Rational Bankroll Management – Part 2

We study the principles underlying bankroll management, using statistics and poker modeling techniques
1. Introduction
This is the second part in an article series about bankroll management (BRM) in poker. In Part 1 we defined expected value (EV), variance, and standard deviation.

For a game with n outcomes, each associated with a probability p and a value x, EV, variance and standard deviation are defined as:

EV

Variance

Standard deviation

To illustrate these concepts, we defined two simple dice games with the same EV, but different variance. Then we calculated EV, variance and standard deviation for both games:

Dice Game 1
We wager $1 per throw in the following dice game:

  • Our wager is returned on 1, 2, 3 and 4
  • We lose our wager on 5
  • We win twice our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =0.8056
  • the standard deviation of the game is SD =0.8975

Dice Game 2
We wager $1 per throw in the following dice game:

  • We lose our wager on 1, 2, 3, 4, and 5
  • We win 6 times our wager on 6
  • The expected value of the game is EV =0.1667
  • The variance of the game is V =6.8056
  • the standard deviation of the game is SD =2.6087

We did this to show how the definitions of EV, variance and standard deviation can be used in practice for simple games. What we have done so far shows that we in principle can calculate these properties, based on the definitions and the rules of the game we are studying.

2. EV, variance and standard deviation for poker games
Using the definitions of EV, variance and standard deviation directly for a real poker game is in practice impossible, since we would have to list all outcomes of the games together with their probabilities and values.

These quantities are too complicated to calculate exactly in real poker games. And even if we could, the number of outcomes we would have to list is astronomical. For example, in heads-up limit Hold’em there are an estimated 10^18 game states, and this is one of the simplest games to study theoretically.

Specifying complete strategies for all players, and then calculating probabilities and values for billions of outcomes is impossible in practice. We bypass this problem by extracting estimates of EV and variance/standard deviation from PokerTracker or HoldemManager. These estimates are based on the sample of hands we have played, and they are our best estimates.

Before we move on to real poker, we’ll study a simple poker model to show how the definitions of EV,variance and standard deviation can be used in poker games.

3. EV, variance and standard deviation for the AKQ game over 1/2 street with fixed-limit betting
We’ll now calculate EV, variance, and standard deviation for the “AKQ game over 1/2 street with fixed-limit betting”. This is a simple poker toy game that we have studied extensively in the article series “Modeling Poker”.

3.1 Definition of the game

  • We have two players: Alice (out of position) and Bob (in position)
  • At the beginning of the game both players put an ante of n bb to build a pot of P =2n bb
  • Both players get dealt one random card from the AKQ deck
  • Alice checks “in the dark”
  • Bob can now check and see a showdown, or he bets 1 bb
  • If Bob bets, Alice can fold, or she can call and see a showdown
  • When the betting round is over, and nobody has folded, the highest card wins in a showdown

We found the following solution to the game, under the assumption that both players are playing optimally:

3.2 Solution for the AKQ game over 1/2 street with fixed-limit betting

Alice

  • Always check-calls A
  • Check-calls K (P – 1)/(P + 1) of the time
  • Always check-folds Q

Bob

  • Always bets A for value
  • Always checks behind K
  • Bluffs Q 1/(P + 1) of the time

Bob’s EV for the game is:

EV =(1/6)(P-1)/(P+1)

In this article we’ll only look at the case where the ante is 1 bb and the pot is P =2 bb before the betting starts.

3.3 Calculating EV, variance and standard deviation for the AKQ game
We’ll now give the AKQ game the same treatment we gave the two dice games in Part 1. We list all outcomes, find the corresponding probabilities and values, and then we plug these numbers into the formulas for EV, variance and standard deviation. We calculate all values relative to Bob.

When this work is done, we’ll have exact solutions for a simple poker toy game, and this will illustrate how the statistical properties we have defined work in poker games. Real poker is much more complicated to work with, but the underlying principles are the same.

We start by observing that there are 6 possible scenarios in the game:

  • Scenario 1: Alice has A and Bob has K
  • Scenario 2: Alice has A and Bob has Q
  • Scenario 3: Alice has K and Bob has A
  • Scenario 4: Alice has K and Bob has Q
  • Scenario 5: Alice has Q and Bob has A
  • Scenario 6: Alice has Q and Bob has K

All scenarios are equally probable, and the probability is 1/6 for each of them. What remains is to calculate the value associated with each scenario, relative to Bob. How Alice and Bob play the various scenarios is given in the solution to the game, defined previously.

We write out the EV equations for each scenario and calculate Bob’s win or loss for each of them. Note that we are calculating the EV for the whole game, and not only the betting round:

Scenario 1: Alice has A and Bob has K

EV1 =(1/6){-1}

Bob always checks behind and loses his ante for -1 bb total.

Scenario 2: Alice has A and Bob has Q

EV2
=(1/6){(2/3)(-1) + (1/3)(-2)}
=(1/6){-4/3}

Bob checks Q 2/3 of the time and loses his 1 bb ante. 1/3 of the time he bluffs and loses the ante plus a bet for -2 bb total.

Scenario 3: Alice has K and Bob has A

EV3
=(1/6){(1/3)(+2) + (2/3)(+1)}
=(1/6){4/3}

Bob always bets. Alice calls 1/3 of the time and Bob then wins Alice’s ante plus a bet for +2 bb total. The remaining 2/3 of the time Alice folds, and Bob wins her ante for +1 bb total.

Scenario 4: Alice has K and Bob has Q

EV4
=(1/6){(2/3)(-1) + (1/3){(1/3)(-2) + (2/3)(+1)}}
=(1/6){-2/3 + (1/3){0}}
=(1/6){-2/3}

2/3 of the time Bob checks behind and loses his ante for -1 bb total. 1/3 of the time he bluffs. Alice then calls 1/3 of the time, and Bob loses his ante plus one bet for -2 bb total. The remaining 2/3 of the time Alice folds, and Bob wins her ante for +1 bb total.

Scenario 5: Alice has Q and Bob has A

EV5 =(1/6){+1}

Bob always bets and Alice always folds. Bob wins Alice’s ante for +1 bb total.

Scenario 6: Alice has Q and Bob has K

EV6 =(1/6){+1}

Bob always checks behind K and wins Alice’s ante for +1 bb total.

This gives us the following list of outcomes for the AKQ game:

  • Scenario 1: Alice has A and Bob has K
    • Probability: 1/6
    • Value for Bob: -1 bb
  • Scenario 2: Alice has A and Bob has Q
    • Probability: 1/6
    • Value for Bob: -4/3 bb
  • Scenario 3: Alice has K and Bob has A
    • Probability: 1/6
    • Value for Bob: +4/3 bb
  • Scenario 4: Alice has K and Bob has Q
    • Probability: 1/6
    • Value for Bob: -2/3 bb
  • Scenario 5: Alice has Q and Bob has A
    • Probability: 1/6
    • Value for Bob: +1 bb
  • Scenario 6: Alice has Q and Bob has K
    • Probability: 1/6
    • Value for Bob: +1 bb

Now we plug these numbers into the formulas defining EV, variance and standard deviation:

Calculating EV

EV
=(1/6){-1} + (1/6){-4/3} + (1/6){4/3}
+ (1/6){-2/3} + (1/6){+1} + (1/6){+1}
=(1/6){-1 -4/3 +4/3 -2/3 + 1 + 1}
=(1/6){1/3}
=1/18

Expressed in the unit “per 100 hands” (standard unit for poker win rate) we get EV =100/18 bb/100 =5.56 bb/100.

Then we plug EV into the formula for the variance and calculate this property:

Calculating variance

V
=(1/6){-1 - 1/18}^2 + (1/6){-4/3 - 1/18}^2
+ (1/6){4/3 - 1/18}^2 + (1/6){-2/3 - 1/18}^2
+ (1/6){1 - 1/18}^2 + (1/6){1 - 1/18}^2
=1.1636

For our purpose, variance is only a property we calculate in order to find the standard deviation (the property we usually use in statistical analysis of poker games):

Calculating standard deviation

SD =sqrt(1.1636) =1.0787

Then we express standard deviation in units “bb/100” according to this formula:

SD(100 hands) =sqrt(100)SD(1 hand) =10(1.0787) =10.787

4. Variance simulations for the AKQ game over 1/2 street with fixed-limit betting
We now do the same type of simulation that we did for the two dice games in Part 1. We use an online variance simulator and plug in EV =5.56 bb/100 and standard deviation SD =10.787 bb/100.

First we let 10 players (let “Num. of trials to run” =10) play 1000 hands each (let “Num. of hands” =1000), and plot the profit graph:

The expected profit for Bob is 1000 x 5.56 bb/100 =55.6 bb (the dotted line). All 10 players ended up with a profit, but the spread is large after only 1000 hands. The spread in observed profit is between +16 bb for the unluckiest player to +98 bb for the luckiest one.

Even for a game where Bob is guaranteed a theoretical profit of +5.56 bb/10 (which would be a fantastic guaranteed win rate in any poker game), there is no guarantee that a single session will be profitable even with 1000 hands played per session.

If we increase the number of players to 100, we’ll get a better picture of the expected spread:

The 100 players in this simulation are spread out between about -50 bb and +140 bb for a session of 1000 hands. Losing 50 bb in 1000 hands is a pretty big loss in a fixed-limit game. When we see that a guaranteed +5.56 bb/100 winner can have such a session, we can easily imagine the kind of losses that are possible for a moderate winner (for example, 2 bb/100) in a fixed-limit game. Especially for games with high variance, like 2-7 triple draw, 7-card stud and heads-up limit Hold’em.

Then we increase the number of hands to 10,000 and do another simulation:

The expected profit after 10,000 hands is 10,000 x 5.56 bb/100 =556 bb. The 10 players now spread out between about 280 to 710 bb. All have a solid profit, but the luckiest player has won about 2.5 times more than the unluckiest one, even if they both have the same theoretical win rate. So we can’t really say that we have reached the long run in the AKQ game, even after 10,000 hands.

If we increase the number of players to 100 for this simulation, we get:

The spread now goes from about 300 bb to about 800 bb. Being a guaranteed winner in the game does not guarantee you get to realize all of your expected profit over a few thousand hands.

The simulation for the simple AKQ game draws a somewhat glum picture. A serious poker player who is playing for income would like his profit curve to be smooth and predictable, but this is not possible due to the nature of poker. This becomes clear after an analysis of a simple poker model game, and in practice it’s even worse for real poker.

In real poker we not only have to take worse variance into account, we also suffer from win rates that are moderate compared to the one Bob had in the AKQ game. When variance increases and win rate decreases, our bankroll experiences much bigger swings. To protect ourselves against these swings, we need to operate with a larger bankroll.

5. Summary
In this article we have continued with the definitions of EV, variance and standard deviation and shown how they can be applied to poker games. We used the AKQ game as a model of real poker, and did variance simulations for this game.

We have now most of the concepts and tools we need to estimate bankroll requirements for real poker games. What we lack at this point is a formula that ties together the concepts we have learned so far. What we want is the answer to the following question:

When we know our EV and standard deviation for a poker game, what does this tell us about the bankroll requirement for the game?

In the next article we’ll present the risk-of-ruin formula. This formula lets us calculate the bankroll requirements for a game when we know the EV and standard deviation. Since we can always estimate these properties from PokerTracker or HoldemManager, we now have everything we need to design bankroll management schemes adapted to our own win rate and standard deviation. We’ll also see that there is another subjective component involved, namely our tolerance for risk.

Good luck!
Bugs

– See more at: http://en.donkr.com/Articles/rational-bankroll-management—part-2-806#sthash.gArItYAk.dpuf

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