Static Stability in Games

Static stability of equilibrium in strategic games differs from dynamic stability in not being linked to any particular dynamical system. In other words, it does not make any assumptions about off-equilibrium behavior. Examples of static notions of stability include evolutionarily stable strategy (ESS) and continuously stable strategy (CSS), both of which are meaningful or justifiable only for particular classes of games, namely, symmetric multilinear games or symmetric games with a unidimensional strategy space, respectively. This paper presents a general notion of local static stability, of which the above two are essentially special cases. It is applicable to virtually all n-person strategic games, both symmetric and asymmetric, with non-discrete strategy spaces.Stability of equilibrium, static stability

Equilibrium Computation and Robust Optimization in Zero Sum Games with Submodular Structure

We define a class of zero-sum games with combinatorial structure, where the best response problem of one player is to maximize a submodular function. For example, this class includes security games played on networks, as well as the problem of robustly optimizing a submodular function over the worst case from a set of scenarios. The challenge in computing equilibria is that both players' strategy spaces can be exponentially large. Accordingly, previous algorithms have worst-case exponential runtime and indeed fail to scale up on practical instances. We provide a pseudopolynomial-time algorithm which obtains a guaranteed $(1 - 1/e)^2$-approximate mixed strategy for the maximizing player. Our algorithm only requires access to a weakened version of a best response oracle for the minimizing player which runs in polynomial time. Experimental results for network security games and a robust budget allocation problem confirm that our algorithm delivers near-optimal solutions and scales to much larger instances than was previously possible.Comment: 20 pages, 8 figures. A shorter version of this paper appears at AAAI 201

Mixed Strategy Constraints in Continuous Games

Equilibrium problems representing interaction in physical environments typically require continuous strategies which satisfy opponent-dependent constraints, such as those modeling collision avoidance. However, as with finite games, mixed strategies are often desired, both from an equilibrium existence perspective as well as a competitive perspective. To that end, this work investigates a chance-constraint-based approach to coupled constraints in generalized Nash equilibrium problems which are solved over pure strategies and mixing weights simultaneously. We motivate these constraints in a discrete setting, placing them on tensor games ($n$-player bimatrix games) as a justifiable approach to handling the probabilistic nature of mixing. Then, we describe a numerical solution method for these chance constrained tensor games with simultaneous pure strategy optimization. Finally, using a modified pursuit-evasion game as a motivating examples, we demonstrate the actual behavior of this solution method in terms of its fidelity, parameter sensitivity, and efficiency

On the Interplay between Social Welfare and Tractability of Equilibria

Computational tractability and social welfare (aka. efficiency) of equilibria are two fundamental but in general orthogonal considerations in algorithmic game theory. Nevertheless, we show that when (approximate) full efficiency can be guaranteed via a smoothness argument \`a la Roughgarden, Nash equilibria are approachable under a family of no-regret learning algorithms, thereby enabling fast and decentralized computation. We leverage this connection to obtain new convergence results in large games -- wherein the number of players $n \gg 1$ -- under the well-documented property of full efficiency via smoothness in the limit. Surprisingly, our framework unifies equilibrium computation in disparate classes of problems including games with vanishing strategic sensitivity and two-player zero-sum games, illuminating en route an immediate but overlooked equivalence between smoothness and a well-studied condition in the optimization literature known as the Minty property. Finally, we establish that a family of no-regret dynamics attains a welfare bound that improves over the smoothness framework while at the same time guaranteeing convergence to the set of coarse correlated equilibria. We show this by employing the clairvoyant mirror descent algortihm recently introduced by Piliouras et al.Comment: To appear at NeurIPS 202

Beyond Monotone Variational Inequalities: Solution Methods and Iteration Complexities

In this paper, we discuss variational inequality (VI) problems without monotonicity from the perspective of convergence of projection-type algorithms. In particular, we identify existing conditions as well as present new conditions that are sufficient to guarantee convergence. The first half of the paper focuses on the case where a Minty solution exists (also known as Minty condition), which is a common assumption in the recent developments for non-monotone VI. The second half explores alternative sufficient conditions that are different from the existing ones such as monotonicity or Minty condition, using an algorithm-based approach. Through examples and convergence analysis, we show that these conditions are capable of characterizing different classes of VI problems where the algorithms are guaranteed to converge.Comment: 29 page

Time and Space in Video Games: A Cognitive-Formalist Approach

Video games are temporal artifacts: They change with time as players interact with them in accordance with rules. In this study, the author investigates the formal aspects of video games that determine how these changes are produced and sequenced. Theories of time perception drawn from the cognitive sciences lay the groundwork for an in-depth analysis of these features, making for a comprehensive account of time in this novel medium. This book-length study dedicated to time perception and video games is an indispensable resource for game scholars and game developers alike. Its reader-friendly style makes it readily accessible to the interested layperson

Directed Emergent Drama

A fully interactive drama, where the player interacts with intelligent Non- Player Characters (NPCs), can revolutionise entertainment, gaming, education, and therapy. Creating such a genuinely interactive drama that is entertaining and gives players a sense of coherency as active participants in the unfolding drama has seen a substantial research effort. Authors have the power to shape dramatised stories for theatre or television at will. Conversely, the authors' ability to shape interactive drama is limited because the drama emerges from players' and NPCs actions during game-play, which significantly limits authoring control. A coherent drama has a recognisable dramatic structure. One philosophy is to use planning algorithms and narrative structures to reduce required authoring. However, planning algorithms are intractable for the large state-spaces intrinsic to interactive dramas, and they have not reduced the authoring problem sufficiently. A more straightforward and computationally feasible method is emergent interactive drama from players' and NPCs' actions. The main difficulty with this approach is maintaining a drama structure and theme, such as a mystery theme or a training scenario, that the player experiences while interacting with the game world. Therefore, it is necessary to impose some form of structure to guide or direct the unfolding drama. The solution introduced in this thesis is to distribute the computation among autonomous actors that are guided by goals and drama structures which a centralised autonomous director agent distributes among the actors, which comprises the following four main elements: a) autonomous rational actor agents that know they are acting and can negotiate dialogues between them to remain realistic while simultaneously progressing the drama, without the player knowing, b) Bayesian network to model the actors reasoning, including beliefs about other actors' mental states c) an autonomous director agent uses "schemas", conceptual structures based on motifs, to guide the actors