We study network connection games where the nodes of a network perform edge swaps in order to improve their communication costs. For the model proposed by Alon et al. (2010), in which the selfish cost of a node is the sum of all shortest path distances to the other nodes, we use the probabilistic method to provide a new, structural characterization of equilibrium graphs. We show how to use this characterization in order to prove upper bounds on the diameter of equilibrium graphs in terms of the size of the largest $k$-vicinity (defined as the the set of vertices within distance $k$ from a vertex), for any $k \geq 1$ and in terms of the number of edges, thus settling positively a conjecture of Alon et al. in the cases of graphs of large $k$-vicinity size (including graphs of large maximum degree) and of graphs which are dense enough.
Next, we present a new swap-based network creation game, in which selfish costs depend on the immediate neighborhood of each node; in particular, the profit of a node is defined as the sum of the degrees of its neighbors. We prove that, in contrast to the previous model, this network creation game admits an exact potential, and also that any equilibrium graph contains an induced star. The existence of the potential function is exploited in order to show that an equilibrium can be reached in expected polynomial time even in the case where nodes can only acquire limited knowledge concerning non-neighboring nodes.
Quitting games are one of the simplest stochastic games in which at any stage each player has only two possible actions, continue and quit. The game ends as soon as at least one player chooses to quit. The players then receive a payoff, which depends on the set of players that did choose to quit. If the game never ends, the payoff to each player is zero. For analysis of quitting games the so called one-step games are used. Important properties of the expected payoff and of equilibria in one-step games are stated. Furthermore some relations between equilibria in one-step games and equilibria in…
Many models from a variety of areas involve the computation of an equilibrium or fixed point of some kind. Examples include Nash equilibria in games; market equilibria; computing optimal strategies and the values of competitive games (stochastic and other games); stable configurations of neural networks; analysing basic stochastic models for evolution like branching processes and for language like stochastic context-free grammars; and models that incorporate the basic primitives of probability and recursion like recursive Markov chains. It is not known whether these problems can be solved in p…
This note, suitable for a lecture in an advanced undergraduate or basic graduate course on the economic theory of networks, exposits basic ideas of linear best-response games and their equilibria.
In security games, the solution concept commonly used is that of a Stackelberg equilibrium where the defender gets to commit to a mixed strategy. The motivation for this is that the attacker can repeatedly observe the defender's actions and learn her distribution over actions, before acting himself. If the actions were not observable, Nash (or perhaps correlated) equilibrium would arguably be a more natural solution concept. But what if some, but not all, aspects of the defender's actions are observable? In this paper, we introduce solution concepts corresponding to this case, both with and wi…
We present a simple primal-dual algorithm for computing approximate Nash-equilibria in two-person zero-sum sequential games with incomplete information and perfect recall (like Texas Hold'em Poker). Our algorithm is numerically stable, performs only basic iterations (i.e matvec multiplications, clipping, etc., and no calls to external first-order oracles, no matrix inversions, etc.), and is applicable to a broad class of two-person zero-sum games including simultaneous games and sequential games with incomplete information and perfect recall. The applicability to the latter kind of games is th…
A recent body of experimental literature has studied empirical game-theoretical analysis, in which we have partial knowledge of a game, consisting of observations of a subset of the pure-strategy profiles and their associated payoffs to players. The aim is to find an exact or approximate Nash equilibrium of the game, based on these observations. It is usually assumed that the strategy profiles may be chosen in an on-line manner by the algorithm. We study a corresponding computational learning model, and the query complexity of learning equilibria for various classes of games. We give basic res…
We show that a Bayesian strategy map profile is a Bayesian Nash Equilibrium independent of any prior if and only if the Bayesian strategy map profile, evaluated at any type profile, is the Nash equilibrium of the so-called local deterministic game corresponding to that type profile. We call such a Bayesian game type-regular. We then show that an m-dimensional n-agent Bayesian game whose utilities are linearly dependent on the types of the agents is equivalent, following a normalisation of the type space of each agent into the (m-1)-simplex, to a simultaneous competition in nm so-called basic n…
In this paper, we present a unifying framework for analyzing equilibria and designing interventions for large network games sampled from a stochastic network formation process represented by a graphon. We first introduce a new class of infinite population games, termed graphon games, where a continuum of heterogeneous agents interact according to a graphon. After studying properties of equilibria in graphon games, we show that graphon equilibria can approximate equilibria of large network games sampled from the graphon. We next show that, under some regularity assumptions, the graphon approach…
We introduce and study a group formation game in which individuals/agents, driven by self-interest, team up in disjoint groups so as to be in groups of high collective strength. This strength could be group identity, reputation, or protection, and is equally shared by all group members. The group's access to resources, obtained from its members, is traded off against the geographic dispersion of the group: spread-out groups are more costly to maintain. We seek to understand the stability and structure of such partitions. We define two types of equilibria: Acceptance Equilibria (AE), in which n…
A growing body of literature in networked systems research relies on game theory and mechanism design to model and address the potential lack of cooperation between self-interested users. Most game-theoretic models applied to system research only describe competitive equilibria in terms of pure Nash equilibria, that is, a situation where the strategy of each user is deterministic, and is her best response to the strategies of all the other users. However, the assumptions necessary for a pure Nash equilibrium to hold may be too stringent for practical systems. Using three case studies on comput…
Dynamics and Convergences for Markov Coevolutionary Opinion Formation Games in Dynamic Social Networks
National Yang Ming Chiao Tung University（国立阳明交通大学）；Academia Sinica（中央研究院）；National Taiwan Ocean University（国立台湾海洋大学）；New Jersey Institute of Technology（新泽理工学院）；National Chung Hsin University（国立中兴大学）
We investigate the existence of certain types of equilibria (Nash, $\varepsilon$-Nash, subgame perfect, $\varepsilon$-subgame perfect, Pareto-optimal) in multi-player multi-outcome infinite sequential games. We use two fundamental approaches: one requires strong topological restrictions on the games, but produces very strong existence results. The other merely requires some very basic determinacy properties to still obtain some existence results. Both results are transfer results: starting with the existence of some equilibria for a small class of games, they allow us to conclude the existence…
A natural goal in multiagent learning besides finding equilibria is to learn rationalizable behavior, where players learn to avoid iteratively dominated actions. However, even in the basic setting of multiplayer general-sum games, existing algorithms require a number of samples exponential in the number of players to learn rationalizable equilibria under bandit feedback. This paper develops the first line of efficient algorithms for learning rationalizable Coarse Correlated Equilibria (CCE) and Correlated Equilibria (CE) whose sample complexities are polynomial in all problem parameters includ…
In this paper, a gentle introduction to Game Theory is presented in the form of basic concepts and examples. Minimax and Nash's theorem are introduced as the formal definitions for optimal strategies and equilibria in zero-sum and nonzero-sum games. Several elements of cooperative gaming, coalitions, voting ensembles, voting power and collective efficiency are described in brief. Analytical (matrix) and extended (tree-graph) forms of game representation is illustrated as the basic tools for identifying optimal strategies and "solutions" in games of any kind. Next, a typology of four standard n…
The common sense suggests that networks are not random mazes of purposeless connections, but that these connections are organised so that networks can perform their functions well. One function common to many networks is targeted transport or navigation. Using game theory, here we show that minimalistic networks designed to maximise the navigation efficiency at minimal cost share basic structural properties with real networks. These idealistic networks are Nash equilibria of a network construction game whose purpose is to find an optimal trade-off between the network cost and navigability. We …
We study the structure and evolution of the Internet's Autonomous System (AS) interconnection topology as a game with heterogeneous players. In this network formation game, the utility of a player depends on the network structure, e.g., the distances between nodes and the cost of links. We analyze static properties of the game, such as the prices of anarchy and stability and provide explicit results concerning the generated topologies. Furthermore, we discuss dynamic aspects, demonstrating linear convergence rate and showing that only a restricted subset of equilibria is feasible under realist…
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…
We study the complexity of computing equilibria in two classes of network games based on flows - fractional BGP (Border Gateway Protocol) games and fractional BBC (Bounded Budget Connection) games. BGP is the glue that holds the Internet together and hence its stability, i.e. the equilibria of fractional BGP games (Haxell, Wilfong), is a matter of practical importance. BBC games (Laoutaris et al) follow in the tradition of the large body of work on network formation games and capture a variety of applications ranging from social networks and overlay networks to peer-to-peer networks. The centr…
When comparing new wireless technologies, it is common to consider the effect that they have on the capacity of the network (defined as the maximum number of simultaneously satisfiable links). For example, it has been shown that giving receivers the ability to do interference cancellation, or allowing transmitters to use power control, never decreases the capacity and can in certain cases increase it by $Ω(\log (Δ\cdot P_{\max}))$, where $Δ$ is the ratio of the longest link length to the smallest transmitter-receiver distance and $P_{\max}$ is the maximum transmission power. But there is no re…
Strategic network formation arises where agents receive benefit from connections to other agents, but also incur costs for forming links. We consider a new network formation game that incorporates an adversarial attack, as well as immunization against attack. An agent's benefit is the expected size of her connected component post-attack, and agents may also choose to immunize themselves from attack at some additional cost. Our framework is a stylized model of settings where reachability rather than centrality is the primary concern and vertices vulnerable to attacks may reduce risk via costly …
Learning and equilibrium computation in games are fundamental problems across computer science and economics, with applications ranging from politics to machine learning. Much of the work in this area revolves around a simple algorithm termed \emph{randomized weighted majority} (RWM), also known as "Hedge" or "Multiplicative Weights Update," which is well known to achieve statistically optimal rates in adversarial settings (Littlestone and Warmuth '94, Freund and Schapire '99). Unfortunately, RWM comes with an inherent computational barrier: it requires maintaining and sampling from a distribu…
Social networks on the Internet have seen an enormous growth recently and play a crucial role in different aspects of today's life. They have facilitated information dissemination in ways that have been beneficial for their users but they are often used strategically in order to spread information that only serves the objectives of particular users. These properties have inspired a revision of classical opinion formation models from sociology using game-theoretic notions and tools. We follow the same modeling approach, focusing on scenarios where the opinion expressed by each user is a comprom…
Computing Equilibria in Integer Programming Games with Shared Constraints
Chang’an University（长安大学）；Grado Department of Industrial and Systems Engineering, Virginia Tech（弗吉尼亚理工大学格雷多工业与系统工程系）；Department of Industrial and Systems Engineering, University of Wisconsin–Madison（威斯康星大学麦迪逊分校工业与系统工程系）
We study network formation with the bilateral link formation rule (Jackson and Wolinsky 1996) with $n$ players and link cost $α>0$. After the network is built, an adversary randomly destroys one link according to a certain probability distribution. Cost for player $v$ incorporates the expected number of players to which $v$ will become disconnected. This model was previously studied for unilateral link formation (K. 2011). We prove existence of pairwise Nash equilibria under moderate assumptions on the adversary and $n\geq 9$. As the main result, we prove bounds on the price of anarchy for two…
We investigate a non-cooperative game-theoretic model for the formation of communication networks by selfish agents. Each agent aims for a central position at minimum cost for creating edges. In particular, the general model (Fabrikant et al., PODC'03) became popular for studying the structure of the Internet or social networks. Despite its significance, locality in this game was first studied only recently (Bilò et al., SPAA'14), where a worst case locality model was presented, which came with a high efficiency loss in terms of quality of equilibria. Our main contribution is a new and more op…
Regret Minimization in Population Network Games: Vanishing Heterogeneity and Convergence to Equilibria
School of Mechanical Engineering, Northwestern Polytechnical University（西北工业大学机械工程学院）；Department of Computing, Hong Kong Polytechnic University（香港理工大学计算机学院）；Shanghai Artificial Intelligence Laboratory（上海人工智能实验室）；School of Statistics and Mathematics, Yunnan University of Finance and Economics（云南财经大学统计与数学学院）；School of Artificial Intelligence, Optics and Electronics (iOPEN), Northwestern Polytechnical University（西北工业大学人工智能、光学与电子学院（iOPEN））；School of Cybersecurity, Northwestern Polytechnical University（西北工业大学网络安全学院）
We propose a new class of game-theoretic models for network formation in which strategies are not directly related to edge choices, but instead correspond more generally to the exertion of social effort. The observed social network is thus a byproduct of an expressive strategic interaction, which can more naturally explain the emergence of complex social structures. Within this framework, we present a natural network formation game in which agent utilities are locally defined and that, despite its simplicity, produces a rich class of equilibria that exhibit structural properties commonly obser…
We introduce two min-max problems: the first problem is to minimize the supremum of finitely many rational functions over a compact basic semi-algebraic set whereas the second problem is a 2-player zero-sum polynomial game in randomized strategies and with compact basic semi-algebraic pure strategy sets. It is proved that their optimal solution can be approximated by solving a hierarchy of semidefinite relaxations, in the spirit of the moment approach developed in Lasserre. This provides a unified approach and a class of algorithms to approximate all Nash equilibria and min-max strategies of m…
Analysis of Equilibria and Strategic Interaction in Complex Networks
GRASP Laboratory, School of Engineering and Applied Science, University of Pennsylvania（宾夕法尼亚大学工程与应用科学学院）；Department of Operations and Information Management, Wharton School, University of Pennsylvania（宾夕法尼亚大学沃顿商学院运营管理系）
Modeling traffic in road networks is a widely studied but challenging problem, especially under the assumption that drivers act selfishly. A common approach is the deterministic queuing model, for which the structure of dynamic equilibria has been studied extensively in the last couple of years. The basic idea is to model traffic by a continuous flow that travels over time through a network, in which the arcs are endowed with transit times and capacities. Whenever the flow rate exceeds the capacity the flow particles build up a queue. So far it was not possible to represent spillback or kinema…
