Convex preferences as aggregation of orderings

Arthaniti: Journal Of Economic Theory And Practice, 2018

In this paper, we show that there does not exist any triple acyclic preference aggregation rule that satisfies Majority property, weak Pareto criterion and a version of a property due to Alan Taylor. We also show that there are non-dictatorial preference aggregation rules and in particular non-dictatorial social welfare functions which satisfy the weak Pareto criterion and Taylor's Independence of Irrelevant Alternatives. Further, we are able to obtain analogous results for preference aggregation functionals by suitably adjusting the desired properties to fit into a framework which uses individual utility functions rather than individual preference orderings. Our final result is a modest generalisation of Sen's version of Arrow's impossibility theorem which is shown to hold under our mild domain restriction.

Economic Theory Bulletin, 2018

We construct a complete space of smooth strictly convex preference relations defined over physical commodities and monetary transfers. This construction extends the classical one by assuming that preferences are monotone in transfers, but not necessarily in all commodities. We thereby provide a natural framework to perform genericity analyses in situations involving inventory costs or decisions under risk. The space of preferences we construct is contractible, which allows for a natural aggregation procedure in collective decision situations.

2005

We provide a general theorem on the aggregation of preferences under uncertainty. We study, in the Anscombe-Aumann setting a wide class of preferences, that includes most known models of decision under uncertainty (and state-dependent versions of these models). We prove that aggregation is possible and necessarily linear if (society's) preferences are "smooth". The latter means that society cannot have a non-neutral attitude towards uncertainty on a subclass of acts. A corollary to our theorem is that it is not possible to aggregate maxmin expected utility maximizers, even when they all have the same set of priors. We show that dropping a weak notion of monotonicity on society's preferences allows one to restore the possibility of aggregation of non-smooth preferences.

Journal of Mathematical Economics, 1987

It is shown that if a consumer's preference ordering is strictly convex and is representable by means of a concave, twice continuously differentiable utility function, then the partial derivative of a demanded commodity with respect to its price is bounded from above in a neighborhood of a price vector at which the demand fails to be differentiable.

Working paper series (RGEA), 2008

We consider a continuum of commodities represented by a real interval or, more generally, by a convex set K of any Euclidean space. A consumption plan is a real function defined on the set of commodities K. We show that strictly monotonic preferences on the set of consumption plans always exist. However, they are not representable by utility functions. Moreover, if we consider a consumption set with topological structure, as the positive cone of the Banach space of bounded functions on K, we show that continuous preferences cannot be strictly monotonic.

Order, 2009

We find conditions on the order on subsets of a finite set which are necessary and sufficient for the relative ranking of any two subsets in this order to be determined by their extreme elements relative to an abstract convex geometry. It turns out that this question is closely related to the rationalisability of path independent choice functions by hyper-relations.

Social Choice and Welfare, 1999

An Excess-Voting Function relative to a pro®le p assigns to each pair of alternatives xY y, the number of voters who prefer x to y minus the number of voters who prefer y to x. It is shown that any non-binary separable Excess-Voting Function can be achieved from a preferences pro®le when individuals are endowed with separable preferences. This result is an extension of Hollard and Le Breton (1996). Soc Choice Welfare (1999) 16: 159±167 Many thanks are due to Jean-FrancË ois Laslier and to two referees for their valuable remarks to improve this paper. I would like to thank Basudeb Chaudhuri for his careful reading.

Springer eBooks, 2006

A direct construction of concave utility functions representing convex preferences on finite sets is presented. An alternative construction in which at first directions of supergradients ("prices") are found, and then utility levels and lengths of those supergradients are computed, is exhibited as well. The concept of a least concave utility function is problematic in this context.

Journal of Mathematical Psychology, 2003

We study the topological properties of aggregation maps combining individuals' preferences over n alternatives, with preference expressed by a real-valued, n-dimensional utility vector u defined on an interval scale. Since any such utility vector is specified only up to arbitrary affine transformations, the space of utility vectors R n may be partitioned into equivalence classes of the form fau þ b1 j aAR þ 0 ; bARg: The quotient space, denoted T; is shown to be the union of the n À 2-dimensional sphere S ¼ S nÀ2 with the singleton f0g; which corresponds to indifference or null preference. The topology of T is non-Hausdorff, placing it outside the scope of most existing theory (e.g., J. Econom. Theory 31 (1983) 68). We then investigate the existence and nature of continuous aggregation maps under the four scenarios of allowing or disallowing null preference both in individual and in social choice, i.e. maps f : P Â ? Â P-Q with P; QAfT; Sg: We show that there exist continuous, anonymous, unanimous aggregation maps iff the outcome space includes the null point ðQ ¼ TÞ; and provide a simple well-behaved example for the case f : S Â ? Â S-T: Similar examples exist for f : T Â ? Â T-T; but these and all other maps have a property of always either over-or under-allocating influence to each voter (in a specific manner). We conclude that there exist acceptable aggregation rules if and only if null preference is allowed for the society but not for the individual. r

Mathematical Social Sciences, 1999

A main aim of this paper is to make connections between two well-but up to now independently-developed theories, the theory of choice functions and the theory of closure operators. It is shown that the classes of ordinally rationalizable and path independent choice functions are related to the classes of distributive and anti-exchange closures.