382 research outputs found
Sharing Non-Anonymous Costs of Multiple Resources Optimally
In cost sharing games, the existence and efficiency of pure Nash equilibria
fundamentally depends on the method that is used to share the resources' costs.
We consider a general class of resource allocation problems in which a set of
resources is used by a heterogeneous set of selfish users. The cost of a
resource is a (non-decreasing) function of the set of its users. Under the
assumption that the costs of the resources are shared by uniform cost sharing
protocols, i.e., protocols that use only local information of the resource's
cost structure and its users to determine the cost shares, we exactly quantify
the inefficiency of the resulting pure Nash equilibria. Specifically, we show
tight bounds on prices of stability and anarchy for games with only submodular
and only supermodular cost functions, respectively, and an asymptotically tight
bound for games with arbitrary set-functions. While all our upper bounds are
attained for the well-known Shapley cost sharing protocol, our lower bounds
hold for arbitrary uniform cost sharing protocols and are even valid for games
with anonymous costs, i.e., games in which the cost of each resource only
depends on the cardinality of the set of its users
Computing Approximate Equilibria in Weighted Congestion Games via Best-Responses
We present a deterministic polynomial-time algorithm for computing
-approximate (pure) Nash equilibria in weighted congestion games
with polynomial cost functions of degree at most . This is an exponential
improvement of the approximation factor with respect to the previously best
deterministic algorithm. An appealing additional feature of our algorithm is
that it uses only best-improvement steps in the actual game, as opposed to
earlier approaches that first had to transform the game itself. Our algorithm
is an adaptation of the seminal algorithm by Caragiannis et al. [FOCS'11, TEAC
2015], but we utilize an approximate potential function directly on the
original game instead of an exact one on a modified game.
A critical component of our analysis, which is of independent interest, is
the derivation of a novel bound of for the
Price of Anarchy (PoA) of -approximate equilibria in weighted congestion
games, where is the Lambert-W function. More specifically, we
show that this PoA is exactly equal to , where
is the unique positive solution of the equation . Our upper bound is derived via a smoothness-like argument,
and thus holds even for mixed Nash and correlated equilibria, while our lower
bound is simple enough to apply even to singleton congestion games
Approximate Pure Nash Equilibria in Weighted Congestion Games: Existence, Efficient Computation, and Structure
We consider structural and algorithmic questions related to the Nash dynamics
of weighted congestion games. In weighted congestion games with linear latency
functions, the existence of (pure Nash) equilibria is guaranteed by potential
function arguments. Unfortunately, this proof of existence is inefficient and
computing equilibria is such games is a {\sf PLS}-hard problem. The situation
gets worse when superlinear latency functions come into play; in this case, the
Nash dynamics of the game may contain cycles and equilibria may not even exist.
Given these obstacles, we consider approximate equilibria as alternative
solution concepts. Do such equilibria exist? And if so, can we compute them
efficiently?
We provide positive answers to both questions for weighted congestion games
with polynomial latency functions by exploiting an "approximation" of such
games by a new class of potential games that we call -games. This allows
us to show that these games have -approximate equilibria, where is the
maximum degree of the latency functions. Our main technical contribution is an
efficient algorithm for computing O(1)-approximate equilibria when is a
constant. For games with linear latency functions, the approximation guarantee
is for arbitrarily small ; for
latency functions with maximum degree , it is . The
running time is polynomial in the number of bits in the representation of the
game and . As a byproduct of our techniques, we also show the
following structural statement for weighted congestion games with polynomial
latency functions of maximum degree : polynomially-long sequences of
best-response moves from any initial state to a -approximate
equilibrium exist and can be efficiently identified in such games as long as
is constant.Comment: 31 page
Cost-sharing in generalised selfish routing
© Springer International Publishing AG 2017. We study a generalisation of atomic selfish routing games where each player may control multiple flows which she routes seek-ing to minimise their aggregate cost. Such games emerge in various set-tings, such as traffic routing in road networks by competing ride-sharing applications or packet routing in communication networks by competing service providers who seek to optimise the quality of service of their cus-tomers. We study the existence of pure Nash equilibria in the induced games and we exhibit a separation from the single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees it. We also prove that the price of anarchy and price of stability is no larger than in the single-commodity model for general cost-sharing methods and general classes of convex cost functions. We close by giving results on the existence of pure Nash equilibria of a splittable variant of our model
Pure Nash equilibria in restricted budget games
In budget games, players compete over resources with finite budgets. For every resource, a player has a specific demand and as a strategy, he chooses a subset of resources. If the total demand on a resource does not exceed its budget, the utility of each player who chose that resource equals his demand. Otherwise, the budget is shared proportionally. In the general case, pure Nash equilibria (NE) do not exist for such games. In this paper, we consider the natural classes of singleton and matroid budget games with additional constraints and show that for each, pure NE can be guaranteed. In addition, we introduce a lexicographical potential function to prove that every matroid budget game has an approximate pure NE which depends on the largest ratio between the different demands of each individual player
Non-cooperative queueing games on a Jackson network
In this paper we introduce non-cooperative games on a Jackson network. A player has a set of routes available and has to decide which routes to use for its customers. Each player's goal is to minimize the expected sojourn time of its customers. We consider two cases. First, each player is allowed to divide his customers over multiple routes. This results in a game for which it can be shown that a unique pure-strategy Nash equilibrium exists. This Nash equilibrium can be found by using a best-response algorithm. Second, each player may only select a single route for its customers. This results in a game with finite strategy spaces. In general, such games need not have a pure-strategy Nash equilibrium, as shown by an example. We show the existence of pure-strategy Nash equilibria for four subclasses of games on a Jackson network: (i) N-player games with equal arrival rates for the players, (ii) 2-player games with identical service rates for all nodes, (iii) 2-player games on a 2 x 2-grid, and (iv) 2-player games on an A x B-grid with small differences in the service rates
Existence and Efficiency of Equilibria for Cost-Sharing in Generalized Weighted Congestion Games
This work studies the impact of cost-sharing methods on the existence and efficiency of (pure) Nash equilibria in weighted congestion games. We also study generalized weighted congestion games, where each player may control multiple commodities. Our results are fairly general; we only require that our cost-sharing method and our set of cost functions satisfy certain natural conditions. For general weighted congestion games, we study the existence of pure Nash equilibria in the induced games, and we exhibit a separation from the standard single-commodity per player model by proving that the Shapley value is the only cost-sharing method that guarantees existence of pure Nash equilibria. With respect to efficiency, we present general tight bounds on the price of anarchy, which are robust and apply to general equilibrium concepts. Our analysis provides a tight bound on the price of anarchy, which depends only on the used cost-sharing method and the set of allowable cost functions. Interestingly, the same bound applies to weighted congestion games and generalized weighted congestion games. We then turn to the price of stability and prove an upper bound for the Shapley value cost-sharing method, which holds for general sets of cost functions and which is tight in special cases of interest, such as bounded degree polynomials. Also for bounded degree polynomials, we provide a somewhat surprising result, showing that a slight deviation from the Shapley value has a huge impact on the price of stability. In fact, for this case, the price of stability becomes as bad as the price of anarchy. Again, our bounds on the price of stability are independent on whether players are single or multi-commodity
Equilibrium Analysis of Customer Attraction Games
We introduce a game model called "customer attraction game" to demonstrate
the competition among online content providers. In this model, customers
exhibit interest in various topics. Each content provider selects one topic and
benefits from the attracted customers. We investigate both symmetric and
asymmetric settings involving agents and customers. In the symmetric setting,
the existence of pure Nash equilibrium (PNE) is guaranteed, but finding a PNE
is PLS-complete. To address this, we propose a fully polynomial time
approximation scheme to identify an approximate PNE. Moreover, the tight Price
of Anarchy (PoA) is established. In the asymmetric setting, we show the
nonexistence of PNE in certain instances and establish that determining its
existence is NP-hard. Nevertheless, we prove the existence of an approximate
PNE. Additionally, when agents select topics sequentially, we demonstrate that
finding a subgame-perfect equilibrium is PSPACE-hard. Furthermore, we present
the sequential PoA for the two-agent setting
Designing Cost-Sharing Methods for Bayesian Games
We study the design of cost-sharing protocols for two fundamental resource allocation problems, the Set Cover and the Steiner Tree Problem, under environments of incomplete information (Bayesian model). Our objective is to design protocols where the worst-case Bayesian Nash equilibria have low cost, i.e. the Bayesian Price of Anarchy (PoA) is minimized. Although budget balance is a very natural requirement, it puts considerable restrictions on the design space, resulting in high PoA. We propose an alternative, relaxed requirement called budget balance in the equilibrium (BBiE). We show an interesting connection between algorithms for Oblivious Stochastic optimization problems and cost-sharing design with low PoA. We exploit this connection for both problems and we enforce approximate solutions of the stochastic problem, as Bayesian Nash equilibria, with the same guarantees on the PoA. More interestingly, we show how to obtain the same bounds on the PoA, by using anonymous posted prices which are desirable because they are easy to implement and, as we show, induce dominant strategies for the players
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