[EM] Introduction to "Distribution of power under stochastic social choice rules" Pattanaik and Peleg, Econometrica Vol 54 No 4

[David Catchpole]

Distribution of power under stochastic social choice rules
Prasanta K. Pattanaik and Bezalel Peleg
Econometrica Vol 54 No 4

Abstract: This paper considers stochastic social choice rules which, for
every feasible set of alternatives and every profile of individual
orderings, specify social choice probabilities for the feasible
alternatives. It is shown that if such a social choice rule satisfies:

	(i)	a probabilistic counterpart of Arrow's independence of
		irrelevant alternatives;
	(ii)	ex-post Pareto optimality; and
	(iii)	"regularity" (a "rationality" property postulating that
		given the individual preference orderings, if the feasible set of
		alternatives is expanded, then the social choice probability for an
		initially feasible alternative cannot increase);

then the power structure under it is almost completely characterised by
weighted random dictatorship.

Introduction:

Probabilistic social choice rules have recieved a considerable amount of
attention in the theory of social choice. Partly, this has been motivated
by the desire to escape the impossibility results generated in the
deterministic framework of social choice theory. However, the
probabilistic framework also provides many plauisible rules for
aggregating individual preferences which are interesting in their own
right, and which provide scope for incorporating certain notions of
fairness and reasonable compromise. For example, in a situation of
conflict of preferences, one could use a lottey under which the different
individual's preferences have equal probability of emerging as the
society's preference. This is, of course, a simple version of the well
known method of random dictatorship. Despite the name, it is arguable that
random dictatorship avoids the undesireable features of dictatorship of a
deterministic variety. Indeed, on closer scrutiny random dictatorship
seems to have several attractive features some of which are connected with
our intuitive notion of fairness. One can think of many other interesting
probabilistic group decision rules. For example, Barbera (1978,
1979)[Review of Economic Studies, 46 - Journal of Economic Theory, 18] has
discussed what he calls point voting decision schemes and supporting size
schemes. Under the point voting decision scheme an alternative recieves a
certain score gk whenever it receives the kth position in some
individual's ordering. One then calculates the total weight assigned to an
alternative by summing up all the scores received by it. The scores, i.e.
the gk's, are chosen in such a way that the total weights assigned to
different alternatives constitute a probability distribution over the
alternatives, and this probability distribution constitutes the
probabilistic basis of social choice. Under the supporting size scheme an
alternative x is given a score hk when k individuals prefer x to some
other alternative y. Then each alternative is assigned a total weight
equal to the sum of all the scores received by that alternative in the
different pairwise comparisons where it is involved. The scores, i.e. the
hk's, are chosen in such a way that the total weights assigned to
different alternatives constitute a probability distribution on the basis
of which social choice is made. The point voting decision scheme and the
supporting size scheme can be regarded as embodying, respectively, the
positional voting principle and the majority principle in the
probabilistic formulation of the problem of social choice. These as well
as other examples to be found in the literature show that the class of
probabilistic group decision rules have considerable richness and appeal,
and deserve detailed exploration.

The purpose of this paper is to investigate the structure of probabilistic
social decision procedures which satisfy three plausible properties:
independence of irrelevant alternatives, ex-post Pareto optimality and
regularity. Independence of irrelevant alternatives here requires that
given a feasible set B, if the individual preference orderings over B
remain the same, then the lottery on the basis of which the society makes
choice from B should also remain the same even though individual
preferences may have changed otherwise. This is clearly the probabilistic
counterpart of Arrow's (1964)[I think you all know the one] independence
of irrelevant alternatives. Ex-post Pareto optimality requires that if
everybody prefers x to y, then given that x is available, y must have zero
probability of being socially chosen. Regularity implies that given the
profile of individual preferences, if one enlarges the feasible set of
alternatives by adding more alternatives, then the probability of the
society's choosing any of the alternatives figuring in the original
feasible set cannot increase after the feasible set is enlarged. This
seems to be the natural probabilistic counterpart of Sen's
(1970)[_Collective Choice and Social Welfare_ San Fransisco: Holden Day] 
condition alpha in the deterministic framework, which requires that
anything rejected in a smaller feasible set cannot be selected in a larger
feasible set. We show that these three assumptions, together with the
assumption that the universal set of alternatives has at least four
elements and that individual preference orderings are strict, imply that
for every proper subset of the universal set of alternatives, the
probabilistic social decision procedure must take the form of random
dictatorship in the following sense: for each individual i, there will be
a "weight" or a nonnegative number alphai (the sum of all these numbers
being equal to 1) such that for every proper subset A of the universal set
of alternatives and for every profile of individual preferences, the
probability of the society's choosing an alternative x from A will be the
sum of the weights attached to all individuals for whom x is best in A in
terms of their preferences in the given preference profile. This random
dictatorship result can be extended to the universal set of alternatives
as well as every proper subset of it, if the cardinality of the universal
set of alternatives exceeds the number of individuals by at least two.
While our assumptions lead to random dictatorship, we do not view our
results as "impossibility results." This is because, as we have remarked
earlier, it is not at all clear that random dictatorship is necessarily
undesirable in a way in which dictatorship of a deterministic variety, for
example, can be considered undesireable.

It is of interest to view our results from the perspective of a line of
investigation intitiated by Barbera and Sonnenschein (1978)[Oh, I give
up...] and subsequently pursued by a number of writers (Bandyopadhyay, Deb
and Pattanaik, 1982; Barbera and Valenciano, 1983; Heiner and Pattanaik,
1983; and McLennan, 1980). While these contributions of Barbera and
Sonnenschein and others, which investigate the structure of coalitional
power under probabilistic social decision rules, vary somewhat in their
formal structures and their assumptions, one basic feature is common to
all of them. The restrictions on coalitional power derived in all of them
relate to the power of coalitions to influence social choice probabilities
over two-element feasible sets only. In the classical deterministic
framework of Arrow (1964), also, the power of "decisiveness" of coalitions
referred to the ability of coalitions to influence social preferences or
choice over pairs of alternatives but in Arrow's framework, given the
assumption of collective rationality, restrictions on coalitional power
over sets with more than two alternatives follow naturally from the
relevant restrictions for pairs of alternatives. This, however, is not
true of the probabilistic social decision rules considered in the papers
referred to above. Within the framework adopted by these writers, it is
not clear what if any restrictions on the power of coalitions to influence
social choice probabilities for large sets can be derived from the
restrictions on coalitional power established for two-element feasible
sets. Therefore, the question naturally asires whether it is possible to
characterise the structure of coalitional power under probabilistic social
decision rules without confining the analysis to two-element feasible sets
only. The results of our paper, which we have outlined above, show that
this is indeed possible. For the case of linear individual orderings our
results provide an almost complete characterisation of the power structure
for probabilistic social decision procedures under assumptions which are
similar to (though not identical with) the assumptions used in many of the
contributions cited earlier (formally, the main price paid for our
characterisation result is the use of the general version of independence
of irrelevant alternatives instead of the weaker pairwise version used in
the literature cited earlier).

In Section 2 [S. 1 was the introduction] we introduce the notion of
decision schemes and some properties of decision schemes. In Section 3 we
discuss the properties of what we call probabilistic voting procedures. In
Section 4 we prove our main results (Theorems 4.11 and 4.14) and in
Section 5 we give some examples to illustrate the relevance of certain
assumptions figuring in our main results. We conclude in Section 6.

[Whew!]

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