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Experiment 17 Coalitions and Shared Value

Who Deserves the Credit?

Compare equal shares with average contributions, then see whether a subgroup could gain by leaving.

5–10 minutes to explore Prototype Updated

Three colored wooden supports hold up one arch, with a playing piece and a bowl of rewards for each contributor.
About these models Math step by step For experts Math symbol guide

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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.

1. Predict, Then Choose a Sharing Rule

How Should a Team Divide Its Credit?

You are A on a three-person project with B and C. A coalition is any group of teammates who work together, including one person working alone. Each group can create the number of points shown below. Points can be divided among members.

Equal shares divide the full project’s points into three identical amounts. The Shapley value pays each person their average added contribution across all six possible joining orders. “Added contribution” means how many more points the team can create when that person joins.

2. Reveal the Incentives

Check the Shares and the Alternatives

Both rules use the same team values; the last column is your chosen rule.
MemberEqual shareShapley shareChosen share

Could a Subgroup Gain by Leaving?

    All Six Joining Orders

      We compute every order exactly. This is not a random estimate. Displayed shares round to two decimal places; the checks use the unrounded numbers.

      3. Change One Assumption

      Change What Each Team Can Create

      These are created values, before anyone divides them. The empty team creates zero. A group may be worth less than one of its smaller groups; that is allowed here and can produce a negative added contribution or share. A negative share means the member would owe points.

      Try an additive project: A alone 2, B alone 3, C alone 1, A and B 5, A and C 3, B and C 4, everyone 6. Here each person adds the same amount in every joining order.

      Work It Out, Step by Step

      1. With the starting values, consider the joining order A, then B, then C.
      2. A arrives at an empty team: A alone creates zero, so A adds zero points.
      3. A and B together can make 12. Before B arrived the team could make zero, so B adds 12.
      4. The full team still makes 12. C adds zero in this order.
      5. Repeat for the other five orders, then average each person’s six added amounts.
      \[\text{B adds }12-0=12.\]

      In words: subtract the team’s value before B joined from its value afterward. An average here means add the six amounts and divide by six.

      The starting Shapley shares are A 5, B 5, C 2. A and B get 10 together but could make 12 alone. They have 2 extra points to divide if they leave. This shows why a contribution rule and a rule that keeps every subgroup satisfied can disagree.

      For experts: formal model and assumptions

      Contribution and the Core

      A transferable-value game means a team’s value can be split freely among its members. The core is the set of allocations that distribute the whole team’s value and give every subgroup at least its own value. An allocation is one proposed list of shares. Checking one allocation outside the core does not prove the core is empty.

      \(\phi_A\)In words: A’s Shapley share.
      The Greek letter phi is just the name for this share; the small A says whose share. Each \(c_1\) through \(c_6\) below is A’s added contribution in one joining order. The small numbers label the orders.
      \(v(\{A,B\})\)In words: the value A and B can make together.
      Curly braces list the members of a group, called a set. The letter v names the value rule. Parentheses hold the group supplied to that rule. The share \(x_A\) is how much A is given; \(x_B\) is B’s share.
      \[\phi_A=\frac{c_1+c_2+c_3+c_4+c_5+c_6}{6}.\]
      \[x_A+x_B\geq v(\{A,B\}).\]

      In words: average A’s contribution in all six orders. Then, to test the subgroup A and B, check that their shares together are greater than or equal to what they could make alone. The bar in a fraction means division. The symbol \(\geq\) means “greater than or equal to.” The app checks all three individuals, all three pairs, and the full-team total.

      For more examples of the notation, use the math reading guide.

      4. Transfer the Lesson

      Credit for a Shared Tool

      A designer, programmer, and trainer build a useful tool. The designer and programmer could release it without training. Should the trainer’s share depend on total project value, joining order, or what smaller groups could accomplish?

      Compare your reasoning

      Shapley contributions average over joining orders so one chosen order does not decide the result. A core check asks a different question: can a subgroup make more by leaving? Neither calculation decides which real-world values or fairness goals you should use.

      Concept reference: TU Delft’s readings on the core and Shapley value.

      A Model Is a Place to Start

      These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.

      Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.