Example 19

EXAMPLE 19 Minimal Winning Coalitions in the Four-Shareholder Corporation

Example 6 (page 470) was about the Shapley-Shubik index of the corporation with four shareholders, $A, B, C, D$ . it uses the voting system $[51 : 49, 48, 2, 1]$ , and Table 11.7 lists the winning coalitions and their extra votes.

The minimal winning coalitions are those in which each voter is a critical voter: ${A, B}$ , ${A, C}$ , and ${B, C, D}$ . These minimal winning coalitions are displayed in Figure 11.1.

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Table 11.11: **TABLE 11.7** Winning Coalitions in the Four-Shareholder Corporation
Winning Coalition	Extra Votes	Critical Voters
Grand coalition	49	None
${A, B}$	46	$A, B$
${A, B, C}$	48	A
${A, B, D}$	47	$A, B$
${A, C}$	0	$A, C$
${A, C, D}$	1	$A, C$
${B, C, D}$	0	$B, C, D$

Figure 11.1 Each oval surrounds a minimal winning coalition for the four-shareholder corporation.