We start by listing all winning coalitions. If you aren’t sure how to do this, you can list all coalitions, then eliminate the non-winning coalitions. No player is a dictator, so we’ll only consider two and three player coalitions.

[latex]\{P_1,P_2\}[/latex]: Total weight: [latex]9[/latex]. Meets quota. [latex]\{P_1,P_3\}[/latex]: Total weight: [latex]8[/latex]. Meets quota. [latex]\{P_2,P_3\}[/latex]: Total weight: [latex]5[/latex]. Does not meet quota. [latex]\{P_1,P_2,P_3\}[/latex]: Total weight: [latex]11[/latex]. Meets quota.

Next we determine which players are critical in each winning coalition. In the winning two-player coalitions, both players are critical since no player can meet quota alone. Underlining the critical players to make it easier to count:

[latex]\left\{\underline{P}_{1}, \underline{P}_{2}\right\}[/latex] [latex]\left\{\underline{P}_{1}, \underline{P}_{3}\right\}[/latex]

In the three-person coalition, either [latex]P_2[/latex] or [latex]P_3[/latex] could leave the coalition and the remaining players could still meet quota, so neither is critical. If [latex]P_1[/latex] were to leave, the remaining players could not reach quota, so [latex]P_1[/latex] is critical.

[latex]\left\{\underline{P}_{1}, P_{2}, P_{3}\right\}[/latex]

Altogether, [latex]P_1[/latex] is critical [latex]3[/latex] times, [latex]P_2[/latex] is critical [latex]1[/latex] time, and [latex]P_3[/latex] is critical [latex]1[/latex] time.

Converting to percents:

[latex]P_{1}=3 / 5=60 \%[/latex] [latex]P_{2}=1 / 5=20 \%[/latex] [latex]P_{3}=1 / 5=20 \%[/latex]

The voting system tells us that the quota is [latex]36[/latex], that Player 1 has [latex]20[/latex] votes (or equivalently, has a weight of [latex]20[/latex]), Player 2 has [latex]17[/latex] votes, Player 3 has [latex]16[/latex] votes, and Player 4 has [latex]3[/latex] votes.

A coalition is any group of one or more players. What we’re looking for is winning coalitions – coalitions whose combined votes (weights) add to up to the quota or more. So the coalition [latex]\{P3,P4\}[/latex] is not a winning coalition because the combined weight is [latex]16+3=19[/latex], which is below the quota.

So we look at each possible combination of players and identify the winning ones:

[latex]\begin{array} {ll} {\{\mathrm{P}_1, \mathrm{P}_2\}(\text { weight }: 37)} & {\{\mathrm{P}_1, \mathrm{P}_3\} \text { (weight: } 36)} \\ {\{\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3\} \text { (weight: } 53)} & {\{\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_4\} \text { (weight: } 40)} \\ {\{\mathrm{P}_1, \mathrm{P}_3, \mathrm{P}_4\} \text { (weight: } 39)} & {\{\mathrm{P}_1, \mathrm{P}_2, \mathrm{P}_3, \mathrm{P}_4\} \text { (weight: } 56)} \\ {\{\mathrm{P}_2, \mathrm{P}_3, \mathrm{P}_4\}(\text { weight: } 36)} \end{array}[/latex]

From the list of winning coalitions, we can see that each player is part of certain winning coalitions. The Banzhaf power index measures the ability of each player to change a losing coalition into a winning one by joining it. In other words, it counts the number of times a player is a “critical” player in a winning coalition.

To find the Banzhaf power index, we need to identify the critical players in each winning coalition:

[latex]\begin{array}{ll} {P_1, P_2} & \text{Critical:} P_1, P_2 \\ {P_1, P_3} & \text{Critical:} P_1, P_3 \\ {P_1, P_2, P_3} & \text{Critical: None} \\ {P_1, P_2, P_4} & \text{Critical:} P_4 \\ {P_1, P_3, P_4} & \text{Critical:} P_4 \\ {P_1, P_2, P_3, P_4} & \text{Critical: None} \\ {P_2, P_3, P_4} & \text{Critical:} P_2, P_3, P_4 \end{array}[/latex]

Now, we count the number of times each player is critical:

[latex]P_1[/latex] is critical [latex]2[/latex] times [latex]P_2[/latex] is critical [latex]3[/latex] times [latex]P_3[/latex] is critical [latex]3[/latex] times [latex]P_4[/latex] is critical [latex]3[/latex] times

The Banzhaf power index is then calculated by taking the number of times each player is critical and dividing it by the total number of critical positions:

[latex]\text{Total Critical Positions} = 2 + 3 + 3 + 3 = 11[/latex] [latex]\text{Banzhaf Power Index for} P_1 = \frac{2}{11}[/latex] [latex]\text{Banzhaf Power Index for} P_2 = \frac{3}{11}[/latex] [latex]\text{Banzhaf Power Index for} P_3 = \frac{3}{11}[/latex] [latex]\text{Banzhaf Power Index for} P_4 = \frac{3}{11}[/latex]

Listing all sequential coalitions and identifying the pivotal player:

[latex]\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}[/latex]

[latex]P_1[/latex] is pivotal [latex]3[/latex] times, [latex]P_2[/latex] is pivotal [latex]3[/latex] times, and [latex]P_3[/latex] is pivotal [latex]0[/latex] times.

[latex]\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}[/latex]