The Risk-Reward Concept in poker

Introduction

In this article:

• How the risk-reward concept aids decision-making in poker
  
• Why you use bankroll management
  
• How you will find the most suitable style of poker for yourself
  

In this article you will learn about a concept that many poker players
already use subconsciously: the risk-reward concept. This simple
concept is a very useful tool. It provides us with theoretical
explanations for many basic ideas underlying a successful poker
strategy.

The risk-reward concept (RRC) can help you in the following areas:

• It illustrates the need for bankroll management.
• It highlights advantages and disadvantages of various styles of play.
• It provides you with a thorough explanation of the trade-off between expected value and variance.

That's enough chit-chat, let's move on! The RRC is based on one
basic idea. Poker players look for situations with a positive expected
value and try to avoid variance. The potential gain for a poker player
can be reduced to a simple risk/reward function:

profit = EV – a * variance

The parameter 'a' describes to what extent variance has a detrimental
effect on your game. A larger 'a' value means the harder it'll be to
deal with variance. The size of 'a' depends on many separate factors
like your
psychological robustness and your bankroll management.

Using the formula is simple. Your gains increase
as EV rises, while growing variance reduces your profits. We can test the
RRC using two case studies.

Case No. 1: bankroll management

Say you have a bankroll of $1,000. You have the
opportunity to invest your entire bankroll in an all-in situation with
pocket aces. Your opponent has a stack of equal size and holds
pocket kings. Your chances of winning are about 80%. This is obviously
a
profitable opportunity for investment.

Your EV is calculated the following way: potsize x probability of winning:

In this case: $2,000 x 80% = $1,600

Calculating the variance is more complicated.
Variance measures the spread of the actual result around the expected
value. High variance means individual results will, on average, vary
significantly from your EV.

This is how you proceed with your calculations:

• We know your EV is $1,600.
• You will either win $2000 (case no. 1: your aces win) or you will win nothing (case no. 2): your aces get beaten.
• If you win your deviation from the EV is +$400. If you lose your deviation from the EV is: -$1,600.
• Square
  your results to to make sure you have a positive value. In the first case your
  squared result is: $400 x $400 = 160,000$². In the second case the
  result is: (-$1,600) x (– $1,600) = +2,560,000$².
• To get the
  variance you need to multiply the deviation with the probability with
  which it will occur. The first case will occur 80% of the time and the second 20% of the time.
• This gives you: 80% x 160,000$² = 128,000$², and: 20% x 2,560,000$² = 512,000$².
• The last step is to add both numbers to give you your final variance: variance = 128,000$² + 512,000$² = 640,000$²

The entire calculation in one line looks like this:

80% ($2,000-$1,600)² + 20% ($0-$1,600)² = 640,000$²

How do you interpret your variance? Why is the result such a big number? This is the answer:

• Your variance is not a sum of money. You
  have to think of your variance as a measure with no real unit. The
  greater the number the further, on average, your individual results
  will vary from your EV.
• The scale of the number is a result
  of squaring your deviations. It is possible to calculate a number from
  your variance that you can interpret as a realistic amount of real
  money measured in dollars; it is called standard deviation.
• The standard deviation is the square root of the variance.
• The square root of your variance (640,000$²) is $800.
• The
  standard deviation tells you to what extent the result of your play
  will usually differ from the EV. In this case you can expect a
  deviation of $800.

Now imagine you use your bankroll of $1,000 to invest it into 20
reccurring all-in situations where your aces run against kings. You bet $50 every time.

Your EV does not change: $100 x 80% x 20 spots = $1,600

What happens to the variance? The variance of reccurring situations can be calculated the following way:

• You calculate the EV of each all-in situation ($80).
• You square the deviation from your EV in case no. 1 (you win) and case no. 2 (you lose).
• You multiply the squared deviation with the probability with which it will occur.
• You add the two values, which will give you the variance of each all-in situation, which is 1,600$².
• Because
  you recreate the all-in situation 20 times you have to multiply the
  variance by 20. This gives you the total variance of 32,000$². The
  standard deviation (square root of 32,000$²) is roughly $179.

The entire calculation in one line looks like this:

20 x [80% x ($100$-$80)² + 20% x ($0-$80)²] = 32,000$²

Now let's compare both options:

The result speaks for itself. Repeating the scenario 20 times will
result in a lower variance. Spreading your bankroll over a higher
number of plays will reduce the extent to which your winnings will
vary. This spreading of risk is called diversification.

In accordance with the risk and reward function,
diversification is clearly desirable, because you reduce your variance
without reducing your EV. A sensible bankroll management will increase
your profits.

Hence the RRC provides a theoretical explanation for bankroll management and its advantages.

Case No. 2: different styles of play

Given a suitable sample size, EV and variance are
related. Increasing EV is usually linked with rising variance. For
greater winnings, greater risks need to be taken. This is as true for
poker as it is forthe stock exchange.

Let's compare three stereotypical poker players. We
will look at our favourite player, the fish, as well as tight-aggressive players (TAGs) and loose-aggressive players (LAGs).

The fish usually plays poker for fun and acts in
irrational ways. EV-maximisation is not even taken into consideration by him. Risking the
entire chip stack in ludicrous moves or crazy calldowns are some of the
favourite plays of the fish. That is why in the long run, fish make a
loss and suffer from high variance.

• The risk and reward function shows that a combination of negative EV and high variance is the worst possible strategy.

TAGs are solid winners. They extract maximum value
from strong hands and avoid tricky situations. They avoid busting moves
as much as possible, resulting in positive EV and low variance.

• The combination of positive EV and low variance leads to profit in the long run.

LAGs jump on any spot with a positive EV. This
helps them maximise their EV. However, this style of play often lands the LAG in difficult and marginal situations which increase
variance.

• The combination of very high EV and high variance also leads to profits in the long run.

The following graphic illustrates all three
styles. EV is on the horizontal axis and variance on the vertical axis.
Every style of play is the result of a combination of both factors.

• The fish has a negative EV of -1 and a variance of 3.
• The TAG has a positive EV of 1.5 and a variance of 1.
• The LAG has a positive EV of 2.5 and a variance of 3.

Now you can compare different styles of play
using the risk and reward function and the graph. The function and the
graph clearly show that fish are inferior to LAGs and TAGs for the following reasons:

• TAGs have lower variance and a higher EV.
• LAGs have the same variance but a higher EV.

Hence LAGs as well as TAGs earn more money than
fish. Fish get the worst of both worlds. As a result of the minus EV
play fish do not make profit in the long run which is exacerbated by
the high
variance they achieve.

It has been proven that LAGs and TAGs do better than fish. Can you
find the ultimate style of playing by comparing TAGs with LAGs?

• The TAG style has the advantage of resulting in low variance but achieves a lower EV than the LAG style.
• The LAG style benefits from high EV but suffers from high variance.

Both styles have their unique advantages and
disadvantages. What consequences does this have on their long-term profit? You can find out now:

• Put the values for variance and EV into the
  risk and reward function. Do this for the LAG style and TAG style and
  take the parameter 'a' (= 0.5) as given.
• TAGs have an EV of
  1.5 and a variance of 1. The profit for TAGs is "EV – a * variance".
  Hence TAGs make a profit of 1.5 – 0.5*1 = 1.
• LAGs have an EV of 2.5 and a variance of 3. The profit for LAGs is 2.5 – 3*0.5 = 1.

It is, for now, impossible to determine whether
LAGs or TAGs make more profit because according to our calculations
they both make a profit of 1.

However, we used the simplifying assumption that a
= 0.5 for both styles. A quick reminder: the parameter a determines to
what extent variance has an impact on your profits. The value of 0.5
shows an average relationship of variance and risk.

Assuming one player particularly dislikes variance
as a result of his somewhat delicate mindset then the parameter 'a'
will rise in value. Say
in this case, a = 0.75 so the TAG style is the obvious choice. Using
the TAG style the profits are equal to: 1.5 – 0.75*1 = 0.75. For the
same player the LAG style would yield the following results: 2.5 –
0.75*3 = 0.25.

If on the other hand a player can handle variance
particularly well, then 'a' will decrease. Say in this case a = 0.25
which makes the LAG style the play of choice. The profits using the LAG
style in this case will be: 2.5 – 0.25*3 = 1.75. The profits using the
TAG style will only be: 1.5 – 0.25*1 = 1.25.

Any player with a large parameter 'a' will tend to do better
playing the TAG style. A player with low 'a' can use the LAG style to
maximise profits.

Dan Harrington came up with an interesting theory in his book
„Harrington on Cashgames“. In his opinion LAGs and TAGs enter an
implicit contract. TAGs agree to let the LAGs have higher EV in turn
for receiving lower variance. This increased EV for LAGs is the reward
for agreeing to play with higher variance. You can decide for yourself
which part of the deal you would like to sign up for.

It is impossible to empirically determine the
exact value of the parameter 'a'. You know yourself best. You have to be
honest with yourself about how well you can handle variance. Do you start
tilting after you lose two coin flips? Or do bad beats barely make you
raise an eyebrow? Maybe you are halfway in between the two? If you can
put yourself into one of the three groups, the RRC will help you find your optimal playing style.

Summary

The risk-reward concept is a tool that can be used to analyse the basic results of playing poker: EV and Variance.

Anyone who invests money has to deal with the same
basic conflict. This holds true for stockbrokers, businesses, and poker
players in the same way. Increasing EV is always accompanied by higher
variance over the long run. Your ability to deal with variance
determines how much risk you should take when investing your money.

A TAG style stockbroker will invest in government
bonds and safe shares whilst the stock exchange LAG will trade in
derivatives and high risk shares. You can come up with an infinite
number of examples.

It is important that you understand the
relationship between variance and EV and that you find a way to deal
with it. The RRC is meant to help you with that task. Several classic
debates can be looked at with a new, more analytical angle using the
RRC. These are, for instance, some of the recurring questions:

• Why do you use BRM? How aggressive should your BRM be?
• Which is the better style? TAG or LAG?

The RRC can also be used in other areas. For
example it can be used to analyse individual hands. Especially spots
with high variance like way ahead / way behind or combo draws can be
looked at from a different angle. It is these situations with marginal
+EV that can lead to losses for players who do not deal well with
variance.

The risk-reward concept can be used to analyse
your game in theoretical and real life scenarios. At the end of the
day, EV and your profits are not everything. Variance and how you deal
with it are just as important. You can only find the game most suitable
to your personality if you take all factors into consideration. This is
where the RRC can come in handy.

LINKS
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