Analytics Blog

Feb 9, 2021

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The Favourite-Longshot Bias is not a bias

The so-called "favourite-longshot (FL) bias" is the well-documented observation from
betting markets that bets placed at higher offered odds (longshots) yield
worse rates of return than bets placed at lower odds (favourites).
This empirical regularity has received
attention from academics and intellectually-inclined gamblers for decades. It has been a
very robust finding: few betting markets have been studied that did not
exhibit lower rates of return at longer odds.

Why do academics, specifically economists, care so much about this pattern? The favourite-longshot bias has often been interpreted as a real-world example of irrational behaviour and market inefficiency. Indeed, we should ask, if the rate of return is worse on bets with longer odds, why do bettors take those bets? Why don't market forces, in the form of bettor demand, act on these offered prices until their rates of return are equal? Viewed through the lens of standard economics, the favourite-longshot pattern seemingly requires a non-standard explanation.

As its title suggests, the thesis put forward here is that the favourite-longshot bias is not a "bias"; it is not the result of bettor irrationality or non-standard preferences, nor is it an example of market inefficiency. We'll proceed as follows. The first section makes the intuitive case for the**inevitability** of lower rates
of return at longer odds relative to returns at shorter odds.
At very short odds, e.g. events that have "true", or objective, probabilities greater than 99%,
a bookmaker can only apply so much margin
without making the offered price absurd (i.e. exceeding 100%).
A proportional allocation of the margin — which is required to equalize
the rate of return across all offered bets — forces the bookmaker
to reduce the total margin to keep the favourite's price below 100%, which in
turn creates an essentially margin-free price on the longshot. Neither of
these options — offering a price in excess of 100% on a favourite, or a
margin-free price on the longshot — could possibly be optimal from the bookmaker's
perspective, and yet one of them is required if all offered bets are to have equal expected returns.
The second section
highlights a simple but important point, and one that appears to have been overlooked:
in research on this subject, the FL bias is defined differently
in "traditional" gambling markets (e.g. bookmaker, parimutuel) from how it is defined in
prediction markets. More specifically, in traditional betting markets the favourite-longshot
bias is defined as I have introduced it here: worse rates of returns at longer odds. However, in
prediction markets, researchers define the
favourite-longshot bias as a scenario in which the midpoint of the bid-ask spread overestimates the objective
outcome probability at longer odds.
A small puzzle in the literature
has been why the favourite-longshot bias
seems to be less prevalent in prediction markets than traditional betting markets.
The answer is simple: *the absence of a bias
in the midpoint of the bid-ask spread implies returns will decrease as odds lengthen*.
That is, the absence of an FL pattern using the prediction market definition implies
there will be an FL pattern using the traditional definition. The one exception to this is a market
where there are zero transcation costs (i.e. no margin in a bookmaker's odds, or no
spread in the bid-ask); in this case the two definitions of the favourite-longshot
bias are equivalent. While all of the traditional betting markets that have been empirically studied and
shown to exhibit the FL bias have non-zero margin,
many of the theoretical models used to explain
the bias exclusively consider the special case of zero transaction costs. This is problematic.
The third section presents a basic economic model of a
betting market. Both the bookmaker and the bettors in this model are behaving rationally,
but equilibrium offered odds will be such that worse returns for bettors are realized at longer odds.
The *only* twist in this model is that bettors
are heterogeneous: they
disagree on the probability of the event occurring. I explain why the standard "representative bettor" model is
not useful for describing a betting market, and argue that the model I present is the simplest
and most intuitive description of how bettors and "sharp" bookmakers interact.
The fourth section
first analyzes data from soccer markets at a sharp bookmaker
(Pinnacle). These are markets that see very high volume
and have many sophisticated participants; I document the extent of the favourite-longshot bias
and compare it to the predictions of the model
presented in Section 3. I then take a closer look at the motivating
empirical evidence from a well-known paper on the FL bias. The
final section concludes.

Why do academics, specifically economists, care so much about this pattern? The favourite-longshot bias has often been interpreted as a real-world example of irrational behaviour and market inefficiency. Indeed, we should ask, if the rate of return is worse on bets with longer odds, why do bettors take those bets? Why don't market forces, in the form of bettor demand, act on these offered prices until their rates of return are equal? Viewed through the lens of standard economics, the favourite-longshot pattern seemingly requires a non-standard explanation.

As its title suggests, the thesis put forward here is that the favourite-longshot bias is not a "bias"; it is not the result of bettor irrationality or non-standard preferences, nor is it an example of market inefficiency. We'll proceed as follows. The first section makes the intuitive case for the

Just give me the intuition

The market setup I use throughout this post is as follows.
All contracts have only two possible outcomes: they pay out 1 unit if the event occurs and 0 otherwise.
The market will be described
only by "win" contracts: for example, if we consider a two-golfer contest
between Golfer A and Golfer B,
the two contracts offered will be "Golfer A wins" and "Golfer B wins" (and not "Golfer A does not win",
"Golfer B does not win"). The price of these 1-unit-win contracts is the implied probability
of the contract — that is, the required probability for the contract to have an expected value of zero.
This formulation is essentially how traditional bookmaker markets are presented, except
instead of the payout being fixed at 1 unit, books list as odds the payout you receive
from a winning 1 unit bet.
For example, a bet with European
odds of 3.0 means you receive a payout of 3 units
on a 1 unit bet (for a profit of 2 units) if the event occurs; this is equivalent
to a 1-unit-win contract with a price of 1/3.
Conveniently, this formulation can be easily related to a prediction market
setting, with the added advantage that it is easy to calculate the expected rate of return on 1-unit-win contracts (in contrast
to a bid-ask or back-lay setup — both common in prediction markets — which makes
calculating rates of return less obvious, as
Section 2 highlights).
To understand why it would be very unusual for a betting market to

One way to model a betting market is to assume that all bettors are identical, risk-neutral, utility-maximizing agents. Equivalently, in this model, we can think of there only being a single "representative" bettor, instead of many identical ones. The fact that the bettor is risk-neutral means she only cares about expected returns (and not the variance in returns). The only equilibrium that can be supported in this model is one where expected returns are equalized across all offered bets; if returns weren't equal, our representative bettor would put all her money on the higher expected value bet. This model provides a rationale for the proportional allocation of a bookmaker's margin, as it is the only allocation that equalizes the rate of return across bets. Under this model of the betting market, the empirical finding of lower returns at longer odds is viewed as a bias: how can our representative bettor be indifferent to all offered bets if the expected returns are not equal? The two most common early answers to this question were: 1) people who gamble are not risk-neutral, but risk-loving; this means they are willing to take lower expected value propositions for higher upside (and downside), and 2) people over-estimate small probabilities and under-estimate large probabilities, so they perceive expected returns to be equalized across longshots and favourites even though they aren't in reality. Much of economic theory is built on the assumption that people are risk-averse and rational (a loaded term, but for our purposes this means they can assess probabilities in an unbiased fashion); the favourite-longshot bias appears to fly in the face of this, which is why it has received a lot of attention.

It is an intriguing line of reasoning. However, modelling a betting market where there is non-zero margin added to the prices with a representative bettor does not make much sense. Under that setup, we immediately have the troubling question of why this bettor is participating in the market to begin with; if they are indifferent to all offered odds, and the odds sum to more than 100% (due to a bookmaker's margin), then the bettor

An important difference in definition

Lower expected returns at longer odds is how the favourite-longshot bias is typically defined
(see [1], [2]). In an otherwise excellent paper on
prediction markets by Justin Wolfers and
Eric Zitzewitz [3], they document the favourite-longshot bias from a traditional fixed odds
betting market for horse racing by examining rate of return as a function of
odds (p.12); however,
on the very next page they investigate whether a bias is present in a
prediction market by examining the midpoint
of the bid-ask spread, failing to notice that these two examples are not conveying the same information.
In the second example, there is in fact a very substantial and monotonic decline in the rate of return
as odds lengthen, however the authors state that there is only a bias in the region of
odds between 20%-30% because they are using the "midpoint of the bid-ask" definition of the FL bias.
This difference in definition goes unnoticed by the authors.
The absence of a bias in the midpoint of the bid-ask spread implies that the rate of return declines as odds lengthen. Continuing with Golfers A and B from the introduction, suppose now that Golfer A has an 80% chance of winning (and Golfer B a 20% chance). In a prediction market setting, suppose that a market-maker sets the ask at 0.81 for the contract "Golfer A wins" (this is the price at which the bettor can buy a contract from the market-maker) and sets the bid at 0.79 (this is the price at which the bettor can sell a contract to the market-maker). Given that the true probability of Golfer A winning is 80%, it's clear that the midpoint of the bid-ask spread is unbiased. However, what is the bettor's expected rate of return from buying versus selling a "Golfer A wins" contract? Let's convert the bid-ask formulation to my preferred notation where everything is framed in terms of 1-unit-win contracts. We already have a win contract for Golfer A with a price of 0.81. For Golfer B, consider a win contract priced at 0.21. As you might expect, this contract is identical — in the sense that it has the same payouts in every possible state of the world — to selling the "Golfer A wins" contract. To (short-)sell the Golfer A win contract, you would first borrow it from someone else with the requirement you return it to them after the market ends. So, first you borrow it and then sell it for 0.79 units to the market-maker. If Golfer A loses, the contract is now worth nothing, so you get the contract back for free, for a profit of 0.79 units; if Golfer B wins, the contract is now worth 1 unit, so it costs you 1 unit to get the contract back and return it to your lender, for a profit of -0.21 units. Evidently, these are the same profits that would be realized in each state when buying the "Golfer B wins" contract for 0.21 units. Given this equivalence, we can just focus on the win contracts and easily calculate their respective rates of return: for Golfer A it is equal to:

$$ \frac{0.80 \cdot 0.19 + 0.20 \cdot -0.81}{0.81} = -1.23\% $$

$$ (0.80 \cdot 0.19 + 0.20 \cdot -0.81)/0.81 = -1.23\% $$

and for Golfer B the analogous calculation looks like:
$$ \frac{0.20 \cdot 0.79 + 0.80 \cdot -0.21}{0.21} = -4.76\% $$

$$ (0.20 \cdot 0.79 + 0.80 \cdot -0.21)/0.21 = -4.76\% $$

Given that Golfer B is the Longshot here (1-unit-win contract priced at 21%), we can see
that it's the favourite-longshot bias as it is traditionally defined! As stated from the outset,
when there is no bias
in the midpoint of the bid-ask spread If you are already satisfied with my claim that the rate of return declines at longer odds (or lower probabilities) when the midpoint of the bid-ask spread is unbiased, you can skip this paragraph. Otherwise, I am going to walk through the same expected return calculations using the "Back-Lay" terminology common at betting exchanges (e.g. Betfair). Starting with the same offered prices as before for Golfer A, the offered Back odds would be \( 1/0.81 \) = 1.235 and the offered Lay odds would be 1.266. Back odds work the same as how odds are presented by bookmakers: if you back Golfer A for 1 unit, you receive 1.266 units if she does in fact win (for a profit of 0.226) and lose a unit otherwise; conversely, if you lay 1 unit on Golfer A to win, you profit 1 unit if A loses and profit -0.266 units if A wins. The interesting question here is this: how many units did I "commit" when laying Golfer A to win? That is, what should I put in the denominator for the rate of return calculation? The answer, it turns out, is 0.266 units. One way to think about it is that the Backer commits 1 unit and the Layer commits 0.266 units; after the market is resolved, this 1.266 units is then paid out to the winning party. If you are skeptical, we can just convert this to our preferred 1-unit-win-contract formulation: laying 0.79 units at odds of 1.266 on Golfer A is equivalent to buying a 1-unit-win contract on Golfer B for 0.21 units (you can check that the payouts are the same in each state). As calculated above, we know that the rate of return on the 'Golfer B wins' contract is -4.76%. To confirm using the Lay formulation, rate of return can be written as:

$$ \frac{0.20 \cdot 1 + 0.80 \cdot (-0.266)}{0.266} = -4.76\% $$

$$ (0.20 \cdot 1 + 0.80 \cdot -0.266)/(0.266) = -4.76\% $$

To summarize, the absence of a bias in the midpoint of the bid-ask spread requires that the margin be added

Let's talk economics

Section 1 made the intuitive case for why lower average returns at longer
odds should be expected in any betting market with non-zero margin.
This section formalizes that intuition in the form of a simple model of the betting market, with a bookmaker
setting odds and many The model outlined here is essentially the same one that has been used to model prediction markets [4], with a couple minor differences. As Section 2 illustrated, there has been an odd compartmentalization of the research done on traditional gambling markets from that done on prediction markets. Most researchers seems to agree that it would be a fruitless task to model a prediction market with homogeneous bettors; why that insight hasn't been applied to models of traditional bookmaker markets is not clear.

The market we consider is based on an underlying event that has only two possible outcomes; to focus ideas, we'll again consider a contest between golfers A and B, with the two outcomes being "A wins" and "B wins". A market-maker, who we'll refer to as the bookmaker, has knowledge of the true probability of Golfer A winning, and has the task of publishing prices for two 1-unit-win contracts. There are a large number of

More formally: bettor

$$ U_{A,ir} = \beta_{A,i} + \epsilon_{A,ir} \\ U_{B, ir} = \beta_{B,i} + \epsilon_{B,ir} $$

where \( \beta_{A,i} = \beta + \eta_{A,i} \) and \( \beta_{B,i} = \beta + \eta_{B,i} \), with
\( \eta_{g,i} \sim \mathcal{N}(0, \sigma_2) \) for \( g=A,B \) and \( \beta \) as the true skill gap,
and \( \epsilon_{g,ir} \sim \mathcal{N}(0, \sigma_1) \) for \( g=A,B \). In a single round,
Golfer A beats Golfer B if \( U_{A} > U_{B} \), which leads bettor An interesting point to note here is the shape of the belief distribution. While the belief distribution for the difference in golfer skill is symmetric and centered at the true skill difference, the win probability belief distributions will not quite be symmetric.

How do bettors and bookmakers make their respective decisions? Each bettor

Suppose bettor

$$ x_{i} = Max(y_{i} \cdot \frac{q_{i} - \pi}{\pi (1-\pi)}, 0) $$

$$ x_{i} = Max(y_{i} \cdot (q_{i} - \pi)/\left(\pi (1-\pi) \right), 0) $$

(This demand equation is the solution to a straightforward
utility maximization problem.) In this setup it's not possible to demand negative assets,
hence the non-negative restriction (this doesn't affect things, it is just a simpler
exposition).
The bookmaker

The bookmaker knows the shape of the belief distribution, but they do not know any individual bettor's beliefs about event outcomes. Bookmaker utility is a function of their belief about the probability that the event will occur,

$$ p \cdot log(b - x_{A} * (1-\pi_{A}) + x_{B} * \pi_{B}) + \\
(1-p) \cdot log(b - x_{B} * (1-\pi_{B}) + x_{A} * \pi_{A}) $$

A key point here is that for any given pair of prices they set, the bookmaker
knows what aggregate demand will be for the two contracts.
To solve this model, we need to find a set of prices for the 1-unit-win contracts such that
the bookmaker and bettors are behaving optimally (given the information they possess and their
utility functions). Suppose that bettor beliefs
about Golfer A's win probability range from 70% to 90% (and are centered at the correct
probability of 80%). If the bookmaker sets prices for A and B equal
to 0.31 and 0.91, no bettors will participate in the market as they all
have negative (subjective) expected value from betting on either golfer.
Therefore the bookmaker will
have an expected (and actual) profit of 0 from setting these prices.
Consider a different set of prices: 0.23 and 0.83.
All bettors with beliefs on Golfer A below 77% or above 83% will participate in the market by buying
contracts on golfers B and A, respectively. With these prices, the bookmaker would
expect to make a profit. Therefore the price pair (0.23, 0.83) is preferred by the bookmaker
to (0.31, 0.91). To find the solution we continue like this,
considering all possible price pairs, and finding the one that maximizes
bookmaker utility.
The solution will of course depend on the specific parameters we choose to generate
bettor beliefs, bettor utility, and bookmaker utility. In the model parameterization
outlined so far we've assumed risk-averse bettors and bookmakers (i.e. log utility).
Here is the bookmaker's optimal margin allocation for golfers A and B, as well as
the corresponding expected returns for
bettors at these prices (price equals the true win probability plus the margin),
as a function of the skill difference between A and B:
skill gap |
true win prob Golfer A |
true win prob Golfer B |
margin A | margin B | exp. return Golfer A |
exp. return Golfer B |
---|---|---|---|---|---|---|

0 | 0.500 |
0.500 |
0.012 | 0.012 | -2.3% | -2.3% |

1 | 0.399 | 0.601 | 0.012 | 0.012 | -2.9% | -2% |

2 | 0.304 | 0.696 | 0.012 | 0.012 | -3.8% | -1.7% |

3 | 0.220 | 0.780 | 0.01 | 0.01 | -4.3% | -1.3% |

4 | 0.152 | 0.848 | 0.008 | 0.008 | -5% | -0.9% |

5 | 0.099 | 0.901 | 0.006 | 0.006 | -5.7% | -0.7% |

6 | 0.061 | 0.939 | 0.004 | 0.004 | -6.1% | -0.4% |

7 | 0.036 | 0.964 | 0.002 | 0.002 | -5.3% | -0.2% |

8 | 0.020 | 0.980 | 0.002 | 0.002 | -9.2% | -0.2% |

Second, under any reasonable parameterization of bookmaker and bettor utility,

I want to focus on the intuition behind the equal (absolute) allocation of margin. The bookmaker has two competing interests when they set prices: they would like to maximize expected profit, but also — depending on their preferences — want to minimize the difference in their profit from A winning or B winning. In the specific model formulation above we assumed the bookmaker was very risk-averse (i.e. log utility) which means they will care a lot about equalizing the number of contracts bought on A and B.

But what if our bookmaker is risk-neutral? After all, with the size of their bankroll, this could be a more reasonable assumption. A risk-neutral bookmaker only cares about expected profit, which means we can consider their optimal price-setting procedure separately for Golfer A's price and Golfer B's price. Maximizing expected profit has the following fundamental tradeoff: a higher price builds in a larger advantage, but it also induces fewer bettors to participate. Thinking back to the first section, this is the formal reason for why a bookmaker would never offer a price above 1: there will be zero bettor demand, which cannot be optimal for a bookmaker looking to maximize profit. It turns out that this tradeoff for a bookmaker is roughly equivalent at prices

At a more basic level, the key fact that drives these implications is that bettor beliefs are approximately symmetric around the true probability. A utility-maximizing bookmaker would only set prices with a proportional margin if the distribution of bettor beliefs was such that there was a long right tail on the favourite and a short right tail on the longshot (e.g. for true probabilities of 10% and 90%, belief distributions with ranges of 5%-11% and 89% to 95%). But, as alluded to earlier, as fair probabilities become more extreme

The model detailed in this section is not particularly original. As was mentioned earlier, it has already been applied to prediction markets and many of the implications that result from tweaking the various parameters of the model have been explored. For the purposes of this blog post, the key takeaway is that there is no reasonable parameterization of this model that results in a proportional allocation of margin being the optimal decision for a bookmaker, and there are many that result in an equal (absolute) allocation being optimal.

While simple, I think this model does a good job of capturing how sharp bettors and bookmakers interact. For example, our betting strategy at Data Golf fits perfectly with the bettor behaviour described in the model: we have our subjective assessment of the fair probability, and participate in markets whenever that fair probability exceeds the offered price. At a bookmaker like Pinnacle, which has high limits and responsive price-setting, it's not unreasonable to assume that the bookmaker "knows" the true probability when the market is near closing. In our analysis of golf betting markets we found that prices from other bookmakers add no predictive value to Pinnacle's closing price. Similarly, various articles have shown that Pinnacle's closing line in soccer markets is very difficult to beat.

To finish this section, a random thought: consider how this framework can help us understand why betting markets on events with many participants (e.g. golf tournaments with 156 players) have such high total margins. The prices in golf win markets tend to add up to anywhere from 120-150%; contrast this with a two-way market on a single golfer, e.g. Golfer A vs. The Field (i.e. not Golfer A), which tend to have total margin around 5-7%. Why such a large difference for two markets based on the same underlying event? Most of these golfers will have true win probabilities below 1%; for any given golfer, there will be some bettors who think his win probability is greater than its correct value. Consequently optimal price-setting by the bookmaker might result in a margin of 0.1-0.2% applied to these golfers' prices. With over 100 golfers, this quickly adds up to 10-20% margin. However, bettor beliefs on the probability of The Field winning (i.e. anyone but Golfer A), which might have a fair probability around 90-95%, will be capped at 100% (and likely well below). In the two-way market there is simply nowhere for the 20%-50% margin to go and still induce non-zero demand. This may seem like a trivial empirical fact to explain, but the representative bettor model has nothing to say about why these two markets would have different total margins.

Into the wild: Analyzing real-world markets

In this section I first do a brief analysis of Pinnacle's soccer markets.
The data is taken from Joseph Buchdahl's
excellent website and consists of Pinnacle's closing odds for all the main
soccer leagues from 2012-2020. The full sample consists of 27,150 matches,
with each match including odds for Away Team Win, Home
Team Win, and Draw. (Therefore, unlike the model in the previous section,
we have 3 outcomes here not 2; the basic intuitions still apply.)
This first plot shows the average implied margin — equal to the
implied probability minus our estimate of the fair probability — as a
function of implied probability. To make things transparent, I've simply binned
the data (~4000 data points per bin) and calculated implied margin as the
average implied probability in that bin minus the average result in that bin.
For example, the 4000 longest
odds in the data, captured by the left-most data point in the plot below,
had an average implied probability of 0.0391, and 0.0263 of these
events in fact occurred; this implies an average margin of 0.0128.
The average total margin for matches in this data — i.e. the sum of
Home, Away, and Draw implied probabilities minus 1 — is 0.027.
There is no relationship between
the total margin of a bet and price in this data, which means that an
equal allocation would result in a margin of 0.009 being applied to all prices.
This is basically
what the above plot shows (along with some statistical noise, as 4000 bets is still
a small sample size). While it is true that betting on the 4000 shortest odds in this data
would actually have turned a profit, as indicated by the negative implied margin
in the rightmost data point, the next 4000 shortest odds had the largest
implied margin. Statistical noise seems a likely explanation.
Given the approximately equal margin allocation,
we will observe lower average returns
as odds lengthen.
Equal allocation of margin fits with the model of the previous section, however
the constant margin level across the entire price range does not.
In the model of Section 3,
as prices moved towards 0 and 1 margin declined, while in Pinnacle's markets no such decline is observed.
Recall that, in the model setup, we assumed
the bookmaker knows the true probability of the event; if we relax this assumption
and instead assume that the bookmaker observes a noisy, but unbiased, signal
of the true probability, equilibrium prices won't exhibit the declining
total margin. The intuition is that without exact knowledge of the correct price,
a (risk-averse) bookmaker has to be more cautious about not accidentally underpricing extreme
longshots.
Next, we'll look at some of the data from one of the better-known papers on the FL bias: Wolfers and Snowberg 2010 (W&S). They motivate their paper with a plot of average returns as a function of the odds level, using data from over 5 million horse races in the US (p.1 of the linked pdf). As odds lengthen, the average rate of return declines drastically. The authors state that this illustrates that market prices are providing biased estimates of the probability of a horse winning. To make a statement like this requires some assumption about how the margin in the market should be removed. If you assume proportional margin allocation, as a risk-neutral representative bettor model predicts, then the claim follows. However, as is hopefully clear at this point, there is no reason to expect a margin to be allocated proportionally, and therefore no reason to conclude that the market provides biased probability estimates simply because returns decline as odds lengthen.

The plot below displays average returns as a function of odds (using the same log scale as W&S for the purpose of comparability) from simulated data

odds | implied probability |
rate of return |
implied true probability |
implied margin |
---|---|---|---|---|

1/3 | 0.750 | -0.09 | 0.683 | 0.067 |

1/2 | 0.667 | -0.10 | 0.600 | 0.067 |

1 | 0.500 | -0.15 | 0.425 | 0.075 |

2 | 0.333 | -0.17 | 0.276 | 0.057 |

5 | 0.167 | -0.19 | 0.135 | 0.032 |

10 | 0.091 | -0.20 | 0.073 | 0.018 |

20 | 0.048 | -0.23 | 0.037 | 0.011 |

50 | 0.020 | -0.40 | 0.012 | 0.008 |

100 | 0.010 | -0.58 | 0.004 | 0.006 |

200 | 0.005 | -0.64 | 0.002 | 0.003 |

In Pinnacle's soccer markets we observed a relatively constant margin allocation across the entire range of prices, while in W&S there is a sharp decline in margin at extreme probabilities. Therefore, in this respect, these margins are more consistent with the model presented in Section 3 than Pinnacle's. Even though we didn't observe it with Pinnacle's prices, it must be the case that the margin

My takeaways (and hopefully yours)

The empirical evidence for declining rates of return
at longer odds in gambling markets is
strikingly robust; indeed, it's so strong
that it should give you pause. If this pattern was the product of irrational
behaviour or non-standard preferences, it seems unlikely that it would be
as ubiquitous as it is.
From a theoretical standpoint, the finding of lower average returns at longer odds
is not interesting;
the simplest heterogeneous agent models that have been
used to model prediction markets can account for it.
The only class of model that seems capable of predicting equal rates of returns
across the range of possible prices
is that of the representative bettor. But, as I've argued in the preceding
sections, representative bettor models
are not suitable for modelling markets with non-zero margin.
A related empirical pattern is one typically associated with prediction markets: the midpoint of the bid-ask spread sometimes overestimates objective probabilities for low-probability events and underestimates it for high-probability events. In the framing of a traditional betting market, a bias in the midpoint of the bid-ask is equivalent to more margin being allocated to longshots than favourites

The most important takeaway from this blog post is a simple one: the two definitions of the favourite-longshot bias just described have been conflated by researchers. Most of the motivating empirical evidence for papers on the FL bias comes in the form of declining returns at longer odds, while most of the proposed theory is attempting to explain why a bias in the bid-ask spread might arise. The one setting where these two definitions of the FL bias are equivalent is when there is no margin in the market, and this is often the only case considered by researchers when developing a theoretical framework to rationalize the bias. This is one potential reason why this insight has slipped through the cracks. A second reason might be that research on bookmaker markets and prediction markets has been siloed to a large degree. As Section 2 showed, a binary prediction market can always be reframed as a traditional bookmaker market with only "win" contracts. This reframing makes it clear that the absence of a bias in the bid-ask spread implies that returns will decline as odds lengthen. The key implication of this conflation of definitions is that the evidence for the bias-in-the-midpoint version of the FL bias is not anywhere near as strong as we've presumed it is, because most of the empirical evidence is for the lower-returns-at-long-odds version of the bias. It seems everyone has taken evidence for the latter to be evidence for the former, when in fact that is not the case.

There is no question that declining returns at longer odds feels like a bias that needs explaining. Taking a broader view, the key characteristic of betting markets is that they are, in the aggregate, negative expected value for prospective bettors. As a result, the usual intuitions about risk-neutral arbitrage, i.e. if asset A returns more than asset B arbitrageurs will buy up A, which representative bettor models rely on, don't apply. For example, if a bettor came along who, having read the literature on the favourite-longshot bias, understood that better returns could be had by betting only on heavy favourites, would she employ that strategy? The answer is no, because the bettor, if risk-averse and rational, would be better off by simply not participating. On the other hand, suppose there was a betting market that had

To conclude, consider the question to the answer in the title: does the fact that expected returns are lower at longer odds represent a market inefficiency? Market efficiency is an illusive concept. As stated in the seminal 1970 paper by Eugene Fama [5], an efficient market is one where prices "fully reflect" the available information. Differing degrees of efficiency are then defined on the basis of what is considered "available information". As Fama explains, to go from a claim about market efficiency to a claim about expected returns requires specifying the process of price formation in the market. When using a representative bettor framework, which is a particular model of price formation, prices must be set such that the bettor is indifferent to all offered bets, making lower returns at longer odds a sign of some inefficiency. Conversely, using a simple heterogeneous bettor model, which outlines a different process for price formation, declining returns at long odds says nothing about the efficiency of market prices. In the version of the model outlined in this post, the bookmaker knew the true event probability, while the bettors' beliefs were only correct on average. Whether this constitutes "full information" is up for debate; if both bookmakers and bettors knew the objective probabilities, a market with transaction costs could not exist. Therefore it seems this might be as close to full information as a real-world betting market could get. If you concede this, and agree that the heterogeneous bettor model is the most straightforward representation of a betting market, then it follows that the favourite-longshot bias — as it's traditionally defined — is not a market inefficiency or a bias.