# What is equilibrium dependent upon in Generalized Second-Price Auctions?

Theory states that GSP auctions induce truthful bidding. Is it the case that this is true ONLY IF

a) each of the bidders truthfully bids their value ($$b_i = v_i$$) (each bidder's optimal strategy) AND

b) each bidder assumes that all of the other bidders are also truthfully employing their own optimal strategy ?

My question arises out of the following possible example:

Two bidders bidding for a single item, Vickery sealed-bid second-price auction.

The bidders have equal values for the same item (say $$v_1 = v_2 = 10$$)

$$b_1 = 7$$

$$b_2 = 9$$

$$b_2$$ will win the auction and pay $$7$$; with a surplus payoff of $$2= (9-7)$$. This outcome is equivalent to the example where:

$$v_1 = b_1 = 7$$

$$v_2 = b_2 = 9$$

$$b_2$$ wins in either example but neither bidder is using the optimal strategy in the first example. What logic am I missing?

• In all of the examples I have seen in my research, the numbers given for values and bids always seem to conveniently prove the theory--- but none of them show the case where both / all bidders are NOT using their optimal strategies, and none show examples where the bidders' values are equal to each other. Nov 5 '19 at 22:04
• Which theory states "that GSP auctions induce truthful bidding"? Non-truthfulness is one of the main properties of generalized second-price auction. Nov 5 '19 at 22:57
• en.wikipedia.org/wiki/… Nov 6 '19 at 3:16
• coursera.org/learn/game-theory-1 Nov 6 '19 at 3:16
• All I can tell from those sources is that truthful bidding is weakly dominant in second price auctions, but the generalized second price auction, contrary to what its name may suggest, is NOT a generalization of the second price auction, and so does not possess the truthful bidding property of the latter auction. Nov 6 '19 at 5:43

## 2 Answers

The surplus is the value of the object obtained minus the price paid. In your first example, the surplus would be $$10-7 = 3$$. The reason why you don't find examples where BOTH players are not using optimal strategies is because $$b=v$$ is weakly dominant. Unlike a lot of games where you need to consider best response functions, this one can be solved simply by removing dominated strategies.

Using your first example, there exists no bids that will give a higher payoff than $$b_i=10$$ regardless of what $$b_{-i}$$. Suppose we look at player 1, then consider all cases:

$$b_2 < b_1 < v_1$$: surplus is the same as $$b_1=v_1$$

$$b_1 < b_2 < v_1$$: surplus is 0, $$b_1=v_1$$ is profitable deviation

$$b_1 < v_1 < b_2$$: surplus is the same as $$b_1=v_1$$

$$v_1 < b_1$$ surplus is negative or 0 (depending on $$b_2$$), $$b_1=v_1$$ is profitable deviation

Also, auctions are usually used for a non dividable object, which means that examples with multiple players with the same value require a tie breaking rule. Whether or not this weak dominance survives depends on this tie breaking rule.

As Herr K pointed out your premise is not exact, it is not about GSPs but rather SPs. But even for SPs, I do not understand your examples.

What you show is that if the bidders choose the same strategies in example 1 and example 2, the same bidder wins and pays the same amount. This is not surprising: the outcome (who wins and pays how much) depends on the bids. For most (all?) auction formats, as long as the bids remain the same, the outcome will be the same.

You are not checking whether what the bidders did was optimal for them. In example 1 it was not: bidder 1 would have been better off by upping her bid to 10.