How Uncertain Game Probabilities Can Skew Series Betting Odds
Summary
The document compares fixed odds for a three-match rugby series with probabilities implied by a simple binomial model. It backs out a per-game win probability from the market’s estimated probability of a South African series win, then uses that probability to estimate the chances of 2–1 and 3–0 outcomes. The market quotes assign relatively more probability to the extreme 3–0 result than the model does, while the model gives a higher probability to the 2–1 result.
The author suggests that uncertainty in the underlying per-game probability could produce a skew in the distribution of series outcomes, by analogy with volatility-driven skew in option prices. The comparison is illustrative rather than conclusive: the odds have wide spreads, draws are excluded, and cancellations could affect outcomes. The document poses the idea as an open question and provides no developed model or evidence that distinguishes probability uncertainty from market pricing effects.
Key ideas
- A binomial model can translate a per-game win probability into probabilities for series outcomes.
- The market quotes differ from the series probabilities implied by a single estimated game probability.
- Uncertainty in the per-game probability may increase the weight assigned to lower-probability extreme outcomes.
- Wide odds spreads, draws, cancellations, and bettor behavior limit what can be inferred from the comparison.
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# is the concept of skew observed in fixed odds betting markets?
# is the concept of skew observed in fixed odds betting markets?
Bear with me if this sounds a little flippant, but this has got me curious. I know "sports arbitrage" is an active economic activity, although the arbitrage arguments, I think, are not published on as widely as financial markets!
For the upcoming Lions vs. South Africa rugby tour, (which includes 3 test matches), I was consulting the betfair site, and came across the following betting markets, (& as such the use the Betfair odds convention: 1/odds = probability).
Odds of a Lions Series victory 1.85/ 1.94
Odds of a 2-1 South Africa Series Victory 2.64 / 3.5
Odds of a 3-0 South Africa Series Victory 5.5 / 7.6
I don't think it changes the argument when you work it out in detail but let's ignore draws for the following exposition.
THe mid market probabilities come out as: SA series win: 47.2%. SA series win 2-1: 33.2% SA series win 3-0: 15.6%
If you construct a little binomial model and back out a per game probability win from the SA series win (e.g. $p^{series}_{SA}=[p^{game}_{SA}]^{3} +3[p^{game}_{SA}]^{2}[1-p^{game}_{SA}] $) then uses that to infer the 2-1 series win prob$ (3[p^{game}_{SA}]^{2}[1-p^{game}_{SA}] $), and 3-0 series win $[p^{game}_{SA}]^{3} $ then the model predicts 36.9% for 2-1 and 11.2% for 3-0 but 48.1% from a win from either 2-1 or 3-0. The model underestimates low prob events and overestimates high prob.
Now there is noise from wide spreads and obviously a sensibile bettor would acknowledge the possibly of game cancellation from Covid above and beyond regular draws (cf. Lions vs. NZ).
However, nonetheless, it strikes me that in an exactly analagous way to how stochastic volatility models give rise to option skews, if the single game probabilities are (surely correctly) considered as uncertain and to be taken from a distribution, then the nature of the binomial formulae will create a larger expected probabilities of lower probability events (e.g. 3-0 vs 2.1).
And it will also put contraints on the "smiles" to prevent an arbitrage.
Has anyone seen a developed theory of this? One can image the problem gets quite involved when, say, there are many teams involved in a competition and one only observes odds for each time to win the whole thing.
(I still thing the 3-0 number was due to irrational South African hubris, however! :) )Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.