Learning Convex Partitions and Computing Game-theoretic Equilibria from Best Response Queries

Suppose that an $m$-simplex is partitioned into $n$ convex regions having disjoint interiors and distinct labels, and we may learn the label of any point by querying it. The learning objective is to know, for any point in the simplex, a label that occurs within some distance $\epsilon$ from that point. We present two algorithms for this task: Constant-Dimension Generalised Binary Search (CD-GBS), which for constant $m$ uses $poly(n, \log \left( \frac{1}{\epsilon} \right))$ queries, and Constant-Region Generalised Binary Search (CR-GBS), which uses CD-GBS as a subroutine and for constant $n$ uses $poly(m, \log \left( \frac{1}{\epsilon} \right))$ queries. We show via Kakutani's fixed-point theorem that these algorithms provide bounds on the best-response query complexity of computing approximate well-supported equilibria of bimatrix games in which one of the players has a constant number of pure strategies. We also partially extend our results to games with multiple players, establishing further query complexity bounds for computing approximate well-supported equilibria in this setting.Comment: 38 pages, 7 figures, second version strengthens lower bound in Theorem 6, adds footnotes with additional comments and fixes typo

Multiobjective analysis for the design and control of an electromagnetic valve actuator

The electromagnetic valve actuator can deliver much improved fuel efficiency and reduced emissions in spark ignition (SI) engines owing to the potential for variable valve timing when compared with cam-operated, or conventional, variable valve strategies. The possibility exists to reduce pumping losses by throttle-free operation, along with closed-valve engine braking. However, further development is required to make the technology suitable for accept- ance into the mass production market. This paper investigates the application of multiobjective optimization techniques to the conflicting objective functions inherent in the operation of such a device. The techniques are utilized to derive the optimal force–displacement characteristic for the solenoid actuator, along with its controllability and dynamic/steady state performance

An Empirical Study of Finding Approximate Equilibria in Bimatrix Games

While there have been a number of studies about the efficacy of methods to find exact Nash equilibria in bimatrix games, there has been little empirical work on finding approximate Nash equilibria. Here we provide such a study that compares a number of approximation methods and exact methods. In particular, we explore the trade-off between the quality of approximate equilibrium and the required running time to find one. We found that the existing library GAMUT, which has been the de facto standard that has been used to test exact methods, is insufficient as a test bed for approximation methods since many of its games have pure equilibria or other easy-to-find good approximate equilibria. We extend the breadth and depth of our study by including new interesting families of bimatrix games, and studying bimatrix games upto size $2000 \times 2000$. Finally, we provide new close-to-worst-case examples for the best-performing algorithms for finding approximate Nash equilibria

Not Just a Theory—The Utility of Mathematical Models in Evolutionary Biology

Models have made numerous contributions to evolutionary biology, but misunderstandings persist regarding their purpose. By formally testing the logic of verbal hypotheses, proof-of-concept models clarify thinking, uncover hidden assumptions, and spur new directions of study. thumbnail image credit: modified from the Biodiversity Heritage Librar

Evidence of Skyrmion excitations about ν=1\nu =1 in n-Modulation Doped Single Quantum Wells by Inter-band Optical Transmission

We observe a dramatic reduction in the degree of spin-polarization of a two-dimensional electron gas in a magnetic field when the Fermi energy moves off the mid-point of the spin-gap of the lowest Landau level, $\nu=1$. This rapid decay of spin alignment to an unpolarized state occurs over small changes to both higher and lower magnetic field. The degree of electron spin polarization as a function of $\nu$ is measured through the magneto-absorption spectra which distinguish the occupancy of the two electron spin states. The data provide experimental evidence for the presence of Skyrmion excitations where exchange energy dominates Zeeman energy in the integer quantum Hall regime at $\nu=1$

Randomized Revenue Monotone Mechanisms for Online Advertising

Online advertising is the main source of revenue for many Internet firms. A central component of online advertising is the underlying mechanism that selects and prices the winning ads for a given ad slot. In this paper we study designing a mechanism for the Combinatorial Auction with Identical Items (CAII) in which we are interested in selling $k$ identical items to a group of bidders each demanding a certain number of items between $1$ and $k$. CAII generalizes important online advertising scenarios such as image-text and video-pod auctions [GK14]. In image-text auction we want to fill an advertising slot on a publisher's web page with either $k$ text-ads or a single image-ad and in video-pod auction we want to fill an advertising break of $k$ seconds with video-ads of possibly different durations. Our goal is to design truthful mechanisms that satisfy Revenue Monotonicity (RM). RM is a natural constraint which states that the revenue of a mechanism should not decrease if the number of participants increases or if a participant increases her bid. [GK14] showed that no deterministic RM mechanism can attain PoRM of less than $\ln(k)$ for CAII, i.e., no deterministic mechanism can attain more than $\frac{1}{\ln(k)}$ fraction of the maximum social welfare. [GK14] also design a mechanism with PoRM of $O(\ln^2(k))$ for CAII. In this paper, we seek to overcome the impossibility result of [GK14] for deterministic mechanisms by using the power of randomization. We show that by using randomization, one can attain a constant PoRM. In particular, we design a randomized RM mechanism with PoRM of $3$ for CAII

Paroxysmal eye–head movements in Glut1 deficiency syndrome

Objective:To describe a characteristic paroxysmal eye–head movement disorder that occurs in infants with Glut1 deficiency syndrome (Glut1 DS).Methods:We retrospectively reviewed the medical charts of 101 patients with Glut1 DS to obtain clinical data about episodic abnormal eye movements and analyzed video recordings of 18 eye movement episodes from 10 patients.Results:A documented history of paroxysmal abnormal eye movements was found in 32/101 patients (32%), and a detailed description was available in 18 patients, presented here. Episodes started before age 6 months in 15/18 patients (83%), and preceded the onset of seizures in 10/16 patients (63%) who experienced both types of episodes. Eye movement episodes resolved, with or without treatment, by 6 years of age in 7/8 patients with documented long-term course. Episodes were brief (usually &lt;5 minutes). Video analysis revealed that the eye movements were rapid, multidirectional, and often accompanied by a head movement in the same direction. Eye movements were separated by clear intervals of fixation, usually ranging from 200 to 800 ms. The movements were consistent with eye–head gaze saccades. These movements can be distinguished from opsoclonus by the presence of a clear intermovement fixation interval and the association of a same-direction head movement.Conclusions:Paroxysmal eye–head movements, for which we suggest the term aberrant gaze saccades, are an early symptom of Glut1 DS in infancy. Recognition of the episodes will facilitate prompt diagnosis of this treatable neurodevelopmental disorder.</jats:sec

Design of a fault tolerant airborne digital computer. Volume 2: Computational requirements and technology

This final report summarizes the work on the design of a fault tolerant digital computer for aircraft. Volume 2 is composed of two parts. Part 1 is concerned with the computational requirements associated with an advanced commercial aircraft. Part 2 reviews the technology that will be available for the implementation of the computer in the 1975-1985 period. With regard to the computation task 26 computations have been categorized according to computational load, memory requirements, criticality, permitted down-time, and the need to save data in order to effect a roll-back. The technology part stresses the impact of large scale integration (LSI) on the realization of logic and memory. Also considered was module interconnection possibilities so as to minimize fault propagation

Speeding up shortest path algorithms

Given an arbitrary, non-negatively weighted, directed graph $G=(V,E)$ we present an algorithm that computes all pairs shortest paths in time $\mathcal{O}(m^* n + m \lg n + nT_\psi(m^*, n))$, where $m^*$ is the number of different edges contained in shortest paths and $T_\psi(m^*, n)$ is a running time of an algorithm to solve a single-source shortest path problem (SSSP). This is a substantial improvement over a trivial $n$ times application of $\psi$ that runs in $\mathcal{O}(nT_\psi(m,n))$. In our algorithm we use $\psi$ as a black box and hence any improvement on $\psi$ results also in improvement of our algorithm. Furthermore, a combination of our method, Johnson's reweighting technique and topological sorting results in an $\mathcal{O}(m^*n + m \lg n)$ all-pairs shortest path algorithm for arbitrarily-weighted directed acyclic graphs. In addition, we also point out a connection between the complexity of a certain sorting problem defined on shortest paths and SSSP.Comment: 10 page