On hedging your bets

What is hedging?

Hedging a bet means to remove its risk, by placing other bets that are in some sense “opposite”.

Let’s be more precise. We have a space of possible outcomes \( x_i \) with probability \( p_i \), in arbitrary order. Then for a set of bets \( B \), define the return \( r_i \) as \[ r_i(B) = \sum_{b \in B} b(x_i) \]

I will call a set of bets hedged if all returns are the same \[ r_0(B) = r_i(B) \]

To hedge a set of bets is then to find a different set of bets \( H \) such that \( B \cup H \) is hedged.

Note: This definition is very strict and binary. In practice we might want to maximize some utility function \( U(B) \). One possibility is to minimize potential losses with \( U(B) = \min r_i(B) \). Another is to maximize long term growth of our wealth \( W \) by using log-utility \( U(B) = \sum (W + r_i(B))p_i \), which introduces the problem of figuring out the true probabilities.

Bets on single outcomes

Let’s define a bet on a single outcome $j$ with stake \( S \) and odds \( O \) as \[ B_j(S)(x_i) = \begin{cases} O_j S_j - S_j & \text{ if } i = j \\ -S_j & \text{ if } i \ne j \end{cases} \]

Note that the odds are fixed but we can choose the stake. Given a starting bet \( B_0(S_0) \) we can hedge this by matching the return for each possible outcome: \[ H = \{ B_1(O_0 S_0 / O_1), … , B_{n-1}(O_0 S_0 / O_{n-1}) \} \] resulting in a fixed return of \[ O_0 S_0 - \sum S_i \]

Bets on multiple outcomes

If we consider everything that can be bet on a different outcome, then most bets are actually bets on multiple outcomes. Let’s define a bet on a set of outcomes \( E \) as \[ B_E(S)(x_i) = \begin{cases} O_E S_E - S_E & \text{ if } x_i \in E \\ -S_E & \text{ if } x_i \notin E \end{cases} \]

To hedge a bet on multiple outcomes \( E_0 \), we can partition the set of outcomes \( X \) as \( X = \{ E_0, E_1, …, E_n \} \). This gives basically the same hedge as in the single outcome case.

One thing to note is that it doesn’t necessarily have to be a partition, it’s also fine to cover all outcomes multiple times. For example for a 1X2 bet, we can hedge 1X with 2, but another hedge is 2X and 12.

The cost of hedging

If we normalize the potential returns \( S_0 O_0 \) to 1 by division we get that the final returns after hedging are \[ 1 - \sum 1 / O_i \] Genarally the sum is greater than 1 (overround). If the sum is less than 1, we can make a risk-free profit (arbitrage).

Maximizing our final return is equivalent to minimizing the overround. Is there anything more we can say? Empirically, more popular and competitive betting markets have less overround. So digging around in secondary markets to find better odds usually doesn’t help. Different providers have consistently different overround in different categories. Also different providers can be slightly out of line, so combining them can reduce overround. Generally you don’t want to hedge everything with one provider anyway.