Game Theoretic Equivalence
Posted by AF3
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AF3
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Game Theoretic Equivalence
I'm trying to understand why it is valid to use "ex-showdown value" to determine the value of various strategies, as used in MOP. In other words, why are we justified in ignoring the situations where no betting takes place?
Could somebody confirm (or deny) the following thoughts on the topic:
I see that the idea is not to show equality between various methods of calculating values, but "game theoretic"
equivalence (i.e. two values are equal to within a constant that is
independent of strategic variables so that unilaterally improving one of
the values does so for the other).
Would it be that the value of game situations where no betting takes
place is somehow a constant that is independent of strategic variables?
On the one hand, I don't see how this is true, since the strategic
variable of betting frequencies, for example, determine how often we
have a situation where no betting takes place (but on the other hand,
they do not influence the value).
If it was true that we could somehow treat "non-bet" situations as a constant, then we would have:
Value of game (for one player) = Value of game situations where betting
takes place + Value of game situations where no betting takes place
==>
Value of game ~ Value of game situations where betting takes place
Where ~ denotes game theoretic equivalence (i.e. if we can unilaterally
improve the right side with a different strategy, then we can
unilaterally improve the right side)
Am I on the track and just wrong about not being able to treat situations where no betting takes a place as a game-theoretical constant?
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I'm not quite sure if I understand what you are talking about. Is there some reading I could do to get the foundation to start being able to discuss these topics? Recommended books?
Thanks,
Mush
I have no formal background in game theory and am unfamiliar with much of the terminology they use in MOP (from the general standpoint of "what is the basic concept here that carries over to all games), but I know enough about mathematics to know when somebody is trying to pull the wool over your eyes to avoid explaining things in detail for the sake of "accessibility" (which is obviously what the authors in MOP decided to do with many of their examples).
For example, the basic idea behind "ex-showdown" value is that we can use the definition of "overall expectation" to cleverly break up the total game value of a given strategy into the sum of it's various parts, and then if we argue correctly we can show that it is sufficient to only pay attention to specific parts of the equation in our calculations.
It then becomes a matter of the most convenient way to break things up and which part to choose.
They recommend some books on Game Theory at the back of MOP.
Okay, I can show that overall expectation is proportional to the value gained or lost by betting, and therefore it suffices to only consider ex-showdown value in deciding whether or not we are equilibrium.
However, I'm still struggling with something basic:
Let ~ mean "is proportional to" (i.e. equal to within a multiplicative constant)
Let V = Overall expectation (for a given player)
Let x = Ex-Showdown Value (the value due to betting or calling)
Let c = Showdown value (this is just the expectation of having the hand checked down)
Let Pb = The probability that there will be a bet made during game-play
After a little work, we have the following:
V = Pb*(x-c)
==> V ~ (x-c)
Okay, so if we raise the value gained by betting (relative to checking), then we will certainly raise our overall expectation, and thus we can use ex-showdown value for calculating possible improvements to our strategy.
However, if we change our strategy, we also change Pb. This is what's confusing me, because it seems that the examples in MOP are treating the amount we gain with a certain hand by betting it hand as if it's the exact same amount we gain for our overall expectation. For example:
In the full-street spread-limit AKQ game, Player X gains 1/6 of a unit with a king by betting the smaller amount with a king. We then use that figure of 1/6 to determine how often Player Y should bluff raise a queen.
Edit: I guess it doesn't matter because we have:
Ex showdown value due to betting - Ex showdown value due to checking King = 1/6
==> Overall gains in our total game value ~ 1/6 (but do not necessarily equal 1/6)
Thus, Y counters in such a way we do NOT gain by betting a King and therefore we have:
Overall gains in total game value ~ 0 ==> Overall gains in total game value = 0
By the way, I don't really care if anybody responds, since it's kind of a free-roll as to whether or not they will (as I'm writing these things down anyways), but if somebody like user "Game Theory" or anybody with a real background in mathematical game theory could come in and let me know that I'm on the right track (or not), I would highly appreciate it.
It´s a bit difficult to follow you because you don't put your questions in a way that everybody can precisely understand what you want - at least it´s my feeling. :)
OK, so I´m on the travel and don´t have MOP at hand, which makes it even more difficult. But as far as I understood, you were wondering, why we treat Pb as a constant, when it seems to not be constant, right? And if it wouldn´t be, but differ with our strategy (as the bettor), there´s no way to calculate Pb(x-c), because Pb is variable, right?
Your misunderstanding - if I´m correct - is that you expect Pb to be dependent on (from?!) our strategy, but it clearly isn´t! You see that in the formula, Pb neither depends on x, nor on c - so it´s not variable. Villain should have a constant "bluff%", because he has to balance his A - which he will value bet all time, so his bluff% is not variable and it does not depend on our betting frequency. That way we can treat Pb as constant and in fact treat the value of betting as a measure for overall profit.
Agree?
Your misunderstanding - if I´m correct - is that you expect Pb to be
dependent on (from?!) our strategy, but it clearly isn´t! You see that
in the formula, Pb neither depends on x, nor on c - so it´s not
variable. Villain should have a constant "bluff%", because he has to
balance his A - which he will value bet all time, so his bluff% is
not variable and it does not depend on our betting frequency.
Oh, so that's why they calculate the game value for the second player in all these models?
It's easy to calculate the second player's value using ex-showdown value because the second player has a constant bluffing frequency, and to calculate the first player's we just use the fact that these games are constant sum and take the negative of the second player's (ex-showdown) value for each situation?
My question below about comparing strategies still stands, though. I don't see how we're showing that our overall equity has improved because we're not able to directly compare how often we are obtaining a situation where bets go in, we are only able to show that we improve (or lessen) the EV of situations where betting takes place.
Thanks for answering.
It´s a bit difficult to follow you because you don't put your questions
in a way that everybody can precisely understand what you want
1) Precisely define ex-showdown value in mathematical terms. It is total horseh*t that they didn't do this, and then they relied on bullsh*t arguments for the rest of these chapters without even showing that it was valid to use their method in the first place. Which brings me to....
2) Prove (mathematically) that it is sufficient to consider only our ex-showdown value when calculating the value of certain plays.
3) Is the value of our strategy obtained by "dilating" the final pot size by the (-) value of checking down the same as if we didn't do this?
For example, let's say we bet the Ace our opponent folds the King. When we say that "intuitively" we gain 0 units by betting the Ace and having our opponent folds the Queen, what we're really saying is:
Checkdown value (of the Ace) = (Equity)(Pot Size) = (100%)(P)
==> <"Value" gained by betting Ace when our opponent has a Queen> =
(PrWin)*(Final Pot Size - Bet) - Checkdown Value = (100%)*(P + s - s) - (100%)*P) = P - P = 0
What I'm asking is why it's valid to use this method?
Your misunderstanding - if I´m correct - is that you expect Pb to be
dependent on (from?!) our strategy, but it clearly isn´t! You see that
in the formula, Pb neither depends on x, nor on c - so it´s not
variable.
I'm not wondering why you treat Pb as a constant. This is what lets you claim that the value is given strategy is proportional to (x-c), which I think is what they're trying to call ex-showdown value.
However, Pb is a parameter which changes when we select a different strategy.
It isn't a variable in the sense of being directly dependent on x, but it does vary across our responses.
If V is our value in the game (for a given strategy) then we have:
V = Pb (x - c)
If we change our strategy and let Vn be the value of our new strategy, we have:
Vn - V = (Pb (new strat)) * (x - c)' - (Pb (old strat))*(x-c)
where (x-c)' denotes our new ex-showdown value
What I'm asking is: How do we know that we have improved our overall expectation by increasing our ex-showdown value when we are changing other parameters as well?
I know I'm making some big mistakes here, but I haven't figured out what they are yet.
This looks silly and ill-defined. For instance take 6-max NLHE, without calling or raising, everyone has to fold and the BB gets a walk. So the game value for the BB is +0.5, the SB -0.5 and the rest 0.
Now we add the betting, for instance give everyone 10bb stacks and force push/fold, the SB and the BB are now losing, and all other positions are winning. So the value of the betting is worst for the BB and best for the SB?
It's very silly and ill-defined, I've been trying out how. I understand it now, though. The point is that when we express the value of a certain as the sum overall the individual values of all possible situations, what we end up with is being able to express the value in units of
(value obtained when betting takes place - value of checking down) or (x-c) as I called it
When we do this we end up with exactly what they're doing in the book.
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