The Mathematical Case for Action Reverse Bets
As sports betting continues its push into the mainstream, an increasing number of market participants are seeking analytical methods for establishing statistical edges. One approach involves quantitative models used to predict the outcome of sporting events. Another strategy exploits arbitrage opportunities created by marketing promotions.
This article, however, will examine at a tactic leveraged to improve expected profitability, simply using the menu of choices offered at a typical online sportsbook.
Specifically, we will discuss the economics of action reverse bets.
This article assumes a basic understanding of sports betting math, all of which was covered in our discussion on Bookmaking Economics.
Consider the two following NFL games and corresponding odds:
Further imagine that we have a high conviction that both favorites will cover their respective spreads, Ravens -7.5 / Giants -3, and wish to risk a total of $100 on those outcomes. There are a number of different ways in which we can wager, each with different risk profiles and levels of variability.
Straight Bets
The most intuitive way to wager on two event outcomes is to place straight bets on each. Assuming that there is a 50% likelihood of any team covering any spread, we can quickly put together a table explaining the Expected Value (EV) of our wagers.
We split the $100 we wish to risk into two $50 wagers on each of the spread bets. We find the expected value for each bet to be identical, producing a long-run loss of $4.55 per $100 wagered.
- Maximum Profit: $90.90
- Expected Loss: $4.55
- Maximum Loss: $100.00
Parlay
A parlay, often referred to as an “accumulator” in other markets, is a wager that consists of multiple predictions, all of which need to be correct in order for the bettor to win.
A 2-leg parlay composed of -110 wagers will pay +265 at DraftKings. Therefore, we can similarly model our EV as follows:
- Maximum Profit: $265.00
- Expected Loss: $8.75
- Maximum Loss: $100.00
As we can see, parlays increase maximum possible profit at the expense of the expected loss.
If Bet
An “If Bet” is a multi-leg wager wherein game results determines whether subsequent legs have action, regardless of the sequence in which the actual events occur.
For example, we could place a bet with the logic:
IF(Ravens -7.5) → THEN(Giants -3.0)
We would be risking $100 on the Ravens -7.5 and then, only if the Ravens cover, will we risk another $100 on the Giants -3. We can visualize the possible outcomes with a tree diagram:
The first bet is placed on the Ravens. If they fail to cover, we lose $100 and the process terminates. If they succeed in covering, we will win our bet and then risk another $100 on the Giants to cover the spread.
Because each team is expected to cover with a probability of 50%, we can quickly assess the EV:
- Maximum Profit: $181.80
- Expected Loss: $6.83
- Maximum Loss: $100.00
We can see that an If Bet serves as something of a middle ground between straight bets and parlays, in terms of maximizing potential upside while preserving expected profitability.
Action Reverse
An action reverse is accomplished by combining two if bets made simultaneously on the same set of outcomes. In our case, we will split our maximum loss of $100 into two If Bets, each with a maximum loss of $50.
One of the bets will be identical to the example shown in the previous section, while the other will be the exact opposite:
- Bet #1: IF(Ravens -7.5) → THEN(Giants -3.0)
- Bet #2: IF(Giants -3.0) → THEN(Ravens -7.5)
With two tree diagrams, we can visualize these bets:
We can quickly put together a matrix that lays out the results in each potential scenario:
We can then aggregate the outcomes to summarize our results:
We can now assemble an EV table as we have for the other bet types:
- Maximum Profit: $181.80
- Expected Loss: $6.83
- Maximum Loss: $100.00
Compared to the single if bet, the action reverse appears to provide a similar return profile, at least from an expected value standpoint*.
Volatility
Expected Value, however, does not tell the full story. Just as is the case with financial markets, the variability of expected returns can be just as important as the expectation itself.
As we discussed in much greater detail in An Intro to Monte Carlo Simulation for Risk Management, we can use a random number generator in order to derive further insight into this economic decision.
We can iterate through 10,000 simulations and look at the profit or loss in each:
This allows us to calculate the Standard Deviation of the expected payouts, and add a final column to our table:
Through this final table, we can see why some bettors are attracted to action reverses. They resemble parlays in that they offer significantly more upside when compared to straight bets at the same level of maximum loss. In comparison with the other multi-leg options, action reverses are expected to be the most profitable (i.e. least unprofitable) and also the least volatile.
While action reverse bets are likely a poor substitute for straight bets, given the materially different risk profile, they demonstrably offer compelling economic benefits when compared to parlays, despite the latter being far more popular.
*A previously published version of this article contained a typo that resulted in an incorrect EV for the If Bet, which has since been corrected.
