Christian Kroer

Extensive-form game abstraction with bounds

Abstraction has emerged as a key component in solving extensive-form games of incomplete information. However, lossless abstractions are typically too large to solve, so lossy abstraction is needed. All prior lossy abstraction algorithms for extensive-form games either 1) had no bounds on solution quality or 2) depended on specific equilibrium computation approaches, limited forms of abstraction, and only decreased the number of information sets rather than nodes in the game tree. We introduce a theoretical framework that can be used to give bounds on solution quality for any perfect-recall extensive-form game. The framework uses a new notion for mapping abstract strategies to the original game, and it leverages a new equilibrium refinement for analysis. Using this framework, we develop the first general lossy extensive-form game abstraction method with bounds. Experiments show that it finds a lossless abstraction when one is available and lossy abstractions when smaller abstractions are desired. While our framework can be used for lossy abstraction, it is also a powerful tool for lossless abstraction if we set the bound to zero. Prior abstraction algorithms typically operate level by level in the game tree. We introduce the extensive-form game tree isomorphism and action subset selection problems, both important problems for computing abstractions on a level-by-level basis. We show that the former is graph isomorphism complete, and the latter NP-complete. We also prove that level-by-level abstraction can be too myopic and thus fail to find even obvious lossless abstractions.

Feature Filtering for Instance-Specific Algorithm Configuration

Instance-Specific Algorithm Configuration (ISAC) is a novel general technique for automatically generating and tuning algorithm portfolios. The approach has been very successful in practice, but up to now it has been committed to using all the features it was provided. However, traditional feature filtering techniques are not applicable, requiring multiple computationally expensive tuning steps during the evaluation stage. To this end, we show three new evaluation functions that use precomputed runtimes of a collection of untuned solvers to quickly evaluate subsets of features. One of our proposed functions even shows how to generate such an effective collection of solvers when only one highly parameterized solver is available. Using these new functions, we show that the number of features used by ISAC can be reduced to less than a quarter of the original number while often providing significant performance gains. We present numerical results on both SAT and CP domains.

Imperfect-Recall Abstractions with Bounds in Games

Imperfect-recall abstraction has emerged as the leading paradigm for practical large-scale equilibrium computation in imperfect-information games. However, imperfect-recall abstractions are poorly understood, and only weak algorithm-specific guarantees on solution quality are known. We develop the first general, algorithm-agnostic, solution quality guarantees for Nash equilibria and approximate self-trembling equilibria computed in imperfect-recall abstractions, when implemented in the original (perfect-recall) game. Our results are for a class of games that generalizes the only previously known class of imperfect-recall abstractions for which any such results have been obtained. Further, our analysis is tighter in two ways, each of which can lead to an exponential reduction in the solution quality error bound. We then show that for extensive-form games that satisfy certain properties, the problem of computing a bound-minimizing abstraction for a single level of the game reduces to a clustering problem, where the increase in our bound is the distance function. This reduction leads to the first imperfect-recall abstraction algorithm with solution quality bounds. We proceed to show a divide in the class of abstraction problems. If payoffs are at the same scale at all information sets considered for abstraction, the input forms a metric space, and this immediately yields a

Power napping with loud neighbors: optimal energy-constrained jamming and anti-jamming

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Faster First-Order Methods for Extensive-Form Game Solving

-approximation algorithm for abstraction. Conversely, if this condition is not satisfied, we show that the input does not form a metric space. Finally, we provide computational experiments to evaluate the practical usefulness of the abstraction techniques. They show that running counterfactual regret minimization on such abstractions leads to good strategies in the original games.

Robust Optimization of Recommendation Sets with the Maximin Utility Criterion

The openness of wireless communication and the recent development of software-defined radio technology, respectively, provide a low barrier and a wide range of capabilities for misbehavior, attacks, and defenses against attacks. In this work we present finite-energy jamming games, a game model that allows a jammer and sender to choose (1) whether to transmit or sleep, (2) a power level to transmit with, and (3) what channel to transmit on. We also allow the jammer to choose on how many channels it simultaneously attacks. A major addition in finite-energy jamming games is that the jammer and sender both have a limited amount of energy which is drained according to the actions a player takes. We develop a model of our system as a zero-sum finite-horizon stochastic game with deterministic transitions. We leverage the zero-sum and finite-horizon properties of our model to design a simple polynomial-time algorithm to compute optimal randomized strategies for both players. The utility function of our game model can be decoupled into a recursive equation. Our algorithm exploits this fact to use dynamic programming to construct solutions in a bottom-up fashion. For each state of energy levels, a linear program is solved to find Nash equilibrium strategies for the subgame. With these techniques, our algorithm has only a linear dependence on the number of states, and quadratic dependence on the number of actions, allowing us to solve very large instances. By computing Nash equilibria for our game models, we explore what kind of performance guarantees can be achieved both for the sender and jammer, when playing against an optimal opponent. We also use the optimal strategies to simulate finite-energy jamming games and provide insights into robust communication among reconfigurable, yet energy-limited, radio systems. To test the performance of the optimal strategies we compare their performance with a random and adaptive strategy. Matching our intuition, the aggressiveness of an attacker is related to how much of a discount is placed on data delay. This results in the defender often choosing to sleep despite the latency implication, because the threat of jamming is high. We also present several other findings from simulations where we vary the strategies for one or both of the players.

Symbolic Configuration for Interactive Container Ship Stowage Planning

We study the problem of computing a Nash equilibrium in large-scale two-player zero-sum extensive-form games. While this problem can be solved in polynomial time, first-order or regret-based methods are usually preferred for large games. Regret-based methods have largely been favored in practice, in spite of their theoretically inferior convergence rates. In this paper we investigate the acceleration of first-order methods both theoretically and experimentally. An important component of many first-order methods is a distance-generating function. Motivated by this, we investigate a specific distance-generating function, namely the dilated entropy function, over treeplexes, which are convex polytopes that encompass the strategy spaces of perfect-recall extensive-form games. We develop significantly stronger bounds on the associated strong convexity parameter. In terms of extensive-form game solving, this improves the convergence rate of several first-order methods by a factor of O((#information sets ⋅ depth ⋅ M)/(2depth)) where M is the maximum value of the l1 norm over the treeplex encoding the strategy spaces. Experimentally, we investigate the performance of three first-order methods (the excessive gap technique, mirror prox, and stochastic mirror prox) and compare their performance to the regret-based algorithms. In order to instantiate stochastic mirror prox, we develop a class of gradient sampling schemes for game trees. Equipped with our distance-generating function and sampling scheme, we find that mirror prox and the excessive gap technique outperform the prior regret-based methods for finding medium accuracy solutions

Trembling-Hand Perfection in Extensive-Form Games with Commitment

We investigate robust decision-making under utility uncertainty, using the maximin criterion, which optimizes utility for the worst case setting. We show how it is possible to efficiently compute the maximin optimal recommendation in face of utility uncertainty, even in large configuration spaces. We then introduce a new decision criterion, setwise maximin utility SMMU, for constructing optimal recommendation sets: we develop algorithms for computing SMMU and present experimental results showing their performance. Finally, we discuss the problem of elicitation and prove analogously to previous results related to regret-based and Bayesian elicitation that SMMU leads to myopically optimal query sets.

Smoothing Method for Approximate Extensive-Form Perfect Equilibrium

Low‐cost containerized shipping requires high‐quality stowage plans. Scalable stowage planning optimization algorithms have been developed recently. All of these algorithms, however, produce monolithic solutions that are hard for stowage coordinators to modify, which is necessary in practice owing to exceptions and operational disruptions. This article introduces an approach for modifying a stowage plan interactively without breaking its constraints. We focus on rearranging the containers in a single‐bay section and show two approaches for providing complete and backtrack‐free decision support using symbolic configuration techniques, one based on binary decision diagrams and one based on DPLL solvers. We show that binary decision diagrams can be used to solve real‐world sized instances of a single bay, and that search‐based solvers can be used to solve simplified instances going beyond a single bay.

Theoretical and Practical Advances on Smoothing for Extensive-Form Games

We initiate the study of equilibrium refinements based on trembling-hand perfection in extensiveform games with commitment strategies, that is, where one player commits to a strategy first. We show that the standard strong (and weak) Stackelberg equilibria are not suitable for trembling-hand perfection, because the limit of a sequence of such strong (weak) Stackelberg commitment strategies of a perturbed game may not be a strong (weak) Stackelberg equilibrium itself. However, we show that the universal set of all Stackelberg equilibria (i.e., those that are optimal for at least some follower response function) is natural for tremblinghand perfection: it does not suffer from the problem above. We also prove that determining the existence of a Stackelberg equilibrium—refined or not—that gives the leader expected value at least ν is NP-hard. This significantly extends prior complexity results that were specific to strong Stackelberg equilibrium.