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Multiple Attribute Decision Making (MADM) problem is a management science technique, which is popularly used to rank the priority of alternatives with respect to their competing attributes. It is popularly used in diverse fields such as engineering management, portfolio selection, transportation planning, and performance evaluation. Weights form the core of MADM: it is obvious that different weights lead to various evaluation results and decisions. Several approaches have been developed for assessing the weights of MADM problems, e.g., the eigenvector method, ELECTRE, and TOPSIS. However, an assessment approach of weights in MADM, which meets both the need of simplicity interface for practitioners and concrete theory for scholars is not easy, and balancing these two aspects is a challenging and tough task. Since the pay-off matrix in game theory could be regarded as a simple interface for data input/output, and very few scholars had ever explored the two-person zero-sum game on MADM problems. In this paper, the weights of a MADM problem are obtained by formulating it as a two-person zero-sum game with multiple decision makers. The group equilibrium solution, i.e., consensus of weights and the resolution steps for such a group MADM game has also been originally developed and validated in this study. Finally, an actual case of selecting the appropriate portfolio decision for a paper company is illustrated.

In this paper, we consider a scheduling problem with $m$ uniform parallel-batching machines $\{M_1,M_2,\dots ,M_m\}$ under game situation. There are $n$ jobs, each of which is associated with a load. Each machine $M_i(1\le i\le m)$ has a speed $s_i$ and can handle up to $b$ jobs simultaneously as a batch. The load of a batch is the load of the longest job in the batch. All the jobs in a batch start and complete at the same time. Each job is owned by an agent and its individual cost is the completion time of the job. The social cost is the largest completion time over all jobs, i.e., the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of Nash Equilibrium and offer an upper bound on the price of anarchy (POA) of the coordination mechanism. We present a greedy algorithm and show that: (i) under the coordination mechanism, any instance of the scheduling game problem has a unique Nash Equilibrium and it is precisely the schedule returned by the greedy algorithm; (ii) the mechanism has a POA no more than $1+\frac{s_{max}}{\bar{s}}\left(\right.1-\frac{1}{max\{m,b\}}\left)\right.+\delta$, where $s_{max}=max\{s_1,s_2,\dots ,s_m\}$, $\bar{s}=(s_1+s_2+\cdots +s_m)/m$, and $\delta$ is a small positive number that tends to 0.