Loreto Llorente

Josemari Aizpurua

Universidad Pública de Navarra, Pamplona, Spain

A Betting Market: Description and a theoretical explanation of

bets in Pelota Matches

Objective

• Study the Pelota betting system– Description of the betting system

• The game

• The betting system

– Explain theoretically the existence of a bet

– Study empirically this betting market: field data analysis

Introduction

Financial markets

Betting markets

- Odds systems

- Pari-mutuel betting

- Odds offered by bookmakers

- Point spread offered by bookmakers

Sauer 1998

The Pelota betting system

• Two teams, the reds and the blues play by taking turns to hit a ball against a wall in a place called fronton.

54 m

12 m11m

The Pelota betting system

THE GAME

• When a team makes an error, the opponent scores one point.

• The team that accumulates a fixed number of points (40) wins the match.

Jai Alai game

• Bettors bet one against another

THE BETTING SYSTEM

• The middleman gets a commission

• Bets can be place at any time

• Odds vary but are fixed in a bet

MIDDLEMAN(16% of the earnings)

YOUYOUR OPONENT

“6 TO 100”

Two teams playing

• Throughout the whole game you can see on a screen the effective odds in the market and the score at the moment

sr = red team’s score

sb = blue team’s scoreR srB sb

Odds Scores

R = Amount of money you risk if you bet on red team

B = Amount you risk if you bet on blue team

• The odds consist of two numbers

• The higher number is always the same (100) and the other varies as points are played

The odds

The game has just started

– The score is zero - zero

– One bet on the reds: you play the lottery

• 84 if reds win, -100 if blues win

– One bet on blue

• 84 if blues win, -100 if reds win

Example:100 0100 0

sr = red team’s score

sb = blue team’s scoreR srB sb

Odds Scores

R = Quantity of money you risk if you bet on red teamB = Quantity you risk if you bet on blue team

• Reds score 15 point. – The score is 15 zero to reds.

– The odds are 100 to 2 on reds

– One bet on blues

– One bet on reds

• 1,68 if reds win, -100 if blues win

• 84 if blue win, -2 if reds win

100 152 0

• Reds score 1 point. – The score is 1 - zero to reds. – The odds are 100 to 90 on reds– One bet on reds: 75,6 if reds win, -100 if blues win– One bet on blues: -90 if reds win, 84 if blues win

100 190 0

• Reds score 15 points. – The score is 15 zero to reds.

– The odds are 100 to 2 on reds

– One bet on reds

• 1,68 if reds win, -100 if blues win

– One bet on blues

• 84 if blue win, -2 if reds win

100 152 0

Odds Scores

• Near the end – The score is 39 to 38 to the reds

– The odds are 100 to 40 on the reds

– One bet on reds: 33,6 if reds win, -100 if blues win

– One bet on blues: -40 if reds win, 84 if blues win

100 3940 38

• At the end of the game.

The middleman

– The middleman is paid by the people who have lost.

– He gets 16% (commission).

– The middleman pays people the amount won.

Odds Scores

Theoretical explanation of a bet

Assume all individuals are equal

• Expected utility theory (EU)– Risk-averse individuals there are no bets– Risk-neutral individuals when commissions,

there are no bets– Risk-taking individuals they decide to bet

all their wealth

We look for theoretical explanation of bets in the fronton and we find it

• Rank dependent expected utility model (RDEU) by Quiggin.

Wi-OR

OB

Wi+OR

Sb=1

Sr=1

S0

Final wealth if r

Final wealth if b

OR / OB

Wi

OB

Wi

Consumption set without commissions

risk averse’s IC

when pr/pb= OR/OBwhen pr/pb> OR/OBwhen pr/pb< OR/OB

Under EU there are no bets!

Theoretical explanation of a bet (2)

EU

r = the reds winb = the blues winWi = i’s wealthSr = #bets on r

Assuming equal individuals

– Under EU there are no possible bets• Sr EU ({(W - Sr OR, W + Sr OB); (1-pr, pr)}) = (1-

pr) u(W - Sr OR) + pr u(W + Sr OB)

• Sb EU ({(W + Sr OR, W - Sr OB); (1-pr, pr)}) = (1-pr) u(W + Sr OR) + pr u(W - Sr OB)

• There are no possible bets :– If OR/OB > pr /(1-pr) everyone willing to bet on the blues

– If OR/OB < pr /(1-pr) everyone willing to bet on the reds

– If OR/OB < pr /(1-pr) everyone bets 0

– RDEU: possible find bets (interior solution)

• Sr RDEU ({(W - Sr OR, W + Sr OB); (1-pr, pr)}) = q(1-pr) u(W - Sr OR) + (1- q(1-pr)) u(W +Sr OB)

• Sb RDEU ({(W - Sb OB, W + Sb OR); (pr, 1-pr)}) = q(pr) u(W - Sr OR) + (1- q(pr)) u(W +Sr OB)

• Existence of a bet requires

)('

)('

1 xU

yU

p

p

O

O

r

r

B

R

Optimum s.t.

Theoretical explanation of a bet (3)

x = final wealth if r

y = final wealth if b

Optimistic bettors!

Decreasing MgU(W)

)('

)('

1 yU

xU

p

p

O

O

r

r

B

R

)('

)('

)1(

)1(1

xU

yU

pq

pq

O

O

r

r

B

R

)('

)('

)(1

)(

yU

xU

pq

pq

O

O

r

r

B

R

)1()(1 rr pqpq

Wi-OR

OB

Wi+OR

Sb=1

Sr=1

S0

Final wealth if r

Final wealth if b

OR / OB

Wi

OB

Wi

Under RDEU there are possible bets

between optimistic bettors!

Under RDEU individual’s probability of outcome is weighted depending on the outcome’s rank, thus we find possible bets when individuals are optimistic.

optimistic’s IC:

Theoretical explanation of a bet (4)

)1()(1 rr pqpq )(1

)(

r

r

pq

pq

)1(

)1(1

r

r

pq

pq

Consumption set with commissions

CS’s slope betting on reds =

OR/OB(1-t)

•Wi

OB (1-t)OB

Wi-OR

Sb=1

Sr=1

S0

•

•

Final wealth if r

Final wealth if b

Wi

CS’s slope betting on blues =

OR(1-t)/OB

Wi+(1-t)OR

Under RDEU with optimistic individuals there are possible bets even with commissions!

Theoretical explanation of a bet (5)

Betting on f:

Empirical analysis

Odds;   favorite favorite  

(Of , Ol) Cases winnes lf losses f

l

(1000, 900) 121 62 (0,51) 0,48 59 (0,49) 0,43

(1000, 800) 68 29 (0,43) 0,51 39 (0,57) 0,4

(1000, 700) 39 25 (0,64) 0,55 14 (0,36) 0,37

(1000, 600) 51 38 (0,75) 0,58 13 (0,25) 0,34

(1000, 500) 48 41 (0,85) 0,63 7 (0,15) 0,3

(1000, 400) 33 21 (0,64) 0,68 12 (0,36) 0,25

(1000, 300) 28 20 (0,71) 0,74 8 (0,29) 0,2

(1000, 250) 12 8 (0,67) 0,77 4 (0,33) 0,17

(1000, 200) 28 27 (0,96) 0,81 1 (0,04) 0,14

(1000, 150) 16 16 (1) 0,85 0 (0) 0,11

(1000, 120) 11 11 (1) 0,88 0 (0) 0,09

(1000, 100) 20 20 (1) 0,89 0 (0) 0,08

(1000, 80) 9 9 (1) 0,91 0 (0) 0,06

(1000, 60) 3 3 (1) 0,93 0 (0) 0,05

(1000, 50) 2 2 (1) 0,94 0 (0) 0,04

• Data from 27 matches f = favourite l = long-shot

Betting on l:

This upper bound on the worst outcome: subjective probability weight (worst outcome)

lf

ff

l

OtO

tO

1

1

0

fl

ffl

l tOO 11 0

ll

ll

f

OtO

tO

1

1

lf

llf

f tOO 11

Worst outcome's probabilities weights

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 0,2 0,4 0,6 0,8 1

Real frequency of the worst outcome

Weig

ht

att

ach

ed

(in

ferr

ed

fro

m o

dd

s)

(frequency "favorite w ins", a non-favorite bettor's w eight)(frequency "favorite losses", a favorite bettor's w eight)

Empirical Analysis (2)

p72.0104.0 865.02 R

Long-shot bias

cc

c

pbp

bp

1

839.02 R

Bettors

- Optimistic

- Overestimate low probabilities and underestimate high ones

55.0 84.0 cb

• Description of the betting system• Theoretical support for the existence of a bet• Empirical study

Summary and Conclusions

Opinions are welcome!

loreto.llorente@unavarra.es