Hi,
Mathematica finds numerical solutions to your problem. I pick only the 1st in the list of solutions.
sol2 = Solve[{v1n == ArgMax[f1[r, v1, v2n, x1star, x2star], v1],
v2n == ArgMax[f2[r, v1n, v2, x1star, x2star], v2]}, {v1n,
v2n}][[1, 1]]
Since I am not familiar with the Nash equation I took arbitrary numerical values (my choice probably does not make sense in the context of your problem):
sol2 /. { sk -> 2, sk2 -> 3.7, x1star -> 3, x2star -> 4, sa -> 0.7,
sa2 -> 0.9, beta -> 0.6, r -> 1, Km -> 1.}
which results in:
v1n -> 1.16268
In order to explore a larger space you can use this:
prmLst = { sk, sk2, x1star, x2star, sa , sa2, beta , r, Km};
tab = Map[
Flatten[{#,
Quiet[Check[sol2 /. Thread[prmLst -> #],
Text["Convergence failed",
BaseStyle -> {Red, FontFamily -> "Times", Italic}]]]}] &,
RandomReal[{-1, 1}, {50, Length[prmLst]}]];
TableForm[tab, TableHeadings -> {None, Join[prmLst, {"solution"}]},
TableDepth -> 2]
For some parameter sets Solve does not converge (see table entries)