Abstract Bisimulations are a key notion to study the expressive power of a modal language. This paper studies the expressivity of Game Description Language (GDL) and its epistemic extension Epistemic GDL (EGDL) through a bisimulation approach. We first define a notion of bisimulation for GDL and prove that it coincides with the indistinguishability of GDL formulas. Based on it, we establish a characterization of the definability of GDL in terms of $k$-bisimulations. Then we design novel notions of bisimulation for EGDL and obtain characterizations of the expressive power of EGDL in terms of them. These characterizations provide a powerful tool to identify the expressive power of game description languages. Finally, we demonstrate with real games that bisimulation can be generalized to capture a wide range of game equivalence.

Judgment aggregation deals with the problem of how collective judgments on logically connected propositions can be formed based on individual judgments on the same propositions. The existing literature on judgment aggregation mainly focuses on the anonymity condition requiring that individual judgments be treated equally. However, in many real‐world situations, a group making collective judgments may assign individual members or subgroups different priorities to determine the collective judgment. Based on this consideration, this article relaxes the anonymity condition by giving a hierarchy over individuals so as to investigate how the judgment from each individual affects the group judgment in such a hierarchical environment. Moreover, we assume that an individual can abstain from voting on a proposition and the collective judgment on a proposition can be undetermined, which means that we do not require completeness at both individual and collective levels. In this new setting, we first identify an impossibility result and explore a set of plausible conditions in terms of abstentions. Secondly, we develop an aggregation rule based on the hierarchy of individuals and show that the aggregation rule satisfies those plausible conditions. The computational complexity of this rule is also investigated. Finally, we show that the proposed rule is (weakly) oligarchic over a subset of agenda. This is by no means a negative result. In fact, our result reveals that with abstentions, oligarchic aggregation is not necessary to be a single‐level determination but can be a multiple‐level collective decision making, which partially explains its ubiquity in the real world.

This paper proposes a logical model of multi-demand bargaining with integrity constraints. We also construct a simultaneous concession solution to bargaining games of this kind and show that the solution is uniquely characterized by a set of logical properties. Moreover, we prove that the solution also satisfies the most fundamental game theoretic properties such as symmetry and Pareto optimality. In addition, by lots of simulation experiments we study how the number of conflicting demands, bargainers' risk attitude, and bargainer number influence the bargaining success rate and efficiency as well as the agreement quality.

This paper introduces a modal logic for reasoning about game strategies. The logic is based on a variant of the well-known game description language for describing game rules and further extends it with two modalities for reasoning about actions and strategies. We develop an axiomatic system and prove its soundness and completeness with respect to a specific semantics based on the state transition model of games. Interestingly, the completeness proof makes use of forgetting techniques that have been widely used in the KR&R literature. We demonstrate how general game-playing systems can apply the logic to develop game strategies.

This paper presents a framework for relevance-based belief change in propositional Horn logic. We firstly establish a parallel interpolation theorem for Horn logic and show that Parikh's Finest Splitting Theorem holds with Horn formulae. By reformulating Parikh's relevance criterion in the setting of Horn belief change, we construct a relevance-based partial meet Horn contraction operator and provide a representation theorem for the operator. Interestingly, we find that this contraction operator can be fully characterised by Delgrande and Wassermann's postulates for partial meet Horn contraction as well as Parikh's relevance postulate without requiring any change on the postulates, which is qualitatively different from the case in classical propositional logic.

Studies on mechanism design mostly focus on a single market where sellers and buyers trade. This paper examines the problem of mechanism design for capacity allocation in two connected markets where a supplier allocates products to a set of retailers and the retailers resale the products to end-users in price competition. We consider the problems of how allocation mechanisms in the upstream market determine the behaviors of markets in the downstream market and how pricing policy in the downstream market influences the properties of allocation mechanisms. We classify an effective range of capacity that influences pricing strategies in the downstream market according to allocated quantities. Within the effective capacity range, we show that the retailers tend to inflate orders under proportional allocation, but submit truthful orders under uniform allocation. We observe that heterogeneous allocations results in greater total retailer profit which is a unique phenomenon in our model. The results would be applied to the design and analysis of Business-to-Business (B2B) marketplaces and supply chain management.

Shapley's impossibility result indicates that the two-person bargaining problem has no non-trivial ordinal solution with the traditional game-theoretic bargaining model. Although the result is no longer true for bargaining problems with more than two agents, none of the well known bargaining solutions are ordinal. Searching for meaningful ordinal solutions, especially for the bilateral bargaining problem, has been a challenging issue in bargaining theory for more than three decades. This paper proposes a logic-based ordinal solution to the bilateral bargaining problem. We argue that if a bargaining problem is modeled in terms of the logical relation of players' physical negotiation items, a meaningful bargaining solution can be constructed based on the ordinal structure of bargainers' preferences. We represent bargainers' demands in propositional logic and bargainers' preferences over their demands in total preorder. We show that the solution satisfies most desirable logical properties, such as individual rationality (logical version), consistency, collective rationality as well as a few typical game-theoretic properties, such as weak Pareto optimality and contraction invariance. In addition, if all players' demand sets are logically closed, the solution satisfies a fixed-point condition, which says that the outcome of a negotiation is the result of mutual belief revision. Finally, we define various decision problems in relation to our bargaining model and study their computational complexity.

This paper addresses a problem in supply chain management that how scarce resources can be efficiently allocated among competing interests. We present a formal model of allocation mechanisms for such settings that a supplier with limited production capacity allocates its products to a set of competitive retailers. In contrary to the existing allocation mechanisms in which retailers are local monopolists, the new model exhibits much more complicated market behaviors. We show that the widely-used proportional allocation mechanism is no longer necessarily Pareto optimal, even if all retailers are in a symmetric situation. A necessary and sufficient condition for the proportional allocation to be Pareto optimal is given. We propose a truth-inducing allocation mechanism based on our capacity allocation model, which is more intuitive and applicable than the existing truth-inducing mechanisms.