Arbitrage in Betting Markets
Three bookmakers offer decimal odds on a tennis match between Player A and Player B:
- Bookmaker 1: A wins at 2.1, B wins at 1.8
- Bookmaker 2: A wins at 1.9, B wins at 2.0
- Bookmaker 3: A wins at 2.2, B wins at 1.7
Decimal odds work as follows: a bet of $\
- Is there an arbitrage opportunity? How do you check?
- If so, how should you allocate your bets to guarantee a risk-free profit, and what is the guaranteed return as a percentage of your total stake?
Hints
- For each outcome, find the best odds offered across all bookmakers -- you are shopping for the most generous line on each side.
- Convert decimal odds to implied probabilities via /d$. If the best implied probabilities sum to less than 1, there is an arbitrage.
- To lock in a guaranteed profit, allocate your stake so that $d_A^{*} \cdot x_A = d_B^{*} \cdot x_B$, i.e., the payout is the same regardless of which side wins.
Worked Solution
How to Think About It: In any two-outcome event, each bookmaker's odds imply a probability for each side. If you shop across bookmakers and find the best odds on each outcome, you are combining the most generous line on A from one shop with the most generous line on B from another. If those implied probabilities sum to less than 1, you have found an arbitrage -- you can bet both sides and lock in a guaranteed profit regardless of who wins. This is exactly the same logic as triangular arbitrage in FX: the "round trip" costs less than 1, so the remainder is your free money.
Quick Estimate: Scan the odds. Best price on A is 2.2 (Bookmaker 3), best price on B is 2.0 (Bookmaker 2). Implied probabilities:
/2.2 \approx 0.455$ and/2.0 = 0.50$. Sum $\approx 0.955$. That is less than 1, so arbitrage exists. The profit margin is roughly.0 \times 0.5238C = 1.04762C$. Either way:- 0.955 = 4.5\%$ of stake. Already a solid back-of-envelope answer..2 \times 0.4762C = 1.04762C$. If B wins, you receiveApproach: We formalize the allocation that equalizes returns across outcomes, then compute the exact guaranteed profit.
Formal Solution:
Let $d_A^{*} = 2.2$ (best odds on A, from Bookmaker 3) and $d_B^{*} = 2.0$ (best odds on B, from Bookmaker 2).
Step 1 -- Arbitrage condition. An arbitrage exists when:
$\frac{1}{d_A^{*}} + \frac{1}{d_B^{*}} < 1$
Plugging in:
$\frac{1}{2.2} + \frac{1}{2.0} = 0.4545 + 0.5000 = 0.9545 < 1 \quad \checkmark$
The sum is less than 1, so arbitrage exists.
Step 2 -- Bet allocation. Let $C$ be the total capital to invest. We bet $x_A$ on A at odds 2.2 and $x_B$ on B at odds 2.0, with $x_A + x_B = C$. To guarantee the same payout regardless of who wins, set:
$d_A^{*} \cdot x_A = d_B^{*} \cdot x_B$
$2.2 \, x_A = 2.0 \, x_B$
The standard formula for the allocation that equalizes payouts is:
$x_A = \frac{C / d_A^{*}}{1/d_A^{*} + 1/d_B^{*}} = \frac{C \times 0.4545}{0.9545} = 0.4762 \, C$
$x_B = \frac{C / d_B^{*}}{1/d_A^{*} + 1/d_B^{*}} = \frac{C \times 0.5000}{0.9545} = 0.5238 \, C$
Step 3 -- Guaranteed return. If A wins, you receive
$\text{Guaranteed profit} = 1.04762C - C = 0.04762C$
Equivalently, the guaranteed profit rate is:
$r = \frac{1}{1/d_A^{*} + 1/d_B^{*}} - 1 = \frac{1}{0.9545} - 1 \approx 4.76\%$
Answer: Yes, arbitrage exists. Bet $47.62\%$ of your stake on A at Bookmaker 3 (odds 2.2) and $52.38\%$ on B at Bookmaker 2 (odds 2.0). The guaranteed risk-free profit is $\approx 4.76\%$ of your total stake.
Intuition
The core idea is identical to covered interest arbitrage or triangular FX arbitrage: when different market makers quote prices that are inconsistent with each other, you can "round-trip" across them and extract risk-free profit. Each bookmaker's odds encode an implied probability for each outcome, but each also bakes in a margin (the "overround"). By cherry-picking the best price on each outcome from different bookmakers, you can sometimes assemble a synthetic portfolio where the implied probabilities sum to less than 1 -- meaning you are paying less than a dollar for a contract that always pays a dollar. The leftover is your arbitrage profit.
In practice, sports betting arbitrage is real but margins are thin (typically 1-5%), execution risk is significant (odds change fast, bookmakers may limit or ban sharp bettors), and you need capital spread across multiple accounts. The mathematical structure, though, is exactly what quant traders exploit in any fragmented market: when the same asset or event is priced differently across venues, the law of one price is violated, and there is money on the table.