48,173 research outputs found
Evolutionary stability on graphs
Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree k > 2. Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth-death (BD), death-birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs
Stabilization of Capacitated Matching Games
An edge-weighted, vertex-capacitated graph G is called stable if the value of
a maximum-weight capacity-matching equals the value of a maximum-weight
fractional capacity-matching. Stable graphs play a key role in characterizing
the existence of stable solutions for popular combinatorial games that involve
the structure of matchings in graphs, such as network bargaining games and
cooperative matching games.
The vertex-stabilizer problem asks to compute a minimum number of players to
block (i.e., vertices of G to remove) in order to ensure stability for such
games. The problem has been shown to be solvable in polynomial-time, for
unit-capacity graphs. This stays true also if we impose the restriction that
the set of players to block must not intersect with a given specified maximum
matching of G.
In this work, we investigate these algorithmic problems in the more general
setting of arbitrary capacities. We show that the vertex-stabilizer problem
with the additional restriction of avoiding a given maximum matching remains
polynomial-time solvable. Differently, without this restriction, the
vertex-stabilizer problem becomes NP-hard and even hard to approximate, in
contrast to the unit-capacity case.
Finally, in unit-capacity graphs there is an equivalence between the
stability of a graph, existence of a stable solution for network bargaining
games, and existence of a stable solution for cooperative matching games. We
show that this equivalence does not extend to the capacitated case.Comment: 14 pages, 3 figure
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Evolutionary Stability on Graphs
Evolutionary stability is a fundamental concept in evolutionary game theory. A strategy is called an evolutionarily stable strategy (ESS), if its monomorphic population rejects the invasion of any other mutant strategy. Recent studies have revealed that population structure can considerably affect evolutionary dynamics. Here we derive the conditions of evolutionary stability for games on graphs. We obtain analytical conditions for regular graphs of degree . Those theoretical predictions are compared with computer simulations for random regular graphs and for lattices. We study three different update rules: birth–death (BD), death–birth (DB), and imitation (IM) updating. Evolutionary stability on sparse graphs does not imply evolutionary stability in a well-mixed population, nor vice versa. We provide a geometrical interpretation of the ESS condition on graphs.MathematicsOrganismic and Evolutionary Biolog
Computing the Least-core and Nucleolus for Threshold Cardinality Matching Games
Cooperative games provide a framework for fair and stable profit allocation
in multi-agent systems. \emph{Core}, \emph{least-core} and \emph{nucleolus} are
such solution concepts that characterize stability of cooperation. In this
paper, we study the algorithmic issues on the least-core and nucleolus of
threshold cardinality matching games (TCMG). A TCMG is defined on a graph
and a threshold , in which the player set is and the profit of
a coalition is 1 if the size of a maximum matching in
meets or exceeds , and 0 otherwise. We first show that for a TCMG, the
problems of computing least-core value, finding and verifying least-core payoff
are all polynomial time solvable. We also provide a general characterization of
the least core for a large class of TCMG. Next, based on Gallai-Edmonds
Decomposition in matching theory, we give a concise formulation of the
nucleolus for a typical case of TCMG which the threshold equals . When
the threshold is relevant to the input size, we prove that the nucleolus
can be obtained in polynomial time in bipartite graphs and graphs with a
perfect matching
Celebrity games
We introduce Celebrity games, a new model of network creation games. In this model players have weights (W being the sum of all the player's weights) and there is a critical distance ß as well as a link cost a. The cost incurred by a player depends on the cost of establishing links to other players and on the sum of the weights of those players that remain farther than the critical distance. Intuitively, the aim of any player is to be relatively close (at a distance less than ß ) from the rest of players, mainly of those having high weights. The main features of celebrity games are that: computing the best response of a player is NP-hard if ß>1 and polynomial time solvable otherwise; they always have a pure Nash equilibrium; the family of celebrity games having a connected Nash equilibrium is characterized (the so called star celebrity games) and bounds on the diameter of the resulting equilibrium graphs are given; a special case of star celebrity games shares its set of Nash equilibrium profiles with the MaxBD games with uniform bounded distance ß introduced in Bilò et al. [6]. Moreover, we analyze the Price of Anarchy (PoA) and of Stability (PoS) of celebrity games and give several bounds. These are that: for non-star celebrity games PoA=PoS=max{1,W/a}; for star celebrity games PoS=1 and PoA=O(min{n/ß,Wa}) but if the Nash Equilibrium is a tree then the PoA is O(1); finally, when ß=1 the PoA is at most 2. The upper bounds on the PoA are complemented with some lower bounds for ß=2.Peer ReviewedPostprint (author's final draft
Testing Stability Properties in Graphical Hedonic Games
In hedonic games, players form coalitions based on individual preferences
over the group of players they belong to. Several concepts to describe the
stability of coalition structures in a game have been proposed and analyzed.
However, prior research focuses on algorithms with time complexity that is at
least linear in the input size. In the light of very large games that arise
from, e.g., social networks and advertising, we initiate the study of sublinear
time property testing algorithms for existence and verification problems under
several notions of coalition stability in a model of hedonic games represented
by graphs with bounded degree. In graph property testing, one shall decide
whether a given input has a property (e.g., a game admits a stable coalition
structure) or is far from it, i.e., one has to modify at least an
-fraction of the input (e.g., the game's preferences) to make it have
the property. In particular, we consider verification of perfection, individual
rationality, Nash stability, (contractual) individual stability, and core
stability. Furthermore, we show that while there is always a Nash-stable
coalition (which also implies individually stable coalitions), the existence of
a perfect coalition can be tested. All our testers have one-sided error and
time complexity that is independent of the input size
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Chris Cannings: A Life in Games
Chris Cannings was one of the pioneers of evolutionary game theory. His early work was inspired by the formulations of John Maynard Smith, Geoff Parker and Geoff Price; Chris recognized the need for a strong mathematical foundation both to validate stated results and to give a basis for extensions of the models. He was responsible for fundamental results on matrix games, as well as much of the theory of the important war of attrition game, patterns of evolutionarily stable strategies, multiplayer games and games on networks. In this paper we describe his work, key insights and their influence on research by others in this increasingly important field. Chris made substantial contributions to other areas such as population genetics and segregation analysis, but it was to games that he always returned. This review is written by three of his students from different stages of his career
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