Nash Sets Research Articles

Data Poisoning to Fake a Nash Equilibria for Markov Games

We characterize offline data poisoning attacks on Multi-Agent Reinforcement Learning (MARL), where an attacker may change a data set in an attempt to install a (potentially fictitious) unique Markov-perfect Nash equilibrium for a two-player zero-sum Markov game. We propose the unique Nash set, namely the set of games, specified by their Q functions, with a specific joint policy being the unique Nash equilibrium. The unique Nash set is central to poisoning attacks because the attack is successful if and only if data poisoning pushes all plausible games inside it. The unique Nash set generalizes the reward polytope commonly used in inverse reinforcement learning to MARL. For zero-sum Markov games, both the inverse Nash set and the set of plausible games induced by data are polytopes in the Q function space. We exhibit a linear program to efficiently compute the optimal poisoning attack. Our work sheds light on the structure of data poisoning attacks on offline MARL, a necessary step before one can design more robust MARL algorithms.

Unique jet determination of CR maps into Nash sets

Let M⊂CN be a real-analytic CR submanifold, M′⊂CN′ a Nash set and EM′ the set of points in M′ of D'Angelo infinite type. We show that if M is minimal, then, for every point p∈M, and for every pair of germs of C∞-smooth CR maps f,g:(M,p)→M′, there exists an integer k=kp such that if f and g have the same k-jets at p, and do not send M into EM′, then necessarily f=g. Furthermore, the map p↦kp may be chosen to be bounded on compact subsets of M. As a consequence, we derive the finite jet determination property for pairs of germs of CR maps from minimal real-analytic CR submanifolds in CN into Nash subsets in CN′ of D'Angelo finite type, for arbitrary N,N′≥2.

Equisingular algebraic approximation of real and complex analytic germs

We show that a Cohen-Macaulay analytic singularity can be arbitrarily closely approximated by germs of Nash sets which are also Cohen-Macaulay and share the same Hilbert-Samuel function. We also prove that every analytic singularity is topologically equivalent to a Nash singularity with the same Hilbert-Samuel function.

Differentiable approximation of continuous semialgebraic maps

In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are differentiable of class~${\mathcal C}^\nu$ for a fixed integer $\nu\geq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $\nu=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $\nu \geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${\mathcal C}^\nu$ semialgebraically equivalent to a polyhedron, for instance when $T$ is a compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for instance when $T$ is a Nash set. Our density results are based on a recent ${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.

Algebraicity of Nash sets and of their asymmetric cobordism

This paper deals with the existence of algebraic structures on compact Nash sets. We introduce the algebraic-topological notion of asymmetric Nash cobordism between compact Nash sets, and we prove that a compact Nash set is semialgebraically homeomorphic to a real algebraic set if and only if it is asymmetric Nash cobordant to a point or, equivalently, if it is strongly asymmetric Nash cobordant to a real algebraic set. As a consequence, we obtain new large classes of compact Nash sets semialgebraically homeomorphic to real algebraic sets. To prove our results, we need to develop new algebraic-topological approximation procedures. We conjecture that every compact Nash set is asymmetric Nash cobordant to a point, and hence semialgebraically homeomorphic to a real algebraic set.

On the irreducible components of globally defined semianalytic sets

t In this work we present the concept of amenable C-semianalytic subset of a real analytic manifold M and study the main properties of this type of sets. Amenable C-semianalytic sets can be understood as globally defined semianalytic sets with a neat behavior with respect to Zariski closure. This fact allows us to develop a natural definition of irreducibility and the corresponding theory of irreducible components for amenable Csemianalytic sets. These concepts generalize the parallel ones for: complex algebraic and analytic sets, C-analytic sets, Nash sets and semialgebraic sets.

Simultaneous resolution of singularities in the Nash category: finiteness and effectiveness

In this paper we present new proofs using real spectra of the finiteness theorem on Nash trivial simultaneous resolution and the finiteness theorem on Blow-Nash triviality for isolated real algebraic singularities. That is, we prove that a family of Nash sets in a Nash manifold indexed by a semialgebraic set always admits a Nash trivial simultaneous resolution after a partition of the parameter space into finitely many semialgebraic pieces and in the case of isolated singularities it admits a finite Blow-Nash trivialization. We also complement the finiteness results with recursive bounds.

Rétraction capsulaire totale chez un patient avec un syndrome de Steinert

We prove that every complex analytic set X in a Runge domain Ω can be approximated by Nash sets on any relatively compact subdomain Ω0 of Ω. Moreover, for every Nash subset Y of Ω with Y⊂X, the approximating sets can be chosen so that they contain Y∩Ω0. As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.

On Nash approximation of complex analytic sets in Runge domains

We prove that every complex analytic set X in a Runge domain Ω can be approximated by Nash sets on any relatively compact subdomain Ω0 of Ω. Moreover, for every Nash subset Y of Ω with Y⊂X, the approximating sets can be chosen so that they contain Y∩Ω0. As a consequence, we derive a necessary and sufficient condition for a complex analytic set X to admit a Nash approximation which coincides with X along its arbitrary given subset.

Approximation on Nash sets with monomial singularities

This paper is devoted to the approximation of differentiable semialgebraic functions by Nash functions. Approximation by Nash functions is known for semialgebraic functions defined on an affine Nash manifold M, and here we extend it to functions defined on Nash setsX⊂Mwhose singularities are monomial. To that end we discuss first finiteness and weak normality for such sets X. Namely, we prove that (i) X is the union of finitely many open subsets, each Nash diffeomorphic to a finite union of coordinate linear varieties of an affine space, and (ii) every function on X which is Nash on every irreducible component of X extends to a Nash function on M. Then we can obtain approximation for semialgebraic functions and even for certain semialgebraic maps on Nash sets with monomial singularities. As a nice consequence we show that m-dimensional affine Nash manifolds with divisorial corners which are class k semialgebraically diffeomorphic, for k>m2, are also Nash diffeomorphic.

Finiteness problems on Nash manifolds and Nash sets

We study here several finiteness problems concerning affine Nash manifolds M and Nash subsets X . Three main results are: (i) A Nash function on a semialgebraic subset Z of M has a Nash extension to an open semialgebraic neighborhood of Z in M , (ii) A Nash set X that has only normal crossings in M can be covered by finitely many open semialgebraic sets U equipped with Nash diffeomorphisms (u_1,\dots,u_m):U\to\mathbb R^m such that U\cap X=\{u_1\cdots u_r=0\} , (iii) Every affine Nash manifold with corners N is a closed subset of an affine Nash manifold M where the Nash closure of the boundary \partial N of N has only normal crossings and N can be covered with finitely many open semialgebraic sets U such that each intersection N\cap U=\{u_1\ge0,\dots u_r\ge0\} for a Nash diffeomorphism (u_1,\dots,u_m):U\to\mathbb R^m .

Algebraic models of symmetric Nash sets

The aim of this paper is to prove the existence of algebraic models for Nash sets having suitable symmetries. Given a Nash set $$M \subset {\mathbb {R}}^n$$ , we say that $$M$$ is specular if it is symmetric with respect to an affine subspace $$L$$ of $${\mathbb {R}}^n$$ and $$M \cap L=\emptyset $$ . If $$M$$ is symmetric with respect to a point of $${\mathbb {R}}^n$$ , we call $$M$$ centrally symmetric. We prove that every specular compact Nash set is Nash isomorphic to a specular real algebraic set and every specular noncompact Nash set is semialgebraically homeomorphic to a specular real algebraic set. The same is true replacing “specular” with “centrally symmetric”, provided the Nash set we consider is equal to the union of connected components of a real algebraic set. Less accurate results hold when such a union is symmetric with respect to a plane of positive dimension and it intersects that plane. The algebraic models for symmetric Nash sets $$M$$ we construct are symmetric. If the local semialgebraic dimension of $$M$$ is constant and positive, then we are able to prove that the set of birationally nonisomorphic symmetric algebraic models for $$M$$ has the power of continuum.

Rational Generating Functions and Integer Programming Games

We explore the computational complexity of computing pure Nash equilibria for a new class of strategic games called integer programming games, with differences of piecewise-linear convex functions as payoffs. Integer programming games are games where players' action sets are integer points inside of polytopes. Using recent results from the study of short rational generating functions for encoding sets of integer points pioneered by Alexander Barvinok, we present efficient algorithms for enumerating all pure Nash equilibria, and other computations of interest, such as the pure price of anarchy and pure threat point, when the dimension and number of “convex” linear pieces in the payoff functions are fixed. Sequential games where a leader is followed by competing followers (a Stackelberg–Nash setting) are also considered.

A Positive Theory of Income Taxation

We explore the consequences of electoral competition for nonlinear income taxation. Our model is a dynamic version of the standard two-party electoral competition model adapted to nonlinear income taxation. The theory has a number of desirable features. First, equilibria always exist, even though the set of admissible tax policies is multidimensional. Second, the Nash set can be characterized generically, and its components give sharp predictions. Third, the features of equilibrium tax policies depend only on empirically meaningful fundamentals.Equilibrium tax schedules benefit the more numerous income groups and place the burden of taxation on income groups with fewer voters. For empirical income distributions, the features of an equilibrium tax schedule are reminiscent of Director's law of public income redistribution (Stigler [39]).

Games With Complementarities

We introduce a class of games with complementarities that has the quasisupermodular games, hence the supermodular games, as a special case. Our games retain the main property of quasisupermodular games: the Nash set is a nonempty complete lattice. We use monotonicity properties on the best reply that are weaker than those in the literature, as well as pretty simple and linked with an intuitive idea of complementarity. The sufficient conditions on the payoffs are weaker than those in quasisupermodular games. We also separate the conditions implying existence of a greatest and a least Nash equilibrium from those, stronger, implying that the Nash set is a complete lattice.

Some remarks on global real Nash sets

It is proved that every global real Nash set which is not pure dimensional is isomorphic to a real Nash set which is not global. Mathematics Subject Classification: 32C07, 14P20, 32B15

Approximation of analytic sets with proper projection by Nash sets

Let X be an analytic subset of U × C k of pure dimension such that the projection π : X → U is surjective and proper, where U is a Runge domain. We show that X can be approximated by Nash sets. To cite this article: M. Bilski, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Zeta functions and blow-Nash equivalence

We propose a refinement of the notion of blow-Nash equivalence between Nash function germs, which is an analog in the Nash setting of the blow-analytic equivalence defined by T.-C. Kuo. The new definition is more natural and geometric. Moreover, this equivalence relation still does not admit moduli for a Nash family of isolated singularities. Some previous invariants are no longer invariants for this new relation, however, thanks to a Denef & Loeser formula coming from motivic integration in a Nash setting, we managed to derive new invariants for this equivalence relation.

On approximation of analytic sets

We give conditions sufficient for the existence of a sequence of Nash sets convergent to a given analytic set. Moreover, for limit sets which are k-sheeted analytic covers we additionally require that sets in approximating sequence are also k-sheeted analytic covers. Then we present examples of applications of the developed theory.

Uniform bounds on complexity and transfer of global properties of Nash functions

We show that the complexity of semialgebraic sets and mappings can be used to parametrize Nash sets and mappings by Nash families. From this we deduce uniform bounds on the complexity of Nash functions that lead to first-order descriptions of many properties of Nash functions and a good behaviour under real closed field extension (e.g. primary decomposition). As a distinguished application, we derive the solution of the extension and global equations problems over arbitrary real closed fields, in particular over the field of real algebraic numbers. This last fact and a technique of change of base are used to prove that the Artin-Mazur description holds for abstract Nash functions on the real spectrum of any commutative ring, and solve extension and global equations in that abstract setting. To complete the view, we prove the idempotency of the real spectrum and an abstract version of the separation problem. We also discuss the conditions for the rings of abstract Nash functions to be noetherian.