Finite Choice, Convex Choice and Sorting

Kihara, Taro; Pauly, Arno

doi:10.1007/978-3-030-14812-6_23

Cited by 3 publications

(2 citation statements)

References 28 publications

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“…Since the degrees of AoUC * [0,1] and AoUC ⋄ [0,1] are very similar in many ways, few of the established techniques are available to resolve this situation. The use of the recursion theorem as demonstrated in [24,26] might be possible, but certainly seems very challenging.…”

Section: An Open Question and A Remarkmentioning

confidence: 99%

The Weihrauch degree of finding Nash equilibria in multiplayer games

Crook,

Pauly

2021

Preprint

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Is there an algorithm that takes a game in normal form as input, and outputs a Nash equilibrium? If the payoffs are integers, the answer is yes, and lot of work has been done in its computational complexity. If the payoffs are permitted to be real numbers, the answer is no, for continuity reasons. It is worthwhile to investigate the precise degree of non-computability (the Weihrauch degree), since knowing the degree entails what other approaches are available (eg, is there a randomized algorithm with positive success change?). The two player case has already been fully classified, but the multiplayer case remains open and is addressed here. Our approach involves classifying the degree of finding roots of polynomials, and lifting this to systems of polynomial inequalities via cylindrical algebraic decomposition.ACM classification: Theory of computation-Computability; Mathematics of computing-Topology; Mathematics of computing-Nonlinear equations * This work is supported by the UKRI AIMLAC CDT, cdt-aimlac.org, grant no. EP/S023992/1. 1 The PPAD-completeness result from [20] is for ε-Nash equilibria, not for actual Nash equilibria.

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Section: An Open Question and A Remarkmentioning

confidence: 99%

The Weihrauch degree of finding Nash equilibria in multiplayer games

Crook,

Pauly

2021

Preprint

Self Cite

View full text Add to dashboard Cite

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“…Since the degrees of AoUC * [0,1] and AoUC ⋄ [0,1] exhibit significant similarities in various aspects, only a few of the established techniques are available to resolve this situation. While the use of the recursion theorem as demonstrated in [86,88] might be possible, it certainly presents a considerable challenge.…”

Section: An Open Question and A Remarkmentioning

confidence: 99%

Computable Analysis and Game Theory: From Foundations to Applications

Crook

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This body of research showcases several facets of the intersection between computer science and game theory. On the foundational side, we explore the obstructions to the computability of Nash equilibria in the setting of computable analysis. In particular, we study the Weihrauch degree of the problem of ﬁnding a Nash equilibrium for a multiplayer game in normal form. We conclude that the Weihrauch degree Nash for multiplayer games lies between AoUC∗[0,1] and AoUC⋄[0,1] (Theorem 5.3). As a slight detour, we also explore the demarcation between computable and non-computable computational problems pertaining to the veriﬁcation of machine learning. We demonstrate that many veriﬁcation questions are computable without the need to specify a machine learning framework (Section 7.2). As well as looking into the theory of learners, robustness and sparisty of training data. On the application side, we study the use of Hypergames in Cybersecurity. We look into cybersecurity AND/OR attack graphs and how we could turn them into a hypergame (8.1). Hyper Nash equilibria is not an ideal solution for these games, however, we propose a regret-minimisation based solution concept. In Section 8.2, we survey the area of Hypergames and their connection to cybersecurity, showing that even if there is a small overlap, the reach is limited. We suggest new research directions such as adaptive games, generalisation and transferability (Section 8.3).

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Parallelizations in Weihrauch Reducibility and Constructive Reverse Mathematics

Fujiwara

2020

Lecture Notes in Computer Science

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Finite Choice, Convex Choice and Sorting

Cited by 3 publications

References 28 publications

The Weihrauch degree of finding Nash equilibria in multiplayer games

The Weihrauch degree of finding Nash equilibria in multiplayer games

Computable Analysis and Game Theory: From Foundations to Applications

Parallelizations in Weihrauch Reducibility and Constructive Reverse Mathematics

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