The Use of Symmetry for Models with Variable-size Variables

Takeshi Fukasawa

Papers from arXiv.org

Abstract: This paper shows the universal representations of symmetric functions with multidimensional variable-size variables, which help assessing the justification of approximation methods aggregating the information of each variable by moments. It then discusses how the results give insights into game theoretic applications, including two-step policy function estimation, Moment-based Markov Equilibrium (MME), and aggregative games. Regarding policy function estimations, it is justifiable to estimate a common policy function as a function of own firm's states and the sums of polynomial terms (moments) of competitors' states under some conditions, regardless of the number of firms in each market, as long as the number of moments is sufficiently large. Concerning the MME, this study shows that MME is equivalent to the Markov Perfect Equilibrium if the number of moments reaches a certain level and regularity conditions are satisfied. Regarding aggregative games, we can easily show that any games satisfying a condition of symmetry and continuity of payoff functions can be represented as multidimensional generalized aggregative, which introduces multidimensional aggregates in the generalized (fully) aggregative games previous studies have intensively studied.

Date: 2023-11, Revised 2024-10
New Economics Papers: this item is included in nep-gth
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