Six card deck consisting of two Aces, two Kings and two Queens. Hand rankings are pairs beating high cards. The small blind posts 0.5 units, the big blind 1 unit, with stack sizes of N. Each player is dealt a single hole card, then there is a no limit betting round. After this has concluded, a single community card is dealt. There is another no limit betting round (with the big blind acting first) followed by a showdown if necessary.

Questions:

Can you prove or disprove -

a) That a strategy involving a strictly positive amount of limping is optimal for the small blind at some stack size?
b) That the small blind's optimal strategy never has a negative expected value for any stack size?

If the continuous nature of no limit betting makes it too complicated, consider solving this with discrete units (e.g. betting increments of one small blind or one big blind except for all-in) instead.

The answers to these questions could be quite helpful for HUNLHE, especially for small stack sizes.

the game works like this:

If you bet pot, (and you really cant bet more in this game because of GT reasons) i get 33% on my call. I will pair up about 1/6. Are the implied odds justify a call?

Everytime i hit i need to get a 2/3 PSB in on average assume im not bluffing. That should be very possible.

So you can call pre allways.

The maximum postflop you can bet is 3x pot. That will obviously just get a call from a pair. If you bet pot you will get called by ace high too.

so limping pre witha shortstack might be profitable because you can bet more profitable postflop. And you are oop Pre, what makes you playing only pre -ev (of course you can create a bigger pot for post as long as your stack is big enough).

Another question:

Particularly if you find the optimal strategy never involves limping or the game always has a positive expected value for the small blind, do you think this may be simply a result of the peculiarities of this game and may not tell us much about the answers for other games? Like if you modified the game slightly (e.g. expanded the deck to include three of each card) do you think it could then involve limping, or did your working give strong evidence or perhaps even a proof that limping could never be optimal (at least within some constraint)?

Don't you get the same answers (with regards to HUNLHE) if there is an infinite deck with 1/3 of each rank?

This will get rid of the effects of missing cards and makes the whole thing a lot easyer to solve.

I suppose, but card removal effects are relevant in Hold'em, too.

On p. 171 of MoP, Chen summarizes how to solve a no-limit AKQ game with a 3-card deck and antes. Have you tried adapting this for your 6-card deck and board?

Yes, but much less so.

In real poker hand strengths are much more fluid and ranges ranges larger, so equity changes only a little with card removal effects compared to this game.

Here taking one card out of the deck changes the probability of the other having that same card from 1/3 to 1/5. And having on of the other cards from 1/3 to 2/5.

On the second street this goes up to [0, 1/2, 1/2] or [1/4, 1/4, 1/2].

Since this is an AKQ or nuts/bluffcatcher/air game this has a huge impact on the play on later streets and each player's equity.

What do you think of a variant where the hole cards and community card are dealt from separate 3 card decks? Since the community card is dealt independently, it removes a lot of the complications due to card removal effects. It also removes the complexity of ties, which has a nice effect of ensuring preflop equities are nice and round (higher hand has 2/3 equity versus lower hand).

If each deal has it's own deck this is equivalent to dealing from an infinite deck. The chance distributions don't change.

This is unless you mean that the holecards are dealt from one deck and the board cards are dealt from an equally distributed deck. In this case his distribution remains in the form [1/5, 2/5, 2/5]

I don't really see the point of this. Your questions were about limping. To answer this for specific holdem situations, you'll need holdem distributions. To answer this in general, you don't need to simulate card removal effects.

Each player gets dealt a card at random from a deck consisting of {A, K, Q}. The hole card is then deal at a random from another deck consisting of {A, K, Q}. The lower ranked hand has 33.3% equity preflop, thus Q has 33.3% preflop equity, K has 50% and A has 66.7%. Since the community card is independent, the distribution of hands remains identical post-flop. There are no ties unlike in the other games, which may make calculations simpler. (Also, the infinite deck's preflop equities are 38.9%, 50% and 61% as a result of these ties).

I'm not too concerned about about the exact game. I'm more concerned with the questions "Does a toy game with the NLHE betting system exist where limping is part of an optimal strategy, or where the small blind has a negative expectation at some point?" The game was just an example. If you think you can construct a game where either of those might be true that is still in the spirit of some normal poker game, that'd be great.

Okay, this game is bad for testing the second question since it's trivial to find a 0EV small blind strategy for any stack size. Below 3BB, the BB will always call a shove, so you can guarantee 0EV by shoving everything. Above 3BB, shoving only Aces guarantees you 1BB 1/3 a time, while losing 0.5BB 2/3 of the time, for an EV of 0.

This is more complex that I thought it would be, so here's some simpler games:

- [0,1] no-limit betting, but with a small and big blind of 0.5 and 1 respectively.
- Two street static version of the above game, small blind has position in the second street.

I'm fairly certain that there is a limping range for the SB in GTO HU NLTH. Is that the question you really want to get to? Are there any theorists that believe there is not? Or are you just genuinely interested in the toy games?

You mentioned small stack sizes. Obv that's important. How small is small? If stacks are small enough, there will certainly be no limping range.

Pretty sure limping is part of the strategy.

Think of small blind as having a two part decision
1. decide whether to fold or limp+
2. if limp+ then decide whether to check or raise

Then it becomes equivalent to an ante game with ranges (high good)
BB [0..1]
SB [L..1]
and SB must decide whether to check or raise.

For a 1 street game SB will definitely check middling hands even if L is large, and that's the same as limping in the game you decribed. If SB never limps then his raising range is a continuum of value hands (with no bluff hands) and the bottom of that range will never get called by worse.

Not sure about 2+ streets, but I don't think it matters.

I'm unable to prove your complete statement, it's an interesting question, looking forward to seeing a proof of it. We can easily prove something weaker :

Regardless of the game, and with the extra assumption that position is decided by a coinflip for later streets, if BB has non-negative expectation at stack size S then the SB has a non-negative expectation at stack size S/2. This shows that BB doesn't have strictly positive expectation for some [M,+infinity[ interval. So with the idea in mind that there should not be an infinite number of "state changes" (like SB has strictly negative expectation for stacks of size in [n,n+1/2] for any n> some fixed N), it should mean that SB has non-negative expectation when stacks tend to infinity. As it also does when stacks are short (jam or fold game basically), that's a start.


So assume that stack size is s, and BB has a non-negative expectation strategy. In particular it has one against any strategy of SB starting with minraising 100% of the hands at 2BB. Now we have a game with new SB = former BB and new BB = 2*former BB. That's the game where roles have been reverted and stack size divided by 2. So for this s/2 game, SB has a non-negative expectation strategy.

And as SB's excpectation cannot decrease when it always have position on later streets, this shows that BB doesn't have strictly positive expectation for some [M,+infinity[ interval in your setting.