Since bets are made simultaneously, bluffing is far less valuable.

]]>It was counterintuitive to me that you would just raise randomly below 1/2, instead of bluffing the worst cards in the range (i.e. everything below 1/6).

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Full text of Von Neumann and Morgenstern is online at the Internet Archive / Million Books Project http://archive.org/details/theoryofgamesand030098mbp.

]]>Actually, you are indifferent between the two actions if you draw a 1 (in the limiting case). You get 100 either way, since nobody else is raising.

But you’re right that it’s intuitive once you think about it. I just wouldn’t have guessed it. Von Neumann and Morgenstern argue that there are bounds to raising in real world poker because unbounded raising would eliminate bluffing. But it turns out that unbounded raising also eliminates raising.

]]>Rajiv, I find the limiting case intuitive. You clearly bluff less as the cost of bluffing rises. Also, in the limiting case you’re close to the game restricted to pure strategies, in which case you always benefit from raising slightly less aggressively than your opponent (you save a first-order amount from avoiding losing when raising, while forgoing the wins when raising is 2nd order.)

And to be clear, you do still raise when you draw a 1, though it’s a measure zero outcome.

]]>Yes, this is the von-Neumann Morgenstern solution for the parameters specified here. And as the high bid rises (holding the low bid constant) the threshold above which you raise with probability 1 rises, and the probability of raising below the threshold (bluffing) falls. The limiting solution is a bit counterintuitive: all types bid low with probability 1.

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