2,381 research outputs found
Connectivity and equilibrium in random games
We study how the structure of the interaction graph of a game affects the
existence of pure Nash equilibria. In particular, for a fixed interaction
graph, we are interested in whether there are pure Nash equilibria arising when
random utility tables are assigned to the players. We provide conditions for
the structure of the graph under which equilibria are likely to exist and
complementary conditions which make the existence of equilibria highly
unlikely. Our results have immediate implications for many deterministic graphs
and generalize known results for random games on the complete graph. In
particular, our results imply that the probability that bounded degree graphs
have pure Nash equilibria is exponentially small in the size of the graph and
yield a simple algorithm that finds small nonexistence certificates for a large
family of graphs. Then we show that in any strongly connected graph of n
vertices with expansion the distribution of the number
of equilibria approaches the Poisson distribution with parameter 1,
asymptotically as .Comment: Published in at http://dx.doi.org/10.1214/10-AAP715 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Monotone methods for equilibrium selection under perfect foresight dynamics
This paper studies equilibrium selection in supermodular games
based on perfect foresight dynamics. A normal form game is played
repeatedly in a large society of rational agents. There are frictions:
opportunities to revise actions follow independent Poisson processes.
Each agent forms his belief about the future evolution of action distribution
in the society to take an action that maximizes his expected
discounted payo�. A perfect foresight path is de�ned to be a feasible
path of the action distribution along which every agent with a revision
opportunity takes a best response to this path itself. A Nash
equilibrium is said to be absorbing if there exists no perfect foresight
path escaping from a neighborhood of this equilibrium; a Nash equilibrium
is said to be globally accessible if for each initial distribution,
there exists a perfect foresight path converging to this equilibrium.
By exploiting the monotone structure of the dynamics, a unique Nash
equilibrium that is absorbing and globally accessible for any small degree
of friction is identi�ed for certain classes of supermodular games.
For games with monotone potentials, the selection of the monotone
potential maximizer is obtained. Complete characterizations of absorbing
equilibrium and globally accessible equilibrium are given for
binary supermodular games. An example demonstrates that unanimity
games may have multiple globally accessible equilibria for a small
friction
Pure Nash Equilibria and Best-Response Dynamics in Random Games
In finite games mixed Nash equilibria always exist, but pure equilibria may
fail to exist. To assess the relevance of this nonexistence, we consider games
where the payoffs are drawn at random. In particular, we focus on games where a
large number of players can each choose one of two possible strategies, and the
payoffs are i.i.d. with the possibility of ties. We provide asymptotic results
about the random number of pure Nash equilibria, such as fast growth and a
central limit theorem, with bounds for the approximation error. Moreover, by
using a new link between percolation models and game theory, we describe in
detail the geometry of Nash equilibria and show that, when the probability of
ties is small, a best-response dynamics reaches a Nash equilibrium with a
probability that quickly approaches one as the number of players grows. We show
that a multitude of phase transitions depend only on a single parameter of the
model, that is, the probability of having ties.Comment: 29 pages, 7 figure
A cognitive hierarchy model of games
Players in a game are “in equilibrium” if they are rational, and accurately predict other players' strategies. In many experiments, however, players are not in equilibrium. An alternative is “cognitive hierarchy” (CH) theory, where each player assumes that his strategy is the most sophisticated. The CH model has inductively defined strategic categories: step 0 players randomize; and step k thinkers best-respond, assuming that other players are distributed over step 0 through step k − 1. This model fits empirical data, and explains why equilibrium theory predicts behavior well in some games and poorly in others. An average of 1.5 steps fits data from many games
A cognitive hierarchy theory of one-shot games: Some preliminary results
Strategic thinking, best-response, and mutual consistency (equilibrium) are three
key modelling principles in noncooperative game theory. This paper relaxes mutual
consistency to predict how players are likely to behave in in one-shot games before they
can learn to equilibrate. We introduce a one-parameter cognitive hierarchy (CH) model
to predict behavior in one-shot games, and initial conditions in repeated games. The CH
approach assumes that players use k steps of reasoning with frequency f (k). Zero-step
players randomize. Players using k (≥ 1) steps best respond given partially rational
expectations about what players doing 0 through k - 1 steps actually choose. A simple
axiom which expresses the intuition that steps of thinking are increasingly constrained by
working memory, implies that f (k) has a Poisson distribution (characterized by a mean
number of thinking steps τ ). The CH model converges to dominance-solvable equilibria
when τ is large, predicts monotonic entry in binary entry games for τ < 1:25, and predicts
effects of group size which are not predicted by Nash equilibrium. Best-fitting values of
τ have an interquartile range of (.98,2.40) and a median of 1.65 across 80 experimental
samples of matrix games, entry games, mixed-equilibrium games, and dominance-solvable
p-beauty contests. The CH model also has economic value because subjects would have
raised their earnings substantially if they had best-responded to model forecasts instead
of making the choices they did
Discretized Multinomial Distributions and Nash Equilibria in Anonymous Games
We show that there is a polynomial-time approximation scheme for computing
Nash equilibria in anonymous games with any fixed number of strategies (a very
broad and important class of games), extending the two-strategy result of
Daskalakis and Papadimitriou 2007. The approximation guarantee follows from a
probabilistic result of more general interest: The distribution of the sum of n
independent unit vectors with values ranging over {e1, e2, ...,ek}, where ei is
the unit vector along dimension i of the k-dimensional Euclidean space, can be
approximated by the distribution of the sum of another set of independent unit
vectors whose probabilities of obtaining each value are multiples of 1/z for
some integer z, and so that the variational distance of the two distributions
is at most eps, where eps is bounded by an inverse polynomial in z and a
function of k, but with no dependence on n. Our probabilistic result specifies
the construction of a surprisingly sparse eps-cover -- under the total
variation distance -- of the set of distributions of sums of independent unit
vectors, which is of interest on its own right.Comment: In the 49th Annual IEEE Symposium on Foundations of Computer Science,
FOCS 200
A Size-Free CLT for Poisson Multinomials and its Applications
An -Poisson Multinomial Distribution (PMD) is the distribution of the
sum of independent random vectors supported on the set of standard basis vectors in . We show
that any -PMD is -close in total
variation distance to the (appropriately discretized) multi-dimensional
Gaussian with the same first two moments, removing the dependence on from
the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is
obtained by bootstrapping the Valiant-Valiant CLT itself through the structural
characterization of PMDs shown in recent work by Daskalakis, Kamath, and
Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS
for approximate Nash equilibria in anonymous games, significantly improving the
state of the art, and matching qualitatively the running time dependence on
and of the best known algorithm for two-strategy anonymous
games. Our new CLT also enables the construction of covers for the set of
-PMDs, which are proper and whose size is shown to be essentially
optimal. Our cover construction combines our CLT with the Shapley-Folkman
theorem and recent sparsification results for Laplacian matrices by Batson,
Spielman, and Srivastava. Our cover size lower bound is based on an algebraic
geometric construction. Finally, leveraging the structural properties of the
Fourier spectrum of PMDs we show that these distributions can be learned from
samples in -time, removing
the quasi-polynomial dependence of the running time on from the
algorithm of Daskalakis, Kamath, and Tzamos.Comment: To appear in STOC 201
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