4 research outputs found
On the Convergence of Learning Algorithms in Bayesian Auction Games
Equilibrium problems in Bayesian auction games can be described as systems of
differential equations. Depending on the model assumptions, these equations
might be such that we do not have a rigorous mathematical solution theory. The
lack of analytical or numerical techniques with guaranteed convergence for the
equilibrium problem has plagued the field and limited equilibrium analysis to
rather simple auction models such as single-object auctions. Recent advances in
equilibrium learning led to algorithms that find equilibrium under a wide
variety of model assumptions. We analyze first- and second-price auctions where
simple learning algorithms converge to an equilibrium. The equilibrium problem
in auctions is equivalent to solving an infinite-dimensional variational
inequality (VI). Monotonicity and the Minty condition are the central
sufficient conditions for learning algorithms to converge to an equilibrium in
such VIs. We show that neither monotonicity nor pseudo- or quasi-monotonicity
holds for the respective VIs. The second-price auction's equilibrium is a
Minty-type solution, but the first-price auction is not. However, the
Bayes--Nash equilibrium is the unique solution to the VI within the class of
uniformly increasing bid functions, which ensures that gradient-based
algorithms attain the {equilibrium} in case of convergence, as also observed in
numerical experiments
On the Existence of Nash Equilibrium in Bayesian Games *
Abstract We furnish conditions on the primitives of a Bayesian game that guarantee the existence of a BayesNash equilibrium. We consider spaces of distributional and behavioral strategies, and highlight the relationship between their respective topologies, which implies that the two spaces are mutually interchangeable throughout the analysis. By allowing for payoff discontinuities in actions, we cover various applications that cannot be handled by extant results
A Service of zbw Leibniz-Informationszentrum Wirtschaft Leibniz Information Centre for Economics On the existence of Nash Equilibrium in Bayesian Games On the Existence of Nash Equilibrium in Bayesian Games *
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On the Existence of Nash Equilibrium in Bayesian Games Keywords: Bayesian game • discontinuous game • infinite game of incomplete information • behavioral strategy • distributional strategy • payoff security
Abstract. We furnish conditions on the primitives of a Bayesian game that guarantee the existence of a Bayes-Nash equilibrium. By allowing for payoff discontinuities in actions, we cover various applications that cannot be handled by extant results