A Betting Market: Description and a theoretical explanation of bets in Pelota Matches

1. A Betting Market: Description and a theoretical explanation of bets in Pelota Matches Loreto Llorente Josemari Aizpurua Universidad Pública de Navarra, Pamplona, Spain

2. 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

3. 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

4. The Pelota betting system 12 m 11m 54 m THE GAME • Two teams, the reds and the blues play by taking turns to hit a ball against a wall in a place called fronton. Jai Alai 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.

5. THE BETTING SYSTEM YOUR OPONENT YOU • Bettors bet one against another “6 TO 100” MIDDLEMAN (16% of the earnings) • The middleman gets a commission Two teams playing • Bets can be place at any time • Odds vary but are fixed in a bet

6. Odds Scores sr= red team’s score sb= blue team’s score The odds • Throughout the whole game you can see on a screen the effective odds in the market and the score at the moment • The odds consist of two numbers R = Amount of money you risk if you bet on red team B = Amount you risk if you bet on blue team • The higher number is always the same (100) and the other varies as points are played

7. Odds Scores sr= red team’s score sb= blue team’s score Example: • 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 R = Quantity of money you risk if you bet on red team B = Quantity you risk if you bet on blue team Odds Scores • 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 • 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 • 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

8. 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 • 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.

9. 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.

10. Theoretical explanation of a bet (2) Consumption set without commissions Final wealth if b EU risk averse’s IC Sb=1 Wi+OR when pr/pb> OR/OB when pr/pb= OR/OB when pr/pb< OR/OB S0 Wi Sr=1 Under EU there are no bets! Wi-OR OR / OB Wi Final wealth if r r = the reds win b = the blues win Wi = i’s wealth Sr = #bets on r OB OB

11. Theoretical explanation of a bet (3) Optimum s.t. 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 x = final wealth if r y = final wealth if b Decreasing MgU(W) Optimistic bettors!

12. Theoretical explanation of a bet (4) Final wealth if b Sb=1 optimistic’s IC: Wi+OR S0 Under RDEU there are possible bets between optimistic bettors! Wi Sr=1 Wi-OR OR / OB Wi Final wealth if r OB OB Under RDEU individual’s probability of outcome is weighted depending on the outcome’s rank, thus we find possible bets when individuals are optimistic.

13. Theoretical explanation of a bet (5) Consumption set with commissions Final wealth if b CS’s slope betting on blues = OR(1-t)/OB Sb=1 Wi+(1-t)OR • S0 Wi • Sr=1 Wi-OR • CS’s slope betting on reds = OR/OB(1-t) Wi Final wealth if r OB (1-t)OB Under RDEU with optimistic individuals there are possible bets even with commissions!

14. Empirical analysis • Data from 27 matches f = favourite l = long-shot Betting on l: This upper bound on the worst outcome: subjective probability weight (worst outcome) Betting on f:

15. Empirical Analysis (2) Long-shot bias • Bettors • Optimistic • Overestimate low probabilities and underestimate high ones

16. Summary and Conclusions • Description of the betting system • Theoretical support for the existence of a bet • Empirical study Opinions are welcome! loreto.llorente@unavarra.es