
# 18.3: Weighted Voting


1.

1. 9 players
2. $$10+9+9+5+4+4+3+2+2 = 48$$
3. 47

3.

1. 9, a majority of votes
2. 17, the total number of votes
3. 12, which is 2/3 of 17, rounded up

5.

1. P1 is a dictator (can reach quota by themselves)
2. P1, since dictators also have veto power
3. P2, P3, P4

7.

1. none
2. P1
3. none

9.

1. 11+7+2 = 20
2. 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.

1. $$\{\underline{\mathrm{P} 1}\}\{\mathrm{P} 1, \mathrm{P} 2\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}$$ P1: 100%, P2: 0%, P3: 0%, P4: 0%
2. $$\{\underline{\mathrm{P} 1, \mathrm{P} 2}\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}\}\{\underline{\mathrm{P} 1, \mathrm{P} 4}\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 3, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}$$ P1: 7/10 = 70%, P2: 1/10 = 10%, P3: 1/10 = 10%, P4: 1/10 = 10%
3. $$\{\underline{\mathrm{P} 1, \mathrm{P} 2}\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\}\{\underline{\mathrm{P} 1, \mathrm{P} 2}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1, \mathrm{P} 3}, \mathrm{P} 4\}\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3, \mathrm{P} 4\}$$ 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.

1. 6, 7
2. 8, given P1 veto power
3. 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.

1. $$\left\{\underline{\mathrm{P} 1}, \underline{\mathrm{P} 2}\right\}\left\{\mathrm{P} 1, \underline{\mathrm{P} 3}\right\}\left\{\underline{\mathrm{P} 1}, \mathrm{P} 2, \mathrm{P} 3\right\}$$. P1: 3/5, P2: 1/5, P3: 1/5
2. $$<\mathrm{P} 1, \underline{\mathrm{P} 2}, \mathrm{P} 3><\mathrm{P} 1, \underline{\mathrm{P} 3}, \mathrm{P} 2><\mathrm{P} 2, \underline{\mathrm{P} 1}, \mathrm{P} 3><\mathrm{P} 2, \mathrm{P} 3, \underline{\mathrm{P} 1}><\mathrm{P} 3, \underline{\mathrm{P} 1}, \mathrm{P} 2><\mathrm{P} 3, \mathrm{P} 2, \underline{\mathrm{P} 1}>$$

$$\mathrm{P} 1: 4 / 6, \quad \mathrm{P} 2: 1 / 6, \quad \mathrm{P} 3: 1 / 6$$

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

18.3: Weighted Voting is shared under a CC BY-SA 3.0 license and was authored, remixed, and/or curated by David Lippman via source content that was edited to conform to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.