On Nash-Solvability of Finite Two-Person Tight Vector Game Forms

TL;DR

This paper explores the properties of finite two-person game forms, establishing new equivalences involving Nash-solvability and tightness, and introduces a new class of vector game forms ($v^+$-forms) with implications for longstanding conjectures.

Contribution

It introduces $v^+$-tightness for vector game forms and proves its equivalence to Nash-solvability under certain conditions, extending the theory of game form properties.

Findings

$v^+$-tightness is equivalent to Nash-solvability for weakly rectangular forms with positive costs.

The results connect to the bi-shortest path conjecture, reducing it to $v^+$-tightness.

Established that tightness and Nash-solvability are equivalent in specific game form classes.

Abstract

We consider finite two-person normal form games. The following four properties of their game forms are equivalent: (i) Nash-solvability, (ii) zero-sum-solvability, (iii) win-lose-solvability, and (iv) tightness. For (ii, iii, iv) this was shown by Edmonds and Fulkerson in 1970. Then, in 1975, (i) was added to this list and it was also shown that these results cannot be generalized for $n$-person case with $n > 2$. In 1990, tightness was extended to vector game forms ($v$-forms) and it was shown that such $v$-tightness and zero-sum-solvability are still equivalent, yet, do not imply Nash-solvability. These results are applicable to several classes of stochastic games with perfect information. Here we suggest one more extension of tightness introducing $v^+$-tight vector game forms ($v^+$-forms). We show that such $v^+$-tightness and Nash-solvability are equivalent in case of weakly rectangular game forms and positive cost functions. This result allows us to reduce the so-called bi-shortest path conjecture to $v^+$-tightness of $v^+$-forms. However, both (equivalent) statements remain open.