Expected Value (EV) of Poker Staking with makeup

The whole business of staking, in a staker’s perspective, centers around one question: is this staking deal profitable for me? To know this, you have to know, or at least to have an approximation, of the expected value of investing in a horse. This isn’t as straightforward as it may appear at first glance, because in staking you pay all the horse’s losses but only get a percentage of his profits. The staker’s EV (abbreviated as EV(Staker) from now on) will depend on the horse’s EV (abbreviated as EV(Horse) from now on) for that given game, the game’s variance,  the % of the profit you take and the number of games played. Notice that there is a subtle difference between the EV of a staker and the EV of a poker player: the EV of a poker player never depends on the variance of the game, while the EV of the staker depends on the variance of the game!

To visualize how these variables affect the staker’s EV, imagine the extreme example that a horse’s ROI of a given SNG is 10%, but you only take 0.001% of his winnings. How will EV(Staker) be? It’ll be negative obviously, even though the EV(Horse) is positive for the game!

To get the formula for the EV(Staker) requires the use of somewhat advanced probability theory. For a given game, this is the expected value of the staker:

$\displaystyle EV(staker)=\sqrt{\frac{N}{2\pi}}\sigma e^{-(N\mu )^{2}/(2N\sigma^{2})}(f-1)+N\mu(0.5+0.5erf(\frac{-N\mu}{\sigma\sqrt{2}}) +f(0.5-0.5 erf(\frac{-N\mu}{\sigma\sqrt{2}})) )$

Where:

• N – number of games/events.
• σ – Standard deviation of the game (square root of variance)
• μ – Expected value of the horse in 1 game; has to be measured in the same unit as σ
• f – fraction of the profits the staker gets
• erf – Error function. Its value for a given input can be calculated here.

Don’t be scared though! You can calculate it automatically using this spreadsheet, you only need to supply the variables values: Staking EV Spreadsheet

Example 1:

A horse proposes a staking that divides the profits by 60/40 (60% to the staker), of 200 45-man SNGs. The horse has an estimated 5% ROI in 45-man SNGs, which means the expected value is 0.05 buyins. So:

• N = 2; You should always agglomerate the number of games in groups of 100. This has to do with the normality assumptions of the returns, which I explain below
• σ = 45; This is a fixed value, which depends on the type of game.
• μ = 0.05*100=5 (since σ is expressed in buyins, μ has to be expressed in buyins as well!)
• f = 0.6

$\displaystyle EV(staker)=\sqrt{\frac{2}{2\pi}}45e^{-(2*5 )^{2}/(2*2*45^{2})}(0.6-1)+2*5(0.5+0.5erf(\frac{-2*5}{45\sqrt{2}}) +0.6(0.5-0.5 erf(\frac{-2*5}{45\sqrt{2}})) ) = -2.38 buyins$

This means that in the total of 200 games, the expected profit of the staker is -2.38 buyins. That means that the staker’s EV for each SNG is -1.28/200=-0.0119 buyins. Notice how the EV of the horse is positive for the game, but the EV for this staking is negative!

Example 2:

A staking is proposed, in which the horse plays 10k hands of NL 10 SH, and the profits will be divided 70/30. The horse has an estimated winrate of 4bbs/100. The typical standard deviation for this type of game is 80bbs/100. So the variables values will be:

• N = 100; It’s 100 and not 10k, because the winrate is being measured in 100 hands, and not in 1 hand. There are 100 events of 100 hands (100*100 = 10k).
• σ = 80
• μ = 4
• f = 0.7

Using the excel spreadsheet, the value is EV(Staker) =196bbs. That means that in 100 hands, the “winrate” of the staker is 196/100 = 1.96bbs/100. Notice that even though the staker gets 70% of the profits, the EV is close to half of the player’s EV.

Assumptions and limitations of this formula

This formula assumes the horse’s returns we’re dealing with follow a normal distribution. For this to happen, we have to agglomerate the data in sets of 100 hands/tournaments or more, so that the Theorem of the Central Limit is satisfied and we can be sure we’re dealing with a normal distribution. If those conditions aren’t met, the formula will give higher values for the staker’s EV than the real value.

In the next post I’ll explain the correct way to estimate the horse’s EV (μ) by building a confidence interval, and how to estimate standard deviation (σ) for each game.

Appendix

For N = 100, here’s the graph of EV(Staker) for varying μ and σ. It increases in μ, for a fixed σ, and decreases in σ for a fixed μ. The lowest values are in a high variance and low horse’s EV conditions, and the highest values are in a low variance and high horse’s EV conditions, as one would expect.