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Does anybody know how to interpret the sentence: For any set $T$ of mixed strategies, let $D[T]$ denote the set of probability distributions over the elements of $T$, each expressed as vector, conformable to a mixed strategy, that gives the ultimate distribution of pure strategies.

This sentence appeared at p.139 of the paper

Crawford, V.P. Equilibrium without independence, Journal of Economic Theory 50 (1990), 127-154.

The pdf is available online (just google).

In particular, what I really don't understand is the last part: What's the meaning of being conformable to a mixed strategy? More seriously: what's the ultimate distribution of pure strategies?

Many thanks in advance for any help,


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My interpretation: You identify a finite set of strategies $S$ with the unit vectors in $\mathbb{R}^{|S|}$, so that the simplex spanned becomes the space of mixed strategies. So every set of mixed strategies is a subset of this simplex. A probability distribution over $T$ induces again a probability distribution over $S$, so the set of probability distributions over the elements can again be identified with a subset of the simplex.

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Yes I think I got it! Thank you. – Valerio Capraro Oct 18 '12 at 18:47

Ultimately gives a distribution on pure strategies. Papers in economics can be wordy. A mixed strategy over mixed strategies of course is a mixed strategy. A probability distribution over probability distributions over a set of states induces probability distribution on those states.

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