Suppose you were the nonvoting chair of a committee...

Suppose you were the nonvoting chair of a committee consisting of three additional members (Smith, Jones, and Yang) responsible for allocating $$\$ 100,000$$ each to two out of three projects based on majority vote by the committee. You have a strong preference for option $\mathrm{C}$. What options are funded if the committee votes to allocate the first $$\$ 100,000$$ to the choice between A and B and the second $$\$ 100,000$$ between the remaining two projects? How would you have structured the voting agenda? Explain your reasoning. $$ \begin{array}{llll} \hline \begin{array}{l} \text { Ranking of } \\ \text { preferences of voters: } \end{array} & \begin{array}{l} \text { Option A: } \\ \text { cost }=\$ 100,000 \end{array} & \begin{array}{l} \text { Option B: } \\ \text { cost }=\$ 100,000 \end{array} & \begin{array}{l} \text { Option C: } \\ \text { cost }=\$ 100,000 \end{array} \\ \hline \text { Smith } & 1 & 3 & 2 \\ \text { Jones } & 3 & 2 & 1 \\ \text { Yang } & 2 & 1 & 3 \\ \hline \end{array} $$

Suppose you were the nonvoting chair of a committee consisting of three additional members (Smith, Jones, and Yang) responsible for allocating $$\$ 100,000$$ each to two out of three projects based on majority vote by the committee. You have a strong preference for option $\mathrm{C}$. What options are funded if the committee votes to allocate the first $$\$ 100,000$$ to the choice between A and B and the second $$\$ 100,000$$ between the remaining two projects? How would you have structured the voting agenda? Explain your reasoning.
$$
\begin{array}{llll}
\hline \begin{array}{l}
\text { Ranking of } \\
\text { preferences of voters: }
\end{array} & \begin{array}{l}
\text { Option A: } \\
\text { cost }=\$ 100,000
\end{array} & \begin{array}{l}
\text { Option B: } \\
\text { cost }=\$ 100,000
\end{array} & \begin{array}{l}
\text { Option C: } \\
\text { cost }=\$ 100,000
\end{array} \\
\hline \text { Smith } & 1 & 3 & 2 \\
\text { Jones } & 3 & 2 & 1 \\
\text { Yang } & 2 & 1 & 3 \\
\hline
\end{array}
$$

Key Concepts

Majority Voting

Majority voting is a decision rule in which the option that receives more than half of the votes wins. When committee members vote based on their ranked preferences, the majority rule ensures that the option with support from most of the voters passes the decision. However, when the process is sequential, the outcome may differ from cases where all options are considered simultaneously, and strategic structuring becomes important.

Strategic Vote Ordering

Strategic vote ordering involves designing the sequence of voting in such a way that it leads to the elimination of less preferred alternatives and the survival of the more desired ones. By carefully choosing which pair of options is presented first, a decision maker can influence the voters’ choices in subsequent rounds, thereby channeling the outcome toward a preferred alternative—even if it is not the top choice of the majority of voters.

Sequential Voting

Sequential voting refers to a decision process where votes are taken in a series of steps rather than all at once. The order of the votes matters because earlier decisions can shape the choices available in later rounds. In settings with sequential voting, the sequence can be manipulated so that certain alternatives are eliminated early, thus bolstering the chances for a preferred alternative to win in the final round.

Agenda Setting

Agenda setting is the process by which the order and structure of votes or issues is determined. In collective decision-making settings, the chair or organizer can use agenda control to influence the final outcome by strategically sequencing the decisions. This is particularly important when the individuals voting have different preference orders or when sequential voting can lead to different outcomes based on the order in which alternatives are considered.

Transcript

00:01 For this problem, we are told that a condorcet committee of, say, three members is a committee that is preferred by voters over any other committee of three members.

00:09 We're told that a scoring function was used to determine condorset committees in mathematical social sciences.

00:16 We're asked to consider a committee with members a, b, and c.

00:19 We are to suppose there are 10 voters who each have a preference for a three -member committee.

00:25 For example, one voter may prefer a committee member or made up of members a, c, and g.

00:30 Then this voter's preference score for the abc committee is two because two of the members are on this voter's preferred list.

00:39 We have that for a three -member committee, voter preference scores range from zero, no members on the preferred list, to three, all members on the preferred list.

00:48 We have the table below shows the preferred lists of ten voters for a three -member committee selected from potential members.

00:56 In part a, we are asked to find the preference score for committee a, b, c, for each voter.

01:03 So voter 1 has a, c, and d, so the score is going to be 2 because of a and d.

01:10 Voter 2 has b and c, so that's another score of 2.

01:14 Voter 3 has a and b, so that's another 2.

01:17 Voter 4 has b and c, so that's another 2.

01:20 Voter 5 has d -e -f, so that is 0.

01:24 6 has c -e -g, so that is 1...