[Games_] In Von Neumann's analysis of a simplified poker game he considers the hands to be numbers between 0 and 1. The player with the higher number wins at the end if nobody folds. The real poker hands are different types of objects, however. For example in holdem you start with two cards, which can later develop into a variety of hands. Thus the most natural representation of a hand is not a number but a probability distribution of possible resulting hand values they can turn into.

Can't we just map these distributions into numbers based on which is better? How do we decide whose hand is better? Is it the one whose expected value is higher? It turns out the answer is not that simple. Probability distributions cannot be compared as easily as numbers. Specifically let X and Y be two independent random variables. E[X] > E[Y] does not imply X > Y with higher than 1/2 probability. In fact probability distributions cannot even be ordered. So the events X > Y and Y > Z each having Pr > 1/2 does not imply X > Z with Pr > 1/2. Here is an example:

X: 0.5=>20 0.5=>30

Y: 0.6=>10 0.4=>100

Here X beats Y, i.e. P(X>Y)=0.6 even though E[X]=25 and E[Y]=46. Well maybe the mean is misleading and it is the median which decides who's better. Let's add another variable to the example:

Z: 0.4=>0 0.6=>50

Compared to X, Z is better because it wins 0.6 of the time. Compared to Y, Z is worse because it wins 0.36 of the time. So we have three hands with X > Y, Y > Z but Z > X. This means in general distributions cannot be ordered and it is not possible to find a single number (mean median etc.) that will decide which hand beats which.

Also note that the real poker hand distributions are not independent in holdem. Since everybody shares the cards after the first two, the outcomes cannot be represented by independent distributions.

Whether these two facts really matter, or hands can be well represented with an order depends on the details of the poker deck. Must write program to see.

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