969 research outputs found
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
Price Competition in Online Combinatorial Markets
We consider a single buyer with a combinatorial preference that would like to
purchase related products and services from different vendors, where each
vendor supplies exactly one product. We study the general case where subsets of
products can be substitutes as well as complementary and analyze the game that
is induced on the vendors, where a vendor's strategy is the price that he asks
for his product. This model generalizes both Bertrand competition (where
vendors are perfect substitutes) and Nash bargaining (where they are perfect
complements), and captures a wide variety of scenarios that can appear in
complex crowd sourcing or in automatic pricing of related products.
We study the equilibria of such games and show that a pure efficient
equilibrium always exists. In the case of submodular buyer preferences we fully
characterize the set of pure Nash equilibria, essentially showing uniqueness.
For the even more restricted "substitutes" buyer preferences we also prove
uniqueness over {\em mixed} equilibria. Finally we begin the exploration of
natural generalizations of our setting such as when services have costs, when
there are multiple buyers or uncertainty about the the buyer's valuation, and
when a single vendor supplies multiple products.Comment: accept to WWW'14 (23rd International World Wide Web Conference
Computing Equilibria in Markets with Budget-Additive Utilities
We present the first analysis of Fisher markets with buyers that have
budget-additive utility functions. Budget-additive utilities are elementary
concave functions with numerous applications in online adword markets and
revenue optimization problems. They extend the standard case of linear
utilities and have been studied in a variety of other market models. In
contrast to the frequently studied CES utilities, they have a global satiation
point which can imply multiple market equilibria with quite different
characteristics. Our main result is an efficient combinatorial algorithm to
compute a market equilibrium with a Pareto-optimal allocation of goods. It
relies on a new descending-price approach and, as a special case, also implies
a novel combinatorial algorithm for computing a market equilibrium in linear
Fisher markets. We complement these positive results with a number of hardness
results for related computational questions. We prove that it is NP-hard to
compute a market equilibrium that maximizes social welfare, and it is PPAD-hard
to find any market equilibrium with utility functions with separate satiation
points for each buyer and each good.Comment: 21 page
Computational Complexity in Additive Hedonic Games
We investigate the computational complexity of several decision problems in hedonic coalition formation games and demonstrate that attaining stability in such games remains NP-hard even when they are additive. Precisely, we prove that when either core stability or strict core stability is under consideration, the existence problem of a stable coalition structure is NP-hard in the strong sense. Furthermore, the corresponding decision problems with respect to the existence of a Nash stable coalition structure and of an individually stable coalition structure turn out to be NP-complete in the strong sense.Additive Preferences, Coalition Formation, Computational Complexity, Hedonic Games, NP-hard, NP-complete
Complexity results for some classes of strategic games
Game theory is a branch of applied mathematics studying the interaction of self-interested entities, so-called agents. Its central objects of study are games, mathematical models of real-world interaction, and solution concepts that single out certain outcomes of a game that are meaningful in some way. The solutions thus produced can then be viewed both from a descriptive and from a normative perspective. The rise of the Internet as a computational platform where a substantial part of today's strategic interaction takes place has spurred additional interest in game theory as an analytical tool, and has brought it to the attention of a wider audience in computer science.
An important aspect of real-world decision-making, and one that has received only little attention in the early days of game theory, is that agents may be subject to resource constraints. The young field of algorithmic game theory has set out to address this shortcoming using techniques from computer science, and in particular from computational complexity theory. One of the defining problems of algorithmic game theory concerns the computation of solution concepts. Finding a Nash equilibrium, for example, i.e., an outcome where no single agent can gain by changing his strategy, was considered one of the most important problems on the boundary of P, the complexity class commonly associated with efficient computation, until it was recently shown complete for the class PPAD. This rather negative result for general games has not settled the question, however, but immediately raises several new ones: First, can Nash equilibria be approximated, i.e., is it possible to efficiently find a solution such that the potential gain from a unilateral deviation is small? Second, are there interesting classes of games that do allow for an exact solution to be computed efficiently? Third, are there alternative solution concepts that are computationally tractable, and how does the value of solutions selected by these concepts compare to those selected by established solution concepts?
The work reported in this thesis is part of the effort to answer the latter two questions. We study the complexity of well-known solution concepts, like Nash equilibrium and iterated dominance, in various classes of games that are both natural and practically relevant: ranking games, where outcomes are rankings of the players; anonymous games, where players do not distinguish between the other players in the game; and graphical games, where the well-being of any particular player depends only on the actions of a small group other players. In ranking games, we further compare the payoffs obtainable in Nash equilibrium outcomes with those of alternative solution concepts that are easy to compute. We finally study, in general games, solution concepts that try to remedy some of the shortcomings associated with Nash equilibrium, like the need for randomization to achieve a stable outcome
Robust Stackelberg Equilibria
This paper provides a systematic study of the robust Stackelberg equilibrium
(RSE), which naturally generalizes the widely adopted solution concept of the
strong Stackelberg equilibrium (SSE). The RSE accounts for any possible
up-to- suboptimal follower responses in Stackelberg games and is
adopted to improve the robustness of the leader's strategy. While a few
variants of robust Stackelberg equilibrium have been considered in previous
literature, the RSE solution concept we consider is importantly different -- in
some sense, it relaxes previously studied robust Stackelberg strategies and is
applicable to much broader sources of uncertainties.
We provide a thorough investigation of several fundamental properties of RSE,
including its utility guarantees, algorithmics, and learnability. We first show
that the RSE we defined always exists and thus is well-defined. Then we
characterize how the leader's utility in RSE changes with the robustness level
considered. On the algorithmic side, we show that, in sharp contrast to the
tractability of computing an SSE, it is NP-hard to obtain a fully polynomial
approximation scheme (FPTAS) for any constant robustness level. Nevertheless,
we develop a quasi-polynomial approximation scheme (QPTAS) for RSE. Finally, we
examine the learnability of the RSE in a natural learning scenario, where both
players' utilities are not known in advance, and provide almost tight sample
complexity results on learning the RSE. As a corollary of this result, we also
obtain an algorithm for learning SSE, which strictly improves a key result of
Bai et al. in terms of both utility guarantee and computational efficiency
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