What odds on the other side create an arb?

Question

A two-outcome tennis match. Bookmaker A is offering decimal odds of $2.50$ on Player X. You are watching Bookmaker B, who is about to post odds on Player Y.

Above what decimal odds on Player Y does a riskless arbitrage open up? If B posts $1.80$ on Y, what is the guaranteed return?

Accepted Answer

Book A's price on X implies \frac{1}{2.50} = 0.40. For a two-way arbitrage you need the two implied probabilities to sum below 1: \frac{1}{2.50} + \frac{1}{dY} < 1 \quad\Longrightarrow\quad \frac{1}{dY} < 0.60 \quad\Longrightarrow\quad dY \frac{1}{0.60} \approx 1.667. So any price on Y above 1.667 hands you a Dutch book. Exactly 1.667 is the break-even (fair complement), where the two prices are jointly consistent and the edge is zero. With B at 1.80 on Y: \frac{1}{2.50} + \frac{1}{1.80} = 0.40 + 0.5556 = 0.9556 < 1. Stake proportional to 1/di. On a \1000 bankroll, \418.6 on X and \581.4 on Y…

What odds on the other side create an arb?

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