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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! :) )

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  • $\begingroup$ Betfair's book or exchange? $\endgroup$ Jul 1 at 19:32
  • $\begingroup$ @RodrigodeAzevedo: it was on the exchange $\endgroup$
    – Tom Weston
    Jul 1 at 20:52

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