A FURTHER GENERALIZATION OF THE KAKUTANI FIXED POINT THEOREM, WITH APPLICATION TO NASH EQUILIBRIUM POINTS

By I. Glicksberg. 1952

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Citations in the corpus

Randomized Prediction Games for Adversarial Machine Learning (2016, methodology): “Proof: The result follows directly from the DebreuGlicksberg-Fan Theorem [29].”
A Decentralized Market with Common Values Uncertainty: Non-Steady States (2001, unknown): “Hence θ meets the conditions of the Fan–Glicksberg fixed point theorem (Fan (1952), Glicksberg (1952)), and there exists φ∈Φ 6 such that φ∈ θ(φ).”
Two-Player Games for Efficient Non-Convex Constrained Optimization (2018, background): “Frank-Wolfe, as in Hazan and Kale (2012); Jaggi (2013); Garber and Hazan (2013)).”
Price-Matching Guarantees (2006, unknown): “A mixed-strategy equilibrium exists per Glicksberg's (1952) theorem, since the payoff functions of”
Non-Cooperative Games with Many Players (2002, background): “…theorem on the compactness of the product of an arbitrary set of compact spaces, and rely on a suitable extension of Kakutani’s fixed point theorem.17 The upper semicontinuity result in Fan (1952), and the fixed point theorem in Fan (1952) and Glicksberg (1952) furnish these technical supplements.”
Game Theory in Wireless and Communication Networks (2008, background): “2 [167, 141, 125] Given a non-cooperative game in strategic form (N , (Si )i∈N , (ui )i∈N ), if ∀i ∈N , every strategy setSi is compact and convex, ui (si ,s−i ) is a continuous function in the profile of strategies s ∈ S and quasi-concave in si , then the game has at least one pure-strategy Nash eq”
Probabilistic Voting in the Spatial Model of Elections: The Theory of Office-motivated Candidates (2005, methodology): “Proof. Continuity of the candidates' objective functions follows as in the proof of Theorem 1. Since the objective functions are quasi-concave, Ghckberg's [ 14 ] theorem can be applied in the space of platform pairs to deliver a pure strategy equilibrium.”
What is Local Optimality in Nonconvex-Nonconcave Minimax Optimization? (2019, background): “Unlike pure strategy Nash equilibrium, the existence of mixed strategy Nash equilibrium in this setting is always guaranteed Glicksberg [1952]. One challenge for finding mixed strategy equilibria is that it requires optimizing over a space of probability measures, which is of infinite dimension.”
A Bargaining Model of Collective Choice (2000, unknown): “Define the correspondence B: [P(X)]n →→ [P(X)]n by B(π) = (B1(π), . . . , Bn(π)); since [P(X)]n is compact and convex, by Glicksberg (1952) B has a fixed point, say π∗ = (π∗1, . . . , π ∗ n).”
On the Characterization of Local Nash Equilibria in Continuous Games (2014, background): “In particular, the mixed strategies are probability measures over the pure strategies [28].”
Power control in two-tier femtocell networks (2008, methodology): “We now employ the following theorem from Glicksberg [38], Rosen [39] and Debreu [40]: Theorem 2: A Nash equilibrium exists in gameG = [N , {Pi}, {Ui(.)}] if, for all i = 0, 1, . . . , N , 1) Pi is a nonempty, convex and compact subset of some Euclidean spaceRN+1.”
Minmax Optimization: Stable Limit Points of Gradient Descent Ascent are Locally Optimal (2019, unknown): “Proposition 12 (Glicksberg [1952]) .”