Antoine Billot

Axiomatization of an Exponential Similarity Function

An agent is asked to assess a real-valued variable y based on certain characteristics x=(x^{1},...,x^{m}), and on a database consisting of n observations of (x^{1},...,x^{m},y). A possible approach to combine past observations of x and y with the current values of x to generate an assessment of y is similarity-weighted averaging. It suggests that the predicted value of y, y_{n+1}^{s}, be the weighted average of all previously observed values y_{i}, where the weight of y_{i} is the similarity between the vector x_{n+1}^{1},...,x_{n+1}^{m}, associated with y_{n+1}, and the previously observed vector, x_{i}^{1},...,x_{i}^{m}. This paper axiomatizes, in terms of the prediction y_{n+1}, a similarity function that is a (decreasing) exponential in a norm of the difference between the two vectors compared.

Aggregation of preferences: The fuzzy case

The basic purpose of this paper is to link both theorems of impossibility and existence by introducing fuzzy relations of preference and an exogeneous requirement, the planners one, and then proving the fundamental part played by the extremist agents, leximin and leximax. In other words, to bring out the link between the planners requirement and the difficulty of the transition from individual to collective, as well as the theoric relation between this requirement and the extremist agents, we define a fuzzy behavior of preference which allows us to build up two determinant fuzzy coalitions. These coalitions will be the base of the planners requirement and the link between pessimistic results (Arrows impossibility) and optimistic ones (Mays theorem of majority choice).

From fuzzy set theory to non-additive probabilities: how have economists reacted?

Abstract In this paper, we are going to try to conduct a comparative review of the two programmes (devoted to the application of fuzzy sets theory in economics for the first and to nonadditive probabilities for the second) while putting forward some explanations and conjecture as to their development and their future. In Section 1, we defend the thesis which claims that the first programme has only had a global focus on the approach in terms of generalization and we shall go deeper into the analysis of the particular area of General Equilibrium as it is the programme on which Ponsard worked the most. In Section 2, we shall seek to show that the second programme, devoted to the introduction of nonadditive probabilities in decision theory, on the contrary, has attempted to come up with independent findings. This would explain why this last programme has managed to fit into topics at a standstill or in crisis easily (which is undeniably the case of the standard theory of decision making) while the first programme was only envisaged as a kind of mathematical game involving the application of new rules to an old model. In the case of the first programme, one might thus speak of an extension principle and in the second case of a resolution principle .

Epistemic properties of knowledge hierarchies

Abstract After proving correspondence theorems between syntactic and semantic properties of a Kripke Structure, these properties are combined in order to generate two polar semantic structures, that is, partitions and nestings. Then, the relevant properties are extended to Probabilistic Kripke Structures and the concept of a Mixed Knowledge Hierarchy is introduced and associated with two dual measures of events. Finally, an application to Decision Theory is provided for the particular case of a 2-level hierarchy.

How Liberalism Kills Democracy or Sen's Theorem Revisited

First, a general qualification rule based on individualpreferences is proposed which allows any given coalition todistinguish among its members some individuals who are said tobe qualified by it since sensitive in preferences withall individuals of this coalition. A particular qualificationrule, the liberal one, is then introduced as a ruleconferring the power to qualify or disqualify any individualon the individual himself. Now, since each preference propertycorresponds to a qualification property, the liberalqualification rule is precisely characterized by individualpreferences. Second, a delegator, that is a mappingdescribing a subset of delegates within the society, isdefined to justify the standard notion of decisiveness.The idea of delegation allows us to generalize Arrow-Senframework. This is done by means of a collective rationalitypostulate such that a coalition must be competent, i.e.includes all delegates, to be decisive. Then, we prove that(1) there exists a delegate who is qualified by any coalitionhe belongs to, (2) such a delegate is a dictator iff thequalification rule is liberal, (3) the qualification rule isliberal iff the preferences are selfish.

Cognitive Rationality and Alternative Belief Measures

In this article, we first examine the various criticisms of the probabilistic model. Then we introduce capacities in order to show that if a probability measure corresponds to anesthetizing the belief of the agents knowledge, it is then possible to suggest another type of rationality—namely, being able to describe a wise and a rash behavior when facing risk—and therefore another model of belief under uncertainty. While trying to specify various alternative measures, possibility, necessity, and measures resulting from a triangular norm or from a triangular conorm, we finally try to define the field of application of the probabilistic model as well as a sign of the rationality choice: constraint of mass-unity for traditional rationality, and constraint of duality for the one we present.

Fuzzy Convexity and Peripherial Core of an Exchange Economy Represented as a Fuzzy Game

One of the theory of games miracles is to have converted the problem of non emptiness of the core into the convexity of the model (sets and functions). But, what happens if the agents’ fuzzy preferences are not convex? The peripherial core existence is a possible answer. Defined from a new and weaker notion of convexity, it allows to balance an exchange economy approximately if the usual core in empty and it also permits to distinguish betweeen the usual balancing allocations (when they exist), the one which corresponds to the higher social level of satisfaction.

The Deep Side of Preference Theory

An individual is said to have a taste for a particular menu, i.e. a subset of available commodities, if he is indifferent between all commodity bundles that contain the same quantity for each commodity which actually is in the menu, whatever the rest of the bundle. Then, a taste is directly defined as a deep property of preferences. As a first result, it is shown that a complete and transitive preference relation over the commodity bundles is equivalent to regular tastes where regularity means that tastes can be derived from a pure qualitative relation between the different commodities. Besides, a preference family based on preference relations corresponding to each particular commodity is said to be rationalizable if there exists a metapreference over commodity bundles which consistently summarizes the preference family and then allows to decide. As a second result, it is shown that if a preference family is rationalizable, then the tastes are organized thanks to a reflexive and transitive qualitative relation between the different commodities.

Informative Contests and the Efficient Selection of Agents

This paper is based on the tournament model proposed by Arai, Billot & Lanfranchi 1995 [ABL]. Following ABL, a model of candidate selection is constructed in which there exists only limited information about their abilities. In particular, it is assumed that only the reputation earned by candidates in competition with one another is known. Competition is here formalized in terms of contests (or tournaments) which may involve individual and/or coalitions of individuals, and in which only winners are observed. The specific nature of ‘reputation’ is left unspecified, and is treated simply as an abstract measure which is increased monotonely for winners and left unaltered for all others. Here, our key assumption is that the winners’ gain increases with the reputations of the losers.

Aggregation of Paretian preferences for independent individual uncertainties

This paper considers situations of social choice where the resolution of the uncertainty affecting each individual is independent of the resolution of the uncertainty affecting all other individuals. Individuals as well as society itself have Subjective Expected Utility preferences, and society conforms to a set of Pareto-like requirements. In this case, the social utility function must be a convex combination of individual utility functions, thereby extending the logic of Harsanyi’s (JPE 63:309–321, 1955) seminal aggregation theorem. Moreover, the social probabilistic beliefs must be the independent product of individual probabilistic beliefs.