How Arbitrage Keeps Mutually Exclusive Betting Prices Consistent
Summary
The document asks how prices for mutually exclusive outcomes on a betting exchange stay consistent with probabilities summing to one. For an event with one winner, each outcome contract pays one unit if that competitor wins and zero otherwise. The answer explains that buying every outcome costs less than the guaranteed total payout when their prices sum below one. If the sum exceeds one, selling every outcome can create the corresponding arbitrage, subject to the exchange allowing those trades and settlement terms matching the assumed payoffs.
The proposed mechanism is market activity rather than an exchange rule that mechanically adjusts related prices. Traders monitor outcome prices and place orders when they identify a profitable discrepancy; this activity tends to remove the opportunity. The explanation is a simplified model of a market with mutually exclusive, collectively exhaustive outcomes. It does not discuss fees, liquidity constraints, order delays, unmatched exposure, or how prices should be interpreted when they include bookmaker margins or other market frictions.
Key ideas
- Contracts on mutually exclusive outcomes should collectively pay one unit when exactly one outcome wins.
- When the sum of outcome prices falls below the guaranteed payout, buying all outcome contracts creates an arbitrage before costs.
- When the sum exceeds the payout, selling all outcomes may create the reverse arbitrage under suitable market rules.
- The answer attributes price consistency to traders acting on arbitrage opportunities rather than automatic exchange adjustments.
- Fees, liquidity, execution risk, and settlement details can affect whether the simplified arbitrage is available.
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Full text
# How is the "probabilities sum to $1$" rule enforced in betting exchanges? # How is the "probabilities sum to $1$" rule enforced in betting exchanges? Suppose that I am interested in a market on a betting exchange for the outright winner of some event, with three competitors, $A, B$ and $C$ with corresponding probabilities of winning $a, b$ and $c$. We expect that the sum $a + b + c = 1$. Now suppose that the probability of $A$ winning changes to some $a'$ (as defined by that bet's current price). We expect $b$ and $c$ to change to $b'$ and $c'$ such that $a' + b' + c' = 1$. By what mechanism do $b$ and $c$ change? I see two options: - They move by market forces, since if at any instant $a + b + c < 1$ people will take this arbitrage opportunity which will push $b$ and $c$ up. - The probabilities sum to one rule is enforced 'externally' by the exchange through some mechanism. I am interested in how such a mechanism could work. I'm aware that there are probably different ways of doing this, but I'm interested in any ideas people can come up with! ## Answer by amdopt (score 3) https://quant.stackexchange.com/a/33995 > By what mechanism do b and c change? I see two options: They move by market forces, since if at any instant a+b+c<1 people will take this arbitrage opportunity which will push b and c up. The probabilities sum to one rule is enforced 'externally' by the exchange >through some mechanism. I am interested in how such a mechanism could work. If participants A, B and C are all competitors in the same contest and there can only by 1 winner then the sum of all 3 contracts should = 1 at any point in time. When the contest ends, the winning contract is equal to 1, the others zero. While betting is open, if the odds of one competitor is increasing, then the odds of the other competitors must be decreasing because the sum of all three binary events must equal 1. If the sum is greater or less than 1, an arbitrage exists. If it is less than 1, you could buy all 3 contracts knowing that the winning contract will be equal to 1 when the contest is over. The opposite would be true if the sum is greater than 1--you could sell all three knowing that you will only owe 1 when the contest closes. Either way, you win. Exchanges don't need to enforce this with a set of rules. Market participants watch for arbitrages continuously and if/when one does exist, orders are entered as quickly and with as much size as possible until the arbitrage no longer exists. For this reason, they rarely exist.
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