Second-Price Auction (SPA)

In a second-price auction, too, bidders submit sealed bids. However, the bid is understood to be the maximum price that a bidder is willing to pay. The auctioneer again arranges the bids in a descending order of magnitude and awards the object to the bidder who submits the highest bid.

The price that the winning bidder pays in this case is equal to the second-highest (i.e., the highest losing) bid. This is where the second-price auction differs from the first-price auction. 

Now you may wonder if it makes sense to ask the winning bidder to pay the second highest bidder's bid as price instead of her own bid if the auctioneer cares about the revenue from the auction. Perhaps, in your mind you are probably thinking that a bidder bids the same way regardless of whether she is in a first-price or second-price auction. Think again, if you have a value of $10 for an object would you bid $10 in a FPA? You should not, because if you win your profit will be 0. It is not that bad under SPA, if you bid $10 and win, the second highest bid is almost surely less than $10, so you may not mind bidding $10 in SPA. In fact, it turns out that in our IPV environment you will bid $10.

That puts the auctioneer in a dilemma, in FPA each bidder submits a lower bid than she would under SPA, but in FPA the highest bidder's bid sets the price instead of the second highest bidder's bid. It is not at all obvious which is better for the seller if it is trying to get the highest price.

But first thing first. Let's verify that the bidders will indeed bid their values in a SPA so that we can calculate the price that the seller should expect in a SPA.

Bidding true value as dominant bidding strategy

Let’s say that you have a value of $1000 for a painting and you are wondering what bid to submit in a SPA. You could bid your true value $1000, or you could consider a lower or higher bid.

What do you gain/lose by bidding a lower value, say, $800 instead of $1000?

With such a bid you could encounter different situations, but all those situations can be described by the bid b from your highest rival bidder. For instance, there would be one situation where b is larger than $1000. In that situation, regardless of whether you bid $1000 or $800 you lose the item and get a zero payoff. So, bidding $800 does not make you better off.

It could also be that b < $800 (or that b = $800 and the tie is broken in your favor). In that situation, regardless of whether you bid $1000 or $800 you win the item and you get a payoff of $1000 – b. Again, bidding $800 does not make you better off.

It could be that $1000 > b > $800 (or that b = $800 and the tie is broken against you). In that situation, bidding $1000 would have given you $1000 – b, but bidding $800 would result in a zero payoff. So bidding $800 made you worse off.

This is the idea behind the statement that bidding true value is a dominant strategy in a SPA. If you bid something different that your true value then in every situation that arises you find yourself making either the same payoff or a lower payoff, never more!

Dominant Strategy Equilibrium and Nash Equilibrium

In the language of game theory dominant strategy equilibrium (where each bidder bids her dominant strategy) is considered a stronger equilibrium concept than Nash equilibrium. What this means is that an equilibrium where the bidders use dominant strategies is also a Nash equilibrium. The converse is not true, strategies may be in Nash equilibrium but that does not mean that the strategies are dominant strategies for all the players. Game theorists argue (with support from experimental studies) that dominant strategy equilibrium is a more “accurate” predictor of the outcome of the game. Unfortunately, most games (e.g., FPA) do not have equilibrium in dominant strategies. So we have to be happy with using Nash equilibrium to predict the outcomes.