The Encyclopedia of Economic Principals

Chapter 30

Social Choice Mechanisms and Strategy-Proofness

Rules that make honest reporting safe, and what they cost.

Four of the chapter's worked examples, made interactive: a median rule with phantoms, Clarke pivot payments that fall short of cost, log-score peer prediction, and quadratic voting.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Phantoms pin the median

Can a district move a generalized median toward its own peak by misreporting?

A generalized median takes the median of the reports and fixed phantoms. With single-peaked preferences, a report above the outcome cannot raise it past the next pivot, and a report below it can only pull the outcome away from the true peak.

Equation, written in LaTeX: 15,35,40,50,55,60,65,75,90,

Equation, written in LaTeX: f(r)=55.

Equation, written in LaTeX: |60-55|=5

Scroll sideways for the whole equation

Five districts report peaks on [0, 100] and the authority adds four fixed phantoms at 40, 50, 55 and 65. The outcome f(r) is the fifth of the nine ordered points.

Predict first. Can the district with peak 60 pull the outcome toward 60 by exaggerating upward?

Your prediction

Choose an example

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Figure: Phantoms pin the median. Number line with the five reports, four phantoms and the outcome 55 when the district with peak 60 reports 60.
Report of the district with true peak 60: 60
Constructed example: the chapter's hypothetical water authority (peaks 15, 35, 60, 75, 90; phantoms 40, 50, 55, 65; reports 45 and 100); the report 30 is added.

Calculated values

Report of the peak-60 district
60
Outcome (5th of 9)
55
Distance from 60
5
Better than truthful
no

The nine points in order are 15, 35, 40, 50, 55, 60, 65, 75, 90; the fifth is 55. The district's distance is |60 - 55| = 5. This is the truthful report. No report brings the outcome closer to 60, because the phantom at 55 caps upward moves and lower reports pull the median down.

Worked steps

  1. Sorted: 15, 35, 40, 50, 55, 60, 65, 75, 90
  2. Fifth value: 55
  3. Distance = |60 - 55| = 5

Use the idea

Phantoms let a designer bias a median rule toward a policy range without giving anyone a reason to misreport.

Where the conclusion applies

Single-peaked preferences with distance loss. If a district prefers 50 to 55 for a non-distance reason, as in the chapter's grant example, truthful reporting can fail.

Check your understanding: If the district reports 30, what is the outcome and its distance from 60?
Sorted 15, 30, 35, 40, 50, 55, 65, 75, 90; the fifth value is 50 and the distance |60 - 50| = 10, worse than 5.

Chapter 30 source: section "Generalized-median voter schemes".

Demonstration 2 of 4

Efficient, truthful, and still short of money

Do Clarke pivot payments pay for an efficient public project?

Clarke payments make truthful reporting a dominant strategy and pick the efficient decision, but the payments are set by pivotality, not by cost. Green and Laffont show no such mechanism can balance the budget in general.

Equation, written in LaTeX: 70+55+15=140,

Equation, written in LaTeX: 140-110=30.

Equation, written in LaTeX: 0-(-40)=40.

Equation, written in LaTeX: 40+25+0=65,

Equation, written in LaTeX: 110-65=45.

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Three owners value a flood barrier at 70, 55 and 15 thousand dollars; it costs 110. Each owner pays the harm their presence does to the others' welfare, project cost included.

Predict first. Does raising the third owner's value raise or lower total Clarke revenue?

Your prediction

Choose an example

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Figure: Efficient, truthful, and still short of money. Bars of the three Clarke payments 40, 25, 0 and revenue 65 against the cost line at 110, with a gap of 45.
Barrier cost ($ thousand): 110, Third owner's value ($ thousand): 15
Constructed example: the chapter's hypothetical flood barrier (cost 110, values 70, 55, 15); costs 100 and 120 and third values 5 and 25 are added.

Calculated values

Total benefit
140
Net value
30
Payments
40, 25, 0
Revenue
65
Financing gap
45
Utilities
30, 30, 15

Total benefit 70 + 55 + 15 = 140 covers the cost 110, so the barrier is built with net value 140 - 110 = 30. Owner 1: others 70 < 110, pays 0 - (70 - 110) = 40. Owner 2: others 85 < 110, pays 0 - (85 - 110) = 25. Owner 3: others 125 >= 110, still built, pays (125 - 110) - (125 - 110) = 0. Revenue 40 + 25 + 0 = 65 leaves a gap of 110 - 65 = 45: efficient and truthful, but not self-financing.

Worked steps

  1. Total = 70 + 55 + 15 = 140
  2. Net = 140 - 110 = 30
  3. Owner 1: others 70 < 110, pays 0 - (70 - 110) = 40
  4. Owner 2: others 85 < 110, pays 0 - (85 - 110) = 25
  5. Owner 3: others 125 >= 110, still built, pays (125 - 110) - (125 - 110) = 0
  6. Revenue = 40 + 25 + 0 = 65
  7. Gap = 110 - 65 = 45

Use the idea

Budget for an outside subsidy, like the chapter's grant of 45, when truthful efficient provision is the goal.

Where the conclusion applies

Quasilinear values, a single binary project and private values known to each owner. Collusion between owners can break the Clarke incentives.

Check your understanding: At cost 110 and third value 25, what is the financing gap?
Payments 30, 15 and 0 (owner 3 is not pivotal: others 125 >= 110). Revenue 45, gap 110 - 45 = 65.

Chapter 30 source: section "Green-Laffont impossibility".

Demonstration 3 of 4

Paying for honest peer reports

How much does a log-score peer payment reward an honest label?

When your signal predicts your peer's, an honest forecast matches your belief, and the log score rewards an accurate forecast. The advantage grows with the correlation and vanishes when signals are independent.

Equation, written in LaTeX: 0.75\ln(0.75)+0.25\ln(0.25)\approx -0.562.

Equation, written in LaTeX: 0.75\ln(0.25)+0.25\ln(0.75)\approx -1.112.

Equation, written in LaTeX: \frac{1}{2}\ln(3)\approx 0.549.

Equation, written in LaTeX: \ln(0.5)\approx -0.693.

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q is the probability a truthful peer reports the same label you saw. A report is scored by the log of the probability its implied forecast gives the peer's label. A constant may be added to every score.

Predict first. What happens to the truth-telling advantage as q falls to 0.5?

Your prediction

Choose an example

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Figure: Paying for honest peer reports. Two bars: truthful expected score -0.5623 and false -1.1116 at agreement probability 0.75, gap 0.5493.
Probability a peer shares your signal: 0.75, Constant added to scores: 0
Constructed example: the chapter's hypothetical review system (q = 0.75, shift 1.2, failure test 0.5); q = 0.6 and 0.9 are added. Scores show four decimals, as the book's gap uses unrounded scores.

Calculated values

Truthful score
-0.5623
False score
-1.1116
Truthful advantage
0.5493
Strictly truthful
yes

Truthful: 0.75 ln 0.75 + 0.25 ln 0.25 = -0.5623. False: 0.75 ln 0.25 + 0.25 ln 0.75 = -1.1116. Advantage -0.5623 - (-1.1116) = 0.5493. Truth telling is strictly better.

Worked steps

  1. Truthful = 0.75 ln 0.75 + 0.25 ln 0.25 = -0.5623
  2. False = 0.75 ln 0.25 + 0.25 ln 0.75 = -1.1116
  3. Advantage = -0.5623 - (-1.1116) = 0.5493

Use the idea

Check that reviewers' signals are informative about one another before relying on peer scores.

Where the conclusion applies

A common, correct model, truthful peers and no coordination. A population that swaps the labels can form another equilibrium, as the chapter warns.

Check your understanding: At q = 0.9, what is the truthful advantage?
Truthful 0.9 ln 0.9 + 0.1 ln 0.1 = -0.3251; false 0.9 ln 0.1 + 0.1 ln 0.9 = -2.0829; advantage 1.7578.

Chapter 30 source: section "Peer-prediction mechanism".

Demonstration 4 of 4

Intensity versus numbers under quadratic voting

When does one intense supporter outvote a less intense majority?

Quadratic credits make each extra vote dearer, so votes rise with intensity but influence costs the square. An intense minority wins only when its intensity outweighs the numbers against it.

Equation, written in LaTeX: v_S=\frac{24}{2}=12

Equation, written in LaTeX: v_O=\frac{-4}{2}=-2

Equation, written in LaTeX: 12+4(-2)=4,

Equation, written in LaTeX: 7-8=-1.

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V is a participant's normalized value of influence. With credit cost v^2, each buys v = V/2 votes; the proposal passes when the sum of votes is positive.

Predict first. What supporter value exactly ties four opponents?

Your prediction

Choose an example

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Figure: Intensity versus numbers under quadratic voting. Vote bars: supporter 12.0 votes against 4 opponents at -2 each, net 4.0; the proposal passes.
Supporter's normalized value: 24, Number of opponents: 4
Constructed example: the chapter's hypothetical bus vote (V = 24 and 14, four opponents at -4); values 10 and 30 and three opponents are added.

Calculated values

Supporter votes
12.0
Supporter credits
144.00
Opposition votes
-8
Opposition credits
16
Net tally
4.0
Outcome
passes

The supporter buys 24/2 = 12.0 votes for 12.0^2 = 144.00 credits. Each of the 4 opponents buys -4/2 = -2 votes for 4 credits, 16 in all. Net tally 12.0 + 4(-2) = 4.0, so the proposal passes. With 4 opponents a supporter value of 16 ties.

Worked steps

  1. Supporter votes = 24/2 = 12.0, credits = 144.00
  2. Each opponent: -4/2 = -2 votes, 4 credits
  3. Tally = 12.0 + 4(-2) = 4.0
  4. Outcome: passes

Use the idea

Use quadratic voting across a portfolio of issues where credits have a real opportunity cost.

Where the conclusion applies

The continuous benchmark v = V/2, verified identities and no coordinated blocs. Pivotal beliefs and integer votes change the mapping.

Check your understanding: With V = 14 and three opponents, does the proposal pass?
7 + 3(-2) = 1 > 0, so it passes; with four opponents 7 - 8 = -1 and it fails. The tie with four opponents is V = 16.

Chapter 30 source: section "Quadratic voting".