# 3.5: Calculating Power- Shapley-Shubik Power Index

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The **Shapley-Shubik** power index was introduced in 1954 by economists Lloyd Shapley and Martin Shubik, and provides a different approach for calculating power.

In situations like political alliances, the order in which players join an alliance could be considered the most important consideration. In particular, if a proposal is introduced, the player that joins the coalition and allows it to reach quota might be considered the most essential. The Shapley-Shubik power index counts how likely a player is to be **pivotal**. What does it mean for a player to be pivotal?

First, we need to change our approach to coalitions. Previously, the coalition \(\left\{P_{1}, P_{2}\right\}\) and \(\left\{P_{2}, P_{1}\right\}\) would be considered equivalent, since they contain the same players. We now need to consider the *order* in which players join the coalition. For that, we will consider **sequential coalitions** – coalitions that contain all the players in which the order players are listed reflect the order they joined the coalition. For example, the sequential coalition

\(<P_{2}, P_{1}, P_{3}>\) would mean that \(P_2\) joined the coalition first, then \(P_1\), and finally \(P_3\). The angle brackets < > are used instead of curly brackets to distinguish sequential coalitions.

A sequential coalition lists the players in the order in which they joined the coalition.

A **pivotal player** is the player in a sequential coalition that changes a coalition from a losing coalition to a winning one. Notice there can only be one pivotal player in any sequential coalition.

In the weighted voting system \([8: 6, 4, 3, 2]\), which player is pivotal in the sequential coalition \(<P_{3}, P_{2}, P_{4}, P_{1}>\)?

**Solution**

The sequential coalition shows the order in which players joined the coalition. Consider the running totals as each player joins:

\(\begin{array}{lll}P_{3} & \text { Total weight: } 3 & \text { Not winning } \\ P_{3}, P_{2} & \text { Total weight: } 3+4=7 & \text { Not winning } \\ P_{3}, P_{2}, P_{4} & \text { Total weight: } 3+4+2=9 & \text { Winning } \\ R_{2}, P_{3}, P_{4}, P_{1} & \text { Total weight: } 3+4+2+6=15 & \text { Winning }\end{array}\)

Since the coalition *becomes* winning when \(P_4\) joins, \(P_4\) is the pivotal player in this coalition.

To calculate the Shapley-Shubik Power Index:

- List all sequential coalitions
- In each sequential coalition, determine the pivotal player
- Count up how many times each player is pivotal
- Convert these counts to fractions or decimals by dividing by the total number of sequential coalitions

How many sequential coalitions should we expect to have? If there are *N* players in the voting system, then there are \(N\) possibilities for the first player in the coalition, \(N – 1\) possibilities for the second player in the coalition, and so on. Combining these possibilities, the total number of coalitions would be:\(N(N-1)(N-2)(N-3) \cdots(3)(2)(1)\). This calculation is called a **factorial**, and is notated \(N!\) The number of sequential coalitions with \(N\) players is \(N!\)

How many sequential coalitions will there be in a voting system with 7 players?

**Solution**

There will be \(7!\) sequential coalitions. \(7 !=7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5040\)

As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small.

Consider the weighted voting system \([6: 4, 3, 2]\). We will list all the sequential coalitions and identify the pivotal player. We will have 3! = 6 sequential coalitions. The coalitions are listed, and the pivotal player is underlined.

\(\begin{aligned}

&<P_{1}, \underline{P}_{2}, P_{3}>\quad<P_{1}, \underline{P}_{3}, P_{2}>\quad<P_{2}, \underline{P}_{1}, P_{3}>\\

&<P_{2}, P_{3}, \underline{P}_{1}>\quad<P_{3}, P_{2}, \underline{P}_{1}>\quad<P_{3}, \underline{P}_{1}, P_{2}>

\end{aligned}\)

**Solution**

\(\mathrm{P}_{1}\) is pivotal 4 times, \(\mathrm{P}_{2}\) is pivotal 1 time, and \(\mathrm{P}_{3}\) is pivotal 1 time.

\(\begin{array}{|l|l|l|}

\hline \textbf { Player } & \textbf { Times pivotal } & \textbf { Power index } \\

\hline P_{1} & 4 & 4 / 6=66.7 \% \\

\hline P_{2} & 1 & 1 / 6=16.7 \% \\

\hline P_{3} & 1 & 1 / 6=16.7 \% \\

\hline

\end{array}\)

For comparison, the Banzhaf power index for the same weighted voting system would be \(\mathrm{P}_{1}: 60 \%, \mathrm{P}_{2}: 20 \%, \mathrm{P}_{3}: 20 \%\). While the Banzhaf power index and Shapley-Shubik power index are usually not terribly different, the two different approaches usually produce somewhat different results.

Find the Shapley-Shubik power index for the weighted voting system \(\bf{[36: 20, 17, 15]}\).

**Answer**-
Listing all sequential coalitions and identifying the pivotal player:

\(\begin{array} {lll} {<P_{1}, \underline{P}_{2}, P_{3}>} & {<P_{1}, P_{3}, \underline{P}_{2}>} & {<P_{2}, \underline{P}_{1}, P_{3}>} \\ {<P_{2}, P_{3}, \underline{P}_{1}>} & {<P_{3}, P_{2}, \underline{P}_{1}>} & {<P_{3}, P_{1}, \underline{P}_{2}>} \end{array}\)

\(\mathrm{P}_{1}\) is pivotal 3 times, \(\mathrm{P}_{2}\) is pivotal 3 times, and \(\mathrm{P}_{3}\) is pivotal 0 times.

\(\begin{array}{|l|l|l|}

\hline \textbf { Player } & \textbf { Times pivotal } & \textbf { Power index } \\

\hline P_{1} & 3 & 3 / 6=50 \% \\

\hline P_{2} & 3 & 3 / 6=50 \% \\

\hline P_{3} & 0 & 0 / 6=0 \% \\

\hline

\end{array}\)