Description
Return withValue minus withoutValue and weight 1/[players × binomial(players-1, coalitionSize)], then their product.
Cooperative-game calculations compare coalition values, allocated payoffs or voting weights. A single coalition term or party share can be useful in a larger calculation without establishing a complete power index, core allocation or bargaining solution.
Return withValue minus withoutValue and weight 1/[players × binomial(players-1, coalitionSize)], then their product.
When to use this tool
Use Shapley One Coalition Marginal Term when you need one explicit coalition, voting or bargaining term.
How to use the tool and how it works
- Enter players, coalitionSize, withoutValue, withValue. Use the units and conventions below.
- Run the tool. Correct any format, range, or cross-field input error before interpreting the output.
- Read marginalContribution, permutationWeight, weightedContribution. Use the output convention below.
Require players 1-100 and coalitionSize 0 through players-1. Return withValue minus withoutValue and weight 1/[players × binomial(players-1, coalitionSize)], then their product.1
| Input field | Meaning and units | Accepted input |
|---|---|---|
| players | Players | Required. Safe integer; minimum 1; maximum 100. |
| coalitionSize | Coalition Size | Required. Safe integer; minimum 0. |
| withoutValue | Without Value | Required. Finite number. |
| withValue | With Value | Required. Finite number. |
| Output field | Interpretation | Returned shape |
|---|---|---|
| marginalContribution | Marginal Contribution | Required. Finite number. |
| permutationWeight | 1/[n·binomial(n-1,k)] for this one subset, not an aggregate over all subsets of size k. | Required. Finite number. |
| weightedContribution | Weighted Contribution | Required. Finite number. |
{
"input": {
"players": 4,
"coalitionSize": 1,
"withoutValue": 10,
"withValue": 22
},
"output": {
"marginalContribution": 12,
"permutationWeight": 0.08333333333333333,
"weightedContribution": 1
}
}Limitations and assumptions
- This is one coalition's Shapley contribution, not the sum over coalitions or a complete allocation. Values must represent the same coalition with and without the focal player.
- Inputs must satisfy both the field types and the cross-field conditions above. Invalid input causes an error; a successful false flag or documented placeholder is a different result.
- Numeric calculations use binary floating point unless exact decimal-integer arithmetic is explicitly stated. No hidden measurement tolerance or decimal rounding is added; nonfinite or out-of-contract results are rejected. Numeric signed-zero results are exported as zero.
Alternative or Complementary methods
Evaluate all required coalitions and allocation constraints for the chosen solution concept. Keep payoff units, quota conventions, feasibility and normalization explicit.
References
Similar or alternative tools
- Core Coalition Excess One Subset
Sum it, subtract the sum from coalitionValue for excess, and flag blocked for strictly positive excess.
- Banzhaf One Swing Indicator Term
Compare coalition weight before and after adding the player, and flag swing only for a losing-to-winning change.