# Vickrey Auction (WS 2017 /2018)

## Contents

## Vickrey Auction Definition and Concept

Vickrey auction is a specific type of silent auction, where the goal is to attain the highest price possible for the seller by encouraging bidders to bid truthfully. The highest bid wins in vickrey auction, but the final price paid for item is equivalent to the price paid. Therefore, vickrey auction can be defined as a sealed-bid auction in which each bidder submits a bid just like sealed-bid auction. However, the second price auction is the price paid for the exchanged item.

This type of auction is named after William Vickrey who was a Canadian professor of economic and Nobel Laurent. He has first described this theory in a paper he researched and wrote in 1961, where he explained in details the benefits of the vickrey auction and he received a Nobel Prize for his work.

## History

The idea of vickery’s auction existed long before William Vickery’s research. A similar concept was used by stamp collectors and auctioneers in the 1870s when the first postage stamp known as the Penny Black appeared in England in 1840, and eventually, just like other valuable items, such as: paintings, silver, and ceramics, stamps were presented for auctions.

During the period of the 1870s, there existed standard auctions allowing absentee bidders to submit maximum bid amounts, and awarding goods to winning absentee bids at the price of one increment over the second highest bid. There also existed sales which accepted only sealed bids. The Vickrey auction combines these two ideas: holding a sealed-bid-only sale, and awarding the good at the price of one bid increment over the second-highest bid.

The earliest Vickery auction has been identified so far was in 1893, when two journalists from Northampton, Massachusetts announced their sales and was published in Golden Star newspaper in March 1893, and it reads as the following:

*“Catalogue of a Collection of U.S. and Foreign Stamps*

*To be sold WITHOUT RESERVE except where noted.*

*Bids will be received up to 4 P.M., May 15, 1893.*

*Bids are for the LOT, and, contrary to the usual custom in sales of this kind, we shall make this a genuine AUCTION sale; that is to say, each lot will be sold at an advance of from 1c to 10c above the second highest bidder. Address all bids to*

*Wainwright & Lewis, Northampton, Mass.”''*

The phrase “an advance of from 1c to 10c above the second highest bidder” indicates that the final price would be one minimum bid increment above the second-highest bid. They are indicating their recognition that “selling at one advance over the second highest bid” is the key feature of a standard English auction. This is the same insight as Vickrey’s.

## Properties and Execution

As it is mentioned previously, in this type of auction bidders are asked to submit sealed bids b1, ..., b(n). The bidder who submits the highest bid is awarded the object, and pays the amount of the second highest bid.

**Proposition:** In a second price auction, it is a weakly dominant strategy to bid one’s value, bi (si) = si.

**Proof:** Suppose i’s value is si, and she considers bidding bi > si.

Let ˆb denote the highest bid of the other bidders j ≠ i (from i’s perspective this is a random variable).

There are three possible outcomes from i’s perspective: (1) ˆb>bi, si; (ii) bi > ˆb>si; or (iii) bi, si > ˆb. In the event of the first or third outcome, i would have done equally well to bid si rather than bi > si. In (i) she won’t win regardless, and in (2) she will win, and will pay ˆb regardless. However, in case (2), i will win and pay more than her value if she bids ˆb, something that won’t happen if she bids si. Thus, i does better to bid si than bi > si. A similar argument shows that i also does better to bid si than to bid bi < si.

Since each bidder will bid their value, the seller’s revenue (the amount paid in equilibrium) will be equal to the second highest value. Let S^i:n denote the ith highest of n draws from distribution F (so S^i:n is a random variable with typical realization s^i:n). Then the seller’s expected revenue is E£ S^2:n¤.

The truthful equilibrium described in the proposition is the unique symmetric Bayesian Nash equilibrium of the second price auction. There are also asymmetric equilibria that involve players using weakly dominated strategies. One such equilibrium is for some player i to bid bi (si) = v and all the other players to bid bj (sj) = 0.

While Vickrey auctions are not used very often in practice, open ascending (or English) auctions are used frequently. One way to model such auctions is to assume that the price rises continuously from zero and players each can push a button to “drop out” of the bidding.

In an independent private values setting, the Nash equilibria of the English auction are the same as the Nash equilibria of the Vickrey auction. In particular, the unique symmetric equilibrium (or unique sequential equilibrium) of the English auction has each bidder drop out when the price reaches his value. In equilibrium, the auction ends when the bidder with the second-highest value drops out, so the winner pays an amount equal to the second highest value.

## Example

An easy and a simple example of Vickrey Auction is Google AdWords. eBay's system of proxy bidding is also similar to a Vickrey auction. For instance, there is an auction of 3 people: Joe bids $10, George bids $20, and Bill bids $30. In this case, Bill wins the bid but pays only $20.

## Benefits

Vickrey auction mechanism reduces the likelihood that a bidder will overpay for an item. Vickrey auction also increases the likelihood that the seller will get the most he can get for his item.

## Weaknesses

It is vulnerable to the bidder’s collusion. If all bidders in Vickrey auction get together, they can get an idea from each other what everyone thinks the true value of the auctioned item is. This could result in lowering their valuation. It is also susceptible to shill bidding. If there is one bidder with two accounts, he can use one account to bid high to secure the winning position and then use the other account to minimize the price he will pay for the auctioned item.

## Sources

Davidreileycom. (2018). Davidreileycom. Retrieved 24 January, 2018, from http://www.davidreiley.com/papers/oldVickreyHistory.pdf

Stanfordedu. (2018). Stanfordedu. Retrieved 24 January, 2018, from https://web.stanford.edu/~milgrom/publishedarticles/Lovely but Lonely Vickrey Auction-072404a.pdf

Psuedu. (2018). Psuedu. Retrieved 24 January, 2018, from http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.201.7899

Maxi-pediacom. (2018). Maxi-pediacom. Retrieved 24 January, 2018, from http://www.maxi-pedia.com/vickrey auction

Stanfordedu. (2018). Stanfordedu. Retrieved 24 January, 2018, from https://web.stanford.edu/~jdlevin/Econ 286/Auctions.pdf