The Encyclopedia of Economic Principals

Chapter 29

Mechanism Design, Revelation, and Implementation

Posted prices, weighted rules, scoring rules and VCG payments in small numbers.

Four of the chapter's worked examples, made interactive: the gains a posted price cannot reach, how weights change an affine maximizer's choice, why quadratic scores reward honest forecasts, and VCG payments as displaced alternatives. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

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

No posted price captures every gain from trade

Can one fixed price let every profitable trade happen when values and costs are private?

A price high enough to include the high-cost seller excludes the low-value buyer, and the reverse. Because the efficient decision depends on both private types, no balanced rule that respects participation captures all three profitable pairs.

Equation, written in LaTeX: \frac14(30+0+80+30)=35

Equation, written in LaTeX: \frac14(80)=20.

Equation, written in LaTeX: \frac14(30+80)=27.5.

Scroll sideways for the whole equation

A buyer's value is 50 or 100 and a seller's cost is 20 or 70 (thousand dollars), each equally likely and independent. At a posted price p, trade happens when value >= p >= cost.

Predict first. Which price does better than 60, and can any price reach 35?

Your prediction

Choose an example

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Figure: No posted price captures every gain from trade. Four bars for the type pairs. At price 60 the pairs (100, 20) trade; the efficient pairs (50, 20) and (100, 70) do not. Expected gains are 20 of 35.
Posted price ($ thousand): 60
Constructed example: the chapter's hypothetical buyer and seller (values 50 and 100, costs 20 and 70, prices 40, 60 and 80); a price of 50 is added.

Calculated values

Pairs that trade
(100, 20)
Efficient pairs lost
(50, 20) and (100, 70)
Expected gains at this price ($000)
20
First-best expected gains ($000)
35
Shortfall ($000)
15

At a posted price of 60, a buyer accepts when its value is at least 60 and a seller when its cost is at most 60. Trade happens at (100, 20), so expected gains are 1/4 x (80) = 20 thousand, against the first best 1/4 x (30 + 0 + 80 + 30) = 35. The price loses (50, 20) and (100, 70), a shortfall of 35 - 20 = 15.

Worked steps

  1. Trades at p = 60: (100, 20)
  2. Gains = 1/4 x (80) = 20
  3. First best = 1/4 x (30 + 0 + 80 + 30) = 35
  4. Shortfall = 35 - 20 = 15

Use the idea

When a bilateral deal fails, check whether the value and cost ranges overlap: if they do, some lost trade is the price of voluntary, balanced bargaining.

Where the conclusion applies

A finite illustration of the continuous-type theorem with one fixed price and no subsidy from a third party.

Check your understanding: At p = 50, which pairs trade and what are expected gains?
(50, 20) with gain 30 and (100, 20) with gain 80 trade; (100, 70) fails since 70 > 50. Expected gains are (30 + 80)/4 = 27.5, a shortfall of 7.5.

Chapter 29 source: section "Myerson-Satterthwaite theorem".

Demonstration 2 of 4

Weights change which project wins

How does a fixed weight on one district's reports change the chosen project?

Roberts' theorem says that on an unrestricted domain, dominant-strategy rules choose by a fixed weighted sum plus constants. The weights are a distributive choice made before reports arrive; changing them changes the winner.

Equation, written in LaTeX: S_F=90+1.5(25)-60=67.5,

Equation, written in LaTeX: S_G=55+1.5(70)-50=110, S_N=0.

Equation, written in LaTeX: S_F=90+0.5(25)-60=42.5,

Equation, written in LaTeX: S_G=55+0.5(70)-50=40, S_N=0.

Scroll sideways for the whole equation

District A's values (in $ million) are 90, 55 and 0 for F, G and N; district B's are 25, 70 and 0. Weights are w_A = 1 and w_B; constants are -60, -50 and 0. The rule picks the largest score S.

Predict first. At what weight on district B do flood defense and grid tie?

Your prediction

Choose an example

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Figure: Weights change which project wins. Three bars: S_F = 67.5, S_G = 110, S_N = 0. Grid G is highlighted as the choice.
District B weight: 1.5
Constructed example: the chapter's hypothetical municipality (weights 1.5 and 0.5); weights 1 and 2 are added.

Calculated values

S_F
67.5
S_G
110
S_N
0
Choice
Grid G
Weight at which F and G tie
0.556

With w_B = 1.5, S_F = 90 + 1.5 x 25 - 60 = 67.5 and S_G = 55 + 1.5 x 70 - 50 = 110, while S_N = 0. The rule picks Grid G. F and G tie where 30 + 25w = 5 + 70w, so w = 25/45 = 0.556; above that weight the grid wins.

Worked steps

  1. S_F = 90 + 1.5 x 25 - 60 = 67.5
  2. S_G = 55 + 1.5 x 70 - 50 = 110
  3. S_N = 0
  4. Tie: 30 + 25w = 5 + 70w, w = 25 / 45 = 0.556

Use the idea

When a public body weights some groups more, write the score for each option and find the weight at which the decision flips.

Where the conclusion applies

Quasilinear preferences, at least three alternatives and weights fixed in advance; weights that react to reports break the dominant-strategy property.

Check your understanding: At w_B = 1, which project is chosen?
S_F = 90 + 25 - 60 = 55 and S_G = 55 + 70 - 50 = 75, so the grid wins.

Chapter 29 source: section "Roberts affine-maximizer theorem".

Demonstration 3 of 4

Quadratic score rewards honesty, linear rewards bravado

Which scoring rule makes a forecaster report the probability it actually believes?

A strictly proper rule makes the expected score peak exactly at the forecaster's belief. The quadratic rule's penalty grows with the square of the error; the linear rule's does not, so it pays for overconfidence.

Equation, written in LaTeX: 0.65(0.8775)+0.35(0.5775)=0.7725.

Equation, written in LaTeX: 0.65(0.96)+0.35(0.36)=0.75.

Equation, written in LaTeX: 0.65q+0.35(1-q)=0.35+0.30q.

Scroll sideways for the whole equation

The planner believes the event has probability p = 0.65 and reports q. Y is 1 if the event occurs and 0 if not. Quadratic score S(q, Y) = 1 - (q - Y)^2; linear score S(q, Y) = Yq + (1 - Y)(1 - q). One score unit pays $100.

Predict first. Under the linear rule, what report maximizes expected score?

Your prediction

Choose an example

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Figure: Quadratic score rewards honesty, linear rewards bravado. Expected quadratic score against the report q for belief 0.65; it peaks at 0.65. The current report 0.65 scores 0.7725.
Reported probability: 0.65, Scoring rule: quadratic
Constructed example: the chapter's hypothetical demand forecast (belief 0.65, reports 0.50, 0.65 and 0.80, quadratic and linear rules); a report of 1 is added.

Calculated values

Score if the event occurs
0.8775
Score if it does not
0.5775
Expected score
0.7725
Best report under this rule
0.65
Shortfall from the best report
0.0000

Under the quadratic rule with belief 0.65, a report of 0.65 scores 0.8775 if the event occurs and 0.5775 if not, so the expected score is 0.65 x 0.8775 + 0.35 x 0.5775 = 0.7725. The quadratic rule peaks at the true belief 0.65, so honesty is optimal; this report is the best one.

Worked steps

  1. S(q, 1) = 0.8775, S(q, 0) = 0.5775
  2. E[S] = 0.65 x 0.8775 + 0.35 x 0.5775 = 0.7725
  3. Best report 0.65: 0.7725, gap 0.7725 - 0.7725 = 0.0000

Use the idea

Before paying forecasters by accuracy, check that the expected payment peaks at the honest probability.

Where the conclusion applies

A risk-neutral forecaster who cares only about the score and has no stake in the outcome.

Check your understanding: Under the quadratic rule, what is the expected score at q = 1?
0.65 x 1 + 0.35 x 0 = 0.65, a loss of 0.35^2 = 0.1225 against the truthful 0.7725.

Chapter 29 source: section "Strictly proper scoring rules".

Demonstration 4 of 4

VCG payments are displaced alternatives

What does each winner pay under VCG, and why is it not tied to its own bid?

A VCG payment is the value the winner's presence takes from everyone else. Raising a loser's bid raises that harm, so the winners pay more, while their own bids only decide whether they win.

Equation, written in LaTeX: 80+75=155>140.

Equation, written in LaTeX: p_Y=140-75=65.

Equation, written in LaTeX: p_Z=140-80=60.

Equation, written in LaTeX: p_X=155-0=155

Scroll sideways for the whole equation

Firm X values both maintenance windows A and B together at x; Y values A at y; Z values B at 75 (thousand dollars). Each winner pays the others' best value without it minus the others' value in the chosen allocation.

Predict first. If X's value rises to 150, do Y and Z pay more or less?

Your prediction

Choose an example

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Figure: VCG payments are displaced alternatives. Bars: Y pays 65, Z pays 60, Revenue 125, Value 155.
X's package value ($ thousand): 140, Y's value for A ($ thousand): 80
Constructed example: the chapter's hypothetical maintenance windows (X 140, counterfactual 160, Y 80, Z 75); X values 130 and 150 and a Y value of 90 are added.

Calculated values

Winners
Y (window A) and Z (window B)
Payments
Y 65, Z 60
Utilities
Y 15, Z 15
Revenue
125
Allocation value
155

Y and Z create 80 + 75 = 155, more than X's 140, so they win. Removing Y makes X's 140 the best alternative, and Z gets 75 in the chosen allocation, so p_Y = 140 - 75 = 65. Likewise p_Z = 140 - 80 = 60. Utilities are 80 - 65 = 15 and 75 - 60 = 15; revenue 65 + 60 = 125 against value 155.

Worked steps

  1. 80 + 75 = 155 > 140
  2. p_Y = 140 - 75 = 65
  3. p_Z = 140 - 80 = 60
  4. Revenue = 65 + 60 = 125

Use the idea

To price a shared resource by VCG, compute for each winner the best allocation without it and subtract what the others receive with it.

Where the conclusion applies

Quasilinear values, no budget limits and truthful reports; the chapter notes that a binding budget breaks the benchmark.

Check your understanding: With x = 160 and y = 90, who wins and what are the payments?
Y + Z = 165 > 160, so Y and Z win; p_Y = 160 - 75 = 85 and p_Z = 160 - 90 = 70; revenue 155.

Chapter 29 source: section "Vickrey-Clarke-Groves mechanism".