A profit-maximizing oligopolist, like any profit-maximizing firm, should choose its input bundle to minimize the cost of producing its output. How should it choose that output?
The models of a competitive industry and of a monopoly both consider firms in isolation. In the model of a competitive industry, each firm takes the price as given; it ignores any impact its actions may have on the price. In the model of a monopoly, the firm has no rivals to worry about. Consequently, we can use the tool of decision theory to analyze both models: in each case, the problem of a firm is simply a maximization problem.
The essence of oligopoly is that the firms interact. Thus to gain an understanding of oligopoly we need tools suitable for analyzing situations in which decision-makers interact. Game theory provides such tools.