SimuVeda

Operations · Simulator 079

Auction Mechanisms

Several bidders want one item and each has a private value for it. Set those values and the bids, then compare four auction rules: who wins, what they pay, and how much they gain. Plot bidder 1's payoff for every possible bid, and run thousands of random auctions to see why very different rules raise the same money on average.

How the model works

First-price sealed bid: the highest bid wins and pays its own bid. Bidding your full value guarantees zero gain, so bidders shade their bids below value.

Second-price sealed bid (Vickrey): the highest bid wins but pays the second-highest bid. Your bid only decides whether you win, not what you pay, so bidding exactly your value is a dominant strategy: it is never worse than any other bid.

English (ascending): the price rises until only one bidder is left. Each bidder stays in until the price reaches their value, so the winner pays (about) the second-highest value, like Vickrey.

Dutch (descending): the price falls until someone accepts. You must decide in advance at what price to jump in, which is the same choice as a first-price sealed bid.

With values drawn independently and uniformly from 0 to 100 and n bidders, the symmetric first-price equilibrium bid is b = v (n − 1) / n. Then both sealed-bid rules give the seller the same expected revenue:

E[revenue] = 100 × (n − 1) / (n + 1)

This is the revenue equivalence theorem (Vickrey 1961; Myerson 1981).

One item, independent private values, risk-neutral bidders, no reserve price and no collusion. Ties go to the lower-numbered bidder. The English auction is shown with a continuous price, so its winner pays exactly the second-highest value.

Guided experiments

  1. Everyone bids their value. Values 80, 65 and 50; all bids equal values. Prediction: bidder 1 wins every format. First-price and Dutch: pays 80 and gains nothing. Second-price and English: pays 65 and gains 15.
  2. Bidder 1's best bid. Same values; rivals bid their values. Look at the payoff plot. Prediction: under second-price, every bid above 65 pays 15 and every bid below loses, so bidding 80 is as good as anything. Under first-price, payoff peaks just above 65 (about 15) and falls to zero at 80.
  3. Rivals shade their bids. Rivals bid 66% of their value (about 43 and 33). Prediction: under first-price, bidder 1's best bid drops to just above 43, for a gain of about 37. Under second-price, any bid above 43, including the truthful 80, wins and pays 43. English still ends at 65: in an open auction rivals stay in until their full value./span>
  4. Revenue equivalence. Many random auctions with 3 bidders, then 5. Prediction: average revenue settles at 50 under both rules for 3 bidders, and about 66.7 for 5, although single auctions differ a lot.

Challenge

In a second-price auction your value is 70, and the highest rival bid (unknown to you) turns out to be 75. Compare bidding 70 with bidding 80. Then suppose the rival's bid was 60 instead.

Reveal the explanation

Rival at 75: bidding 70 loses (gain 0); bidding 80 wins but pays 75 for something worth 70, a loss of 5. Rival at 60: both bids win and pay 60, a gain of 10 either way. Overbidding can only add losses and never adds gains, and underbidding can only lose profitable wins, so bidding your value is best whatever the rivals do.

References

  • Vickrey, W. (1961), "Counterspeculation, auctions, and competitive sealed tenders", Journal of Finance 16(1): 8–37.
  • Myerson, R. B. (1981), "Optimal auction design", Mathematics of Operations Research 6(1): 58–73.
  • Krishna, V., Auction Theory, 2nd ed. (2010), ch. 2–3.