceed directly to a spin-off). We have the following representation: S 21 = T 1 and SP 1 > SP 2-1 has a spin total less than or equal to 1.00 and player 2 s first spin is lower than 1 s total. 2 spins again and obtains a total equal to 1 s amount. In the spin-off, 1 defeats 2. This is illustrated by: S 21 < T 1 and S 21 + S 22 = T 1 and SP 1 > SP 2 We wish to remark upon an additional assumption in our model. Specifically, we are only conducting one round of spin-offs. There exists a possibility, albeit a very small one, that 1 and 2 could both tie after their spin-off, thus necessitating an additional spin of the wheel for each player. Incorporating the event of a second spin-off would have enhanced the 6

7 realism of our approach, at the expense of an increase in the number of terms in our expression in cell B4. We are confident that only considering one round of spin-offs (in essence, giving the victory to 2 in the event of a tie in the first spin-off) will not dramatically affect our model results. This, obviously, could generate an interesting set of classroom discussions. Tradeoffs between model realism and complexity become apparent. Victory by player 1 can be represented in cell B4 by an OR statement with a series of AND expressions. To wit, we have: OR (AND(C2<=1,D2<C2,F2>1),AND(C2<=1,D2<C2,F2<C2), AND(C2<=1,D2=C2,G2>H2),AND(C2<=1,D2<C2,F2=C2,G2>H2)) Each of the four possibilities for 1 winning is represented in the expression. The OR statement implies that so long as one of these conditions holds, player 1 wins. This expression will return the term TRUE if player 1 emerges victorious, and FALSE if player 1 loses. In cell C4, we included the following IF statement: IF (B4 = TRUE,1,0) Cell C4 will be used as our output cell For each of our 10,000 replications for the different first spin will keep track of the number of times 1 prevailed in the Showcase Showdown. An average probability of victory can be calculated by dividing the number of victories by total replications. We will run analyses for either using the second spin, or declining this opportunity. The 3-contestant game can be modeled in a similar fashion. Table 2 illustrates a sample spreadsheet for the 3-player version. As before, cell A2 refers to the particular score obtained by 1 after their first spin. In this particular numerical example, we are illustrating the case of 1 foregoing their 7

8 second spin of the wheel; hence the value of 0 in cell B2. Cell C2 uses the same probability distribution expression as in the 2-player game; namely, ROUNDUP(RiskUniform(0,20),0)/20. Cells D2 through I2 represent spin amounts and resulting totals for players 2 and 3. Finally, cells J2 through K2 contain spin-off values for the three contestants. Our numerical example lists a particular simulation run in which the first player, although they recorded a value of $0.75 on their initial (and only) spin, did not emerge victorious. 3 won with their total of $0.85 (player 2 had been eliminated by going over $1.00). Again, for sake of completeness, the spin-off values are listed for each participant. We can use similar logic as that adopted in the 2-player game to determine the events that result in victory for player 1. In all, there are 16 such events as illustrated in Table 3. The cells at the intersection of a particular row for 2 and a specific column for 3 correspond to these winning scenarios. Cell B4 in Table 2 uses an OR statement with several AND expressions to illustrate winning for player 1 (due to its length, this expression is not given in this manuscript). As before, the OR statement suggests that should one of the 16 events occur, player 1 wins. It will return TRUE and FALSE readings for victory and loss, respectively. An IF statement in cell C4 quantifies the TRUE and FALSE results. RESULTS we performed 10,000 replications for each initial spin amount. Table 4 presents the results of our modeling efforts with shaded cells corresponding to the better decision in each case. In-class discussion could comment on the change in probabilities as the total after the first spin increases. As it climbs, the probability of winning should the contestant spin again 8

9 drops, while staying results in a greater and greater likelihood of victory. Moreover, as one would expect, the probabilities of victory in the 3-player game are always less than those observed in a 2-player contest for all initial spin values. In the 2-player game, the first contestant ought to spin again as long as their first spin was $0.50 or less (although the probabilities of victory were very similar for each strategy when the first spin was $0.55). When there are 3 players, the first participant should spin one more time if their initial attempt scored $0.65 or less. Even scoring $1.00 on the first spin (and thus staying) still does not guarantee victory. The probabilities are about 5% (in the 2-player game) and roughly 10% (in a 3-player event) that a subsequent contestant will also score $1.00 in their spin(s), then defeat player 1 in a spin-off. DISCUSSION We have developed a spreadsheet model to simulate the spinning of the wheel during the Showcase Showdown, a popular game conducted twice during each taping of the awardwinning game show The Price is Right. This model can be used in an instructor s efforts to innovatively introduce the important topic of simulation. We expect to use this model as an in-class example during our sections of Decision Sciences in the Fall 2004 semester. We are confident that students will appreciate this decisionmaking scenario and gain an enhanced understanding of the applicability of spreadsheet simulation modeling. Discuss how we used this exercise during the Fall 2004 semester 9

10 REFERENCES Albright, S. Christian; Winston, Wayne L.; and Zappe, Christopher J. (2003). Data Analysis & Decision Making with Microsoft Excel. Pacific Grove, CA: Thomson Brooks/Cole. Coe, Paul R. and Butterworth, William. (1995). Optimal Stopping in The Showcase Showdown, The American Statistician 49 (3), pp Kohn, A. (1997). Students Don t Work They Learn. Education Week, Sept. 3. Ragsdale, Cliff. (2004). Spreadsheet Modeling and Decision Analysis. Cincinnati: Thomson South-Western. Author s vita Keith A. Willoughby teaches operations management and decision sciences at Bucknell University. Prior to joining Bucknell, he taught at the University of Saskatchewan. Upon completion of his M.Sc. in Management Science from the University of British Columbia, he was employed as a transportation analyst with the Vancouver Regional Transit System. His scholarship interests include materials management and logistics, transportation analysis, college operations, and pedagogical innovation. He has published articles in scholarly journals such as The European Journal of Operational Research, Interfaces, OMEGA, and The International Journal of Production Economics. He earned his Ph.D. at the University of Calgary. 10

11 column/ row 1 Table 1 Spreadsheet model for 2-player game A B C D E F G H 1 after 1 st spin 1 2 nd spin 1 total 2 1 st spin 2 2 nd spin 2 total 1 spin-off 2 spin-off wins TRUE 1 11

12 column/ row 1 Table 2 Spreadsheet model for 3-player game A B C D E F G H I J K L 1 after 1 st spin 1,2 nd spin 1 total 2, 1 st spin 2, 2 nd spin 2 total 3, 1 st spin 3, 2 nd spin 3 total 1 spinoff 2 spinoff spinoff 4 1 wins FALSE 0 12

13 Table 3 Winning scenarios for 1 in 3-person game 2 S 21 < T 1 and T 2 > 1.00 S 21 < T 1 and S 21 + S 22 < T 1 S 21 = T 1 and SP 1 > SP 2 S 21 < T 1 and S 21 + S 22 = T 1 and SP 1 > SP 2 S 31 < T 1 and T 3 > 1.00 S 31 < T 1 and S 31 + S 32 < T 1 3 S 31 = T 1 and SP 1 > SP 3 S 31 < T 1 and S 31 + S 32 = T 1 and SP 1 > SP 3 13

14 Table 4 Model Results Total after 1 st 2-player 3-player spin P(Win Spin) P(Win Stop) P(Win Spin) P(Win Stop)