77 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Approximating Nash Equilibria in Normal-Form Games via Stochastic Optimization
We propose the first, to our knowledge, loss function for approximate Nash
equilibria of normal-form games that is amenable to unbiased Monte Carlo
estimation. This construction allows us to deploy standard non-convex
stochastic optimization techniques for approximating Nash equilibria, resulting
in novel algorithms with provable guarantees. We complement our theoretical
analysis with experiments demonstrating that stochastic gradient descent can
outperform previous state-of-the-art approaches
Geometry of the sets of Nash equilibria in mixed extensions of finite games
The theory of strategic games in normal form is a part of game theory. The most important solution concept for them is the notion of a Nash equilibrium. Nash defined it and proved that the mixed extension of any finite game has Nash equilibria. Here the space in which the Nash equilibria live is a product of simplices, namely a product of spaces of probability distributions, each over a finite set of pure strategies. The existence leads to questions on the shape of the set of all Nash equilibria for a given game. In this thesis we concentrate on generic games. There it is well-known that the number of Nash equilibria is finite and odd. It is interesting to think about the maximal number of Nash equilibria in generic games with fixed number of players and fixed finite sets of pure strategies. In general, the precise number is unknown. But in the case of 2 players, there are
good upper and lower bounds, which are not so far apart. In the case of m ≥ 3 players, up to now only an upper bound was known. In the case of m players each of whom has exactly two pure strategies, we present a lower bound, which is surprisingly close to the known upper bound. It is more than half of the upper bound. This result was the outcome of a mixture of conceptual and calculational steps. We present more calculational results for such games. We also study with computer a certain 2-person game where each player has six pure strategies. One chapter recalls a good part of the history of the problem. The penultimate chapter works out an old foundational result on the union of the sets of mixed Nash equilibria for all games with fixed player set and fixed finite sets of pure strategies. The second chapter presents a stronger result on generic games than can be found in the literature. The product of simplices embeds naturally into a product of real projective spaces. The equalities and inequalities for Nash equilibria make sense in this bigger space. In the case of generic games all involved hypersurfaces are smooth and maximally transversal in this bigger space
Tropical Positivity and Semialgebraic Sets from Polytopes
This dissertation presents recent contributions in tropical geometry with a view towards positivity, and on certain semialgebraic sets which are constructed from polytopes.
Tropical geometry is an emerging field in mathematics, combining elements of algebraic geometry and polyhedral geometry. A key in establishing this bridge is the concept of tropicalization, which is often described as mapping an algebraic variety to its 'combinatorial shadow'. This shadow is a polyhedral complex and thus allows to study the algebraic variety by combinatorial means. Recently, the positive part, i.e. the intersection of the variety with the positive orthant, has enjoyed rising attention. A driving question in recent years is: Can we characterize the tropicalization of the positive part?
In this thesis we introduce the novel notion of positive-tropical generators, a concept which may serve as a tool for studying positive parts in tropical geometry in a combinatorial fashion. We initiate the study of these as positive analogues of tropical bases, and extend our theory to the notion of signed-tropical generators for more general signed tropicalizations. Applying this to the tropicalization of determinantal varieties, we develop criteria for characterizing their positive part. Motivated by questions from optimization, we focus on the study of low-rank matrices, in particular matrices of rank 2 and 3. We show that in rank 2 the minors form a set of positive-tropical generators, which fully classifies the positive part. In rank 3 we develop the starship criterion, a geometric criterion which certifies non-positivity. Moreover, in the case of square-matrices of corank 1, we fully classify the signed tropicalization of the determinantal variety, even beyond the positive part.
Afterwards, we turn to the study of polytropes, which are those polytopes that are both tropically and classically convex. In the literature they are also established as alcoved polytopes of type A. We describe methods from toric geometry for computing multivariate versions of volume, Ehrhart and h^*-polynomials of lattice polytropes. These algorithms are applied to all polytropes of dimensions 2,3 and 4, yielding a large class of integer polynomials. We give a complete combinatorial description of the coefficients of volume polynomials of 3-dimensional polytropes in terms of regular central subdivisions of the fundamental polytope, which is the root polytope of type A. Finally, we provide a partial characterization of the analogous coefficients in dimension 4.
In the second half of the thesis, we shift the focus to study semialgebraic sets by combinatorial means. Intersection bodies are objects arising in geometric tomography and are known not to be semialgebraic in general. We study intersection bodies of polytopes and show that such an intersection body is always a semialgebraic set. Computing the irreducible components of the algebraic boundary, we provide an upper bound for the degree of these components. Furthermore, we give a full classification for the convexity of intersection bodies of polytopes in the plane.
Towards the end of this thesis, we move to the study of a problem from game theory, considering the correlated equilibrium polytope of a game G from a combinatorial point of view. We introduce the region of full-dimensionality for this class of polytopes, and prove that it is a semialgebraic set for any game. Through the use of oriented matroid strata, we propose a structured method for classifying the possible combinatorial types of , and show that for (2 x n)-games, the algebraic boundary of each stratum is a union of coordinate hyperplanes and binomial hypersurfaces. Finally, we provide a computational proof that there exists a unique combinatorial type of maximal dimension for (2 x 3)-games.:Introduction
1. Background
2. Tropical Positivity and Determinantal Varieties
3. Multivariate Volume, Ehrhart, and h^*-Polynomials of Polytropes
4. Combinatorics of Correlated Equilibri
Computer Aided Verification
This open access two-volume set LNCS 13371 and 13372 constitutes the refereed proceedings of the 34rd International Conference on Computer Aided Verification, CAV 2022, which was held in Haifa, Israel, in August 2022. The 40 full papers presented together with 9 tool papers and 2 case studies were carefully reviewed and selected from 209 submissions. The papers were organized in the following topical sections: Part I: Invited papers; formal methods for probabilistic programs; formal methods for neural networks; software Verification and model checking; hyperproperties and security; formal methods for hardware, cyber-physical, and hybrid systems. Part II: Probabilistic techniques; automata and logic; deductive verification and decision procedures; machine learning; synthesis and concurrency. This is an open access book
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum
Probabilistic model checking for strategic equilibria-based decision making:advances and challenges
Game-theoretic concepts have been extensively studied in economics to provide insight into competitive behaviour and strategic decision making. As computing systems increasingly involve concurrently acting autonomous agents, game-theoretic approaches are becoming widespread in computer science as a faithful modelling abstraction. These techniques can be used to reason about the competitive or collaborative behaviour of multiple rational agents with distinct goals or objectives. This paper provides an overview of recent advances in developing a modelling, verification and strategy synthesis framework for concurrent stochastic games implemented in the probabilistic model checker PRISM-games. This is based on a temporal logic that supports finite- and infinite-horizon temporal properties in both a zero-sum and nonzero-sum setting, the latter using Nash and correlated equilibria with respect to two optimality criteria, social welfare and social fairness. We summarise the key concepts, logics and algorithms and the currently available tool support. Future challenges and recent progress in adapting the framework and algorithmic solutions to continuous environments and neural networks are also outlined
Essays on Economics and Computer Science
146 pagesThis dissertation considers a number of problems in pure and applied game theory. The first chapter considers the problem of how the introduction of fines and monitoring affects welfare in a routing game. I characterize equilibria of the game and discuss network topologies in which the introduction of fines can harm those agents which are not subject to them. The second, and primary, chapter considers the computational aspects of tenable strategy sets. I characterize these set-valued solution concepts using the more familiar framework of perturbed strategies, introduce strong alternatives to the problems of verifying whether a strategy block satisfies the conditions of tenability, and provide some hardness results regarding the verification of fine tenability. Additionally, I show an inclusion relation between the concept of coarse tenability and the notion of stability introduced by Kohlberg and Mertens (1986). Finally, I show how the methods developed for tenability provide an alternative characterization for proper equilibria in bimatrix games. This characterization gives a bound on the perturbations required in the definition of proper equilibria, though such bounds cannot be computed efficiently in general. The third, and final, chapter develops a model of contracting for content creation in an oligopolistic environment of attention intermediaries. I characterize symmetric equilibria in single-homing (exclusive) and multi-homing regimes. The focus is on the trade-off between reductions in incentives offered by intermediaries and the benefits of access to additional content for consumers. I show that when the extent of multi-homing is exogenous in the absence of exclusivity clauses, consumer surplus is always higher with multi-homing than under exclusivity, despite weaker incentives offered by platforms to content creators
Search and optimization with randomness in computational economics: equilibria, pricing, and decisions
In this thesis we study search and optimization problems from computational economics with primarily stochastic inputs. The results are grouped into two categories: First, we address the smoothed analysis of Nash equilibrium computation. Second, we address two pricing problems in mechanism design, and solve two economically motivated stochastic optimization problems.
Computing Nash equilibria is a central question in the game-theoretic study of economic systems of agent interactions. The worst-case analysis of this problem has been studied in depth, but little was known beyond the worst case. We study this problem in the framework of smoothed analysis, where adversarial inputs are randomly perturbed. We show that computing Nash equilibria is hard for 2-player games even when input perturbations are large. This is despite the existence of approximation algorithms in a similar regime. In doing so, our result disproves a conjecture relating approximation schemes to smoothed analysis. Despite the hardness results in general, we also present a special case of co-operative games, where we show that the natural greedy algorithm for finding equilibria has polynomial smoothed complexity. We also develop reductions which preserve smoothed analysis.
In the second part of the thesis, we consider optimization problems which are motivated by economic applications. We address two stochastic optimization problems. We begin by developing optimal methods to determine the best among binary classifiers, when the objective function is known only through pairwise comparisons, e.g. when the objective function is the subjective opinion of a client. Finally, we extend known algorithms in the Pandora's box problem --- a classic optimal search problem --- to an order-constrained setting which allows for richer modelling.
The remaining chapters address two pricing problems from mechanism design. First, we provide an approximately revenue-optimal pricing scheme for the problem of selling time on a server to jobs whose parameters are sampled i.i.d. from an unknown distribution. We then tackle the problem of fairly dividing chores among a collection of economic agents via a competitive equilibrium, which balances assigned tasks with payouts. We give efficient algorithms to compute such an equilibrium
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