A stationary equilibrium is a Nash equilibrium in which players may condition their actions only on the current state of the game, not on past actions.
(Remark: If all states are payoff-relevant, a stationary equilibrium is often also called Markov perfect equilibrium.)
More formally, a stationary strategy \sigma_i(s) for player i is a function \sigma_i: S \rightarrow \Delta(A_{si}) on the domain of states, mapping state s to a probability distribution \mathbb{P} over state-specific actions A_{si} such that \sigma_i(s,a_{si})=\mathbb{P}(a_{si}|s). A stationary equilibrium is a Nash equilibrium in stationary strategies.
Due to Bellman's principle of optimality, stationary equilibria admit a recursive representation. A stationary strategy profile \boldsymbol{\sigma}=(\sigma_{sia})_{s\in S,i\in I, a\in A_{si}} together with state-player values \boldsymbol{V}=(V_{si})_{s\in S,i\in I} constitutes a stationary equilibrium if and only if
\sigma_{si} \; \in \; \underset{\sigma_{si}\in\Delta(A_{si})}{\arg\max} \;\; V_{si}
\text{s.t. } \quad V_{si} \; = \; u_{si}(\boldsymbol{\sigma}_s) + \delta_i \sum\limits_{s'\in S} \phi_{s\rightarrow s'}(\boldsymbol{\sigma}_s) \, V_{s'i}
for all states s\in S and players i\in I.
Finding a stationary equilibrium amounts to solving the above maximization (which is generally difficult) for equilibrium strategies \boldsymbol{\sigma} (and corresponding values \boldsymbol{V}). The corresponding the necessary and sufficient conditions can be expressed as a (potentially high-dimensional and nonlinear) system of equations. To solve it, sGameSolver relies on a solution method called :doc:`homotopy continuation <homotopy_continuation>`.