Vickrey Auctions

Vickrey auctions were so named after Canadian economist William Vickrey who won the Nobel Prize in Economics in 1996 for his ground-breaking work on the economic theory of incentives under asymmetric information. These auctions also go by the name of Second-Price sealed bid auctions.

Let’s start by understanding what a sealed-bid auction is. Suppose there is a prized object under auction. The object has n suitors, each one has a certain perceived value of the object in mind. All of them are required to submit their bids in sealed envelopes to the auctioneer who then decides who gets the prize. The bids are not revealed at any point during the course of the auction and hence the name.

The auctioneer is charged with the duty of deciding who is the rightful owner of the prize. Obviously he wants the prize to go to the suitor who values it the most. He has no clue who that person is. But can he design the auction such that the suitors are forced to reveal their values to him? Turns out, he can.

Before the start of the auction, the auctioneer declares that the person with the highest bid would win the prize, but he would have to pay only as much as the second-highest bid. This is the essence of Vickrey’s much vaulted Second-Price auction. But what is so special about the second price? Let’s inspect it mathematically.

Let the n suitors have perceived values v_1, v_2, …..v_n. Their respective bids are b_1, b_2,……b_n. Without loss of generality, let us assume that

v_1 > v_2 > v_3………> v_n

The payoffs are defined in the following way :

p_i = v_i — b, if b_i is the winning bid and b is the second-highest bid

p_i = 0, if b_i is not the winning bid

In case of a tie where two suitors i and j submit the same bid, i wins the prize if v_i > v_j and vice-versa.

Each suitor has 3 choices, either he can bid equal to his value or more or less than it. We will now investigate how the 3 different strategies would fare against each other.

Table 1: Payoff matrix

From the above table, we observe that the strategy b_i = v_i fares at least as good as the other two strategies with at least one case each where it performs strictly better. Hence, the strategy of bidding equal to the perceived value is a weakly dominant strategy for all suitors. Since all players are assumed to be rational, they will always go for this strategy which means that the bids reveal their perceived values of the prize. And the auctioneer’s job is straightforward. Uncanny!

The bidding profile (v_1, v_2,……v_n) is called a weakly dominant Nash equilibrium because at this profile, all players act on their weakly dominant strategies. Similarly, a strongly dominant NE would be composed of all strongly dominant player strategies. It is obvious why a strongly dominant and a weakly dominant NE cannot co-exist in the same game. However, note that a strongly dominant NE is unique while a game with a weakly dominant NE can have other NEs as well. Let’s see if we can find some more NEs for the Vickrey auction.

On examination, we easily find 2 other Nash equilibria. Actually there are many.

So far, in all the Nash equilibria we have found, the player with the highest valuation always ends up winning the prize. But, is it possible to construct a NE where somebody else wins it? Actually, it is possible.

Consider the following bidding profile (v_2, v_1,……..v_n). So player 1 bids player 2’s valuation and vice-versa. Here, player 2 wins the prize. Let’s check why this is a NE. Player 1 currently has a payoff of zero as his bid is not the highest. With an even lower bid, his payoff remains at zero. With a higher bid, either he loses the object and gets zero payoff (v_2 < b_1 < v_1) or he wins the object and still gets zero payoff (b_1 > v_1). Thus player 1 has no incentive to deviate. For player 2, his current payoff is zero(second highest bid is equal to his own valuation). If he goes higher, he still gets a payoff of zero. Going lower than v_2, he loses the object to player 1 and gets zero payoff. Going lower than v_1 but higher than v_2 is not better off either because the second-highest bid does not change and so does the payoff. Hence, this profile is a Nash equilibrium.

For n-player games, often we have to resort to visual inspection to identify Nash equilibria. It might seem daunting initially, but once you get a hang of it, it is kinda fun.

Note : This article is influenced by the coursework I completed at IITD. I was lucky to be taught by one hell of a teacher and I felt that I should pass on the knowledge. Cheers !

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