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+ 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
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 group’s
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 winner’s 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 (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
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  pN
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)= 1–G(1–p|x=1), then pN =1/2.
17
Equilibrium: Brief Derivation
•
•
WN (b|x) is expected payout to b-bet given x-win
With π 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 
N 1
•
Arbitrage condition for the marginal belief pN
pNWN (b=1|x=1)=(1– pN)WN (b=–1|x=–1), or
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
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 (1–G(p|x=1))
• –1 wins with chance 1–p & 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 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
 G (1/ 2 | x  1) 
 1
log     N log 

 1
 G (1/ 2 | x  1) 
then the Bayesian odds ratio is longer than the
market’s:
 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
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


 

k
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 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)
22
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)
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 τ of total amount bet
31
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).
32
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=a1≡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.
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 deviator’s 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
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
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 winner’s curse54