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In this paper, we consider a new class of generalized Convex structure and we investigate their tropical limits. Some properties are pointing out such that translation homotheticity and others ones allowing to consider the case of discrete production sets that are related to some specific dual forms. Along this line a general class of mathematical programs are derived and it is shown that they can be computed using standard methods. The proposed approach allows to deal with efficiency measures (output oriented or input oriented) on continuous and discrete data.

This paper proposes to estimate the returns-to-scale of production sets by considering the individual return of each observed firm through the notion of $\Lambda$-returns to scale assumption. Along this line, the global technology is then constructed as the intersection of all the individual technologies. Hence, an axiomatic foundation is proposed to present the notion of $\Lambda$-returns to scale. This new characterization of the returns-to-scale encompasses the definition of $\alpha$-returns to scale, as a special case as well as the standard non-increasing and non-decreasing returns-to-scale models. A non-parametric procedure based upon the goodness of fit approach is proposed to assess these individual returns-to-scale. To illustrate this notion of $\Lambda$-returns to scale assumption, an empirical illustration is provided based upon a dataset involving 63 industries constituting the whole American economy over the period 1987-2018., Comment: 40 pages, 18 figures

This paper introduces in production theory a large class of efficiency measures that can be derived from the notion of utility function. This article also establishes a relation between these distance functions and Stone-Geary utility functions. More specifically, the paper focusses on new distance function that generalizes several existing efficiency measures. The new distance function is inspired from the Atkinson inequality index and maximizes the sum of the netput expansions required to reach an efficient point. A generalized duality theorem is proved and a duality result linking the new distance functions and the profit function is obtained. For all feasible production vectors, it includes as special cases most of the dual correspondences previously established in the literature. Finally, we identify a large class of measures for which these duality results can be obtained without convexity.

This paper considers an idempotent and symmetrical algebraic structure as well as some closely related concept. A special notion of determinant is introduced and a Cramer formula is derived for a class of limit systems derived from the Hadamard matrix product and we give the algebraic form of a sequence of hyperplanes passing through a finite number of points. Thereby, some standard results arising for Max-Times systems with nonnegative entries appear as a special case. The case of two sided systems is also analyzed. In addition, a notion of eigenvalue in limit is considered. It is shown that one can construct a special semi-continuous regularized polynomial to find the eigenvalues of a matrix with nonnegative entries.

$\mathbb B$-convexity was defined in [7] as a suitable Kuratowski-Painlev\'e upper limit of linear convexities over a finite dimensional Euclidean vector space. Excepted in the special case where convex sets are subsets of $\mathbb R^n_ +$, $\mathbb B$-convexity was not defined with respect to a given explicit algebraic structure. This is done in that paper, which proposes an extension of $\mathbb B$-convexity to the whole Euclidean vector space. An unital idempotent and non-associative magma is defined over the real set and an extended $n$-ary operation is introduced. Along this line, the existence of the Kuratowski-Painlev\'e limit of the convex hull of two points over $\mathbb R^n$ is shown and an explicit extension of $\mathbb B$-convexity is proposed.

We prove that, under appropriate conditions, an abstract game with quasi-Leontief payoff functions $u_i : \prod_{j=1}^nX_j\to\mathbb{R}$ has a Nash equilibria. When all the payoff functions are globally quasi-Leontief, the existence and the characterization of efficient Nash equilibria mainly follows from the analysis carried out in part I. When the payoff functions are individually quasi-Leontief functions the matter is somewhat more complicated. We assume that all the strategy spaces are compact topological semilattices, and under appropriate continuity conditions on the payoff functions, we show that there exists an efficient Nash equilibria using the Eilenberg-Montgomery Fixed Point Theorem for acyclic valued upper semicontinuous maps defined on an absolute retract and some non trivial properties of topological semilattices. The map in question is defined on the set of Nash equilibria and its fixed points are exactly the efficient Nash equilibria.

A function $u: X\to\mathbb{R}$ defined on a partially ordered set is quasi-Leontief if, if for all $x\in X$, the upper level set $\{x^\prime\in X: u(x^\prime)\geqslant u(x)\} $ has a smallest element. A function $u: \prod_{j=1}^nX_j\to\mathbb{R}$ whose partial functions obtained by freezing $n-1$ of the variables are all quasi-Leontief is an individually quasi-Leontief function; a point $x$ of the product space is an efficient point for $u$ if it is a minimal element of $\{x^\prime\in X: u(x^\prime)\geqslant u(x)\} $. Part I deals with the maximisation of quasi-Leontief functions and the existence of efficient maximizers. Part II is concerned with the existence of efficient Nash equilibria for abstract games whose payoff functions are individually quasi-Leontief. Order theoretical and algebraic arguments are dominant in the first part while, in the second part, topology is heavily involved. In the framework and the language of tropical algebras, our quasi-Leontief functions are the additive functions defined on a semimodule with values in the semiring of scalars.