16 research outputs found
Uniform Proofs of Order Independence for Various Strategy Elimination Procedures
We provide elementary and uniform proofs of order independence for various
strategy elimination procedures for finite strategic games, both for dominance
by pure and by mixed strategies. The proofs follow the same pattern and focus
on the structural properties of the dominance relations. They rely on Newman's
Lemma established in 1942 and related results on the abstract reduction
systems.Comment: 48 page
Direct Proofs of Order Independence
We establish a generic result concerning order independence of a dominance
relation on finite games. It allows us to draw conclusions about order
independence of various dominance relations in a direct and simple way.Comment: 9 page
Direct proofs of order independence
We establish a generic result concerning order independence of a dominance relation on finite games. It allows us to draw conclusions about order independence of various dominance relations in a direct and simple way.dominance relations,order independence, hereditarity, monotonicity
On Iterated Dominance, Matrix Elimination, and Matched Paths
We study computational problems arising from the iterated removal of weakly
dominated actions in anonymous games. Our main result shows that it is
NP-complete to decide whether an anonymous game with three actions can be
solved via iterated weak dominance. The two-action case can be reformulated as
a natural elimination problem on a matrix, the complexity of which turns out to
be surprisingly difficult to characterize and ultimately remains open. We
however establish connections to a matching problem along paths in a directed
graph, which is computationally hard in general but can also be used to
identify tractable cases of matrix elimination. We finally identify different
classes of anonymous games where iterated dominance is in P and NP-complete,
respectively.Comment: 12 pages, 3 figures, 27th International Symposium on Theoretical
Aspects of Computer Science (STACS
A Generic Approach to Coalition Formation
We propose an abstract approach to coalition formation that focuses on simple
merge and split rules transforming partitions of a group of players. We
identify conditions under which every iteration of these rules yields a unique
partition. The main conceptual tool is a specific notion of a stable partition.
The results are parametrized by a preference relation between partitions of a
group of players and naturally apply to coalitional TU-games, hedonic games and
exchange economy games.Comment: 21 pages. To appear in International Game Theory Review (IGTR
Iterated elimination of weakly dominated strategies in well-founded games
Recently, in [3], we studied well-founded games, a natural extension of finite extensive games with perfect information in which all plays are finite. We extend here, to this class of games, two results concerned with iterated elimination of weakly dominated strategies, originally established for finite extensive games. The first one states that every finite extensive game with perfect information and injective payoff functions can be reduced by a specific iterated elimination of weakly dominated strategies to a trivial game containing the unique subgame perfect equilibrium. Our extension of this result to well-founded games admits transfinite iterated elimination of strategies. It applies to an infinite version of the centipede game. It also generalizes the original result to a class of finite games that may have several subgame perfect equilibria. The second one states that finite zero-sum games with n outcomes can be solved by the maximal iterated elimination of weakly dominated strategies in n - 1 steps. We generalize this result to a natural class of well-founded strictly competitive games
The order independence of iterated dominance in extensive games
Shimoji and Watson (1998) prove that a strategy of an extensive game is rationalizable
in the sense of Pearce if and only if it survives the maximal elimination of conditionally dominated strategies. Briefly, this process iteratively eliminates
conditionally dominated strategies according to a specific order, which is also the
start of an order of elimination of weakly dominated strategies. Since the final
set of possible payoff profiles, or terminal nodes, surviving iterated elimination of
weakly dominated strategies may be order-dependent, one may suspect that the
same holds for conditional dominance.
We prove that, although the sets of strategy profiles surviving two arbitrary
elimination orders of conditional dominance may be very different from each
other, they are equivalent in the following sense: for each player i and each pair
of elimination orders, there exists a function φi mapping each strategy of i surviving
the first order to a strategy of i surviving the second order, such that, for every
strategy profile s surviving the first order, the profile (φi(si))i induces the same
terminal node as s does.
To prove our results, we put forward a new notion of dominance and an elementary
characterization of extensive-form rationalizability (EFR) that may be
of independent interest. We also establish connections between EFR and other
existing iterated dominance procedures, using our notion of dominance and our
characterization of EFR
The Order Independence of Iterated Dominance in Extensive Games, with Connections to Mechanism Design and Backward Induction
Shimoji and Watson (1998) prove that a strategy of an extensive game is rationalizable in the sense of Pearce if and only if it survives the maximal elimination of conditionally dominated strategies. Briefly, this process iteratively eliminates conditionally dominated strategies according to a specific order, which is also the start of an order of elimination of weakly dominated strategies. Since the final set of possible payoff profiles, or terminal nodes, surviving iterated elimination of weakly dominated strategies may be order-dependent, one may suspect that the same holds for conditional dominance. We prove that, although the sets of strategy profiles surviving two arbitrary elimination orders of conditional dominance may be very different from each other, they are equivalent in the following sense: for each player i and each pair of elimination orders, there exists a function phi_i mapping each strategy of i surviving the first order to a strategy of i surviving the second order, such that, for every strategy profile s surviving the first order, the profile (phi_i(s_i))_i induces the same terminal node as s does. To prove our results we put forward a new notion of dominance and an elementary characterization of extensive-form rationalizability (EFR) that may be of independent interest. We also establish connections between EFR and other existing iterated dominance procedures, using our notion of dominance and our characterization of EFR