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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.

  • Your understanding of Nash equilibrium is different than mine. – paparazzo Dec 11 '18 at 20:55
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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.

  • Right answer to the wrong question – David Sep 9 at 6:59
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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.

  • Your own bankroll is not hidden information! – David Sep 9 at 6:59
  • @david What part of "your opponents' ..." did you fail to understand? :) – Confused-cius Sep 9 at 7:19
  • Your opponentes' bankroll is irrelevant to a utility function like the one described by the OP – David Sep 9 at 7:24
  • @david Why is optimal tournament play different from optimal cash game play? – Confused-cius Sep 9 at 7:31
  • Optimal tournament play looks for maximizing money won, just as cash game strategy does – David Sep 9 at 7:35
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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.

  • Although I think you place the boundary between the "game" and the "metagame" differently than I, we both realize that the broader view provides important context that should not be ignored. However, I do take issue with the statement "... a dollar should be assumed to be worth a dollar to all players - otherwise how could the theory make sense?" I would argue that mathematicians and economists are quite able to model the equilibrium interactions of agents with radically different utility functions without much difficulty. (Insurance, math finance, auction theory, public choice, etc.) – Confused-cius Sep 9 at 22:15
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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

  • A linear relationship only holds in the limit of an infinitely long cash game with infinite bankrolls. The instant either of these conditions are violated, then the boundary conditions will induce risk averse behaviour, which will shift the location of the Nash equilibrium. If you don't believe me, we can resolve it with a coin flip for your entire bankroll ... :) – Confused-cius Sep 9 at 7:25
  • @Confused-cius Please show me one single example of a GTO strategy that does not assume a linaer relationship between the utility function and money won – David Sep 9 at 7:26
  • How good is your math? Can I attach papers from the literature, or do I have to try to simplify it for you? – Confused-cius Sep 9 at 7:29
  • @Confused-cius I am a professional mathematician, so papers are fine, but I don't see why it's needed. It's enough with saying. "In spot X, this hand would be a call if the utility function depended linearly on money won, but since it does not, GTO suggests a fold" – David Sep 9 at 7:34
  • Ok, I'll attempt to walk you through this. Let's consolidate discussion under my answer. – Confused-cius Sep 9 at 7:40

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