Late Informed Betting and the FavoriteLongshot Bias

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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 markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing incentives
• 4. Implications for market designs

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Parimutuel Betting

• Betting format used at horse-racing tracks
  worldwide
• Bets on horses are placed over time
• Tote board shows current bets, regular updates
• Betting is closed and race run
• 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 ?i
  where
• ?i(N(1- t)ki)/ki
• is the market odds ratio for horse i
• Budget balance always pay out ki(1?i)(1- t)N
• Expected payoffs equalized across horses when win
  probability is the implied market probability
• ri(1- t)/(1?i)ki/N

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Talk Plan

• 1. Parimutuel betting markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing incentives
• 4. 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 groups empirical
  winning probability
• Compare market probability with empirical winning
  probability

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Asch, Malkiel and Quandt (82) Data
empirical probability
market probability
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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)

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Ziemba and Hausch (1986)
Favorite-Longshot Bias Empirically
favorites
longshots
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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 Plan

• 1. Parimutuel betting markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing incentives
• 4. 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 Theories

• Overestimation of low probabilities Griffith
  (1949)
• Risk (or skewness) loving behavior Weitzman
  (1965), Ali (1977), Golec and Tamarkin (1998)
• Monopoly power of insider Isaacs (1953)
• Limited arbitrage due to positive track take
  Hurley and McDonough (1995)
• Response of uninformed bookmaker to market with
  some insiders Shin (1991, 1992)
• Behavioral misunderstanding of the winners
  curse Potters and Wit (1995)

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Talk Plan

• 1. Parimutuel betting markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing incentives
• 4. Implications for market designs

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Simplest Setting

• Two horses 1,1 prior win chance qPr(x1)
• No prior bets (a-1a10), no track take (t0)
• N risk-neutral bettors with private posterior
  belief pi (that 1 wins), continuous cdf G(px)
• All bettors must make unit bet simultaneously
• Derivation of equilibrium betting behavior
• Compare market odds to Bayesian empirical odds

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Equilibrium Betting Characterization

• Proposition 1 There exists a unique symmetric
  equilibrium, where ppN bet on 1 pN solves
• As N tends to infinity, pN tends to the unique
  solution to
• Example Fair prior and symmetric signal
  G(px1) 1G(1px1), then pN 1/2.

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Equilibrium Brief Derivation

• WN (bx) is expected payout to b-bet given x-win
• With p chance of any opponent winning
• Arbitrage condition for the marginal belief pN
• pNWN (b1x1)(1 pN)WN (b1x1), or

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Equilibrium Derivation for Large N

• Perfectly competitive limit, N8
• Indifferent bettor thinks 1 wins with chance p
• Horse 1 attracts bets from all bettors with
  posterior above p, for the mass (1G(px1))
• 1 wins with chance 1p has G(px1) bets
• Indifference holds at

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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 q1/2, symmetric signal, informative
  G(1/2x1)/G(1/2x1)1
• Proposition 3 For any long market odds ratio
  ?1, if
• (i) Informativeness G(1/2x-1)/G(1/2x1) is
  large,
• or (ii) there are many insiders N, so that
• then the Bayesian odds ratio is longer than the
  markets

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Proof of Proposition 3

• Market odds ratio shorter than Bayesian odds if
• Taking logarithms and rearranging, we get
• Since ?1 and G(1/2x1)G(1/2x1), all terms
  are positive.
• Generalize to asymmetric prior q?1/2 (Prop. 2)

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Intuition

• The bet chance for every bettor is
• 1G(1/2x1) G(1/2x1) on winner horse x in
  state x
• G(1/2x1) 1 G(1/2x1) on loser horse -x
  in state x
• Market odds converge to G(1/2x1)/G(1/2x1) or
  its reciprocal (depending on the state) as N is
  large
• Since G(1/2x1)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 q1/2, G(1/2x1)1/4, G(1/2x1)3/4, N4
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 Plan

• 1. Parimutuel betting markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing incentives
• Bet late to free-ride on others private
  information
• Bet early to pre-empt others bets on public
  information
• 4. Implications for market designs

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Timing Incentives

• There are two forces at play
• Bet late, to conceal private information and
  maybe observe others (like open auction with
  fixed deadline)
• Bet early, to capture a good market share of
  profitable bets (as in Cournot oligopoly)
• First force isolated with small bettors with
  private information
• Second force isolated with large bettors sharing
  the same information

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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

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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(px)
• Track takes proportion t of total amount bet

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Equilibrium in the Last Period

• Assume (i) belief distribution unbounded
  (0
• Proposition 6 There is a unique symmetric BNE
  All players use interior thresholds 0Bet on -1 when 0pp-1 and on 1 when p1p1.
• Proposition 7 A greater prior q implies larger
  thresholds p 1 and p1, weakly smaller W(11) 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 (q1/2 and
  a1a-1a0), last-period equilibrium has
  symmetric thresholds p-11p1. Fewer initial bets
  a/N, or lower track take t, imply more extreme
  market odds and so reduce the favorite-longshot
  bias.

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Timing

• Proposition 9 Given above assumptions.
  Postponing all bets to last period is a perfect
  Bayesian equilibrium.
• Proof After any history, 2 cases
• 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.
• Belief distribution still unbounded If player
  deviates by betting now on 1, q goes weakly up,
  W(11) weakly down (Prop 7), reducing deviators
  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 N1 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 N1 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

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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)
• First players announce at which time they wish to
  choose action (and are committed to this choice)
• 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 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

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Talk Plan

• 1. Parimutuel betting markets
• 2. Empirical facts
• Favorite-longshot bias
• Informed betting at the end
• 3. Theoretical model
• Equilibrium with simultaneous betting
• Timing 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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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

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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 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

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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(sx1)2s
f(sx-1)2(1-s) with s in 0,1
cutoff s
N1
N2
N8
Prior q
For N1, optimal to bet according to posterior
s1-q For N1, bet more on ex-ante longshot
because of winners curse