The Complexity of All-switches Strategy Improvement

Strategy improvement is a widely-used and well-studied class of algorithms for solving graph-based infinite games. These algorithms are parameterized by a switching rule, and one of the most natural rules is "all switches" which switches as many edges as possible in each iteration. Continuing a recent line of work, we study all-switches strategy improvement from the perspective of computational complexity. We consider two natural decision problems, both of which have as input a game $G$, a starting strategy $s$, and an edge $e$. The problems are: 1.) The edge switch problem, namely, is the edge $e$ ever switched by all-switches strategy improvement when it is started from $s$ on game $G$? 2.) The optimal strategy problem, namely, is the edge $e$ used in the final strategy that is found by strategy improvement when it is started from $s$ on game $G$? We show $\mathtt{PSPACE}$-completeness of the edge switch problem and optimal strategy problem for the following settings: Parity games with the discrete strategy improvement algorithm of V\"oge and Jurdzi\'nski; mean-payoff games with the gain-bias algorithm [14,37]; and discounted-payoff games and simple stochastic games with their standard strategy improvement algorithms. We also show $\mathtt{PSPACE}$-completeness of an analogous problem to edge switch for the bottom-antipodal algorithm for finding the sink of an Acyclic Unique Sink Orientation on a cube

Morphology of coronal mass ejections between the sun and the earth

The theme of my PhD has been to investigate the global shape and size of coronal mass ejections, or CMEs, as they propagate from the Sun towards the Earth. CMEs are large eruptive events originating from previously magnetically confined structures in the solar atmosphere. These phenomena are the single biggest drivers for geomagnetic disturbances at Earth. My research is focused on analysing spacecraft data obtained both by imaging observations and in situ instrumentation. The three pieces of work presented in this thesis are summarised below: Using the NASA STEREO mission, launched in 2006, I have analysed data from the Heliospheric Imager (HI) instruments. This new instrument is uniquely positioned to observe CMEs as they propagate away from the Sun into the inner heliosphere between 0.1 and 1 AU. Using this data I have been able to estimate the radial expansion of a single CME as it propagates in the inner heliosphere. Investigating another case study event seen by STEREO-B in November 2007, I have been able to show that the distortion of a CME can be directly attributed to a structured solar wind. By using a 3D MHD simulation of the solar wind in the vicinity of the CME, it has been shown that a bimodal velocity structure within this solar wind was driving the CME from behind and distorting it from a circular to a concave morphology. Using in situ data, I have also attempted to deduce the shape of CMEs in the inner heliosphere. To do this I analysed the shock wave driven ahead of the propagating CME, applying a technique previously used to predict the distance of the shock upstream of Earth’s magnetosphere - this distance can be predicted when the object’s shape (Earth) is known. I have carried out a statistical survey of many CMEs over a range of distances from the Sun, and compared them to theoretical predictions of their shape based on geometry

Computing Approximate Nash Equilibria in Polymatrix Games

In an $\epsilon$-Nash equilibrium, a player can gain at most $\epsilon$ by unilaterally changing his behaviour. For two-player (bimatrix) games with payoffs in $[0,1]$, the best-known$\epsilon$ achievable in polynomial time is 0.3393. In general, for $n$-player games an $\epsilon$-Nash equilibrium can be computed in polynomial time for an $\epsilon$ that is an increasing function of $n$ but does not depend on the number of strategies of the players. For three-player and four-player games the corresponding values of $\epsilon$ are 0.6022 and 0.7153, respectively. Polymatrix games are a restriction of general $n$-player games where a player's payoff is the sum of payoffs from a number of bimatrix games. There exists a very small but constant $\epsilon$ such that computing an $\epsilon$-Nash equilibrium of a polymatrix game is \PPAD-hard. Our main result is that a $(0.5+\delta)$-Nash equilibrium of an $n$-player polymatrix game can be computed in time polynomial in the input size and $\frac{1}{\delta}$. Inspired by the algorithm of Tsaknakis and Spirakis, our algorithm uses gradient descent on the maximum regret of the players. We also show that this algorithm can be applied to efficiently find a $(0.5+\delta)$-Nash equilibrium in a two-player Bayesian game

The Complexity of the Homotopy Method, Equilibrium Selection, and Lemke-Howson Solutions

We show that the widely used homotopy method for solving fixpoint problems, as well as the Harsanyi-Selten equilibrium selection process for games, are PSPACE-complete to implement. Extending our result for the Harsanyi-Selten process, we show that several other homotopy-based algorithms for finding equilibria of games are also PSPACE-complete to implement. A further application of our techniques yields the result that it is PSPACE-complete to compute any of the equilibria that could be found via the classical Lemke-Howson algorithm, a complexity-theoretic strengthening of the result in [Savani and von Stengel]. These results show that our techniques can be widely applied and suggest that the PSPACE-completeness of implementing homotopy methods is a general principle.Comment: 23 pages, 1 figure; to appear in FOCS 2011 conferenc

Finding Nash equilibria of bimatrix games

This thesis concerns the computational problem of finding one Nash equilibrium of a bimatrix game, a two-player game in strategic form. Bimatrix games are among the most basic models in non-cooperative game theory, and finding a Nash equilibrium is important for their analysis. The Lemke—Howson algorithm is the classical method for finding one Nash equilib-rium of a bimatrix game. In this thesis, we present a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labelled facets in d-space. The construc-tion is extended to two classes of non-square games where, in addition to exponentially long Lemke—Howson computations, finding an equilibrium by support enumeration takes exponential time on average. The Lemke—Howson algorithm, which is a complementary pivoting algorithm, finds at least one solution to the linear complementarity problem (LCP) derived from a bimatrix game. A closely related complementary pivoting algorithm by Lemke solves more general LCPs. A unified view of these two algorithms is presented, for the first time, as far as we know. Furthermore, we present an extension of the standard version of Lemke's algorithm that allows one more freedom than before when starting the algorithm

Web Developing with Block Bins

Block Bins is a compost collection company based in Chicago that creates food scrap drop off points using shared bins, providing an inexpensive solution to people who want to compost their food scrap. This service is enabled with a website, automation, and data collection & analysis, which were developed and improved over the course of this internship. The goal of this project was to improve the current website with an updated front-end that uses modern web frameworks and a revised back-end that is better tailored to the Block Binsuse-case. These improvements would reduce reliance on third-party software and developers, and automate more of Block Bins’ workflows. Working on the front-end of the website gave me the opportunity to learn Vue.js, a front-end JavaScript framework, and to utilize the Leaflet API in order to render maps.Developing with Vue.js was a significant aspect of this internship as the framework is increasing quickly in popularity due to all the functionality it provides. Back-end work entailed using Node.js to write scripts that compile collection statistics for each bin, and then analyze the statistics to determine optimal service intervals, and detect overuse, theft, and service issues. Through this process we created a JSON data model for Block Bins to structure its back-end and front-end development around, which will provide guidance for future web development.https://digitalcommons.imsa.edu/intern_reports_2020/1004/thumbnail.jp