Game-theoretic Characterization Research Articles

Some Game-Theoretic Characterizations for Rapid Filters on ω

Some Game-Theoretic Characterizations for Rapid Filters on ω

Characterizing Omega-Regularity through Finite-Memory Determinacy of Games on Infinite Graphs

We consider zero-sum games on infinite graphs, with objectives specified as sets of infinite words over some alphabet of colors. A well-studied class of objectives is the one of $\omega$-regular objectives, due to its relation to many natural problems in theoretical computer science. We focus on the strategy complexity question: given an objective, how much memory does each player require to play as well as possible? A classical result is that finite-memory strategies suffice for both players when the objective is $\omega$-regular. We show a reciprocal of that statement: when both players can play optimally with a chromatic finite-memory structure (i.e., whose updates can only observe colors) in all infinite game graphs, then the objective must be $\omega$-regular. This provides a game-theoretic characterization of $\omega$-regular objectives, and this characterization can help in obtaining memory bounds. Moreover, a by-product of our characterization is a new one-to-two-player lift: to show that chromatic finite-memory structures suffice to play optimally in two-player games on infinite graphs, it suffices to show it in the simpler case of one-player games on infinite graphs. We illustrate our results with the family of discounted-sum objectives, for which $\omega$-regularity depends on the value of some parameters.

Codensity Games for Bisimilarity

Bisimilarity as an equivalence notion of systems has been central to process theory. Due to the recent rise of interest in quantitative systems (probabilistic, weighted, hybrid, etc.), bisimilarity has been extended in various ways, such as bisimulation metric between probabilistic systems. An important feature of bisimilarity is its game-theoretic characterization, where Spoiler and Duplicator play against each other; extension of bisimilarity games to quantitative settings has been actively pursued too. In this paper, we present a general framework that uniformly describes game characterizations of bisimilarity-like notions. Our framework is formalized categorically using fibrations and coalgebras. In particular, our characterization of bisimilarity in terms of fibrational predicate transformers allows us to derive what we call codensity bisimilarity games: a general categorical game characterization of bisimilarity. Our framework covers known bisimilarity-like notions (such as bisimulation metric and bisimulation seminorm) as well as new ones (including what we call bisimulation topology).

The Combinatorics and Absoluteness of Definable Sets of Real Numbers

AbstractThis thesis divides naturally into two parts, each concerned with the extent to which the theory of $L(\mathbf {R})$ can be changed by forcing.The first part focuses primarily on applying generic-absoluteness principles to how that definable sets of reals enjoy regularity properties. The work in Part I is joint with Itay Neeman and is adapted from our paper Happy and mad families in $L(\mathbf {R})$ , JSL, 2018. The project was motivated by questions about mad families, maximal families of infinite subsets of $\omega $ of which any two have only finitely many members in common. We begin, in the spirit of Mathias, by establishing (Theorem 2.8) a strong Ramsey property for sets of reals in the Solovay model, giving a new proof of Törnquist’s theorem that there are no infinite mad families in the Solovay model.In Chapter 3 we stray from the main line of inquiry to briefly study a game-theoretic characterization of filters with the Baire Property.Neeman and Zapletal showed, assuming roughly the existence of a proper class of Woodin cardinals, that the boldface theory of $L(\mathbf {R})$ cannot be changed by proper forcing. They call their result the Embedding Theorem, because they conclude that in fact there is an elementary embedding from the $L(\mathbf {R})$ of the ground model to that of the proper forcing extension. With a view toward analyzing mad families under $\mathsf {AD}^+$ and in $L(\mathbf {R})$ under large-cardinal hypotheses, in Chapter 4 we establish triangular versions of the Embedding Theorem. These are enough for us to use Mathias’s methods to show (Theorem 4.5) that there are no infinite mad families in $L(\mathbf {R})$ under large cardinals and (Theorem 4.9) that $\mathsf {AD}^+$ implies that there are no infinite mad families. These are again corollaries of theorems about strong Ramsey properties under large-cardinal assumptions and $\mathsf {AD}^+$ , respectively. Our first theorem improves the large-cardinal assumption under which Todorcevic established the nonexistence of infinite mad families in $L(\mathbf {R})$ . Part I concludes with Chapter 5, a short list of open questions.In the second part of the thesis, we undertake a finer analysis of the Embedding Theorem and its consistency strength. Schindler found that the the Embedding Theorem is consistent relative to much weaker assumptions than the existence of Woodin cardinals. He defined remarkable cardinals, which can exist even in L, and showed that the Embedding Theorem is equiconsistent with the existence of a remarkable cardinal. His theorem resembles a theorem of Harrington–Shelah and Kunen from the 1980s: the absoluteness of the theory of $L(\mathbf {R})$ to ccc forcing extensions is equiconsistent with a weakly compact cardinal. Joint with Itay Neeman, we improve Schindler’s theorem by showing that absoluteness for $\sigma $ -closed $\ast $ ccc posets—instead of the larger class of proper posets—implies the remarkability of $\aleph _1^V$ in L. This requires a fundamental change in the proof, since Schindler’s lower-bound argument uses Jensen’s reshaping forcing, which, though proper, need not be $\sigma $ -closed $\ast $ ccc in that context. Our proof bears more resemblance to that of Harrington–Shelah than to Schindler’s.The proof of Theorem 6.2 splits naturally into two arguments. In Chapter 7 we extend the Harrington–Shelah method of coding reals into a specializing function to allow for trees with uncountable levels that may not belong to L. This culminates in Theorem 7.4, which asserts that if there are $X\subseteq \omega _1$ and a tree $T\subseteq \omega _1$ of height $\omega _1$ such that X is codable along T (see Definition 7.3), then $L(\mathbf {R})$ -absoluteness for ccc posets must fail.We complete the argument in Chapter 8, where we show that if in any $\sigma $ -closed extension of V there is no $X\subseteq \omega _1$ codable along a tree T, then $\aleph _1^V$ must be remarkable in L.In Chapter 9 we review Schindler’s proof of generic absoluteness from a remarkable cardinal to show that the argument gives a level-by-level upper bound: a strongly $\lambda ^+$ -remarkable cardinal is enough to get $L(\mathbf {R})$ -absoluteness for $\lambda $ -linked proper posets.Chapter 10 is devoted to partially reversing the level-by-level upper bound of Chapter 9. Adapting the methods of Neeman, Hierarchies of forcing axioms II, we are able to show that $L(\mathbf {R})$ -absoluteness for $\left |\mathbf {R}\right |\cdot \left |\lambda \right |$ -linked posets implies that the interval $[\aleph _1^V,\lambda ]$ is $\Sigma ^2_1$ -remarkable in L.Abstract prepared by Zach Norwood.E-mail: zachnorwood@gmail.com

The Price of Defense

We consider a game on a graph \(G=\langle V, E\rangle \) with two confronting classes of randomized players: \(\nu \) attackers, who choose vertices and seek to minimize the probability of getting caught, and a single defender, who chooses edges and seeks to maximize the expected number of attackers it catches. In a Nash equilibrium, no player has an incentive to unilaterally deviate from her randomized strategy. The Price of Defense is the worst-case ratio, over all Nash equilibria, of \(\nu \) over the expected utility of the defender at a Nash equilibrium.We orchestrate a strong interplay of arguments from Game Theory and Graph Theory to obtain both general and specific results in the considered setting:(1) Via a reduction to a Two-Players, Constant-Sum game, we observe that an arbitrary Nash equilibrium is computable in polynomial time. Further, we prove a general lower bound of \(\frac{\textstyle |V|}{\textstyle 2}\) on the Price of Defense. We derive a characterization of graphs with a Nash equilibrium attaining this lower bound, which reveals a promising connection to Fractional Graph Theory; thereby, it implies an efficient recognition algorithm for such Defense-Optimal graphs.(2) We study some specific classes of Nash equilibria, both for their computational complexity and for their incurred Price of Defense. The classes are defined by imposing structure on the players’ randomized strategies: either graph-theoretic structure on the supports, or symmetry and uniformity structure on the probabilities. We develop novel graph-theoretic techniques to derive trade-offs between computational complexity and the Price of Defense for these classes. Some of the techniques touch upon classical milestones of Graph Theory; for example, we derive the first game-theoretic characterization of König-Egerváry graphs as graphs admitting a Matching Nash equilibrium.

Set-Menger and related properties

In this work, we consider new types of covering properties called set-Menger, set-Rothberger, set-Hurewicz and set-Gerlits-Nagy. We investigate the relationships between set-Mengerness and some other Menger-type covering properties (such as Mengerness, weak Mengerness, quasi-Mengerness) and study topological properties of set-Menger and related spaces. In particular, we give a game-theoretic characterization of set-Menger spaces.

GAMES AND RAMSEY-LIKE CARDINALS

AbstractWe generalise the α-Ramsey cardinals introduced in Holy and Schlicht (2018) for cardinals α to arbitrary ordinals α, and answer several questions posed in that paper. In particular, we show that α-Ramseys are downwards absolute to the core model K for all α of uncountable cofinality, that strategic ω-Ramsey cardinals are equiconsistent with remarkable cardinals and that strategic α-Ramsey cardinals are equiconsistent with measurable cardinals for all α > ω. We also show that the n-Ramseys satisfy indescribability properties and use them to provide a game-theoretic characterisation of completely ineffable cardinals, as well as establishing further connections between the α-Ramsey cardinals and the Ramsey-like cardinals introduced in Gitman (2011), Feng (1990), and Sharpe and Welch (2011).

Dynamic Power Distribution System Management With a Locally Connected Communication Network

Coordinated optimization and control of distribution-level assets can enable a reliable and optimal integration of massive amount of distributed energy resources (DERs) and facilitate distribution system management (DSM). Accordingly, the objective is to coordinate the power injection at the DERs to maintain certain quantities across the network, e.g., voltage magnitude, line flows, or line losses, to be close to a desired profile. By and large, the performance of the DSM algorithms has been challenged by two factors: i) the possibly non strongly connected communication network over DERs that hinders the coordination; ii) the dynamics of the real system caused by the DERs with heterogeneous capabilities, time-varying operating conditions, and real-time measurement mismatches. In this paper, we investigate the modeling and algorithm design and analysis with the consideration of these two factors. In particular, a game-theoretic characterization is first proposed to account for a locally connected communication network over DERs, along with the analysis of the existence and uniqueness of the Nash equilibrium (NE) therein. To achieve the equilibrium in a distributed fashion, a projected-gradient-based asynchronous DSM algorithm is then advocated. The algorithm performance, including the convergence speed and the tracking error, is analytically guaranteed under the dynamic setting. Extensive numerical tests on both synthetic and realistic cases corroborate the analytical results derived.

Algebra, selections, and additive Ramsey theory

Hindman's celebrated Finite Sums Theorem, and its high-dimensional version due to Milliken and Taylor, are extended from covers of countable sets to covers of arbitrary topological spaces with Menger's classic covering property. The methods include, in addition to Hurewicz's game theoretic characterization of Menger's property, extensions of the classic idempotent theory in the Stone--Czech compactification of semigroups, and of the more recent theory of selection principles. This provides strong versions of the mentioned celebrated theorems, where the monochromatic substructures are large, beyond infinitude, in an analytic sense. Reducing the main theorems to the purely combinatorial setting, we obtain nontrivial consequences concerning uncountable cardinal characteristics of the continuum. The main results, modulo technical refinements, are of the following type (definitions provided in the main text): Let $X$ be a Menger space, and $\mathcal{U}$ be an infinite open cover of $X$. Consider the complete graph, whose vertices are the open sets in $X$. For each finite coloring of the vertices and edges of this graph, there are disjoint finite subsets $\mathcal{F}_1,\mathcal{F}_2,\dots$ of the cover $\mathcal{U}$ whose unions $V_1 := \bigcup\mathcal{F}_1, V_2 := \bigcup\mathcal{F}_2,\dots$ have the following properties: 1. The sets $\bigcup_{n\in F}V_n$ and $\bigcup_{n\in H}V_n$ are distinct for all nonempty finite sets $F<H$. 2. All vertices $\bigcup_{n\in F}V_n$, for nonempty finite sets $F$, are of the same color. 3. All edges $\bigl\{\,\bigcup_{n\in F}V_n, \bigcup_{n\in H}V_n\,\bigr\}$, for nonempty finite sets $F<H$, have the same color. 4. The family $\{V_1,V_2,\dots\}$ is an open cover of $X$. A self-contained introduction to the necessary parts of the needed theories is provided.

Game-theoretic characterization of the Gurarii space

We present a simple and natural infinite game building an increasing chain of finite-dimensional Banach spaces. We show that one of the players has a strategy with the property that, no matter how the other player plays, the completion of the union of the chain is linearly isometric to the Gurariĭ space.

Deciding First-Order Properties of Nowhere Dense Graphs

Nowhere dense graph classes, introduced by Nešetřil and Ossona de Mendez [2010, 2011], form a large variety of classes of “sparse graphs” including the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs and graph classes of bounded expansion. We show that deciding properties of graphs definable in first-order logic is fixed-parameter tractable on nowhere dense graph classes (parameterized by the length of the input formula). At least for graph classes closed under taking subgraphs, this result is optimal: it was known before that for all classes C of graphs closed under taking subgraphs, if deciding first-order properties of graphs in C is fixed-parameter tractable, then C must be nowhere dense (under a reasonable complexity theoretic assumption). As a by-product, we give an algorithmic construction of sparse neighborhood covers for nowhere dense graphs. This extends and improves previous constructions of neighborhood covers for graph classes with excluded minors. At the same time, our construction is considerably simpler than those. Our proofs are based on a new game-theoretic characterization of nowhere dense graphs that allows for a recursive version of locality-based algorithms on these classes. On the logical side, we prove a “rank-preserving” version of Gaifman’s locality theorem.

Complexity and monotonicity results for domination games

In this paper we study Domination Games, a class of games introduced by Fomin, Kratsch, and Müller in [8]. Domination games are a variant of the well-known graph searching games (also called cops and robber games), where a number of cops tries to capture a robber hiding on the vertices of a graph. Variants of these games are often used to provide a game-theoretic characterization of important graph parameters such as pathwidth, treewidth, and hypertreewidth.We are primarily interested in questions concerning the complexity and monotonicity of these games. We show that dominating games are computationally much harder than standard cops and robber games and establish strong non-monotonicity results for various notions of monotonicity that arise naturally in the context of domination games. Answering a question of [8], we show that there are graphs where the shortest winning strategy for a minimal number of cops must necessarily be of exponential length.

Games for query inseparability of description logic knowledge bases

We consider conjunctive query inseparability of description logic knowledge bases with respect to a given signature—a fundamental problem in knowledge base versioning, module extraction, forgetting and knowledge exchange. We give a uniform game-theoretic characterisation of knowledge base conjunctive query inseparability and develop worst-case optimal decision algorithms for fragments of Horn-ALCHI, including the description logics underpinning OWL 2 QL and OWL 2 EL. We also determine the data and combined complexity of deciding query inseparability. While query inseparability for all of these logics is P-complete for data complexity, the combined complexity ranges from P- to ExpTime- to 2ExpTime-completeness. We use these results to resolve two major open problems for OWL 2 QL by showing that TBox query inseparability and the membership problem for universal conjunctive query solutions in knowledge exchange are both ExpTime-complete for combined complexity. Finally, we introduce a more flexible notion of inseparability which compares answers to conjunctive queries in a given signature over a given set of individuals. In this case, checking query inseparability becomes NP-complete for data complexity, but the ExpTime- and 2ExpTime-completeness combined complexity results are preserved.

Specifying Peirce's law in classical realizability

This paper deals with the specification problem in classical realizability (such as introduced by Krivine (2009 Panoramas et synthéses27)), which is to characterize the universal realizers of a given formula by their computational behaviour. After recalling the framework of classical realizability, we present the problem in the general case and illustrate it with some examples. In the rest of the paper, we focus on Peirce's law, and present two game-theoretic characterizations of its universal realizers. First, we consider the particular case where the language of realizers contains no extra instruction such as ‘quote’ (Krivine 2003 Theoretical Computer Science308 259–276). We present a first game $\mathds{G}$0 and show that the universal realizers of Peirce's law can be characterized as the uniform winning strategies for $\mathds{G}$0, using the technique of interaction constants. Then we show that in the presence of extra instructions such as ‘quote’, winning strategies for the game $\mathds{G}$0 are still adequate but no more complete. For that, we exhibit an example of a wild realizer of Peirce's law, that introduces a purely game-theoretic form of backtrack that is not captured by $\mathds{G}$0. We finally propose a more sophisticated game $\mathds{G}$1, and show that winning strategies for the game $\mathds{G}$1 are both adequate and complete in the general case, without any further assumption about the instruction set used by the language of classical realizers.

Game-theoretic characterization of antidegradable channels

We introduce a guessing game involving a quantum channel, three parties—the sender, the receiver, and an eavesdropper, Eve—and a quantum public side channel. We prove that a necessary and sufficient condition for the quantum channel to be antidegradable, is that Eve wins the game. We thus obtain a complete operational characterization of antidegradable channels in a game-theoretic framework.

On the existence of pure-strategy equilibria in games with private information: A complete characterization

This paper reports a definitive resolution to the question of the existence of a pure-strategy Bayesian–Nash equilibrium in games with a finite number of players, each with a compact metric action set and private information. The resolution hinges on saturated spaces. If the individual spaces of information are saturated, there exists a pure-strategy equilibrium in such a game; and if there exists a pure-strategy equilibrium for the class of games under consideration and with uncountable action sets, the spaces of private information must be saturated. As such, the paper offers a complete characterization of a longstanding question, and offers another game-theoretic characterization of the saturation property, one that complements a recent result of Keisler–Sun (2009) on large non-anonymous games with complete information.

Tree projections and structural decomposition methods: Minimality and game-theoretic characterization

Tree projections provide a mathematical framework that encompasses all the various (purely) structural decomposition methods that have been proposed in the literature to single out classes of nearly-acyclic (hyper)graphs, such as the tree decomposition method, which is the most powerful decomposition method on graphs, and the (generalized) hypertree decomposition method, which is its natural counterpart on arbitrary hypergraphs.The paper analyzes this framework, by focusing in particular on “minimal” tree projections, that is, on tree projections without useless redundancies. First, it is shown that minimal tree projections enjoy a number of properties that are usually required for normal form decompositions in various structural decomposition methods. In particular, they enjoy the same kind of connection properties as (minimal) tree decompositions of graphs, with the result being tight in the light of the negative answer that is provided to the open question about whether they enjoy a slightly stronger notion of connection property, defined to speed-up the computation of hypertree decompositions. Second, it is shown that tree projections admit a natural game-theoretic characterization in terms of the Robber and Captain game. In this game, as for the Robber and Cops game characterizing tree decompositions, the existence of winning strategies implies the existence of monotone ones. As a special case, the Robber and Captain game can be used to characterize the generalized hypertree decomposition method, where such a game-theoretic characterization was missing and asked for. Besides their theoretical interest, these results have immediate algorithmic applications both for the general setting and for structural decomposition methods that can be recast in terms of tree projections.

Modeling international climate change negotiations more responsibly: Can highly simplified game theory models provide reliable policy insights?

In a recent article in this journal entitled “Game Theory and Climate Diplomacy”, DeCanio and Fremstad (2013) provide an interesting treatment of a range of simple game theoretic characterizations of international climate negotiations. The authors use the Nash and Maxi-min stability definitions to analyze 25 two-by-two ordinal games, which they recognize as “possible game-theoretic characterizations of climate negotiations between two players (e.g., Great Powers or coalitions of states)”. The authors' main conclusion that the Prisoner's Dilemma might not be the best description of climate negotiations game is consistent with the findings of others who have studied two-by-two conflicts over natural commons (Bardhan, 1993; Madani, 2010; Sandler, 1992; Taylor, 1987). Nevertheless, given the importance of the climate change issue, as well as the potential effects of our actions on the state of the environment and the well-being of future generations, I would like to address some gaps in their analysis, which result in it having limited usefulness for policy purposes. Of course, all models are simplified representations of reality, full of limitations. “Essentially, all models are wrong, but some are useful” (Box and Draper, 1987). So, “the practical question is how wrong do they have to be to not be useful” (Box and Draper, 1987). Models' limitations need to be carefully considered when interpreting them or applying their results to policy but some models are too simple to provide useful policy advice.

Explaining Quantity Implicatures

We give derivations of two formal models of Gricean Quantity implicature and strong exhaustivity in bidirectional optimality theory and in a signalling games framework. We show that, under a unifying model based on signalling games, these interpretative strategies are game-theoretic equilibria when the speaker is known to be respectively minimally and maximally expert in the matter at hand. That is, in this framework the optimal strategy for communication depends on the degree of knowledge the speaker is known to have concerning the question she is answering. In addition, and most importantly, we give a game-theoretic characterisation of the interpretation rule Grice (formalising Quantity implicature), showing that under natural conditions this interpretation rule occurs in the unique equilibrium play of the signalling game.

Ideal games and Ramsey sets

It is shown that Matet’s characterization of the Ramsey property relative to a selective co-ideal H \mathcal {H} , in terms of games of Kastanas, still holds if we consider semiselectivity instead of selectivity. Moreover, we prove that a co-ideal H \mathcal {H} is semiselective if and only if Matet’s game-theoretic characterization of the H \mathcal {H} -Ramsey property holds. This lifts Kastanas’s characterization of the classical Ramsey property to its optimal setting, from the point of view of the local Ramsey theory, and gives a game-theoretic counterpart to a theorem of Farah, asserting that a co-ideal H \mathcal {H} is semiselective if and only if the family of H \mathcal {H} -Ramsey subsets of N [ ∞ ] \mathbb {N}^{[\infty ]} coincides with the family of those sets having the abstract H \mathcal {H} -Baire property. Finally, we show that under suitable assumptions, for every semiselective co-ideal H \mathcal H all sets of real numbers are H \mathcal H -Ramsey.