# The Economics of Parimutuel Sports Betting

Many sports bettors are familiar only with the fixed-odds style of wagering, which has seemingly always been ubiquitous in the United States, whether in regulated markets like Nevada or elsewhere. In places outside of the US, there are also markets at various stages of maturity insofar as exchange wagering is concerned. However, aside from fans of horse racing, few sports fans tend to be familiar with a third type of betting market structure known as a ** parimutuel betting**.

It may be helpful to first recall how the economics of traditional bookmakers function, and then discuss where this other form comes in.

# Parimutuel Betting Overview

The most prominent shared feature between exchange wagering and fixed-odds wagering platforms is that bettors know with certainty what the payout will be on any particular proposition before placing the wager. Of course, a line can move between the time when a user decides to place a bet and the time when they actually submit it, just as can happen with the price of a stock or any other asset. However, barring such a market movement, the odds and payouts are locked in. Certainly, once the bet has been accepted, there will be no changes in the payout relative to the stake.

This is fortunate, because if we sought to bet on our team to win the division at +600, only to find out that so many other people placed the same wager, forcing the operator to decrease our payout on a $10 bet from $60 to $20, we would be less than thrilled. But, that is exactly how horse racing and, more broadly, parimutuel betting functions.

# The Math of Parimutuel Horse Racing

Consider any event (e.g. a horse race) with *n *possible single-winner outcomes. There will be total amount wagered…

…on each outcome…

…such that the total amount of money available in the pool is represented by…

Suppose this event is a horse race and imagine a hypothetical distribution of wagers placed on the proposition that a respective horse wins the race, prior to the race starting, as follows:

In other words:

And:

Once the event or race begins, no further wagers are taken. Forget, for a moment, about the track operator’s need to build a profit margin into the business model.

Suppose Horse #2 wins the race. We have a total of $100.00 to be distributed to those who wagered on Horse #2 in the proportion of…

Thus, if you wagered $3.00 on the proposition that Horse #2 wins the race, you would receive 6.67x your money, or $20.00. If the $15.00 risked on Horse #2 was all wagered by one customer, they would receive $15.00*6.67=$100.00. We quickly realize that the amount returned includes the original stake, which makes sense because total winnings can never exceed the net losses aggregated across all other customers (without additional outside influence).

We can generalize to see that, in an *n*-participant parimutuel horse race with no commission reserved for the operator, the amount returned to those who successfully wagered on the winning horse, denoted Horse #X will be proportional to the amount wagered, specifically in the ratio of:

Accordingly, we can establish a payout table for the race with our initial configuration as follows:

We recall that we just noticed how these multiples include the amount staked in the payout multiple. Some may wish to remove this, which can be done easily by adjusting the formula to…

…so that our payout table now looks as follows…

What we actually just did here, perhaps without even realizing, was provide one column (“*Payout Multiple*”) that provides odds in “** Decimal**” notation and one (“

*Net Payout Multiple*”) that provides odds in “

**” notation. Just as Americans insist on using feet and pounds instead of meters and kilograms, we also insist on our own odds notation in the gaming world. Fortunately, it is relatively easy to convert between the notation sets, as we discuss in further detail in**

*Fractional***An Intro to Odds & Notation**.

We can also add in a 3rd column (“*American Payout Multiple*”) that will hold the “** American**”

**odds, or what many reading this will simply call, “odds”.**

*For anyone following along at home, this work was all done in Google Sheets. Specifically, the formula found in cell F3 in the chart above uses the formula:*

=IF(D3>2,

CONCATENATE(“+”,ROUND((D3–1)*100,0)),

CONCATENATE(“-”,ROUND(100/(D3–1),0)))

The operators, of course, want to make a profit from their activity. Specifically, they want to be compensated for the business risk they took along with the costs associated with providing the platform through which liquidity is created, even if in a somewhat synthetic fashion.

Suppose, for any total amount wagered on an event, the operator wishes to take a profit margin of…

In other words, for every $100 wagered, the operator wishes to remove 8% or $8 from the pool, in the form of guaranteed profit, before distributing winnings to customers. Now, we start building our payout table with a slightly different formula. Instead of the payout multiple on the winning Horse #X being calculated as…

…it will be calculated as…

And then we can build out just as last time…

We can do a quick sanity check here. If $10.00 is risked on Horse #1 to pay out +820 in winnings, that would produce $82.00 to be paid in the event that Horse #1 in fact wins. Combined with the $10.00 initially staked by customers who wagered on Horse #1, that is $92.00 returned out of a total $100.00 wagered, for a hold or profit of $8.00 or 8%, just as expected. We can further verify this to be the case for any event outcome.

# Parimutuel Sports Betting

Horse tracks have functioned like this for years, but how can the parimutuel betting style be applied to sports ? The key to understanding is to first realize that the success of a parimutuel horse race is based on viewing the single race, composed of each individual horse and customer betting on that horse, as a discrete event that will support its own liquidity pool from which money risked on horses that do not win is used to pay those who successfully wager on the horse that does win.

When applying the model to sports betting, it is not a specific sporting event that must be looked at as this type of discrete event. Rather, the discrete events that create their own liquidity pools are defined purely by time periods — typically weeks or months. Occasionally, however, an entire tournament like March Madness will represent its own risk pool.

Parimutuel betting is sometimes referred to as “

pool betting”

With our horseracing example, there were no odds or payouts indicated whatsoever. We simply calculated final payouts from the final distribution of wagers placed. However, at any given time leading up to the race, customers would have been able to see the payout multiples that describe the way money would be distributed if no additional wagers were placed. So, while a bet placed at those odds is not locked in or fixed, as it would be with *fixed*-odds operators, we at least can get an indication of who the favorite is and who the longshot is, and my roughly what margin.

This is another portion of the model that will need to be slightly reimagined in order to be adapted to a sports betting use case.

# An Example: March Madness

As this article is, quite devastatingly, being written during the week following the Final Four, had it not been for the cancellation of the 2020 NCAA Men’s Basketball Tournament, we can use March Madness as an illustrative example.

Imagine a group of customers who wish to participate with one another in a parimutuel betting pool surrounding March Madness. Throughout the duration of the tournament, they may wager on any game they wish using odds and spreads provided by the operator of the parimutuel platform. A standard house edge will be applied, with spread bets and totals typically charging -110 and a similar expected profit built into moneyline markets. No money will actually change hands, but the “gains” and “losses” will be tracked and recorded on paper. The operator may choose to set minimums and maximum on amounts wagered, include or exclude parlays and other longshot bets, and any number of other constraints intended to maintain parity across the league. However, this is merely to maximize the experience of the users, as the math will work itself out regardless.

Suppose that over the course of the tournament, each customer in an *n-*customer parimutuel betting pool wagers an amount…

…for…

…and the total amount risked is calculated…

In our case, suppose there is a 6-customer league and the amounts risked over the course of March Madness for each are as follows:

In other words:

And:

Each customer should also have what amounts to an unrealized gain or loss indicated as…

…for…

with the total net gain or loss calculated…

If the structure and mechanics seem similar to our horse racing example, that’s because they are virtually identical. Using randomly generated values, we can imagine that over the course of a tournament where customers 1 through 6 risk the amount indicated in the table, each of them, respectively, win and lose the following amounts:

In other words:

And:

In our horse racing example, we looked as an individual race as the event. Within that event, we took the money from those who were unsuccessful (customers who wagered on non-winning horses) and use that money to reward those who were successful (customers who wagered on winning horses).

In the March Madness implementation, we will look at the entire tournament as the event. Within that event, we will take the money from those who were unsuccessful (customers who finished the tournament with a net loss from their wagers, on paper) and use that money to reward those who were successful (customers who finished the tournament with a net gain from their wagers, on paper).

The difference, which will add a complicating step, is that the horse racing structure guarantees the amount lost by losers is enough to pay winners before any events take place, whereas the sports betting one requires a bit more math.

First, let’s continue with our current hypothetical where the customers, in aggregate, accrued a net loss. In this case, a net loss of $262.00.

As will become more clear momentarily, the lifeblood of any partimutuel betting operation is the critical mass of users or customers needed to add enough “liquidity” to make the operation run smoothly.

Each customer’s contribution to the total risk pool is denoted

Where

We can see that Customer #1, #3, and #5 won a total of $158 ($100, $23, $35) while Customer #2, #4, and #6 lost a total of $420 ($150, $195, $75). In this case, the losses are more than enough to pay out the winnings.

Specifically, they exceed the amount necessary by $262, which is also the net amount lost by the group as a whole,

The final step involved in a parimutuel sports betting league is to now redistribute that excess $262 in proportion to the amount risked by each customer. In other words, since each customer contributed

…as a percentage of the total amount risked, they will receive

…out of the “Adjustment Allocation.” For example, Customer #1 risked $100 out of a total $1,000 risked, or 10% of the total risk pool. Thus,

Since the total Adjustment Allocation was $262,

We can calculate this for all of the remaining customers and make sure that the total amount of the aggregated Adjustment Allocation is exactly the inverse of the net winnings/losses,

Finally, we can add a last column with the net winnings/losses for each customer, after having completed the adjustment.

We can confirm that the math has worked because the total amount won and lost by the group is $0.00, meaning no additional cash was needed from outside sources nor was there any leftover. Of course, a mechanism needs to exist (typically provided or facilitated by the operator of the parimutuel league) for collecting and distributing funds. Among friends, an organizer can use a relatively basic algorithm to suggest the optimal payout flow in order to square everyone with each other.

For example, in our 6-customer March Madness league, we had the following net winning/loss values:

- Customer 1: $126.20
- Customer 2: $(110.70)
- Customer 3: $49.20
- Customer 4: $(129.50)
- Customer 5: $74.30
- Customer 6: $(9.50)

We can actually simplify all of these balances with just 5 payments:

**Payment 1 — Customer #6 pays $9.50 to Customer #3****Payment 2 — Customer #2 pays $39.70 to Customer #3****Payment 3 — Customer #2 pays $71.00 to Customer #5****Payment 4 — Customer #4 pays $3.30 to Customer #5****Payment 5 — Customer #4 pays $126.20 to Customer #1**

In order to remove the need to rely on participants to post funds once the event has ended, an operator can require each participant to buy a minimum number of credits or tokens in a ratio of 1:1. For example, suppose each participant spent $250 on 250 credits to be used for wagering, where a credit is simply a virtual representation of a dollar.

If $250.00 is collected from each of the 6 customers, the operator will have custody of $1,500.00. Therefore, as long as the total to be redistributed at the end of the event is equal to (or less than) this amount, the league will remain self-sustaining. Each user, at the end of the “event” will receive back their initial stake of $250.00, plus or minus their net winnings or losses, after having applied the Adjustment Allocation.

Of course, building the infrastructure and acquiring the users to make this type of league a reality can require expenditures that may warrant a commission being taken, which we can easily build into our model. Suppose, as with the horse racing example, that the operator desires a margin of

The operator could calculate the commission to be charged as a percentage of the total amount risked and then distribute that pro rata according to each customer’s contribution to the risk pool. Along with the allocation adjustment, the commission adjustment is also applied to each customer’s net winnings or losses, in order to determine the amount of funds that need to be returned.

# When Losses Don’t Exceed Gains

In our set up to the sports betting example, we had each customer randomly assigned an amount won or lost which, in that particular case, resulted in a net loss across all participants. It was easy to essentially provide this as a “refund” to all users on the platform, either increasing their winnings or lessening the magnitude of their losses.

And, since all bets are placed at odds that have some degree of house edge built in, we expect that this will generally be the case, particularly as the sample size contained within the event increases in terms of number of customers, number of sporting events, number of wagers, etc.

But, outlier events do occur, especially over smaller samples. As such, we should be prepared for instances in which customers win more than they lose, sometimes by an enormous margin.

First, we can imagine that Customer #4, instead of losing $195.00 on $250.00 wagered, ends up winning $250.00, thereby swinging the total net winnings of the group to a positive $183.00:

In our previous case, the Adjustment Allocation took the aggregated losses and redistributed them among the customer base. In this case, the same thing will happen. Unfortunately, it will decrease the winnings (and increase the losses) in proportion to each customer’s contribution to the risk pool.

Over the long-run, Expected Value is generally enhanced by using a low-commission parimutuel betting pool. However, over certain timeframes, it can be costly. To illustrate the need for parity among customers, imagine Customer #4 actually risked his $250.00 on a 4-team parlay paying 10:1 odds and that the parlay was successful.

Very quickly, we can see that in case like this, there will be a disproportionate amount of money that needs to be clawed back from each customer in order to bring the pool into equilibrium after Customer #4’s outsized victory…

We see that Customer #6, for example, accrued an unrealized net loss of $75.00 from his betting activity on $250.00 wagered. Under the original scenario where the group lost more than it won, this $75.00 loss was adjusted to just a $9.50 loss. Under the scenario where the group won more than it lost, but only by a little, the $75.00 loss was adjusted to a $120.75 loss.

Though this latter scenario would likely not make that customer happy, if they wagered $250.00 over the course of the tournament, it’s reasonable to suppose they would or should be comfortable with a loss of less than 50% of that amount.

However, in the case where another customer hits a large bet with an outsized payout, we see the $75.00 loss is adjusted to $433.25, over 170% of the amount risked. If we add on the 8% commission, this increases to an adjusted loss of $453.25, over 180% of the amount risked.

# Closing Thoughts

The parimutuel betting structure offers a number of potential advantages to exchanges and fixed-odds operations.

## More Favorable Odds (Maybe)

Similar to an exchange, the operator of the partimutuel league or pool has no exposure to the outcome of any particular sporting event, and thus is not exposed to information asymmetries and liquidity challenges in the way that a fixed-odds operator is. This is typically passed on to the user in the form of more favorable odds.

In a parimutuel pool with no fee charged by the operator, we know that the total amount won is exactly equal to the total amount lost, thereby making the net proceeds of all wagers (after adjustments) equal to $0.00. As such, no matter how many customers we have, the average loss will be equal to $0.00. In other words, absent transaction costs, a customer’s long-run expected profit would be $0.00. Compared to negative expected profits for virtually all other forms of wagering, this can be an attractive value proposition. Of course, the operator will want to be compensated and will be economically incentivized to do so at least to the extent that any fees charged bring the long-run expected value in line with the next most competitive substitute product.

Generally, the operator of a pariumutuel pool enjoys similar luxuries as that of an exchange, particularly by way of negating the need for most forms of risk management, a task that amounts to a significant undertaking for many stakeholders.

## Failure To Capture Consumer Surplus

One flaw of the parimutuel format, as we saw, is the way in which a small minority of disproportionately high-volume users can destroy the parity and, in turn, user experience of the greater majority. This can be mitigated by setting limitations around the allowable bet types and amounts, but at the expense of likely foregone revenue. If a platform’s revenue is based on a percentage of total transaction volume, and that volume is artificially limited because of operational constraints, something known as ** consumer surplus** is created. Generally, this is defined as the amount a person is willing to pay for something in excess of the price they actually pay. Economists generally view it as a measure of consumer satisfaction and an indication of future spending habits. However, for businesses, consumer surplus is profit not captured.

The practice of price discrimination is technically defined as the one by which a producer captures all possible consumer surplus by charging each customer the maximum price they are willing to pay for an item, also known as a ** reservation price**. If Yankee Stadium could assess the maximum that each fan is willing to pay for a given seat and somehow charge them that amount, they would attain optimal profitability. However, this historically has been impossible, so leagues and teams spend the off-season strategically setting ticket prices in order to maximize both expected attendance and average ticket price.

More recently, firms like Ticketmaster have deployed “Dynamic Ticket Pricing” where real-time algorithms are measuring the supply and demand for each type of ticket, and making adjustments when appropriate. Though one might hope that this would result in a large volume of ultra-cheap tickets that aren’t supported by strong demand, most anecdotal reports have suggested these adjustments are predominantly shifting up the average ticket price on the primary market in order to capture the consumer surplus, which is currently accruing to secondary market operators and participants like StubHub.

For example, at the time of writing, seats in Section 126 Row 2 for the New York Red Bulls vs. Atlanta United FC game, scheduled to take place on May 16, are selling for the respective prices found below:

Assuming the StubHub listing results in a sale, and excluding the difference in fees charged on the different platforms, there will be $30.00 of consumer surplus that Ticketmaster failed to capture by not selling the original ticket for $130.00 to the customer who ultimately purchases it on StubHub.

Similarly, if our parimutuel betting pool has 100 participants that are all limited to wagering $5,000 over the course of the event period, the operator’s maximum possible profit is a percentage of the 100 * $5,000 = $500,000 wagered. However, if a single customer was interested in placing $10,000,000 worth of wagers, the operator would likely not be able to enjoy a percentage commission on that entire amount, because it would have a negative impact on the experience and retention of other users. The amount that such a customer was willing to wager but was unable to as a result of artificial limits can be thought of as ** consumer surplus** uncaptured by the parimutuel betting structure.

Tangentially, in more efficient markets, arbitrage mechanisms would quickly bring these prices into equilibrium. However, exorbitant service fees, among other factors, typically leave at least some disparity in the ticketing industry.

## Flexibility

Given sufficient liquidity or activity in the pool, the partimutuel structure can enable increased flexibility in terms of betting options for participants. For example, some operators heavily limit the number of moneyline underdog bets that can be combined into a parlay or accumulator, and virtually all of them prohibit “correlated parlays” such as betting on a team to cover in the first half as well as for the entire game.

Within parimutuel pools, however, individuals as well as groups of participants often have flexibility to calibrate what is and is not allowed to their specific needs and risk appetites.

## Complexity

The biggest flaw of the partimutuel structure lies in its own complexity. Even avid gamblers often find it difficult to understand exactly how the underlying mathematics will work themselves out, much less impact their profitability over the long-run. Particularly in the US, where there is a base level of market education that still needs to occur, this will likely prove quite challenging when it comes to user acquisition. However, there are undoubtedly pockets of more sophisticated, passionate, and curious early-adopters to whom such a structure would currently be found appealing. For instance, those involved in the Financial Productization vertical could easily be imagined to have interest in this format.

# Conclusion

Parimutuel betting pools can be operated casually within peer groups as well in a more commercialized, enterprise capacity. There are some incredible benefits available to a number of user bases and stakeholders, but the perceived complexity of the model is likely to serve as a barrier to entry and adoption in the near-term.