Strong Nash Equilibria in Games with the Lexicographical Improvement Property

We introduce a class of finite strategic games with the property that every deviation of a coalition of players that is profitable to each of its members strictly decreases the lexicographical order of a certain function defined on the set of strategy profiles. We call this property the Lexicographical Improvement Property (LIP) and show that it implies the existence of a generalized strong ordinal potential function. We use this characterization to derive existence, efficiency and fairness properties of strong Nash equilibria. We then study a class of games that generalizes congestion games with bottleneck objectives that we call bottleneck congestion games. We show that these games possess the LIP and thus the above mentioned properties. For bottleneck congestion games in networks, we identify cases in which the potential function associated with the LIP leads to polynomial time algorithms computing a strong Nash equilibrium. Finally, we investigate the LIP for infinite games. We show that the LIP does not imply the existence of a generalized strong ordinal potential, thus, the existence of SNE does not follow. Assuming that the function associated with the LIP is continuous, however, we prove existence of SNE. As a consequence, we prove that bottleneck congestion games with infinite strategy spaces and continuous cost functions possess a strong Nash equilibrium

A Game-Theoretical Model of the Landscape Theory

In this paper we examine a game-theoretical generalization of the landscape theory introduced by Axelrod and Bennett (1993). In their two-bloc setting each player ranks the blocs on the basis of the sum of her individual evaluations of members of the group. We extend the Axelrod-Bennett setting by allowing an arbitrary number of blocs and expanding the set of possible deviations to include multi-country gradual deviations. We show that a Pareto optimal landscape equilibrium which is immune to profitable gradual deviations always exists. We also indicate that while a landscape equilibrium is a stronger concept than Nash equilibrium in pure strategies, it is weaker than strong Nash equilibrium

Coalition Resilient Outcomes in Max k-Cut Games

We investigate strong Nash equilibria in the \emph{max $k$-cut game}, where we are given an undirected edge-weighted graph together with a set $\{1,\ldots, k\}$ of $k$ colors. Nodes represent players and edges capture their mutual interests. The strategy set of each player $v$ consists of the $k$ colors. When players select a color they induce a $k$-coloring or simply a coloring. Given a coloring, the \emph{utility} (or \emph{payoff}) of a player $u$ is the sum of the weights of the edges $\{u,v\}$ incident to $u$, such that the color chosen by $u$ is different from the one chosen by $v$. Such games form some of the basic payoff structures in game theory, model lots of real-world scenarios with selfish agents and extend or are related to several fundamental classes of games. Very little is known about the existence of strong equilibria in max $k$-cut games. In this paper we make some steps forward in the comprehension of it. We first show that improving deviations performed by minimal coalitions can cycle, and thus answering negatively the open problem proposed in \cite{DBLP:conf/tamc/GourvesM10}. Next, we turn our attention to unweighted graphs. We first show that any optimal coloring is a 5-SE in this case. Then, we introduce $x$-local strong equilibria, namely colorings that are resilient to deviations by coalitions such that the maximum distance between every pair of nodes in the coalition is at most $x$. We prove that $1$-local strong equilibria always exist. Finally, we show the existence of strong Nash equilibria in several interesting specific scenarios.Comment: A preliminary version of this paper will appear in the proceedings of the 45th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM'19

Routing Games with Progressive Filling

Max-min fairness (MMF) is a widely known approach to a fair allocation of bandwidth to each of the users in a network. This allocation can be computed by uniformly raising the bandwidths of all users without violating capacity constraints. We consider an extension of these allocations by raising the bandwidth with arbitrary and not necessarily uniform time-depending velocities (allocation rates). These allocations are used in a game-theoretic context for routing choices, which we formalize in progressive filling games (PFGs). We present a variety of results for equilibria in PFGs. We show that these games possess pure Nash and strong equilibria. While computation in general is NP-hard, there are polynomial-time algorithms for prominent classes of Max-Min-Fair Games (MMFG), including the case when all users have the same source-destination pair. We characterize prices of anarchy and stability for pure Nash and strong equilibria in PFGs and MMFGs when players have different or the same source-destination pairs. In addition, we show that when a designer can adjust allocation rates, it is possible to design games with optimal strong equilibria. Some initial results on polynomial-time algorithms in this direction are also derived

Approximate Equilibrium and Incentivizing Social Coordination

We study techniques to incentivize self-interested agents to form socially desirable solutions in scenarios where they benefit from mutual coordination. Towards this end, we consider coordination games where agents have different intrinsic preferences but they stand to gain if others choose the same strategy as them. For non-trivial versions of our game, stable solutions like Nash Equilibrium may not exist, or may be socially inefficient even when they do exist. This motivates us to focus on designing efficient algorithms to compute (almost) stable solutions like Approximate Equilibrium that can be realized if agents are provided some additional incentives. Our results apply in many settings like adoption of new products, project selection, and group formation, where a central authority can direct agents towards a strategy but agents may defect if they have better alternatives. We show that for any given instance, we can either compute a high quality approximate equilibrium or a near-optimal solution that can be stabilized by providing small payments to some players. We then generalize our model to encompass situations where player relationships may exhibit complementarities and present an algorithm to compute an Approximate Equilibrium whose stability factor is linear in the degree of complementarity. Our results imply that a little influence is necessary in order to ensure that selfish players coordinate and form socially efficient solutions.Comment: A preliminary version of this work will appear in AAAI-14: Twenty-Eighth Conference on Artificial Intelligenc

A new model for coalition formation games

We present two broad categories of games, namely, group matching games and bottleneck routing games on grids. Borrowing ideas from coalition formation games, we introduce a new category of games which we call group matching games. We investigate how these games perform when agents are allowed to make selfish decisions that increase their individual payoffs versus when agents act towards the social benefit of the game as a whole. The Price of Anarchy (PoA) and Price of Stability (PoS) metrics are used to quantify these comparisons. We show that the PoA for a group matching game is at most kα and the PoS is at most k/α where k is the maximum size of a group and α is a switching cost. Furthermore we show that the PoA and PoS of the games do not change significantly even if we increase γ, the number of groups that an agent is allowed to join. We also consider routing games on grid network topologies. The social cost is the worst congestion in any of the network edges (bottleneck congestion). Each player\u27s objective is to find a path that minimizes the bottleneck congestion in its path. We show that the price of anarchy in bottleneck games in grids is proportional to the number of bends β that the paths are allowed to take in the grids\u27 space. We present games where the PoA is O(β). We also give a respective lower bound of Ω(β) which shows that our upper bound is within only a poly-log factor from the best achievable price of anarchy. A significant impact of our analysis is that there exist bottleneck routing games with small number of bends which give a poly-log approximation to the optimal coordinated solution that may use an arbitrary number of bends. To our knowledge, this is the first tight analysis of bottleneck games on grids

Packing, Scheduling and Covering Problems in a Game-Theoretic Perspective

Many packing, scheduling and covering problems that were previously considered by computer science literature in the context of various transportation and production problems, appear also suitable for describing and modeling various fundamental aspects in networks optimization such as routing, resource allocation, congestion control, etc. Various combinatorial problems were already studied from the game theoretic standpoint, and we attempt to complement to this body of research. Specifically, we consider the bin packing problem both in the classic and parametric versions, the job scheduling problem and the machine covering problem in various machine models. We suggest new interpretations of such problems in the context of modern networks and study these problems from a game theoretic perspective by modeling them as games, and then concerning various game theoretic concepts in these games by combining tools from game theory and the traditional combinatorial optimization. In the framework of this research we introduce and study models that were not considered before, and also improve upon previously known results.Comment: PhD thesi