5.4 Discussion
5.4.2 Explanations for the Favorite-Longshot Bias
A number of theories have been proposed to explain FLB. Ottaviani and Sørensen (2008) summarize them into seven major categories: (i) misestimation of proba- bilities, (ii) preference for risk, (iii) heterogeneous beliefs, (iv) market power by informed bettors, (v) market power by uninformed bookmakers, (vi) limited arbi- trage by informed bettors, and (vii) timing of bets. Of those, we are particularly interested in (i), (ii), (iii), and to some extent (vii). We do not cover explanations based on market power or arbitrage here, primarily because those theories con- sider an environment with fixed-odds betting, while our data are from a betting exchange market.15
Weitzman (1965), who estimates utility function using over 12,000 horse races, suggests that local risk loving (i.e., convex utility function) is consistent with ob- served FLB. Quandt (1986) proves that FLB is a necessary condition for equi- librium in the parimutuel betting market with risk-loving traders. Jullien and Salanié (2000) later use 34,000 horse races in the U.K. and estimate expected util- ity model with constant absolute risk aversion (CARA) utility function. Their estimation result indicates risk loving. Next, they estimate cumulative prospect theory and find that utility function is convex, probability weighting function for gains is convex but not significant, and probability weighting function for losses
14Hartzmark and Solomon (2012) relate this finding with thedisposition effect, the tendency of investors to sell stocks trading at a gain relative to purchase price, rather than stocks trading at a loss (Odean, 1998; Shefrin and Statman, 1985). Their reasoning goes as follows. When the price is above the pre-game price and there is positive news pushing up the price (including touchdowns and intercepts), there will be an excess supply, causing short-term negative returns and prices being pushed below their fundamental value. This will be followed by positive long-term returns as trading prices return to their true equilibrium probability. They find patterns of apparent overreaction followed by underreaction that depend on whether the price is above or below the pre-game price.
15For example, Shin (1991, 1992) explain FLB based on the response of an uninformed book- maker to insider’s private information.
is concave, a rejection of expected utility maximization. 16
The earliest explanation for the FLB can be found in Griffith (1949), who ar- gues that a simple psychology of overestimation of low probabilities can explain the bias. Using data from 6.4 million horse races started in the U.S. between 1992 and 2001, Snowberg and Wolfers (2010) conduct a crucial test of distinguishing preference for risk and probability misperceptions. Their innovation is in the use of the exotic bets (compound lotteries) to differentiate two theories, while most other studies focus only on thewin bets. They find that misperceptions of proba- bility explain FLB better than preferences for risk.
Ali (1977) proves that FLB can be explained with bettors that are risk-neutral expected utility maximizers but have heterogeneous risk perceptions and capi- tal constraints. Based on this idea, Gandhi and Serrano-Padial (2015) show that differences in agents’ beliefs lead to a pricing pattern consistent with FLB in a competitive market for Arrow-Debreu securities. Using data from 176,000 U.S.
horse races (and assuming that bettors are risk neutral), they estimate that about 70% of the bettors have roughly correct beliefs and the remaining 30% have dis- persed beliefs. 17 In a similar spirit, Ottaviani and Sørensen (2015) theoretically examine how the market price aggregates the bettors’ posterior beliefs and how the equilibrium price reacts to information that becomes publicly available to all bettors in the environment with heterogeneous prior beliefs, common knowledge of information structure, and wealth effects. They find that the price underreacts to new information, which implies FLB: outcomes favored by the market occur more often than probability implied by market prices and, conversely, longshots win less frequently than the price indicates. They further show that wider disper- sion of beliefs corresponds to more pronounced FLB.
In addition to those relatively standard set of explanations, Ottaviani and Sørensen (2009, 2010) advance an additional theoretical explanation based on in- formation and timing of bets in parimutuel markets. In Ottaviani and Sørensen’s (2009) model, a large number of privately informed bettors who share a common prior belief take simultaneous positions just before post time (i.e., simultaneous
16Jullien and Salanié (2000) estimate three functional forms for the probability weighting func- tion, power function, a specification due to Cicchetti and Dubin (1994):
w(q) 1−w(q) =
q 1−q
γ q0
1−q0
1−γ
,
whereγcontrols curvature andq0specifies the crossing point, and the one proposed by Lattimore et al. (1992). They obtain similar results from all of those specifications.
17Chiappori et al. (2012) estimate heterogeneity in preferences, rather than beliefs, from aggre- gated betting data.
Table5.6: Potential explanations of the evolution of FLB.
In-play
Explanation horse race betting
Misestimation of probabilities
Preference for risk
×
Heterogeneous beliefs
×
Timing of bets
×
move game). FLB arises because bettors are not allowed to condition their behav- ior on the final odds: they are statistically “surprised” ex post, by which favorite emerges when the results of the simultaneous betting are announced. Bettors would prefer to bet more on the revealed favorite and cancel bets on the revealed longshot, but simultaneous-move assumption prohibits them from doing so. Ot- taviani and Sørensen (2010) extend the analysis and show that the direction and magnitude of FLB depend on the signal-to-noise ratio of private information in the market.
What insights do those explanations provide in interpreting our results on the observed evolution of FLB over time? We informally argue that misperception of probabilities is the only likely candidate of explanation to our findings (Table 5.6).
First thing to note is that stronger FLB is observed during the last 40 seconds of the races, the time window in which bettors’ beliefs would become increas- ingly aligned. Thus, heterogeneous beliefs story a la Ali (1977) and Gandhi and Serrano-Padial (2015) cannot explain our findings.
Ottaviani and Sørensen (2009, 2010) obtain their results on FLB as an ex-post
“surprise” mainly from the simultaneous-move assumption that implies inability of changing trades after observing market odds. It is clear that traders in betting exchanges are clearly not moving simultaneously and are able to make counter- trades anytime if they wish to do so. Furthermore, information regarding which horse is currently favored (and which ones are longshots) are continually updated on the limit-order book, leaving no room for traders to experience “surprise.”
Therefore, this line of explanation is also unlikely to hold.
Preferences for risk is an often-cited explanation for FLB as described above, but this will not be the full story. Golec and Tamarkin (1998), for example, show that FLB is in fact consistent with risk averse traders with preferences for skew- ness. Camerer (1998) points out that anecdotal evidence on bettors’ typical pref- erence for a “dead heat,” in which two horses finish the race almost exactly tied (and the track stewards cannot declare a single winner) and people who bet on
either horse share the total pool of money, suggests that their wagering is not due to convex utility per se (if they have convex utility, they should prefer to flip a coin and either win alone or lose than to a dead heat declared).
Furthermore, if we try to explain our findings on the dynamics of FLB using risk-loving framework, it is required to assume that either (i) bettors become in- creasingly risk-loving over time or (ii) less risk-loving bettors gradually exit from the market and only risk-loving ones remain. Further elaboration of data analysis is necessary, although nontrivial given the aggregated nature of our data, to prove or disprove this line of explanation.
One fruitful avenue for future analysis is to test if difference in race types explains our finding on the dynamics of FLB. A maiden race is a race for horses that have never won before (including those who are racing for the first time) and a non-maiden race is a race for horses with a racing (and winning) history.
Thus, there is much less information about horses in maiden races as compared to non-maiden races (Camerer, 1998). This analysis, if we are in fact able to observe differences, potentially serves as a test discriminating belief-based story and preference-based story, since the amount of information change beliefs but not preferences for risk.18
Misperception of probabilities is a likely explanation to our findings. In par- ticular, we speculate that an affect-based story proposed by Rottenstreich and Hsee (2001), who argue that the probability weighting function becomes more inverse-Sshaped for lotteries involving affect-rich than affect-poor outcomes, is a key driver of the dynamics of FLB. 19 They propose that some outcomes are rel- atively affect-rich and others are relatively affect-poor, even when the monetary values associated with those outcomes are controlled. An example of affect-rich outcomes is a $100 coupon redeemable for payment toward dinner for two at a fancy French restaurant. They claim that this coupon is likely to evoke relatively strong emotional reactions compared with a $100 coupon redeemable for payment toward one’s phone bill.
Horse racing is a particularly affect-rich environment. 20 Wulfert et al. (2005),
18Exploiting this natural variation in race types, Gandhi and Serrano-Padial (2015) in fact ob- serve that the magnitude of FLB is much more pronounced in maiden compared to non-maiden races. They do not look at the dynamics, since their data are from parimutuel markets.
19Barberis (2013) points out that “it is interesting to think about the psychological foundations of probability weighting. Tversky and Kahneman (1992) and Gonzalez and Wu (1999) offer an interpretation based on the principle of diminishing sensitivity, while Rottenstreich and Hsee (2001) give an affect-based interpretation. More recently, Bordalo et al. (2012) argue that salience is an important driver of probability weighting.”
20One source of affective experiences may be “suspense” and “surprise” from dynamics of
for example, measure experimental subjects’ heart rates while showing them a videotaped horse race with an exciting neck-to-neck finish. Half of the subjects bet $1 for a chance of winning $7 if they picked the winning horse while the other half only predicted the winning horse. They find that subjects with a chance to win money exhibited greater heart rate elevations and reported more subjective excitement while watching the race compared with those who did not wager.
Furthermore, Coventry and Norman (1997) find that the average heart rate was at a peak during the last 30 seconds of the race. Those observations support our view that increasing affective reactions, especially during the late in the races, are partly responsible for the observed dynamics of FLB.
One direction for future research is to test this affect-based interpretation di- rectly, using a controlled laboratory experiment in which subjects make trades in a short-horizon laboratory prediction market (as in Sonnemann et al., 2013) while their physiological reactions, such as skin conductance responses, are recorded (as in Kang et al., 2012).