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Register a new userA Primal Condition for Approachability with Partial Monitoring
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Added by
Gilles Stoltz
Publication date
2013
fields
Informatics Engineering
and research's language is
English
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In approachability with full monitoring there are two types of conditions that are known to be equivalent for convex sets: a primal and a dual condition. The primal one is of the form: a set C is approachable if and only all containing half-spaces are approachable in the one-shot game; while the dual one is of the form: a convex set C is approachable if and only if it intersects all payoff sets of a certain form. We consider approachability in games with partial monitoring. In previous works (Perchet 2011; Mannor et al. 2011) we provided a dual characterization of approachable convex sets; we also exhibited efficient strategies in the case where C is a polytope. In this paper we provide primal conditions on a convex set to be approachable with partial monitoring. They depend on a modified reward function and lead to approachability strategies, based on modified payoff functions, that proceed by projections similarly to Blackwells (1956) strategy; this is in contrast with previously studied strategies in this context that relied mostly on the signaling structure and aimed at estimating well the distributions of the signals received. Our results generalize classical results by Kohlberg 1975 (see also Mertens et al. 1994) and apply to games with arbitrary signaling structure as well as to arbitrary convex sets.
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A dynamical system is defined in terms of the gradient of a payoff function. Dynamical variables are of two types, ascent and descent. The ascent variables move in the direction of the gradient, while the descent variables move in the opposite direction. Dynamical systems of this form or very similar forms have been studied in diverse fields such as game theory, optimization, neural networks, and population biology. Gradient descent-ascent is approximated as a Newtonian dynamical system that conserves total energy, defined as the sum of the kinetic energy and a potential energy that is proportional to the payoff function. The error of the approximation is a residual force that violates energy conservation. If the residual force is purely dissipative, then the energy serves as a Lyapunov function, and convergence of bounded trajectories to steady states is guaranteed. A previous convergence theorem due to Kose and Uzawa required the payoff function to be convex in the descent variables, and concave in the ascent variables. Here the assumption is relaxed, so that the payoff function need only be globally `less convex or `more concave in the ascent variables than in the descent variables. Such relative convexity conditions allow the existence of multiple steady states, unlike the convex-concave assumption. When combined with sufficient conditions that imply the existence of a minimax equilibrium, boundedness of trajectories is also assured.
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