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An auction is a game with a prize and a rule for who pays what. Change the rule and rational bidders change how they bid — sometimes honestly, sometimes shading below their value. Auction theory turns those rules into predictions, and it often brings students to a microeconomics tutor in Los Angeles. This page covers the four standard formats, why one makes honesty optimal, and how far to shade in another.

1 · The four standard formats

Four designs cover almost everything you will meet. Two are open — bidders watch each other — and two are sealed, each bid private.

• English (ascending). The price rises, bidders drop out, and the last one left wins, paying about where the second-last quit.
• Dutch (descending). The price starts high and falls; the first to accept wins and pays that price.
• First-price sealed-bid. One secret bid each; the highest wins and pays its own bid.
• Second-price sealed-bid (Vickrey). One secret bid each; the highest wins but pays the second-highest bid.

Two are secretly the same game. A Dutch auction reveals nothing as the price falls, so your only choice is the price at which to jump in — a sealed bid by another name. Dutch and first-price sealed-bid are thus strategically equivalent. And under private values the English auction tracks the second-price: stay in until the price reaches your value, so the top-value bidder wins and pays about the second-highest value.

2 · Why the second-price rule rewards the truth

Here is the surprise that made the Vickrey auction famous. In a second-price auction, bidding exactly your own value is a dominant strategy — best whatever anyone else does. (A dominant strategy beats your alternatives regardless of rivals’ choices; page 25 builds that idea in full.)

The intuition is one sentence: your bid decides only whether you win, while the price you pay is fixed by the highest rival bid. Bid above your value and you only risk overpaying for the prize; bid below it and you only risk losing a bargain. The worked example proves it case by case.

3 · Bid shading in the first-price auction

The first-price auction is different: you pay your own bid, so bidding your true value v nets zero even when you win. You must shade — bid below v — trading a lower win chance for a positive margin.

By how much? Take n bidders, values drawn independently and uniformly on [0, V], all using the same increasing rule. You win only when all n − 1 rivals bid less, and with uniform values that chance climbs with your bid b. Writing expected surplus as (v − b) times the win probability and maximising over b gives a clean symmetric equilibrium:

   

You bid the fraction (n − 1)/n of your value and keep v/n as shading. It moves with competition: two bidders and you bid half your value; five and you bid four-fifths; as n grows the shading v/n shrinks toward nothing. More rivals, less room to lowball. The diagram plots these lines.

4 · Revenue equivalence, the winner’s curse, and design

Revenue equivalence. The first-price winner shades but has the highest value; the second-price winner is honest but pays only the runner-up. These pull revenue opposite ways — and they cancel. Under private values, risk-neutral bidders and a symmetric equilibrium, all four formats hand the seller the same expected revenue. For n uniform bidders on [0, V] it is

   

the expected second-highest value — exact, but leaning on those assumptions.

The winner’s curse. With common values — an oil tract or a spectrum licence, worth the same to all but estimated with noise — winning is bad news: you win because you were the most optimistic, so your estimate was probably too high. Shade extra, in advance, for that.

Mechanism design. The auction is itself a game you design: choose the rule — who wins, who pays — and you choose the behaviour it induces. The Vickrey rule is built so truth-telling is dominant, and revenue equivalence says the format rarely changes takings — so design turns on other goals: simplicity, honesty, resistance to collusion.

Worked example — the second-price auction rewards the truth

You bid in a sealed-bid second-price auction for a rare first-edition novel. Your value is v = 80. Let m be the highest rival bid — the price you pay if you win.

Step 1 — The rule. You win if your bid is highest, and then pay m, the second-highest bid overall. Your own bid never enters the price.

Step 2 — So your bid does one job. It decides whether you clear m and win. Nothing else.

Step 3 — Bid your value, b = 80. Suppose m = 60. You clear 60, win, and pay 60: surplus 80 − 60 = 20. Honest and profitable.

Step 4 — Overbid, b = 100. This changes nothing unless a rival bid lands between your value and your bid. Say m = 90. Truthful, you lose — rightly, the prize is worth 80 and the price is 90. Bid 100 and you win, paying 90: surplus 80 − 90 = −10, a loss.

Step 5 — Underbid, b = 50. This matters only when m lands between your bid and your value. Say m = 65. Truthful, you win and net 80 − 65 = 15; shaded to 50 you lose, netting 0 — a profitable win thrown away.

Step 6 — Resolve. Against every possible m, bidding 80 is never worse than another bid and sometimes strictly better — the definition of a weakly dominant strategy.

Step 7 — Interpret. Your bid moves the win-or-lose line, never the price, so there is nothing to gain by shading it up or down. The second-price rule is designed to make honesty optimal — mechanism design in miniature.

Could you prove that bidding your value is weakly dominant — case by case, against any rival bid? That argument, plus the first-price shading rule b = (n − 1)v/n, is exactly what auction questions reward. A one-on-one microeconomics tutor works the four formats and the two equivalences with you until the logic is second nature. Book a trial session.

Practice

Q1. A first-price sealed-bid auction has n = 4 bidders, values uniform on [0, 100]. Your value is 90. What is your equilibrium bid, and how much do you shade?

Q2. Five bidders have values uniform on [0, 120]. What expected revenue does the seller earn, and does switching from a first-price to a second-price auction change it?

Q3. In a second-price auction your value is 50. Find your surplus if (a) the top rival bid is 40 and you bid your value; (b) the top rival bid is 40 and you bid 45; (c) the top rival bid is 55 and you bid your value.

Answers. Q1: bid = (n − 1)v/n = (3/4)(90) = 67.5; shading v/n = 90/4 = 22.5. Q2: revenue = (n − 1)V/(n + 1) = (4)(120)/6 = 80, the same under both formats — revenue equivalence. Q3: (a) win, pay 40, surplus 10; (b) still win, still pay 40, surplus 10 — the bid does not move the price; (c) you lose, surplus 0 — rightly, since 55 exceeds your value 50.

Key takeaways

• Four formats, two pairs. Dutch and first-price sealed-bid are strategically identical; the English auction mirrors the second-price under private values.
• Second-price bidding is honest by design. Your bid sets only whether you win, not what you pay, so bidding your value v weakly dominates every alternative.
• First-price bidding is shaded: b(v) = (n − 1)v/n, keeping v/n, which shrinks as bidders n rise.
• Revenue equivalence. Under private values and risk-neutral bidders, all four formats raise the same expected revenue — the expected second-highest value.
• Winning can be bad news. With common values the winner most overestimated, so shade extra for the winner’s curse.

Why Los Angeles students choose our microeconomics tutoring

• Derivations you can reproduce: sessions rebuild the first-price bid function and second-price dominance from scratch, so you can generate them under exam pressure, not memorise them.
• The distinctions examiners reward: open versus sealed, first- versus second-price, private versus common values, strategic versus revenue equivalence.
• One-on-one and matched to your course: a tutor works from your own notation and past papers, whether your module follows Varian, Osborne, or its own problem sets.

FAQ

Q: What is the difference between a first-price and a second-price auction?
A: In both, the highest sealed bid wins. The first-price winner pays their own bid; the second-price (Vickrey) winner pays the second-highest bid.

Q: Why should I bid my true value in a second-price auction?
A: Your bid decides only whether you win, never the price — that is the highest rival bid. Bidding your value is weakly dominant: never worse, sometimes better, whatever others do.

Q: How much should I shade in a first-price auction?
A: With n bidders and values uniform on [0, V], the equilibrium bid is (n − 1)/n of your value. You shave off v/n, which shrinks as more bidders join.

Q: What is the winner’s curse?
A: In a common-value auction the winner is usually whoever most overestimated the item’s worth, so they overpay. The remedy is to shade down before you win.

Q: Does the auction format change how much the seller earns?
A: Often not. Revenue equivalence: under private values and risk-neutral bidders, all four standard formats yield the same expected revenue.

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Auction theory rewards a clear method: name the format, ask who pays what, then derive the bid. One-on-one sessions build that fluency on the exact problems your course sets — Vickrey truth-telling, first-price shading, revenue equivalence, the winner’s curse. Tell us your university and exam date, and we will match you with the right tutor this week.