Approximating n-player behavioural strategy nash equilibria using coevolution

Coevolutionary algorithms are plagued with a set of problems related to intransitivity that make it questionable what the end product of a coevolutionary run can achieve. With the introduction of solution concepts into coevolution, part of the issue was alleviated, however efficiently representing and achieving game theoretic solution concepts is still not a trivial task. In this paper we propose a coevolutionary algorithm that approximates behavioural strategy Nash equilibria in n-player zero sum games, by exploiting the minimax solution concept. In order to support our case we provide a set of experiments in both games of known and unknown equilibria. In the case of known equilibria, we can confirm our algorithm converges to the known solution, while in the case of unknown equilibria we can see a steady progress towards Nash. Copyright 2011 ACM

In Honor of Matthew Rabin: Winner of the John Bates Clark Medal

Although there is some evidence that Matthew Rabin existed before 1990, we had the pleasure of discovering him for ourselves when, in the early 1990s, he sent each of us a copy of his manuscript "Incorporating Fairness into Game Theory and Economics" [2]. Matthew was, at this time, an assistant professor in Berkeley's economics department, having recently finished his graduate training at MIT. The paper was remarkable in many ways, and it induced us both to call around and ask: "Who is this guy Rabin?" Now, just a decade later, we find ourselves writing an article in honor of his winning the John Bates Clark award. So, who is this guy

Analysis and Optimization of Deep Counterfactual Value Networks

Recently a strong poker-playing algorithm called DeepStack was published, which is able to find an approximate Nash equilibrium during gameplay by using heuristic values of future states predicted by deep neural networks. This paper analyzes new ways of encoding the inputs and outputs of DeepStack's deep counterfactual value networks based on traditional abstraction techniques, as well as an unabstracted encoding, which was able to increase the network's accuracy.Comment: Long version of publication appearing at KI 2018: The 41st German Conference on Artificial Intelligence (http://dx.doi.org/10.1007/978-3-030-00111-7_26). Corrected typo in titl

A case study of the integration of ICT in teaching and learning in a smart school in Sabah

This research investigates teachers’ views of their use of ICT in teaching and learning (T&LICT). The objective of this research was to study in depth the thoughts, beliefs and opinions of the teachers’ attempt towards pedagogical improvement as part of the Smart School Project. Specifically this research examines and describes the teachers’ implementation of T&LICT in the classroom in terms of the instructional practice, the instructional roles and the instructional environment. A case study research methodology is employed. The case is Sekolah Menengah Bestari (a psuedonym), which is a Smart School in Sabah. Analysis of data from 52 survey questionnaires complemented the qualitative data from the 13 interviews and 3 observations, as well as document analysis. Findings indicated that hardware and software technology infrastructure were available to support the T&LICT implementation. Nevertheless, the teachers felt it was not enough to implement T&LICT effectively. It was estimated that about half of Sekolah Menengah Bestari staff, mainly Bestari and ETeMS teachers, implemented T&LICT. Findings indicated that teacher practices were little changed. IT was used mainly to support the existing teacher-directed and teacher-centered approach. The role of the teacher extended to that of facilitating without releasing control of lesson to the students

Multiparty Selection

Given a sequence A of n numbers and an integer (target) parameter 1 ? i ? n, the (exact) selection problem is that of finding the i-th smallest element in A. An element is said to be (i,j)-mediocre if it is neither among the top i nor among the bottom j elements of S. The approximate selection problem is that of finding an (i,j)-mediocre element for some given i,j; as such, this variant allows the algorithm to return any element in a prescribed range. In the first part, we revisit the selection problem in the two-party model introduced by Andrew Yao (1979) and then extend our study of exact selection to the multiparty model. In the second part, we deduce some communication complexity benefits that arise in approximate selection. In particular, we present a deterministic protocol for finding an approximate median among k players

Single-Elimination Brackets Fail to Approximate Copeland Winner

Single-elimination (SE) brackets appear commonly in both sports tournaments and the voting theory literature. In certain tournament models, they have been shown to select the unambiguously-strongest competitor with optimum probability. By contrast, we reevaluate SE brackets through the lens of approximation, where the goal is to select a winner who would beat the most other competitors in a round robin (i.e., maximize the Copeland score), and find them lacking. Our primary result establishes the approximation ratio of a randomly-seeded SE bracket is 2^{- Theta(sqrt{log n})}; this is underwhelming considering a 1/2 ratio is achieved by choosing a winner uniformly at random. We also establish that a generalized version of the SE bracket performs nearly as poorly, with an approximation ratio of 2^{- Omega(sqrt[4]{log n})}, addressing a decade-old open question in the voting tree literature