Second Price Auction Dynamics

Introduction

Do you guys know, every-time you see an Advertisement in your Facebook, Instagram, snapchat, google search or for that matter any website is the result of a Second Price Auction. A Second Price auction is also known as Vickrey Auction as it was first described academically by Columbia University professor William Vickrey in 1961.

First Price vs Second Price Auction

• First-Price Auction – Digital buying model where if your bid wins, you pay exactly what you bid. This maximizes revenue potential for the seller. So if there are 3 bidders A($3), B($5), C($4), B will win the auction and will need to pay $5 which is also called as the Closing Price of Auction.
• Second-Price Auction – Digital buying model where if your bid wins, you pay $0.01 above the second highest bid in the auction. In this type of auction, it is in your best interest to bid the highest amount you are willing to pay, knowing that often you will end up paying less than that amount. So if there are 3 bidders A($3), B($5), C($4), B will win the auction and will need to pay $4.01 as the Closing Price.

By the first look, it looks very strange as why would any seller conduct Second Price auction and leave some money on the table. It is true when you are conducting a single auction and not dealing with the bidders in a sequence of auctions. The auction dynamics changes completely in a RTB (Real Time Bidding) world, where thousands of auctions are conducted with the same set of bidders every second. Now let us revisit the example of 3 bidders A($3), B($5), C($4), here B will win the auction and will need to pay $5 in the first auction and then he might get a buyer's remorse "Did I just pay too much ?". So in the second auction he will bid $4.5 and still win, next try he bids $3.9 and alas, he loses but that doesn't harm him too much as the scale we are talking is of millions of auction. In time with multiple tries he will find the sweet spot of $4.01 which is theoretically a Second Price auction. So what we can see in here is that, in a First price Auction some smart bidder can always outsmart other bidders and the auction ceases to be fair anymore. While on the other hand Second Price auction maintains the fairity of the Auctions by compelling each bidder to reveal and bid their “Private Value” for the item, which in theory is the maximum they would want to pay for it. Private value is the value that this item might bring to the bidder, which is not publicly known. In the advertising space, it might be the value in incremental sales the ad would generate or the brand awareness value they would gain by showing the Ad.

In a second price auction, each bidder’s best strategy is to bid their individual private value for an item, which is ultimately what every seller wants. Let’s use some basic Game Theory to understand why. Before we start, just a few definitions on notation –

• B - is the Bid that an individual makes
• V - is their Private Value
• U - is defined as the individual’s Utility of winning (V - Closing Price)

Now let us discuss discuss why bidding their Private Value is the best strategy in a Second Price auction. We will try to prove this by contradiction i.e. by proving why does not it makes sense for bidders to either bid below or more than the private value.

Case 1: Bidding more than Private Value (B > V)

Withing Case 1, there are three possibilities with regards to B' (Bid from a competitor)

1. B’ < V < B : You win & pay B’ with Positive Utility. But the results would have been same for B=V
2. V < B’ < B : You win & pay B’ with Negative Utility. Had B=V you would have lost which is better than having negative utility.
3. V < B < B’ : You loose but the results would have been the same for B=V . Negative utility avoided in both cases.

Case 2: Bidding less than Private Value (B < V)

Withing Case 2, there are three possibilities with regards to B' (Bid from a competitor)

1. B’ < B < V : You win & pay B’ with Positive Utility. But the results would have been same for B=V
2. B < B’ < V : You loose. Had B=V you would have won & payed B’ with Positive Utility
3. B < V < B’ : You loose but the results would have been the same for B=V, Negative utility avoided in both cases

If we collate the information above we see bidding more than Private Value is not a good idea and nor is bidding below. So the only choice the bidders are left is to bid on the Private Value. Now comes an edge case "What happens if there is only one bidder ?" The seller solves this issue by introducing and communicating a floor with buyer. A floor is the lowest allowed bid, any bid lower than the floor will be rejected and in cases where only one buyer bids floor is treated as the second bid.