Does Heads Up Hold ’em GTO assume a linear relationship between utility and money?

Question

Would it be possible to exploit a GTO poker player if you recognized that you have different values than that player? For instance, if there are decreasing returns to money (such that your first million don’t make as much of a difference to your quality of life as your second million etc). I’m guessing differences of this kind would likely just shift the Nash equilibrium for a particular game but I’m not sure.

Answer 1

I am going to assume this player uses perfect GTO.

The short answer is no. No perfect GTO player is exploitable in any way (that is the definition of GTO). GTO means that the player is making the most optimal decision for long term profit every time.

Most players that play a GTO style tend to be focused on the poker, rather than the money (or other factors outside of the game for that matter). This is especially true at the highest levels (since you mentioned a million dollars in the question). Any player who is factoring in money into their decision making is not playing GTO, and therefore is exploitable.

Hopefully that makes sense, that is how I understand your question.

Answer 2

You're asking two questions here.

Are dollars always worth a dollar in theory? Yes, a dollar should be assumed to be worth a dollar to all players - otherwise how could the theory make sense?

Can you exploit a "GTO" player who doesn't understand utility theory? HAHA! Yes! I remember training a successful pro years ago. He wouldn't believe me that no two sane pros would ever play heads up except as a stunt. The reality is, you don't have to play anything NEAR GTO to make playing heads up against you a negative proposition due to the rake and tips [let alone your TIME].

PROS DON'T PLAY HEADS UP EXCEPT AS A STUNT OR A SCAM.

So to answer you're question... YES. If I'm playing a "GTO" player heads up, that's because I have marked cards in the deck. Or a hooker is robbing his hotel room while I play, or I know he has a drug habit and I'm waiting for the "yay-yo" guy to arrive at the Belagio.

Nits are going to reply "what about tournaments!?"... OK. the five time you'll final table at a major tourney in your life.. yes there are GTO considerations. However, since you only have 5 samples of experience in your life , this is a pretty stupid discussion. And YOU'RE the one being exploited. Since you're doing something important and it's only 5 times in your life, you better believe your own utility function is skewed.

And on top of that, the GTO strategy for almost all poker tournaments is to NOT PLAY. See: All Railbirds throughout gambling history.

This is why one of the best places to make money in poker is to offer a prick deal when you're heads up at the final table. At that point, most player's "utility function" is broken, and they'll take a poor deal.

Answer 3

A player employing a game theory optimal (GTO) strategy is implementing the equilibrium solution of a game. However, the precise definition of the game being played matters!

The utility functions of the two players will indeed influence the resulting Nash equilibrium solution of the game.

As a simple example, consider the difference between a heads-up tournament match, and a heads-up cash game. The existence of a terminal boundary condition in the tournament -- there is no further play once one player has all the chips -- manifests as a non-linear value function between your chips and the expected money you will win; ICM and FGS are commonly used approximations of this effect.

This, in turn, leads to different play at the same effective stack depth depending on whether you hold the larger or the smaller of the two stacks. The same does not hold in an infinite duration cash game between two opponents with infinite bankrolls; i.e., there are direct linear relationships between chips, money, and utility.

So long as both players employ the Nash equilibrium solution for the game they are actually playing, then neither is exploitable. However, if one player employs the Nash equilibrium solution from a different (but similar) game, then that player can be exploited.

For example, if one player in a heads-up tournament plays the cash game equilibrium, while the other employs the correct tournament equilibrium, the tournament player gains a slight edge. Naturally, if the tournament player discovers his opponent's error, he can switch to an exploitative strategy and do even better.

Even in cash games, a similar effect can arise, albeit of a much lesser magnitude. If one player is playing with a 40 buy-in bankroll, while the other has a 100 buy-in bankroll, the corresponding Nash equilibrium will show evidence of differing levels of risk aversion between the two -- the player with the smaller bankroll will tend to select lower variance lines slightly more often and push small edges slightly less aggressively. (And I do mean slightly.)

In practical terms, though, as your opponents' utilities, stochastic discount factors, and bankroll sizes are hidden information, any additional edge you might gain is almost surely purely theoretical.

Answer 4

Yes! GTO play assumes a linaer relationship between money and the utility function. Indeed, money won is the utility function itself. There is no other way it could be defined otherwise.

Still, I don't see how you having a, let's say, logarythmic utility function would make you a "log-winner" against a GTO player