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Network slicing is a key capability for next generation mobile networks. It enables one to cost effectively customize logical networks over a shared infrastructure. A critical component of network slicing is resource allocation, which needs to ensure that slices receive the resources needed to support their mobiles/services while optimizing network efficiency. In this paper, we propose a novel approach to slice-based resource allocation named Guaranteed seRvice Efficient nETwork slicing (GREET). The underlying concept is to set up a constrained resource allocation game, where (i) slices unilaterally optimize their allocations to best meet their (dynamic) customer loads, while (ii) constraints are imposed to guarantee that, if they wish so, slices receive a pre-agreed share of the network resources. The resulting game is a variation of the well-known Fisher market, where slices are provided a budget to contend for network resources (as in a traditional Fisher market), but (unlike a Fisher market) prices are constrained for some resources to provide the desired guarantees. In this way, GREET combines the advantages of a share-based approach (high efficiency by flexible sharing) and reservation-based ones (which provide guarantees by assigning a fixed amount of resources). We characterize the Nash equilibrium, best response dynamics, and propose a practical slice strategy with provable convergence properties. Extensive simulations exhibit substantial improvements over network slicing state-of-the-art benchmarks.