is lottery demand driven by price or long odds event ... · introductionindustry...

Introduction Industry Background Examination of Lottery Price Data and Analysis Monte Carlo Simulation Conclusion

Is Lottery Demand Driven by Price or LongOdds Event? Evidence from China Lottery

Industry

Jason GaoCarleton University

Jia YuanUniversity of Macau

September 20, 2012

1


Motivation

Debate on which approach explains the gambling behaviorbetter: Expected Utility Model or the Prospect TheoryWhat influences lottery demandEffective Price Approach

The expected loss of each lottery ticket as lottery priceCook and Clotfelter (1993), Scoggins (1995), Forrest,Gulley, and Simmons (2001), Fink, Marco and Rork (2004),Lin and Lai (2006), Lee, Lin and Lai (2010)

Small odds Approach

Lottery demand depends more on jackpot size, or smallodds event, than expected valuesForrest, Simmons and Chesters (2002), Papachristou(2006) and Geronikoulaou and Papachristou (2007),Arobua and Kearney (2011)

2


Idea

Hard to disentangle these two different approachesThe variation of small odds events usually is alsoassociated with the variation of lottery priceTo exploit a unique lottery game policy in the most popularlottery game in ChinaThe cap policy on the grand jackpot prize, which limitseach single winner’s reward.

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Findings

This apparently complicated cap policy makes this lotterygame structure much simplerThe lottery price is almost fixed all the timeThis almost constant lottery price cannot explain theobserved lottery sale variationLottery sale is highly correlated with rollover moneyMonte Carlo simulations provide evidence showing that theestimation method in the effective price literature may givespurious estimation result

4


Road Map

Introduction of lottery industry and the SSQ lotteryModel setup and Examination of the lottery priceData and AnalysisMonte Carlo SimulationConclusion

5


China Lottery Industry

In 2011, the lottery sale in China was over 214 billion RMB,around 35 billion USDThere are around 160 million lottery players in ChinaBicolor Ball(SSQ) lotto is the most popular lotteryIn each round, SSQ can sell out over 150 million tickets onaverage

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The growth of SSQ

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The rules of SSQ Game

Similar to other popular lotteries such as the Powerball ofUSA, the LottoMax of CanadaNominal price: 2RMBPick numbers in two groups

In the first group, to pick 6 numbers from 1 to 33, called rednumbersIn the second group, pick 1 number out of 1 to 16, calledblue numbers

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Table 1: Policies of Bicolor Ball Lotto

Award level

Winning conditions Prize distribution Explanation

Red balls Blue ball

First prize

If the rollover money from the last jackpot is less than 100 million RMB, then the grand jackpot winners will split the rollover from the previous draw and the 70% from the “high prize pool”. If the prize is more than 5 million RMB, each winning ticket will only be worth 5 million RMB. If the rollover money from the last jackpot is at least 100 million RMB or more, there is a two part prize package. The winners split the rollover money from the previous draw and 50% from the “high prize pool”, as well as 20% from the “high prize pool”. With each prize, a maximum of 5 million RMB is paid (total of 10 million RMB).

Select 6+1 win 6+1

Second prize

30% of current grand prize

Select 6+1, and win 6+0

Third prize

Fixed amount of 3000 RMB per winning lottery ticket

Select 6+1, and win 5+1

Fourth prize


Select 6+1, and win 5+0 or 4+1

Fifth prize


Select 6+1, and win 4+0 or 3+1

Sixth prize


Select 6+1, and win 2+1 or 1+1 or 0+1

Figure 2: Policies of Bicolor Ball Lotto

If: R < 100,000,000

50% of S goes to the government, 1% of S goes to the adjustment fund.

Fixed prizes (3rd to 6th prizes) take prize money from S

Grand Prize of the Jackpot = Min{5m, (50% of H+R) / N. of Winners} + Min{5m, (20% of H ) / N. of Winners} There is a cap of 10 m for the grand prize.

S: Lottery Sales in Current Jackpot: S = 2 * N. of Tickets Sold

30% of H goes to 2nd prize: winners split prize

H = the remaining value of S: high prize pool

R: Rollover from the previous draw

If: R > 100,000,000

Grand Prize of the Jackpot = Min{5m, (70% of H+R) / N. of Winners}. There is a cap of 5 m for the grand prize

R_New: New Rollover to the Next Jackpot R_New = R + S – Total Payout


The Special Cap Policy

If the prize pool is less than 100 million RMB, the jackpot isno more than 5 million RMBIf the prize pool is more than 100 million RMB, the jackpotis no more than 10 million RMB

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Model Setup

Traditional Effective price approach calculates theexpected loss and studies the reactions of players to thiseffective priceWe will prove the effective price is constant in SSQ due tothe special policy.

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Model Setup

N tickets, therefore 2N is the revenueLet qj denotes the probability of winning the jth prize

Each qj can be exactly calculated

Ej is the expected return from the jth prizeR denotes the rollover from the previous roundWe assume that the players choose their numbersindependently

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Decomposition of E

E = E1 + E2 + E3 + E4 + E5 + E6

Fixed prize: Ef = E2 + E3 + E4 + E5 + E6 ≈ 0.486

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Table 2: The probabilities and expected returns for SSQ

Probability Expected return

�� =1

16 �336 �≈ 5.643 × 10−8

�� =15

16 �336 �≈ 8.464 × 10−7

�� =�65� �

271 �

16 �336 �≈ 9.142 × 10−6

�� = 3000�� ≈ 0.0274

�� =15 �65� �

271 �+ �64� �

272 �

16 �336 �≈ 4.342 × 10−4

�� = 200�� ≈ 0.0868

�� =15 �64� �

272 �+ �63� �

273 �

16 �336 �≈ 7.758 × 10−3

�� = 10�� ≈ 0.0776

�� =�62� �

274 �+ �61� �

275 �+ �276 �

16 �336 �≈ 5.889 × 10−2

�� = 5�� ≈ 0.2945


Calculation of E2

Claim 1: The expected return of 2nd prize is E2 is almost fixed.E2 ≈ 0.148 if N is over 11,000,000.

Then, E2 = 0.148NN (1 − (1 − q2)

N) ≈ 0.148

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Calculation of E1

Case 1: R ≤ 100000000Claim 2: If N is around between 120,000,000 and 230,000,000,and R is more than 30,000,000, E1 ≈ 0.282.

We prove E1 ≈ q1 * 5,000,000 = 0.282That is, the grand prize is just 5 million RMBThe intuition: the grand prize pool is, for example, morethan 65 million. The event of winning a prize less than 5million is that around 12 other people hit the jackpotsimultaneously, which is almost impossible.

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Table 3: The Expected Return E� for Different Possible Values of Sale and Rollover Money

Rollover Money in 107

3 4 5 6 7 8 9

N=1.1*108 E�=.2819765 .2821275 .2821473 .2821494 .2821496 .2821496 .2821496

N=1.2*108 .2819075 .2821143 .2821454 .2821492 .2821496 .2821496 .2821496

N=1.3*108 .2819053 .2821118 .2821448 .2821491 .2821495 .2821496 .2821496

N=1.4*108 .2819120 .2821110 .2821444 .2821490 .2821495 .2821496 .2821496

N=1.5*108 .2818505 .2820960 .2821415 .2821486 .2821495 .2821496 .2821496

N=1.6*108 .2818516 .2820939 .2821408 .2821484 .2821495 .2821496 .2821496

N=1.7*108 .28186241 .2820940 .2821405 .2821483 .2821495 .2821496 .2821496

Note: N is the number of players in the game. Sale = N * 2 as the lottery price is 2 RMB.


Calculation of E1

Case 2: R ≥ 100000000Claim 3: If N is around between 120,000,000 and 230,000,000,E1 ≈ 0.282 + 0.0987 = 0.381.

E1 ≈ 0.282 + 0.0987 = 0.381

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Theorem

Theorem 1

The effective price of the SSQ lottery is 1.09 if R ≤ 108; or 0.99if R ≥ 108

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Data

The whole data set of SSQ in 2010 and 2011It includes sale, prize payout, rollover, number of winnersTotal 321 observations

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Table 3: Simple Statistics for SSQ

Mean S.D. Min Max

Sales in 108 2.91 0.48 2.14 4.34

Rollover in 108 2.61 1.85 0.28 7.83

Sales Change from Previous Draw

(Absolute Value)

.56%

7.94%

9.97%

6.04%

-35.1%

0

71.6%

71.6%

Lottery Price 1.00 0.037 0.9845 1.0837

Share of p = 0.9845

Share of p = 1.0837

82.5%

17.5%

N=320


Lottery Price vs Sale

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Rollover vs Sale

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Rollover vs Sale

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The Traditional 2SLS Estimation Method inEffective Price Literature

First Stage:p = β0+β1∗Trend+β2∗T 2+β3∗R+β4∗R2+β5∗p−i+β6∗S−i+εSecond Stage:S = α0+γ∗p̂+α1∗T+α2∗T 2+α3∗R+α4∗R2+α5∗p−i+α6∗S−i+ε

Following the effective price literatureUsing rollover and its square as instruments

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Table 4: 2SLS Regression Estimation for the Lottery Demand Elasticity of Price

(1) (2) (3) First Stage Second

Stage First Stage Second

Stage First Stage Second

Stage trend 0.0000472 0.000932** 0.0000059

9 0.000138 0.0000041

8 0.0000999

(1.02) (3.24) (0.12) (0.50) (0.08) (0.36) trend2 -9.75e-08 0.0000014

0 -9.46e-08 -

0.000000492

-9.15e-08 -0.00000033

3 (-0.71) (1.73) (-0.70) (-0.69) (-0.68) (-0.47) log_p_1 0.116 0.107 0.116 0.105 (1.65) (0.53) (1.65) (0.60) log_p_2 -0.00704 -0.0353 -0.00502 0.0825 (-0.12) (-0.12) (-0.09) (0.37) log_p_3 -0.0367 -0.132 -0.0371 0.183 (-0.74) (-0.56) (-0.75) (1.16) log_r -0.660*** -0.639*** -0.605*** (-10.94) (-9.92) (-8.76) log_r2 0.0166*** 0.0159*** 0.0151*** (10.45) (9.20) (8.20) log_p -0.652** -0.355*** -0.538*** (-2.90) (-3.45) (-4.42) log_s_1 0.0148 0.368*** 0.0165 0.360*** (1.07) (3.95) (1.18) (3.89) log_s_2 0.0146 0.000011

7 0.0163 0.00406

(1.10) (0.00) (1.21) (0.05) log_s_3 0.00961 0.632*** 0.00513 0.636*** (0.54) (6.70) (0.28) (6.80) _cons 6.539*** 19.28*** 5.633*** 0 5.307*** 0 (11.41) (868.83) (6.75) (.) (6.16) (.) N 317 317 317 317 317 317 t statistics in parentheses. * p < 0.05, ** p < 0.01, *** p < 0.001


Monte Carlo Simulation

The basic idea is to generate sales which are independentof lottery priceThe data generating process (DGP) is the following

Let R0 be the initial rollover money.St = β0 + β1 ∗ Rt−1 + εt , here εt ∼ NID(0, δ)Rt = Rt−1 + St − payoutt , here payoutt is calculated by therules of SSQ

Calibration by simple OLS regressionPick a range of values for the initial rollover money R0

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Monte Carlo Simulation

We run 6 Monte Carlo experiments, varying the parametervalue of initial rollover at each experimentThe initial rollover money takes the following numbers: 50million, 100 million, 150 million, 200 million, 250 million and300 millionFor each of the 6 simulations, we simulate 320 rounds oflottery sales and rollovers, following the SSQ rule

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Table 5: Monte Carlo Simulations: Estimation for the Lottery Demand Elasticity of Price

�� = 50��

(1) (2) (3) Elasticity -0.6668**

(0.2550) -0.2786 (0.2309)

-0.4687 (0.3027)

�� = 100��

(1) (2) (3) Elasticity -0.6693***

(.2535) -0.1552 (.2119)

-0.5106* (.2666)

�� = 150��

(1) (2) (3) Elasticity -0.8696***

(.2216) -0.4368* (.2335)

-0.7391*** (.2571)

�� = 200��

(1) (2) (3) Elasticity -0.427*

(.2842) -0.0187 (.2522)

-0.1433 (.3393)

�� = 250��

(1) (2) (3) Elasticity -0.7516**

(.2703) -0.1926 (.2507)

-0.5470* (.3213)

�� = 300��

(1) (2) (3) Elasticity -0.4800 -0.0081 -0.0891


Robustness Check

Lottery players may purchase several lottery tickets for thesame combination of numbers in the same drawNeed to examine the distribution of the lottery purchaseNeed to examine how this distribution changes in differentdraws

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Individual Data

Obtain the data directly from the lottery companyThe dataset contains 30,366 SSQ lottery players and3,557,606 purchased ticketsIt is a record of all the lottery number selection informationfrom draw 2011141 to 2011149 in 2001Table 4 illustrates the distribution of lottery purchasestrategies and the distribution variation for each draw

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Table 4: Distribution of Lottery Purchase Strategies

Shares (in %) N of Tickets N of Players

Round= Single Double More than 3

2011141 98.2% 1.23% 0.56% 360,432 12,968

2011142 98.0% 1.45% 0.55% 368,265 13,462

2011143 98.4% 1.04% 0.59% 371,667 12,653

2011144 98.2% 1.2% 0.65% 393,255 12,620

2011145 98.1% 1.2% 0.64% 395,660 12,866

2011146 98.0% 1.2% 0.71% 397,808 12,150

2011147 98.2% 1.2% 0.65% 398,376 12,450

2011148 98.2% 1.2% 0.61% 423,067 12,857

2011149 98.5% 0.9% 0.56% 449,076 12,074

Total Number of Tickets

3,557,606

Total Number of Players

30,366


Robustness Check

Around 98% of the lottery players purchased one ticket forthe same combination of lottery numbersAround 1% of the players purchased two tickets for thesame combination of numbersThe distribution does not significantly vary

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Conclusion

Effective price approach seems not enough to explain thelottery sale demand”Dreaming to win big” seems consistent with the dataThe standard 2SLS method may provide spuriousestimation results

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