Bill Chen - "The Mathematics Of Poker" Study Group

[rhombus]

To give an example how to calculate dEV, I made a simple spreadsheet to quickly calculate EVs in Roshambo . Suppose we don't know what the optimal strategy pair for this game looks like so we take a guess (rock%, paper%, scissors%):

A: (10, 30, 60)
B: (50, 20, 30)

1) EV of A's strategy against B's strategy is: EV(A) = -$0.11
2) Best response (ME strategy) for A against B is to start playing only Paper (becuse B plays rock "too much"): EV(A') = $0.20
3) Best response for B against A is to start playing only Rock (because A plays Scissors a lot): EV(A'') = -$0.30

dEV = $0.33 (it's the longer distance from those ME strategies to the original one: dEV = max(EV(A') - EV(A); EV(A'') - EV(A))). Next guess:

A: (35, 25, 40)
B: (45, 20, 35)

1) EV(A) = -$0.02
2) EV(A') = $0.10
3) EV(A'') = -$0.15

dEV = $0.13 -> it's closer to dEV = 0 then the previous guess so...we are closer to Equilibrium

not sure how dEV works
you say dEV = max(EV(A') - EV(A); EV(A'') - EV(A))) whats the ; in the equation
also wasnt sure about the second part after the semi colon EV(A'') - EV(A)))

1st one 1st part of equation
dEV = max(EV(A') - EV(A)
0.20 - (-0.11) = $0.31

2nd one
dEV = max(EV(A') - EV(A)
= .15 - (0.02) = $0.17

 

[Fknife]

I used ';' to separate both arguments of the max() function but I guess people are more used to have ',' for that. Sorry about that.

 

[rhombus]

I used ';' to separate both arguments of the max() function but I guess people are more used to have ',' for that. Sorry about that.

so
dEV = max(EV(A') - EV(A); EV(A'') - EV(A)))

is the distance between max(EV(A') - EV(A) and EV(A'') - EV(A)))

whats the difference between A, A' and A''

 

[Fknife]

whats the difference between A, A' and A''

When you have two players A and B, I just marked A's strategy as...A A' is A's maximally exploitive (ME) strategy against B and lets change A'' to B'. B' is B's ME strategy against A.

is the distance between max(EV(A') - EV(A) and EV(A'') - EV(A)))

It's a maximal distance between ME strategies and A.

You "assume" some A (and B) and you calculate its EV(A). Then you calculate A' and B' and their EVs from A's "perspective". So you will get some EV(A') and because I am interested in A's EV against B', I can't write: EV(B') so let's call it: EV(A vs B').

If EV(A') and EV(A vs B') are equal (or very close) to EV(A) that means that neither player can improve their EV by deviating from their previously assumed A and B strategies -> dEV = 0 and we have an equlibrium. On the other hand, you might get that A can improve his EV by switching to A'. Some goes with B switching to B' (in which case A's EV will drop). You calculate those "changes" as distances to the original EV(A), then you pick the larger one and this is called dEV. Clear?

 

[rhombus]

When you have two players A and B, I just marked A's strategy as...A A' is A's maximally exploitive (ME) strategy against B and lets change A'' to B'. B' is B's ME strategy against A.


It's a maximal distance between ME strategies and A.

You "assume" some A (and B) and you calculate its EV(A). Then you calculate A' and B' and their EVs from A's "perspective". So you will get some EV(A') and because I am interested in A's EV against B', I can't write: EV(B') so let's call it: EV(A vs B').

If EV(A') and EV(A vs B') are equal (or very close) to EV(A) that means that neither player can improve their EV by deviating from their previously assumed A and B strategies -> dEV = 0 and we have an equlibrium. On the other hand, you might get that A can improve his EV by switching to A'. Some goes with B switching to B' (in which case A's EV will drop). You calculate those "changes" as distances to the original EV(A), then you pick the larger one and this is called dEV. Clear?

clear as MUD LOL

Must be missing something


in the 2 examples you gave above I still dont get you dEVs to match (see screenshot below of the two examples )

1st one you said dEV = $0.33
I get the equation
dEV = max(EV(A') - EV(A); EV(A'') - EV(A)))

= 0.2 - (-0.11); -0.3 - (-0.11)
= 0.31; -0.19

2nd one you said dEV = $0.13
I get the equation
dEV = max(EV(A') - EV(A); EV(A'') - EV(A)))

= 0.15 - (-0.02); -0.15 - (-0.02)
= 0.17; -0.13


also in the 2nd example you said
1) EV(A) = -$0.02
2) EV(A') = $0.10
3) EV(A'') = -$0.15

Shouldn't EV(A') be 0.15 instead of 0.10

 

[Lofwyr]

Wow, great to see a group working their way through this book! Loved it when I read it a couple years ago...definitely provides a lot of great food for poker thought.

My first post back in these forums parts in quite some time...but I think I'll keep an eye on this thread.

*so this post is not totally superfluous...I think everyone should be prepared to not learn a bunch of direct application concepts from this book. The information presented is on a much more abstract level, so trying to fit each of the games into a particular HE situation can often be an exercise in futility (or at least frustration).

 

[Fknife]

Sorry rhom, I started by job recently and got lots things to catch up on there. Will try to get back to your post this weekend. Dont lose hope!

@Lofwyr
Welcome.

 

[iamjokerface]

Sorry for reviving this thread. I recently got around purchasing this book and made an account here just so that I can ask this one question. And hopefully will be able to contribute some . It's interesting how he uses the maths that I am learning as a second year physics student.
On page 78 (around third of page), in the situation that Y calls X's bet on flop, why does the calculation of <x,bet; Y call | Y misses> use the new pot value of 900? There was initially 100 in the pot and x bet 100 and since Y called, there should be 300? 900 is when they are all- in with their effective stacks of 400. Am I reading the play wrong?
And when he said "At this point, X will bet $300" is he referring to the turn?
I am a bit confused so if someone can lead me through the calculation of case 2: mid stacks, it would be greatly appreciated!

 

[Fknife]

It has been a while since I did any poker related stuff (especially math) so I might be a bit rusty tbh... From what I see you're calculating EV of X's line here - the case when X bets and Y calls, to be more specific. Two essential points to keep in mind here:

1) You can't calculate the EV of a certain action at a certain point without "knowing" (calculating) your decisions at later points.
2) All the hole cards are exposed in the example you are refering to, therefore it's quite easy to fulfil 1).

Going back to your question. X bets, Y calls. Two possible outcomes:

1. Y hits - in which case X won't put any more money into the pot because he knows he can't win (X sees Y's cards!).
2. Y misses - in which case X can shove (he has a pot sized bet behind) and given the odds Y will have to call (because again - they can see each others cards).

So both <X, bet; Y call | Y hits> and <X, bet; Y call | Y hits> account for Turn and River play.

Don't know what's the point of the next 3 equations though (they are basically the same as the 3 previous ones). It might be an error especially that there is a "known" one in one of the equations on the bottom of that page (http://www.conjelco.com/mathofpoker/mathofpoker-errata.pdf) (or maybe I simply forgot about sth).

Hope it helps.

 

[Chazzz801]

Showdown & ex-showdown equity

I'm on page 97, and am struggling with the notion of 'ex-showdown equity'.

It says B's ex-showdown equity is 3x when he folds (where x is A's bluffing frequency). How can equity be anything other than zero when you fold? I expect it has something to do with me not fully understanding what ex-showdown equity is.

Thanks for any help

 

[GLS6474]

page 97

I am also having problems with page 97 - any help appreciated !

 

[Sneaky Feet]

Subscribed! I just got the book so I have to catch up but I've read through the thread and am looking forward to following along.

 

[Chazzz801]

I am also having problems with page 97 - any help appreciated !

Incase you haven't resolved this yet, I have found the best way to think of ex-showdown equity is as follows (assume the bet sizes are 1 unit and the pot is P)

If you value bet and your opponent calls with a worse hand; +1

If you value bet and your opponent calls with a better hand; -1

If you bluff, and your opponent calls (with a better hand); -P

If you bluff, and your opponent folds (with a better hand); +P

If you call a bluff; P+1

Ex-showdown equity is basically what you have gained/lost because of the action you took, and your opponents response. You don't lose the pot when you are not bluffing because you could have a better hand than your opponent. When you read on to part 3 on optimal strategies it all starts to fall in place. Hope that helps.

 

[MMoyle]

I own the book but after seeing the calculus I got a bit of a flashback to when I was attempting to do my degree in Mechanical Engineering and gave up. I know it's a bit late, but if I have questions regarding the book can I still post it now?

 

[Fknife]

I own the book but after seeing the calculus I got a bit of a flashback to when I was attempting to do my degree in Mechanical Engineering and gave up. I know it's a bit late, but if I have questions regarding the book can I still post it now?

Of course

 

[MMoyle]

Thanks you

 

[cwdignus]

a

:joyman:complicated

 

[deeznuts70]

Will I need the book

I'm all in for learning poker strategy. But will I need a copy of the book?