# 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.

#1 saxemephone, 14 Nov 09 16:51

Just to give an extreme example to give people an idea of how variance evens out over time:Example: Playstyle A has EV = 5bb/100hands with a standard deviation of 50bb/100 hands, Playstyle B has EV = 6bb/100hands with a standard deviation of 2000bb/100 hands. Playstyle C has EV = 20bb/100 hands with the SD of B. The variance in style B is WAY higher for only a small increase in winrate. If we compare the EV within 2 standard deviations (95% of the time it will be within 2 SD):

Hands EV range for A EV range for B EV range for C

100 -95bb to 105bb -3094bb to 4006bb -3080 to 4020bb

10,000 -500bb to 1500bb -39.4Kbb to 40.6Kbb -38Kbb to 42Kbb

1,000,000 40Kbb to 60Kbb -140Kbb to 260Kbb 0bb to 400Kbb

100M 4.9Mbb to 5.1Mbb 2Mbb to 10Mbb 16Mb to 24Mbb

10Billion 499Mbb to 501Mbb 560M to 640M 1.96Bbb to 2.04Bbb

(If the standard deviation for something is S and you do that n times adding the results, the total standard deviation is:

SD = sqrt(Variance) = sqrt(n*Variance for one time) = sqrt(n)*S, while the EV is just n*(EV for one time) )

In this extreme example, playstyle B is virtually guaranteed to be more profitable than A after a whopping 10 Billion hands, and has a reasonable chance of being far below 0 after 1M hands. Obviously almost anyone would prefer playstyle A over B here, but if the difference in variance wasn't so big you might expect for your -2SD EV to catch up after 100K-1M hands. Notice that when the winrate becomes significantly higher (playstyle C) the huge increase in variance takes less time to settle down.

#2 PavelM11, 15 Nov 09 02:55

omg! I've just looked into the article and want to say it really makes no sense to me. Just take a look the main formula:profit = EV – a * variance

As far as I understand, profit - is a random variable which cannot be directly expressed in any way (well it's possible to do it with a distribution function, but not the given formula). The EV everybody is talking about is actually the EV of profit per hand or winrate. Variance is the measure of deviation of a random variable (winrate in our case) from it's EV. Variance is actually calculated based on a series of experiments (in our case the actual played hands). The most basic model looks like this:

EV = (w1 + w2 + ... + wn)/n where wi - the actual winning in the i-th hand

SO (standart deviation) is calculated as RMS (root main square) of dwi where dwi = EV - wi

As I said earlier, winnings (profit)is the actual data and variance and EV are calculated based on them, not otherwise. This "a" as far as I understood is the measure of tilt. Tilt affects your winnings and as a result BOTH you EV and variance.

It's pretty late in my country at the moment, so I might have missed something. Gonna go through the article again when as soon as I have some free time.

links to definitions:

http://en.wikipedia.org/wiki/Random_variable

http://en.wikipedia.org/wiki/Expected_value

http://en.wikipedia.org/wiki/Experiment

http://en.wikipedia.org/wiki/Probability_distribution

http://en.wikipedia.org/wiki/Root_mean_square

http://en.wikipedia.org/wiki/Standard_deviation

#3 PavelM11, 15 Nov 09 03:23

Oh! I just forgot to say.EV can also mean the expected value of the winnings in THE PARTICULAR situation.

For example: a player holds AsKs, board is JsTs2c, there is $100 in the pot, player bets 70, opp reraises all-in for 270. His range (to make things simple) is TT+, JT, every J with Q+ kicker.

So in this situation players equity is 51%.

So EV = 51%*440 - 49%*200 = 224,4 - 98 = 126.4 ($)

This EV value is the EV of winnings in the current situation. The total EV should be something like RMS of each hand's EV i guess.

#4 excelgeo, 27 Nov 09 15:43

i need to improve my maths#5 Koshburger, 16 Jan 10 10:54

ok#6 Rap1d007, 29 Mar 10 20:10

Nice comparision of TAG and LAG style...#7 pogodon, 13 Aug 10 17:33

im a tagy fish lag#8 KINGFLX, 21 Sep 10 22:08

is the variance the chances of him having u beat basically at this point? having more outs or hit better on flop? regardless of the bet?#9 fabio4711, 12 Mar 11 13:13

TAG 45% lag 55%#10 lollo1588, 27 Jul 13 11:51

ok#11 biggood, 07 Sep 14 19:00

ok bro thanks#12 toske1, 13 Mar 15 18:37

ok#13 LyToLV, 13 Jun 15 16:51

no need no maths to improve just need to playe and anderstend when bluff when fold when calll all this maths are for them hu wants to make up more money and fast.#14 vinkojudi, 31 Jul 15 11:43

nice#15 vinkojudi, 31 Jul 15 11:43

nice#16 Leonventer112, 17 Oct 15 03:34

This is FAR too complicated for begginers#17 hassux, 25 Jan 16 21:34

badi nik kiss emou#18 bubamarasr, 20 Feb 16 21:04

Read it. Thank you!#19 TetonSouix, 16 May 16 02:45

sweet,,,#20 Nizamakadic, 20 Jun 16 04:36

no need no maths to improve just need to playe and anderstend game..