Oh, the Possibilties! (HeadsUP Holdem)

Goal: To determine the possible number of states in a 2-player game of limit texas holdem.

Note: In order to determine the magnitude of states, a finite number of raises must be established, so I have set this limit to match what is typically found with online poker sites (a bet and four raises).

1st Objective:
Calculating the number of betting routes before the flop.

(Remember, the button acts first before the flop, and is acting as the small blind.)

c=check f=fold r=raise

f

c c c c c c c c c
c r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

So, there is one course of action when the button starts off by folding, 9 when he starts of by calling, and 8 more when he starts off by raising, totaling 18 courses of action before the flop. (Yes, I realize that some of the lines make little poker sense: such as reraising, then folding, but all lines must be included to allow for a complete model).

2nd Objective:
Calculating the number of betting routes on each street after the flop.

c
c

c c c c c c c c c c
b b b b b b b b b b
c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

b b b b b b b b b b
c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

So, there is one course of action when they both check, and ten courses each when the 1st player to act checks or bets totaling 21 courses of action on each street (flop, turn, river).

3rd Objective:
Multiplying the possible hand combinations by the possible betting routes.

There are 1,624,350 (1,326 * 1,225) starting hand combinations possible in a heads up holdem match.
There are 18 different ways the betting could go.
So, there are 29,238,300 ways the preflop action could go.

There are 17,296 { (48*47*46)/6 } different flops in a heads up holdem match (knowing all four hole cards).
There are 21 different ways the betting could go.
So, there are 363,216 ways the flop action could go.

There are 45 different turn cards possible.
There are 21 different ways the betting could go.
So, there are 945 ways the turn action could go.

There are 44 different river cards possible.
There are 21 different ways the betting could go.
So, there are 924 ways the river action could go.

So, when you multiply all the ways the action could go on each street you reach the enormous number of 9,273,013,006,761,504,000 ...or 9.273 quintillion. Is this correct? I’ve read the that the possible number of states is just over one quintillion, am I off by a factor of nine???

Did I make any silly mistakes that account for this?

Thanks in advance to all those who math ability is greater than my own (so, I guess, I’ll be thanking a lot of people).

Rick.

How can there be a sequence between 2 players with a fold followed by a raise (c f r r)?

[Guernica4000]

c=check f=fold r=raise

c f r r r r r r
c f r r r r
c f r r

c f r r r r r r
c f r r r r
c f r r

c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r

c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r

opps Filled beat me to it

I don't think it is a mistake, just bad formatting on my part.

All the action lists go top to bottom, not side to side, and c=check, and (this was my goof up), c=call as well. Obviously, check can only be used when acting first, or only after the first person has checked (only one route).

F=Fold, C=Call/Check, R=Raise, B=Bet

Before the Flop:

f

c c c c c c c c c
c r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

Betting After the Flop:

c
c

c c c c c c c c c c
b b b b b b b b b b
c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

b b b b b b b b b b
c f r r r r r r r r
c f r r r r r r
c f r r r r
c f r r
c f

The University of Alberta is pretty well-known for poker research:
http://www.cs.ualberta.ca/~games/poker/FAQ.html#holdemsize

...
Let's take a quick gander at this:
Pre-flop:
fold (1)
+
call -> check (1)
+
call -> 1-4 raises -> call | fold (8)
raise -> 0-3 raises -> call | fold (8)
For a total of 18 pre-flop possibilities, of which 9 are terminal.

On each round of betting we have:
check->check (1)
check->1 through 4 raises -> call|fold (8)
raise->0-3 raises->call|fold (8)
So, 17 possibilities of which 8 are terminal.
So we have:
9+9(8+9(8+9(8+9)))=13123
betting trees as far as I can tell.
Now, there are 52 chose 2 * 50 chose 2 * 48 chose 3 * 45 * 44 possible hands
That's 55,627,620,048,000 as far as I can tell.
So I come up with:
729,945,630,269,856,000
possibilities in total. That's roughly (2^64) which is too big to do brute force work on.

In practice, the tree is smaller:
There are only 89,998 possible starting combinations of hole cards if we use suit equivalence. (That saves a factor of about 18.) It's probably possible to squeeze out another factor of 18 or so for all of the betting rounds - which puts the total closer to:
7,000,000,000,000 - approximately 2^44

So, if you do manage to generate a perfect poker player, you could store it in petabytes of disk space.

Doh! I can't believe I forgot that there were nine responses in which wouldn't carry over onto the other streets, what an oversight on my part. Thanks for that. Does anyone know if the 'suit equivalence' factor applies to all streets as rufus suggests, or does it only apply in reducing the possibilites in preflop play, or is the math more complicated than that?