On the saddle-point stability for a class of dynamic games

How to do comparative dynamics on the back of an envelope for open‐loop Nash equilibria in differential game theory

“…As α∈B α ∘ ; ε ð Þis arbitrary, as noted in the proof of Theorem 1, (19) and (20) hold for all α∈B α ∘ ; ε ð Þ. Noting this and substituting (20) in (19) yields (17).…”

“…Dockner and Takahashi [, p. 252], on the other hand, obtained intrinsic results for open‐loop Nash equilibria, as they employed only basic existence and smoothness assumptions in proving that in a wide class of capital accumulation games ‘… saddle‐point stability is the best that we can expect’. The results to follow are equally as basic as those in Dockner and Takahashi , but rather than focusing on the local stability of a steady state, they give the intrinsic comparative dynamics of open‐loop Nash equilibria of a capital accumulation game.…”

The Qualitative Properties of Symmetric Open‐Loop Nash Equilibria in the State‐Control Dynamic System in Differential Games

“…Another sufficient condition for assumption (A.6) to hold is the Hadamard diagonal dominance , namely, , following and is typically assumed to hold globally in the stability literature. As is easily verified, it implies assumption (A.6), but not the converse, thus implying that assumption (A.6) is the weaker of the two.…”

“…Specifically, the investment rate will fall (rise) over time if and only if is less (greater) than zero. Moreover, it is worth mentioning that the term was typically assumed to be less than zero in the capital accumulation literature, for example, , suggesting that the investment rate is to fall at the moment of the discount rate increase, and then gradually decrease to the new steady state.…”

“…, following [24] and is typically assumed to hold globally in the stability literature. As is easily verified, it implies assumption (A.6), but not the converse, thus implying that assumption (A.6) is the weaker of the two.…”