1,740 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
Polynomial-time Computation of Exact Correlated Equilibrium in Compact Games
In a landmark paper, Papadimitriou and Roughgarden described a
polynomial-time algorithm ("Ellipsoid Against Hope") for computing sample
correlated equilibria of concisely-represented games. Recently, Stein, Parrilo
and Ozdaglar showed that this algorithm can fail to find an exact correlated
equilibrium, but can be easily modified to efficiently compute approximate
correlated equilibria. Currently, it remains unresolved whether the algorithm
can be modified to compute an exact correlated equilibrium. We show that it
can, presenting a variant of the Ellipsoid Against Hope algorithm that
guarantees the polynomial-time identification of exact correlated equilibrium.
Our new algorithm differs from the original primarily in its use of a
separation oracle that produces cuts corresponding to pure-strategy profiles.
As a result, we no longer face the numerical precision issues encountered by
the original approach, and both the resulting algorithm and its analysis are
considerably simplified. Our new separation oracle can be understood as a
derandomization of Papadimitriou and Roughgarden's original separation oracle
via the method of conditional probabilities. Also, the equilibria returned by
our algorithm are distributions with polynomial-sized supports, which are
simpler (in the sense of being representable in fewer bits) than the mixtures
of product distributions produced previously; no tractable algorithm has
previously been proposed for identifying such equilibria.Comment: 15 page
Efficiency in Multi-objective Games
In a multi-objective game, each agent individually evaluates each overall
action-profile on multiple objectives. I generalize the price of anarchy to
multi-objective games and provide a polynomial-time algorithm to assess it.
This work asserts that policies on tobacco promote a higher economic
efficiency
Query Complexity of Approximate Equilibria in Anonymous Games
We study the computation of equilibria of anonymous games, via algorithms
that may proceed via a sequence of adaptive queries to the game's payoff
function, assumed to be unknown initially. The general topic we consider is
\emph{query complexity}, that is, how many queries are necessary or sufficient
to compute an exact or approximate Nash equilibrium.
We show that exact equilibria cannot be found via query-efficient algorithms.
We also give an example of a 2-strategy, 3-player anonymous game that does not
have any exact Nash equilibrium in rational numbers. However, more positive
query-complexity bounds are attainable if either further symmetries of the
utility functions are assumed or we focus on approximate equilibria. We
investigate four sub-classes of anonymous games previously considered by
\cite{bfh09, dp14}.
Our main result is a new randomized query-efficient algorithm that finds a
-approximate Nash equilibrium querying
payoffs and runs in time . This improves on the running
time of pre-existing algorithms for approximate equilibria of anonymous games,
and is the first one to obtain an inverse polynomial approximation in
poly-time. We also show how this can be utilized as an efficient
polynomial-time approximation scheme (PTAS). Furthermore, we prove that
payoffs must be queried in order to find any
-well-supported Nash equilibrium, even by randomized algorithms
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