drothe72 asked me about explaining some of this game theory stuff in greater detail, and I thought it would perhaps bet better to do it in the forum, since I think some others are interested as well--and may also know some things about other parts of game theory that I don't.

I'm also posting it here, because I think the interest in it is mainly for a NLHE player (I think it's also interesting for PLO, though), and I think this board gets the most traffic. We can move it to math and odds if others like, but I was thinking that leaving it here might be good so that others might follow along as the thread(s) progress(es) and could then also look through the background explanations.

To those have no interest: I'd suggest just not reading posts in the thread if you're not interested. It will be immediately recognizable as game theory, and if you don't think it's relevant, just don't waste your time.

Anyhow, now to the substantive part on the one part of game theory where I have at least some understanding. The general game is this: Various players are dealt a real number on the interval [0,1]. At showdown, low number wins. It's the same game if you do the high number winning, but calculations are usually easier if you take 0 as the nuts and 1 as the worst possible "hand."

In principle, one can even do it for multiple players (and I even tried one), but it gets EXTREMELY complicated, so complicated that I couldn't solve even the simplest of games. So, let's stick with HU situations to begin with (maybe we'll reach levels where we can try something with multiple players, but HU already gets rough pretty fast).

Now, in order to solve these games, you have to set up various rules for betting. Again, in principle, you could have a pot of 4, stack-depths of 100, and allow any bet, raise, re-raise for either player in increments of 1 (betting 1, 2, ..., 100). But that would again be horribly complicated.

And it's really not worth the trouble because usually in NLHE we're only worried about a few possible betting options.

One other note before going to an example for solving methods: The solution is always "optimal play." I'm really convinced that in poker, you don't want to play optimally but rather do things that get your opponent to play sub-optimally, then shift gears so as to capitalize much more extremely on that sub-optimal play. So, imo anyway, when it comes to applications, the optima are important mainly for giving one some orientation about what exactly IS suboptimal, then it usually follows fairly clearly what strategy will do better.

I'll attempt to clarify that at least a bit in abstract terms. Let's say that player A's optimal strategy is a, and player B's optimal strategy is b. But player B is playing a suboptimal strategy b'. Since B is playing b' rather than B, A does know that if he sticks with strategy a, he'll be doing at least as well as if B were playing b, and probably a bit better. He also knows that when B is playing b, he can do know better than a. But against b', particularly if it's a really bad strategy, A can probably increase EV significantly by playing some strategy a', which will certainly do no better than a vs. strategy b, but which will capitalize on b' much faster than a would. Example: On b', B calls way too much. Well, A eliminates the bluffs entirely, which are now losing money (compensated by the increased value he gets on VBs) and possibly adjusts his VB range.

Ok, so much for the general idea here. I'll analyse an example in a separate post to show how one solves these things and sets them up.

Examples just for fun to see how to set things up and solve the game.

There's a pot of 1. We have to define that because if there's no money in there, whoever acts has no motivation to bet without the nuts.

Now we have to define the allowed betting sequences, and it's a really good idea to start off very simple at first. When you allow complex betting sequences, it gets really complicated really fast.

I'll do one that's kind of weird just to get the idea. Player A as first to act is allowed only to check or to bet 8. If A bets, then B can call or fold. If A checks, hands are simply shown down (I'm trying to keep it simple).

Here's the way I go about solving this:

First question: How often does B have to call so that it is unprofitable (or break-even) for A to bluff with his worst possible hand, namely 1?

It's costing A 8 for a chance to win 1. So, B has to call on [0,1/9] for bluffing at 1 to be break-even. If we run it 9 times on that call range, A loses 8 one time and wins 1 eight times, so that's break-even.

So, we have solved the first part: On optimal play, B will call on [0,1/9]. That's going to make A indifferent to bluffing on his bad hands.

Now, finding A's optimal VB range is always just the top half of B's optimal call range, namely [0,1/18]. Again, at 1/18, A is indifferent to value-betting or checking, because if B calls, he wins exactly half the time.

One can also set this up in the form of indifference equations. Given B's call range, when is A indifferent to VBing or checking at the point x?

Assume A has x. What we do here is compare the EV of a bet vs. a check at x given B's call range.

Suppose A bets at x (obviously, x < 1/9).

If B has [0,x], then A loses 8. EV = -8*x
If B has [x,1/9], then A wins 9. EV = 9*(1/9-x) = 1 - 9*x
If B has [1/9,1], then A wins 1. EV = 8/9.

Total EV is -8*x + 1 - 9*x + 8/9 = 17/9 - 17*x

Now suppose A checks at x.

If B has [0,x], then A wins nothing (EV 0).
If b has [x,1], then A wins 1 for a toal of 1-x.

A is indifferent to betting or checking when these are equal.

So, we have 1 - x = 17/9 - 17*x
16*x = 8/9
x = 1/18.

Clearly, against this range, if x <1/18, A isn't indifferent to betting but prefers to bet rather than check.

Now let's look at bluffs. Clearly, if A were never bluffing but only made these value bets, B would be better off calling only at some values < 1/18.

It's the bluffs that are going to make B indifferent to calling or folding at 1/9, where he always loses to the VBs but always wins against the bluffs.

B is calling 8 for a chance at 9, so to be indifferent to calling, there must be 8 bluffs for every 9 VBs. (8/9)*(1/18) = 4/81. So, A bluffs [77/81,1].

So, we have:

A VB: [0,1/18]
A bluff: [77/81,1]
B call: [0,1/9]

Now to finish it off, let's see how the pot of 1 is really divided up in this game on optimal play:

I'm going to use Excel to multiply it out because we get into very small fractions. I'll also figure it for both players to check my math (the sum should equal 1).

EV for A:
A[0,1/18], B[0,1/18]. Split the initial pot for .50. Total is (1/18)*(1/18)*.50 = .001543
A[0,1/18], B[1/18,1/9]. A wins 9. Total is 9*(1/18)*(1/18) = .027778
A[0,1/18], B[1/9,1]. A wins 1. Total is (1/18)*(8/9) = .049383
A[1/18,77/81], B[1,1/18]. 0
A[1/18,77/81], B[1/18,77/81]. Split. EV is .400568
A[1/18,77/81], B[77/81,1]. A wins 1. EV .044201.
We set up the bluffs so that they break even, so EV there is 0.
A's EV is: .523472.

Not surprizingly, since A has more options, A wins a bit more than half the pot on this game.

Now let's check by doing EV for B.
B[0,1/18], A[0,1/18]. Split for .50. Total is .001543
B[0,1/18], A[1/18,77/81]. B wins 1. EV .049726
B[0,1/18]. A[77/81,1]. B wins 9. EV .024691
B[1/18,1/9], A[0,1/18]. B loses 8. EV -.02469
B[1/18,1/9], A[1/18,1/9]. Split. EV .001543
B[1/18,1/9], A[1/9,77/81]. B wins 1. EV .046639
B[1/18,1/9], A[77/81,1]. B wins 9. EV .024691
B[1/9,77/81], A[0,1/9]. 0
B[1/9,77/81], A[1/9,77/81]. Split. EV .352385
B[1/9,77/81], A[77/81,1]. 0
B[77/81,1]. 0 Regarldess of A's hand.
Total: .476528

And they add up to 1, so we must have done it correctly.

Anyhow, that's the way it works. The EV calculation is rather tedious, so I usually leave that out, but I thought I'd throw it in for completeness on the initial example.

i think i'm going to have to print this out and study it.

One other note: I just noticed today that this month's 2+2 magazine has a nice Sklansky article on game theory and applications--a somewhat more formalized version of what I was saying about not playing optimally but shifting gears. And also not going for bluffing always or never to capitalize but being more selective. I thought the article was pretty good.

C'mon, Excession, you're a smarty. It ain't that hard. Don't go discouraging Ais.