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 Post subject: Building a Champion Level Computer Poker Player, 2007
PostPosted: Sat Oct 20, 2007 11:55 am 
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After some quiet period (along with events like Polaris-Laak matches) here comes the latest UofA master thesis:

Robust Strategies and Counter-Strategies: Building a Champion Level Computer Poker Player

Name of Author: Michael Bradley Johanson
Department of Computing Science
University of Alberta
M.Sc. thesis, October 2007

Abstract

Poker is a challenging game with strong human and computer players. In this thesis, we will explore
four approaches towards creating a computer program capable of challenging these poker experts.
The first approach is to approximate a Nash equilibrium strategy which is robust against any opponent.
The second approach is to find an exploitive counter-strategy to an opponent. We will show
that these counter-strategies are brittle: they can lose to arbitrary other opponents. The third approach
is a compromise of the first two, to find robust counter-strategies. The four approach is to
combine several of these agents into a team, and learn during a game which to use. As proof of the
value of these techniques, we have used the resulting poker programs to win an event in the 2007
AAAI Computer Poker Competition and play competitively against two human poker professionals
in the First Man-Machine Poker Championship.


Full text of paper can be downloaded here:
http://www.cs.ualberta.ca/~games/poker/publications/johanson.msc.pdf

----

Don't expect easy reading. It will take me also some time to read and digest the 110 pages but at first glance it looks really comprehensive and good work. Loved that, btw: "Too much chaos, nothing gets finished. Too much order, nothing gets started. — Hexar’s Corollary"

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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Sat Oct 20, 2007 5:14 pm 
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Actually one of the things I have not noticed so far, reading briefly the paper is that the Polaris bot actually WON against Laak/Eslami in 2007 ...

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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Sat Oct 20, 2007 11:39 pm 
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Table of Contents

1 Introduction 1
1.1 Playing Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Beating humans at their own games . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Texas Hold’em Poker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3.1 Poker and Heads-Up Texas Hold’em . . . . . . . . . . . . . . . . . . . . . 5
1.3.2 Variants of Texas Hold’em Poker . . . . . . . . . . . . . . . . . . . . . . 7
1.3.3 Poker Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3.4 Poker Features . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Contributions of This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.5 Author’s Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Background and RelatedWork 11
2.1 Types of poker strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Evaluating a poker program . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.1 Duplicate games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 DIVAT Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Benchmark programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.1 Best Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Poker Academy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 CPRG Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.4 2006 AAAI Computer Poker Competition Programs . . . . . . . . . . . . 17
2.3.5 2007 Computer Poker Competition . . . . . . . . . . . . . . . . . . . . . 17
2.3.6 First Man-Machine Poker Championship . . . . . . . . . . . . . . . . . . 17
2.4 Extensive Form Games and Definitions . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4.2 Nash Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.4.3 Sequence Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.5 Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.1 Card Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.2 Action Abstraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.5.3 Bucketing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.4 PsOpti Bucketing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.5.5 More Advanced Bucketing . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.6 Related approaches to creating poker agents . . . . . . . . . . . . . . . . . . . . . 28
2.6.1 Simulation Based Systems . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6.2 -Nash Equilibria Strategies . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.6.3 Best Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.6.4 Adaptive Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.7 Teams of programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.7.1 Hyperborean06 and Darse’s Rule . . . . . . . . . . . . . . . . . . . . . . 35
2.7.2 UCB1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3 Playing to Not Lose: Counterfactual Regret Minimization 38
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Formal Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3.1 -Nash Equilibria, Overall Regret, and Average Strategies . . . . . . . . . 40
3.3.2 Counterfactual Regret . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.3 Minimizing Immediate Counterfactual Regret . . . . . . . . . . . . . . . . 42
3.3.4 Counterfactual Regret Minimization Example . . . . . . . . . . . . . . . . 42
3.3.5 Bounds on Regret . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Applying Counterfactual Regret Minimization to Poker . . . . . . . . . . . . . . . 44
3.4.1 General Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.2 Poker Specific Implementation . . . . . . . . . . . . . . . . . . . . . . . . 47
3.4.3 Optimizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.5.1 Convergence to a Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . 50
3.5.2 Comparison to existing programs . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Playing toWin: Frequentist Best Response 54
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.2 Best Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3 Frequentist Best Response . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3.1 Obtaining the training data . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.3.2 Creating the opponent model . . . . . . . . . . . . . . . . . . . . . . . . . 58
4.3.3 Finding a best response to the model . . . . . . . . . . . . . . . . . . . . . 59
4.4 Choosing the Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.4.1 Parameter 1: Collecting Enough Training data . . . . . . . . . . . . . . . . 59
4.4.2 Parameter 2: Choosing An Opponent For opp . . . . . . . . . . . . . . . 60
4.4.3 Parameter 3: Choosing the Default Policy . . . . . . . . . . . . . . . . . . 60
4.4.4 Parameter 4: Choosing the Abstraction . . . . . . . . . . . . . . . . . . . 62
4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.5.1 Comparisons against benchmark programs . . . . . . . . . . . . . . . . . 63
4.5.2 Comparisons against BRPlayer . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

5 Playing toWin, Carefully: Restricted Nash Response 66
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Formal Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4.1 Choosing p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4.2 Comparison to benchmark programs . . . . . . . . . . . . . . . . . . . . . 70
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6 Managing a Team of Players: Experts Approaches 73
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.2 Choosing the team of strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.3 Using DIVAT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

7 Competition Results 77
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2 The 2007 AAAI Computer Poker Competition . . . . . . . . . . . . . . . . . . . . 77
7.2.1 Heads-Up Limit Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . 77
7.2.2 Heads-Up Limit Online . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.2.3 No-Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 The First Man-Machine Poker Competition . . . . . . . . . . . . . . . . . . . . . 79
7.3.1 Session 1: Monday July 23rd, Noon . . . . . . . . . . . . . . . . . . . . . 80
7.3.2 Session 2: Monday July 23rd, 6pm . . . . . . . . . . . . . . . . . . . . . 81
7.3.3 Session 3: Tuesday July 24th, Noon . . . . . . . . . . . . . . . . . . . . . 83
7.3.4 Session 4: Tuesday July 24th, 6pm . . . . . . . . . . . . . . . . . . . . . . 83
7.3.5 Man-Machine Match Conclusions . . . . . . . . . . . . . . . . . . . . . . 84

8 Conclusion 91
8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.1.1 Improved Parallelization . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.1.2 No Limit Texas Hold’em . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
8.1.3 Dynamic Opponent Modeling . . . . . . . . . . . . . . . . . . . . . . . . 93
8.1.4 Imperfect Recall Abstractions . . . . . . . . . . . . . . . . . . . . . . . . 93
8.1.5 Equilibrium Strategies in Perturbed Abstractions . . . . . . . . . . . . . . 93
8.1.6 Improved Abstractions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
8.2 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Bibliography

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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Thu Jan 24, 2008 11:51 am 
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I got a great deal out of this paper:
    There is a lot of maths, but it is possible to understand it without the maths
    Counterfactual Regret Minimisation is a very elegant algorithm for finding Nash Equilibria. Shame it needs 64GB ram
    Polaris is a world class player
    Polaris doesn't use mixed strategies
    Polaris doesn't deal with board texture well
    Polaris doesn't narrow his opponents range of cards
    It's possible to write a bot that performs close to Nash Equilibrium, that can exploit opponent weakness, without being significantly exploited itelf


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Wed Sep 17, 2008 11:15 am 
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I misunderstood part of this paper. Polaris does use mixed strategies. Sorry for any confusion caused.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Thu Sep 18, 2008 11:58 am 
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This is very good paper. If you are inclined to academic reading it is solid foundation to start to build own ideas on top.

One question which is not studied well (independant of this paper) is when to play optimal and when to be exploitive. This question, and largely the theory behind exploitive play as well, becomes fundamental in pokerbot development in real money environments. There is much more to it, than one would forsee apriory.

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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Fri Dec 19, 2008 2:54 pm 
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spears wrote:
I got a great deal out of this paper:
    There is a lot of maths, but it is possible to understand it without the maths
    Counterfactual Regret Minimisation is a very elegant algorithm for finding Nash Equilibria. Shame it needs 64GB ram
    Polaris is a world class player
    Polaris doesn't use mixed strategies
    Polaris doesn't deal with board texture well
    Polaris doesn't narrow his opponents range of cards
    It's possible to write a bot that performs close to Nash Equilibrium, that can exploit opponent weakness, without being significantly exploited itelf

Counterfactual Regrest Minimization is really elegant, and you can do it with much less than 64 GB of ram. Using an 8-bucket abstraction, I was able to run it all within 4.5 GB... What have others tried? And have people played around with metrics other than HS and HS^2?


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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Fri Jan 02, 2009 1:15 pm 
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Pyrho wrote:
Counterfactual Regrest Minimization is really elegant, and you can do it with much less than 64 GB of ram.
Agreed. Since I wrote that summary, I've discovered ways to reduce the memory too.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Tue Oct 06, 2009 12:35 pm 
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The papaer was moved to this link:
http://www.cs.ualberta.ca/~games/poker/ ... on.msc.pdf

Link in OP doesn't work anymore.

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 Post subject: Re: Building a Champion Level Computer Poker Player (M. Johanson
PostPosted: Tue Oct 06, 2009 1:52 pm 
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Pyrho wrote:
Counterfactual Regrest Minimization is really elegant, and you can do it with much less than 64 GB of ram. Using an 8-bucket abstraction, I was able to run it all within 4.5 GB... What have others tried? And have people played around with metrics other than HS and HS^2?


Was this with 4 bets/street and no other abstractions than 8 buckets of (HS, HS^2) per street?
How long did it take? Haven't looked at the algorithm but impressive results.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Tue Apr 13, 2010 11:26 pm 
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I read about half of the paper (to counter factual regression), but I didn't feel comfortable enough with game theory to continue. As someone just starting to create a NL poker bot, is an understanding of this paper necessary to create a winning agent?


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Wed Apr 14, 2010 7:26 am 
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You could read another paper, e.g. about monte carlo tree search. I guess the ideas there are easier to understand.


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 Post subject: Johanson, 2007: History Bucketing
PostPosted: Tue Dec 14, 2010 3:47 am 
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Hi all,
I'm reading "Robust Strategies and Counter-Strategies: Building a Champion Level Computer Poker Player" (Johanson, 2007) and could use a little clarification on the History Bucketing section (pp. 38-39).

I'm not following his reasoning for this statement:
"Since very few low-ranked cards might progress to such a high hand strength, the size of the 1:5 bucket sequence is large to accommodate the top 20% of hands."

I've made a diagram in an attempt to understand what's going on here. I'm ranking the cards from 0 to 10 instead of 0 to 1 to not deal with decimal points in the diagram. The percentage shown on each edge is the chance of going to that node. The numbers in brackets is the range of hand strengths placed in that bucket.

I still don't understand why the bucket sequence 1:5 should warrant such a large range of hand strengths. Let's look at a few examples:

PF: 7c2c (suppose we're in the big blind)
F1: 7h7s2d
F2: 3c 9c Jh
F3: 7h2h9s

Preflop, we are obviously in bucket 1, the weakest bucket.
F1 we hit a full house, we're clearly in bucket 5.
F2 we've got a weak flush draw
F3, weak two pair.

The paper gives an example range of [.4, 1] for the bucket sequence 1:5. F2 and F3 probably both fall within this range, so all three of these hands would be in the same bucket. However, these hands probably warrant different strategies.

I'd greatly appreciate it if someone would help to elucidate this for me.


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 Post subject: Re: Johanson, 2007: History Bucketing
PostPosted: Tue Dec 14, 2010 8:49 am 
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He could have been more clear on this particular section. My read of it is that the 1:5 bucket must be enlarged because hand strength is relatively unlikely to go from very weak to very strong from one street to the next (that's why they're weak hands). Therefore, the bucket size is increased so that the bucket is meaningful. He probably wants roughly equal frequency among buckets (implied by "the 1:5 bucket sequence is large to accommodate the top 20% of hands"). And because these 1:5 hands are rare, the top bucket range has to be increased to balance frequencies among buckets.


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 Post subject: Re: Johanson, 2007: History Bucketing
PostPosted: Wed Dec 15, 2010 1:41 am 
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Ahh, OK. Thanks for the clarification.

So in my diagram above, the percentages should all be 20%. The large range for bucket 1:5 is necessary to implement the 'Percentile Bucketing' technique he discusses above (in the paper).

The percentages given in the diagram would be indicative of the simple bucketing strategy used in PsOpti4.

Thanks again.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Wed May 09, 2012 7:43 pm 
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they mention briefly the idea of bucketing based on the texture of the board. has anyone played around with this?

one way to do it would be to have some rules which defines the texture. for example, first separating the flophands into 3 buckets, whether the flop is all suited, two suited, or rainbow. then separating each bucket into more buckets with ehs2. maybe a bucket for flops where a straight is floppable. (89T, 79J, 78T, etc).

a more universal way do to it would be to give each flop (and possibly turn), a "drawiness"-value. something like:

sum of all preflopholdings ehs-value on the given board
divided by (or maybe subtracted by?)
sum of all preflopholdings ehs2-value on the given board

then we can first bucket the hands based on the boards drawiness into N equal sized buckets, then each bucket into M buckets based on ehs2 for a total of N*M buckets.

does anyone have a better way of identifying the board texture?


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Wed May 09, 2012 9:45 pm 
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somehomelessguy wrote:
Does anyone have a better way of identifying the board texture?


I used to calculate the strength of the flop in combination with the top 20% preflop hands and then used that as a parameter in my opponent modelling. That was reasonably successful and more parsimonious than lots of piecemeal rules about the composition of the flop.

I reckon ehs2 rolls two parameters, strength ("nuttiness") and variance ("drawiness"), into one. So why not try using them separately? Not tried it though.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Fri May 11, 2012 7:12 am 
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spears wrote:
I used to calculate the strength of the flop in combination with the top 20% preflop hands and then used that as a parameter in my opponent modelling. That was reasonably successful and more parsimonious than lots of piecemeal rules about the composition of the flop.

top 20% preflop hands on the given board, or just the same top 20% on every board?

spears wrote:
I reckon ehs2 rolls two parameters, strength ("nuttiness") and variance ("drawiness"), into one. So why not try using them separately? Not tried it though.

not sure what you mean by this.


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Fri May 11, 2012 10:33 am 
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somehomelessguy wrote:
top 20% preflop hands on the given board, or just the same top 20% on every board?
The same top 20% preflop on every board

somehomelessguy wrote:
not sure what you mean by this.

Which part don't you understand?


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 Post subject: Re: Building a Champion Level Computer Poker Player, 2007
PostPosted: Fri May 11, 2012 8:21 pm 
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spears wrote:
Which part don't you understand?

no part.


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