Rational Addiction
in Lotto Play
Ian Walker and Rhys Wheeler
Princeton U and LUMS
Thanks to Gambling Commission, the British Academy, and EU
Framework 7 ALICE-RAP project
Preview
• Gambling is a challenge to economists
– No convincing general theory
– Lotto especially bad bet but most prevalent
form of gambling
• Rational Addiction (Becker/Murphy JPE 88)
– Highlights the effect of expected future prices
• Lotto is a unique product for testing for RA
– pre-announced price changes NOT required
• RA restrictions satisfied, MA rejected
• “Price” elasticity suggests large CS
Outline of talk
• Provide a simple analytical model of lotto
– Use it to estimate lotto demand elasticity
• Find backward looking behavior
– Strong habituation
• Find forward looking behavior
– Rational addiction?
• Find LR demand elasticity about -1
– Takeout rate is about right?
• Lots to do yet
Lotto literature
• Clotfelter and Cook
– JEP 90, AER 93
– Peculiar economies of scale
• Walker et al
– Econ Policy 98, OxBull 99, JPubEcon 00, EJ 01
• Myopic models
• Guryan and Kearney
– JPubEcon 05, AER 08, AEJ EP 10
• Lucky stores
• Tests for persistence after demand shocks but doesn’t
directly test the rational addiction model.
Lotto background
• Most prevalent form of gambling
– Still biggest revenue, definitely biggest losses
• Major contributor to state revenues
• US annual sales about $70 billion pa!
– Ca Lottery $5b in 2013/4. Illinois $2.5 b
– UK Lotto sales £2.5b ($4b) pa (£7.5b ($11b) all
games)
• “Takeout” about 50% (but its taxable in US)
Long history
• Lottery funding popular
- When hard to raise revenue elsewhere
• Often used to fund conflict/defence
- Fighting the English (US War of
Independence)
- Fighting each other (US Civil War)
- Fighting the Spanish (Armada)
• First modern Lotto game in NJ 1970
- Quickly spread to many states
• UK game 1994+
Lotto history
Public good provision
• Lotteries associated with the supply of public
goods more generally
– Construction of the SOH
– Harvard, Yale, and …. Princeton
• Illinois State Lottery only funds k-12
• California funds k-12 and beyond ($1.4b)
• UK game fund various “causes”
– “additionality” constraint
• No evidence that it matters who you lose to!
This talk
• Properties of “addictive” goods
– inter-temporal complementarity (reinforcement)
– inter-temporal diminishing MU (tolerance)
• “Rational addiction” sales model estimated
– Becker and Murphy, JPE 1988
– aggregate time series sales data for UK lotto
• RA assumes players know addiction occurs
– But still ‘choose’ to do become addicted
• Existing research relies on responses to preannounced price changes
– Lotto does not need this
Lotto
• Players choose n numbers from N
– Recorded on slip of paper and scanned
• n winning numbers are randomly chosen
from N without replacement
• Rotating drum or similar
4.8
4.6
4.4
4.2
4
3.8
3.6
3.4
3.2
3
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49
UK Lotto
• Total prize pool defined by (1-τ)S
– τ = ½ = 12% tax, 28% “good causes”, 5% store,
and 5% for operator
• Tickets that match all of the n balls drawn
share the n-ball prize pool (jackpot).
– Could be several who share jackpot, or none
– Jackpot “rolls over” to next draw if no
winners
– Players whose tickets match 5+B, 5, 4 balls
win (smaller) shares of total prize pool
• But 3-ball prize is fixed at £10 (paid first)
UK Lotto
• UK game licensed private sector “Camelot”
to run UK game in successive beauty
contests since 1994
• UK prizes paid in full, up front. No tax
• UK winners confidential
– Can relinquish anonymity
– No “lucky” stores in UK
• Free advice
Lotto players
• HSE/SHS 2012 and GPS 2010
– 8291, 4815, and 7666 obs
• Lotto prevalence correlation with X’s similar to
most other gambling
– Men > women, middle aged > old/young
– Catholic > other > muslim
– Increasing in BMI, BP, drinking, smoking
– No income/education effects
• Problem gambling (DSM-IV and PGSI)
– Underpowered to analyse tiny numbers
– But problem gamblers seem to play everything
Lotto players
• GPS 2010
• Problem gambling strongly associated with
low “happiness”
• Income effect on happiness small
• Problem gambling a big problem for a small
slice of the population
• But vast majority of lotto players only play
lotto, and in small amounts
• Lotto widely regarded as “soft” gambling
UK Lotto
• Sales (originally) very high by international
standards
– n = 6, N = 49, τ = ½
• Prob win 6-ball pool = n!/N!(N-n)! ≈ 1/14m
• And 3-ball prize is fixed - guaranteed £10
– Probability ≈ 1.75%
• J6t ≈ St /2 + Jt-1 – 3-ball winnerst x 10
– If St = 30m then average number of winners ≈2
– If J6t-1= 0 and St=30m then J6t ≈ £5m
Simplest possible lotto model
Old Lotto Extra game
• Imagine only 6-ball prize, J6 = S/2 since τ =½
– Chance not skill - so winnings untaxed
– UK game more generous (to players) than US
– And more efficiently operated
• In reality, jackpot gets about 15% of stakes,
3-ball get about 17.5% and other pools get
about 17.5% between them
– But only 6-ball pool matters for our modelling
– Because only jackpot rolls over
Lotto model
• Peculiar economies of scale (Clotfelter and
Cook AER paper)
– Higher S, lowers rollover probability, raises
expected value of ticket in current draw
– Players exert positive externalities
• Following a rollover in t-1, Jt= St/2 + St-1/2
– St-1/2 like a “raffle” element of prize in draw t
since St-1 is fixed at t
– But the higher is St the less Jt is (usually) worth
– So negative externality in rollover draw
Lotto model
• Double, treble, …. rollovers possible
• Lotto seldom generates fair bets
- Occasional examples (Dublin, BC, MIT, Extra)
Operator’s supply curve Rollover probability
1.400
1.200
1.000
0.6
R=0
0.5
R=5
R=10
0.800
0.4
0.3
0.600
0.2
0.400
0.1
0.200
0
0.000
10 20 30 40 50 60 70 80 90 100 110 120
70 65 60 55 50 45 40 35 30 25 20 15 10
Overall lottery sales
• Early UK time series
160
140
Weds Regular
Sat Regular
Sat Extra
Weds Extra
Scratch
£ millions/week
120
100
80
60
40
20
0
1
27
53
79
105
131
157
183
209
Weeks
235
261
287
313
339
365
391
Lotto sales
• Overall UK Lotto time series
Time series properties
• Strong negative time trend – near loglinear
– From Weds draw introduction in early 1996
– Until game redesigned - in early 2014
• Large rollover effects
– Proportionately larger midweek
– Midweek and weekend draw linked by rollovers
• “Halo” effect
– Rollovers have lagged as well as current effects
• Conscious selection
- Too many rollovers, too high 3-ball winners var
Conscious selection
• Prob of rollover determined by number of
unique tickets sold
– if numbers randomly picked
• Otherwise Prob(Rollover) = (1-1/14m)ϕS
– where ϕ≤1 is proportion of unique tickets
• Estimate ϕ comparing actual rollover
frequency with hypothetical
– Or compare variance in number of winners of
each prize pool with hypothetical
Rollovers
– And vice versa
• Also “superdraws”
– We ignore these
90
80
Percent of draws
• 1800 draws from intro
of Weds game in 1996
to the 2014 game
reform
• Weds rolls over into
Sat
70
60
50
40
30
Weds
Sat
20
10
0
Non Single Double Triple
rollover
Conscious selection
Too many rollovers
Conscious selection
Match 3 variance
“Lucky” numbers
• ML estimates of effect of winning
numbers on distn of the number
of winners of each prize pool
0.03
- See Walker et al JBES 2000
0.028
0.026
0.024
0.022
0.02
0.018
0.016
0.014
0.012
0.01
1 2 3 4 5 6 7 8 9 10111213141516171819202122232425262728293031323334353637383940414243444546474849
Rollover probability
• Rollover probability depends on sales
- And on number of winning numbers 31+
- For example – Saturday draws …..
Myopic model
• Cwt= αw +βwPwt +γwPst +δwwCwt-1+δwsCst-1 + uwt
• Cst = αs + βs Pst +γsPwt +δssCSt-1+δswCwt-1 + ust
– ust probably correlated with uwt so use SUR
– Include trend decline
• But P’s depend on sales, so endogenous
– And sales are probably serially correlated
• with last draw sales and in same game last week
• Include lagged C to capture “halo” effect(s)
– But lagged sales makes Pts endogenous
– So cannot use lagged P’s (ie rollovers) as IVs
RA model
• “Addictive” goods feature inter-temporal
complementarity (reinforcement) and intertemporal diminishing MU (tolerance)
• Max Σ βtU(Ct , Ct-1 , Xt) st lifetime budget
• Assuming U quadratic yields linear model
– Ct = θ Ct−1 + βθ Ct+1 + α1Pt + α2et + α3et+1
• RA implies that effect of Ct-1 same as
(discounted) effect of Ct+1
• But we need expected future C’s
Endogenity
• Previous work instruments Ct+1 with Pet+1
– Pre-announced tax changes
• But announcements may not be credible
• Tax may be driven by fear of expected rises in C
• Here, players can form expectation of
– price in the next draw if no rollover in current
draw
– price in the next draw if rollover in current
draw
– the probability of rollover in current draw
– And the level of a rollover in current draw
Expectations
1.400
Weds
Sat
1.200
R=0
R=5
R=10
1.000
0.800
0.600
0.400
0.200
0.000
10
20
30
40
50
60
70
80
90
100
110
120
Expectations
• E(Pt+1| Dayt , Rt-1 ) depends on
– Rollover probability at t
– and the size of the rollover at t
• Rollover probability at t depends on the
winning numbers at t-1 and sales at t
• Size of rollover depends on number of 3
ball winners at t-1 and sales at t
– Which depends on the winning numbers at t-1
• Use winning numbers at t-1 as IVs
– Evaluate P at Camelot’s own jackpot forecast
Preliminary estimates
Saturday sales
Wednesday sales
OLS
IV
OLS
IV
St-1s
0.42
(0.06)
0.34
(0.12)
0.17
(0.04)
0.14
(0.05)
St+1s
0.38
(0.06)
0.32
(0.11)
0.13
(0.03)
0.10
(0.05)
Pts
-31.9
(4.27)
-25.8
(7.22)
-39.29
(2.79)
-30.5
(10.6)
St-1w
0.25
(0.07)
0.24
(0.11)
0.15
(0.02)
0.14
(0.05)
St+1w
0.21
(0.05)
0.20
(0.07)
0.11
(0.03)
0.10
(0.05)
-
613
-
177
Ist stage F
Interpretation
• Reject exogeneity but results close to OLS
– Expected value is close enough to being flat
• Lotto demand ε ≈ -1.3 (-1.1) Sat (Weds)
• Lotto widely regarded as harmless fun
– Conventional CS calculation suggests £1.3b of fun
• Reject myopic addiction. Some evidence of
rationality in addiction
– Which maybe why we think its harmless fun?
– But implied discount rate very high
– Maybe we need to think about exponential
discounting?
Further work
• Improve precision by exploiting structure
– ML estimates of probability and 3-ball matches
• Model underpredicts effect of reform on S
– Effect of temporary P change (rollover)
different from permanent (design) P change
• Interesting design change
– Midweek now more valuable than weekend
– Changes dynamics of rollovers
• Also look at Euromillions game
– Game design change in middle of long series
Further work
• Short panel micro dataset – UK LCF
– Potential to estimate RA model using
microdata
• Rollovers may not load onto expected value
(P) alone
– Convenient assumption because P is linear in R
– But “gamblers love skewness”
– Variance and skewness might matter too for
sales
• Highly nonlinear in rollover prob and rollover size