Can Lotteries Exhibit Temporal Correlations?

Can Lotteries Exhibit Temporal
Correlations?
Fernando José Antonio
Universidade Estadual de Maringá
13th November 2012
Maringá
2 /23
Tropics of Capricorn passes through Maringá
3 /23
In this strip, the Sun reaches a point directly overhead at
least once during the solar year.
⇒ At the December solstice, the southern hemisphere is
tilted towards the Sun.
Who are we?
4 /23
My grant
5 /23
Brazilian government offers a large variety of grants for
students to study abroad.
Mine is provided by the sandwich PhD program.
Fondation pour la Coordination du
Perfectionnement du Personnel de
Niveau Supérieur
My grant
5 /23
Brazilian government offers a large variety of grants for
students to study abroad.
Mine is provided by the sandwich PhD program.
Fondation pour la Coordination du
Perfectionnement du Personnel de
Niveau Supérieur
⇒ Ciência sem Fronteiras
Mega-Sena
6 /23
Mega-Sena, the major and most famous lottery in Brazil.
Mega-Sena
6 /23
Mega-Sena, the major and most famous lottery in Brazil.
Data are held by Caixa Econômica Federal bank, an institution
which maintains online informations related to the draws (number
of winners, prizes, etc.)
We analysed the data since the
beginning of Mega-Sena on March
1996 up to 06 October 2012.
The Game
7 /23
It’s a 6/60 gamble game. Sena, quina, and quadra are the
prizes for players hitting 6, 5, or 4 numbers, respectively.
The Game
7 /23
It’s a 6/60 gamble game. Sena, quina, and quadra are the
prizes for players hitting 6, 5, or 4 numbers, respectively.
Players may bet from 6 to 15 numbers.
Bet prices escalate depending on all the permutations with
6 numbers within the choices.
So, it varies between R$ 2.00 for 6 numbers and R$ 10,010.00
for 15 numbers (15!/(6!9!) = 5,005 games possible).
The Game
7 /23
It’s a 6/60 gamble game. Sena, quina, and quadra are the
prizes for players hitting 6, 5, or 4 numbers, respectively.
Players may bet from 6 to 15 numbers.
Bet prices escalate depending on all the permutations with
6 numbers within the choices.
So, it varies between R$ 2.00 for 6 numbers and R$ 10,010.00
for 15 numbers (15!/(6!9!) = 5,005 games possible).
.
Distinctive feature from other random systems: a jackpot
prize can be ‘rolled-over’ to the same tier in the next draw.
.
The Odds of Winning
8 /23
The probability of hitting exactly i numbers, out of k selected by the player among N is given by
(k)(N−k)
P[X = i] =
i
(Nk−i
) .
k
For a bet with 6 numbers the chances of winning sena,
quina and quadra are 1 in 50,063,860, 1 in 154,518.1 and 1
in 2,332.35 respectively.
Whilst the odds of not hitting any number is 1 in 1.9!
The Odds of Winning
8 /23
The probability of hitting exactly i numbers, out of k selected by the player among N is given by
(k)(N−k)
P[X = i] =
i
(Nk−i
) .
k
For a bet with 6 numbers the chances of winning sena,
quina and quadra are 1 in 50,063,860, 1 in 154,518.1 and 1
in 2,332.35 respectively.
25% of Brasilian population (∼ 2 × 108 )
Usually, there are ∼ 106 bets per draw.
Prizes
9 /23
The net value for prizes is actually 32.2% of the income:
35% are distributed to the winners of sena.
19% are distributed to the winners of quina.
19% are distributed to the winners of quadra.
22% will accumulate for the next draw with final 0 or 5.
5% will accumulate for Mega Da Virada.
■
■
■
■
■
The other part of the income is spent on costs, social programmes and income tax.
.
R$ 177.62 millions was shared among 5 winners of sena
(in that day, R$ 1.87 ≈ 1 US$) in the last Mega Da Virada.
.
How many bets per draw?
10 /23
Differences between Saturdays draws or Wednesdays draw?
7
Number of bets (10 )
30
20
10
0
17 NOV 2010
29 OCT 2011
Draw
06 OCT 2012
How many players win a prize?
a
11 /23
18
Number of winners
9
0
4x103
2x103
0
1x105
5x104
0
11 MAR 1996
19 NOV 2005
Draw
06 OCT 2012
How many players win a prize?
a
11 /23
18
Number of winners
9
0
4x103
2x103
0
1x105
5x104
0
11 MAR 1996
19 NOV 2005
06 OCT 2012
Draw
What a correlation analysis can tell us about these time
series?
Conclusions from DFA Method:
12 /23
b 103
a
0.5
0.6
10
1
2
10
s
F (s)
10
Quadra
F (s)
F (s)
0
10
c
Quina
Sena
4
10
0.7
2
1
2
10
10
s
1
2
10
10
s
Typically, time series exhibiting fractal properties are characterized by a power-law behaviour like
F(s) ∝ sh ,
.
where h gives a measure of the autocorrelation degree.
.
DFA Method of order l:
13 /23
Given a time series u(j), j = 1, 2, · · · , N:
t
∑
■ reduce fluctuations: y(t) =
u(j);
j=1
■
■
divide y(t) in windows;
remove polynomial trends of degree l for each window;
5
0.4
4
0.2
3
0.0
2
-0.2
1
-0.4
0
0
■
■
20
40
60
80
100
0
20
40
60
80
100
compute the variance in detrended window;
compute F(s), the root square of the average of these
variances over all the windows of size s.
Memory of the Past
14 /23
Autocorrelations in a time series are connected (mainly) to
a memory effect.
‘How can an unbiased system exhibit long range memory?’
Memory of the Past
14 /23
Autocorrelations in a time series are connected (mainly) to
a memory effect.
‘How can an unbiased system exhibit long range memory?’
Assuming unbiased odds of winning (there is no cheating)
the memory of the past has to be connected somehow to
the dynamics of the bets.
So, if ‘something’ happens, the number of bets should increase more and more. Else, it should decrease.
Memory of the Past
14 /23
Autocorrelations in a time series are connected (mainly) to
a memory effect.
‘How can an unbiased system exhibit long range memory?’
Assuming unbiased odds of winning (there is no cheating)
the memory of the past has to be connected somehow to
the dynamics of the bets.
So, if ‘something’ happens, the number of bets should increase more and more. Else, it should decrease.
In this case, ‘something’ = a jackpot (from rollovers)
Why Money?
15 /23
The reason is simple. How many people do not want to
obtain easily a hefty amount of money?
Money enables us to achieve goals more easily, conducting
to a state of self-sufficiency.
It acts as an incentive or a reinforcer of actions that in the
past led or not to a reward.
In this sense, money is a motivator.
.
In Psychology, motivator is something that can change
the way we think, behave, and act.
.
Why Money?
15 /23
.
In Psychology, motivator is something that can change
the way we think, behave, and act.
.
Motivators Throughout the History
16 /23
The demand for food, power, sex, fame, and wealth has
played a central role in the social dynamics.
⇒
Decisions and choices related to chance and necessity.
Motivators Throughout the History
16 /23
The demand for food, power, sex, fame, and wealth has
played a central role in the social dynamics.
⇒
Decisions and choices related to chance and necessity.
Our life is full of
decision-makings!

 common everyday situations

highly consequential ones
Motivators Throughout the History
16 /23
The demand for food, power, sex, fame, and wealth has
played a central role in the social dynamics.
⇒
Decisions and choices related to chance and necessity.
Our life is full of
decision-makings!

 common everyday situations

highly consequential ones
Humans adapt the degree of effort they make according to
the magnitude of the expected reward. We always attempt
to optimize the outcomes.
How to measure the desire for something?
b
7
A(t+1) (10 Reais)
A tricky quantitative
measure:
After a rollover, how
the jackpot size grow
compared to its previous value?
17 /23
12
10
8
6
4
2
0
0
2
4
6
7
8
A(t) (10 Reais)
10
12
How to measure the desire for something?
b
7
A(t+1) (10 Reais)
A tricky quantitative
measure:
After a rollover, how
the jackpot size grow
compared to its previous value?
17 /23
12
10
8
6
4
2
0
0
2
4
6
8
10
12
7
A(t) (10 Reais)
Lottery is a random scenario driven by unbiased drawings
strongly influenced by an external agent, the greed.
.
Mechanism: the desire for wealth causes players to bet
more depending on how big the offered prize is.
.
Design of Lotteries
18 /23
Most of the modern lottery games are variations of a unique
design:
■
players buy tickets choosing m different numbers from
a set of n options;
■
when a draw takes place, m numbered balls are selected
randomly from the set n;
■
prizes are awarded to players hitting all the m winning
numbers.
Modelling Results
19 /23
■
N bets are performed. In each bet, 6 different numbers
are randomly selected in the range 01 – 60;
■
A random draw is held by drawing 6 numbers in the
same range;
■
The bets are compared to the draw and the number of
bets scoring n numbers are stocked;
■
If no bet hits all the 6 numbers drawn, N increases a
percentage r for the next step. Else, N receives its
initial value N0 .
Output from Model
b
20 /23
18
Number of winners
9
0
8x102
4x102
0
2x104
1x104
0
11 MAR 1996
19 NOV 2005
Draw
06 OCT 2012
Data vs Model
21 /23
b 103
a
Quina
F (s)
0
10
0.5
0.6
2
10
1
2
10
10
10
s
1
2
10
s
c
Quadra
F (s)
F (s)
Sena
10
4
0.7
1
10
10
s
2
Conclusions
22 /23
The prospect of being awarded acts on cognitive processes
and mental rules associated with our decisions.
⇒
It’s a feedback loop, where the system outputs feed
back into the system changing the input, which can either
ramp up or down the system output.
Conclusions
22 /23
The prospect of being awarded acts on cognitive processes
and mental rules associated with our decisions.
⇒
It’s a feedback loop, where the system outputs feed
back into the system changing the input, which can either
ramp up or down the system output.
This ‘external agent’ could be related to the necessity or
desire for food, social status, or anything else.
Example: the desire for food acting on a hunting process
could generate autocorrelation, explaining the ubiquitous
presence of 1/f α noise in Biology.
.
Thank you for your attention!
.