On an approach to constructing a characteristic function in cooperative differential games

We propose a novel approach to constructing characteristic functions in cooperative differential games. A characteristic function of a coalition S is computed in two stages: first, optimal control strategies maximizing the total payoff of the players are found, and next, these strategies are used by the players from the coalition S, while the other players, those from N S, use strategies minimizing the total payoff of the players from S. The characteristic function obtained in this way is superadditive. In addition, it possesses a number of other useful properties. As an example, we compute values of a characteristic function for a specific differential game of pollution control.     

Guaranteed approach with the farthest of the runaways

Games of the family {Λ _N } _N?2 are formulated and studied with the application of generalized Isaacs’s approach. The game Λ _N is a simplest model of the counteraction of one persecutor P and coalition N of E ^N runaways for the case when the payoff is the distance up to the coalition of E ^N equal to the Euclidean distance between P and the farthest from the runaways; P is in command of the termination moment. Moreover, an approach within the limits of which in games with a smooth terminal payoff are generated strategies prescribing players’ motions in the directions of local gradients of the payoff is described. The approach is used for constructing pursuit strategies in games in which smooth approximations of the maximum of Euclidean distances up to the runaways are in place of payoffs. Pursuit strategies prescribing the motion in the direction of the farthest of the runaways are studied. A numerical simulation of the development of the games Λ₂ and Λ₃ is conducted in using different strategies by the players.     

Majority bargaining for resource division

We address the problem of how a set of agents can decide to share a resource, represented as a unit-sized pie. The pie can be generated by the entire set but also by some of its subsets. We investigate a finite horizon non-cooperative bargaining game, in which the players take it in turns to make proposals on how the resource should for this purpose be allocated, and the other players vote on whether or not to accept the allocation. Voting is modelled as a Bayesian weighted voting game with uncertainty about the players’ weights. The agenda, (i.e., the order in which the players are called to make offers), is defined exogenously. We focus on impatient players with heterogeneous discount factors. In the case of a conflict, (i.e., no agreement by the deadline), no player receives anything. We provide a Bayesian subgame perfect equilibrium for the bargaining game and conduct an ex-ante analysis of the resulting outcome. We show that the equilibrium is unique, computable in polynomial time, results in an instant Pareto optimal outcome, and, under certain conditions provides a foundation for the core and also the nucleolus of the Bayesian voting game. In addition, our analysis leads to insights on how an individual’s bargained share is influenced by his position on the agenda. Finally, we show that, if the conflict point of the bargaining game changes, then the problem of determining the non-cooperative equilibrium becomes NP-hard even under the perfect information assumption. Our research also reveals how this change in conflict point impacts on the above mentioned results.     

Novel steganographic method based on generalized <Emphasis Type="Italic">K</Emphasis>-distance <Emphasis Type="Italic">N</Emphasis>-dimensional pixel matching

In this paper, a steganographic scheme adopting the concept of the generalized K _d -distance N-dimensional pixel matching is proposed. The generalized pixel matching embeds a B-ary digit (B is a function of K and N) into a cover vector of length N, where the order-d Minkowski distance-measured embedding distortion is no larger than K. In contrast to other pixel matching-based schemes, a N-dimensional reference table is used. By choosing d, K, and N adaptively, an embedding strategy which is suitable for arbitrary relative capacity can be developed. Additionally, an optimization algorithm, namely successive iteration algorithm (SIA), is proposed to optimize the codeword assignment in the reference table. Benefited from the high dimensional embedding and the optimization algorithm, nearly maximal embedding efficiency is achieved. Compared with other content-free steganographic schemes, the proposed scheme provides better image quality and statistical security. Moreover, the proposed scheme performs comparable to state-of-the-art content-based approaches after combining with image models.     

<Emphasis Type="Italic">N</Emphasis>-person credibilistic strategic game

This paper enlarges the scope of fuzzy-payoff game to n-person form from the previous two-person form. Based on credibility theory, three credibilistic approaches are introduced to model the behaviors of players in different decision situations. Accordingly, three new definitions of Nash equilibrium are proposed for n-person credibilistic strategic game. Moreover, existence theorems are proved for further research into credibilistic equilibrium strategies. Finally, two numerical examples are given to illustrate the significance of credibilistic equilibria in practical strategic games.     

Chain equilibria in secure strategies

In this paper we introduce a modification of the concept of Equilibrium in Secure Strategies (EinSS), which takes into account the non-uniform attitudes of players to security in non-cooperative games. In particular, we examine an asymmetric attitude of players to mutual threats in the simplest case, when all players are strictly ordered by their relation to security. Namely, we assume that the players can be reindexed so that each player i in his behavior takes into account the threats posed by players j > i but ignores the threats of players j < i provided that these threats are effectively contained by some counterthreats. A corresponding equilibrium will be called a Chain EinSS. The conceptual meaning of this equilibrium is illustrated by two continuous games that have no pure Nash equilibrium or (conventional) EinSS. The Colonel Blotto two-player game (Borel 1953; Owen 1968) for two battlefields with different price always admits a Chain EinSS with intuitive interpretation. The product competition of many players on a segment (Eaton, Lipsey 1975; Shaked 1975) with the linear distribution of consumer preferences always admits a unique Chain EinSS solution (up to a permutation of players). Finally, we compare Chain EinSS with Stackelberg equilibrium.     

Altruistic behavior in a nonantagonistic positional differential game

We consider a two-person nonantagonistic positional differential game (NPDG) whose dynamics is described by an ordinary nonlinear vector differential equation. Constraints on values of players’ controls are geometric. Final time of the game is fixed. Payoff functionals of both players are terminal. The formalization of positional strategies in an NPDG is based on the formalization and results of the general theory of antagonistic positional differential games (APDGs) (see monographs by N.N. Krasovskii and A.I. Subbotin [3, 4]). Additionally, in the present paper we assume that each player, together with the usual, normal (nor), type of behavior aimed at maximizing his own functional, can use other behavior types introduced in [2, 5]. In particular, these may be altruistic (alt), aggressive (agg), and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. Using the possibility of such switches in a repeated bimatrix 2 × 2 game in [5, 6] allowed to obtain new solutions of this game. In the present paper, extension of this approach to NPDGs leads to a new formulation of the problem. In particular, of interest is the question of how players’ outcomes at Nash solutions are transformed. An urgent problem is minimizing the time of “abnormal” behavior while achieving a good result. The paper proposes a formalization of an NPDG with behavior types (NPDGwBT). It is assumed that in an NPDGwBT each player, simultaneously with choosing a positional strategy, chooses also his own indicator function defined on the whole game horizon and taking values in the set {normal, altruistic, aggressive, paradoxical}. The indicator function of a player shows the dynamics of changes in the behavior type demonstrated by the player. Thus, in this NPDGwBT each player controls the choice of a pair {positional strategy, indicator function}. We define the notion of a BT-solution of such a game. It is expected that using behavior types in the NPDGwBT which differ from the normal one (so-called abnormal types) in some cases may lead to more favorable outcomes for the players than in the NPDG. We consider two examples of an NPDGwBT with simple dynamics in the plane in each of which one player keeps to altruistic behavior type over some time period. It is shown that in the first example payoffs of both players increase on a BT-solution as compared to the game with the normal behavior type, and in the second example, the sum of players’ payoffs is increased.     

Partitions of the set of selected unknowns in linear differential–algebraic systems

As was shown earlier, for a linear differential–algebraic system A ₁ y′ + A ₀ y = 0 with a selected part of unknowns (entries of a column vector y), it is possible to construct a differential system ?′ = B ?, where the column vector ? is formed by some entries of y, and a linear algebraic system by means of which the selected entries that are not contained in ? can be expressed in terms of the selected entries included in ?. In the paper, sizes of the differential and algebraic systems obtained are studied. Conditions are established under the fulfillment of which the size of the algebraic system is determined unambiguously and the size of the differential system is minimal.     

Mixed <Emphasis Type="Italic">H</Emphasis><Subscript>2</Subscript>/<Emphasis Type="Italic">H</Emphasis><Subscript>∞</Subscript> control for linear infinite-dimensional systems

In this paper, we consider mixed H ₂/H _∞ control problems for linear infinite-dimensional systems. The first part considers the state feedback control for the H ₂/H _∞ control problems of linear infinite-dimensional systems. The cost horizon can be infinite or finite time. The solutions of the H ₂/H _∞ control problem for linear infinitedimensional systems are presented in terms of the solutions of the coupled operator Riccati equations and coupled differential operator Riccati equations. The second part addresses the observer-based H ₂/H _∞ control of linear infinite-dimensional systems with infinite horizon and finite horizon costs. The solutions for the observer-based H ₂/H _∞ control problem of linear infinite-dimensional systems are represented in terms of the solutions of coupled operator Riccati equations. The first-order partial differential system examples are presented for illustration. In particular, for these examples, the Riccati equations are represented in terms of the coefficients of first-order partial differential systems.     

When Can Limited Randomness Be Used in Repeated Games?

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the n-stage repeated version of the game exist if and only if both players have Ω(n) random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the n-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on “random-like” sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators do not exist, then Ω(n) random bits remain necessary for equilibria to exist.     

Mean field games based on stable-like processes

We investigate the mean field games of N agents based on the nonlinear stable-like processes. The main result of the paper is that any solution of the limiting mean field consistency equation generates a 1/N-Nash equilibrium for the approximating game of N agents.     

Protecting query privacy with differentially private <Emphasis Type="Italic">k</Emphasis>-anonymity in location-based services

Nowadays, location-based services (LBS) are facilitating people in daily life through answering LBS queries. However, privacy issues including location privacy and query privacy arise at the same time. Existing works for protecting query privacy either work on trusted servers or fail to provide sufficient privacy guarantee. This paper combines the concepts of differential privacy and k-anonymity to propose the notion of differentially private k-anonymity (DPkA) for query privacy in LBS. We recognize the sufficient and necessary condition for the availability of 0-DPkA and present how to achieve it. For cases where 0-DPkA is not achievable, we propose an algorithm to achieve ??-DPkA with minimized ??. Extensive simulations are conducted to validate the proposed mechanisms based on real-life datasets and synthetic data distributions.     

<Emphasis Type="Italic">N</Emphasis>-body gravitational and contact dynamics for asteroid aggregation

The development of dedicated numerical codes has recently pushed forward the study of N-body gravitational dynamics, leading to a better and wider understanding of processes involving the formation of natural bodies in the Solar System. A major branch includes the study of asteroid formation: evidence from recent studies and observations support the idea that small and medium size asteroids between 100 m and 100 km may be gravitational aggregates with no cohesive force other than gravity. This evidence implies that asteroid formation depends on gravitational interactions between different boulders and that asteroid aggregation processes can be naturally modeled with N-body numerical codes implementing gravitational interactions. This work presents a new implementation of an N-body numerical solver. The code is based on Chrono::Engine (2006). It handles the contact and collision of large numbers of complex-shaped objects, while simultaneously evaluating the effect of N to N gravitational interactions. A special case of study is considered, investigating the relative dynamics between the N bodies and highlighting favorable conditions for the formation of a stable gravitationally bound aggregate from a cloud of N boulders. The code is successfully validated for the case of study by comparing relevant results obtained for typical known dynamical scenarios. The outcome of the numerical simulations shows good agreement with theory and observation, and suggests the ability of the developed code to predict natural aggregation phenomena.     

On Supersymmetric M-brane configurations with an R<Stack><Subscript>*</Subscript><Superscript>1,1</Superscript></Stack> /Z<Subscript>2</Subscript> submanifold

We obtain new examples of partly supersymmetric M-brane solutions defined on products of Ricci-flat manifolds, which contain a two-dimensional Lorentzian submanifold R _* ^1,1 /Z ₂ with one parallel spinor. The examples belong to the following configurations: M2, M5, M2 ∩M5 and M5 ∩M5. Among them, an M2 solution with N = 1/32 fractional number of preserved supersymmetries is presented.     

Power and welfare in bargaining for coalition structure formation

We investigate a noncooperative bargaining game for partitioning n agents into non-overlapping coalitions. The game has n time periods during which the players are called according to an exogenous agenda to propose offers. With probability \(\delta \), the game ends during any time period \(t<n\). If it does, the first t players on the agenda get a chance to propose but the others do not. Thus, \(\delta \) is a measure of the degree of democracy within the game (ranging from democracy for \(\delta =0\), through increasing levels of authoritarianism as \(\delta \) approaches 1, to dictatorship for \(\delta =1\)). We determine the subgame perfect equilibrium (SPE) and study how a player’s position on the agenda affects his bargaining power. We analyze the relation between the distribution of power of individual players, the level of democracy, and the welfare efficiency of the game. We find that purely democratic games are welfare inefficient and that introducing a degree of authoritarianism into the game makes the distribution of power more equitable and also maximizes welfare. These results remain invariant under two types of player preferences: one where each player’s preference is a total order on the space of possible coalition structures and the other where each player either likes or dislikes a coalition structure. Finally, we show that the SPE partition may or may not be core stable.     

On the <Emphasis Type="Italic">k</Emphasis>-accessibility of cores of <Emphasis Type="Italic">TU</Emphasis>-cooperative games

This paper proposes a strengthening of the author’s core-accessibility theorem for balanced TU-cooperative games. The obtained strengthening relaxes the influence of the nontransitivity of classical domination αv on the quality of the sequential improvement of dominated imputations in a game v. More specifically, we establish the k-accessibility of the core C(α _v ) of any balanced TU-cooperative game v for all natural numbers k: for each dominated imputation x, there exists a converging sequence of imputations x₀, x₁,..., such that x₀ = x, lim x _r ∈ C(α _v ) and x_r?m is dominated by any successive imputation x _r with m ∈ [1, k] and r ≥ m. For showing that the TU-property is essential to provide the k-accessibility of the core, we give an example of an NTU-cooperative game G with a ”black hole” representing a nonempty closed subset B ? G(N) of dominated imputations that contains all the α _G -monotonic sequential improvement trajectories originating at any point x ∈ B.     

The bounded core for games with restricted cooperation

A game with restricted (incomplete) cooperation is a triple (N, v, Ω), where N represents a finite set of players, Ω ? 2^N is a set of feasible coalitions such that N ∈ Ω, and v: Ω → R denotes a characteristic function. Unlike the classical TU games, the core of a game with restricted cooperation can be unbounded. Recently Grabisch and Sudhölter [9] proposed a new solution concept—the bounded core—that associates a game (N, v,Ω) with the union of all bounded faces of the core. The bounded core can be empty even if the core is nonempty. This paper gives two axiomatizations of the bounded core. The first axiomatization characterizes the bounded core for the class G^r of all games with restricted cooperation, whereas the second one for the subclass G_bc^r ? G^r of the games with nonempty bounded cores.     

Pseudo-Randomness of Certain Sequences of <Emphasis Type="Italic">k</Emphasis> Symbols with Length <Emphasis Type="Italic">pq</Emphasis>

The theory of finite pseudo-random binary sequences was built by C. Mauduit and A. Sárközy and later extended to sequences of k symbols (or k-ary sequences). Certain constructions of pseudo-random sequences of k symbols were presented over finite fields in the literature. In this paper, two families of sequences of k symbols are constructed by using the integers modulo pq for distinct odd primes p and q. The upper bounds on the well-distribution measure and the correlation measure of the families sequences are presented in terms of certain character sums over modulo pq residue class rings. And low bounds on the linear complexity profile are also estimated.     

Stochastic coalitional better-response dynamics and stable equilibrium

We consider coalition formation among players in an n-player finite strategic game over infinite horizon. At each time a randomly formed coalition makes a joint deviation from a current action profile such that at new action profile all the players from the coalition are strictly benefited. Such deviations define a coalitional better-response (CBR) dynamics that is in general stochastic. The CBR dynamics either converges to a K-stable equilibrium or becomes stuck in a closed cycle. We also assume that at each time a selected coalition makes mistake in deviation with small probability that add mutations (perturbations) into CBR dynamics. We prove that all K-stable equilibria and all action profiles from closed cycles, that have minimum stochastic potential, are stochastically stable. Similar statement holds for strict K-stable equilibrium. We apply the CBR dynamics to study the dynamic formation of the networks in the presence of mutations. Under the CBR dynamics all strongly stable networks and closed cycles of networks are stochastically stable.     

The formula of the solution for some classes of initial boundary value problems for the hyperbolic equation with two independent variables

A new representation is proved of the solutions of initial boundary value problems for the equation of the form u _xx (x, t) + r(x)u _x (x, t) ? q(x)u(x, t) = u _tt (x, t) + μ(x)u _t (x, t) in the section (under boundary conditions of the 1st, 2nd, or 3rd type in any combination). This representation has the form of the Riemann integral dependent on the x and t over the given section.