Quasi-Leontief utility functions on partially ordered sets I: efficient points

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.