Prices of Anarchy of Selfish 2D Bin Packing Games

2019 ◽  
Vol 30 (03) ◽  
pp. 355-374
Author(s):  
Cristina G. Fernandes ◽  
Carlos E. Ferreira ◽  
Flávio K. Miyazawa ◽  
Yoshiko Wakabayashi
Keyword(s):  
Nash Equilibrium ◽  
Bin Packing ◽  
Price Of Anarchy ◽  
Dimensional Case ◽  
Theoretical Problem ◽  
Single Item ◽  
Strong Nash Equilibrium ◽  
Square Packing ◽  
The Cost

We consider a game-theoretical problem called selfish 2-dimensional bin packing game, a generalization of the 1-dimensional case already treated in the literature. In this game, the items to be packed are rectangles, and the bins are unit squares. The game starts with a set of items arbitrarily packed in bins. The cost of an item is defined as the ratio between its area and the total occupied area of the respective bin. Each item is a selfish player that wants to minimize its cost. A migration of an item to another bin is allowed only when its cost is decreased. We show that this game always converges to a Nash equilibrium (a stable packing where no single item can decrease its cost by migrating to another bin). We show that the pure price of anarchy of this game is unbounded, so we address the particular case where all items are squares. We show that the pure price of anarchy of the selfish square packing game is at least [Formula: see text] and at most [Formula: see text]. We also present analogous results for the strong Nash equilibrium (a stable packing where no nonempty set of items can simultaneously migrate to another common bin and decrease the cost of each item in the set). We show that the strong price of anarchy when all items are squares is at least [Formula: see text] and at most [Formula: see text].

THE PRICE OF NASH EQUILIBRIA IN MULTICAST TRANSMISSION GAMES

2010 ◽  
Vol 11 (03n04) ◽  
pp. 97-120 ◽  
Author(s):  
VITTORIO BILÒ
Keyword(s):  
Nash Equilibrium ◽  
Cost Sharing ◽  
Price Of Anarchy ◽  
Optimal Solution ◽  
Minimum Cost ◽  
Price Of Stability ◽  
Common Source ◽  
Cost Share ◽  
Worst Case ◽  
The Cost

We consider the problem of sharing the cost of multicast transmissions in non-cooperative undirected networks where a set of receivers R wants to be connected to a common source s. The set of choices available to each receiver r ∈ R is represented by the set of all (s, r)-paths in the network. Given the choices performed by all the receivers, a public known cost sharing method determines the cost share to be charged to each of them. Receivers are selfish agents aiming to obtain the transmission at the minimum cost share and their interactions create a non-cooperative game. Devising cost sharing methods yielding games whose price of anarchy (price of stability), defined as the worst-case (best-case) ratio between the cost of a Nash equilibrium and that of an optimal solution, is not too high is thus of fundamental importance in non-cooperative network design. Moreover, since cost sharing games naturally arise in socio-economical contests, it is convenient for a cost sharing method to meet some constraining properties. In this paper, we first define several such properties and analyze their impact on the prices of anarchy and stability. We also reconsider all the methods known so far by classifying them according to which properties they satisfy and giving the first non-trivial lower bounds on their price of stability. Finally, we propose a new method, namely the free-riders method, which admits a polynomial time algorithm for computing a pure Nash equilibrium whose cost is at most twice the optimal one. Some of the ideas characterizing our approach have been independently proposed in Ref. 10.

scholarly journals Cost of cooperation for scheduling meetings

2010 ◽  
Vol 7 (3) ◽  
pp. 551-568 ◽  
Author(s):  
Alon Grubshtein ◽  
Amnon Meisels
Keyword(s):  
Nash Equilibrium ◽  
Equilibrium Point ◽  
Price Of Anarchy ◽  
Distributed Optimization ◽  
Objective Functions ◽  
Game Theoretic ◽  
The Cost ◽  
Global Objective

Scheduling meetings among agents can be represented as a game - the Meetings Scheduling Game (MSG). In its simplest form, the two-person MSG is shown to have a price of anarchy (PoA) which is bounded by 0.5. The PoA bound provides a measure on the efficiency of the worst Nash Equilibrium in social (or global) terms. The approach taken by the present paper introduces the Cost of Cooperation (CoC) for games. The CoC is defined with respect to different global objective functions and provides a measure on the efficiency of a solution for each participant (personal). Applying an ?egalitarian? objective, that maximizes the minimal gain among all participating agents, on our simple example results in a CoC which is non positive for all agents. This makes the MSG a cooperation game. The concepts are defined and examples are given within the context of the MSG. Although not all games are cooperation games, a game may be revised by adding a mediator (or with a slight change of its mechanism) so that it behaves as a cooperation game. Rational participants can cooperate (by taking part in a distributed optimization protocol) and receive a payoff which will be at least as high as the worst gain expected by a game theoretic equilibrium point.

A bin packing game with cardinality constraints under the best cost rule

2019 ◽  
Vol 11 (02) ◽  
pp. 1950022
Author(s):  
Qingqin Nong ◽  
Jiapeng Wang ◽  
Suning Gong ◽  
Saijun Guo
Keyword(s):  
Cooperative Game ◽  
Bin Packing ◽  
Social Cost ◽  
Price Of Anarchy ◽  
Packing Problem ◽  
Cardinality Constraints ◽  
Bin Packing Problem ◽  
The Social ◽  
The Cost

We consider the bin packing problem with cardinality constraints in a non-cooperative game setting. In the game, there are a set of items with sizes between 0 and 1, and a number of bins each of which has a capacity of 1. Each bin can pack at most [Formula: see text] items, for a given integer parameter [Formula: see text]. The social cost is the number of bins used in the packing. Each item tries to be packed into one of the bins so as to minimize its cost. The selfish behaviors of the items result in some kind of equilibrium, which greatly depends on the cost rule in the game. We say a cost rule encourages sharing if for an item, compared with sharing a bin with some other items, staying in a bin alone does not decrease its cost. In this paper, we first show that for any bin packing game with cardinality constraints under an encourage-sharing cost rule, the price of anarchy of it is at least [Formula: see text]. We then propose a cost rule and show that the price of anarchy of the bin packing game under the rule is [Formula: see text] when [Formula: see text].

Applications to Economics

2016 ◽  
Author(s):  
Jacob K. Goeree ◽  
Charles A. Holt ◽  
Thomas R. Palfrey
Keyword(s):  
Nash Equilibrium ◽  
Private Information ◽  
Price Competition ◽  
Quantal Response ◽  
Unique Nash Equilibrium ◽  
Common Effort ◽  
The Cost ◽  
All Pay Auction ◽  
Individual Effort ◽  
Applications To Economics

This chapter explores whether the equilibrium effects of noisy behavior can cause large deviations from standard predictions in economically relevant situations. It considers a simple price-competition game, which is also partly motivated by the possibility of changing a payoff parameter that has no effect on the unique Nash equilibrium, but which may be expected to affect quantal response equilibrium. In the minimum-effort coordination game studied, any common effort in the range of feasible effort levels is a Nash equilibrium, but one would expect that an increase in the cost of individual effort or an increase in the number of players who are trying to coordinate would reduce the effort levels observed in an experiment. The chapter presents an analysis of the logit equilibrium and rent dissipation for a rent-seeking contest that is modeled as an “all-pay auction.” The final two applications in this chapter deal with auctions with private information.

Classes of Potential Games

2017 ◽  
Author(s):  
João P. Hespanha
Keyword(s):  
Resource Allocation ◽  
Nash Equilibrium ◽  
Nash Equilibria ◽  
Potential Games ◽  
Congestion Games ◽  
Fictitious Play ◽  
Resource Allocation Problem ◽  
Distributed Resource Allocation ◽  
Symmetric Games ◽  
The Cost

This chapter discusses several classes of potential games that are common in the literature and how to derive the Nash equilibrium for such games. It first considers identical interests games and dummy games before turning to decoupled games and bilateral symmetric games. It then describes congestion games, in which all players are equal, in the sense that the cost associated with each resource only depends on the total number of players using that resource and not on which players use it. It also presents other potential games, including the Sudoku puzzle, and goes on to analyze the distributed resource allocation problem, the computation of Nash equilibria for potential games, and fictitious play. It concludes with practice exercises and their corresponding solutions, along with additional exercises.

Markets versus Negotiations: The Predominance of Centralized Markets

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Zvika Neeman ◽  
Nir Vulkan
Keyword(s):  
Nash Equilibrium ◽  
Exchange Mechanism ◽  
Future Period ◽  
Strong Nash Equilibrium ◽  
Heterogenous Group ◽  
Informed Traders ◽  
Decentralized Bargaining

The paper considers the consequences of competition between two widely used exchange mechanisms, a “decentralized bargaining'' market, and a “centralized'' market. In every period, members of a large heterogenous group of privately-informed traders who each wish to buy or sell one unit of some homogenous good may opt for trading through one exchange mechanism. Traders may also postpone their trade to a future period. It is shown that trade outside the centralized market completely unravels. In every strong Nash equilibrium, all trade takes place in the centralized market. No trade ever occurs through direct negotiations.

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