Late Informed Betting and the Favorite-Longshot Bias Marco Ottaviani London Business School Peter Norman Sorensen University of Copenhagen http://www.london.edu/faculty/mottaviani/flb.pdf 1 Talk Plan 1. Parimutuel betting markets 2. Empirical facts: I. II. Favorite-longshot bias
Informed betting at the end 3. Theoretical model: I. II. Equilibrium with simultaneous betting Timing incentives 4. Implications for market designs 2 Parimutuel Betting Betting format used at horse-racing tracks worldwide 1. Bets on horses are placed over time 2. Tote board shows current bets, regular updates
3. Betting is closed and race run 4. Pool of money bet (minus track take) is shared among winning bettors, in proportion to bets Variants used in other sports, lotto, hedging markets 3 Parimutuel vs. Fixed Odd Betting Parimutuel betting: Return to a bet depends on other bets placed Bets are placed before knowing the payoff Fixed odd betting (not in our paper): Bookmakers accept bets at quoted (and possibly changing) odds Return is not affected by later bets 4
From Market Odds to Probabilities If horse i wins, and ki out of N dollars were bet on it, every dollar on that horse gets 1+ i where i=(N(1- )ki)/ki is the market odds ratio for horse i Budget balance: always pay out ki(1+i)=(1- )N Expected payoffs equalized across horses when win probability is the implied market probability ri=(1- )/(1+i)=ki/N 5
Talk Plan 1. Parimutuel betting markets 2. Empirical facts: I. II. Favorite-longshot bias Informed betting at the end 3. Theoretical model: I. II. Equilibrium with simultaneous betting Timing incentives
4. Implications for market designs 6 Using Outcomes for Rationality Test Many horse races, each with several horses To each racing horse i associate corresponding market probabilities ki/N in that race Group horses with same market probability
From outcomes of races compute groups empirical winning probability Compare market probability with empirical winning probability 7 Asch, Malkiel and Quandt (82) Data empirical market probability probability 8
Favorite-Longshot Bias Market odds very correlated with empirical odds But too many bets on longshots, the horses unlikely to win! Sometimes an expected profit from favorite bets Anomaly, challenges orthodox economic theory Griffith (1949), Weitzman (1965), Rosett (1965), Ali (1977), Thaler and Ziemba (1988) 9 Favorite-Longshot Bias Empirically favorites longshots Ziemba and Hausch (1986)
10 Evidence on Timing Asch, Malkiel and Quandt (1982) Late odds changes predict the order of finish very well, better than actual odds Informally: bettors who have inside information would prefer to bet late in the period so as to minimize the time that the signal is available to the general public Late informed betting 11 Talk Plan 1. Parimutuel betting markets 2. Empirical facts:
I. II. Favorite-longshot bias Informed betting at the end 3. Theoretical model:: I. II. Equilibrium with simultaneous betting Timing incentives 4. Implications for market designs 12
Our Theory: Preview Partially informed bettors, wait till the end Simultaneous bets reveal information that is not properly incorporated in the final odds Many (few) bets placed on a horse indicate private information for (against) this horse If market were allowed to revise odds after last minute bets, market odds would adjust against longshots 13 Other Theories 1. Overestimation of low probabilities: Griffith (1949) 2. Risk (or skewness) loving behavior: Weitzman (1965), Ali (1977), Golec and Tamarkin (1998) 3. Monopoly power of insider: Isaacs (1953)
4. Limited arbitrage due to positive track take: Hurley and McDonough (1995) 5. Response of uninformed bookmaker to market with some insiders: Shin (1991, 1992) 6. Behavioral misunderstanding of the winners curse: Potters and Wit (1995) 14 Talk Plan 1. Parimutuel betting markets 2. Empirical facts: I. II. Favorite-longshot bias Informed betting at the end
3. Theoretical model: I. II. Equilibrium with simultaneous betting Timing incentives 4. Implications for market designs 15 Simplest Setting
Two horses 1,1; prior win chance q=Pr(x=1) No prior bets (a-1=a1=0), no track take (=0) N risk-neutral bettors with private posterior belief pi (that 1 wins), continuous cdf G(p|x) All bettors must make unit bet simultaneously 1. Derivation of equilibrium betting behavior 2. Compare market odds to Bayesian = empirical odds 16 Equilibrium Betting: Characterization Proposition 1: There exists a unique symmetric equilibrium, where p>pN bet on 1; pN solves pN 1 G ( p N | x 1) 1 1 G ( p N | x 1)
1 p N G ( p N | x 1) 1 G ( p N | x 1) N N As N tends to infinity, pN tends to the unique solution to p 1 G ( p | x 1) 1 p
G ( p | x 1) Example: Fair prior and symmetric signal G(p| x=1)= 1G(1p|x=1), then pN =1/2. 17 Equilibrium: Brief Derivation WN (b|x) is expected payout to b-bet given x-win With chance of any opponent winning: N N N 1 k 1
(1 ) N 1 k WN (b x | x) (1 ) k 0 k 1 k
N1 Arbitrage condition for the marginal belief pN pNWN (b=1|x=1)=(1 pN)WN (b=1|x=1), or N p N 1 G ( p N | x 1) 1 1 G ( p N | x 1) 1 p N G ( p N | x 1) 1 G ( p N | x 1) N 18 Equilibrium: Derivation for Large N Perfectly competitive limit, N=
Indifferent bettor thinks 1 wins with chance p Horse 1 attracts bets from all bettors with posterior above p, for the mass (1G(p|x=1)) 1 wins with chance 1p & has G(p|x=1) bets Indifference holds at p 1 G ( p | x 1) 1 p G ( p | x 1) 19 Market, Bayesian and Empirical Odds
Bayesian posterior odds: Given the observed bets, what is the posterior odds ratio for horse 1 Bayesian odds are natural estimates of empirical odds, as they incorporate the information revealed in the bets and adjust for noise We uncover a systematic relation between Bayesian and market odds depending on noise and information 20 Comparing Market & Bayesian Odds Fair prior q=1/2, symmetric signal, informative G(1/2|x=1)/G(1/2|x=1)>1
Proposition 3: For any long market odds ratio >1, if: (i) Informativeness G(1/2|x=-1)/G(1/2|x=1) is large, or (ii) there are many insiders N, so that G (1/ 2 | x 1) 1 log N log 1 G (1/ 2 | x 1) then the Bayesian odds ratio is longer than the markets: 1 G 1/ 2 | x 1 N 2k
1 G 1/ 2 | x 1 21
Proof of Proposition 3 Market odds ratio shorter than Bayesian odds if k N k 1 G 1 / 2 | x 1 1 G 1 / 2 | x 1 = k 1 G 1 /
2 | x 1 1 G 1 / 2 | x
1 N k 1 G 1 / 2 | x 1 = 1 G 1
/ 2 | x 1 N 2k Taking logarithms and rearranging, we get G (1/ 2 | x 1) 1 log N log
1 G (1/ 2 | x 1) Since >1 and G(1/2|x=1)>G(1/2|x=1), all terms are positive.
Generalize to asymmetric prior q1/2 (Prop. 2) 22 Intuition The bet chance for every bettor is 1G(1/2|x=1) = G(1/2|x=1) on winner [horse x in state x] G(1/2|x=1) = 1 G(1/2|x=1) on loser [horse -x in state x] Market odds converge to G(1/2|x=1)/G(1/2|x=1) or its reciprocal (depending on the state) as N is large
Since G(1/2|x=1)< G(1/2|x=1) and signals are i.i.d., by the LLN the bets reveal x for large N Bayesian odds tend to the extremes as N is large (Logic applies also to few well informed bettors) 23 Verbal Intuition Consider case with large number of bettors Bayesian odds are extreme (close to 0 or infinity) provided signals are informative If less than 50% bet on a horse, it is most
likely to lose; Bayesian odds are very long Market odds are less extreme one would always observe too many bets on longshot 24 Interplay of Noise & Information If the signals contain little information, Bayesian odds are close to prior odds, even with extreme market odds Deviation of market odds from prior odds are then mostly due to the noise contained in the signal Reversed favorite-longshot bias when signals contain little information: Long market odds too long! As N increases, realized bets contain more information and less noise so that Bayesian odds are more accurate than market odds, resulting in favorite-longshot bias 25
Predicted Expected Payoff as Function of LogOdds With q=1/2, G(1/2|x=1)=1/4, G(1/2|x=1)=3/4, N=4 informed bettors 26 Bias and Rationality The market odds test of rationality assumes too much information to bettors As is they know the final bet distribution
which they do not with simultaneous betting If betting were to reopen, market odds could adjust to eliminate the puzzle Theory predicts reverse bias with few poorly informed bettors e.g. lotto (no private info) 27 Talk Plan 1. Parimutuel betting markets 2. Empirical facts: I. II. Favorite-longshot bias Informed betting at the end
3. Theoretical model: I. II. Equilibrium with simultaneous betting Timing incentives: A. B. Bet late to free-ride on others private information Bet early to pre-empt others bets on public information 4. Implications for market designs 28 Timing Incentives
There are two forces at play: 1. Bet late, to conceal private information and maybe observe others (like open auction with fixed deadline) 2. Bet early, to capture a good market share of profitable bets (as in Cournot oligopoly) A. First force isolated with small bettors with private information B. Second force isolated with large bettors sharing the same information 29 Extreme Timing A] With no market power, bettors wait till the end in order to conceal information B] Large bettors with no private information
bet early to preempt competitors, but this is incompatible with favorite longshot bias if small bettors can bet after them informative last minute betting 30 A] Model with Small Private Info Pre-existing noise initial bets, a-1 and a1 Size-N continuum of small bettors, individually not affecting odds Private beliefs, distributed G(p|x) Track takes proportion of total amount bet 31
Equilibrium in the Last Period Assume: (i) belief distribution unbounded (00), last-period equilibrium has symmetric
thresholds p-1=1p1. Fewer initial bets a/N, or lower track take , imply more extreme market odds and so reduce the favorite-longshot bias. 33 Timing Proposition 9: Given above assumptions. Postponing all bets to last period is a perfect Bayesian equilibrium. Proof: After any history, 2 cases: 1. Belief distribution no longer unbounded: the state has been revealed, and all remaining players bet on winning outcome & are indifferent regarding the timing might as well postpone. 2. Belief distribution still unbounded: If player deviates by betting now on 1, q goes weakly up, W(1|1) weakly down (Prop 7), reducing deviators payoff.
34 B] Competition Among Large Bettors N large bettors share the same (superior) information We review how bets affect odds and show isomorphism with Cournot model In equilibrium bets are placed early, contrary to the empirical observation that late betting contains lots of information 35
How Betting Affects Odds Consider N=1 bettor with superior information who believes that horse 1 is very likely to win The more this bettor bets on horse 1, the lower the payout per dollar if that horse wins! Standard monopoly tradeoff Last bet has payout above marginal cost market odds not equal to posterior belief Consider the case with N>1 bettors who share the same superior information 36 ax is pre-existing bets on x bx is the total amount bet by rational bettors on x If x wins, every dollar bet on outcome x receives the payout
ax bx a - x b - x (1 - ) ax bx If the rational bettors bet only on x, this is a Cournot model with unit production cost and inverse demand curve ax a - x b p(b) = Pr(x)(1- ) ax b 37
Endogenous Timing Hamilton and Slutsky (1990) A. Extended game with action commitment Player can only play early by selecting action to which one is then committed B. Extended game with observable delay (not here) 1. First players announce at which time they wish to choose action (and are committed to this choice) 2. After announcement, players select their actions knowing when others make choice 38 Large Bettors w/ Common Information Proposition 5: With N large bettors, there are 2 types of equilibrium. In the first, all move early.
In the second, all but one move early. Proof: Appeals directly to Matsumura (1999). 39 Timing Incentives: Summary A] Late betting with small bettors possessing private information, due to incentive to conceal private information from the other bettors and maybe observe others B] Early betting with large bettors sharing common information, due to incentive to capture market share of profitable bets 40 Talk Plan
1. Parimutuel betting markets 2. Empirical facts: I. II. Favorite-longshot bias Informed betting at the end 3. Theoretical model: I. II. Equilibrium with simultaneous betting Timing incentives 4. Implications for market designs
41 Information Aggregation and Market Micro-Structure In parimutuel betting all trades are executed at the same final price so small traders have an incentive to postpone trade to the last minute In regular financial markets (Kyle (1985)) competition among traders drive them to trade early, so information is revealed early (Holden and Subrahmanyam (1992)) subsequent arbitrage trading would eliminate favorite-longshot bias 42
Parimutuel Market Structure Advantage: Intermediary bears no risk Disadvantage: Poor information aggregation Peculiar Feature: If you buy an asset, you dislike being followed by more buyers 43 Shins Explanation with Fixed Odds Monopolist bookmaker in fixed odds betting Some bettors are uninformed and others informed
Bookmaker with no private information sets odds Odd on each horse set according to inverse elasticity rule Demand for longshots is more inelastic because it is made up by relatively more uninformed bettors Bookmaker chooses shorter odds on longshots Favorite-longshot bias results from the bookmaker's market power 44 Our Explanation: Summary Some partially informed bettors, wait till the end Late simultaneous bets reveal information that is not properly incorporated in the final odds Many (few) bets placed on a horse indicate private information for (against) this horse
Horses obtaining lots (few) of late bets are more (less) likely to win than according to final market odds, as posterior odds are more extreme 45 F-L Bias and Market Structure Persistent cross countries differences in the observed biases could be attributed to different patterns in the coexistence of parallel (fixed odd and parimutuel) betting schemes amount of randomness in the closing time in parimutuel markets. Bettors might have different incentives to place their bets on the parimutuel system rather than with the bookmakers depending on the quality
of their information. 46 Conclusion The final bet distribution reflects equilibrium betting and so differs from the posterior beliefs We can explain both bias and timing with simple model with initial bets from uninformed bettors late bets from small (liquidity constrained) profit maximizing privately informed bettors 47 48
NCAA Basketball Metrick (1999) finds too much betting on the favorites in NCAA sweeps With little private information and some noise on the distribution of bets, our theory predicts the reverse favorite-longshot bias If bettors do not know the distribution of bets, they tend to bet too much on the some outcomes 49 Experimental Evidence Plott, Wit and Yang (2003) Consider setting with limited budget, private information, and random termination
Find two puzzles: (i) favorite-longshot bias and (ii) not all profitable bets are made Argue against individual decision biases because subjects were explained Bayes' rule Random termination time gives an additional incentive to the bettors to move early in order to reduce the termination risk, so we can explain both favorite-longshot bias & bettors taking risk by waiting to place their bets later 50 Market Manipulation Field experiment by Camerer (1998) Bets moved odds visibly and had slight tendency to draw money toward the horse that was temporarily bet
Net effect close to zero and statistically insignificant Some bettors inferred information from bets and others did not their reaction roughly cancelled out. 51 Horse Races v. Lotteries In lotto, typically outcomes are equally likely punters do not know the distribution of other bets (no tote board!) no private information! Observe too many bets on lucky numbers Rarely possible to make money betting on
unpopular numbers because of large take 52 Hotelling Location Games Competitors (politicians) take positions Objective to maximize market (vote) share Incentive to be close to consumers (voters) but far from competitors Parimutuel betting and forecasting contests are Hotelling location games with private information This work is also relevant for many other applications of Hotelling location game 53 Equilibrium: Example
f(s|x=1)=2s & f(s|x=-1)=2(1-s) with s in [0,1] cutoff s N=2 N=1 N= Prior q For N=1, optimal to bet according to posterior s=1-q For N>1, bet more on ex-ante longshot because of winners curse54
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