Shapley value vs voting power: sharing credit is different from being decisive

Choose what you want the number to represent

An allocation of value needs a model of what coalitions can create. A voting-power measure needs a rule for which coalitions can pass a decision. A payment mechanism needs preferences, an outcome rule and incentives for reporting information. Substituting one kind of number for another can give a precise-looking answer to the wrong question.

For a real collaboration, begin by describing the decision the calculation is meant to support. Dividing revenue, recognising effort, assigning responsibility and predicting bargaining influence are distinct tasks. Even a mathematically well-defined allocation does not remove the need to justify its inputs or explain whose values the rule is intended to represent.

The Shapley value averages marginal contributions

Shapley's original value for n-person games provides an allocation rule for a specified cooperative game. The Shapley-value experiment uses three participants. Each creates zero alone; AB creates sixty, AC twenty, BC forty and the full group ninety. The table is authored so the calculation can be inspected without uncertainty about the inputs.

A participant's contribution depends on which others are already present. Joining an empty group can add zero, while joining a pair can add substantial value. Averaging over every possible order avoids privileging one arbitrary sequence of arrival. It does not prove that the coalition estimates reflect actual effort, or that averaging marginal contributions is the only defensible fairness principle.

Worked example: two joining orders and their average

Keep the coalition values fixed while switching order. This separates a change in marginal contribution from a change in the collaboration itself. The six permutations are equally weighted by the Shapley rule, not sampled according to a story about who actually arrived first.

1. Inspect A→B→C. A adds zero, B increases value from zero to sixty, and C increases it from sixty to ninety, receiving thirty marginal units.
2. Inspect B→C→A. B adds zero, C creates forty with B, and A increases the pair's forty to ninety, adding fifty.
3. Average all six orders. The final allocation is A=30, B=40 and C=20, summing to the full ninety. Explain the coalition table before treating these numbers as a real compensation proposal.

Vote weight is different from pivotal opportunity

Banzhaf's voting-power account asks when adding a voter changes a coalition from losing to winning. In the Banzhaf experiment, weights are four, three and two with quota six. The four-vote member has three critical joins; the others have one each. Normalising gives sixty, twenty and twenty percent.

Those values differ from the shares of vote weight. Change the quota and the critical-coalition structure can change while every weight stays fixed. This index treats possible coalitions equally; real alliances or agenda control can make some coalitions much more likely than others. It is therefore a defined structural measure, not a complete forecast of political influence.

An incentive-compatible payment is another object again

The VCG public-project game asks how reported benefits determine construction and pivot payments. At benefits forty-five, thirty-five and ten, a sixty-cost project is built. Payments of fifteen, five and zero do not cover the cost. Positive reported surplus and a truthfulness property do not automatically imply budget balance.

The game assumes an outside budget covers the forty-unit gap. That assumption should remain visible whenever the mechanism is discussed. A payment rule can be useful for eliciting information while being unsuitable as a self-financing institution. Ask separately about efficiency, incentives, funding, distribution and the reliability of the reported valuations rather than bundling them into the word fair.

A model audit before using a fairness number

Write the assumptions next to the result, especially when the inputs are difficult to estimate. A clean arithmetic calculation can conceal a contestable valuation table. Sensitivity to plausible alternative inputs may be more informative than adding decimal places to a single allocation.

• State whether the question concerns value, power, payments or responsibility.
• Explain how coalition values or voting weights were obtained.
• Specify the quota and treatment of possible coalitions.
• Check whether assigned values sum to the intended total.
• Identify any outside funding required by the rule.
• Separate mathematical properties from a claim of moral fairness.

Precision does not remove the choice of a principle

A real team may create non-transferable benefits, disagree about counterfactual coalition output or face contractual rights that take priority over a modelled division. A board may have stable alliances that violate equal-coalition assumptions. These are not minor details to add after presenting the number; they help determine which model, if any, fits the decision.

Use the experiments to sharpen a discussion rather than end it prematurely. Ask what would change the allocation, which member is pivotal under which quota, and who bears a funding shortfall. A useful quantitative rule makes those tradeoffs explicit. Its strength comes from answering a clearly specified question, not from transforming every difficult question about cooperation into the same formula.