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We consider finite two-person normal form games. The following fourproperties 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 begeneralized for $n$-person case with $n > 2$. In 1990, tightness was extendedto vector game forms ($v$-forms) and it was shown that such $v$-tightness andzero-sum-solvability are still equivalent, yet, do not imply Nash-solvability.These results are applicable to several classes of stochastic games withperfect information. Here we suggest one more extension of tightnessintroducing $v^+$-tight vector game forms ($v^+$-forms). We show that such$v^+$-tightness and Nash-solvability are equivalent in case of weaklyrectangular game forms and positive cost functions. This result allows us toreduce the so-called bi-shortest path conjecture to $v^+$-tightness of$v^+$-forms. However, both (equivalent) statements remain open.

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