The Shapley-Shubik power index<\/strong> was introduced in 1954 by economists Lloyd Shapley and Martin Shubik, and provides a different approach for calculating power.<\/p>\r\n 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?<\/p>\r\n First, we need to change our approach to coalitions. Previously, the coalition [latex]\\{P_1,P_2\\}[\/latex] and [latex]\\{P_2,P_1\\}[\/latex] 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<\/strong> \u2013 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 [latex]< P_2,P_1,P_3>[\/latex] would mean that [latex]P_2[\/latex] joined the coalition first, then [latex]P_1[\/latex], and finally [latex]P_3[\/latex]. The angle brackets [latex]< >[\/latex] are used instead of curly brackets to distinguish sequential coalitions.<\/p>\r\n A sequential coalition<\/strong> lists the players in the order in which they joined the coalition.<\/p>\r\n <\/p>\r\n A pivotal player<\/strong> 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.<\/p>\r\n<\/div>\r\n<\/section>\r\n The sequential coalition shows the order in which players joined the coalition. Consider the running totals as each player joins:<\/p>\r\n [latex]\\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}[\/latex]<\/p>\r\n Since the coalition becomes winning when [latex]P_4[\/latex] joins, [latex]P_4[\/latex] is the pivotal player in this coalition.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n How To: Calculating Shapley-Shubik Power Index<\/strong><\/p>\r\n To calculate the Shapley-Shubik Power Index:<\/p>\r\n How many sequential coalitions should we expect to have? If there are [latex]N[\/latex] players in the voting system, then there are [latex]N[\/latex] possibilities for the first player in the coalition, [latex]N\u20131[\/latex] possibilities for the second player in the coalition, and so on. Combining these possibilities, the total number of coalitions would be: [latex]N(N\u22121)(N\u22122)(N\u22123)\u22ef(3)(2)(1)[\/latex]. This calculation is called a factorial<\/strong>, and is notated [latex]N![\/latex] . The number of sequential coalitions with [latex]N[\/latex] players is [latex]N![\/latex] .<\/p>\r\n Factorial<\/strong><\/p>\r\n\r\nCalculating the factorial <\/strong>is a way to calculate the product of all positive whole numbers up to a given number. Notation:<\/strong> A factorial is represented by an exclamation mark [latex](!)[\/latex] following a number.<\/section>\r\n There will be [latex]7![\/latex] sequential coalitions. [latex]7!=7\u22c56\u22c55\u22c54\u22c53\u22c52\u22c51=5040[\/latex]<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n As you can see, computing the Shapley-Shubik power index by hand would be very difficult for voting systems that are not very small.<\/p>\r\n [latex]\\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} [\/latex]<\/p>\r\n\r\n[reveal-answer q=\"4333\"]Show Solution[\/reveal-answer] [hidden-answer a=\"4333\"]\r\n\r\n [latex]P_1[\/latex] is pivotal [latex]4[\/latex] times, [latex]P_2[\/latex] is pivotal [latex]1[\/latex] time, and [latex]P_3[\/latex] is pivotal [latex]1[\/latex] time.<\/p>\r\n [latex]\\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} [\/latex]<\/p>\r\n For comparison, the Banzhaf power index for the same weighted voting system would be [latex]P_1:60\\%,P_2:20\\%,P_3:20\\%[\/latex]. While the Banzhaf power index and Shapley-Shubik power index are usually not terribly different, the two different approaches usually produce somewhat different results.<\/p>\r\n\r\n[\/hidden-answer]<\/section>\r\n The Shapley-Shubik power index<\/strong> was introduced in 1954 by economists Lloyd Shapley and Martin Shubik, and provides a different approach for calculating power.<\/p>\n 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?<\/p>\n First, we need to change our approach to coalitions. Previously, the coalition [latex]\\{P_1,P_2\\}[\/latex] and [latex]\\{P_2,P_1\\}[\/latex] 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<\/strong> \u2013 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 [latex]< P_2,P_1,P_3>[\/latex] would mean that [latex]P_2[\/latex] joined the coalition first, then [latex]P_1[\/latex], and finally [latex]P_3[\/latex]. The angle brackets [latex]< >[\/latex] are used instead of curly brackets to distinguish sequential coalitions.<\/p>\n A sequential coalition<\/strong> lists the players in the order in which they joined the coalition.<\/p>\n <\/p>\n A pivotal player<\/strong> 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.<\/p>\n<\/div>\n<\/section>\npivotal player<\/h3>\r\n
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Calculating Power: Shapley-Shubik Power Index<\/h2>\n
pivotal player<\/h3>\n