The Use of Symmetry for Models with Variable-size Variables

Abstract: This paper presents a universal representation of symmetric (permutation-invariant) functions with multidimensional variable-size variables. These representations help justify approximation methods that aggregate information from each variable using moments. It further discusses how these findings provide insights into game-theoretic applications, including two-step policy function estimation, Moment-based Markov Equilibrium (MME), and aggregative games. Regarding policy function estimation, under certain conditions, estimating a common policy function as a function of a firm's own state and the sum of polynomial terms (moments) of competitors' states is justified, regardless of the number of firms in a market, provided a sufficient number of moments are included. For MME, this study demonstrates that MME is equivalent to Markov Perfect Equilibrium if the number of moments reaches a certain level and regularity conditions are satisfied. Regarding aggregative games, the paper establishes that any game satisfying symmetry and continuity conditions in payoff functions can be represented as a multidimensional generalized aggregative game. This extends previous research on generalized (fully) aggregative games by introducing multidimensional aggregates.