I'll just ask again then: If the callers EV is higher with 5.7% calling than in your proposed equilibrium, and the pusher's EV is higher as well, why would either of them change and play your strategies instead, where they are both worse off?

You stated that the pusher would somehow adjust against the 5.7% calling range because to remain pushing ATC would be terrible, for some reason. That's the part I quoted above.

Because it is the purpose of a call to deny EV to the bettor, as much as possible, without losing money on the call.

This requires the game to be strictly noncooperative and no collusion between players.

So, in your equilibrium theory, the main objective of players is to deny EV to other players, they are not concerned with their own EV?

The caller should be concerned with the caller's EV, and that's higher at 5.7%.

Then the shover should exploit your overfolding with a better range than ATC.

Only at my equilibrium can neither player improve, if the shover is required to shove ATC or fold, and the GTO caller defends properly.

Ok, let's say the caller plays 5.7%. What range should the pusher play to exploit this? I'm curious how he'll improve his EV.

At what rake, exactly?

At your suggested 26.21% rake.

It would make more sense for me to demonstrate an approximate exploit of your caller in your equilibrium. Which one was it, 17.48% or 20.6197% ?

The exact optimal exploit range requires non-linear regression with hours of data gathering, and a complicated calculus equation.

However, a rough guess should be sufficient as a counterexample.

If the caller plays 5.7% hands at 26.21% rake, you claimed that the pusher should adjust to this nitty range because remaining at ATC would be really bad. I'm curious how the pusher will exploit that 5.7% range, a rough approximation will suffice.

Agreed.

plexiq,

It would seem that your question has proven that my equilibrium is indeed Nash.

At my equilibrium, with pure strategy, the bettor cannot improve his position by altering strategy. He just keeps pushing ATC, never folding any hands. If the bettor folds 72o, he loses EV no matter what the range of the caller is chosen to be.

If the caller stays at equilibrium, then the EV to either player is essentially zero.

If the caller diverges from optimal and calls with 77+, ATs+, AKo, which is 76 combos:


Then the EV of the shover goes from 11 cents to $6.17 bucks.

10*[1-(76/1326)] - 100*(.7153)(76/1326) + 54.959*(.2734)(76/1326) - 22.5205*(.0112)(76/1326) = 6.174

While the EV of the caller goes from zero to 67 cents.

54.959*(76/1326)(.7153) - 100*(76/1326)(.2734) - 22.5205*(.0112)(76/1326) = .6717

So, by overfolding, the caller has awarded the bettor 6.17-.67 = 5.50 in tournament equity.

Thank you plexiq.

Not checking your math at the moment, but if the caller can gain $.67 by changing his strategy then he wasn’t at NE. Why would you skip $.67 in EV just to spite your opponent? Like I said this is not a zero sum game. In a zero sum game the sum of the EVs of each player remains constant across strategies. The game is positive sum because of the $10, but even if that money comes from antes from each player the game is negative sum due to rake. It’s incorrect minimize the opponent’s strategy. We want to maximize our own.

If the game requires rational participants, then a strategy that awards one player 67 cents, and the other player $6.17 would be considered dominated by a strategy that awards both players zero.

Also, you skipped the part about the shover. At true NE, he is fixed, no matter the +EV call range, at shoving ATC. If he folds anything his EV goes down, and the NET INCREASE in his tournament equity is also reduced. He simply keeps shoving and keeps watching the caller overfold. No matter what +EV call range you enter for the caller.

It is quite fun to mess around with it. Find any +EV call range for the caller, and then compare the raw EV to the shover. Shover always shoves.

I can demonstrate that the shover gains EV at the other incorrect equilibriums by folding 72o.

So, turns out the pusher makes ~6$/hand when the caller uses 5.7%, so why would the pusher change his strategy? You agree he won't.

Same question for the caller, his EV is also higher at 5.7% call and you just agreed that the pusher has no reason to play anything other than ATC. So why would the caller play your wider range instead of 5.7%, if he is a rational player and only cares about his own EV?

Because he just gave the pusher 5.50. If the caller does this repetitively, the pusher will win the tournament.

Only at equilibrium can the caller defend his tournament equity according to minimax.

Do you think your equilibrium will pass the same stress tests?

Have you even checked?

This is a toy game with direct payouts. There is no "tournament" in this game, it's only meant to resemble some tournament situations. Maybe re-read the OP.

Exactly. And the winner of a toy game is the player with the most EV.

Your shover is crushing your caller.

My caller stands on a perch and can not be disturbed.

Your shover could improve by folding 72o.

My shover just shoves. Always. No matter what.

Wth are you talking about? They win whatever payouts they get from the game, there is no prize for getting higher payouts. If they both play the ranges you suggested then they simply break even, rohghly. If they play the 5.7% calling range then they make a few bucks per hand on average.

You and some stranger get picked at random for a promo. You are offered a choice:
A) You get 10k cash, stranger gets 20k cash.
B) You get 2$ cash, stranger gets 1$ cash.

You are advocating to pick B) because with option A) the stranger "wins". That's not how rational players are expected to behave though.

I think I can demonstrate that your players can improve by deviating.

Sure, knock yourself out. For the 20.6% rake ranges that is, the 17% rake solution is not an equilibrium, if that wasn't obvious. (You seemed a bit confused on that point earlier.)

Just for the record though, do you still think that your ranges are a Nash Equilibrium?

I will simply demonstrate what happens when the shover folds 72o at 20.6% rake vs your caller, and then demonstrate what happens to a shover that folds 72o at my equilibrium, vs any caller.

You guys are at least approximately right on your shover EV calc.

Is that fair enough for you?

If you still think that your ranges are even remotely close to a Nash Equilibrium then I suggest we don't move on, since you are still missing something really basic in that case.

Is this fair or not?

72o is supposed to turn a profit at both locations.

Heh, you can do whatever you want obviously. But i consider it pointless to go on if we can't agree on the definition of a Nash Equilibrium. Your ranges obviously don't meet the "mainstream" definition of a NE, as the caller can clearly deviate to improve his own EV when the pusher's strategy is treated as fixed. You agree on the later part, so you must be using a different definition. Or you are simply trolling, in which case this is also quite pointless obviously.