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We study the properties of a subclass of stochastic processes calleddiscrete-time nonlinear Markov chains with an aggregator, which naturallyappear in various topics such as strategic queueing systems, inventorydynamics, opinion dynamics, and wealth dynamics. In these chains, the nextperiod's distribution depends on both the current state and a real-valuedfunction of the current distribution. For these chains, we provide conditionsfor the uniqueness of an invariant distribution that do not rely on typicalcontraction arguments. Instead, our approach leverages flexible monotonicityproperties imposed on the nonlinear Markov kernel. We demonstrate the necessityof these monotonicity conditions in proving the uniqueness of an invariantdistribution through simple examples. We also provide existence results andintroduce an iterative computational method that solves a simpler, tractablesubproblem in each iteration and converges to the stationary distribution ofthe nonlinear Markov chain, even in cases where uniqueness does not hold. Weleverage our findings to analyze invariant distributions in strategic queueingsystems, study inventory dynamics when retailers optimize pricing and inventorydecisions, establish conditions ensuring the uniqueness of solutions for aclass of nonlinear equations in $\mathbb{R}^{n}$, and investigate theproperties of stationary wealth distributions in large dynamic economies.

Invariant Distributions in Nonlinear Markov Chains with Aggregators: Theory, Computation, and Applications