Multicast Network Design Game on a Ring

In this paper we study quality measures of different solution concepts for the multicast network design game on a ring topology. We recall from the literature a lower bound of 4/3 and prove a matching upper bound for the price of stability, which is the ratio of the social costs of a best Nash equilibrium and of a general optimum. Therefore, we answer an open question posed by Fanelli et al. in [12]. We prove an upper bound of 2 for the ratio of the costs of a potential optimizer and of an optimum, provide a construction of a lower bound, and give a computer-assisted argument that it reaches $2$ for any precision. We then turn our attention to players arriving one by one and playing myopically their best response. We provide matching lower and upper bounds of 2 for the myopic sequential price of anarchy (achieved for a worst-case order of the arrival of the players). We then initiate the study of myopic sequential price of stability and for the multicast game on the ring we construct a lower bound of 4/3, and provide an upper bound of 26/19. To the end, we conjecture and argue that the right answer is 4/3.Comment: 12 pages, 4 figure

Computing Equilibrium in Matching Markets

Market equilibria of matching markets offer an intuitive and fair solution for matching problems without money with agents who have preferences over the items. Such a matching market can be viewed as a variation of Fisher market, albeit with rather peculiar preferences of agents. These preferences can be described by piece-wise linear concave (PLC) functions, which however, are not separable (due to each agent only asking for one item), are not monotone, and do not satisfy the gross substitute property-- increase in price of an item can result in increased demand for the item. Devanur and Kannan in FOCS 08 showed that market clearing prices can be found in polynomial time in markets with fixed number of items and general PLC preferences. They also consider Fischer markets with fixed number of agents (instead of fixed number of items), and give a polynomial time algorithm for this case if preferences are separable functions of the items, in addition to being PLC functions. Our main result is a polynomial time algorithm for finding market clearing prices in matching markets with fixed number of different agent preferences, despite that the utility corresponding to matching markets is not separable. We also give a simpler algorithm for the case of matching markets with fixed number of different items

Selfishness Level of Strategic Games

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy and is invariant under positive linear transformations of the payoff functions. Also, it naturally applies to other solution concepts and other forms of games. We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate. In particular, the selfishness level of finite ordinal potential games is finite, while that of weakly acyclic games can be infinite. We derive explicit bounds on the selfishness level of fair cost sharing games and linear congestion games, which depend on specific parameters of the underlying game but are independent of the number of players. Further, we show that the selfishness level of the $n$-players Prisoner's Dilemma is $c/(b(n-1)-c)$, where $b$ and $c$ are the benefit and cost for cooperation, respectively, that of the $n$-players public goods game is $(1-\frac{c}{n})/(c-1)$, where $c$ is the public good multiplier, and that of the Traveler's Dilemma game is $\frac{1}{2}(b-1)$, where $b$ is the bonus. Finally, the selfishness level of Cournot competition (an example of an infinite ordinal potential game, Tragedy of the Commons, and Bertrand competition is infinite.Comment: 34 page

Computing Stable Coalitions: Approximation Algorithms for Reward Sharing

Consider a setting where selfish agents are to be assigned to coalitions or projects from a fixed set P. Each project k is characterized by a valuation function; v_k(S) is the value generated by a set S of agents working on project k. We study the following classic problem in this setting: "how should the agents divide the value that they collectively create?". One traditional approach in cooperative game theory is to study core stability with the implicit assumption that there are infinite copies of one project, and agents can partition themselves into any number of coalitions. In contrast, we consider a model with a finite number of non-identical projects; this makes computing both high-welfare solutions and core payments highly non-trivial. The main contribution of this paper is a black-box mechanism that reduces the problem of computing a near-optimal core stable solution to the purely algorithmic problem of welfare maximization; we apply this to compute an approximately core stable solution that extracts one-fourth of the optimal social welfare for the class of subadditive valuations. We also show much stronger results for several popular sub-classes: anonymous, fractionally subadditive, and submodular valuations, as well as provide new approximation algorithms for welfare maximization with anonymous functions. Finally, we establish a connection between our setting and the well-studied simultaneous auctions with item bidding; we adapt our results to compute approximate pure Nash equilibria for these auctions.Comment: Under Revie

Efficient Equilibria in Polymatrix Coordination Games

We consider polymatrix coordination games with individual preferences where every player corresponds to a node in a graph who plays with each neighbor a separate bimatrix game with non-negative symmetric payoffs. In this paper, we study $\alpha$-approximate $k$-equilibria of these games, i.e., outcomes where no group of at most $k$ players can deviate such that each member increases his payoff by at least a factor $\alpha$. We prove that for $\alpha \ge 2$ these games have the finite coalitional improvement property (and thus $\alpha$-approximate $k$-equilibria exist), while for $\alpha < 2$ this property does not hold. Further, we derive an almost tight bound of $2\alpha(n-1)/(k-1)$ on the price of anarchy, where $n$ is the number of players; in particular, it scales from unbounded for pure Nash equilibria ($k = 1)$ to $2\alpha$ for strong equilibria ($k = n$). We also settle the complexity of several problems related to the verification and existence of these equilibria. Finally, we investigate natural means to reduce the inefficiency of Nash equilibria. Most promisingly, we show that by fixing the strategies of $k$ players the price of anarchy can be reduced to $n/k$ (and this bound is tight)