The Complexity of angel-daemons and game isomorphism
Authors:   , García Chacón Alina 
Published: October 2023

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.