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There is no guarantee that symmetric games have symmetric equilibria. See this paper for concrete examples. There is also no guarantee that symmetric games have pure-strategy equilibria. For example, the following game is symmetric and has no pure-strategy equilibria  \begin{array}{cccc} & \mathrm{a} & \mathrm{b} & \mathrm{c}\\ \mathrm{a} &...
Since the choice of $q_i$ can be conditioned on $(s_i,s_j)$, strategies in this game are of the form $(\hat s_i, \hat q_i(s_i,s_j))$. For the given values $\alpha=3$ and $k=1$, the SPNE can be calculated as the profile where $s^*_i=1/3$ and $q^*_i\equiv 1$. Indeed, production levels $q_i^*=1$ are the unique NE in all subgames, independently of the chosen $... 0 I'll assume that$S$is a constant. By the way, the solution for the SPNE is$q_i = \frac{k}{\alpha - 2}$which is derived from$q_i = (q_j + k)/(\alpha - 1) = ((q_i + k)/(\alpha - 1) + k)/(\alpha - 1)\$. To answer your question, one should know what SPNE does. It is actually a refinement of NE by eliminating non-credible threats. Knowing this, the idea ...