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Equilibria, Fixed Points, and Complexity Classes
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 polynomial time. There
are certain common computational principles underlying different types of
equilibria, which are captured by the complexity classes PLS, PPAD, and FIXP.
Representative complete problems for these classes are respectively, pure Nash
equilibria in games where they are guaranteed to exist, (mixed) Nash equilibria
in 2-player normal form games, and (mixed) Nash equilibria in normal form games
with 3 (or more) players. This paper reviews the underlying computational
principles and the corresponding classes
Stochastic Coalitional Better-response Dynamics and Strong Nash Equilibrium
We consider coalition formation among players in an n-player finite strategic
game over infinite horizon. At each time a randomly formed coalition makes a
joint deviation from a current action profile such that at new action profile
all players from the coalition are strictly benefited. Such deviations define a
coalitional better-response (CBR) dynamics that is in general stochastic. The
CBR dynamics either converges to a strong Nash equilibrium or stucks in a
closed cycle. We also assume that at each time a selected coalition makes
mistake in deviation with small probability that add mutations (perturbations)
into CBR dynamics. We prove that all strong Nash equilibria and closed cycles
are stochastically stable, i.e., they are selected by perturbed CBR dynamics as
mutations vanish. Similar statement holds for strict strong Nash equilibrium.
We apply CBR dynamics to the network formation games and we prove that all
strongly stable networks and closed cycles are stochastically stable
A semantical approach to equilibria and rationality
Game theoretic equilibria are mathematical expressions of rationality.
Rational agents are used to model not only humans and their software
representatives, but also organisms, populations, species and genes,
interacting with each other and with the environment. Rational behaviors are
achieved not only through conscious reasoning, but also through spontaneous
stabilization at equilibrium points.
Formal theories of rationality are usually guided by informal intuitions,
which are acquired by observing some concrete economic, biological, or network
processes. Treating such processes as instances of computation, we reconstruct
and refine some basic notions of equilibrium and rationality from the some
basic structures of computation.
It is, of course, well known that equilibria arise as fixed points; the point
is that semantics of computation of fixed points seems to be providing novel
methods, algebraic and coalgebraic, for reasoning about them.Comment: 18 pages; Proceedings of CALCO 200
Metastability of Asymptotically Well-Behaved Potential Games
One of the main criticisms to game theory concerns the assumption of full
rationality. Logit dynamics is a decentralized algorithm in which a level of
irrationality (a.k.a. "noise") is introduced in players' behavior. In this
context, the solution concept of interest becomes the logit equilibrium, as
opposed to Nash equilibria. Logit equilibria are distributions over strategy
profiles that possess several nice properties, including existence and
uniqueness. However, there are games in which their computation may take time
exponential in the number of players. We therefore look at an approximate
version of logit equilibria, called metastable distributions, introduced by
Auletta et al. [SODA 2012]. These are distributions that remain stable (i.e.,
players do not go too far from it) for a super-polynomial number of steps
(rather than forever, as for logit equilibria). The hope is that these
distributions exist and can be reached quickly by logit dynamics.
We identify a class of potential games, called asymptotically well-behaved,
for which the behavior of the logit dynamics is not chaotic as the number of
players increases so to guarantee meaningful asymptotic results. We prove that
any such game admits distributions which are metastable no matter the level of
noise present in the system, and the starting profile of the dynamics. These
distributions can be quickly reached if the rationality level is not too big
when compared to the inverse of the maximum difference in potential. Our proofs
build on results which may be of independent interest, including some spectral
characterizations of the transition matrix defined by logit dynamics for
generic games and the relationship of several convergence measures for Markov
chains
Constrained Cost-Coupled Stochastic Games with Independent State Processes
We consider a non-cooperative constrained stochastic games with N players
with the following special structure. With each player there is an associated
controlled Markov chain. The transition probabilities of the i-th Markov chain
depend only on the state and actions of controller i. The information structure
that we consider is such that each player knows the state of its own MDP and
its own actions. It does not know the states of, and the actions taken by other
players. Finally, each player wishes to minimize a time-average cost function,
and has constraints over other time-avrage cost functions. Both the cost that
is minimized as well as those defining the constraints depend on the state and
actions of all players. We study in this paper the existence of a Nash
equilirium. Examples in power control in wireless communications are given.Comment: 7 pages, submitted in september 2006 to Operations Research Letter
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