Discrete convex analysis: A tool for economics and game theory

A survey of fundamental operations on discrete convex functions of various kinds

“…The reader is referred to [10], [31,Section 3.4], and [36,Section 13] for more about integral convexity, and [24] and [26] for recent developments.…”

On the partnership formation problem

“…We finally present the so-called two-phase algorithms for finding an equilibrium (see Murota et al (2016); see also Murota (2016)). A two-phase algorithm consists of the ascending phase and the descending phase, and the price vector is increasing in the ascending phase and then decreasing in the descending phase.…”

“…For example, the combination of ASCENDMINIMAL and DESCENDMAXIMAL yields the following algorithm (Murota (2016); Murota et al (2016)). …”

“…The ascending phase stops at the vectorq that is the unique minimal minimizer of the Lyapunov function in the region {q | q ≥ q • }, and the descending phase stops at the vectorq that is the unique maximal minimizer of the Lyapunov function in the region {q | q ≤q}, which turns out to be a global minimizer of the Lyapunov function, i.e., an equilibrium price vector (Murota (2016); Murota et al (2016)). A.3 (Murota (2016); Murota et al (2016)).…”

“…It can be shown that the worst case bound for the number of iterations required by the algorithms explained above is proportional to max{w(i, j) | i ∈ A, j ∈ B} (see, e.g., Andersson & Erlanson (2013);Murota (2016); Murota et al (2016)). This means that the number of iterations required by the algorithms is pseudo-polynomial in the terminology of algorithm theory.…”