[jtr]

[ QUOTE ]

Averaging is only useful if the mean of the things you are averaging equals the accurate value you're trying to get. In this case, the value we're trying to get is our expectation with respect to villain's hand.

There's absoloutly no rigorous reason to believe that the expectation for a number of hands averages to the expectation for the hand villain actually has.

So in fact there's a subtle problem with the process above and beyond any problems of execution.

[/ QUOTE ]

This looks pretty dodgy to me. Would you care to give an answer to a simple thought experiment, just to see if we are on the same page?

Suppose you are given an opportunity to choose either

A) a guaranteed payoff of $100

or

B) a chance to choose from one of three boxes, with each box having a different amount of money inside it. Let's say the amounts are $50, $80, and $200.

In fact, if you take option B you don't actually choose the box, but it is chosen for you by someone else having randomly pre-selected one of the three boxes. Your choice is simply to take either A) the $100 payout, or B) one of the three boxes.

Presumably you see where I'm going with this. On the assumption that we are interested in maximizing our expected value, what is wrong with averaging across the payouts of the three boxes in order to figure out whether we prefer option A or option B?

You appear to have a strange take on probability if you want to insist that the only expected value that matters is the one for the opponent's actual holding. Certainly if we knew the opponent's actual holding then we'd make our calculations based on that, but your position equates to someone who insists, in my little experiment, that whether they should choose option A or B depends on which box has been pre-selected under option B. The whole point is that we do not have information that would allow us to distinguish between the $50, $80, and $200 cases for option B. Given this, the best thing we can do is to average over them.

You have built up a bit of a straw man, I think, by suggesting that many people when faced with poker decisions will do the equivalent of throwing in some more possibilities for the cash amounts in the boxes under option B. If I weirdly suppose that instead of the three equiprobable boxes as stated, there is a fourth box that has only $5, or that the box with $50 is much more likely than the others to be chosen, then certainly this will lead to me incorrectly assessing the expected value of option B. However, my error has nothing to do with the process of averaging across expected values, and everything to do with plugging in bad information at the beginning of that process.

In, summary, I can't understand when you've gone after averaging when all you really mean to criticize is poor information gathering and parameter estimation when putting people on hand ranges.