∿ExploreSims
ExploreSims / Economics / Game Theory / Auction Theory, First Price vs Second Price
Premium experiment • bidding rules and winner surplus

ECONOMICS • GAME THEORY

Auction Theory, First Price vs Second Price

Reveal bidder values, compare shaded first-price bids with truthful second-price bids, and inspect how payment rules change winner surplus.

● AUCTION MECHANISMprivate values → bids → payment
First-price winner
First-price payment
Second-price payment
Second-price winner surplus

Reasoning inspector

live model state
First-price winner—
First-price payment—
First-price winner surplus—
Second-price winner—
Second-price payment—
Second-price winner surplus—
Bid shading—

Experiment with the mechanism

Auction rules change strategic incentives. In this classroom comparison, first-price bidders shade their values by a chosen fraction and the winner pays the winning bid, while the second-price auction uses truthful value bids and charges the winner the second-highest bid.

LEARN WITH THE SIMULATION

Understand Auction Theory, First Price vs Second Price by experimenting.

START HERE

Auctions turn private values into bids and payments according to a mechanism. This simulation compares a stylized first-price auction, where each bidder shades the bid below value by the same chosen fraction, with a second-price auction, where bidders are represented as bidding their values and the highest-value bidder pays the second-highest bid.

BUILD THE IDEA

What you should understand first

  • Distinguish a bidder's private value from the submitted bid and the final payment.
  • Compare winner payment and winner surplus under first-price and second-price rules.
  • Explain why second-price incentives differ from the strategic bid shading shown for first-price auctions.

TRY IT YOURSELF

How does stronger first-price bid shading change the winner's payment and surplus?

  1. Record the bidder values and first-price outcome.
  2. Increase the common bid-shading fraction.
  3. Replay the bidder and payment sequence.
  4. Compare the first-price payment and winner surplus with the second-price result.
Read the visual:

With common proportional shading, the highest-value bidder usually remains the highest bidder, but the first-price payment falls and winner surplus rises as shading increases. The second-price payment remains tied to the second-highest value in this simplified comparison.

WHY IT WORKS 01

Why first-price bidders care about the payment directly

In a first-price auction, increasing a bid can improve the chance of winning but also raises the amount paid if the bidder wins. That creates a strategic tradeoff between probability of winning and winner surplus. The common shading control simplifies this reasoning by moving all bids below values proportionally, rather than attempting to solve a full Bayesian bidding equilibrium.

WHY IT WORKS 02

Why the second price changes the incentive problem

In the standard independent private-value second-price auction, the winner's payment is determined by another bidder's bid rather than the winner's own bid. Bidding below value can lose profitable wins, while bidding above value can create wins at a price above the bidder's valuation. Under the standard assumptions this makes truthful bidding a dominant strategy, a very different incentive structure from first-price bidding.

DO NOT MIX THESE UP

Common misconception

The common shading fraction is not a general equilibrium bidding formula. Optimal first-price bids depend on beliefs about rivals and the value distribution. Second-price truthful bidding is a dominant-strategy result under standard private-value assumptions, which may fail when values are interdependent or auction rules differ.

PRACTICE CHALLENGE

Can the same bidder win both auction formats but earn different surplus?

  • Use values with one clear highest bidder.
  • Replay both auction outcomes.
  • Change the first-price shading fraction without changing values.
  • Compare the winner identity, first-price payment, second-price payment and winner surplus.
Reveal the reasoning

Yes. With a clear highest value and common proportional shading, the same bidder can remain the winner under both mechanisms. The payment rules differ, so first-price winner surplus depends on the shaded winning bid while second-price surplus depends on the gap between the highest and second-highest values.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • Five bidders have independent private values entered directly by the user.
  • First-price bids equal value multiplied by one minus a common shading fraction.
  • The second-price comparison uses truthful value bids and charges the winner the second-highest value.

Model limitations

  • Optimal first-price shading depends on beliefs and value distributions, so one common shading parameter is only illustrative.
  • Common values, reserve prices, entry, risk aversion, ties and multi-unit auctions are omitted.

CONNECT IT

Where this matters

  • Auction design
  • Game theory
  • Procurement and online marketplaces
  • Strategic bidding

WORDS THAT MATTER

Key vocabulary

Private value
A bidder's own value for winning the item.
Bid shading
Submitting a first-price bid below private value.
First-price auction
An auction where the highest bidder wins and pays the winning bid.
Second-price auction
An auction where the highest bidder wins but pays the second-highest bid under the simplified rule used here.