Fig. 9: Visualization of the bottom-up statistical payoff calculation when every individual has k = 4 neighbors.

a To calculate the group-based payoff of a focal j-player with neighbor configuration k, we first have the game organized by the j-player itself, with k as the co-player configuration. b Then, we have \({\sum }_{l=1}^{n}{k}_{l}=k\) games organized by the j-player’s neighbors who adopt strategy l = 1, 2, …, n. For the j-player, the co-player configuration contains k − 1 undetermined co-players neighboring the l-player, plus the l-player. c To calculate the group-based payoff of an i-player neighboring a focal j-player, with j-player’s neighbor configuration k, we first have the game organized by the j-player. For the i-player, the co-player configuration is k_−i,+j. d Then, we have the game organized by the i-player itself, with k − 1 undetermined co-players neighboring the i-player plus the j-player. e Finally, we have the games organized by the i-player’s remaining \({\sum }_{l=1}^{n}{k}_{l}^{{\prime} }=k-1\) neighbors. For the i-player, the co-player configuration contains k − 1 undetermined co-players neighboring the l-player, plus the l-player.