GTS Liability Calculations Explanations

Summary

This document explains how the Liability Calculation Engine currently calculates liabilities. In summary, we support the following features :

• Player-level liability

• Market-level liability

• Static Stake Apportionment

• Progressive Multis

• Multiple Number of Winners in a Market

• Dynamic Number of Winners in a Market

• Separate liabilities for PreMatch and InPlay

Stake Apportionment for Multis

Since a multi consists of multiple individual legs, each with individual prices and chances of winning, the assignment of liabilities to individual selections is not as straightforward as dividing the Stake by the number of legs. This is because each leg has different odds of winning so it would not give a true picture of the risk from the traders' perspective if the stake is divided evenly. In fact we want to weight the legs somehow, based on the likelihood of the leg winning. Currently we are using price-based formula to calculated the Apportionment Factor. As the price does not change once the bet has been struck therefore the Apportionment Factor remains constant throughout the lifetime of the bet.

Example - Stake Apportionment (Multis)

Multi Bet

Assuming a Bet Stake of 10 Euro :

Stake Apportionment for System Bets

A System Bet can be thought of as a bundle of individual Multis - there is one Multi for each Combination that makes up the System Bet. The Total Stake for the bet is first divided evenly among the combinations. It is often referred to as unit stake.

• Unit Stake = Total Bet Stake / Number of Combinations

Then for each Combination (i.e. for each individual Multi), its unit stake undergoes the same treatment as Multis above. Once each multi has been independently apportioned, we aggregate the results (we re-combine the system bet) to produce per-leg numbers:

• Stake per Selection = Sum(Combination-level Stakes for that Selection)

Example - Stake Apportionment (System Bet)

First, here is the System Bet and its constituent Combinations, assuming a total stake of 30 Euro

System Bet

Combinations

And here are its combinations, with the apportioned stake calculated. Note:

• there are 3 possible combinations of 2 legs

• the Apportionment Factor (Ak) is calculated for each Combination individually as if it were a Multi.

• thus each combination is given 1/3 of the original 30 Euro stake, giving 10 Euro per combination.

Apportionments

To calculate the final apportionments on a per-leg basis, we simply sum up the individual contributions, grouped by Leg

Takeout Calculation

The takeout for a selection is the amount that must be paid back to the customer if their bet on that selection wins.

• Takeout = Stake * Price

For Multis, note we have previously apportioned the stake between the legs, and so we use the apportioned stake instead

• Takeout = Apportioned Stake * Price

For System Bets, we have multiple combinations of legs generating multiple Multis as sub-bets. In the section above we defined the selection stake as being the sum of all combination-level selection stakes. This is similar when calculating the takeout for legs in a System Bet :

• Takeout = Sum( Apportioned Stake * Price )

Example - Takeout (Multi)

Given the multi in the example above, with a Bet Stake of 10 Euro, the takeout calculations are straightforward:

Example - Takeout (System Bet)

Given the system bet in the example above, and its corresponding apportionments, the takeout calculations are straightforward:

Progressive Takeout Calculation for Multis

The takeout for multis is in fact more complex than the simple example above due to the fact that :

• Multis contain multiple legs

• Their prices are multiplied to give the final price

• The legs are on selections from different markets and thus get settled progressively over time.

• When a multi is first placed, as a whole it is least likely to win, so the corresponding risk assigned to each leg should be relatively low

• As a multi is progressively settled as winning, it becomes more likely to win as a whole, and the effective price increases, so the corresponding risk assigned to each remaining leg should also increase.

For this reason, we implement a progressive takeout calculation throughout the lifetime of a multi in order to reflect the vastly increased risk to the bookmaker assigned to the remaining legs.

We know when a leg is settled, (and if the leg was settled at a reduced price) because the bookmaker provides “PayoutPrice” for any settled legs. Therefore, we should incorporate the PayoutPrice of all settled into the takeout calculation of the remaining active legs.

• Rollup Factor = Product of PayoutPrices from all settled legs

• Takeout (active legs) = Apportioned Stake * Price * Rollup Factor

Note that when a leg is settled as a loser, its PayoutPrice is sent as 0, thus the Rollup Factor for all remaining legs becomes 0 also, and the corresponding takeouts become 0 as well.

Example - Progressive Takeout throughout the lifetime of a Multi

Multi (Initial Placement)

Here we start with the Multi from the examples above having just been placed. The Rollup Factor starts at 1 when all Payout Prices are null. Each leg in the multi acts as though they are independent single bets.

Multi (Partial Winner)

Now what happens if Leg 1 was resulted as a winner? The Rollup Factor is now multiplied by the Payout Price of the settled leg, becoming 1.5. The takeouts on the active legs are multiplied by 1.5

An extra point of note is that we preserve the stake and takeout of settled legs, so that traders can look back on a settled market to see what the liabilities were, and how they relate to the realised profit/loss. These are represented by greyed-out-columns, and you can see the rollup factor is not applied to the takeout on settled rows, so the stakes and takeouts are left in the state they were when the leg was settled.

The figures in bold are have changed from the prior example :

Multi (Partial Winner with a Dead Heat)

Let’s extend the example assuming Leg 2 was resulted. The selection did win, however there was a dead heat. Due to this, the PayoutPrice instead of being 3 was halved by the settlement engine, paying out at 3.25 instead of 6.5. We incorporate this newly settled leg into the rollup factor and this increases the takeout on the remaining active leg.

Multi (Losing)

Instead of the dead heat above, instead assume Leg 2 was resulted as a loser. The rollup factor goes to 0, which zeroes-out the takeout of all remaining legs. Note that since we keep the pre-settlement takeout in the database, the takeout of the losing leg is still reported as 53.63.

Example - Progressive Takeout through the lifetime of System Bets

The takeout is calculated for System Bets in the same way as it is for Multis, the only difference being that we are working on multiple independent multis at once, which are then summed per-leg to provide the final results.

Each example in this section has two tables - the first shows the multis that make up the System Bet, and the second shows the stakes and takeouts summed

System Bet (Initial Placement)

Here is the System Bet, “decombined” into its constituent multis. It is the same System Bet from the apportionment examples above, so we know already the apportioned stakes.

And here we “recombine” the System Bet into legs by summing the stakes and takeouts. The aggregate stake and aggregate takeout are subsequently used as per normal for liability calculations:

System Bet (Partially Winning)

Now, what happens to our System Bet if Leg 1 has been resulted as a winner? The Rollup Factors for all constituent Multis that contain Leg 1 are multiplied by the Payout Price of Selection 1, i.e. 1.5. Therefore those takeouts are increased accordingly.

The figures in bold are the ones that have changed from the prior example. The cells in grey are not updated in the database, thus the takeouts are not recalculated here.

System Bet (Partially Losing)

I won’t go through the dead heat example as it should be fairly obvious based on the dead heat example from multis above how the takeouts in this system bet will work. However, a partially losing system bet behaves slightly differently because although some constituent multis have now lost, there are multis remaining that can still win.

Leg 1 remains a winner, but Leg 2 has now lost. Since leg 1 and leg 2 have now been settled, the aggregate stake and takeouts for those legs remain unchanged. There were previously 2 active multis containing leg 3, now one of those 2 combinations has lost, thus the effect on this system bet is that the takeout for leg 3 is reduced.

Player Level Liability

Player-level liability is calculated for each leg in an incoming bet for assessment purposes. Note the calculation of stakes and takeouts for each bets' legs adheres to the apportionment and progressive takeout principles detailed above.

Market Level Liability

Market-level liability is calculated for each selection in all unresulted markets. Note the calculation of stake and takeouts for each leg of each bet adheres to the apportionment an progressive takeout calculations above.

Example - Market Liability Calculation (Singles)

This example demonstrates the basic market-level liability calculation.

Market

Bets

Liability

Multiple Number of Winner Markets

Some markets have 2 or more winners, and the number of winners is fixed. The classic example is Double Chance, which always has 2 winning selections.

Other markets have a variable number of winners, where the number of winners is unknown beforehand and varies from match to match. For example, Anytime Goalscorer - a 0-0 game has no winning selections, whereas a 4-3 game would have anywhere between 2 and 7 unique goal scorers / winning selections.

Multiple Fixed Number of Winners

In the case of fixed multiple number of winner markets, we divide the Market stake sum by the number of winners before subtracting the takeout:

Dynamic Number of Winners

Here we treat the market as if each selection were its own distinct market where the punter can only bet on the positive result. To demonstrate, an Anytime Goalscorer market containing 22 selections could simply be re-framed as 22 individual markets with 2 selections each - “will player N score a goal” or “won’t player N score a goal”. Thus we calculate liabilities as if each selection were an individual market.

Bet-Related Terminology

Bet - one wager placed on one or more selections in a particular configuration governed by the bet type.

Bet Type - how multiple selections of a bet are combined into one or more combinations

Selection - pertains to Markets / Content - it represents one potential winner in a market.

Leg - pertains to Bets - each leg represents one Selection

Single - a special case of bet that has only one leg

Multi - a bet that has multiple legs, but only one combination. All legs must win.

System Bet - a bet that has multiple Multis, combinatorially generated from the legs of the bet. A System Bet may partially win when some of its multis win but others lose.

Combination - refers to a sub-bet (i.e. multi) within a System Bet.

Unit - another word for a Combination. Often we refer to a ‘unit stake’, meaning the stake per combination in a system bet.

Number of Lines - another phrase for “number of combinations.” Often we refer to the number of lines making up a bet.

Total Stake - the total amount wagered on a bet.

Unit Stake - how much stake is assigned to each constituent Combination within a System Bet.

Liability-Related Terminology

Total Bet Price - the effective “price” or “odds” for a complete bet. In the case of Multis, this means the product of all leg prices. In the case of System Bets, this means the sum of all its constituent combinations' total prices.

Takeout - maximum amount the bet may win, i.e. Stake * Total Bet Price. Note that takeout is not the same as the players' profit on a bet. When a bet is struck, the Stake is taken by the bookmaker. When a bet wins, the takeout is paid to the player. Thus the players' profit on the bet is Takeout - Stake, and not the full takeout.

Liability - The basic definition of liability is similar to the definition of profit / loss, where liability represents the bookmaker’s potential (unrealised) profit. It is defined as unrealised profit, meaning that the liability figures a bookmaker is most concerned about are large negative liabilities.

• liability = stake - takeout

• profit = revenue - cost

There are two types of liability we deal with - player-level liability and selection / market-level liability

Player Level Liability - On a per-selection basis, how much liability is represented by the players' own bets

Market Level Liability - On a per-selection basis, how much liability is represented by all the bets in a market.

Apportionment - Liabilities are calculated and viewed on a per-selection basis. However, bets (esp. multis and system bets) are placed on more than one selection. Apportionment allows us to decide how much stake and therefore takeout and liability to assign to each leg within a bet.