Private-Value, Sealed-Bid Auction

Game Description

Each bidder has a randomly determined private value for one unit of a good. Each submits a bid without knowing the bids of others. The auction ends when all players have submitted bids. The one or more units go the highest bidders, who pay either their own bids or the highest non-winning bid.

Learning Objective 1: Private-Value Sealed-Bid Auction

Demonstrates the workings of a private-value sealed-bid auction.

Learning Objective 2: Equilibrium Bidding Strategies

The equilibrium bidding strategy depends on the pricing rule. When winners pay the highest non-winning bid, the weakly dominant strategy is for each player to bid her value. When winners pay their bid, a bidder has an incentive to shade her bid.

Learning Objective 3: Revenue Equivalence

The second price private-value sealed-bid auction is expected to generate the same seller revenue as a first price private-value sealed-bid auction.

Brief Instructions

By default, there is one item on auction each round (Units=1). Each bidder values one unit, with her private value drawn from uniform distribution with the default endpoints Lowest Bidder Value=$10 and Highest Bidder Value=$100. Each bidder draws a new value each period if Periods > 1.

Equilibrium bidding strategy depends on the pricing rule. With Price Rule=First Price, the winning bidder pays her bid. Assuming bidders are risk-neutral (and the uniform distribution is common knowledge), the Bayesian Nash Equilibrium Bid Function is [(N-1)/N]*Buyer Value, where N=Group Size. With default parameters, the Bayesian Nash Equilibrium Bid Function is 5/6*Buyer Value.

With Price Rule=Second Price, the winning bidder pays the second highest bid. Since each player’s bid only affects whether she wins or loses (but not how much she pays if she wins), each player has a weakly dominant strategy to bid her value.

Key Treatment Variations

Multiple Units

For a multi-unit auction, set Units > 1 (making sure that Units < Group Size). Your options for Price Rule will change. When set to Uniform, each winning bidder pays the highest non-winning bid. For example, if Units=2, then the two winners each pay the third highest bid. When set to Discriminatory, each winning bidder pays her bid.

Reserve Price

Setting a Reserve Price sets the minimum price the seller must receive in order to part with an item. For the second-price (or uniform price, if multi-unit) auction, if the highest non-winning bid is less than the reserve price, then those bidding at or above the reserve price pay the reserve price.


The results highlight the Nash Equilibrium and show how close your students were to equilibrium play. Use the Go To: menu to switch between periods.

Figure 1: Outcome Summary Table

For each round, there is a table summarizing each group’s performance (Figure 1). With Units = X, the first columns display the sum of the X-highest item values (Optimal Surplus) and the sum of the winners' item values (Surplus).1 The final column shows seller Revenue (highest non-winning price × X). Check the radio button to reveal a figure summarizing a group’s bidding.

Figure 2: Participant Bids and Values

By unchecking the first two check-boxes for the Nash Equilibrium Bid Function and Best Fit Line, only markers that indicate the sealed bid and bidder value for each participant remain (Figure 2).

Figure 3: Nash Equilibrium Bid Function and Best Fit Line

To compare participant choices with the Nash equilibrium, use the Nash Equilibrium Bid Function line by clicking the box adjacent to its legend entry. With the Best Fit Line box checked, a linear trend line is fitted to participant bids and values (Figure 3). This line will coincide with the Bid Function if all students played equilibrium strategies.

Robot Play

Our robot (i.e., an automated player) strategy is the following:

First Price

The robot will submit a bid equal to their value multiplied by \(\frac{N-1}{N}\) where \(N\) is the number of bidders in the auction.

Second Price

Robot submits a bid equal to their value.

1. Surplus calculations assume the items have no value to the seller.
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