Soccer Betting Arbitrage Profit

This betting puzzle is about constructing a risk-free position across all possible match outcomes using quoted odds. It sits at the intersection of optimization, basic algebra, and betting markets, and is a canonical example of sports-betting arbitrage that shows up in quant prep material and finance interviews. The setup mimics how a trader might view odds as prices and outcomes as states of the world.

Working through this question trains arbitrage balancing intuition, translating verbal odds into payoffs, and setting up payoff equality conditions across mutually exclusive outcomes. It also reinforces comfort with simple linear relationships, integer constraints, and checking feasibility of a supposed free-money trade.

This matters for quant interviews because it tests how you reason about riskless profit, pricing consistency, and market inefficiencies. It mirrors real-world market making, where quants continuously search for and size arbitrage opportunities under constraints.

What it tests

This problem class is governed by the principle of constructing a system of equations to enforce identical payoffs across mutually exclusive outcomes, a technique known as 'arbitrage balancing.' The core structure is that for $n$ outcomes, each with its own odds and bet amount, the net profit for each outcome must be set equal, leading to $n-1$ independent equations. The reason this works is that by equating the net profits, you are ensuring that no matter which outcome occurs, the payout minus the total amount wagered is constant—eliminating risk. The feasibility of such a strategy depends on the odds: only if the system yields non-negative, integer solutions for the bet amounts can a riskless profit be locked in. This approach generalizes to any scenario where you can bet on all mutually exclusive outcomes and the odds are such that the sum of the reciprocals of the payout odds is less than one, indicating a potential arbitrage opportunity.