The recent surge of interest in reasoning about probabilistic graphical models has led to the development of various techniques for probabilistic reasoning. Of these, techniques based on weighted model counting are particularly interesting since they can potentially leverage recent advances in unweighted model counting and in propositional satisfiability solving. In this paper, we present a new approach to weighted model counting via reduction to unweighted model counting. Our reduction, which is polynomial-time and preserves the normal form (CNF/DNF) of the input formula, allows us to exploit advances in unweighted model counting to solve weighted model counting instances. Experiments with weighted model counters built using our reduction indicate that these counters performs much better than a state-of-the-art weighted model counter

Most models of Stackelberg security games assume that the attacker only knows the defender's mixed strategy, but is not able to observe (even partially) the instantiated pure strategy. Such partial observation of the deployed pure strategy -- an issue we refer to as information leakage -- is a significant concern in practical applications. While previous research on patrolling games has considered the attacker's real-time surveillance, our settings, therefore models and techniques, are fundamentally different. More specifically, after describing the information leakage model, we start with an LP formulation to compute the defender's optimal strategy in the presence of leakage. Perhaps surprisingly, we show that a key subproblem to solve this LP (more precisely, the defender oracle) is NP-hard even for the simplest of security game models. We then approach the problem from three possible directions: efficient algorithms for restricted cases, approximation algorithms, and heuristic algorithms for sampling that improves upon the status quo. Our experiments confirm the necessity of handling information leakage and the advantage of our algorithms.

Shubik's (all-pay) dollar auction is a simple yet powerful auction model that aims to shed light on the motives and dynamics of conflict escalation. Common intuition and experimental results suggest that the dollar auction is a trap, inducing conflict by its very design. However, O'Neill proved the surprising fact that, contrary to the experimental results and the intuition, the dollar auction has an immediate solution in pure strategies, i.e., theoretically it should not lead to conflict escalation. In this paper, inspired by the recent literature on spiteful bidders, we ask whether the escalation in the dollar auction can be induced by meanness. Our results confirm this conjecture in various scenarios.

Cepheus is the first computer program to essentially solve a game of imperfect information that is played competitively by humans. The game it plays is heads-up limit Texas hold'em poker, a game with over 10^14 information sets, and a challenge problem for artificial intelligence for over 10 years. Cepheus was trained using a new variant of Counterfactual Regret Minimization (CFR), called CFR+, using 4800 CPUs running for 68 days. In this paper we describe in detail the engineering details required to make this computation a reality. We also prove the theoretical soundness of CFR+ and its component algorithm, regret-matching+. We further give a hint towards understanding the success of CFR+ by proving a tracking regret bound for this new regret matching algorithm. We present results showing the role of the algorithmic components and the engineering choices to the success of CFR+.

Hedonic games provide a natural model of coalition formation among self-interested agents. The associated problem of finding stable outcomes in such games has been extensively studied. In this paper, we identify simple conditions on expressivity of hedonic games that are sufficient for the problem of checking whether a given game admits a stable outcome to be computationally hard. Somewhat surprisingly, these conditions are very mild and intuitive. Our results apply to a wide range of stability concepts (core stability, individual stability, Nash stability, etc.) and to many known formalisms for hedonic games (additively separable games, games with W-preferences, fractional hedonic games, etc.), and unify and extend known results for these formalisms. They also have broader applicability: for several classes of hedonic games whose computational complexity has not been explored in prior work, we show that our framework immediately implies a number of hardness results for them.

In 1990, Thomas Schwartz proposed the conjecture that every nonempty tournament has a unique minimal TEQ-retentive set (TEQ stands for tournament equilibrium set). A weak variant of Schwartz's Conjecture was recently proposed by Felix Brandt. However, both conjectures were disproved very recently by two counterexamples. In this paper, we prove sufficient conditions for infinite classes of tournaments that satisfy Schwartz's Conjecture and Brandt's Conjecture. Moreover, we prove that TEQ can be calculated in polynomial time in several infinite classes of tournaments. Furthermore, our results reveal some structures that are forbidden in every counterexample to Schwartz's Conjecture.

They concluded that UCT quickly finds Self-play Monte Carlo Tree Search (MCTS) has a good but suboptimal policy, while Outcome Sampling initially been successful in many perfect-information twoplayer learns more slowly but converges to the optimal policy games. Although these methods have been over time. In this paper, we address the question whether the extended to imperfect-information games, so far inability of UCT to converge to a Nash equilibrium can be they have not achieved the same level of practical overcome while retaining UCT's fast initial learning rate. We success or theoretical convergence guarantees focus on the full-game MCTS setting, which is an important as competing methods. In this paper we step towards developing sound variants of online MCTS in introduce Smooth UCT, a variant of the established imperfect-information games. Upper Confidence Bounds Applied to Trees In particular, we introduce Smooth UCT, which combines (UCT) algorithm.

It is known that Nash equilibria and approximate Nash equilibria not necessarily optimize social optima of bimatrix games. In this paper, we show that for every fixed ε > 0, every bimatrix game (with values in [0, 1]) has an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor, (1 − √1 − ε)2, of the optimum. Furthermore, our result can be made algorithmic in the following sense: for every fixed 0 ≤ ε* < ε, if we can find an ε*-approximate Nash equilibrium in polynomial time, then we can find in polynomial time an ε-approximate Nash equilibrium with the total payoff of the players at least a constant factor of the optimum. Our analysis is especially tight in the case when ε ≥ 1/2. In this case, we show that for any bimatrix game there is an ε-approximate Nash equilibrium with constant size support whose social welfare is is at least 2√ε − ε ≥ 0.914 times the optimal social welfare. Furthermore, we demonstrate that our bound for the social welfare is tight, that is, for every ε ≥ 1/2 there is a bimatrix game for which every ε-approximate Nash equilibrium has social welfare at most 2√ε − ε times the optimal social welfare.

In Massively Open Online Courses (MOOCs) TA resources are limited; most MOOCs use peer assessments to grade assignments. Students have to divide up their time between working on their own homework and grading others. If there is no risk of being caught and penalized, students have no reason to spend any time grading others Course staff want to incentivize students to balance their time between course work and peer grading. They may do so by auditing students, ensuring that they perform grading correctly. One would not want students to invest too much time on peer grading, as this would result in poor course performance. We present the first model of strategic auditing in peer grading, modeling the student's choice of effort in response to a grader's audit levels as a Stackelberg game with multiple followers. We demonstrate that computing the equilibrium for this game is computationally hard. We then provide a PTAS in order to compute an approximate solution to the problem of allocating audit levels. However, we show that this allocation does not necessarily maximize social welfare; in fact, there exist settings where course auditor utility is arbitrarily far from optimal under an approximately optimal allocation. To circumvent this issue, we present a natural condition that guarantees that approximately optimal TA allocations guarantee approximately optimal welfare for the course auditors.

Word problems are an established technique for teaching mathematical modeling skills in K-12 education. However, many students find word problems unconnected to their lives, artificial, and uninteresting. Most students find them much more difficult than the corresponding symbolic representations. To account for this phenomenon, an ideal pedagogy might involve an individually crafted progression of unique word problems that form a personalized plot. We propose a novel technique for automatic generation of personalized word problems. In our system, word problems are generated from general specifications using answer-set programming (ASP). The specifications include tutor requirements (properties of a mathematical model), and student requirements (personalization, characters, setting). Our system takes a logical encoding of the specification, synthesizes a word problem narrative and its mathematical model as a labeled logical plot graph, and realizes the problem in natural language. Human judges found our problems as solvable as the textbook problems, with a slightly more artificial language.