Antonio Quesada

Monotonicity and the Hirsch index

The Hirsch index is a number that synthesizes a researchers output. It is the maximum number h such that the researcher has h papers with at least h citations each. Woeginger [Woeginger, G. J. (2008a). An axiomatic characterization of the Hirsch-index. Mathematical Social Sciences, 56(2), 224–232; Woeginger, G. J. (2008b). A symmetry axiom for scientific impact indices. Journal of Informetrics, 2(3), 298–303] characterizes the Hirsch index when indices are assumed to be integer-valued. In this note, the Hirsch index is characterized, when indices are allowed to be real-valued, by adding to Woegingers monotonicity two axioms in a way related to the concept of monotonicity.

More axiomatics for the Hirsch index

The Hirsch index is a number that synthesizes a researcher’s output. It is defined as the maximum number h such that the researcher has h papers with at least h citations each. Woeginger (Math Soc Sci 56: 224–232, 2008a; J Informetr 2: 298–303, 2008b) suggests two axiomatic characterizations of the Hirsch index using monotonicity as one of the axioms. This note suggests three characterizations without adopting the monotonicity axiom.

Further characterizations of the Hirsch index

The Hirsch index is a number that synthesizes a researcher’s output. It is defined as the maximum number h such that the researcher has h papers with at least h citations each. Four characterizations of the Hirsch index are suggested. The most compact one relies on the interpretation of the index as providing the number of valuable papers in an output and postulates three axioms. One, only cited papers can be valuable. Two, the index is strongly monotonic: if output x has more papers than output y and each paper in x has more citations than the most cited paper in y, then x has more valuable papers than y. And three, the minimum amount of citations under which a paper becomes valuable is different for each paper.

Axiomatics for the Hirsch index and the Egghe index

The Hirsch index and the Egghe index are both numbers that synthesize a researchers output. The h-index associated with researcher r is the maximum number h such that r has h papers with at least h citations each. The g-index is the maximum number g of papers by r such that the average number of citations of the g papers is at least g. Both indices are characterized in terms of four axioms. One identifies outputs deserving index at most one. A second one establishes a strong monotonicity condition. A third one requires the index to satisfy a property of subadditivity. The last one consists of a monotonicity condition, for the h-index, and an aggregate monotonicity condition, for the g-index.

Positional independence in preference aggregation

Abstract. If, for strict preferences, a unique choice function (CF) is used to aggregate preferences position-wise then the resulting social welfare function (SWF) is dictatorial. This suggests that the task performed by non-dictatorial SWFs must be “more complex” than just selecting an alternative from a list using a single criterion. This is because the information required by non-dictatorial SWFs to aggregate preferences cannot be compressed into a CF. It is also shown that the attempt to reduce the working of a SWF to the working of a CF involves the adoption of certain positional requirements, whose relationship with the conditions in Arrows theorem is established.

From Common Knowledge of Rationality to Backward Induction

With the only assumption that a player knows the strategy he chooses, it is proved in a generalized version of Aumanns (1995) epistemic model that, in a generic game with perfect information, common knowledge of rationality is equivalent to common knowledge of the fact that the backward induction strategy profile is chosen. This result shows that Aumanns backward induction theorem holds without stipulating partition knowledge structures nor presuming that the epistemic operator defines knowledge in the strict sense.

The majority rule with a chairman

For the case of two alternatives and a given finite set of at least three individuals, seven axioms are shown to characterize the rules that are either the relative majority rule or the relative majority in which a given individual, the chairman, can always break ties. An axiomatization of the relative majority rules with a chairman is suggested that holds for an even number of individuals and that, for an odd number of individuals, characterizes the rules that are either the relative majority rule or a relative majority rule with a chairman.

Monotonicity + efficiency + continuity = majority

Axioms of monotonicity, efficiency and continuity are shown to characterize the relative majority rule when there are only two alternatives. The absolute majority rule and the relative majority rule in which indifferences are resolved following some given tie-breaking rule are also characterized using those axioms. The strategy followed in these two characterizations consists of: (i) identifying a domain D where the relative majority rule coincides with the characterized rule; and (ii) making the value of elements not in D coincide with the value of some element in D.

Abstention as an escape from Arrow's theorem

There are non-dictatorial social welfare functions satisfying the Pareto principle and Arrows independence of irrelevant alternatives when voters can abstain. In particular, with just seven voters, the number of dictatorial social welfare functions satisfying Arrows conditions could be deemed, relative to the total number of social welfare functions satisfying Arrows conditions, negligible.

Belief system foundations of backward induction

Two justifications of backward induction (BI) in generic perfect information games are formulated using Bonannos (1992; Theory and Decision 33, 153) belief systems. The first justification concerns the BI strategy profile and is based on selecting a set of rational belief systems from which players have to choose their belief functions. The second justification concerns the BI path of play and is based on a sequential deletion of nodes that are inconsistent with the choice of rational belief functions.