Mathematical Social Sciences 40 2000 313–325 www.elsevier.nl locate econbase
Maximax, leximax, and the demanding criterion
¨ Jorg Naeve
Institute of Mathematical Economics , Bielefeld University, 70376 Bielefeld, Germany
Accepted 1 October 1999
Abstract
` `
Following the characterizations provided by Barbera and Jackson [Barbera, S., Jackson, M.O., 1988. Journal of Economic Theory 46, 34–44] for the maximin, the leximin, and the protective
criterion, we examine the consequences of replacing convexity in their list of axioms by concavity, which means considering risk-loving instead of risk-averse agents. This yields characterizations of
the maximax, the leximax and a new criterion which we term demanding criterion. We concentrate on the latter and demonstrate the independence of the axioms used for its characterization.
2000 Elsevier Science B.V. All rights reserved.
Keywords : Complete uncertainty; Risk-loving agents; Protective criterion
JEL classification : D81
1. Introduction
` The protective criterion first appears in Barbera and Dutta 1982 where it is used in
implementation theory and compared to Moulin’s concept of prudent behavior cf. Moulin, 1981. The same authors employ the protective criterion in two sequel papers,
clarifying the relation of their approach to the general practice in the implementation `
` literature cf. Barbera and Dutta, 1986 and applying it to matching models cf. Barbera
and Dutta, 1995. Recently Fiestras-Janeiro et al. 1998 examine it in the context of `
finite games in strategic form and their mixed extensions. Barbera and Jackson 1988 give an axiomatic characterization of the protective criterion along with a comparison
with minimax and leximax behavior which they also characterize. Our interest in a characterization of the polar criterion which we term demanding
¨ Corresponding address: Darmstadter Str. 40a, D-70376 Stuttgart, Germany. Tel.: 149-711-500-0211.
E-mail address : naevehotmail.com J. Naeve.
0165-4896 00 – see front matter
2000 Elsevier Science B.V. All rights reserved. P I I : S 0 1 6 5 - 4 8 9 6 9 9 0 0 0 5 2 - 9
314 J
. Naeve Mathematical Social Sciences 40 2000 313 –325
behavior stems from its application to implementation in Merlin and Naeve 1998 who are particularly concerned with the use of their results in the context of voting rules.
` The organization of the paper is quite similar to Barbera and Jackson 1988. We first
introduce some notation and define the three criteria. In listing the axioms, we take care that the definition of the axioms make sense without having to invoke other axioms,
which becomes important when we consider their independence. Then we give the characterization results. Though the propositions and their proofs are basically adapta-
` tions of the theorems of Barbera and Jackson 1988, we chose to present them in full,
both to make the exposition easier to follow and to be able to clarify some minor points that are not made explicit in the original proofs. Given the parallelism of our paper to
` Barbera and Jackson’s, our demonstration of the independence of the axioms immedi-
ately yields the same for theirs. In the concluding remarks, we discuss how our criteria `
compare to the ones presented by Barbera and Jackson 1988. As to the interpretation, we prefer to think of the vectors in 5 describing utility
consequences of an action in different states of the world. Indeed, we feel, that certain axioms are less suited for the alternative interpretation as consequences measured in
real numbers a social arrangement has for different members of a society. We defer a more detailed discussion of this point to the concluding remarks.
2. Notation and definitions
