Let say we play game where rake is 5% and CAP is 10BB(for example NL25 on ACR). There are some postflop spots where GTO play can lose EV compare to EV solver would give you(in sim where rake is included)

Example
SRP UTGvBU 100BB eff
UTG OR for 2.5BB, BTN call everybody, else fold
Flop is Ts Jc Qd
Both players check
Turn Kc
Both players check
River As

Optimal strategy in this spot will depend on rake.

If rake is high enough OOP will jam whole range and IP must fold everything, because IP is not getting right odds to call(due to high rake). EV for OOP player is entire pot (6.175 BB) in this equilibrium. If our opponent decides to call every time, OOP strategy of 100% jam becomes huge -EV play EV=0.5*191.5-97.5=-1.75BB.

If we lower the rake then equilibrium becomes pure checking for both players.

If rake is 0 any pair of strategies that do not involve folding is equilibrium.

In first case rake acts like 3rd player, despite OOP player playing perfect strategy both players are losing EV due to IP mistakes and all that EV is collected by the house. I thought this was an interesting illustration of the idea that a GTO strategy could lose if one of the players makes mistakes, so you cant use it blindly in every situation.

Funny I was literally just studying these prisoner's dilemma situations.

If there's more rake to be collected, but the rake/SPR is low enough, then the GTO strategy involves both players checking.

If the rake is already maxed out (or there's no rake) then the GTO strategy involves always jamming and always calling. Although anything but folding has the same EV, the jam-call line is a weakly dominant strategy since the opponent can only make mistakes against this strategy and can't gain anything.

If the rake is high enough, then you create a prisoner's dilemma. The highest EV combined payoff for both players is to check, however the first player to shove will takedown the pot, winning more than splitting.

You can actually break a solver like this. Here's an example where GTO+ incorrectly range-check-folds instead of jamming. Obviously jamming is the correct strategy, but the algorithm finds OOP checking and IP taking down the pot.

https://drive.google.com/file/d/1t2l...ew?usp=sharing

Pictures: https://imgur.com/a/uuzHEQg

Then the real fun starts. After one player jams the best strategy is to fold everything, but you can spite them by calling instead, hurting both of your EV's lol. It's a strange situaton.

Its similar to prisoners dilemma, but in that game both players make move in same time.

In my sim OOP just jams first



it is interesting that being OOP in a high rake environment can be much higher EV then being IP with no rake, at least in this spot.

Weird. I solved directly from river which maybe causes a difference in the algorithm. Very interesting spot indeed!

2 player 0 sum has some properties such as each equilibrium having the same evs

when you add rake or another player you can get multiple solutions and they can have different evs

the definition of a nash equilibrium is a strategy set such that no player can change only their strategy to increase their expectation

there is another solution in the high rake example you gave - both players check calling 100%. neither player can adjust his own strategy to increase his expectation. changing from calling to folding in a node where the opponent reaches with 0% probability won't increase your expectation. changing from checking to betting will only lower your expectation because the other player is calling 100% when facing a bet. the only way to increase the expectation of a player is to change both sides strategies, so its actually another solution.

I started a thread about a similar spot a while back: https://forumserver.twoplustwo.com/1...brium-1768495/

GTO can lose money in many raked situations. However, it's interesting that there are also multiple ways to make GTO lose money even in games with no rake. You can have fun trying to figure out what those ways are

Cool, reminds me of a 98765 board I was looking at, where the equilibrium was also for OOP to shove entire range and IP to call entire range.

I feel like lots of players are butchering these spots, have definitely seen some checks from OOP, and some folds from IP when OOP jams.

Raked poker is a -EV game so it should be no surprise that GTO loses $; it's supposed to lose money!

If you really want to get in the rabbit hole you can look at multiway spots where one player's -EV actions can cause a GTO strategy to lose, for example, side pots.

This is effectively MW spot 3rd player is the rake. Its true that raked poker is -EV, but post-flop raked poker is still +ev game(both players have some share of the pot).

Not if there's still rake to be taken from the pot lol. Not even postflop spot has already reached the rake cap, especially in lower stakes

I mean you can try and run sims with rake no matter how big the rake is EV is never less then 0. If EV would be less then 0 for some player, then he could improve his EV by always open folding.

The net EV of every player in a GTO sim or whatever, before the blinds are posted, in a raked game is -EV. The only reason the sim doesn't show it as -EV is because the rake is already accounted for; every player is losing

Maybe we are talking about same thing in different terms. I'll try it to clarify

For preflop before blinds are posted poker is -EV game only for two players on the blinds. This is ture for raked or rake-free poker. For rake free pokere adding EVs of all players gives you 0, in raked game it's lowere then 0. So wehen you calculate average EV over all positions its negative if there is rake.

For postflop: EVs are always postive it doesn't matter if there is rake or not.


Reason why in OP one player can lose even playing gto is not because gto is suppose to lose in raked game that's only case for preflop.

The sum of every player's EV and the rake will always add to the size of the pot.

IP EV + OOP EV + Rake EV = Pot

For a given hand you are correct. That’s why there are blinds. Assuming equally skilled players, the EV of an unraked game would be zero. Adding rake makes all EVs negative. Since folding is zero EV, the optimal play is to fold, and there would be essentially no game. The only reason that a player should not automatically fold is that there is money in the pot, namely the blinds, that makes some holdings +EV.

However, over the course of many hands, the blinds even out and, again assuming equally skilled players, each player’s overall EV is negative. That must be true since no player has a skill advantage. In an unraked game, that means each player has zero EV over the long term. The rake must come from somewhere; hence, each (equally skilled) player has negative EV in a raked game.