Special Class Of Games Research Articles

Persuasion with Coarse Communication

How does an expert's ability persuade change with the availability of messages? We study games of Bayesian persuasion the sender is unable to fully describe every state of the world or recommend all possible actions. We characterize the set of attainable payoffs. Sender always does worse with coarse communication and values additional signals. We show that there exists an upper bound on the marginal value of a signal for the sender. In a special class of games, the marginal value of a signal is increasing when the receiver is difficult to persuade. We show that an additional signal does not directly translate into more information and the receiver might prefer coarse communication. Finally, we study the geometric properties of optimal information structures. Using these properties, we show that the sender's optimization problem can be solved by searching within a finite set.

On Fair Cost Sharing Games in Machine Learning

Machine learning and game theory are known to exhibit a very strong link as they mutually provide each other with solutions and models allowing to study and analyze the optimal behaviour of a set of agents. In this paper, we take a closer look at a special class of games, known as fair cost sharing games, from a machine learning perspective. We show that this particular kind of games, where agents can choose between selfish behaviour and cooperation with shared costs, has a natural link to several machine learning scenarios including collaborative learning with homogeneous and heterogeneous sources of data. We further demonstrate how the game-theoretical results bounding the ratio between the best Nash equilibrium (or its approximate counterpart) and the optimal solution of a given game can be used to provide the upper bound of the gain achievable by the collaborative learning expressed as the expected risk and the sample complexity for homogeneous and heterogeneous cases, respectively. We believe that the established link can spur many possible future implications for other learning scenarios as well, with privacy-aware learning being among the most noticeable examples.

The extended Shapley value for generalized cooperative games under precedence constraints

We introduce a new class of games where cooperation among players is restricted by precedence constraints vis a vis the worth of a coalition depends on the order in which the players enter into the coalition. The idea combines two existing classes of cooperative games, namely the cooperative games under precedence constraints due to Faigle and Kern (Int J Game Theory 21(3):249–266, 1992) and the games in generalized characteristic function due to Nowak and Radzik (Games Econ Behav 6(1):150–161, 1994). A Shapley value for this special class of games is proposed, we call it the extended Shapley value to distinguish it for the existing one. Two axiomatic characterizations of the extended Shapley value are given: one uses Efficiency, Null player, and Linearity; the other uses Efficiency, Marginality, and Null game. Some interesting properties of the extended Shapley value are studied. Furthermore, two extensions of the extended Shapley value, called the extended probabilistic value and the extended order value, are proposed and characterized. Our study shows that the results in cooperative games under precedence constraints cannot have a trivial extension to the generalized constrained games and conversely.

Games on Large Networks: Information and Complexity

In this work, we study Static and Dynamic Games on Large Networks of interacting agents, assuming that the players have some statistical description of the interaction graph, as well as some local information. Inspired by Statistical Physics, we consider statistical ensembles of games and define a Probabilistic Approximate equilibrium notion for such ensembles. A Necessary Information Complexity notion is introduced to quantify the minimum amount of information needed for the existence of a Probabilistic Approximate equilibrium. We then focus on some special classes of games for which it is possible to derive upper and/or lower bounds for the complexity. At first, static and dynamic games on random graphs are studied and their complexity is determined as a function of the graph connectivity. In the low complexity case, we compute Probabilistic Approximate equilibrium strategies. We then consider static games on lattices and derive upper and lower bounds for the complexity, using contraction mapping ideas. A LQ game on a large ring is also studied numerically. Using a reduction technique, approximate equilibrium strategies are computed and it turns out that the complexity is relatively low.

Games with a Local Permission Structure: Separation of Authority and Value Generation

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of (weighted) digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games with a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only in order to generate worth, but does not need its predecessors in order to give permission to its own successors. We introduce and axiomatize a Shapley value type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the beta-measure for weighted digraphs.

Stationary Anonymous Sequential Games with Undiscounted Rewards.

Stationary anonymous sequential games with undiscounted rewards are a special class of games that combine features from both population games (infinitely many players) with stochastic games. We extend the theory for these games to the cases of total expected reward as well as to the expected average reward. We show that in the anonymous sequential game equilibria correspond to the limits of those of related finite population games as the number of players grows to infinity. We provide examples to illustrate our results.

Games with a local permission structure: separation of authority and value generation

It is known that peer group games are a special class of games with a permission structure. However, peer group games are also a special class of (weighted) digraph games. To be specific, they are digraph games in which the digraph is the transitive closure of a rooted tree. In this paper we first argue that some known results on solutions for peer group games hold more general for digraph games. Second, we generalize both digraph games as well as games with a permission structure into a model called games with a local permission structure, where every player needs permission from its predecessors only to generate worth, but does not need its predecessors to give permission to its own successors. We introduce and axiomatize a Shapley value-type solution for these games, generalizing the conjunctive permission value for games with a permission structure and the \(\beta \)-measure for weighted digraphs.

On the Complexity of Pareto-Optimal Nash and Strong Equilibria

We consider the computational complexity of coalitional solution concepts in scenarios related to load balancing such as anonymous and congestion games. In congestion games, Pareto-optimal Nash and strong equilibria, which are resilient to coalitional deviations, have recently been shown to yield significantly smaller inefficiency. Unfortunately, we show that several problems regarding existence, recognition, and computation of these concepts are hard, even in seemingly special classes of games. In anonymous games with constant number of strategies, we can efficiently recognize a state as Pareto-optimal Nash or strong equilibrium, but deciding existence for a game remains hard. In the case of player-specific singleton congestion games, we show that recognition and computation of both concepts can be done efficiently. In addition, in these games there are always short sequences of coalitional improvement moves to Pareto-optimal Nash and strong equilibria that can be computed efficiently.

An application of optimization theory to the study of equilibria for games: a survey

This contribution is a survey about potential games and their applications. In a potential game the information that is sufficient to determine Nash equilibria can be summarized in a single function on the strategy space: the potential function. We show that the potential function enable the application of optimization theory to the study of equilibria. Potential games and their generalizations are presented. Two special classes of games, namely team games and separable games, turn out to be potential games. Several properties satisfied by potential games are discussed and examples from concrete situations as congestion games, global emission games and facility location games are illustrated.

TRANSBOUNDARY WATER MANAGEMENT: CAN ISSUE LINKAGE HELP MITIGATE EXTERNALITIES?

Managing transboundary river basins is never easy and usually involves conflicts. This paper introduces a special class of games with externalities and issue linkage to promote cooperation on transboundary water resources. The paper analyzes whether issue linkages can be used as a form of negotiations on sharing benefits and mitigating conflicts. It is shown that whenever opportunities for linkages exist, countries may indeed contribute towards cooperation. In particular, if the linked games are convex, the grand coalition is the only optimal level of social welfare.

Contribution Games in Networks

We consider network contribution games, where each agent in a network has a budget of effort that he can contribute to different collaborative projects or relationships. Depending on the contribution of the involved agents a relationship will flourish or drown, and to measure the success we use a reward function for each relationship. Every agent is trying to maximize the reward from all relationships that it is involved in. We consider pairwise equilibria of this game, and characterize the existence, computational complexity, and quality of equilibrium based on the types of reward functions involved. When all reward functions are concave, we prove that the price of anarchy is at most 2. For convex functions the same only holds under some special but very natural conditions. Another special case extensively treated are minimum effort games, where the reward of a relationship depends only on the minimum effort of any of the participants. In these games, we can show existence of pairwise equilibrium and a price of anarchy of 2 for concave functions and special classes of games with convex functions. Finally, we show tight bounds for approximate equilibria and convergence of dynamics in these games.

Playing monotone games to understand learning behaviors

We deal with a special class of games against nature which correspond to subsymbolic learning problems where we know a local descent direction in the error landscape but not the amount gained at each step of the learning procedure. Namely, Alice and Bob play a game where the probability of victory grows monotonically by unknown amounts with the resources each employs. For a fixed effort on Alice’s part Bob increases his resources on the basis of the results of the individual contests (victory, tie or defeat). Quite unlike the usual ones in game theory, his aim is to stop as soon as the defeat probability goes under a given threshold with high confidence. We adopt such a game policy as an archetypal remedy to the general overtraining threat of learning algorithms. Namely, we deal with the original game in a computational learning framework analogous to the Probably Approximately Correct formulation. Therein, a wise use of a special inferential mechanism (known as twisting argument) highlights relevant statistics for managing different trade-offs between observability and controllability of the defeat probability. With similar statistics we discuss an analogous trade-off at the basis of the stopping criterion of subsymbolic learning procedures. As a conclusion, we propose a principled stopping rule based solely on the behavior of the training session, hence without distracting examples into a test set.

Potential games and well-posedness properties

The aim of this article is to study potential games which are a special class of games, in fact their properties are dictated by a single function called the potential function. We consider Tikhonov well-posedness and other well-posedness properties introduced by the authors in Margiocco et al. (Margiocco, M., Patrone, F. and Pusillo Chicco, L., 1997, A new approach to Tikhonov well–posedness for Nash equilibria. Optimization, 40, 385–400) Margiocco and Pusillo (Margiocco, M. and Pusillo, L., Value bounded approximations for Nash equilibria, Preprint, Submitted). We relate these properties with the Tikhonov well posedness of the potential function as maximum problem.

8.1.2 A Study of Applying Game Theoretic Concepts on Distributed Engineering System Design

AbstractThis paper considers the application of game theoretic concepts to the dynamics of interactions among the designers in an engineering systems design problem under a distributed decision making environment. Among all the possible distributed frameworks, we focus on two types of frameworks in the system design field, the simultaneous and the sequential response frameworks. In the investigation of the simultaneous response framework, we apply the concepts of learning theory to construct insights into the dynamics among the designers and to address convergence and other challenges of the decentralized design problems. We use a special class of games, the potential game, to demonstrate the effectiveness of the sequential response framework which points out a prospective application of game theoretic concepts to future distributed engineering systems problems.

Minimal Inclusive Sets In Special Classes Of Games

A companion paper, Sanchirico (1996), provides a probabilistic theory of learning in games with the convergence property that, almost surely, play will remain almost always (i.e., forever after some point) within one of the stage game's "minimal inclusive sets." This paper investigates the size of minimal inclusive sets in several classes of games, notably, those for which other learning processes have been shown to converge (in various manners weaker than convergence of actual play). These include certain supermodular games, congestion games, potential games, games with identical interests, and games with bandwagon effects. It is shown that in all these classes, if all of a game?s pure equilibria are strict (a fortiori, if its payoffs are generic), then all of its minimal inclusive sets will be singletons consisting of Nash equilibria.

Auctions, public tenders, and fair division games: An axiomatic approach

Auctions, public tenders, and fair division games are considered as special classes of games with incomplete information. The specialty of these games is that choosing a strategy in such a game amounts to displaying (the not necessarily true) preferences. Our main axiom of displayed envy-freeness states that according to his displayed preferences no player should prefer another player's net trade to his own. This axiom and the well-known property of incentive compatibility imply the rules of auctions and public tenders which are originally discussed by Vickrey. We consider our axiomatic characterization as a strong support for the Vickrey-rules. There is no obvious reason why the actually applied rules (e.g. the rules of public tenders in the Federal Republic of Germany) do often violate these rules. For fair division problems the two axioms are shown to be mutually inconsistent. By weakening the requirement of incentive compatibility we, nevertheless, can determine rules for fair division problems which are a reasonable analogue of the Vickrey-rules. Finally, it is discussed how our ideas can be extended to other allocation problems.

Equilibria for far-sighted players

A new equilibrium concept for non-cooperative games, based on the assumptions that players are rational and far-sighted, is examined. An outcome is extended non-myopically (XNM) stable for a player if that player is assured that no movecountermove sequence he could initiate by departing unilaterally from that outcome would benefit him. The extended non-myopic (XNM) equilibria of a game, the outcomes which are XNM stable for each player, therefore model permanent (enduring) equilibria in an ongoing conflict. Algorithms for the identification of XNM equilibria in a 2 × 2 game are presented. The XNM concepts are then applied to three special classes of games (no-conflict games, games of complete opposition, and strict ordinal games) to compare their predictions of long-term stability with the known properties of games in these classes.

Finite Solution Theory for Coalitional Games

In 1944 von Neumann and Morgenstern introduced a theory of solutions (stable sets) for multiperson cooperative games in characteristic function form. Some special classes of games are known to have solutions which are finite sets. These finite solutions give rise to interesting geometrical structures and basic combinatorial patterns. They have provided new insights into problems in the social sciences and they invite additional interpretations and uses in the mathematical and physical sciences. This paper provides an introduction and survey of finite solution theory.

Differential games with open-loop saddle point conditions

Zero-sum differential games are considered in which one or both of the players are restricted to use open-loop control. It is shown that the first-order necessary conditions for such problems are identical to the first-order necessary conditions for the usual form of a differential game, where both players use closed-loop control laws. An investigation of the conjugate point condition for a special class of games shows that this condition is not the same but depends on the type of solution sought. For games where one or both of the players use open-loop control, there are two conjugate point conditions that must be satisfied. This differs from games in which both players use closed-loop control, where there is only one conjugate point necessary condition.