Ok. Wow.

Couldn't help but notice this thread here.
I read through all that and just feel like I was reading a new language.

What kind of statistics is this?

If I had participated in the thread, I would've spent hours putting out the type of info you guys did and it would've felt like homework--> Was this like 10 minute-material or were you guys spending more than an hr at times to calculate this?

Lastly, OP, what conclusions can be made from this to apply to Poker, and was this all in effort to find mathematics behind some sort of strategy?

I've spent many hours thinking about this stuff.

I've been playing with (0..1) toy games looking for algorithms to try to solve them and I noticed that I was making an unwarranted assumption that there were at most 4 intervals used to decide what action to take...

bluff weakest hands,
check middle hands,
bet good hands,
slow play best hands

So I started examining that assumption and concluded it was probably not true. So I put this out there to see if anyone knew or could check my logic.

As far as applicablility to real poker... I guess one lesson is that you can't polarize your ranges too much too early without being exploitable. Another is that decision of how to bet a particular hand is not just a function of my hand against your range. It's my hand and my range against your range (plus all the other factors of course). If my range is too weak, a hand that is quite good against your range can be in trouble, if you know my range.

But for the most part for me this is just theoretical stuff right now. I wasn't really looking at this to try to improve my poker.

A Slave to Variance >> Blog Archive >> The Mathematics of Poker

PM Jerrod Ankenman

It doesn't make sense that x picks stronger hands to k/f he does to b/f

Ok I've not read the whole discussion, seems like you guys would have saved time agreeing on terminology first hand.

So even if there is just one street, but no cap on the betting, the structure of the optimal strategy for first player (same for second) has an infinite number of regions, which first action alternates an infinite number of times (like [bet and raise n times and then call] should be a set of hands worse than [check and raise n+1 times and call] which in turns is a set of hands worse than [bet and raise n+1 times and then call]. So the alternance between checking and calling is infinite.

This is in MOP bobf. [EDIT: ok, I see you realized this in your post #25]

Now if you put a cap on the betting on each street, as is usually done in real poker, your question whether the bound number of regions depends on the number of streets or not becomes relevant, and the answer seems to be yes with the arguments you give (nuts or air situation for the opponent in at least one line of first-round betting).

So here is another (somehow simpler, but perhaps harder to answer) question:

Limit [0,1]-game (or forced pot limit if you prefer, w/e), cap on the number of bets on each street (at 4 bets for instance), with exactly 2 streets of betting.

What is the (or rather a) shape of co-optimal strategies? Now there are just a finite number of strategic regions, but how are they organized?

Bobf, you need to get and read Mathematics of Poker by Chen and Ankenmann straight away. This is discussed and solved by them starting at p. 189 through p. 215. Yes, there is "striping" (that is, c/r'ing with a set of hands stronger than some of which you lead with a bet), and the number of "stripes" is the same as the number of bets allowed on the street.

The parameterization picture that abanger posted is correct (the lowest "b/f" is bluff-fold) for a two-bet single street. There are two regions where you bet, and two where you check, which are interleaved, hence the striping. These four regions are sub-divided for different actions based on how villain responds.

Furthermore, here is a thread that includes the solution to the two-bet 0,1 pot-limit single street game with checkraise (might be helpful to be familiar with MoP's notation, though):

http://forumserver.twoplustwo.com/47...ns-mop-493645/

So, was bobf's original game really just a single-street game masquerading as a multi-street game all along? Is there any purpose (meaning relevant to poker) to analyzing a multi-street 0,1 game where neither hand has even a chance of changing its relative hand strength between streets?

Oh yes, it certainly is relevant. In multi-street game you can't make sure you see the showdown for one more bet, and that makes the difference (hence my question for 2 street games). I know you play a lot of KCL DeuceBuster, not sure whether you play 2-7 (Limit) Triple Draw, but that case happens when 2 players are pat (same in badugi), if they don't break they are facing a multistreet [0,1]-game basically.

You're right, my TD and Badugi play is de minimis. Thanks though, this helps me understand why this might be an interesting problem. Though, say two players start pat before the first draw (in TD or badugi). I'd think that there are going to be betting lines where one player ought to break (instead of folding) before showdown. But we can deal with that later.

Given that this is a hard problem, why don't we simplify to two streets and just two-bets per street? If we make progress there, we can expand.

Chen and Ankenman describe the organization of strategic regions as "parameterizations", so I'll stick with that terminology here. Have you tried parameterizing the two-street two-bet game? What progress did you make? What were the sticky parts?

On my run this morning, it occured to me that a genetic algorithm could probably quickly find the optimal parameterization for any given 0,1 game. But it would probably do a lousy job of zero'ing in on the precise solution. However, once you have the proper parameterization, it's relatively trivial to find the optimal strategy through other means (ie algebraic solution or linear optimization).

Yeah, I do.

Ok so if each player is allowed 2 bets there will be 4 betting regions interleaved with 4 checking regions defining the very first bet?

Thanks I'll take a look.

Yes there was a purpose for multistreet. The difference is that in an N street game a player can force opponent to have to call N raises to reach showdown but in a 1 street game he only needs to call once.

It does when it goes check, check you want the stronger hand.

I assume you meant "the alternance between checking and betting is infinite". Just to make sure: You are talking about an alternance in the regions defining the very first bet (as opposed to subsequence actions) right?

e.g.
Check best hands
Bet very very ... very good hands
Check very ... very good hands
...
Bet very good hands
Check medium hands
Bluff worst hands

This must be for a different reason than I was arguing. Originally I thought this would only happen in a multistreet game.

Yes, I meant betting. Yes, alternance on first bet. Yes, for a different reason (your reason shows why it's the case in multistreet games, and here it's just because when you're following a line for value you should put in more raises with bigger hands).

Yeah, if his hand is breakable (in badugi it's most often not the case, although in TD you could argue that there ar less non-breakable hands). I'm basically nowhere, and we're back to some of our questions, like when first player checks, do we have to slow-play and check back hands that we intend to play for many bets on next street? If so, how the hands we slow play should rank relatively to the hands we bet-fold, we bet-call, we bet-raise, etc? There are already a huge number of lines available in 2 streets of betting, so instead of having 12 thesholds for each player as is the case for single street 4-bet cap, we have, I don't know, a hundred of thresholds perhaps? Or maybe much less, but reasonings like "I'm going to put in the most bets with the best hands" are much harder to get (at least I'm unable to grasp them).

I have found an interesting result relevant to this thread. Chen & Ankenman posit the following parameterization for the first player in the two-bet pot-limit single-street 0-1 game (reading from best hands to worst hands left to right):

Check-Raise, Bet-Call, Bet-Fold, Check-Call, Check-Raise, Check-Fold, Bet-Fold

If we reduce that to first-bet only (of interest to bobf), this reduces to :

Check, Bet, Check, Bet

which has "two stripes" in bobf's terminology.

I have no easy way to solve games with continuous strategy spaces like 0-1, but I can solve many games with discrete strategy spaces. For 0-1, I use the 0-999 game of integers. So far, it appears that strategies from 0-999 discrete appear to be the same or very similar to those of 0-1 continuous.

For the two-bet single-street 0-999 discrete game, I found the following parameterization to provide a co-optimal solution to C&A's parameterization above:

Bet-Call, Chk-Raise, Bet-Call, Check-Call, Bet-Fold, Chk-Call and Check-Raise mixed, Check-Call, Check-Fold, Bet-Fold, Check-Fold

which reduces to this first-bet parameterization:

Bet, Check, Bet, Check, Bet, Check, Bet, Check

Four stripes! Incidentally, I initially found this in the 0-99 discrete game, which is why I expanded to 0-999.

Seeing this result, I wouldn't be surprised if there are actually an infinite number of equilibria in the two-bet single-street potlimit 0-1 continuous game, consisting of an infinite number of stripes. Obviously, I can't prove, it's just a conjecture.

Also, I'm not sure how to adjust my LP's to find the "simplest" solutions (smallest parameterization), I'm still thinking on that one.

hmmm... When I solve the 1-street discrete games I have never seen a situation where bet-call is played with hands strictly better than check-raise. I don't see that when I solve the (0-99) 2 bet game.

With 99 being the best hand I get approx this

Hands (94-99) ~ 65% KR 35% BC
Hand 93 plays ~ 100% BC
Hand 92 plays ~ 55% KC 45% BC
etc.

I'm using ficticious play and 100 discrete hands is about my limit. Maybe I have a bug.

Even though (94-99) appear to be alternating or striping between KR and BC I'm not sure they really are. I think the KR stripes could just as well be shuffled to the top, above the BC hands. I think the striping is just an artifact of ficticious play. I see the same thing with bluffing (striping when I know that striping is not needed).

I haven't seen it either, that's why I thought it was a pretty interesting result. NB: I have no reason to suspect a bug in my code, but it can never be ruled out.

I don't think your results are necessarily indicative of a bug either. There are certainly a large number of equilibria in the game. Even with the same parameterizations, I remember C&A discussing that a certain set of hands (bluff-raising?) could be co-optimally placed at different places in the parameterization, so that alone will provide a lot of different equilibria.

Incidentally, hands played with a mixed strategy as you have indicated is different from "striping" as I've been using that term.

If it's of interest to you, we could swap strategies for the 0-99 game and test for optimality. I'm using strict pot-limit betting (ie each of the two bets must be a pot bet). So first bettor puts 1 chip in pot (to match the 1 chip of dead money), and 2nd bettor puts in 4 chips (call+pot bet) and 1st bettor has to put in 3 more to call.

How can you be sure that the "Bet" with the top of range in your (0-999) game result can not be mixed with the next "Check Raise" hands?

When you solve, do you have to guess and input the parameterization and then see if it works, or does the solving process come up with that?

Yes, that's what I was trying to get at. The way I'm solving these it's not always clear what is striping and what is a mixed strategy.

I'm using the same betting rules.

Does your program output the EV of the game? Mine gives an upper and lower bounds. That might be good for a rough check.

Here are some EV's for the out of position player, assuming each player anted 0.50.

0-9 game: EV = -0.03500 good to 5 decimal places
0-19 game: EV = -0.03750 good to 5 decimal places
0-99 game: EV = somewhere between -0.03992 to -0.03988

I think it might be easier to check a smaller game. Here is player 1 strategy for the (0-9) game.

0 (0.10000) : 0.50012 Raise Fold
0 (0.10000) : 0.48865 Check Fold
0 (0.10000) : 0.01123 Check Raise
1 (0.10000) : 0.97822 Check Fold
1 (0.10000) : 0.02177 Check Raise
1 (0.10000) : 0.00001 Raise Fold
2 (0.10000) : 0.77046 Check Fold
2 (0.10000) : 0.21435 Check Call
2 (0.10000) : 0.01518 Check Raise
3 (0.10000) : 0.73576 Check Fold
3 (0.10000) : 0.24374 Check Call
3 (0.10000) : 0.02050 Check Raise
4 (0.10000) : 0.67883 Check Fold
4 (0.10000) : 0.28806 Check Call
4 (0.10000) : 0.03311 Check Raise
5 (0.10000) : 0.60088 Check Fold
5 (0.10000) : 0.34454 Check Call
5 (0.10000) : 0.05458 Check Raise
6 (0.10000) : 0.49692 Check Fold
6 (0.10000) : 0.40957 Check Call
6 (0.10000) : 0.09350 Check Raise
7 (0.10000) : 0.99993 Check Call
7 (0.10000) : 0.00006 Raise Fold
8 (0.10000) : 0.50007 Check Call
8 (0.10000) : 0.49992 Raise Fold
8 (0.10000) : 0.00001 Check Raise
9 (0.10000) : 0.50026 Raise Call
9 (0.10000) : 0.49974 Check Raise

I am not making that claim. In fact, it's quite likely that they are mixed in some equilibria. The claim I am making is that there are very likely quite a large number of equilibria to a game even as simple as two-bet single street 0,1. Up until now, I had assumed that the solution presented in MoP was the single, unique solution. I am claiming that that isn't even the unique optimal parameterization, let alone solution.

The solution process does it. In fact, the output is a solution similar to what you have shown for the 0,9 game, except that the solutions I am finding, for whatever reason, are usually pure strategies for a given hand (ie hand 492 play certain way 100% of the time) with the odd mixed strategy thrown in at or near the edge of the "stripes".

You are finding uber-mixed strategies, which while I am sure are correct, make analysis difficult. The stripes that exist in my four (eight? depending how you count) stripe solution, I infer them from looking at the strategy, and see that consecutive hands in a certain range are all played the same way, which form a block, or a "stripe".

Good idea. I thought the same after my previous post. I calculate the game value to the first player to act as 0.460001. However, I don't / didn't subtract 1/2 the deadmoney, which you probably did. In that case, our game values coincide and I think we can conclude we have both found equilibria.

Has anyone caculated the gamevalue of the continuous game? I'm going to go out on a limb and guess that it's probably 0.46 (or -0.04). I'm guessing my extra 0.000001 is a floating-point / rounding issue.

Even in the half street (0-1) pot-bet game there are mutliple solutions. The typical solution is
Player 1: bluff worst 1/9, value-bet best 2/9
Player 2: call best 4/9

But player 2 can actually call with top 2/9 union any middling 2/9 hands.

I don't know if that constitutes a different parameterization. Looks like it does to me.

Fold Call vs Fold Call Fold Call Fold Call Fold Call

Ficticious play tends to create uber-mixed strategies. Whenever it doesn't matter which hands you mix with, it mixes them all instead of doing what seems rational to a person. For example with hands (2-6) its mixing check/fold check/call and check/raise. That's because opponent is raising with hands (0-1) and (7+) so hands (2-6) are all equal against that range whether folding, bluff-catching, or bluff-raising. A person would reserve calling for the better hands "just in case" but mathematically it doesn't matter.

Yes, I just wonder if those stripes are required or just an articfact of the solution technique when other valid solutions might have fewer stripes.

Is that for the 0-99 game or the 0-999 game?

I don't know. I don't have a way to solve continous games, though I have a few ideas. I tried to solve it manually by logic and deduction in the general case (1 street, N bets) and made alot of progress on half of the tree (after initial check), but it's tough. I believe it probably can be solved this way and without a system of equations or anything like that.

Ooops.... nevermind this is wrong. That would let player 1 raise with hands just below 7/9 and gain value.

curious, you say there are two solutions to half street games, what do you mean by this?

assuming you don't mean pure vs. mixed, since you said there are two for [0,1] games.

Unless you mean various solutions dependent on pot size?

ty

I meant more than one (actually infinite) optimal pure strategies for player 2, in the single half street (0,1) game where player 1 can make a pot-sized bet or check and player 2 can fold or call.

The strategy I gave was wrong. And I thought my whole idea was wrong. But now I'm thinking my idea was right.

If your familiar with that game the "normal" optimal strategy for player 2 is to call the top 4/9 of his hands:

FFFFFCCCC

But here is another pure strategy (pictured in 1/9ths) for player 2 that I believe is optimal:
FFFCFFCCC

i.e. call with hands (3/9..4/9) union (6/9..1)

The way I checked for optimality is this: Player 1 will bet any hand that does better betting than checking. A bet beats a check with some hand if and only if...

# worse hands that call + # better hands that fold > # better hands that call

I then examined each interval against player 2's modified strategy and found that player 1 will want to bet the exact same hands as before and will get the same results. i.e. he will bet top 2/9 and bottom 1/9.

Yeah, there are uncountably many solutions (and parameterizations). The easiest way to see this is with the bluff-catchers: instead of calling the best ones, you could call any set of admissible bluff catchers (who indeed beat all bluffs) of the same size.

No, that's the same mistake I made above. If you start folding the top of your bluff-catcher range and calling with lesser hands, that invites opponent to start betting the top of your bluff-catcher range so it's not an equilibrium any more.

Suppose some optimal solution you check the river and opponenet raises two-pair+ or high-cards.

Your bluff catchers are:
top pair
second pair
third pair
fourth pair
fifth pair

Suppose your optimal calling % has you calling
top pair
second pair

Now suppose you substitute
second pair
third pair

Opponent can now gain by betting top pair hands since you are now folding top pair.

I agree there are an infinite number of optimal bluff-catching strategies, but you have structure your bluff-catching range appropriately. Any set of the same size with hands between opponents value-bets and bluffs does not necessarily do the trick.