The ZERO Regrets Algorithm: Optimizing over Pure Nash Equilibria via Integer Programming

Designing efficient algorithms to compute Nash equilibria poses considerable challenges in Algorithmic Game Theory (AGT). We shed new light on the intersection between Algorithmic Game Theory and Integer Programming. We introduce ZERO Regrets, a general and efficient cutting plane algorithm to compute, enumerate, and select Pure Nash Equilibria (PNEs) in Integer Programming Games, a class of simultaneous and non-cooperative games. We present a theoretical foundation for our algorithmic reasoning and provide a polyhedral characterization of the convex hull of the Pure Nash Equilibria. We introduce the concept of equilibrium inequality, and devise an equilibrium separation oracle to separate non-equilibrium strategies from PNEs. We evaluate our algorithmic framework on a wide range of problems from the literature and provide a solid benchmark against the existing algorithmic approaches

Learning Rationality in Potential Games

We propose a stochastic first-order algorithm to learn the rationality parameters of simultaneous and non-cooperative potential games, i.e., the parameters of the agents' optimization problems. Our technique combines (i.) an active-set step that enforces that the agents play at a Nash equilibrium and (ii.) an implicit-differentiation step to update the estimates of the rationality parameters. We detail the convergence properties of our algorithm and perform numerical experiments on Cournot and congestion games, showing that our algorithm effectively finds high-quality solutions (in terms of out-of-sample loss) and scales to large datasets

Computing Approximate Nash Equilibria for Integer Programming Games

We propose a framework to compute approximate Nash equilibria in integer programming games with nonlinear payoffs, i.e., simultaneous and non-cooperative games where each player solves a parametrized mixed-integer nonlinear program. We prove that using absolute approximations of the players' objective functions and then computing its Nash equilibria is equivalent to computing approximate Nash equilibria where the approximation factor is doubled. In practice, we propose an algorithm to approximate the players' objective functions via piecewise linear approximations. Our numerical experiments on a cybersecurity investment game show the computational effectiveness of our approach

When Nash Meets Stackelberg

Motivated by international energy trade between countries with profit-maximizing domestic producers, we analyze Nash games played among leaders of Stackelberg games (\NASP). We prove it is both $\Sigma^p_2$-hard to decide if the game has a pure-strategy (\PNE) or a mixed-strategy Nash equilibrium (\MNE). We then provide a finite algorithm that computes exact \MNEs for \NASPs when there is at least one, or returns a certificate if no \MNE exists. To enhance computational speed, we introduce an inner approximation hierarchy that increasingly grows the description of each Stackelberg leader feasible region. Furthermore, we extend the algorithmic framework to specifically retrieve a \PNE if one exists. Finally, we provide computational tests on a range of \NASPs instances inspired by international energy trades.Comment: 40 Pages and a computational appendix. Code is available on gitHu

Merging Combinatorial Design and Optimization: the Oberwolfach Problem

The Oberwolfach Problem $OP(F)$, posed by Gerhard Ringel in 1967, is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In Combinatorial Design Theory, so-called difference methods represent a well-known solution technique and construct solutions in infinitely many cases exploiting symmetric and balanced structures. This approach reduces the problem to finding a well-structured $2$-factor which allows us to build solutions that we call $1$- or $2$-rotational according to their symmetries. We tackle $OP$ by modeling difference methods with Optimization tools, specifically Constraint Programming ($CP$) and Integer Programming ($IP$), and correspondingly solve instances with up to $v=120$ within $60s$. In particular, we model the $2$-rotational method by solving in cascade two subproblems, namely the binary and group labeling, respectively. A polynomial-time algorithm solves the binary labeling, while $CP$ tackles the group labeling. Furthermore, we prov ide necessary conditions for the existence of some $1$-rotational solutions which stem from computational results. This paper shows thereby that both theoretical and empirical results may arise from the interaction between Combinatorial Design Theory and Operation Research

Why Are Outcomes Different for Registry Patients Enrolled Prospectively and Retrospectively? Insights from the Global Anticoagulant Registry in the FIELD-Atrial Fibrillation (GARFIELD-AF).

Background: Retrospective and prospective observational studies are designed to reflect real-world evidence on clinical practice, but can yield conflicting results. The GARFIELD-AF Registry includes both methods of enrolment and allows analysis of differences in patient characteristics and outcomes that may result. Methods and Results: Patients with atrial fibrillation (AF) and ≥1 risk factor for stroke at diagnosis of AF were recruited either retrospectively (n = 5069) or prospectively (n = 5501) from 19 countries and then followed prospectively. The retrospectively enrolled cohort comprised patients with established AF (for a least 6, and up to 24 months before enrolment), who were identified retrospectively (and baseline and partial follow-up data were collected from the emedical records) and then followed prospectively between 0-18 months (such that the total time of follow-up was 24 months; data collection Dec-2009 and Oct-2010). In the prospectively enrolled cohort, patients with newly diagnosed AF (≤6 weeks after diagnosis) were recruited between Mar-2010 and Oct-2011 and were followed for 24 months after enrolment. Differences between the cohorts were observed in clinical characteristics, including type of AF, stroke prevention strategies, and event rates. More patients in the retrospectively identified cohort received vitamin K antagonists (62.1% vs. 53.2%) and fewer received non-vitamin K oral anticoagulants (1.8% vs . 4.2%). All-cause mortality rates per 100 person-years during the prospective follow-up (starting the first study visit up to 1 year) were significantly lower in the retrospective than prospectively identified cohort (3.04 [95% CI 2.51 to 3.67] vs . 4.05 [95% CI 3.53 to 4.63]; p = 0.016). Conclusions: Interpretations of data from registries that aim to evaluate the characteristics and outcomes of patients with AF must take account of differences in registry design and the impact of recall bias and survivorship bias that is incurred with retrospective enrolment. Clinical Trial Registration: - URL: http://www.clinicaltrials.gov . Unique identifier for GARFIELD-AF (NCT01090362)

Mathematical Programming Games

RÉSUMÉ: Dans de nombreux contextes de prise de décision, un agent égoïste cherche à optimiser son bénéfice compte tenu de certaines contraintes situationnelles. Mathématiquement, la tâche du décideur est souvent formulée comme un problème d’optimisation dont la solution fournit une recommandation prescriptive sur la meilleure décision. Cependant, la prise de décision est rarement une tâche individuelle : chaque décideur égoïste interagit souvent avec d’autres décideurs ayant des intérêts similaires. Cette thèse discute et propose une nouvelle perspective pour capturer la dynamique de la prise de décision stratégique impliquant plusieurs agents résolvant des problèmes d’optimisation. Nous explorons les opportunités offertes par l’interaction entre l’optimisation -en nous concentrant sur la programmation en nombres entiers mixtes (MIP) -et la théorie algorithmique des jeux (AGT) en les analysant à travers le prisme d’un cadre unifié, capable d’intégrer des éléments des deux disciplines. Nous introduisons une taxonomie pour les jeux de programmation mathématique (MPGs), des jeux non coopératifs simultanés où le problème de décision de chaque agent est un problème d’optimisation exprimant un ensemble hétérogène et éventuellement complexe de contraintes. Nous développons nos contributions en considérant l’équilibre de Nash comme le principal concept de solution et fondons notre recherche sur le principe suivant : dans les MPGs, la plausibilité des équilibres de Nash découle de la disponibilité d’outils efficaces pour les calculer et les sélectionner. En conséquence, nous fournissons des algorithmes originaux et des cadres théoriques pour caractériser, calculer et sélectionner les équilibres de Nash dans les MPGs. Tout d’abord, nous abordons le problème du calcul et de la sélection des équilibres dans les jeux de programmation en nombres entiers (IPGs), à savoir les MPGs où chaque joueur résout un programme paramétré en nombres entiers. En introduisant des concepts tels que l’inégalité d’équilibre, l’oracle de séparation d’équilibre, et la fermeture d’équilibre, nous permettons à des outils archétypiques de la programmation en nombres entiers d’acquérir un rôle dans la théorie des jeux. Nous concevons ZERO Regrets, un algorithme de plans coupants pour calculer et sélectionner les équilibres dans les IPGs. Nous testons l’algorithme sur un jeu d’AGT et sur une extension multi-agents du problème du sac à dos, et nous fournissons de nouveaux résultats théoriques et informatiques sur l’efficacité de leurs équilibres. Ensuite, nous présentons Cut-and-Play, un algorithme permettant de calculer les équilibres des jeux réciproquement bilinéaires (RBGs), une classe de MPGs où l’objectif de chaque joueur est linéaire par rapport à ses variables et contient des termes bilinéaires entre ses variables et celles de ses adversaires. L’algorithme calcule les équilibres en exploitant une série d’approximations du problème d’optimisation de chaque joueur et en s’appuyant sur des méthodes de branchement et de plans coupants. Notre approche algorithmique est générale, extensible, et elle s’intègre aux solveurs de programmation mathématique existants. En pratique, elle surpasse les meilleurs algorithmes en termes de temps de calcul et d’efficacité des équilibres. Troisièmement, nous analysons une classe de MPGs parmi les leaders des jeux de Stackelberg (c’est-à-dire des jeux séquentiels leader-followers) et leur application aux marchés de l’énergie. Nous prouvons qu’il est Σp2 difficile de décider si le jeu admet un équilibre, et nous introduisons un algorithme pour calculer et sélectionner ces équilibres. De plus, nous fournissons une étude pratique sur le marché de l’énergie chilien-argentin et offrons des perspectives de gestion basées sur les informations fournies par les équilibres. Enfin, nous présentons ZERO, une bibliothèque C++ modulaire et extensible pour expérimenter avec des RBGs. ZERO fournit une boîte à outils complète d’interfaces de modélisation pour concevoir des RBGs, et des algorithmes pour trouver leurs équilibres de Nash. Notre engagement envers le code source ouvert vise à favoriser le développement futur, méthodologique et pratique, dans le domaine des RBGs.----------ABSTRACT: In many decision-making settings, a selfish agent seeks to optimize its benefit given some situational constraints. Mathematically, the decision-maker’s task is often formulated as an optimization problem whose solution provides a prescriptive recommendation on the best decision. However, decision-making is rarely an individual task: each selfish decision-maker often interacts with other similarly self-interested decision-makers. This thesis discusses and proposes a novel perspective to capture the dynamics of multi-agent strategic decision-making involving multiple agents solving optimization problems. We explore the opportunities offered by the interplay of Mathematical Optimization –specifically Mixed-Integer Programming (MIP) –and Algorithmic Game Theory (AGT) by analyzing them through the lenses of a unified framework capable of integrating elements of the two disciplines. We introduce the taxonomy of Mathematical Programming Games (MPGs), simultaneous non-cooperative games where each agent decision problem is an optimization problem expressing a heterogeneous and possibly complex set of constraints. We develop our contributions considering the Nash equilibrium as the primary solution concept and ground our research in the following principle: in MPGs, the plausibility of Nash equilibria stems from the availability of efficient tools to compute and select them. Accordingly, we provide original algorithms and theoretical frameworks to characterize, compute and select Nash equilibria in MPGs. First, we tackle the problem of computing and selecting equilibria in Integer Programming Games (IPGs), namely MPGs where each player solves a parametrized integer program. By devising concepts such as equilibrium inequality, equilibrium separation oracle, and equilibrium closure, we let archetypical tools of integer programming acquire a game-theoretic role. We design ZERO Regrets, a cutting plane algorithm for computing and selecting equilibria in IPGs. We test the algorithm on a game from AGT and a multi-agent extension of the knapsack problem and further provide novel theoretical and computational results on the efficiency of equilibria. Second, we introduce Cut-and-Play, an algorithm to compute equilibria for ReciprocallyBilinear Games (RBGs), a class of MPGs where each player’s objective is linear in its variables and contains bilinear terms among the player’s variables and its opponents’ ones. The algorithm computes equilibria by exploiting a series of approximations of each player’s optimization problem and leveraging branching and cutting plane methods. Our algorithmic approach is general and extensible, and it integrates with existing mathematical programming solvers; in practice, it outperforms the state-of-the-art algorithms in both computing times and equilibria efficiency. x Third, we analyze a class of MPGs among the leaders of Stackelberg Games (i.e., sequential leader-followers games) and their application in energy markets. We prove it is Σp2-hard to decide if the game admits an equilibrium and introduce an algorithm for computing and selecting equilibria. Further, we provide a real-world study on the Chilean-Argentinian energy market and deliver managerial insights based on the information equilibria provide. Finally, we present ZERO, a modular and extensible C++ library for experimenting with RBGs. ZERO provides a comprehensive toolkit of modeling interfaces to design RBGs, and several algorithms to compute their Nash equilibria. Our commitment to open-source aims at fostering methodological and practical advancements in the area of MPG

Merging Combinatorial Design and Optimization: the Oberwolfach Problem

The Oberwolfach Problem $OP(F)$ -- posed by Gerhard Ringel in 1967 -- is a paradigmatic Combinatorial Design problem asking whether the complete graph $K_v$ decomposes into edge-disjoint copies of a $2$-regular graph $F$ of order $v$. In this paper, we provide all the necessary equipment to generate solutions to $OP(F)$ for relatively small orders by using the so-called difference methods. From the theoretical standpoint, we present new insights on the combinatorial structures involved in the solution of the problem. Computationally, we provide a full recipe whose base ingredients are advanced optimization models and tailored algorithms. This algorithmic arsenal can solve the $OP(F)$ for all possible orders up to $60$ with the modest computing resources of a personal computer. The new $20$ orders, from $41$ to $60$, encompass $241200$ instances of the Oberwolfach Problem, which is 22 times greater than those solved in previous contributions.Comment: Pre-print: 31 pages, 6 figures. Code available on gitHu