The explosion of online sports betting in the wake of the Supreme Court’s recent decision is likely to create far more losers than winners. Perhaps one exception to this can be found in the generous signup offers that many sportsbooks have marketed in their efforts to attract new customers. These one-time promotions take several different forms — ranging from risk-free bets and deposit matches on one end, to complimentary credits and bonus wagers on the other — but all tend to improve the expected value of gambles relative to standard betting. In this blog post, we walk through an exercise of how one might exploit these promotions across websites to satisfy different objective functions.
Taken seriously, the strategies we develop are capable of converting the extant complimentary promotions and $2750 of committed capital into more than $1400 of guaranteed profit. In theory, this represents a 50% risk-free return after adjusting for all transaction costs.
Before moving forward, an important disclaimer: as with all my posts, this piece is purely educational and is definitely not investment, financial, or gambling advice of any sort. To adapt an old saying, the best way to make a small fortune in sports betting markets is to start with a large fortune. The purpose of this post is merely to illustrate how one can think about statistical modeling in the institutional setting of sportsbooks.
Suppose a bettor wagers $X$ dollars on a binary outcome event with market-priced probability of success $p$. We can write the expected value (EV) of this bet as:
$$ EV = p X\left(\frac{1}{p}-1\right)+(1-p)(-X) $$
In other words, with probability $p$, the agent earns a payoff of $1/p$ for each dollar invested, less the cost of the wager. With probability $1-p$, the agent loses all $X$ dollars invested. Simplifying terms, we have that $EV=0$, a basic no-arbitrage condition. Put simply, if you were to invest $X$ on two different sides of the same outcome (team A winning, team A losing) you would net 0 profit.
In the real world, the EV is not actually 0 but slightly negative. This is due to the fee the sportsbook collects (the “vig”) that is often on the order of 5%. We can incorporate a vig into our model in several ways; perhaps the easiest is to adjust the odds-implied probability that determines the payoff multiplier. For example, we can write
$$ EV = p X\left(\frac{1}{\tilde{p}}-1\right)+(1-p)(-X) $$
with $\tilde{p} = p+vig$. Simplifying, we have
$$ \begin{align*} EV &= -\left(\frac{vig}{p+vig}\right)X<0\end{align*} $$