Social Choice_Lecture 8_SSI

# First we need to gure out the winning coalitions

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Unformatted text preview: re out the winning coalitions. There are 4 winning coalitions: AB , AC , BC , ABC (Verify) Van Essen (U of A) Power 6 / 22 Shapley-Shubik Index of Power Second, we need to compute the number of possible arrangements of the three voters. How many orderings of A, B, and C are there? Van Essen (U of A) Power 7 / 22 Shapley-Shubik Index of Power Answer: There are 3! = 6 possible orderings ot the three people. ABC ACB BAC BCA CAB CBA . Van Essen (U of A) Power 8 / 22 Shapley-Shubik Index of Power Third, for each ordering, we need determine which voter is pivotal. Starting with the person of the far left we keep adding people to the coalition until it becomes a winning coalition. We put a dot on the pivotal person who causes the coalition to go from losing to winning. ˙ ABC ˙ AC B ˙ B AC ˙ BC A ˙ C AB ˙ C BA Van Essen (U of A) Power 9 / 22 Shapley-Shubik Index of Power ˙ ABC ˙ AC B ˙ B AC ˙ BC A ˙ C AB ˙ C BA Fourth, we count the number of times each person was pivotal: A is pivotal 2 times, B is pivotal 2 times, C is pivotal 2 times. The Shapley-Shubik Index of power for each players is: SSI (A) = SSI (B ) = SSI (C ) = 2 1 = 6 3 They all have equal power! Van Essen (U of A) Power 10 / 22 Shapley-Shubik Index of Power: European Economic Community Consider the EEC set-up in 1958. Suppose we wanted to calculate the SSI for each member. France, Germany, and Italy had 4 votes. Netherlands and Belgium had 2 votes Luxembourg had 1 vote. Passage required 12 or more votes. Let’ compute the SSI for France. s Van Essen (U of A) Power 11 / 22 Shapley-Shubik Index of Power: European Economic Community The total number of orderings is 6! = 720. It would be very tedious to work out all of the orderings and compute for which orderings France is pivotal. Thus, we need to be smart and use some of our counting techniques. Note: France has 4 votes. Thus, France is pivotal if it is preceded by 8, 9, 10, or 11 votes. Why? So we need to …gure out the di¤erent ways to get 8, 9, 10, and 11 and use our counting techniques to …gure out all of the orderings that each case contains. Van Essen (U of A)...
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