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Auction Theory

Posted: Monday, 18 August, 2014. | By: Equipoise Bot

Auction theory is exceptionally relevant today, as it sets the theoretical backbone of auctions with major financial implications like allocation of licenses, or privatization of public sector companies etc.  It analyses various auctioning methods on the basis of their efficiency, the revenue they generate and the equilibrium bidding strategies.
This concept note contrasts the two most commonly used auction designs. Before that, it is essential to know what a sealed bid auction is: the bidder’s identity is kept anonymous, and no one knows what others are bidding. This is important in order to prevent gaming.
First Price Sealed bid Auction: The highest bidder takes the underlying asset and pays the price (s)he quoted
Second Price Sealed bid Auction: The highest bidder takes the underlying asset, and pays the price the second highest bidder quoted
The more commonly used auction is the first price sealed bid auction, but it is inefficient as it does not incentivise the bidder to reveal his true value. Let us investigate this highly loaded statement through a game.
Let us assume that a pen is being auctioned. X values the pen at Rs. 100.

Assumptions:

  • No two bidders bid the same amount.
  • The aim of the auction is to maximise one’s utility (which is defined as the value derived from the pen minus the price one pays to procure it).

Bidder reactions under different auction designs:
First Price Sealed bid Auction:

  • Case 1: If X bids more than 100, then if (s)he wins, (s)he would gain negative utility from winning the auction, as the cost would exceed the return.
  • Case 2: If (s)he bids 100, then (s)he would face a no-profit-no-loss situation on victory.
  • Case 3: If one bids less than Rs. 100, then if one wins, (s)he will get a positive utility. Else, one will get 0 utility (which was also the case had (s)he bid Rs. 100). Thus, the strictly dominant strategy is to shade one’s bid by a small amount (loosely saying, bid Rs. 99). This will imply a return of Re. 1 on victory, or 0 return on loss.

Thus, one does not bid one’s true valuation in case of first price sealed bid auction. One chooses to shade ones bid (i.e. bid 99 instead of 100).

Second Price Sealed bid auction: Recall that in this case, the highest bidder takes the asset at the second highest price.

  • Case 1: If X bids more than Rs. 100, (s)he runs the risk of having a second highest bidder also bidding for more than 100. This would imply that X faces negative utility, as (s)he values the pen at 100.
  • Case 2: If X bids less than 100, then he would lose on to the opportunity of enjoying positive utility in case the highest bidder bids Rs. 100. Thus this is not a good strategy.
  • Case 3: If X bids Rs. 100, then if (s)he wins, X pays a price less than 100 (as no two bidders bid the same amount). This implies that X’s valuation of the pen exceeds his cost, and he enjoys positive utility. Thus, it is his strictly dominant strategy to bid his true valuation (which is 100).

As it can be seen from the above example, it is in the interest of a bidder to reveal his true valuation in case of a second price sealed bid auction visa vis a first price sealed bid auction.
 
On the sideWinners’ Curse
This occurs when the valuation of the underlying asset is very discretionary, as there is heavy asymmetric information (which is usually the case). In such a situation, the winner, who is the highest bidder, always feels that he may have overvalued the asset compared to everyone else.

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