Discussion Papers in Economics Revenue Comparison of Discrete Private-Value Auctions via Weak Dominance

We employ weak dominance to analyze both ﬁrst-price and second-price auctions under the discrete private-value setting. We provide a condition under which the expected revenue from second-price auction is higher than that of ﬁrst-price auction. We also provide implications for large auctions, including the “virtual” revenue equivalence.


Introduction
The comparison of the expected revenues from private-value first-price and secondprice auctions (FPA and SPA henceforth) has been extensively analyzed, including the revenue equivalence result by Riley and Samuelson (1981) and Myerson (1981). It also has been shown that once some underlying assumptions are relaxed, not only the revenue equivalence result does not necessarily hold, but the comparison results 1 become ambiguous. 1 In addition, the analyses often have been limited to the twoplayer case, implying the lack of implications for large auctions.
In this paper, we revisit the revenue comparison of FPA and SPA. There are two departures from the literature. One is the use of the maximal elimination of weakly dominated bids -all weakly dominated bids are eliminated -for both FPA and SPA. 2 It has been typically the case that while SPA is analyzed by the maximal elimination of weakly dominated bids, FPA is analyzed by Bayesian Nash equilibrium. It would be ideal to use the same solution concept to assess the differences purely stemming from the comparison of two distinct institutions. Another departure is that we follow a seminal work by Dekel and Wolinsky (2003), which analyze FPA via rationalizability, and adopt the discrete sets of bids and values. 3 One advantage of the adoption of weak dominance and discrete setting is that we require minimal assumptions. In particular, our analysis allows asymmetry and an arbitrary number of players.
Our main result provides a condition under which SPA generates a higher expected revenue compared to FPA. 4 The key is the comparison of the winning bids in FPA and SPA. The result on the winning bids in FPA is due to Battigalli and Siniscalchi (2003) and Dekel and Wolinsky (2003) who provide upper bounds for bids via weak dominance. The maximal elimination of weakly dominated bids implies that the winning bids in SPA is higher than that of FPA. This leads to the comparison of the highest bid in FPA and the second highest bid in SPA. Our condition concerns the case where the highest and the second highest bids are the same in SPA, in which case the price the winner pays in SPA is higher than that of FPA.
We also show that this result is asymptotically robust. Under the assumptions of (i) independently distributed values and (ii) the same highest value (whose probability is bounded below by an arbitrary small number) for every player's support, the expected revenue from SPA is higher than that of FPA in large auctions. 5 In addition, if the iterative maximal elimination of weakly dominates bids is used with 1 For example, Maskin and Riley (2000) and Kirkegaard (2012) analyzed the case of asymmetry. See Maskin and Riley (2000), Krishna (2010) and Milgrom (2004) for the overview of related studies. 2 We use interim weak dominance. That is, we apply weak dominance for the realization of each value. We hence use "bids" instead of "strategies." Note also that this does not imply an iterative procedure. We use the iterative maximal elimination of weakly dominated bids later.
3 Several other studies also analyze FPA via rationalizability, including Battigalli and Siniscalchi (2003), Cho (2005), and Robles and Shimoji (2012). 4 See also Kim (2013) who compares the revenues under the binary setting. 5 See also Yu (1999) for the equilibrium analysis of the symmetric case.
the additional assumption of (iii) players' risk-aversion, the difference in the expected revenues converges to the smallest monetary unit, which we denote d, as the number of players increases. 6 This implies the virtual revenue equivalence in large auctions for small d.
For asymmetric auctions, Kirkegaard (2012) identified sufficient conditions under which FPA generates a higher expected revenue compared to SPA. Note that our result has a different implication. Both the use of weak dominance for FPA and the discrete setting lead to this difference. We use an example to demonstrate that the discretized version of the condition in Kirkegaard (2012) and ours are not mutually exclusive.

Preliminaries
In this paper, we analyze first-price auction (FPA) and second-price auction (SPA). We utilize the private-value setting. The set of players is N = {1, . . . , n} with n ≥ 2. Player i's utility function is u i : R → R which is assumed to be strictly increasing. Before the auction starts, each player i ∈ N observes her value, Let v be a typical element of V = j∈N V j . We use the subscript "−i" to represent player i's opponents. We assume that each player i ∈ N with any v i ∈ V i assigns a strictly positive probability to every v −i ∈ V −i and that the auctioneer assigns a strictly positive probability to every v ∈ V .
Each player i chooses her bid b i ∈ B i = {0, d, . . . ,b i − d,b i } wherev ≤b i for each i ∈ N and hence V i ⊆ B i . A player wins only if her bid is the highest. If there are multiple players who chose the highest bid, each one of them has an equal chance of winning. If player i ∈ N is the winner, the price she pays, s, is such that s = b i for FPA and s = max j =i {b j } for SPA. Player i's utility is u i (v i − s) if she wins the object while it is u i (0) otherwise.

Maximal Elimination of Weakly Dominated Bids
In this section, we solve both FPA and SPA via the maximal elimination of weakly dominated bids.

Winning Bids in FPA and SPA
In SPA, for each i ∈ N and v i ∈ V i , b i = v i is the only bid surviving the maximal elimination of weakly dominated bids (i.e., weakly dominant bid). Given v ∈ V , the winning bid in SPA is hence max j∈N {v j }.
For FPA, Battigalli and Siniscalchi (2003, p.41) and Dekel and Wolinsky (2003, Subsection 4.3) show that for each i ∈ N and v i ∈ V i , the highest bid which survives the maximal elimination of weakly dominated bids is strictly lower than v i . 7

Lemma 1 (Battigalli and Siniscalchi (2003) and Dekel and Wolinsky (2003))
For each i ∈ N and v i ∈ V i , the highest bid which survives the maximal elimination of weakly dominated bids in FPA is max{v i − d, 0}.
We then have the following result. Note that if v > 0, the expression is simply max j∈N {v j − d}. Corollary 1 leads to the following result.

Lemma 2
Given v ∈ V , the winning bid in SPA is weakly higher than the winning bid in FPA. If max j∈N {v j } > 0, the winning bid in SPA is strictly higher than the winning bid in FPA.
If v > 0, the latter is indeed the case. 7 Battigalli and Siniscalchi (2003) and Dekel and Wolinsky (2003) assume V i = V j for every i, j ∈ N (i.e., identical support) with v = 0 and the former uses the continuous bid and value spaces. Their insight remains valid even with the heterogeneous supports.

Revenue Comparison
Lemma 2 implies that if two highest values are the same and strictly higher than 0, the revenue of SPA is strictly higher than that of FPA. Let P represent the auctioneer's belief over players' values. The following result shows a condition under which this possibility of "ties" outweighs other possibilities, leading to our main result.

Proposition 1
The expected revenue from SPA is strictly higher than that of FPA if The expression on the left-hand side concerns the cases in which the first and second highest values are equal, implying that the revenue from SPA is higher than that of FPA. The expression on the right-hand side concerns the cases where the difference between the first and second highest values are at least 2d, implying that the revenue from FPA can be higher than that of SPA. 8 Given that values are linear in d, note also that the expressions on both sides are linear in d, implying that the size of d does not matter for the result.
To visualize the implication of Proposition 1, Figure 1 plots the combination of two order statistics, the highest and the second highest values, v and v respectively.
A : v = v corresponding to the left-hand side expression of Proposition 1. In this case, (i) SPA leads to a higher revenue than FPA, and (ii) the difference in the revenues is d. C : v − v ≥ 2d corresponding to the expression on the right-hand side. They are the cases where (i) FPA generates a higher revenue than SPA and (ii) the difference of revenues is Figure 1: Visualization of Proposition 1 The expression in Proposition 1 says that if the realizations in A are likely, SPA generates a higher expected revenue than FPA.

Large Auctions
We now provide a condition under which Proposition 1 holds for sufficiently large n. Given n, let • q n (ñ) be the probability thatn =ñ whereñ ∈ {0, . . . ,n}, and We need the following assumptions: Assumption 1 Player's values are independently distributed.
The expression on the left-hand side in Proposition 1 contains the probabilities that the highest and second highest values are the same. The following result identifies a condition under which the expression in Proposition 1 holds as n → ∞. 9 Proposition 2 Given Assumptions 1 and 2, if Proposition 1 holds for sufficiently large n.
The condition implies that as n becomes large, there are a sufficient number of players who hasv in the support and the chance that there is only one player whose value is v diminishes. Note that ifv i =v for each i ∈ N ,n = n and hence q n (n) = 1.
Corollary 2 Given Assumptions 1 and 2, ifv i =v for each i ∈ N , Proposition 1 holds for sufficiently large n.
As an example, consider the case of V i = {0, . . . ,v} for each i ∈ N and each player's value is independently and uniformly distributed. Let |V | = m + 1 (i.e., v = md) and hence the probability attached to each value is 1 m+1 . The expression in Proposition 1 becomes The proof in Yu (1999, Proposition 13) for the symmetric case carries the same observation; i.e., the probability that the first and second highest values coincide converges to one as n → ∞. 7 which can be simplified as: For large m, it is sufficient for the number of players, n, to be approximately 88.2% of m to maintain (1).

Iterative Maximal Elimination of Weakly Dominated Bids
The result for FPA is not as sharp as that of SPA. This is because we have focused on (one round of) the maximal elimination of weakly dominated bids and strictly increasing utility functions. If we use the iterative maximal elimination of weakly dominated bids and weakly concave (still strictly increasing) utility functions, we obtain a condition under which the uniqueness result is achieved for FPA. This is a variant of the results from Dekel and Wolinsky (2003) and Robles and Shimoji (2012) which use rationalizability. This result leads to the virtual revenue equivalence.

Uniqueness in FPA
In this subsection, we show a condition under which each player i ∈ N with v i ∈ V i has a unique bid surviving the iterative maximal elimination of weakly dominated bids. Let • player i ∈ N be such thatv i =v andv = max j =i {v j } (i.e., the second highest upper bound).
We need the following assumption: Assumption 3 For each i ∈ N , u i is weakly concave.
We then have the following result: Then, • ifv −v ≤ d, the only bid which survives iterative weak dominance is b i = max{v i − d, 0} for each i ∈ N and v i ∈ V i , and • ifv −v ≥ 2d, the only bid which survives iterative weak dominance is Note that the right-hand side expression in the condition is at least 1 2 . The result is visualized in Figure 2.
We already know that for v i ∈ {0, d}, b i = 0 is the unique weakly dominant bid. With the assumption that u i is weakly concave, we can also show that for any v i ≥ 2d, b i = 0 is eliminated. Consider v i ≥ 3d and compare b i = d and b i = 2d: 1. b i = 2d may win even if b i = d does not win (but not vice versa). In particular, if the opponents' highest bid is 2d, the expected utility from b i = 2d is strictly positive while it is zero for b i = d.
2. If b i = d wins, the opponents' highest bid is either d or 0. This is the only case where the expected utility from b i = 2d may be lower than that of b i = d. In this particular scenario, given the argument above, • b j = d for every j ∈ N with v j ≥ 2d, and The corresponding utility is u i (v i − d). The left-hand side expression in the condition corresponds to the probability that b i = d wins. Note that b i = 2d wins in this case and the corresponding utility is The condition in Proposition 3 rather states that the expected return from b i = 2d is higher than that of b i = d if the opponents' highest bid is either d or 0. This condition is sufficient as long as u i is weakly concave (Jensen's inequality). The same argument is applied repeatedly to obtain the result.

Large Auctions
Consider again the case of large auctions with independent distributions from the previous section. Remember the definition ofv in the precious subsection: the second highest upper bound. Given n, let • r v i ,n (ñ) be the probability from the view point of player Instead of Assumption 2, we require the following: Assumption 4 There exists τ ∈ (0, 1) such that τ i (v i ≥v) ≥ τ for each i ∈N .
We then have the following result.

Proposition 4 Given Assumptions 1, 3 and 4, if
for each i ∈ N with v i ≥ 3d, the result of Proposition 3 holds for sufficiently large n.
For example, the condition holds if r v i ,n (n) is lower for smallern's. Again, ifv i =v for each i ∈ N (i.e., r v i ,n (n − 1) = 1), the result immediately holds.
Given Corollaries 2 and 3 above, we have the following result. 10 Proposition 5 Given Assumptions 1, 2, 3 and 4, ifv i =v for each i ∈ N , the difference of the expected revenues from FPA and SPA via iterative weak dominance converges to d as n → ∞.
This can be seen as the virtual revenue equivalence result for small d in large auctions.

Discussion
In this section, we discuss (i) the order dependence of weak dominance, (ii) the upper bound identified in Lemma 1 and (iii) the comparison of our result to Kirkegaard (2012).

Order Dependence
The emphasis on "maximal" elimination is due to the possibility of the order dependence of weak dominance. 11 That is, different orders of elimination could lead to different predictions. As an example, consider the two-player SPA where the supports of their values, First, eliminate all bids except b 1 = v 1 for player 1 with v 1 ∈ V 1 at the first step. Then, for player 2, eliminate every b 2 ≥ v 1 at the second step. No further elimination occurs. In this case, the second highest bid can be higher or lower than v 2 , and the price the winner (player 1) pays cannot be uniquely identified for any v 2 ∈ V 2 . Note that this applies not only to our result, but also to previous studies which use weak dominance for SPA.

Upper Bound in FPA
where the second inequality comes from Jensen's inequality, b i = 1 weakly dominates 11 See for example Marx and Swinkels (1997 Given the result from the first step, we consider v i = 3 at the second step. The expected utility from b i = 1 is while the expected utility from b i = 2 is Then, Note that if α + β is close to one, the expression above is strictly positive independent of b j (3). In this case, b i = 1 strictly dominates b i = 2 for v i = 3.

On Kirkegaard (2012)
Kirkegaard (2012) identifies two conditions under which FPA leads to a higher expected revenue than SPA. There are several reasons why our result is different from that of Kirkegaard (2012). One reason is that values and bids are discrete in our setting while they are continuous in Kirkegaard (2012) -a tie is not possible in Kirkegaard (2012). Another reason is that while our focus is on weak dominance, Kirkegaard (2012) uses Bayesian Nash Equilibrium for FPA. We now demonstrate that even if (a discretized version of) a condition in Kirkegaard (2012) is satisfied in our discrete setting, it is possible that our condition still hold. 12 Kirkegaard (2012) considers the case of (i) two players and (ii) the bid and value spaces are continuous. Take V i as a closed interval for i ∈ {1, 2}, i.e,. Kirkegaard (2012) assumes that the supports for values are such that 0 ≤ v 2 ≤ v 1 andv 2 <v 1 (player 1 is strong and player 2 is weak). Let F i (v) be the pdf for player i's value. Kirkegaard (2012) assumes That is, F 1 dominates F 2 in terms of not only the reverse hazard rate but also the hazard rate.
). The sufficient condition in Kirkegaard (2012, Expression (9)) is Note that in the discrete setting, there does not exist the corresponding r(v 2 ) for each v 2 ∈ V 2 generically. Thus, we instead require (5) to hold for every v 1 ∈ {v 2 , . . . ,v 1 }.
We also look at the discretized version of (4). We now turn to an example in the discrete setting. Consider the following example: ,v 2 + d} and where (i) v 2 > 0 and (ii) κ is a positive integer. Note thatv 1 =v 2 + d and hence V 2 ⊂ V 1 . The players' values are independently distributed. Let We also assume that ε is sufficiently small. The first inequality in (4) holds with the equality for v 2 and with the strict inequality for every v ∈ V 2 \{v 2 }. The second inequality in (4) holds with strict inequality for every v ∈ V 2 . The inequality in (5) holds with the strict inequality for v 2 and 12 Kirkegaard (2012) has two sufficient conditions. We only focus on one of them.
14 with the equality for every v 1 ∈ V 1 \{v 2 } and v 2 ∈ V 2 \{v 2 }. Consider the expression in Proposition 1. For sufficiently small ε, the sum of probabilities on the left-hand side is close to one, implying that the inequality is satisfied. 13

A Proof of Proposition 1
The highest possible revenue in FPA is max{max i∈N {v i − d}, 0}. The revenue in SPA is the second highest value. Note that if there are multiple highest values which are strictly higher than 0, the revenue from SPA is strictly higher than that of FPA.
Let v be the highest value and v be the second highest value. The expected revenue from SPA (left-hand side) is strictly higher than that of FPA (right-hand

B Proof for Proposition 2
Given n andn where 2 ≤n ≤ n, take the probability that the second highest value isv:n where {i 1 , . . . , i η } ⊆N . The lower bound of the expression above is Note that the last two expression converge to zero asn → ∞ (Corollary 2). The lower bound for the left-hand expression in Proposition 1 (without d) is hence
Let ln(y) = n−2 i−1 and we have e ln(y) − 1 e ln(y) ≥ 2 ⇔ y 2 − 2y − 1 ≥ 0 where (i) y = 1 + √ 2 if the expression is zero and (ii) the expression is strictly increasing if y > 1. Note that ln (1 + √ 2) ≈ 0.8813. 14 D Proof for Proposition 3 First Step: • For each i ∈ N and v i ∈ V i \{0}, every b i ≥ v i is weakly dominated. None of them leads to a utility strictly higher than u i (0) while b i < v i secures u i (0) or higher (e.g., the opponents' highest bid is equal to b i ).
• For each i ∈ N with v i = 0, every b i ≥ d is weakly dominated. Every b i ≥ d leads to a utility of u i (0) or less (e.g., every opponent bids zero) and b i = 0 guarantees u i (0).
• For each i ∈ N with v i ≥ 2d, b i = d weakly dominates b i = 0. The only way b i = 0 wins is that every opponent bids zero as well. In this case, the expected utility with b i = 0 is 1 n u i (v i ) + n−1 n u i (0) while it is u i (v i − d) with b i = d. Since n ≥ 2, we have for any weakly concave u i . 15 For the other possibilities, b i = d leads to a utility weakly higher than that of b i = 0 (i.e., u i (0)). In particular, if the opponents' highest bid is d, the inequality is strict.
The sets of bids surviving the maximal elimination of weakly dominated bids are (i) {max{v i −d, 0}} for v i ∈ {0, d, 2d} -note that they are unique -and (ii) {d, . . . , v i −d}