Shapley Value Coalition Game
Your allocation
How this teaches game theory
The Shapley value distributes a coalition's total value according to each player's average marginal contribution across every possible joining order. It formalizes a powerful notion of contribution-based fairness in cooperative games.
Complete scenario, mode, level, and concept guide
What is a coalition game?
In a cooperative game, players may work alone or form coalitions. Every possible coalition has a value that represents how much benefit that group can create together. The central question is not only how much value the grand coalition creates, but also how that value should be divided among the participating players.
This simulator uses a characteristic-function game. A coalition's value is built from each player's individual contribution plus pairwise synergy bonuses. A synergy bonus means two players create more value together than they would create separately. Because of these complementarities, simply splitting the final value equally may ignore who actually generated the value.
Scenarios
Startup collaboration
Teams combine product, sales, infrastructure, and data capabilities. A player with a moderate stand-alone value may become extremely important when paired with another complementary team.
What to observe: strong complementarities can make a player's Shapley value much larger than the player's singleton value.
Shared logistics network
Firms pool warehouses, vehicles, routes, and distribution capacity. The coalition value can be interpreted as the total savings created by cooperation.
What to observe: a firm that unlocks savings for several other firms may deserve a larger share even if its own stand-alone savings are small.
Data-sharing consortium
Organizations combine complementary datasets. One dataset may have limited value by itself but become highly useful when matched with another source.
What to observe: the value of information may come from combinations rather than isolated ownership.
How the Shapley value works
The Shapley value asks a simple counterfactual question: how much extra value does a player add when joining a coalition? This extra amount is the player's marginal contribution. The calculation considers every possible order in which players could join and averages that player's marginal contribution across all of those orders.
Marginal contribution
Suppose a coalition is worth 18 units before Player C joins and 27 units after C joins. Player C's marginal contribution in that situation is 9. The same player may have a different marginal contribution in another coalition because the partners already present may change how useful that player is. The Shapley value averages these different contributions with the correct combinatorial weights.
Why the Shapley value is considered fair
Efficiency
The entire value of the grand coalition is distributed. No value is lost or created during allocation.
Symmetry
If two players contribute identically to every coalition, they receive the same allocation.
Dummy player
A player that never increases coalition value receives only the value attributable to its own direct contribution.
Additivity
If two cooperative games are combined, the Shapley allocation of the combined game equals the sum of the allocations from the two games.
Modes
Human guesses allocation
Propose a complete division before seeing the calculated Shapley values. This mode is useful for testing intuitive ideas of fairness.
Try: equal division first, then compare it with the Shapley result.
Human + marginal hints
The simulator reveals each player's singleton value and average pairwise synergy. These clues help you reason about which players generate more value through cooperation without directly revealing the final answer.
Try: identify players with modest singleton values but unusually strong synergy.
Computer calculates Shapley value
The exact Shapley allocation is displayed automatically. Use this mode as a demonstration or to inspect how the result changes across randomly generated games.
Try: create several games and compare equal allocation with contribution-based allocation.
Levels
Level 1 — 3 players
The smallest setting for understanding the idea manually. With only three players, students can inspect most coalitions and reason through several joining orders by hand.
Level 2 — 4 players
More coalitions and more possible joining orders make intuitive allocation harder. This level shows why systematic cooperative-game methods are useful.
Level 3 — 5 players
The coalition space becomes much larger. This level emphasizes the computational cost of exact Shapley-value calculation and motivates approximation methods used in larger systems.
How to read the results
Coalition value is the value created when all players cooperate. Allocation error is the sum of the absolute differences between your proposed allocation and the exact Shapley allocation; smaller values mean your guess is closer. Efficiency compares the total amount you allocated with the coalition's total value. A 100% efficiency score means the full coalition value has been distributed, but this does not by itself mean the division is fair.
Equal division vs contribution-based fairness
An equal split gives every participant the same amount. The Shapley value instead asks how much value each participant makes possible. These two ideas coincide only when players are sufficiently symmetric. In games with strong complementarities, equal division can substantially under-reward pivotal players and over-reward players whose presence adds little.
Suggested classroom experiment
- Start with Level 1 and Human guesses allocation.
- Divide the grand-coalition value equally and evaluate the result.
- Create a new game and switch to Human + marginal hints.
- Use singleton values and synergy hints to make a second allocation.
- Reveal the Shapley values and compare your reasoning with the exact result.
- Move to Levels 2 and 3 and discuss why manual calculation becomes increasingly difficult.
Questions to ask while experimenting
- Does the player with the largest singleton value always receive the largest Shapley value?
- Can a player with a small stand-alone contribution become highly valuable because of synergy?
- When does equal division happen to match the Shapley allocation?
- Can two players with similar singleton values receive different allocations because they interact differently with others?
- Why does exact computation become difficult as the number of players increases?
Model note: this teaching simulator uses additive singleton values plus pairwise synergy terms to keep coalition values understandable. General cooperative games can assign an arbitrary value to every coalition and may contain more complex interactions involving three or more players at once.