892 research outputs found
Finding Any Nontrivial Coarse Correlated Equilibrium Is Hard
One of the most appealing aspects of the (coarse) correlated equilibrium
concept is that natural dynamics quickly arrive at approximations of such
equilibria, even in games with many players. In addition, there exist
polynomial-time algorithms that compute exact (coarse) correlated equilibria.
In light of these results, a natural question is how good are the (coarse)
correlated equilibria that can arise from any efficient algorithm or dynamics.
In this paper we address this question, and establish strong negative
results. In particular, we show that in multiplayer games that have a succinct
representation, it is NP-hard to compute any coarse correlated equilibrium (or
approximate coarse correlated equilibrium) with welfare strictly better than
the worst possible. The focus on succinct games ensures that the underlying
complexity question is interesting; many multiplayer games of interest are in
fact succinct. Our results imply that, while one can efficiently compute a
coarse correlated equilibrium, one cannot provide any nontrivial welfare
guarantee for the resulting equilibrium, unless P=NP. We show that analogous
hardness results hold for correlated equilibria, and persist under the
egalitarian objective or Pareto optimality.
To complement the hardness results, we develop an algorithmic framework that
identifies settings in which we can efficiently compute an approximate
correlated equilibrium with near-optimal welfare. We use this framework to
develop an efficient algorithm for computing an approximate correlated
equilibrium with near-optimal welfare in aggregative games.Comment: 21 page
Computational Complexity of Approximate Nash Equilibrium in Large Games
We prove that finding an epsilon-Nash equilibrium in a succinctly
representable game with many players is PPAD-hard for constant epsilon. Our
proof uses succinct games, i.e. games whose payoff function is represented by a
circuit. Our techniques build on a recent query complexity lower bound by
Babichenko.Comment: New version includes an addendum about subsequent work on the open
problems propose
The Complexity of angel-daemons and game isomorphism
The analysis of the computational aspects of strategic situations is a basic field in Computer
Sciences. Two main topics related to strategic games have been developed. First,
introduction and analysis of a class of games (so called angel/daemon games) designed
to asses web applications, have been considered. Second, the problem of isomorphism
between strategic games has been analysed. Both parts have been separately considered.
Angel-Daemon Games
A service is a computational method that is made available for general use through a
wide area network. The performance of web-services may fluctuate; at times of stress the
performance of some services may be degraded (in extreme cases, to the point of failure).
In this thesis uncertainty profiles and Angel-Daemon games are used to analyse servicebased
behaviours in situations where probabilistic reasoning may not be appropriate.
In such a game, an angel player acts on a bounded number of ¿angelic¿ services
in a beneficial way while a daemon player acts on a bounded number of ¿daemonic¿
services in a negative way. Examples are used to illustrate how game theory can be used
to analyse service-based scenarios in a realistic way that lies between over-optimism and
over-pessimism.
The resilience of an orchestration to service failure has been analysed - here angels and
daemons are used to model services which can fail when placed under stress. The Nash
equilibria of a corresponding Angel-Daemon game may be used to assign a ¿robustness¿
value to an orchestration.
Finally, the complexity of equilibria problems for Angel-Daemon games has been
analysed. It turns out that Angel-Daemon games are, at the best of our knowledge, the
first natural example of zero-sum succinct games.
The fact that deciding the existence of a pure Nash equilibrium or a dominant strategy
for a given player is Sp
2-complete has been proven. Furthermore, computing the value of
an Angel-Daemon game is EXP-complete. Thus, matching the already known complexity
results of the corresponding problems for the generic families of succinctly represented
games with exponential number of actions.
Game Isomorphism
The question of whether two multi-player strategic games are equivalent and the computational
complexity of deciding such a property has been addressed. Three notions
of isomorphisms, strong, weak and local have been considered. Each one of these isomorphisms
preserves a different structure of the game. Strong isomorphism is defined to
preserve the utility functions and Nash equilibria. Weak isomorphism preserves only the
player preference relations and thus pure Nash equilibria. Local isomorphism preserves
preferences defined only on ¿close¿ neighbourhood of strategy profiles.
The problem of the computational complexity of game isomorphism, which depends
on the level of succinctness of the description of the input games but it is independent
of the isomorphism to consider, has been shown. Utilities in games can be given succinctly
by Turing machines, boolean circuits or boolean formulas, or explicitly by tables.
Actions can be given also explicitly or succinctly. When the games are given in general
form, an explicit description of actions and a succinct description of utilities have been
assumed. It is has been established that the game isomorphism problem for general form
games is equivalent to the circuit isomorphism when utilities are described by Turing Machines;
and to the boolean formula isomorphism problem when utilities are described by
formulas. When the game is given in explicit form, it is has been proven that the game
isomorphism problem is equivalent to the graph isomorphism problem.
Finally, an equivalence classes of small games and their graphical representation have
been also examined.Postprint (published version
Query Complexity of Correlated Equilibrium
We study lower bounds on the query complexity of determining correlated
equilibrium. In particular, we consider a query model in which an n-player game
is specified via a black box that returns players' utilities at pure action
profiles. In this model we establish that in order to compute a correlated
equilibrium any deterministic algorithm must query the black box an exponential
(in n) number of times.Comment: Added reference
Pure Nash Equilibria: Hard and Easy Games
We investigate complexity issues related to pure Nash equilibria of strategic
games. We show that, even in very restrictive settings, determining whether a
game has a pure Nash Equilibrium is NP-hard, while deciding whether a game has
a strong Nash equilibrium is SigmaP2-complete. We then study practically
relevant restrictions that lower the complexity. In particular, we are
interested in quantitative and qualitative restrictions of the way each players
payoff depends on moves of other players. We say that a game has small
neighborhood if the utility function for each player depends only on (the
actions of) a logarithmically small number of other players. The dependency
structure of a game G can be expressed by a graph DG(G) or by a hypergraph
H(G). By relating Nash equilibrium problems to constraint satisfaction problems
(CSPs), we show that if G has small neighborhood and if H(G) has bounded
hypertree width (or if DG(G) has bounded treewidth), then finding pure Nash and
Pareto equilibria is feasible in polynomial time. If the game is graphical,
then these problems are LOGCFL-complete and thus in the class NC2 of highly
parallelizable problems
On the Complexity of Nash Equilibria of Action-Graph Games
We consider the problem of computing Nash Equilibria of action-graph games
(AGGs). AGGs, introduced by Bhat and Leyton-Brown, is a succinct representation
of games that encapsulates both "local" dependencies as in graphical games, and
partial indifference to other agents' identities as in anonymous games, which
occur in many natural settings. This is achieved by specifying a graph on the
set of actions, so that the payoff of an agent for selecting a strategy depends
only on the number of agents playing each of the neighboring strategies in the
action graph. We present a Polynomial Time Approximation Scheme for computing
mixed Nash equilibria of AGGs with constant treewidth and a constant number of
agent types (and an arbitrary number of strategies), together with hardness
results for the cases when either the treewidth or the number of agent types is
unconstrained. In particular, we show that even if the action graph is a tree,
but the number of agent-types is unconstrained, it is NP-complete to decide the
existence of a pure-strategy Nash equilibrium and PPAD-complete to compute a
mixed Nash equilibrium (even an approximate one); similarly for symmetric AGGs
(all agents belong to a single type), if we allow arbitrary treewidth. These
hardness results suggest that, in some sense, our PTAS is as strong of a
positive result as one can expect
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