A Slave to Variance

Site down.

Andrew — Mon, 22 Jan 2007 06:46:10 +0000

I’m sure that most of you know this, but the pokerstove.com site has been down for the better part of two weeks. I hope to have this resolved by the end of the month, in the time being, the latest version of the software can be found at:

http://208.109.202.54

Minor PokerStove update

Andrew — Sat, 30 Dec 2006 22:34:44 +0000

I’ve uploaded a very minor update to PokerStove. You can see the changes in the changelog. Several of the changes were to the Monte Carlo evaluation, which was sufficiently neglected so as to have a fair amount of bugs. Also, I’ve begun development on version 2.0 of the software. Hopefully, it will be done sooner than later.

The Mathematics of Poker

Andrew — Tue, 07 Nov 2006 01:08:23 +0000

A very good book has just gone to press:

The Mathematics of Poker, by Bill Chen and Jerrod Ankenman.

I received a review copy of it and found it to be an informative and challenging read. Short review: This is one of the most important books about poker ever published. For a longer review you can read Terrence Chan’s great early review of the book.

Few books on gambling take a truly analytical approach. In fact, the only other book in my library which comes close is The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein. Unfortunatly, that book has very little poker poker content. Chen and Ankenment do the world a service by with their thorough analysis of poker. Their methodology is clear, the problems and solutions are interesting, and their insights into the game are remarkable. They’ve also done an excellent job of keeping most of the mathematics accessible to anyone who has a solid grasp of algebra.

Here is an example of the kinds of insights you can get from reading and understanding this book. Below is a topological representation of the solution to the one street heads-up finite pot game with a maximum of two bets.

The strongest hands are at the top of the chart, and the weakest hands are at the bottom. The action maps as: f=fold, k=check, c=call, b=bet, r=raise. Each column reprsents the optimal strategy for the game. The strategy for the first player is in the X column, and the strategy for the second player is in the Y columns. If X bets, then the Y(b) column is used, if X checks, then the Y(k) column is used. It’s important to note that the actual location of the boundaries are not illustrated here, only their relative ordering.

Some basic insights from the solution: You need a stronger hand to check-raise with than you bet-call with. You should fold some of your value bets. Your bluff raises should be done with hands which are stronger (not weaker) than your bluffs.

An interesting post about stud/8

Andrew — Sun, 01 Oct 2006 19:35:01 +0000

There’s a good post about playing high pairs in stud/8 at the 2+2 magazine. It illustrates a few common scenarios that you might face when you are heads up with a big pair versus someone going low. The basic conclusion is a bit muddied in the article, but can be summarized as follows:

Big pairs get crushed when a single low opponent - with no high draws - catches well on fourth (and fifth).

While a big pair still has an equity edge on 4th street if the low catches, that edge is tiny unless the big pair has not caught a second pair. The big problem is that if the low draw doesn’t brick on 5th the big pair is now caught in a reverse implied odds situation where while they have equity to show the hand down they will be giving up $ on every street from 5th-7th. And this is about as good as it gets for big pairs. If the low draw has as much as a gutshot, ace high, or two backdoor draws for high, the big pair has less than 50% equity.

On that note, this week I will begin working on a version of PokerStove which can handle stud and stud/8 so as to better estimate equity evaluation in those games.

Not doing a very good job at becoming/staying overrated

Andrew — Thu, 24 Aug 2006 19:07:04 +0000

At least I have a (only occasionally updated, but not on MySpace) blog.

[kudos to rmd for grabbing this pic during Phil Gordon’s talk at the BARGE banquet.]

The cost of information, or TANSTAFL

Andrew — Mon, 21 Aug 2006 23:58:54 +0000

I’m back from BARGE, so it’s time for me to fill in a few of the blanks.

So, my last post was a bit cryptic, but the first response is the exact response. Under typical conditions, there is usually a cost to acquire information. In poker, betting patterns are one of the primary conduits of information. The more actions in a typical betting pattern, the more information. For example, a check-raise always carries more information than a check. The crux of the information problem is that when the price of the information is too great you should generally eschew paying for it. This is a bit of inversion of the “Why raise?” question posed by others. Often times you’ll hear players saying that they are raising for information, or raising to “find out where they are at.” In general, most players take this tack far too often.

Both Daniel and David brought this up.

This isn’t so much as a result, or some novel theory. It’s more a recasting of action motivation. Aggression is good because it allows you to get money into the pot, especially if you have the best hand. Passivity is good because it allows you to be less confident about the strength of your opponents hand, and thus promote the relative strength of your own hand.

Of course, the difficulty comes in determining when you should switch from betting to calling. In limit poker, this turns out to be fairly easy to solve for most heads up confrontations.

Summer Vacation + something to think about

Andrew — Wed, 16 Aug 2006 16:01:18 +0000

I’ve been doing all sorts of vacation type activities for a while now, and won’t be back to “work” till the end of August.

For those of you who want something interesting to think about, here is a concept which might seem paradoxical.

Poker is a game of information, usually the more information you have the better. But there are a lot of situations where you are given a choice about how much information you’d like, and it is often better to choose less information. What are some of those situations? What does that suggest about in game tactics?

The Fundamental Flaw of Poker

Andrew — Mon, 15 May 2006 18:30:27 +0000

In The Theory of Poker, David Sklansky introduces his “Fundamental Theorem of Poker”:

Every time you play a hand differently from the way you would have played it if you could see all your opponents cards, they gain; and every time you play your hand the same way you would have played it if you could see all thier cards, they lose. Conversely, every time opponents play their hands differently from the way they would have if they could see all your cards, you gain; and every time they play their hands the same way they would have played if they could see all your cards, you lose.The primary problem with this formulation of poker strategy is that it requires you to step outside the game. Specifially, it measures the appropriate response in terms of parameters which are not available during the game. Once you know what hand your opponent holds, it’s not poker anymore. It might be an interesting game, but by removing the element of the unknown, you’ve removed an essential aspect of poker.

This leads to a vivid contradiction between the FTOP and the practice of playing poker. When you are actually faced with a decision in poker, you must take all the information that you have at hand, and make a decision. There are many ways of formulating the correct action, but none of them allow you access to unknown information (without cheating). In the absense of specific player knowledge, the ideal tool to use is game theory.

Game theory is a branch of mathematical analysis developed to study the decision making in conflict situations. One unique feature of game theory is that when solving a problem, the agents in the conflict are assumed to be fully rational. In other words, your poker opponents play as well as you do. There are no fish in the game theory model, only sharks.

Specifically, once a game theory solution has been found, if both players play according to the strategy, they will break even in the long run, assuming there is no rake. Any deviation from that strategy by either player will result in that players expectation dropping to a negative value.

Using this sort of framework, there are certain classes of poker problems which become solvable. For example, heads up no limit hold’em can be solved in an optimal game theoric manner if you assume the players are restricted to either folding, or going all-in as their first option. Aproximate solutions have been published by several people, including the Jones/Kittock system, and Tysen Streib. And a complete solution has been published by Alex Selby.

As an example of how to use the system, let’s consider the specific scenario of what each player should do when they each have 20 chips. The blinds are 1/2, and the small blind looks down to find 98s. In accordance with the optimal solution, he jams. The big blind looks down to find T8o. Facing the all-in jam by the small blind he consults the optimal solution, and finds that he should fold, and does so correctly, despite the fact that he is better than a 2:1 favorite to win the pot.

Is this a violation of the Fundamental Theory of Poker? Yes.

Is this the optimal play for both players? Yes.

The seeming contradiction comes from the fact that the Fundamental Theory of Poker is a theory which is not about poker. Instead it is a theory about the results of poker. In other words, you cannot use the Fundamental Theorem of Poker to solve any actual poker problems. Instead you can only use it to determine that you were, or were not, unlucky to have been involved in a confrontation where your opponent just happened to be holding a specific hand.

What is unethical

Andrew — Sat, 13 May 2006 00:12:46 +0000

You know, it seems like every month someone is exposed for crossing some sort of line in a poker game and getting into trouble. People have had money confiscated from accounts, and been banned from sites. The most distrubing thing isn’t that this is going on though. It’s the fact that a lot of people, even those who aren’t being caught, don’t think it’s a big deal. That’s just the weird world of poker. It lives in it’s own little grey world, filled with grey people, doing grey things all the time.

Because of this, there is often no consensus on where a line should be drawn. For example, it’s been ruled that playing multiple seats in the same tournament is not ok. But, if you and your friends each individually enter a tournament, but play out of the same bankroll, that is ok. The distinction is subtle, and at a practical level is close to meaningless. Assuming all play is “honest” what is the difference between ten people each entering the same tournament ten times, and those people each entering ten different tournaments and sharing the results? This sort of stuff has been going on forever. Men ‘The Master’ has made a fortune out of doing this, yet he and his horses are still allowed to play at any tournament that they like.

So when it comes to the new breed of players getting caught red handed, I can certainly understand how they slipped down that slope. There are just so many examples of poker biggest and brightest — from authors to champions to internet pros — bending, twisting, and cajoling the rules to suit their own personal needs and desires that it almost seems like it’s part of the game. That doesn’t make it all right though.

The problem is that without clear rules and guidelines, players looking for an edge will do everything they can in order to make it as large as possible. This takes all kinds of forms, from coffeehousing to shorting chips, from active collusion to shared bankrolls. What is fair and what is not when it comes to poker? It’s a hard question to answer, and it’s one that a lot of smart people have come to very different conclusions about.

The cost of the rake

Andrew — Thu, 04 May 2006 21:35:01 +0000

When I first started playing poker, I worried a lot about the rake. The available games were expensive, with 10% up to $5 being a common riverboat rake. One of the first things I did when I had enough data was to do a very remedial rake analysis. I took logged hands from play on IRC, and estimated the cost of the rake based on what would have been taken if there had been a rake. I varied the percentage and the cap. [I just love the old school SGI/Motif window decorations.]

The basic results of that chart indicate that a 5% rake is much worse than a 10% rake. As a result, I made it a point to never play in games that had a 10% rake, no matter how low the cap was. And that was that. Only recently, with the proliferation of rake-back, and other bonuses for online poker have I come back to consider the rake. I used my new simulator to construct a table which looked kind of like something you might find online. I made the players bit loose, and a bit passive, but otherwise good. Of course, the sharks had to be better than the fish to get a clear picture of win rates.

I ran the same exact simulation for 30,000 hands, and varied only the limits at which the players played. The first simulation had no rake, and all the others had a rake of 5% to a max of $3. The rake was taken out in $1 increments at every $20. So a pot which was in the range [$40,$60) would have a $2 rake applied to it. A pot which was $60 or larger would have $3 taken for rake. The results were dramatic.

By the end of 30K hands, the expert player was winning at a rate of 4.77 bb/100 in the unraked game. When the expert was playing in a raked 2/4 game, his earn rate dropped down to 1.05 bb/100. The rake is taking over three-quarters of his winnings. In fact, players don’t begin to escape the gravitational pull of the rake until 5/10.

This goes a long way towards explaining why higher limit games are sometimes easier to beat than lower limit games. Consider how much you have to beat the players for to make the same earn rate. At 2/4, you need to beat the players for about 4.72 bb/100 to take home 1 bb/100. At 15/30, you only need to beat the plays for 2.25 bb/100 to take home 1 bb/100 after the rake. Your edge over the field can be cut in half, and you’ll still acrue big bets at the same rate.

Ditching a bad hand, and pounding nut low

Andrew — Thu, 27 Apr 2006 17:15:19 +0000

hold'em[L] $ 40/80 Paradise Poker - Shoal Bay      Wed Apr 26  ante:0.00 blinds: 20.00 40.00  rake:3.00 pot: 1037 [9h 5c Qc 6d 9s]   Dcifrths      1784  1 Scc  kf                     80   -80   Brentie       1017  0 Br   brc  b    b           400   637 [Qh Ac]   SueHFA        1140  5 cc   cR   c    c           400  -400 [Qd Kd]   mrstyle       5532  4 f                            0     0  *Andrew        3557  3 cc   rf                    160  -160 [Ad 9d]   Dopey         2493  2 f                            0     0

It’s not often you get a chance to raise/fold in a multiway pot on the flop. Usually, this play is reserved for pocket pairs, which are brittle in the sense that once you have a certain amount of information they become almost worthless. Here it’s a bit different, as I’ve got middle pair, top kicker. The problem is that due to the nature of the action, my ace outs are very tainted, so while my effective outs may not be as low as two, they are certainly less than three.

Of course, my two outer would have come in on the river, but I’ll take solace in the fact that when I got out I was crushed.

Here’s another cute hand:

hold'em[L] $ 40/80 Paradise Poker - Onslow         Wed Apr 26  ante:0.00 blinds: 20.00 40.00  rake:3.00 pot:  697 [2c Jd 9c 5c Ad]   mrstyle       2287  1 Sf                          20   -20  *Andrew        1637  0 Bc   kr   b    b           320   377 [3d 4d]   mixfreeze     1197  9 f                            0     0   orkids        1578  8 f                            0     0   Rusty-Nail    3108  7 f                            0     0   jfthomas2     3472  6 f                            0     0   Dcifrths      5235  5 f                            0     0   Mr.Lee        1600  4 Pkf                         40   -40   panman36      1719  3 f                            0     0   SueHFA        2772  2 r    bc   c    c           320  -320 [Qd Qh]

I defend with 43s, and flop nut low. What does one do with nut low? Why raise, and bet, and bet, of course. The river suckout is an added bonus, but by no means necessary.

Red Headed Stepchild (PokerTrace)

Andrew — Sun, 16 Apr 2006 17:49:41 +0000

I originally wrote PokerTrace as a first cut at a hand reviewing tool. I put in a lot of time and energy to make it pretty and to provide the ability to click on any action. The idea was that when you clicked on an action you’d launch a tool which brought up a complete snapshot of all the relevant information you might need to make that specific decision. I never did that. In practice, I’ve found that the lack of navigational tools hampered the utility of the application. The original idea was that you’d be able to use PokerTrace to get a line on how your opponents play. As such, all the filters were geared toward
displaying hands which a specific opponent had shown down. The problem here is that to be effective, you need a lot of hands to get that line.

Ideally, I’d have wired PokerTrace to my database, and used them together to provide a large pool of hands so that you could simply and easily filter through all the hands that you have. Unfortunatly, my database is not in any kind of publicly consumable form, and won’t be ready for quite a while. Without access to a complete history of all hands, any reviewing tool is better suited to session level review. As a session review tool PokerTrace is ok, but with only a crude filter for selecting interesting hands, and no actualy hand/session summary data, it’s doesn’t really do the task justice.

So I’m starting on the slow path to creating a solid session reviewing tool. The first step in the process is to revamp and retool PokerTrace, and make it actually mildly useful for reviewing your own session. If you have any requests or ideas about features you’d like to see in such a tool please post them here.

Thanks,

- Andrew