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In this paper we consider the scheduling problem with parallel-batching machines from a game theoretic perspective. There are m parallel-batching machines each of which can handle up to b jobs simultaneously as a batch. The processing time of a batch is the time required for processing the longest job in the batch, and all the jobs in a batch start and complete at the same time. There are n jobs. Each job is owned by a rational and selfish agent and its individual cost is the completion time of its job. The social cost is the largest completion time over all jobs, the makespan. We design a coordination mechanism for the scheduling game problem. We discuss the existence of pure Nash Equilibria and offer upper and lower bounds on the price of anarchy of the coordination mechanism. We show that the mechanism has a price of anarchy no more than 2 − 2 3b − 1 3 max{m,b} .

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In this paper, we consider a scheduling problem with [Formula: see text] uniform parallel-batching machines [Formula: see text] under game situation. There are [Formula: see text] jobs, each of which is associated with a load. Each machine [Formula: see text] has a speed [Formula: see text] and can handle up to [Formula: see text] 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 [Formula: see text], where [Formula: see text], [Formula: see text], and [Formula: see text] is a small positive number that tends to 0.