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Published in S. Sitharama Iyengar, Richard R. Brooks, Distributed Sensor Networks, 2016
Rajgopal Kannan, Costas Busch, Shuangqing Wei
For an illustration of the aforementioned definitions, Figure 32.3 depicts various outcomes in an atomic bottleneck routing game with four players going from s to t with flow fi = 1. In this game, pci(p) = maxe∊pi also denoted as Ci, which is the flow on the worst congested edge in path pi, where fe denotes the total flow (congestion) on edge e. Figure 32.3b depicts the coordinated optimal solution with SC* = 1. Figure 32.3c depicts a Nash equilibrium where two pairs of two players choose the same path. Each player has cost pci = 2 since each edge of their path is used by two paths. Since no edge is used by more than two paths, the social cost is SC = 2. Since this is the worst Nash equilibrium, the PoA for the case in Figure 32.3c is PoA = 2/1 = 2 (since the social cost of any Nash equilibrium does not exceed 2 and the optimal social cost is 1). The price of stability is PoS = 1, since there is a Nash equilibrium with optimum social cost 1 (the case of Figure 32.3b).
A strategic model of job arrivals to a single machine with earliness and tardiness penalties
Published in IISE Transactions, 2018
Amihai Glazer, Refael Hassin, Liron Ravner
Proposition 3 implies that equilibrium clustering of arrival times is often efficient. Specifically, when n is not very small there is a socially optimal equilibrium for almost all values of β and γ. This means that when the penalty function is fairly symmetric there is a socially optimal equilibrium. However, when the penalty is heavily skewed in one direction the equilibrium arrivals are too early (small (β/(β + γ))) or too late (large (β/(β + γ))). This is illustrated in Figure 4. Furthermore, note that even if for many parameter values there is a socially optimal equilibrium, welfare under most of the equilibria is less than under the socially optimal solution. In particular, the price of stability, which is the ratio between the total costs of the best equilibrium and the social optimum, is typically one, whereas for n > 2 the price of anarchy, which is the ratio between the total costs of the worst equilibrium and the social optimum, exceeds one.
Multi-phase matching mechanism for stable and optimal resource allocation in cloud manufacturing platforms Using IF-VIKOR method and deferred acceptance algorithm
Published in International Journal of Management Science and Engineering Management, 2022
Jalal Delaram, Mahmoud Houshmand, Farid Ashtiani, Omid Fatahi Valilai
Table 5 shows the solution of the proposed mechanism (optimal and stable solution or best equilibrium) with the solution of the optimization problem. The Price of Stability (PoS) of a game is defined as a ratio between the best objective function in its equilibrium state and that of an optimal outcome (Agussurja & Lau, 2009). According to Table 5, the analysis of the PoS reveals that being a private platform yields on average 10% better utility rather than a public platform.