Impossibility Results for Infinite-Electorate Abstract Aggregation Rules

Herzberg, Frederik; Eckert, Daniel

doi:10.1007/s10992-011-9203-5

Cited by 13 publications

(6 citation statements)

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“…A closer study of the relation between finite model properties and the "shape" of some of our axioms might lead to simpler proofs of our axiomatisability results, and might lead to interesting results by using methods from descriptive complexity theory [9]. In this direction, interesting connections with the work of Herzberg and Eckert [17] might be expected.…”

Section: Resultsmentioning

confidence: 88%

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

Grandi

Endriss

2012

J Philos Logic

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In preference aggregation a set of individuals express preferences over a set of alternatives, and these preferences have to be aggregated into a collective preference. When preferences are represented as orders, aggregation procedures are called social welfare functions. Classical results in social choice theory state that it is impossible to aggregate the preferences of a set of individuals under different natural sets of axiomatic conditions. We define a first-order language for social welfare functions and we give a complete axiomatisation for this class, without having the number of individuals or alternatives specified in the language. We are able to express classical axiomatic requirements in our first-order language, giving formal axioms for three classical theorems of preference aggregation by Arrow, by Sen, and by Kirman and Sondermann. We explore to what extent such theorems can be formally derived from our axiomatisations, obtaining positive results for Sen's Theorem and the Kirman-Sondermann Theorem. For the case of Arrow's Theorem, which does not apply in the case of infinite societies, we have to resort to fixing the number of individuals with an additional axiom. In the long run, we hope that our approach to formalisation can serve as the basis for a fully automated proof of classical and new theorems in social choice theory.

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Section: Resultsmentioning

confidence: 88%

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

Grandi

Endriss

2012

J Philos Logic

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“…an expected-utility preference ordering on that set of states of the world), there will be no aggregation rule satisfying the analogues of Arrow's responsiveness axioms, as was shown in Herzberg [23,22]. Note that these impossibility statements can be established both for the case of profiles of a given finite length (the analogue of Arrow's [2] theorem) and for the case of profiles of any given infinite length (the analogue of Campbell's [5] theorem), using a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke [35] and systematically elaborated by Herzberg and Eckert [25,26]. For the special case of the Arrovian finite expected-utility profiles of a given length, this impossibility theorem was proved by Le Breton [36]; an impossibility theorem for the nondictatorial aggregation of expected-utility preferences in a slightly different, yet still very natural setting was established by Hylland and Zeckhauser [27].…”

Section: The Aggregation Of Probability Measuresmentioning

confidence: 90%

Aggregating infinitely many probability measures

Herzberg

2014

Theory Decis

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Abstract. The problem of how to rationally aggregate probability measures occurs in particular (i) when a group of agents, each holding probabilistic beliefs, needs to rationalise a collective decision on the basis of a single 'aggregate belief system' and (ii) when an individual whose belief system is compatible with several (possibly infinitely many) probability measures wishes to evaluate her options on the basis of a single aggregate prior via classical expected utility theory (a psychologically plausible account of individual decisions).We investigate this problem by first recalling some negative results from preference and judgment aggregation theory which show that the aggregate of several probability measures should not be conceived as the probability measure induced by the aggregate of the corresponding expected-utility preferences. We describe how McConway's (Journal of the American Statistical Association, vol. 76, no. 374, pp. 410-414, 1981) theory of probabilistic opinion pooling can be generalised to cover the case of the aggregation of infinite profiles of finitelyadditive probability measures, too; we prove the existence of aggregation functionals satisfying responsiveness axioms à la McConway plus additional desiderata even for infinite electorates. On the basis of the theory of propositional-attitude aggregation, we argue that this is the most natural aggregation theory for probability measures.Our aggregation functionals for the case of infinite electorates are neither oligarchic nor integral-based and satisfy (at least) a weak anonymity condition. The delicate set-theoretic status of integral-based aggregation functionals for infinite electorates is discussed.

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“…The shortest route in proving the above theorems is to invoke recent results from model aggregation theory, due to Herzberg and Eckert (2012a) who generalised previous findings by Lauwers and Van Liedekerke (1995). To employ these results, one needs to reformulate the variational preference aggregation problem as a model aggregation problem (see Appendix B); thereafter, the proofs follow relatively easily from the model aggregation theory in Herzberg and Eckert (2012a) (see Appendix C).…”

Section: Proof Ideamentioning

confidence: 91%

The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences

Herzberg

2013

Econ Theory Bull

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This paper studies collective decision making when individual preferences can be represented by convex risk measures. It addresses the question whether there exist non-dictatorial aggregation functions of convex risk measures satisfying the following Arrow-type rationality axioms: a weak form of universality, systematicity (a strong variant of independence), and the Pareto principle. Herein, convex risk measures are identified with variational preferences on account of the For finite electorates, we prove a variational analogue of Arrow's impossibility theorem. For infinite electorates, the possibility of rational aggregation depends on an equicontinuity condition for the variational preference profiles; we prove a variational analogue of Fishburn's possibility theorem and point to potential analogues of Campbell's impossibility theorem. The proof methodology is based on a model-theoretic approach to aggregation theory inspired by Lauwers and Van Liedekerke (J Math Econ 24(3): 1995). Diverse applications of these results are conceivable, in particular (a) the constitutional design of panels of risk managers and (b) microfoundations for (macroeconomic) multiplier preferences.

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Impossibility Results for Infinite-Electorate Abstract Aggregation Rules

Cited by 13 publications

References 13 publications

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

First-Order Logic Formalisation of Impossibility Theorems in Preference Aggregation

Aggregating infinitely many probability measures

The (im)possibility of collective risk measurement: Arrovian aggregation of variational preferences

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