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Late Informed Betting and the Favorite-Longshot Bias

http://www.london.edu/faculty/mottaviani/flb.pdf

Marco Ottaviani

London Business School

Peter Norman Sorensen

University of Copenhagen

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Talk Plan

1. Parimutuel betting markets2. Empirical facts:

I. Favorite-longshot bias II. Informed betting at the end

3. Theoretical model:I. Equilibrium with simultaneous bettingII. Timing incentives

4. Implications for market designs

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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

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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

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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+ ρi where

ρi=(N(1- τ)–ki)/ki

is the market odds ratio for horse i

• Budget balance: always pay out ki(1+ρi)=(1- τ)N

• Expected payoffs equalized across horses when win probability is the implied market probability

ri=(1- τ)/(1+ρi)=ki/N

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Talk PlanTalk Plan1. Parimutuel betting markets1. Parimutuel betting markets2. Empirical facts:

I. Favorite-longshot bias II. Informed betting at the end

3. Theoretical model:3. Theoretical model:I. Equilibrium with simultaneous bettingEquilibrium with simultaneous bettingII. Timing incentivesTiming incentives

4. Implications for market designs4. Implications for market designs

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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 group’s empirical winning probability

• Compare market probability with empirical winning probability

8“empirical”probability

“market”probability

Asch, Malkiel and Quandt (82) Data

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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)…

10Ziemba and Hausch (1986)

Favorite-Longshot Bias Empirically

longshotsfavorites

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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

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Talk PlanTalk Plan1. Parimutuel betting markets1. Parimutuel betting markets2. Empirical facts: 2. Empirical facts:

I. Favorite-longshot bias Favorite-longshot bias II. Informed betting at the endInformed betting at the end

3. Theoretical model::I. Equilibrium with simultaneous bettingEquilibrium with simultaneous bettingII. Timing incentivesTiming incentives

4. Implications for market designs4. Implications for market designs

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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

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Other Theories1. 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 winner’s curse:

Potters and Wit (1995)

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Talk Plan1. Parimutuel betting markets1. Parimutuel betting markets2. Empirical facts:

I. Favorite-longshot bias II. Informed betting at the endInformed betting at the end

3. Theoretical model:3. Theoretical model:I.I. Equilibrium with simultaneous bettingEquilibrium with simultaneous bettingII. Timing incentivesTiming incentives

4.4. Implications for market designsImplications for market designs

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Simplest Setting

• Two horses –1,1; prior win chance q=Pr(x=1)

• No prior bets (a-1=a1=0), no track take (τ=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

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Equilibrium Betting: CharacterizationProposition 1: There exists a unique symmetric

equilibrium, where p>pN bet on 1; pN solves

As N tends to infinity, pN tends to the unique solution to

• Example: Fair prior and symmetric signal G(p|x=1)= 1–G(1–p|x=1), then pN =1/2.

N

N

NN

N

N

N

N

xpG

xpG

xpG

xpG

p

p

)1|(1

)1|(11

)1|(

)1|(1

1

)1|()1|(1

1

xpGxpG

pp

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Equilibrium: Brief Derivation• WN (b|x) is expected payout to b-bet given x-win• With π chance of any opponent winning:

• Arbitrage condition for the marginal belief pN

pNWN (b=1|x=1)=(1– pN)WN (b=–1|x=–1), or

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0

1 1 (1 )( | ) (1 )

1

NNk N k

Nk

NNW b x x

kk

N

N

NN

N

N

N

N

xpG

xpG

xpG

xpG

p

p

)1|(1

)1|(11

)1|(

)1|(1

1

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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 (1–G(p|x=1))

• –1 wins with chance 1–p & has G(p|x=–1) bets

• Indifference holds at

)1|()1|(1

1

xpGxpG

pp

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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

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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 ρ>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

then the Bayesian odds ratio is longer than the market’s:

1 (1/ 2 | 1)log log

1 (1/ 2 | 1)

G xN

G x

21 1/ 2 | 1

1 1/ 2 | 1

N kG x

G x

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Proof of Proposition 3Market odds ratio shorter than Bayesian odds if

Taking logarithms and rearranging, we get

Since ρ>1 and G(1/2|x=–1)>G(1/2|x=1), all terms are positive.

• Generalize to asymmetric prior q≠1/2 (Prop. 2)

1 (1/ 2 | 1)log log

1 (1/ 2 | 1)

G xN

G x

kNkNk

xGxG

xGxG

xGxG

kkN

2

1|2/111|2/11

=1|2/111|2/11

1|2/111|2/11

=

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Intuition

• The bet chance for every bettor is

1–G(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)

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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

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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

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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

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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)

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Talk Plan1. Parimutuel betting markets1. Parimutuel betting markets2. Empirical facts:

I. Favorite-longshot bias Favorite-longshot bias II. Informed betting at the end

3. Theoretical model:3. Theoretical model:I. Equilibrium with simultaneous bettingEquilibrium with simultaneous bettingII.II. Timing incentives:Timing incentives:

A.A. Bet late to free-ride on others’ private informationBet late to free-ride on others’ private informationB.B. Bet early to pre-empt others’ bets on public information Bet early to pre-empt others’ bets on public information

4. Implications for market designs4. Implications for market designs

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Timing IncentivesThere 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

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Extreme TimingA] 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

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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 τ of total amount bet

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Equilibrium in the Last Period

Assume: (i) belief distribution unbounded (0<G(p)<1) and (ii) track take τ not too large.

Proposition 6: There is a unique symmetric BNE: All players use interior thresholds 0<p -1<p1<1. Bet on -1 when 0≤p≤p-1 and on 1 when p1≤p≤1.

Proposition 7: A greater prior q implies: larger thresholds p –1 and p1, weakly smaller W(1|1) and weakly greater W(-1|-1).

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Favorite-Longshot Bias Revisited

• With some bets on both horses, market odds are not zero/infinite. But the continuum of bets reveals the true state. Extreme form of bias.

Proposition 8: In symmetric case (q=1/2 and a1=a-

1≡a>0), last-period equilibrium has symmetric thresholds p-1=1–p1. Fewer initial bets a/N, or lower track take τ, imply more extreme market odds and so reduce the favorite-longshot bias.

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TimingProposition 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 deviator’s payoff.

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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

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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

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• 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

• If the rational bettors bet only on x, this is a Cournot model with unit production cost and inverse demand curve

•

xx

x-x-xx

bababa

)-(1

babaa

)-Pr(x)(1=p(b) x

x-x

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Endogenous TimingHamilton and Slutsky (1990)

A. Extended game with action commitmentPlayer 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

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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).

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Timing Incentives: SummaryA] 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

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Talk Plan

1. Parimutuel betting markets1. Parimutuel betting markets2. Empirical facts: 2. Empirical facts:

I. Favorite-longshot bias Favorite-longshot bias II. Informed betting at the endInformed betting at the end

3. Theoretical model:3. Theoretical model:I. Equilibrium with simultaneous bettingEquilibrium with simultaneous bettingII. Timing incentivesTiming incentives

4. Implications for market designs

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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

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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

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Shin’s 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

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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

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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.

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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

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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

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Experimental EvidencePlott, 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

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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”.

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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

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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

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Equilibrium: Example f(s|x=1)=2s & f(s|x=-1)=2(1-s) with s in [0,1]

For N=1, optimal to bet according to posterior s=1-qFor N>1, bet more on ex-ante longshot because of winner’s curse

N=1N=2

N=∞

Prior q

cutoff s