4 research outputs found
Selfish Bin Covering
In this paper, we address the selfish bin covering problem, which is greatly
related both to the bin covering problem, and to the weighted majority game.
What we mainly concern is how much the lack of coordination harms the social
welfare. Besides the standard PoA and PoS, which are based on Nash equilibrium,
we also take into account the strong Nash equilibrium, and several other new
equilibria. For each equilibrium, the corresponding PoA and PoS are given, and
the problems of computing an arbitrary equilibrium, as well as approximating
the best one, are also considered.Comment: 16 page
A Best Cost-Sharing Rule for Selfish Bin Packing
In selfish bin packing, each item is regarded as a player, who aims to
minimize the cost-share by choosing a bin it can fit in. To have a least number
of bins used, cost-sharing rules play an important role. The currently best
known cost sharing rule has a lower bound on larger than 1.45, while a
general lower bound 4/3 on applies to any cost-sharing rule under which
no items have incentive unilaterally moving to an empty bin. In this paper, we
propose a novel and simple rule with a matching the lower bound, thus
completely resolving this game. The new rule always admits a Nash equilibrium
and its is one. Furthermore, the well-known bin packing algorithm
(Best-Fit Decreasing) is shown to achieve a strong equilibrium, implying that a
stable packing with an asymptotic approximation ratio of can be produced
in polynomial time
On Colorful Bin Packing Games
We consider colorful bin packing games in which selfish players control a set
of items which are to be packed into a minimum number of unit capacity bins.
Each item has one of colors and cannot be packed next to an item of
the same color. All bins have the same unitary cost which is shared among the
items it contains, so that players are interested in selecting a bin of minimum
shared cost. We adopt two standard cost sharing functions: the egalitarian cost
function which equally shares the cost of a bin among the items it contains,
and the proportional cost function which shares the cost of a bin among the
items it contains proportionally to their sizes. Although, under both cost
functions, colorful bin packing games do not converge in general to a (pure)
Nash equilibrium, we show that Nash equilibria are guaranteed to exist and we
design an algorithm for computing a Nash equilibrium whose running time is
polynomial under the egalitarian cost function and pseudo-polynomial for a
constant number of colors under the proportional one. We also provide a
complete characterization of the efficiency of Nash equilibria under both cost
functions for general games, by showing that the prices of anarchy and
stability are unbounded when while they are equal to 3 for black and
white games, where . We finally focus on games with uniform sizes (i.e.,
all items have the same size) for which the two cost functions coincide. We
show again a tight characterization of the efficiency of Nash equilibria and
design an algorithm which returns Nash equilibria with best achievable
performance
Complementary cooperation, minimal winning coalitions, and power indices
We introduce a new simple game, which is referred to as the complementary
weighted multiple majority game (C-WMMG for short). C-WMMG models a basic
cooperation rule, the complementary cooperation rule, and can be taken as a
sister model of the famous weighted majority game (WMG for short). In this
paper, we concentrate on the two dimensional C-WMMG. An interesting property of
this case is that there are at most minimal winning coalitions (MWC for
short), and they can be enumerated in time , where is the
number of players. This property guarantees that the two dimensional C-WMMG is
more handleable than WMG. In particular, we prove that the main power indices,
i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel
index, and the Deegan-Packel index, are all polynomially computable. To make a
comparison with WMG, we know that it may have exponentially many MWCs, and none
of the four power indices is polynomially computable (unless P=NP). Still for
the two dimensional case, we show that local monotonicity holds for all of the
four power indices. In WMG, this property is possessed by the Shapley-Shubik
index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the
Deegan-Packel index. Since our model fits very well the cooperation and
competition in team sports, we hope that it can be potentially applied in
measuring the values of players in team sports, say help people give more
objective ranking of NBA players and select MVPs, and consequently bring new
insights into contest theory and the more general field of sports economics. It
may also provide some interesting enlightenments into the design of
non-additive voting mechanisms. Last but not least, the threshold version of
C-WMMG is a generalization of WMG, and natural variants of it are closely
related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page