Why Average Contributions Across Entry Orders?
A business unit’s marginal capital depends on which other units are already in the portfolio. Adding the same unit to a small portfolio may require a different increment of capital than adding it to a larger one. Shapley allocation addresses this order dependence by calculating the unit’s incremental capital in every possible entry order and averaging those increments.
This method is useful when coalition capital values are available but a single marginal ordering would be arbitrary. The calculations below are for small portfolios, where the orders and resulting weights can be shown directly. All capital amounts in the examples are hypothetical.
The coalition values must be comparable: they must use a consistent capital measure, confidence level where applicable, valuation date, and set of assumptions. This tutorial takes those values as given; it does not assume a particular loss distribution or specify how the underlying capital measure is calculated. If the coalition values are produced from different bases, the resulting averages do not have a consistent interpretation.
Deriving the Shapley Weights
For a particular ordering of all units, the incremental capital assigned to a unit is the capital after it enters minus the capital of the coalition immediately before it enters. Shapley allocation averages that increment across every ordering. Equivalently, it weights each possible predecessor coalition according to how often that coalition occurs before the unit.
For unit \(i\), take any set \(S\subseteq N\setminus\{i\}\) as the units that enter before \(i\).
The units in \(S\) can be arranged in \(|S|!\) ways before \(i\), and the other \(n-|S|-1\) units can be arranged in \((n-|S|-1)!\) ways after \(i\).
There are \(n!\) total orders. Thus \(S\) is the predecessor coalition with weight \(\frac{|S|!(n-|S|-1)!}{n!}\).
For that coalition, the increment is \(c(S\cup\{i\})-c(S)\). Multiply by its order weight and sum over every possible \(S\).
The counting argument explains why the weights are not generally equal across predecessor coalitions. A particular coalition size can occur before \(i\) in different numbers of orders. For two units, either unit is equally likely to enter first. For three units, a given unit has probability \(1/3\) of entering first, probability \(1/3\) of entering last, and probability \(1/6\) for each of the two possible singleton predecessor coalitions.
Two-Unit Formula
For \(N=\{A,B\}\), there are two orders. Unit \(A\) contributes \(c(A)-c(\varnothing)\) if it enters first and \(c(AB)-c(B)\) if it enters second. Since \(c(\varnothing)=0\), its contribution is the average of those two increments. The same reasoning applies to \(B\).
Worked Example: Allocating Capital Between Two Units
A hypothetical insurer has two units, A and B. The supplied coalition capital values, in thousands of dollars, are \(c(A)=80\), \(c(B)=50\), and \(c(AB)=110\), with \(c(\varnothing)=0\). Calculate the Shapley contributions and check their sum.
Method. Average each unit’s incremental capital over the two possible entry orders. No loss distribution is required for this calculation because all coalition capital values are given in the problem.
If A enters first, its increment is \(80-0=80\). If A enters after B, its increment is \(110-50=60\). Therefore,
If B enters first, its increment is \(50-0=50\). If B enters after A, its increment is \(110-80=30\). Therefore,
The contributions sum to \(70+40=110\), equal to \(c(AB)\). The arithmetic is consistent with both entry orders: the marginal increments are \(80+30=110\) for order A then B, and \(50+60=110\) for order B then A.
Common mistake avoided: Assigning A only its increment when it enters last would give \(110-50=60\), not its Shapley contribution. That uses one order and discards the equally weighted case in which A enters first. The correction is to average both increments.
Three-Unit Formula and Checks
For \(N=\{A,B,C\}\), each unit can enter first, second, or last. The weights for an individual unit’s predecessor coalitions are \(1/3\) for the empty coalition, \(1/6\) for each of the two singleton coalitions, and \(1/3\) for the pair of other units. Applying these weights gives:
The formulas also provide a direct efficiency check for these examples: the contributions should sum to the capital of the full coalition. To see why, fix any entry order. Its successive increments telescope: the first increment starts from \(c(\varnothing)=0\), each later subtraction cancels the preceding coalition’s capital, and the final addition is \(c(ABC)\). Thus the increments sum to \(c(ABC)\) for every order. Averaging the orders preserves that sum.
Symmetry supplies another check. If two units have identical marginal increments whenever they face corresponding predecessor coalitions, swapping their labels leaves the Shapley calculation unchanged. Their contributions must then be equal. This follows from the same weights being applied to matching increments, not from an assumption that standalone capital alone determines allocation.
Worked Example: Checking Symmetry in a Three-Unit Portfolio
A hypothetical portfolio has capital values, in thousands of dollars, \(c(A)=40\), \(c(B)=40\), \(c(C)=30\), \(c(AB)=65\), \(c(AC)=60\), \(c(BC)=60\), and \(c(ABC)=80\). Calculate all Shapley contributions. Notice that A and B have matching coalition values under an exchange of their labels.
Method. Apply the three-unit formula to each unit. The matching values mean A and B face the same weighted increments, so their results should agree.
For A, the four weighted terms are:
For B, the corresponding terms are:
For C:
The unrounded total is \(\frac{175}{6}+\frac{175}{6}+\frac{65}{3}=80\) thousand dollars. In dollars, rounding only at the final step to the nearest cent gives A = $29,166.67, B = $29,166.67, and C = $21,666.67. Their displayed amounts sum to $80,000.01 because each recurring decimal was rounded separately; the exact unrounded contributions sum to $80,000. The equal contributions for A and B confirm symmetry.
Common mistake avoided: Equal standalone capital does not, by itself, prove equal Shapley contributions. Here the equality is justified because the pair and grand-coalition values also match when A and B are exchanged.
Worked Example: Averaging All Six Orders
A hypothetical portfolio has coalition capital values, in thousands of dollars, \(c(A)=60\), \(c(B)=75\), \(c(C)=45\), \(c(AB)=105\), \(c(AC)=85\), \(c(BC)=100\), and \(c(ABC)=130\). Calculate the contribution of each unit by listing the incremental contributions in all six entry orders.
Method. For each order, subtract the capital of the coalition already present from the capital after the next unit enters. Then average each unit’s six increments.
| Entry order | Increment for first unit | Increment for second unit | Increment for last unit |
|---|---|---|---|
| A, B, C | A: 60 | B: 105 − 60 = 45 | C: 130 − 105 = 25 |
| A, C, B | A: 60 | C: 85 − 60 = 25 | B: 130 − 85 = 45 |
| B, A, C | B: 75 | A: 105 − 75 = 30 | C: 130 − 105 = 25 |
| B, C, A | B: 75 | C: 100 − 75 = 25 | A: 130 − 100 = 30 |
| C, A, B | C: 45 | A: 85 − 45 = 40 | B: 130 − 85 = 45 |
| C, B, A | C: 45 | B: 100 − 45 = 55 | A: 130 − 100 = 30 |
A’s six increments total \(60+60+30+30+40+30=250\), so \(\phi_A=250/6=125/3=41.6667\). B’s total is \(45+45+75+75+45+55=340\), so \(\phi_B=340/6=170/3=56.6667\). C’s total is \(25+25+25+25+45+45=190\), so \(\phi_C=190/6=95/3=31.6667\).
The exact contributions sum to \((125+170+95)/3=390/3=130\) thousand dollars, matching the grand coalition. Rounded only at the final step to the nearest cent, the amounts are A = $41,666.67, B = $56,666.67, and C = $31,666.67. Their rounded total is $130,000.01; the exact values sum to $130,000. This calculation also shows why a single chosen entry order is insufficient: the marginal increment for A, for example, ranges from 30 to 60 thousand dollars depending on the order.
Common mistake avoided: The six orders must be given equal weight, not one weight per coalition size or one weight per unit. Listing orders makes the averaging rule explicit and provides a check against omitting an order.
Exam Checks and Interpretation
For small portfolios, it is practical to use either the explicit formula or the order-by-order method. The formula is shorter; the order table is useful for checking which increment belongs to which predecessor coalition. In either method, keep the coalition values and the contribution units consistent, retain exact fractions or unrounded values during the calculation, and round only at the end.
- Check the empty coalition. Use \(c(\varnothing)=0\) under the convention used here; do not treat an unlisted standalone value as the empty coalition.
- Check each increment. A marginal increment is \(c(S\cup\{i\})-c(S)\), not the capital of the enlarged coalition by itself.
- Check the weights. For two units, the two orders each have weight \(1/2\). For three units, the first- and last-entry cases each have weight \(1/3\), while each particular singleton predecessor coalition has weight \(1/6\).
- Check the total and symmetry. Contributions should sum to \(c(N)\), and units with matching marginal contributions under exchange should receive equal allocations.
These checks can catch arithmetic or interpretation errors, but they do not replace the requirement that coalition capital values be calculated on a consistent basis. The Shapley method allocates the supplied coalition values; it does not determine whether the underlying capital model is appropriate.
Check Your Understanding
Use \(c(\varnothing)=0\) in each problem. Capital values are hypothetical and stated in thousands of dollars unless noted otherwise.
- A two-unit portfolio has \(c(A)=48\), \(c(B)=32\), and \(c(AB)=64\). Calculate both Shapley contributions and check the total. (Answer: \(\phi_A=40\), \(\phi_B=24\); total \(=64\).)
- A three-unit portfolio has \(c(A)=50\), \(c(B)=40\), \(c(C)=30\), \(c(AB)=75\), \(c(AC)=65\), \(c(BC)=60\), and \(c(ABC)=90\). Calculate all three contributions using the weighted formula. (Answer: \(\phi_A=115/3\), \(\phi_B=185/6\), \(\phi_C=125/6\); total \(=90\).)
- For three units, let \(c(A)=36\), \(c(B)=36\), \(c(C)=24\), \(c(AB)=60\), \(c(AC)=50\), \(c(BC)=50\), and \(c(ABC)=72\). Calculate the contributions and state what symmetry check applies. (Answer: \(\phi_A=\phi_B=83/3\), \(\phi_C=50/3\); A and B are symmetric and receive equal contributions, and the total is 72.)
- A portfolio has \(c(A)=45\), \(c(B)=55\), \(c(C)=35\), \(c(AB)=85\), \(c(AC)=70\), \(c(BC)=80\), and \(c(ABC)=100\). Calculate each Shapley contribution and verify efficiency. (Answer: \(\phi_A=32.5\), \(\phi_B=42.5\), \(\phi_C=25\); total \(=100\).)
- In a three-unit portfolio, suppose only \(c(ABC)\) increases by 6 while every other coalition value stays fixed. By how much does each unit’s Shapley contribution change, and how does the total change? (Answer: each contribution increases by 2, because the grand-coalition increment has weight \(1/3\) for each unit; the total increases by 6.)