Revenue Comparison in Asymmetric Auctions with Discrete Valuations

Proof of Proposition 1

The complete proof of Proposition 1 is long, mainly because of the proof of essential equilibrium uniqueness. Here we provide a partial proof, in which we verify that (i) if [4] is satisfied, then the strategy profile described by Proposition 1(ii) is a BNE; (ii) if [7] is satisfied, then the strategy profile in Proposition 1(iii) is a BNE.²⁹

The case in which [4] is satisfied

Suppose that the inequalities in [4] are satisfied, and moreover that . Then we obtain that is such that (if , then is equal to , which actually would simplify matters), which implies that the strategies described by Proposition 1(ii) make sense. Now we verify that for each type of each bidder the strategy specified by Proposition 1(ii) is a best reply given the strategies of the two types of the other bidder. We use and to denote the payoff of type and his probability to win – respectively – as a function of his bid b, given the strategies of the two types of the other bidder. Notice that for any and for any ; thus we do not need to consider bids below or above . The same remark applies to the BNE described by Proposition 1(iii).

Type. The strategies of types 2_L and 2_H are such that each type of bidder 2 bids at least with probability one. Therefore, the payoff of 1_L is zero if he bids as specified by Proposition 1(ii), and it is impossible for him to obtain a positive payoff.

Type. For any , the payoff of type is , which is constant and equal to . If , then 1_H loses against and loses also against 2_L unless 2_L bids , in which case 1_H ties with 2_L – an event with probability . Consider the most favorable case for , which means that he wins the tie-break against 2_L with probability one (this occurs if ): his expected payoff from bidding is then , which turns out to be equal to .

Type. For any , the payoff of type is , which is constant and equal to . For bids b in we find , which is decreasing in b, and therefore for any .

Type. For any , the payoff of type is , which is constant and equal to . For bids b in we find and , which is increasing in b and therefore for any .

The case in which [7] is satisfied

Suppose that the inequality in [7] is satisfied. Then in eq. [8] and we verify that for each type of each bidder the strategy specified by Proposition 1(iii) is a best reply given the strategies of the two types of the other bidder. Let .

Type. The same argument given in the proof above for type applies.

Type. For any , the payoff of type is , which is equal to for any .³⁰ If , then ties with type 2_L and loses against 2_H, unless also 2_H bids – an event with probability . Consider the most favorable case for , which means that he wins the tie-break against each type of bidder 2 with probability one (this occurs if ): his expected payoff from bidding is then which turns out to be equal to .

Type. The payoff of type from bidding is . If he bids , then and thus is decreasing in b.

Type. For any , the payoff of type is , which is equal to for any .

Derivation of given the BNE described by Proposition 1

The BNE of Proposition 1(ii) when

We evaluate as the difference between the social surplus generated by the FPA minus the bidders’ rents : . Thus, we need to derive and :

in which def , for , is the probability that wins when he faces type .

In order to derive def , for the case that , we need to evaluate

and using in we find . Hence

In order to derive def , for the case that , we need to evaluate

and using in we find . Hence

Now we can evaluate :

An expression for is found by solving eq. [2]:

[15]

with

The BNE of Proposition 1(ii) when (footnote 10)

In order to derive , for the case that , we need to evaluate

Now we can evaluate :

[16]

The BNE in Proposition 1(iii)

For the case that we need to evaluate def , which is equal to

Now we can evaluate :

Proof of Lemma 1

Given and , when Proposition 1(ii) (footnote 10) applies and reveals that types bid as in the benchmark symmetric setting, whereas and with support , in which . It is simple to see that both and are decreasing with respect to for any , and this implies that and 2_H are both more aggressive, in the sense of first-order stochastic dominance, the larger is in .³¹ Given that

[17]

we infer that is increasing in .

When , Proposition 1(iii) applies and reveals that types bid as in the benchmark symmetric setting, whereas for any . Since is strictly increasing in for any , we infer that 2_H is less aggressive, in the sense of first-order stochastic dominance, the larger is . Using again eq. [17], after replacing with and with , it follows that is strictly decreasing with respect to .

Proof of Proposition 4

The proof when [12] is satisfied

Recall from our final remark in Section 3.2.2 that in the BNE described by Proposition 1(ii) the highest valuation bidder does not always win. Conversely, the efficient allocation is always achieved in the SPA. Therefore a sufficient condition for is that the aggregate bidders’ rents in the FPA, , are (weakly) larger than the rents in the SPA, . Indeed, we show that in region B if by proving that for any profile of values in B. Since , Proposition 1(ii) applies and thus the aggregate bidders’ rents in the FPA are with . On the other hand, the bidders’ rents in the SPA are . Hence, the inequality reduces to . From eq. [15] we obtain with . Therefore boils down to and (after squaring – notice that in B) ultimately to . Given and , we find that , which is positive in region B.

For valuations in region C, that is such that , we show that if [12] is satisfied. As above, the bidders’ rents in the FPA are with . However, the rents in the SPA in region C are , and the inequality reduces to . Using again we see that boils down to and, after squaring – notice that – ultimately to

[18]

We prove that this inequality holds for each by verifying that the left-hand side of [18] is positive both at and at . At , the left-hand side in [18] reduces to which is positive since (i) it is increasing in ; (ii) has value at . At , the left-hand side in [18] reduces to .

Proof for the case of distribution shift

In the case of shift, and . If , then and . As a consequence, reduces to . If , then this inequality is satisfied for any ; if instead , then the inequality is violated for and it holds if and only if .

If , then and . As a consequence, reduces to . In order for this inequality to be satisfied by an larger than it is necessary that .

Proof of the claim in Section 4.3 about the approximation of our discrete setting using a continous model

Suppose that and consider such that ; let be close to zero. If are continuously differentiable c.d.f.s which approximate well our discrete setting, then , , , and by definition of . We show that [14] is violated at by deriving a contradiction if the inequality holds for each . Indeed, if this condition were satisfied then , but we know that .³² In other terms, needs to be small for , and [14] requires that is smaller than for any x between and . But since , it is necessary that grows substantially in , which is impossible given that is small.³³

Proof of Proposition 5

The expression of depends as follows on :

In the case of , for instance, is increasing in , is increasing in , and is increasing in , thus . A similar argument applies in the other cases.

Proof of Proposition 6

Step 1 If , then the optimal reserve price in the FPA belongs to .

First notice that if , then bidding allows bidder 2 to win with certainty. Hence, no type of bidder 2 (or of bidder 1) bids above in equilibrium and for each . Second, if then bidder 1 does not participate in the auction. Bidder 2 participates, and bids r, if and only if . Therefore if (both types of bidder 2 bid r), and if (only type bids r).

Step 2 If , then the optimal reserve price in the FPA belongs to .

The result follows from Steps 2.1 and 2.2 below.

Step 2.1 If we consider only values of r in , then the optimal r is in .

Let bidders i and j be such that (if , then we set , ). Thus is the type of bidder with valuation equal to .

From and it follows that . For r such that the BNE is such that types do not bid; types play mixed strategies, each with support , with , and c.d.f.s

[19]

Since both types bid more aggressively as r increases in , it follows that is increasing with respect to r in the interval , and .

For r such that , only type participates, and bids r. Thus is increasing with respect to r in the interval , and . Hence the optimal r in belongs to ■

Notice that when , Step 2.1 implies immediately that the optimal r belongs to . Therefore in the following of the proof, we assume that

Step 2.2 If we consider only values of r in , then the optimal r is

Given , we need to consider two cases: and .

Step 2.2.1 If , then is increasing with respect to r in the interval .

When the BNE is similar to the BNE described in Proposition 1(ii), with replaced by r: type does not bid; types play mixed strategies with support for , for , for , in which is the smaller solution to

[20]

and . The c.d.f.s for the mixed strategies of are, respectively:

Since is the smaller solution to eq. [20] and the left-hand side in eq. [20] is increasing in r, it follows that is increasing in r and also is so. Hence, it is immediate that an increase in r induces both types and to bid more aggressively. The same result holds for type as well since for , and both and are decreasing in r. Therefore is increasing in r. ■

Step 2.2.2 If , then is increasing with respect to r in the interval .

When the BNE is similar to the BNE described in Proposition 1(iii), with replaced by r: type does not bid; type bids r; types play mixed strategies with the same support , in which and the c.d.f.s are given by eq. [19], with , . Hence is increasing in r. ■

Proof of Proposition 7

For the case of , see the arguments immediately after the statement of Proposition 7. When , first notice that , . For instance, if (similar results are obtained if ) then because given , in both auctions the object is not sold when , and in the other states of the world the winning bidder pays . In case that , we have because in both auctions the object is sold if and only if . On the other hand, () is equal to the expected revenue in the FPA (in the SPA) when , and we know from Proposition 4 (condition [9]) that , unless .