Shapley Value (2-player)

The 2-player Shapley value assigns each player the average of their marginal contribution across the two possible orders in which the players can join a coalition. For player i, that means averaging what they are worth alone, v(i), and what they add when joining the other player, v(N) - v(j). This works because it fairly splits the total surplus while respecting symmetry, so neither player is favored just because of the order in which contributions are counted.

Use this when two agents, firms, or partners jointly create value and you want a principled way to divide the total payoff based on contribution rather than bargaining power. It appears in cooperative game theory, profit-sharing, cost allocation, and any two-player setting where the value of alone vs together can be measured.

• Using the grand coalition value twice instead of averaging each player's standalone and join-in contribution
• Subtracting the wrong player's solo value when computing the marginal contribution term
• Forgetting that the two-player Shapley values should add up exactly to v(N)