8 research outputs found
The Complexity of game isomorphism
We address the question of whether two multiplayer strategic games are equivalent and the computational complexity of deciding such a property. We introduce two notions of isomorphisms, strong and weak. Each one of those isomorphisms preserves a different structure of the game. Strong isomorphisms are defined to preserve the utility functions and Nash equilibria. Weak isomorphisms preserve only the player's preference relations and thus pure Nash equilibria. We show that the computational complexity of the game isomorphism problem depends on the level of succinctness of the description of the input games but it is independent on which of the two types of isomorphisms is considered. 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, we asume a explicit description of actions and a succinct description of utilities. We show that the game isomorphism problem for general form games is equivalent to the circuit isomorphism when utilities are described by TMs and to the boolean formula isomorphism problem when utilities are described by formulas. When the game is given in explicit form, we show that the game isomorphism problem is equivalent to the graph isomorphism problem.Postprint (published version
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
Boolean Game with Prioritized Norms
In this paper we study boolean game with prioritized norms. Norms distinguish illegal strategies from legal strategies. Notions like legal strategy and legal Nash equilibrium are introduced. Our formal model is a combination of (weighted) boolean game and so called (prioritized) input/output logic. After formally presenting the model, we use examples to show that non-optimal Nash equilibrium can be avoided by making use of norms.We study various complexity issues related to legal strategy and legal Nash equilibrium
Partial-order Boolean games: informational independence in a logic-based model of strategic interaction
As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on variable y, then we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly (IF) logic