Transitive matrices, strict preference and intensity operators

Let X be a set of alternatives and a_{ij} a positive number expressing how much the alternative x_{i} is preferred to the alternative x_{j}. Under suitable hypothesis of no indifference and transitivity over the pairwise comparison matrix A= (a_{ij}), the alternatives can be ordered as a chain . Then a coherent priority vector is a vector giving a weighted ranking agreeing with the obtained chain and an intensity vector is a coherent priority vector encoding information about the intensities of the preferences. In the paper we look for operators F that, acting on the row vectors translate the matrix A in an intensity vector

A general unified framework for pairwise comparison matrices in multicriterial methods

In a Multicriteria Decision Making context, a pairwise comparison matrix $A=(a_{ij})$ is a helpful tool to determine the weighted ranking on a set $X$ of alternatives or criteria. The entry $a_{ij}$ of the matrix can assume different meanings: $a_{ij}$ can be a preference ratio (multiplicative case) or a preference difference (additive case) or $a_{ij}$ belongs to $[0,1]$ and measures the distance from the indifference that is expressed by 0.5 (fuzzy case). For the multiplicative case, a consistency index for the matrix $A$ has been provided by T.L. Saaty in terms of maximum eigenvalue. We consider pairwise comparison matrices over an abelian linearly ordered group and, in this way, we provide a general framework including the mentioned cases. By introducing a more general notion of metric, we provide a consistency index that has a natural meaning and it is easy to compute in the additive and multiplicative cases; in the other cases, it can be computed easily starting from a suitable additive or multiplicative matrix

Ashamed to be Selfish

We study a two-stage choice problem, where alternatives are allocations between the decision maker (DM) and a passive recipient. The recipient observes choice behavior in stage two, while stage one choice is unobserved. Choosing selfishly in stage two, in the face of a fairer available alternative, may inflict shame on DM. DM has preferences over sets of alternatives that represent period two choices. We axiomatize a representation that identifies DM’s selfish ranking, her norm of fairness and shame. Altruism is the most prominent motive that can explain non-selfish choice. We identify a condition under which shame to be selfish can mimic altruism, when only stage-two choice is observed by the experimenter. An additional condition implies that the norm of fairness can be characterized as the Nash solution of a bargaining game induced by the second-stage choice problem. The representation is generalized to allow for finitely many recipients and applied to a simple strategic situation, a game of trust.selfishness, fairness, shame, altruism

Ashamed to be Selfish, Second Version

We study a two-stage choice problem. In the first stage, the decision maker (DM) chooses a set of payoff-allocations between herself and a passive recipient. In the second stage, DM chooses an allocation from the set. The recipient is only aware of the second stage choice. Choosing selfishly in the second stage, in the face of a fairer available alternative, may inflict shame on DM. We axiomatize a representation of DM’s preferences over sets that identifies DM’s selfish ranking, her norm of fairness and shame. It has been suggested that altruism is a prominent motive for non-selfish choice. We identify a condition under which shame to be selfish can mimic altruism, when the experimenter only records the second stage choice. An additional condition implies that the norm of fairness can be characterized as the Nash solution of a bargaining game induced by the second-stage choice problem. The representation is applied to a simple strategic situation, a game of trust.Selfishness, Fairness, Shame, Altruism

Compensation and responsibility

This a chapter for the Handbook of Social Choice and Welfare. It deals with the theory of fairness applied to situations when individuals are partly responsible for their characteristics.fairness, responsibility, equal opportunity, compensation, handicap, talent, effort

Searching by questionaire for the meaning of income inequality

Amien and Cowell (1992) have recently performed an experimental test by questionnaire to investigate whether a sample of individuals corroborate the general consensus found in the literature about a number of axioms on the meaning of 'inequality'. They obtained some mixed results. In this article we report on a replica of the experiment with some novelties: we introudce the role of political attitudes toward income redistribution to clarify the interpretation of some results; the questionnaire is enlarged in an attempt to give more room to notions of inequality imtermediate between the relative and absolute polar cases; and we provide a systematic treatment of the degree of consistency exhibited by the respondents

Aggregation theory and the relevance of some issues to others

I propose a general collective decision problem consisting in many issues that are interconnected in two ways: by mutual constraints and by connections of relevance. Aggregate decisions should respect the mutual constraints, and be based on relevant information only. This general informational constraint has many special cases, including premise-basedness and Arrow''s independence condition; they result from special notions of relevance. The existence and nature of (non-degenerate) aggregation rules depends on both types of connections. One result, if applied to the preference aggregation problem and adopting Arrow''s notion of (ir)relevance, becomes Arrow''s Theorem, without excluding indifferences unlike in earlier generalisations.mathematical economics;