Two books, one arbitrage

Question

Two bookmakers price the same two-outcome match. Book A offers $2.05$ on outcome X. Book B offers $2.20$ on outcome Y (the other side). You can bet at both.

Is there an arbitrage, and if so, how do you split a $1{,}000 stake to lock in a guaranteed profit?

Accepted Answer

Compute the implied probabilities of the two prices you would actually take: \frac{1}{2.05} \approx 0.4878, \qquad \frac{1}{2.20} \approx 0.4545. Their sum is 0.9423 < 1. Within a single book the sum exceeds 1 by the vig; here, because the two prices come from different books that disagree, the combined book is under 1. That is an arbitrage: covering both outcomes costs less than the \1 they are guaranteed to pay. Size the stakes for an equal payout regardless of outcome. Split in proportion to the implied probabilities so both bets return the same amount. With a total stake of \1{,}000: 	ex…

Two books, one arbitrage

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