Poker Strategy - Playing with cards up

I've been playing quite a lot of Poker recently, and the more Poker I've played, the more I've come to realise that it's actually a very deep game. One of the interesting features of Poker that makes it so complex is the inability to see each other's cards (this may sound like I'm stating the obvious but bear with me). Let's consider a thought experiment, what if we played Poker against each other without hiding our cards? I show below why it would still be a fairly difficult game to play well. The fact that we also can't see each other's cards  just adds another level of uncertainty to the underlying complexity. 

Incomplete Information
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In technical terms Poker is a game of incomplete information. An example of a deep game which has complete information is Chess. In Chess we can see all our opponents pieces, and they can see all our pieces. The fact that we have perfect information when playing Chess in no way makes the game trivial.

Heads Up No Limit Texas Holdem

Let's suppose we are playing heads up No limit Holdem (No-limit Texas Holdem is the proper name for the game most people think of when they think of Poker, heads up just means that there are only two of playing). Let's first consider the case where we can't see each other's cards.

Suppose we have J♣6♣, and the flop is 10♣ 9♣ 8♠ there is $£100$ in the pot, we have a stack of $£200$, and our opponent also has a stack of $£200$. Our opponent goes all-in, what should we do?

Obviously there is no 100% correct answer to this question, and if we were playing in real life, we'd have to consider a large number of other variables. How likely do we think our opponent is to bluff? What was the betting pre flop? Do we think we are a better player than our opponent overall, and therefore want to reduce the variance of the game? Are we the two short stacked players in a tournament?

All of the above considerations are what make poker such an interesting game.

Playing with Complete Information

Now let's suppose that we can see our opponent's cards. For example, imagine we are in the same situation as above, but now our opponent shows us his cards, they have 5♥ 5♣. What should we do now?

It turns out in this case, there is actually a mathematically correct way of playing, but it is by no means obvious what that is.

At the moment our opponent has a better hand - they have a pair of 5s, and we do not have anything. What we do have though is the possibility of making a better hand.

We can reason as follows. We will win if one of the following happens:

• A Queen or 7 of any suit come on the turn or the river.
• A Club of any value comes on the turn or the river. 
• Two consecutive Jacks of any suit come on the turn and then the river.
• Two consecutive 6s of any suit come on the turn and then the river.

We can calculate the probability of one of these happening, and we also know how much we will win if one of them occurs, we can therefore calculate the expected value of calling our opponents all-in.

It turns out that, the probability of us winning this hand if we call is $68.38$%, the probability of there being a split pot if we call is $1.82$%, and the probability of us losing the pot if we call is $29.08$%. In order to see how to calculate this, we note that there are $46$ cards remaining in the deck, and that we can calculate the probability of getting the cards we need for the $4$ way of winning.

Now that we have the probability of winning, we can compare this to the pay off should we win.

Expected Value of calling

The Expected Value is the average amount we would expect to win over a large sample of hands if we make a given decision.

In this case we calculate the expected value as:

$EV = 0.6838 * 300 - 0.2908 *200 = 146.98$

Therefore, over a large enough number of hands, if we are presented with this exact situation repeatedly and we take the average of how much we win or lose, we would expect the amount we win to converge to $£146.98$

The thing about this I find interesting is that when you are playing in real life, even when you can see your opponent's cards, unless you have access to an EV calculator it is still very difficult to determine the correct way to play, and this difficulty is only compounded by the fact that we also can't see our opponent's card.

In order to play Poker well when playing with Incomplete Information, we have to attempt to reason about who has the better cards, whilst bearing in mind this underlying uncertainty which is present even when playing with Complete Information.