Game Theory: Unusual Question #3 and #4

[well]

[ QUOTE ]
Because the bluff/no-fold strategy is not optimal: you need to add (the point I neglected to mention in my post, because it was so obvious) that B will fold hands below .5 that he doesn't bluff with.

[/ QUOTE ]

Thanks a lot. Obviously he should fold some hands.
Sometimes you just won't see you missed something in the very beginning...

I'll go on now.

[well]

You're right, you don't need fancy math.
But problems are easily solved that way, that is: if you don't mess up...

If A's strategy is:
[0,x] : raise(bluff)
[x,y] : fold
[y,z] : call
[z,1] : raise
And B's strategy is:
[0,a] : fold if raised
[a,1] : call if raised,
where 0<x<y<a<z<1, the EV of B would be z-2*x-y^2+3*x*a+a*z-z^2+y-a.

Solving grad(EV)=(0,0,0,0) gives

x=1/18
y=1/2
z=5/6
a=2/3

And EV becomes 5/18.

Tomorrow I'll look at #4, must sleep now, tomorrow I have an exam...

Next Time.

[Bozeman]

I get the same values for the solution for #4, but I get a game value of 17/72.

Anyone want to check on these?

If this answer is correct, B would rather have the no raising game than the one raise game, even though he acts last when he raises.

Craig

[Aisthesis]

This [0,1] game idea is all new to me, but since it may also be new to others here, I thought I'd post an idea I had as to the analogy to poker.

My first thought was that these numbers between 0 and 1 turn out to be analogous to some kind of hand-ranking. But I think that's not really the idea.

In fact, what one is in a sense dealt pre-flop is simply a probability of winning. Hence, even at a full table with all the positional complexities, given the cards everyone is dealt, your hand actually is nothing more than a PROBABILITY OF WINNING at showdown--hence, it actually IS a number between 0 and 1.

One odd fact, though, is that you don't even know exactly what your own number "objectively" is, because, while it is determinate, it depends on the cards the other people at the table are holding (similarly, the number each of them has is only partially known to them, since it depends on your hand, etc.).

I guess this still all boils down to your having a subjective probability with all the unknowns out there. It's just that the subjective probability you're dealt will in fact be different from the probabilities that could be assigned by someone who actually knew what everyone was holding but just didn't know what cards were going to hit the board on flop, turn, and river.

An extreme example of this type of discrepancy: You are holding KK, which is going to make your subjective probability of winning pretty high. But if someone else at the table has AA, your subjective probability is very far removed from the actual probability you have of winning the hand. From the standpoint of modelling the game, I have no idea whether that type of distinction would prove useful or not.

Anyhow, at each stage of a hand, you are actually holding a subjective probability, and at the next stage (flop, turn, river), you will hold a different subjective probability (unless, for example, you flop a royal flush with AKs in your hand or other similar cases--as soon as you have the nuts and can't be outdrawn, then you have a "1," and once you have a 1 it's not going down) but one that IS NOT RANDOM given your subjective probability at the previous stage.

Anyhow, the one thing I'm seeing at the moment is that, in contrast to the example here, the highest "player's subjective probability" (even with best calculation of this value) does not necessarily win the hand (or does it?).

But if one does introduce the additional value of a "probability from the standpoint of an outside observer," (who is ignorant only as to what cards will subsequently be dealt in the middle but knows the hands of all players as well as the current board) this value does determine who is ahead at any given stage of the hand and will typically be either 1 or 0 on the river (or a fraction of 1 in the case of split pots).

Anyhow, these are just some thoughts for beginners such as myself on how the relationship between poker and this [0,1] game might work.

Am I making any sense? or just completely off-base here? [img]/images/graemlins/smile.gif[/img]

[Aisthesis]

Yeah, the idea for using differentiation over various "decision point" variables for the bluffing solution just turned into a big mess. I couldn't get the equations to break down into anything that looked like it would go anywhere.

[Bozeman]

Basically, the problems tractable to game theory (and also those for which the [0,1] set makes a reasonable approximation) are generally river problems.


Craig

[Jerrod Ankenman]

[ QUOTE ]
Basically, the problems tractable to game theory (and also those for which the [0,1] set makes a reasonable approximation) are generally river problems.

Craig

[/ QUOTE ]

Well, I dunno. The problems that David poses are usually river problems, because those are the ones that he can solve.

Jerrod

[Jerrod Ankenman]

I'm willing to believe that the game value is wrong; I kinda did it in a hurry.

BTW, you folks that are doing these problems by brute force (ie, coming up with huge EV equations and maximizing them) are going to run into trouble if you try to solve more complicated games.

Jerrod

[Aisthesis]

lol... If I'm taking this right, it seems like the river, while already pretty complex, is indeed just going to be the most readily solvable case. But it seems like the model would in principle apply to all streets, if one can eventually get the enormous amount of complexity under control.

Once I've digested at least the "primer material" on this, I'll be very interested to see where you guys are at.

[Jerrod Ankenman]

Sure. I mean, all these games can be turned into gigantic matrices of strategy decisions for the two players and Nature, and you could solve it by brute force. It's just that there are too many combinations and not enough computer power.

The point of the analytical solutions, like the ones Bill and I have worked on, is to find ways to solve entirely games that are really hard to do by brute force - like, for example, no-limit games, as well as effectively approximate solutions in different situations and develop intuition about situations that might not arise from usual analysis.

You guys know about our contest, right? The solution to the no-limit AKQ game, which we have and which is fairly startling, is worth a couple hundred bucks and a cite in our book. I dunno if I'll be censured for advertising, since it isn't a 2+2 book, but whatever.

Jerrod