Mechanism Design & Public Economics
Vickrey-Clarke-Groves (VCG) Pivotal Externality Fee Formula
Make truthful valuation reporting a dominant strategy in shared public-project decisions by computing each participant’s Clarke pivotal externality payment.
Charge each participant ONLY the net social harm when their report flips the group decision!
When 4 departments or districts jointly fund a shared project of cost K = 100 with pre-assigned cost shares s_i (summing to 100) and reported gross benefits v_i, naive cost-sharing invites free-riding (understating v_i) or over-claiming (exaggerating v_i to force construction)! Under the Vickrey-Clarke-Groves (VCG) Clarke Pivotal Mechanism, let each participant’s net reported surplus be w_i = v_i - s_i, total net surplus be W_all = SUM(w), and others’ net surplus be W_minus_i = W_all - w_i. The project is approved iff W_all > 0, and participant i pays a Clarke pivotal fee t_i = MAX(0, IF(W_all > 0, -W_minus_i, W_minus_i)) — positive ONLY when i’s report reverses what the rest of the group would have chosen!
Verify on 4 districts splitting K = 100 equally (s_i = 25 in C2:C5) with gross benefits B2:B5 = {45, 35, 20, 15}: net valuations D2:D5 = {+20, +10, -5, -10} sum to W_all = +15 > 0 (Project Approved!). Without District 1 (+20), others sum to W_-1 = -5 <= 0, so District 1 is PIVOTAL and pays t_1 = 5.00! Districts 2, 3, and 4 have W_-i = {+5, +20, +25} > 0, so none of them flips the outcome and each pays t_i = 0.00! Even if District 1 exaggerates v_1 from 45 to 90, its fee stays fixed at |W_-1| = 5.00 — proving dominant-strategy incentive compatibility!
| Participant (Share s_i = 25) | Gross Benefit v_i & Net w_i | Others’ Net W_-i & Clarke Fee t_i |
|---|---|---|
| District 1 (Pivotal Yes) | 45 -> w_1 = +20 | W_-1 = -5 -> Fee t_1 = 5.00 |
| District 2 (Non-Pivotal Yes) | 35 -> w_2 = +10 | W_-2 = +5 -> Fee t_2 = 0.00 |
| District 3 (Non-Pivotal No) | 20 -> w_3 = -5 | W_-3 = +20 -> Fee t_3 = 0.00 |
| District 4 (Non-Pivotal No) | 15 -> w_4 = -10 | W_-4 = +25 -> Fee t_4 = 0.00 |
Clarke Pivotal Externality Fee t_i = MAX(0, IF(W_all > 0, -W_-i, W_-i)) Formula
Notice how MAX(0, IF(W_all > 0, -W_-i, W_-i)) handles both pivotal directions cleanly: a Yes-voter who flips W_-i <= 0 into W_all > 0 pays -W_-i >= 0, while a No-voter who flips W_-i > 0 into W_all <= 0 pays +W_-i > 0!
=LET(w_all, SUM($D$2:$D$5), w_oth, w_all - D2, ROUND(MAX(0, IF(w_all > 0, -w_oth, w_oth)), 4))Efficient Project Approval Flag and Total Clarke Surplus Collected Formula
Total Clarke fees SUM(E2:E5) must go to an external sink rather than rebating back to participants (Green-Laffont impossibility of simultaneous efficiency, dominant strategies, and exact budget balance).
=IF(SUM(D2:D5) > 0, "Build Project", "Do Not Build")Try the example
Use the original values or change them to test your understanding.