Auction Theory for Markets

Auctions decide who gets a thing and what they pay, and the exact rules quietly change how everyone should behave. Two auctions can hand the item to the same person yet call for completely different bids. Traders care because this isn't an academic sidebar: an order book is literally a running auction, and the interview version — "how would you bid here?" — tests whether you can reason about strategy and selection at once.

Start with the four classic formats and what each one asks of a rational bidder:

Format	How it works	How to bid
English (open, ascending)	price rises, you drop out when it passes your value	stay in until the price hits your value
Dutch (open, descending)	price falls until someone accepts	strategically like first-price — shade below value
First-price sealed	highest sealed bid wins, pays their own bid	shade below your value
Second-price (Vickrey)	highest sealed bid wins, pays the second-highest bid	bid your true value — honesty is optimal

The surprising entry is the last one, so let's see why.

Worked example: why you bid truthfully in a second-price auction

You value an item at $100. In a second-price (Vickrey) sealed-bid auction, if you win you pay not your own bid but the runner-up's. Should you bid your true $100, more, or less?

Say the highest competing bid turns out to be $85. Bid your honest $100: you win and pay $85, a $15 profit. Now check the temptations. Bidding higher, say $110, changes nothing when you'd have won anyway (you still pay the $85 runner-up) but exposes you to paying above $100 if someone bids $105 — you'd "win" a $5 loss. Bidding lower, say $80, only risks losing an auction you'd have profited from (if the runner-up were $82, you'd forfeit $18 of profit) while never lowering the $85 you'd have paid anyway. Every deviation is weakly worse. So bidding your true value is a dominant strategy — the price you pay is decoupled from your bid, which removes any reason to lie.

First-price: shading below value

Change the rule to first-price — you pay your own bid — and truthfulness becomes foolish: bid your full $100 and you win with exactly zero profit. You must shade. The trade-off is clean: bid closer to your value and you win more often but earn thin margins; bid lower and you profit more when you win but win less. With $n$ bidders whose values are spread evenly up to yours, the optimal bid is

$$b = \frac{n - 1}{n}\,v,$$

where $v$ is your value and $n$ the number of bidders. In words: keep the fraction $(n-1)/n$ of your value and shade away the rest. With five bidders and a $100 value, you'd bid $\tfrac{4}{5}\times 100 = 80$. More competitors (bigger $n$) means less room to shade — the fraction creeps toward 1.

A famous result, revenue equivalence, says that under standard assumptions these two formats earn the seller the same expected revenue: the discount buyers enjoy in a second-price auction is exactly cancelled by the shading they do in a first-price one. Different bids, same average outcome.

Common value and the winner's curse

Everything above assumes private values — the item is worth a different, known amount to each bidder. When instead there's one common value nobody knows exactly (a company's true worth, a jar of coins), a new danger appears: the winner's curse. Winning means you were the most optimistic estimator, which means you probably overpaid. In common-value auctions you must shade further than private-value logic alone suggests, and shade more the more rivals you face.

The order book is a continuous double auction

Here's why this belongs in a trading course. A limit order book is a double auction running continuously: buyers post bids, sellers post asks, and a trade happens whenever the best bid meets the best ask. It's "double" because both sides bid at once, and "continuous" because it never closes — orders arrive and match in real time.

Placing a limit order is submitting a bid or ask into this auction; sending a market order (Market vs. Limit Orders) is accepting the best price the other side is currently offering. The market maker's quoting problem is just deciding where to place your bid and ask in this never-ending auction.

In interviews, if you're handed an auction, first name the format: second-price means bid truthfully, first-price means shade by roughly $(n-1)/n$, and any common-value twist means shade further for the winner's curse. Tie it back to markets — the order book is a live double auction — and you've shown you see the game underneath the mechanics.