Why Voting on Public Goods Is Not the Same as Choosing Efficiently
A public-good decision can affect every voter even though voters value the good differently. Majority voting provides a rule for turning those different preferences into a collective choice, but the result depends on how preferences are arranged and which alternatives are compared. A majority-selected quantity need not maximize total surplus, and some voting situations have no stable majority winner at all.
The key case for a predictable result is a choice among levels of one public good, when each voter has a single most-preferred level and preferences decline on either side of it. The median voter’s ideal level then defeats every other level in a pairwise vote. That result is useful, but it is not a general guarantee that majority rule selects an efficient outcome.
For a public-good quantity, the alternatives can be ordered from a lower to a higher level. Voters may differ in their ideal quantities because they value the good differently, face different tax shares, or both. The simplest median-voter result concerns sincere pairwise choices: each voter votes for the alternative they prefer, and a majority is more than half of the voters.
The Median-Voter Result
Suppose there is an odd number of voters, the alternatives are ordered along one dimension, and every voter has single-peaked preferences. Arrange the voters’ ideal points from lowest to highest. The ideal point in the middle is the median ideal point. It is a Condorcet winner: a majority prefers it to any alternative strictly to its left or right.
The argument is direct. Consider any alternative to the right of the median ideal point. At least half of the voters, including the median voter, have ideal points at or to the left of the median. By single-peakedness, each of those voters prefers the median point to an alternative farther to its right. With an odd electorate, that group is a majority. The same reasoning applies to an alternative to the left: at least half of the voters, including the median voter, have ideal points at or to its right, and they prefer the median point. Thus the median point wins either pairwise comparison.
This conclusion depends on the structure of the choice. The alternatives must lie on one dimension, preferences must be single-peaked, and the comparison must be a sincere pairwise majority vote. With an even number of voters, the two middle ideal points can bound a range of majority winners; the exact result depends on the voters’ preferences and the voting procedure. With multiple dimensions or non-single-peaked preferences, the median-voter argument need not apply.
Worked Example: Finding the Majority-Preferred Public-Good Level
Five voters choose one level of a local flood-warning system. Their ideal quantities, measured in standard service units, are 1, 2, 4, 6, and 9. Assume each voter has single-peaked preferences over the quantity. Identify the median-voter outcome and explain why it defeats a proposed quantity of 7.
Method. Order the ideal quantities and select the middle one. The ordered values are already \(1, 2, 4, 6, 9\), so the median ideal quantity is \(4\).
The voters with ideal quantities \(1, 2,\) and \(4\) prefer quantity \(4\) to quantity \(7\). Each has an ideal point at or below \(4\), so single-peakedness means moving farther to the right, from \(4\) to \(7\), cannot improve their preference. These three voters form a majority of the five.
Therefore quantity \(4\) defeats quantity \(7\) in a pairwise vote. The same median-voter result shows that quantity \(4\) defeats every other quantity, so it is the Condorcet winner. Common mistake avoided: choosing the mean ideal quantity, \(4.4\), rather than the median. Majority rule in this model selects the median ideal point, not the average.
Voting Outcomes and Efficient Provision
A median-voter outcome describes what wins a vote; it does not by itself establish that the result maximizes total surplus. To see the distinction, consider voters who receive utility from a public-good level \(G\) and pay taxes that finance its total cost. Suppose total cost is \(C(G)\), each of \(n\) voters pays an equal share \(C(G)/n\), and voter \(i\) has benefit \(B_i(G)\). With private consumption represented in money units, the voter’s relevant utility is \(B_i(G)-C(G)/n\).
If benefits and costs can be measured in the same units, total surplus is the sum of everyone’s benefits minus the total cost. An efficient quantity maximizes that total. For an interior optimum, the sum of the voters’ marginal benefits equals marginal cost. A majority vote instead depends on the ordering of individual preferences: each person counts as one vote, regardless of how large their benefit or loss is.
Worked Example: Comparing the Median Outcome with the Efficient Quantity
Five voters choose a nonnegative quantity \(G\). Total cost is \(C(G)=G^2/2\), divided equally among the voters. Voter \(i\)’s benefit is \(B_i(G)=a_iG-G^2/2\), with \(a_i\) values \(1, 2, 3, 4,\) and \(10\). Assume each voter’s utility is benefit minus their tax share. Find the majority outcome and the efficient quantity.
Step 1: Find each voter’s preferred quantity. Each voter pays \(C(G)/5=G^2/10\), so voter \(i\)’s utility is \(U_i(G)=a_iG-G^2/2-G^2/10=a_iG-3G^2/5\). The derivative is \(a_i-6G/5\), which equals zero at the ideal quantity \(G_i=5a_i/6\). The second derivative is \(-6/5<0\), so this is each voter’s unique maximum.
The ideal quantities, in order, are \(5/6,\ 5/3,\ 5/2,\ 10/3,\ 25/3\). Since utility is strictly concave around each ideal quantity, preferences are single-peaked. The middle ideal quantity is \(5/2\). Thus the majority-voting outcome is \(G=5/2\).
Step 2: Maximize total surplus. Sum the benefit parameters: \(1+2+3+4+10=20\). Total benefits are \(20G-5G^2/2\). Subtracting total cost gives \(W(G)=20G-5G^2/2-G^2/2=20G-3G^2\). The derivative is \(20-6G\), so the maximum occurs at \(G=10/3\). The second derivative is \(-6<0\), confirming a maximum.
The voting quantity \(5/2\) is below the efficient quantity \(10/3\). For a check, total surplus at the voting quantity is \(20(5/2)-3(5/2)^2=125/4=31.25\). At the efficient quantity it is \(20(10/3)-3(10/3)^2=100/3\), which is greater. Common mistake avoided: using the median voter’s benefit parameter as if it determined the sum of all benefits. Voting reflects the median voter’s preferences; efficiency reflects everyone’s benefits and total cost.
When Pairwise Majorities Cycle
The median-voter result does not apply to every collective decision. In particular, preferences over several distinct packages may not be single-peaked along one shared line. In that case, majority comparisons can form a cycle: one option defeats a second, the second defeats a third, and the third defeats the first. There is then no Condorcet winner.
A cycle makes the result sensitive to the agenda—the order in which alternatives are compared. This does not mean that every multi-option vote cycles. It means that majority rule alone, without suitable restrictions on preferences or procedure, does not guarantee one stable winner.
Worked Example: A Majority Cycle Among Public-Good Packages
Three voters compare three packages for a town’s public services. Package A emphasizes a neighborhood alert system, B emphasizes a drainage upgrade, and C emphasizes a cooling center. Their rankings are: voter 1 prefers A to B to C; voter 2 prefers B to C to A; voter 3 prefers C to A to B. Determine the pairwise winners and show how the agenda can change the final outcome.
Method. Count the voters who prefer each option in each pairwise comparison.
In a vote between A and B, voters 1 and 3 prefer A, while voter 2 prefers B. Thus A defeats B, 2–1. In a vote between B and C, voters 1 and 2 prefer B, while voter 3 prefers C. Thus B defeats C, 2–1. In a vote between C and A, voters 2 and 3 prefer C, while voter 1 prefers A. Thus C defeats A, 2–1. The majority relation is \(A>B\), \(B>C\), and \(C>A\), a cycle.
If A and B are compared first, A survives that vote; comparing A with C then produces C as the final winner. If B and C are compared first, B survives; comparing B with A then produces A as the final winner. The pairwise preferences are unchanged, but the agenda changes the outcome. Common mistake avoided: assuming that because each pair has a majority winner, one option must defeat both alternatives. Here, no option does.
Majority Support Does Not Measure Total Surplus
The difference between voting and efficiency also appears in simple approve-or-reject decisions. Suppose a public project has a known cost funded by equal taxes, and each voter’s benefit is measured in money-equivalent units. A voter supports the project if their benefit exceeds their own tax payment. The majority decision counts how many people support it. The efficiency test compares the sum of benefits with the full project cost.
Worked Example: A Project That the Majority Rejects Despite Positive Surplus
Three residents vote on a hypothetical public signal system costing $90, funded equally. Their benefits are $10, $20, and $100. Assume these benefits are money-equivalent and the project has no other costs or benefits. Determine the vote and whether approval would increase total surplus.
Each resident pays \(90/3=\$30\). Their net benefits from approval are \(10-30=-\$20\), \(20-30=-\$10\), and \(100-30=\$70\). The first two residents vote against the project and the third votes for it, so the majority rejects it, 2–1.
Total benefits are \(10+20+100=\$130\). Subtracting the $90 cost gives total surplus of \(\$40\). Under the stated assumptions, approval would increase total surplus even though the majority rejects it. The large benefit to one resident does not count as more than one vote. This illustrates a possible conflict between majority rule and aggregate surplus; it does not, by itself, settle distributional questions about who should bear the cost.
Conditions, Limits, and Exam Traps
When analyzing a collective public-good decision, first identify the alternatives and the voting rule. If voters choose a quantity along one line, ask whether each voter has a single ideal quantity and single-peaked preferences. If so, order the ideal points and find the median. If the choice is between packages with different features, do not presume that a median quantity exists: compare alternatives pairwise and check for a cycle.
Then keep the political and welfare calculations separate. The median-voter theorem predicts a pairwise majority winner under its assumptions. An efficiency calculation instead requires the total benefits and total costs of each alternative. Neither calculation substitutes for the other.
State whether voters choose a single public-good quantity or compare distinct packages.
For a one-dimensional choice, verify single-peaked preferences before invoking the median-voter result. For packages, calculate pairwise majority winners.
Sum benefits and subtract total cost to evaluate surplus; do not infer efficiency from the vote count.
- Using the average instead of the median: single-peaked majority voting selects the median ideal point, not generally the average ideal point.
- Assuming every vote has a stable winner: pairwise majorities can cycle when preferences do not meet the single-peaked condition.
- Treating a majority as a welfare measure: a narrow majority can reject a project with positive total surplus or approve one with negative total surplus.
- Ignoring the financing rule: a voter’s preferred quantity depends on their share of the cost as well as their benefit from the good.
Check Your Understanding
For each problem, distinguish the voting result from any welfare conclusion requested.
- Seven voters have ideal service levels \(1, 2, 2, 4, 5, 7,\) and \(9\), with single-peaked preferences. What level is the median-voter outcome, and why does it defeat level 6? (Answer: Level 4; the four voters with ideal levels 1, 2, 2, and 4 form a majority and prefer 4 to 6.)
- Three voters rank packages as follows: voter 1 has \(X>Y>Z\), voter 2 has \(Y>Z>X\), and voter 3 has \(Z>X>Y\). Find the winner of each pairwise comparison and state whether there is a Condorcet winner. (Answer: X defeats Y, Y defeats Z, and Z defeats X, each 2–1; there is no Condorcet winner.)
- Five voters have utility \(U_i(G)=a_iG-3G^2/5\) for \(G\geq0\), where their ordered \(a_i\) values are \(1,2,4,6,9\). Find the majority outcome under single-peaked preferences. (Answer: The median parameter is 4; setting \(a_i-6G/5=0\) gives \(G=10/3\).)
- A three-person project costs $120 and gives benefits of $15, $35, and $70. It is funded equally, and benefits are measured in money-equivalent units. What is the majority decision, and what is total surplus? (Answer: Each pays $40; net benefits are \(-\$25\), \(-\$5\), and \(+\$30\), so the project is rejected 2–1; total surplus is $0, so approval and rejection yield equal total surplus under these assumptions.)
- Three voters have benefits \(B_i(G)=a_iG-G^2/2\), with \(a_i=1,3,8\). Total cost is \(G^2/2\), divided equally. Find the median-voter quantity and the efficient quantity. (Answer: Individual utility is \(a_iG-2G^2/3\), so the median voter’s ideal quantity is \(G=9/4\); total surplus is \(12G-2G^2\), maximized at \(G=3\).)