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Can someone explain intuitively, the Block-Marschack Polynomials that completely characterize random utility.

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    $\begingroup$ Do you know of the paper where they are first defined? Or, do you have a reference to any paper that uses them? $\endgroup$ – jmbejara Jan 13 '15 at 3:43
  • $\begingroup$ @jmbejara the original work was done in Block, Henry David, and Jacob Marschak. "Random orderings and stochastic theories of responses." Contributions to probability and statistics 2 (1960): 97-132. See my anwer below for a modern notation based on McFadden. $\endgroup$ – TemplateRex Jan 14 '15 at 11:15
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Given a universe of choices $X$, subsets of alternatives $A \subseteq X$, and choice probabilities $\Pi_{A}(x)$ for item $x \in A \subseteq X$, the Block-Marschak polynomials can be defined recursively as (see McFadden, Revealed stochastic preference: a synthesis, 2005) as

$K_{x, \emptyset} = \Pi_{X}(x) \,, \quad K_{x, A} = \Pi_{X\setminus A}(x) - \sum_{C \subset A} K_{x, C}$

When the choice probabilities are the result of utility maximization, Barbara and Pattanaik (Falmagne and the rationalizability of stochastic choices in terms of random orderings, 1986) provide a useful and intuitive interpretation of $K_{x,A}$ as the probability of the event that $x$ is ranked behind the elements of $A$ and ahead of all the remaining elements in $X \setminus A$.

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  • $\begingroup$ Thanks for the nice answer, the interpretation is still difficult since it is still given in terms of the representation. I would have liked an IIA or Sen's $alpha$, $\beta$ axioms/characterization. $\endgroup$ – user157623 Jan 14 '15 at 20:55
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    $\begingroup$ @user157623 I find the probability of being ranked between elements of $A$ and $X \setminus A$ rather intuitive. BTW, the recursive definition can also be solved using Mobius inversion. The algebraic structure is very similar to the Shapley value in cooperative game theory, where instead of choice sets and marginal probabilities you have coalitions and marginal contributions. The Shapley value gives the average marginal value to all coalitions. Perhaps you can interpret these polynomials as the average choice probability among all choice sets. $\endgroup$ – TemplateRex Jan 14 '15 at 21:15

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