Decarbonization via the integration of renewables poses significant challenges for electric power systems, but also creates new market opportunities. Electric energy storage can take advantage of these opportunities while providing flexibility to power systems that can help address these challenges. We propose a solution method for the optimal control of multiple price-maker electric energy storage units that cooperate to maximize their total profit from price arbitrage. The proposed method can tackle the nonlinearity introduced by the price-maker assumption. The main novelty of the proposed method is the combination of a decomposition by unit and a decomposition in time. The decomposition by unit is based on the Alternating Direction Method of Multipliers and breaks the problem into several one-unit subproblems. Every subproblem is solved using an efficient algorithm for one-unit problems from the literature that exploits an on the fly decomposition in time, and this results in a time decomposition for the whole solution method. Our numerical experiments show very promising performance in terms of accuracy and computational time. In particular, they suggest that computational time scales linearly with the number of storage units.