12 research outputs found
Nash Equilibrium in Generalised Muller Games
We suggest that extending Muller games with preference ordering for
players is a natural way to reason about unbounded duration games. In
this context, we look at the standard solution concept of Nash
equilibrium for non-zero sum games. We show that Nash equilibria
always exists for such generalised Muller games on finite graphs and
present a procedure to compute an equilibrium strategy profile. We
also give a procedure to compute a subgame perfect equilibrium when it
exists in such games
Imitation in Large Games
In games with a large number of players where players may have overlapping
objectives, the analysis of stable outcomes typically depends on player types.
A special case is when a large part of the player population consists of
imitation types: that of players who imitate choice of other (optimizing)
types. Game theorists typically study the evolution of such games in dynamical
systems with imitation rules. In the setting of games of infinite duration on
finite graphs with preference orderings on outcomes for player types, we
explore the possibility of imitation as a viable strategy. In our setup, the
optimising players play bounded memory strategies and the imitators play
according to specifications given by automata. We present algorithmic results
on the eventual survival of types
Infinite sequential Nash equilibrium
In game theory, the concept of Nash equilibrium reflects the collective
stability of some individual strategies chosen by selfish agents. The concept
pertains to different classes of games, e.g. the sequential games, where the
agents play in turn. Two existing results are relevant here: first, all finite
such games have a Nash equilibrium (w.r.t. some given preferences) iff all the
given preferences are acyclic; second, all infinite such games have a Nash
equilibrium, if they involve two agents who compete for victory and if the
actual plays making a given agent win (and the opponent lose) form a
quasi-Borel set. This article generalises these two results via a single
result. More generally, under the axiomatic of Zermelo-Fraenkel plus the axiom
of dependent choice (ZF+DC), it proves a transfer theorem for infinite
sequential games: if all two-agent win-lose games that are built using a
well-behaved class of sets have a Nash equilibrium, then all multi-agent
multi-outcome games that are built using the same well-behaved class of sets
have a Nash equilibrium, provided that the inverse relations of the agents'
preferences are strictly well-founded.Comment: 14 pages, will be published in LMCS-2011-65
Multiplayer Cost Games with Simple Nash Equilibria
Multiplayer games with selfish agents naturally occur in the design of
distributed and embedded systems. As the goals of selfish agents are usually
neither equivalent nor antagonistic to each other, such games are non zero-sum
games. We study such games and show that a large class of these games,
including games where the individual objectives are mean- or discounted-payoff,
or quantitative reachability, and show that they do not only have a solution,
but a simple solution. We establish the existence of Nash equilibria that are
composed of k memoryless strategies for each agent in a setting with k agents,
one main and k-1 minor strategies. The main strategy describes what happens
when all agents comply, whereas the minor strategies ensure that all other
agents immediately start to co-operate against the agent who first deviates
from the plan. This simplicity is important, as rational agents are an
idealisation. Realistically, agents have to decide on their moves with very
limited resources, and complicated strategies that require exponential--or even
non-elementary--implementations cannot realistically be implemented. The
existence of simple strategies that we prove in this paper therefore holds a
promise of implementability.Comment: 23 page
From winning strategy to Nash equilibrium
Game theory is usually considered applied mathematics, but a few
game-theoretic results, such as Borel determinacy, were developed by
mathematicians for mathematics in a broad sense. These results usually state
determinacy, i.e. the existence of a winning strategy in games that involve two
players and two outcomes saying who wins. In a multi-outcome setting, the
notion of winning strategy is irrelevant yet usually replaced faithfully with
the notion of (pure) Nash equilibrium. This article shows that every
determinacy result over an arbitrary game structure, e.g. a tree, is
transferable into existence of multi-outcome (pure) Nash equilibrium over the
same game structure. The equilibrium-transfer theorem requires cardinal or
order-theoretic conditions on the strategy sets and the preferences,
respectively, whereas counter-examples show that every requirement is relevant,
albeit possibly improvable. When the outcomes are finitely many, the proof
provides an algorithm computing a Nash equilibrium without significant
complexity loss compared to the two-outcome case. As examples of application,
this article generalises Borel determinacy, positional determinacy of parity
games, and finite-memory determinacy of Muller games
On (Subgame Perfect) Secure Equilibrium in Quantitative Reachability Games
We study turn-based quantitative multiplayer non zero-sum games played on
finite graphs with reachability objectives. In such games, each player aims at
reaching his own goal set of states as soon as possible. A previous work on
this model showed that Nash equilibria (resp. secure equilibria) are guaranteed
to exist in the multiplayer (resp. two-player) case. The existence of secure
equilibria in the multiplayer case remained and is still an open problem. In
this paper, we focus our study on the concept of subgame perfect equilibrium, a
refinement of Nash equilibrium well-suited in the framework of games played on
graphs. We also introduce the new concept of subgame perfect secure
equilibrium. We prove the existence of subgame perfect equilibria (resp.
subgame perfect secure equilibria) in multiplayer (resp. two-player)
quantitative reachability games. Moreover, we provide an algorithm deciding the
existence of secure equilibria in the multiplayer case.Comment: 32 pages. Full version of the FoSSaCS 2012 proceedings pape
Extending finite-memory determinacy to multi-player games
We show that under some general conditions the finite memory determinacy of a class of two-player win/lose games played on finite graphs implies the existence of a Nash equilibrium built from finite memory strategies for the corresponding class of multi-player multi-outcome games. This generalizes a previous result by Brihaye, De Pril and Schewe. We provide a number of example that separate the various criteria we explore.Our proofs are generally constructive, that is, provide upper bounds for the memory required, as well as algorithms to compute the relevant Nash equilibria
Extending Finite Memory Determinacy to Multiplayer Games
We show that under some general conditions the finite memory determinacy of a
class of two-player win/lose games played on finite graphs implies the
existence of a Nash equilibrium built from finite memory strategies for the
corresponding class of multi-player multi-outcome games. This generalizes a
previous result by Brihaye, De Pril and Schewe. For most of our conditions we
provide counterexamples showing that they cannot be dispensed with.
Our proofs are generally constructive, that is, provide upper bounds for the
memory required, as well as algorithms to compute the relevant winning
strategies.Comment: In Proceedings SR 2016, arXiv:1607.0269