The effect of algorithm capabilities on cooperative games

Collaborations lead to cost reductions, both monetary and environmentally. However, it is not immediately clear how multiple companies with a shared optimisation problem should arrive at solutions to this shared problem or a fair allocation of the resulting cost or profit. In contrast to the literature, we assume each company, also referred to as a player, to have access to a potentially heuristic algorithm that is used to determine solutions to this shared optimisation problem. Together, the players can use these algorithms to determine solutions to shared problem instances. We call a cooperative game in which player algorithms are explicitly taken into account an algorithm quality induced game (AQI game). In an AQI game, the cost that is allocated to a player also depends on their algorithmic capabilities, that is, the quality of their algorithms. Moreover, it also allows us to model consultants, i.e., players that do have a good algorithm for the shared optimisation problem, but do not contribute in any other manner to the shared operations. In an AQI game, such players can be allocated a profit. In this paper we describe the core of AQI games and analyse the effects of improving the algorithm of a single player and of adding a consultant to a collaboration. Moreover, we present numerical results for 580,800 instances of the AQI game. We quantify the effect of improving an algorithm on the allocated cost to this player. We show that a player in general is allocated less after improving their algorithm, while in some cases the allocated cost increases. Moreover, we find that in general players with a bad algorithm benefit most from the addition of a consultant while players with a good algorithm may not benefit at all.