The value of an algorithm in a cooperative setting

The characteristic function of a cooperative game typically results from a decision making process involving optimization. In practice, players have individual algorithms to solve the underlying optimization problem, which differ in quality. Therefore, the quality of the algorithm affects the characteristic function. We define the value of an algorithm as the benefit to the player, e.g. a cost reduction. We provide conditions under which an improvement in the algorithm quality yields a benefit, but also show that surprisingly an improved algorithm might be disadvantageous. Our model naturally admits a representation of a consultant, specialized in algorithms while not directly facing the underlying optimization problem. We present conditions under which a profit for the consultant is guaranteed, but also show that only a limited number of consultants can simultaneously make a profit. We illustrate our findings by means of numerical experiments on 580,800 instances with an underlying pickup and delivery problem.