When comparing a [0,1] game solution to the solution of a similar, modified version of One-Card Poker there are some differences that I am not sure how to explain.

The exact game rules are not that important but I used the following:
- 2 players
- 1 full street but raises are not allowed
- Players are dealt one card A-2
- At showdown the highest card wins
- Players can have the same card and chop at showdown
- One player's card does not affect the probability of the other player being dealt a card of the same value
- The pot and bets are 100 units

If we look at the first player's betting range, in the [0,1] game, I think that he vbets 16.67% (the top 16.67% of hands) and bluffs 8.33% (the bottom 8.33% of hands). In my One-Card Poker solution the player vbets 19.22% and bluffs 9.61%. I realize that the two games are not exactly the same so I am ok with this difference. The problem is that the first player's betting strategy looks like this:

A (85.55%), K (50.99%), Q (42.54%), J (70.51%), T (0.31%)
5 (0.18%), 4 (0.04%) 3 (91.92%), 2 (32.78%)

The strategy does not bet the strongest 19.22% of cards (A 100%, K 100%, Q 49.92%). How would you explain this? Is the [0,1] solution only GTO if the players are not allowed to use a mixed strategy? Are both strategies GTO and if so how can they have the same EV against their respective best response strategies (when one is not betting all aces and kings and betting some jacks and tens)?

In a full street game the first player to bet won't be able to bet his strongest range because he will need to slow play some of his value, otherwise his range will be too weak when he checks and the opponent can value bet him a lot.

That's the logical reason for not value betting the top range.
Is it possible that you are comparing your game with a [0,1] game played only on half a street?

Also, although your bluff range may still be right for a Nash equilibrium, notice how you will usually bluff the bottom of your range in such a situation (thus you'd prefer 2 over 3 etc...). As I said, that doesn't mean your range is wrong, but the strategy should be dominated.

I get approx the same results. Similar totals for Ace through Jack betting but with a different mix. Ace 80.0%, King 68.6%, Queen 56.3%, Jack 44.0%. And similar bluffing frequency.

Ace through Jacks are all in against each other no matter what. If we bet A-J he calls with A-J. If we check A-J he bets A-J and we call. If we check A-J he bets with Ten 33.5% and some bluffs... I haven't checked but apparently get as much value from checking and calling against those hands as we get from betting A-J and he calls with a bluff-catcher. So we are indifferent to betting or checking with A-J.

I was trying to solve [0,1] for multiple streets multiple raises one time. I couldn't solve it but one thing I seemed to (maybe be) seeing was that the opening better was totally indifferent to betting or checking every value hand. Maybe this is always true when there are no draws in the game.

Thanks for the responses.

It is possible that the bet region should not be at the top of the range but in all examples that I have seen for this situation (full street) the regions are: bet, check/call, check/fold, bet. How would you setup the action regions? It sounds like you are suggesting (going from strongest to weakest): check/call, bet, check/call, check/fold, bet.

The bolded sentence does not make sense to me if "wrong" means "not GTO". If the strategy is dominated how can it be GTO? Maybe I misunderstood the definitions.

In this situation I don't think that you prefer to bluff with 2 over 3. I haven't tested this yet but I think that bluffing 3 (91.92%), 2 (32.78%) and 3 (32.78%), 2 (91.92%) are both GTO.

Thanks for checking. I can also get a different mix. I think that there are infinite GTO strategies here.

I get the same in my solution and it makes sense why check/calling is the same as betting for those hand matchups.

I checked it and it seems to be only partially true. We are not indifferent to betting or checking with A-J. If we change the strategy to not bet A-J (and also not bluff since it is not betting any hands for value) it is no longer GTO. From what I have seen we are indifferent to betting or checking with A-J as long as we still bet 19.22% of hands for value by betting some mix of A-J. I have also checked that betting only 19.22% of strongest hands is still GTO.

When we bet and the opponent is considering calling with a T or lower he doesn't care what mix of A-J we are betting since they all beat him. Similarly, when we check and the opponent is considering betting with a T or lower he doesn't care what mix of A-J we are checking since they all call and they all beat him. I am still not exactly sure how the overall betting frequency changes things.

This shouldn't be the case. I have looked at some multiple street games but not in the [0,1] form. There are often infinite GTO strategies so there are some values that can be changed but I think that it would be pretty rare for a strategy that checks all hands, a strategy that bets all hands and all strategies in between to all be GTO.

IIRC there are multiple co-optimal strategies with different structures in the full street [0,1] game with one bet left behind. The BB leads with a polar range and the value part of that can come from his very strongest hands but doesn't have to.

It's possible for a GTO strategy to be dominated, not strictly dominated, but still dominated.
Take for example a river half street problem where you have the nuts 100% of the times and your opponent never has it.
The strategic pair "you check/opponent fold to any bet" is GTO as no one will benefit from a change in strategy. But your strategy is dominated by the strategy to always bet, as you will never do worse.

In a similar fashion, when you bluff catch you don't always need to bluff catch with your best bluff catchers, it may be enough to bluff catch with a mix of good hands, but it's dominated by bluff catching with your best ones.

Yes

When I said "indifferent" i meant from an EV standpoint, not from a GTO standpoint. In other words, given that opponent is playing GTO we get the same EV by checking or betting A-J. I didn't mean we would still be playing GTO if we checked A-J.

Again, I didn't mean it would still be GTO. I meant that second player GTO made the first player GTO indifferent from an EV standpoint between betting / checking all value hands... or at least it seemed to be coming out that way.

I think you are confusing Nash Equilibrium from GTO. Its not a huge difference, (and this may just be my interpritation of the terminology) but there may be many Nash Equilibria for the game, but the "best" one would be the GTO solution. And by "best" I mean it would perform the best against non-GTO opponents. (Still a hazy defenition i know) My point is that strategy A (bluffing 3[91.92%] and 2 [32.78%] for example) may perform equally to strategy B (3[32.78%] and 2 [91.92%]) while in a state of Nash equilibrium, and maybe even against most possible opponents, but as soon as it encouneters an opponent who will check back with the 3 too often, Strategy B dominates.

In a full-street [0,1] game with 2 bet cap it makes no theoretical difference which part of our folding region our c/r bluff region is taken from. In this sense there are infinite solutions to the game. However, the GTO solution has us taking those bluffs from the top of our folding range; simply because if opponent decides, for some wierd reason, to call with a weak hand we can still win the pot.

For this reason, I think the best solution to this game would be to bet the best hands 100% and bluff the worst 100% and only have mixed strategies at the thresholds. Thats just my intuition though, I haven't tried to work anything out. But really, how does it make sense to check with the nut low 67.22% of the time?

Thanks, I had no idea that this was the case.

Thanks for the explanation, that makes sense. The strategy is weakly dominated. I've even seen the example that you gave in my own results but somehow forgot about it. In that sense the [0,1] strategy is better because it gives the opponent more opportunity to make a mistake.

I understand now. You were talking about playing against a GTO strategy as a non-GTO opponent. It never occurred to me to check something like this. It's pretty interesting.

I've now tested it in a more complicated two-street game with raises and we do have the same overall EV when checking all hands. My game didn't have any straight or flush draws in it but there were draws in the sense that hand strengths were not static. I'll also have a look at what happens as we start to bet more than we would when playing a GTO strategy.

You are right, I was under the impression that NE and GTO were the same. I normally only use NE but haven't seen much mention of it here and have seen some people use GTO as if it's the same as NE so I decided to go with GTO. I haven't been able to find a clear definition of GTO so I can't check if what you are saying is generally accepted. If I am thinking about this correctly, the GTO strategy would weakly dominate the other NE strategies. It seem like it would be a good idea to have a name for that strategy.

I am not sure if its possible to have multiple competing weakly dominating strategies so that it is not clear which one is "best" and therefore GTO. For example, two strategies that weakly dominate some NE strategies but neither dominates the other. They could both have the same EV depending on which mistakes the opponent makes. Even if one of them has a higher maximum EV I don't know if you can say that it's best without linking it how likely the opponent is to make the mistakes required to reach the maximum EV.

I only had a quick look at Pareto Optimality so I'm not sure it has something to do with GTO.

Yes

If I remember right:
- Betting middle hands that should be checked loses value
- Checking with hands that should be bluffed loses value for the player who is in position because these bluffing hands have +EV equal to the value of checking the best bluffing hand.
- Checking with hands that should be bluffed for the player who is out of position is 0 EV as is bluffing with them so that does not lose value.

I think NE and GTO should be synonyms in this section, your interpretation is interesting but it's not standard.
I also doubt that a strategy which we could define GTO according to your definition has to exist.

The one to rule them all? Isn't it called, well, GTO?

I'm also having a hard time finding a standardized definition for GTO... links appreciated.

They talk about it in MoP a little bit but I can't site exactly where at the moment. I agree that it's a gray area as to exactly which strategy performs the best against non-GTO opponents since one may work best against a certain player but another might work best against another. Also, technically Nash Equilibrium is a game state and refers to a strategy pair, not just a single player's strategy. So technically a player cannot 'play' NE but can play a GTO strategy. It's only a NE when both players are playing optimally and neither can improve their expectation by changing strategy.

http://forumserver.twoplustwo.com/15...ology-1180391/

Yeah, NE needs a strategy for each player. I think it would still be fine to say a player is playing GTO when he's playing a strategy from a NE.

On a side note, I think it may be possible for a dominated strategy to be better than the strategy it's dominated by if the dominated strategy induces mistakes another strategy wouldn't.
For example by stacking off as lightly as possible a dominated strategy may induce a player to value bet when it wouldn't be optimal.

Thanks for the link, although I feel the "standard" is still weakly defined. I would have gone (like Paul Valente) with "the one to rule them all".

I tested points 1 and 3 in the 2-street game and they also seem correct. The reason why I think that this is important is because it simplifies the strategy that one would need to find in order to break even against a NE opponent.

Under the definition that Paul Valente provided, it is. Under the definition where GTO is the same as NE, it is not.

Would you still agree with this definition if it meant that in some cases GTO is either opponent dependent or it does not exist?

I don't see how GTO, or a strategy from NE would be opponent dependent. I thought the general purpose for it's existence was to be non-opponent dependent, contrary to a maximally exploiting strategy.


Please don't say such a thing, even in jest.

[sound] tower crumbling to stones, beasts wailing[/sound]

NE does always exist and so does GTO if GTO = NE.
But if you define GTO as "the best of all strategies you could employ in a NE" that doesn't exist.
That's what we are trying to say: the only optimality which is warranted to be achieved is NE.

Not following... if there are several possible equilibrium strategies (like "which percentage of 2's and 3's to bluff" from the 1CP above), they each perform identically when opponent is also playing his equilibrium strategy (or any of those). One of those performs best against an opponent who doesn't play any of his equilibrium strategies, but say, bluffs too much or too little and calls a bet with Q+ only or alternatively calls a bet with 4+. IP on the river, against villain who plays the italicized strategy, bluffing with a 3 makes little sense, so using all the deuces for bluffs is better.

Or did you mean "doesn't always exist"?
Do you mean something like the bet size in the following?

HU on the river, hero is last to act. Hero has only nuts (and possibly, a tiny fraction of bluffs in his range, never enough to give villain proper odds to bluff catch). It's NE if hero checks all of his range, as against villain playing his NE strategy, hero can't gain anything by changing his strategy (to bet). Against a villain who's off equilibrium, a bet is clearly better. It's still NE of (hero bets all of his range / villain folds all of his range).
So that solves it for limit poker, bet. But if it's PL, how much should be bet?

I mean, if strategy A1 and A2 are both unexploitable ones, but A1 performs best against B1 and A2 performs best against B2, which one is optimal?
None obviously, only thing we know is they are both unexploitable.
I'm not sure if sometimes an optimal "rule them all" strategy will exist, but for sure it won't always exist.

If you can bet any amount, there are tons of unexploitable strategies that will always bet. Deciding which of them is the best is unachievable.

Well, I'm saying that instead of bluffing 3 (91.92%), 2 (32.78%) (which is an NE strategy according to OP) or any other strategy where the percentages sum up to 124.7 , bluffing 3(24.7%), 2(100%) does rule them all.

If you agree with that, can you come up with an A1/A2 (those are NE strategies) pair and B1/B2 (those are not) pair where bet sizes are not involved, i.e. it's fixed limit (say 1 PSB on the river, or bet 1 to P, which ever)? EDIT: meaning a case where a "rule them all" doesn't exist.


Your A1/A2 B1/B2 example fits the bet size problem imo, i.e. we've already decided to bet, A1/A2 are different bet sizes. B1/B2 are the calling percentages a non-NE villain has against the different bet sizes. So to solve, we'd need to know (or estimate) the probabilities of how a non-NE villain picks his strategies? I suppose that does go beyond game theory.

The fact there exists a situation with a single dominating NE, doesn't say anything theoretically speaking. In fact NE are important because they always exist, not because they do exist sometimes.
For example there are spots where a single NE exists, thus obviously it's also the best one.
So, if you found a spot were a best NE qualifies, it's cool, but defining GTO as that kind of NE is not good because it can't always exist.

As for your example, pick an opponent who will always call unless he's holding a 2. The NE bluffing with 3 most of the times is obviously better against this kind of opponent (2 will win a pot when opponent is holding a 2, 3 will win pot when opponent is holding a 2 and split it when he's holding a 3).
So, no, bluffing 2 as much as possible looks fine, but isn't better regardless of opponent's strategy


Best against any opponent's strategy is different from "best against your average opponent". The latter will exist (although many strategies may qualify), but is obviously dependent on the opponent model, and thus beyond pure game theory. If you have a model of your opponent, why should you use a NE strategy when you can use an exploitative one?