Two-Player Shapley Value Calculator

Enter what player one can produce alone, what player two can produce alone, and what they create together. The result gives each player's Shapley allocation, the cooperation surplus or shortfall, and the grand coalition value used for the calculation. It is useful for transparent revenue sharing, cost allocation, partnership analysis, bargaining preparation, and cooperative game theory exercises.

Define the three coalition values consistently

A two-player cooperative game needs three supplied values because the empty coalition is normalized to zero. Enter player one value as the amount player one can create without player two, and enter player two value on the same basis. Then enter the grand coalition value, which is the value created when both participate. All three figures must describe the same outcome, currency, time period, and accounting boundary. For example, do not compare one player's monthly profit with the partnership's annual revenue, or use gross sales for the singleton values and net profit for the joint value. Values may be positive, zero, or negative, so the calculator can represent benefits, costs, losses, or obligations as long as their signs are interpreted consistently. The tool assumes the empty coalition has value zero and that the entire grand coalition value will be allocated between the two players. It does not estimate coalition values from contracts, forecasts, market data, effort, ownership percentages, or bargaining power; those values are inputs supplied by the analyst.

How the two-player Shapley allocation is calculated

The Shapley method asks how much each player adds when participants join the coalition in every possible order, then averages those marginal contributions. With two players there are only two orders. If player one arrives first, player one's contribution is the singleton value v({1}); if player one arrives second, the contribution is the grand coalition value v({1,2}) minus player two's singleton value v({2}). Averaging those two contributions gives player one's Shapley value. Player two is calculated symmetrically. An equivalent shortcut starts with each player's singleton value, computes the cooperation surplus as v({1,2}) - v({1}) - v({2}), and assigns half of that surplus to each player. The two returned allocations therefore add exactly to the grand coalition value before display normalization. If cooperation produces a shortfall instead of a surplus, the same rule divides that negative amount equally. This is not an arbitrary fifty-fifty division of total value: unequal standalone values remain intact, while only the incremental gain or loss from cooperation is shared equally.

Interpret the result and decide whether the model fits

Use each Shapley value as a principled allocation benchmark, not as an automatic legal entitlement or a complete negotiation outcome. The method satisfies efficiency because the allocations exhaust the grand coalition value, and it treats players symmetrically when their coalition contributions are identical. It also recognizes standalone capability: a player who can create more value alone generally receives more, even though the cooperation surplus is split equally in a two-player game. A positive surplus means joint production exceeds the sum of separate production; a negative surplus means the stated joint outcome underperforms that sum, so both allocations absorb half of the shortfall. Before adopting the result, check whether value is transferable between players and whether the three coalition values capture all relevant effects. Contracts may require minimum payments, ownership rights, risk adjustments, taxes, outside options, or constraints that the basic game omits. The calculator also covers exactly two players; games with three or more participants require values for additional coalitions and different combinatorial weights. Document the inputs and assumptions beside the allocation so reviewers can distinguish arithmetic fairness from judgment used to value each coalition.