Second-Price Sealed-Bid Auction

Vickrey · 1961 · doi:10.2307/2977633

In a second-price sealed-bid auction, the highest bidder wins but pays the second-highest bid. The dominant strategy is to bid your true value.

Why truthful bidding is dominant

If you bid above your value and win, you might pay more than the item is worth to you. If you bid below your value, you might lose an auction you would have profited from. Bidding exactly your value is always safe: you win exactly when the price is below your value. This is the foundation of one-shot bidding: when you only get one chance, truthful bidding is the dominant strategy.

Scheme

; Second-price sealed-bid auction
; Winner = highest bidder, pays second-highest bid

(define (second-price-auction bids)
  (let* ((sorted (sort bids >))
         (winner-bid (car sorted))
         (second-bid (cadr sorted)))
    (list winner-bid second-bid)))

; Three bidders with true values 80, 50, 30
; Truthful bidding: each bids their value
(define bids '(80 50 30))
(define result (second-price-auction bids))

(display "Bids: ") (display bids) (newline)
(display "Winner bid: ") (display (car result)) (newline)
(display "Payment (2nd price): ") (display (cadr result)) (newline)
(display "Surplus: ") (display (- (car result) (cadr result))) (newline)

; What if the winner overbids to 100?
; Still pays 50. No advantage.
(display "\nOverbid to 100: ")
(display (second-price-auction '(100 50 30))) (newline)

; What if the winner underbids to 45?
; Loses to bidder at 50. Lost surplus.
(display "Underbid to 45: ")
(display (second-price-auction '(45 50 30)))
; Now the $50 bidder wins and pays $45

; Second-price sealed-bid auction
; Winner = highest bidder, pays second-highest bid

(define (second-price-auction bids)
  (let* ((sorted (sort bids >))
         (winner-bid (car sorted))
         (second-bid (cadr sorted)))
    (list winner-bid second-bid)))

; Three bidders with true values 80, 50, 30
; Truthful bidding: each bids their value
(define bids '(80 50 30))
(define result (second-price-auction bids))

(display "Bids: ") (display bids) (newline)
(display "Winner bid: ") (display (car result)) (newline)
(display "Payment (2nd price): ") (display (cadr result)) (newline)
(display "Surplus: ") (display (- (car result) (cadr result))) (newline)

; What if the winner overbids to 100?
; Still pays 50. No advantage.
(display "\nOverbid to 100: ")
(display (second-price-auction '(100 50 30))) (newline)

; What if the winner underbids to 45?
; Loses to bidder at 50. Lost surplus.
(display "Underbid to 45: ")
(display (second-price-auction '(45 50 30)))
; Now the $50 bidder wins and pays $45

The dominance argument

Truthful bidding weakly dominates every other strategy. No matter what others bid, you never do worse by bidding your true value, and sometimes you do strictly better. This is Vickrey's central result.

Scheme

; Dominance check: is truthful bidding always at least as good?
; Compare payoff of bidding v (truthful) vs bidding b (deviation)

(define (payoff value bid other-bids)
  (let ((max-other (apply max other-bids)))
    (if (> bid max-other)
        (- value max-other)  ; win, pay second price
        0)))                 ; lose, pay nothing

(define my-value 80)
(define others '(50 30))

(display "Truthful (bid 80): ")
(display (payoff my-value 80 others)) (newline)

(display "Overbid  (bid 100): ")
(display (payoff my-value 100 others)) (newline)

(display "Underbid (bid 45):  ")
(display (payoff my-value 45 others)) (newline)

; Truthful never worse. Underbid sometimes worse (loses when it should win).
; Against others = (75):
(display "\nAgainst higher competitor (75):")
(newline)
(display "Truthful (bid 80): ")
(display (payoff my-value 80 '(75))) (newline)
(display "Underbid (bid 60): ")
(display (payoff my-value 60 '(75)))
; Underbid loses, truthful wins with surplus 5

Notation reference

Paper	Scheme	Meaning
v_i	my-value	Bidder i's true value
b_i = v_i	(payoff value value others)	Truthful bidding
p = max(b_j, j != i)	(apply max other-bids)	Payment = second-highest bid

Neighbors

💰 Economics — econ-24: market mechanisms
🎲 Game Theory — nordstrom-06: dominant strategies
💎 VCG 1973 — generalizes second-price to multiple items
🎲 Game Theory Ch.1 — dominant strategies in the game-theoretic frame
💎 VCG Mechanism — the multi-item generalization
🎰 Probability Ch.6 — expected revenue requires integrating over bidder distributions

Ready for the real thing? Read the paper.

by june.kim VCG 1973 · 2 of 8 →