5 research outputs found
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 Pure-Strategy Nash Equilibria in Congestion and Local-Effect Games
doi 10.1287/moor.1080.032
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
Pure Nash equilibria in games with a large number of actions
We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game?, and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in an strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that deciding the existence of a Nash equilibrium in an strategic game is NP-complete when the number of players is large and the number of strategies for each player is constant, while the problem is Sigma_2^p-complete when the number of players is a constant and the size of the sets of strategies is exponential (with respect to the length of the strategies).Postprint (published version
Pure Nash equilibria in games with a large number of actions
We study the computational complexity of deciding the existence of a Pure Nash Equilibrium in multi-player strategic games. We address two fundamental questions: how can we represent a game?, and how can we represent a game with polynomial pay-off functions? Our results show that the computational complexity of deciding the existence of a pure Nash equilibrium in an strategic game depends on two parameters: the number of players and the size of the sets of strategies. In particular we show that deciding the existence of a Nash equilibrium in an strategic game is NP-complete when the number of players is large and the number of strategies for each player is constant, while the problem is Sigma_2^p-complete when the number of players is a constant and the size of the sets of strategies is exponential (with respect to the length of the strategies)