18.3: Weighted Voting
1.
- 9 players
- \(10+9+9+5+4+4+3+2+2 = 48\)
- 47
3.
- 9, a majority of votes
- 17, the total number of votes
- 12, which is 2/3 of 17, rounded up
5.
- P1 is a dictator (can reach quota by themselves)
- P1, since dictators also have veto power
- P2, P3, P4
7.
- none
- P1
- none
9.
- 11+7+2 = 20
- P1 and P2 are critical
11. Winning coalitions, with critical players underlined:
\(\left\{\underline{\mathrm{P} 1}, \underline{\mathrm{P} 2}\right\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1, \mathrm{P} 2}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}, \mathrm{P} 4\}\)
P1: 6 times, P2: 2 times, P3: 2 times, P4: 0 times. Total: 10 times
Power: \(\mathrm{P} 1: 6 / 10=60 \%, \mathrm{P} 2: 2 / 10=20 \%, \mathrm{P} 3: 2 / 10=20 \%, \mathrm{P} 4: 0 / 10=0 \%\)
13.
- \(\{\underline{\mathrm{P} 1}\}\{\mathrm{P} 1, \mathrm{P} 2\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}\) P1: 100%, P2: 0%, P3: 0%, P4: 0%
- \(\{\underline{\mathrm{P} 1, \mathrm{P} 2}\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}\}\{\underline{\mathrm{P} 1, \mathrm{P} 4}\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}\) P1: 7/10 = 70%, P2: 1/10 = 10%, P3: 1/10 = 10%, P4: 1/10 = 10%
- \(\{\underline{\mathrm{P} 1, \mathrm{P} 2}\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1, \mathrm{P} 2}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}\) P1: 6/10 = 60%, P2: 2/10 = 20%, P3: 2/10 = 20%, P4: 0/10 = 0%
15. \(\mathrm{P} 3=5 . \mathrm{P} 3+\mathrm{P} 2=14 . \mathrm{P} 3+\mathrm{P} 2+\mathrm{P} 1=27,\) reaching quota. \(\mathrm{P} 1\) is critical.
17. Sequential coalitions with pivotal player underlined
\(<\mathrm{P} 1, \underline{\mathrm{P} 2}, \mathrm{P} 3><\mathrm{P} 1, \underline{\mathrm{P} 3}, \mathrm{P} 2><\mathrm{P} 2, \underline{\mathrm{P} 1}, \mathrm{P} 3><\mathrm{P} 2, \underline{\mathrm{P} 3}, \mathrm{P} 1><\mathrm{P} 3, \underline{\mathrm{P} 1}, \mathrm{P} 2><\mathrm{P} 3, \underline{\mathrm{P} 2}, \mathrm{P} 1>\)
\(\mathrm{P} 1: 2 / 6=33.3 \%, \mathrm{P} 2: 2 / 6=33.3 \%, \mathrm{P} 3: 2 / 6=33.3 \%\)
19.
- 6, 7
- 8, given P1 veto power
- 9, given P1 and P2 veto power
21. If adding a player to a coalition could cause it to reach quota, that player would also be critical in that coalition, which means they are not a dummy. So a dummy cannot be pivotal.
23. We know P2+P3 can’t reach quota, or else P1 wouldn’t have veto power.
P1 can’t reach quota alone.
P1+P2 and P1+P3 must reach quota or else P2/P3 would be dummy.
- \(\left\{\underline{\mathrm{P} 1}, \underline{\mathrm{P} 2}\right\}\left\{\mathrm{P} 1, \underline{\mathrm{P} 3}\right\}\left\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\right\}\). P1: 3/5, P2: 1/5, P3: 1/5
- \(<\mathrm{P} 1, \underline{\mathrm{P} 2}, \mathrm{P} 3><\mathrm{P} 1, \underline{\mathrm{P} 3}, \mathrm{P} 2><\mathrm{P} 2, \underline{\mathrm{P} 1}, \mathrm{P} 3><\mathrm{P} 2, \mathrm{P} 3, \underline{\mathrm{P} 1}><\mathrm{P} 3, \underline{\mathrm{P} 1}, \mathrm{P} 2><\mathrm{P} 3, \mathrm{P} 2, \underline{\mathrm{P} 1}>\)
\(\mathrm{P} 1: 4 / 6, \quad \mathrm{P} 2: 1 / 6, \quad \mathrm{P} 3: 1 / 6\)
25. \([4: 2,1,1,1]\) is one of many possibilities
27. \([56: 30,30,20,20,10]\)
29. \([54: 10,10,10,10,10,1,1,1,1,1,1,1,1,1,1]\) is one of many possibilities