game theory in poker

2.1. The three card deck

This deck has three cards. The words high, middle and low are written on them. There are two players, Adam and Bill. Each ante one chip. Start with a special case. Bill is dealt the middle card. Adam has equal chances of being dealt one of the two remaining cards. This approximates a common river situation in hold'em. Bill has been betting and a scare card appears on the river. There is only a half a street of action as Bill checks. Adam can bet the pot(two chips) or check. Bill can call or fold.
Questions. Is this a zero sum game? Who does the game favor and what is the expected value of the game?
Adam always bets when he holds the high card. Adam is dealt the low card. Should he bet or check? This is solved by a two by two matrix. Adam uses a mixed strategy. B/B is bet/bet. Bet with both the high and low cards. B/CH is bet/check. Bet with the high card and check with the low card.
With the middle card Bill uses a mixed strategy of sometimes calling and other times folding.

_________BILL
ADAM	Call	Fold
B/B	1/2(4-2)	2
B/CH	1/2(4-0)	1/2(2+0)

Let's do the easy cell first.
Adam bets both the high and low cards. Bill folds. Adam always wins two chips.
B/CH : Fold. Adam bets high and checks low. Half the time Adam wins two chips.
B/B : Call. Adam always bets and Bill always calls. Half the time Adam wins 4 chips and half Adam loses two chips.
B/CH : Call. Adam bets high and checks low. Bill always calls. Half the time Adam wins 4 chips.
The parts within the cells are summed to a single value.

_______BILL
ADAM	Call	Fold
B/B	1	2
B/CH	2	1

optimal strategy
is when you could tell your opponent your plan and there wouldn’t be anything he could do to change your expectation in the game.

How often should Adam choose rows B/B and B/CH? Adam will choose row B/B x of thetime and row B/CH 1 - x of the time.
Adam will randomize between rows B/B and B/CH so that whether Bill calls or folds Bill will have the same expected value. This is done by making the value of the calling column equal to the folding column.
x+2(1-x) = 2x+(1-x)
On the left side of this equation Bill is always calling. On the right he is always folding.x is the first cell on Bill's calling column. 2(1-x) is the second cell. etc.
x+2-2x = 2x+1-x
2-x = x+1
1 = 2x
1/2 = x
Adam choose the B/B line half the time and the B/CH line the other half. By inspection due to the symmetry of the matrix Bill should call the time and fold half the time.
For the value of the game just add the value of the cells.Each cell occurs one quarter of the time. 1+2+2+ 1=6 Divide by 4 = 1.5
This game favors Adam. Adam's expected return is 1.5 chips. He had one chip invested before the betting street. Therefore Adam's net expected return is 0.5 chips.

Additional comments. The math of the matrix supports intuition. If Bill is a calling station, Adam does best by never bluffing. EV of 2. If Bill never calls, Adam should always bet. If Adam always bets, Bill should always call. If either Adam or Bill uses a fixed strategy the other can adjust his strategy to improve on optimal strategy. By using a mixed strategy of properly randomizing their two fixed strategies neither can gain advantage on the other.