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Register a new userOn the robustness of equilibria in generalized aggregative games
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Added by
Filippo Fabiani
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We address the problem of assessing the robustness of the equilibria in uncertain, multi-agent games. Specifically, we focus on generalized Nash equilibrium problems in aggregative form subject to linear coupling constraints affected by uncertainty with a possibly unknown probability distribution. Within a data-driven context, we apply the scenario approach paradigm to provide a-posteriori feasibility certificates for the entire set of generalized Nash equilibria of the game. Then, we show that assessing the violation probability of such set merely requires to enumerate the constraints that ``shape it. For the class of aggregative games, this results in solving a feasibility problem on each active facet of the feasibility region, for which we propose a semi-decentralized algorithm. We demonstrate our theoretical results by means of an academic example.
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This paper shows the existence of $mathcal{O}(frac{1}{n^gamma})$-Nash equilibria in $n$-player noncooperative aggregative games where the players cost functions depend only on their own action and the average of all the players actions, and is lower semicontinuous in the former while $gamma$-H{o}lder continuous in the latter. Neither the action sets nor the cost functions need to be convex. For an important class of aggregative games which includes congestion games with $gamma$ being 1, a proximal best-reply algorithm is used to construct an $mathcal{O}(frac{1}{n})$-Nash equilibria with at most $mathcal{O}(n^3)$ iterations. These results are applied in a numerical example of demand-side management of the electricity system. The asymptotic performance of the algorithm is illustrated when $n$ tends to infinity.
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We prove that every repeated game with countably many players, finite action sets, and tail-measurable payoffs admits an $epsilon$-equilibrium, for every $epsilon > 0$.
We add here another layer to the literature on nonatomic anonymous games started with the 1973 paper by Schmeidler. More specifically, we define a new notion of equilibrium which we call $varepsilon$-estimated equilibrium and prove its existence for any positive $varepsilon$. This notion encompasses and brings to nonatomic games recent concepts of equilibrium such as self-confirming, peer-confirming, and Berk--Nash. This augmented scope is our main motivation. At the same time, our approach also resolves some conceptual problems present in Schmeidler (1973), pointed out by Shapley. In that paper the existence of pure-strategy Nash equilibria has been proved for any nonatomic game with a continuum of players, endowed with an atomless countably additive probability. But, requiring Borel measurability of strategy profiles may impose some limitation on players choices and introduce an exogenous dependence among players actions, which clashes with the nature of noncooperative game theory. Our suggested solution is to consider every subset of players as measurable. This leads to a nontrivial purely finitely additive component which might prevent the existence of equilibria and requires a novel mathematical approach to prove the existence of $varepsilon$-equilibria.
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