Fig 2. Overview of the gameplay.

(I) Each game consists of several turns obeying a common set of rules. First, one or all the players choose a strategy (a direction on the grid) according to the relative probabilities of their choosing each strategy. The score is then updated by one or all the players, based on their distances from the other players, and the probabilities are then updated. Depending on the type of game, three schemes are possible (II): (a) in the AA game, all players apply a strategy and all players then calculate their scores, in two successive steps, (b) in the OA game, all players calculate their score each time a single player plays (the usual total number of turns k is thus divided by the number of players to allow for the same relative number of iterations), (c) in the OO game, each player applies a strategy and the score for that player is calculated before the next player plays.