This paper examines the execution problems of large traders with a generalized price impact. Constructing two related models in a discrete-time setting, we solve these problems by applying the backward induction method of dynamic programming. In the first problem, we formulate the expected utility maximization problem of a single large trader as a Markov decision process and derive an optimal execution strategy. Then, in the second model, we formulate the expected utility maximization problem of two large traders as a Markov game and derive an equilibrium execution strategy at a Markov perfect equilibrium. Both of these two models enable us to investigate how the execution strategies and trade performances of a large trader are affected by the existence of other traders. Moreover, we find that these optimal and equilibrium execution strategies become deterministic when the total execution volumes of non-large traders are deterministic. We also show, by some numerical examples, the comparative statics results with respect to several problem parameters.