On the Efficiency of All-Pay Mechanisms

We study the inefficiency of mixed equilibria, expressed as the price of anarchy, of all-pay auctions in three different environments: combinatorial, multi-unit and single-item auctions. First, we consider item-bidding combinatorial auctions where m all-pay auctions run in parallel, one for each good. For fractionally subadditive valuations, we strengthen the upper bound from 2 [Syrgkanis and Tardos STOC'13] to 1.82 by proving some structural properties that characterize the mixed Nash equilibria of the game. Next, we design an all-pay mechanism with a randomized allocation rule for the multi- unit auction. We show that, for bidders with submodular valuations, the mechanism admits a unique, 75% efficient, pure Nash equilibrium. The efficiency of this mechanism outperforms all the known bounds on the price of anarchy of mechanisms used for multi-unit auctions. Finally, we analyze single-item all-pay auctions motivated by their connection to contests and show tight bounds on the price of anarchy of social welfare, revenue and maximum bid.


Introduction
It is a common economic phenomenon in competitions that agents make irreversible investments without knowing the outcome. All-pay auctions are widely used in economics to capture such situations, where all players, even the losers, pay their bids. For example, a lobbyist can make a monetary contribution in order to influence decisions made by the government. Usually the group invested the most increases their winning chances, but all groups have to pay regardless of the outcome. In addition, all-pay auctions have been shown useful to model rent seeking, political campaigns and R&D races. There is a well-known connection between all-pay auctions and contests [21]. In particular, the all-pay auction can be viewed as a single-prize contest, where the payments correspond to the effort that players make in order to win the competition.
In this paper, we study the efficiency of mixed Nash equilibria in all-pay auctions with complete information, from a worst-case analysis perspective, using the price of anarchy [16] as a measure. As social objective, we consider the social welfare, i.e. the sum of the bidders' valuations. We study the equilibria induced from all-pay mechanisms in three fundamental resource allocation scenarios; combinatorial auctions, multi-unit auctions and single-item auctions.
In a combinatorial auction a set of items are allocated to a group of selfish individuals. Each player has different preferences for different subsets of the items and this is expressed via a valuation set function. A multi-unit auction can be considered as an important special case, where there are multiple copies of a single good. Hence the valuations of the players are not set functions, but depend only on the number of copies received. Multi-unit auctions have been extensively studied since the seminal work by Vickrey [24]. As already mentioned, all-pay auctions have received a lot of attention for the case of a single item, as they model all-pay contests and procurements via contests.

Contribution
Combinatorial Auctions. Our first result is on the price of anarchy of simultaneous all-pay auctions with item-bidding that was previously studied by Syrgkanis and Tardos [23]. For fractionally subadditive valuations, it was previously shown that the price of anarchy was at most 2 [23] and at least e/(e − 1) ≈ 1.58 [8]. We narrow further this gap, by improving the upper bound to 1.82. In order to obtain the bound, we come up with several structural theorems that characterize mixed Nash equilibria in simultaneous all-pay auctions.
Multi-unit Auctions. Our next result shows a novel use of all-pay mechanisms to the multi-unit setting . We propose an all-pay mechanism with a randomized allocation rule inspired by Kelly's seminal proportional-share allocation mechanism [15]. We show that this mechanism admits a unique, 75% efficient pure Nash equilibrium and no other mixed Nash equilibria exist, when bidders' valuations are submodular. As a consequence, the price of anarchy of our mechanism outperforms all current price of anarchy bounds of prevalent multi-unit auctions including uniform price auction [18] and discriminatory auction [9], where the bound is e/(e − 1) ≈ 1.58.
Single-item Auctions. Finally, we study the efficiency of a single-prize contest that can be modeled as a single-item all-pay auction. We show a tight bound on the price of anarchy for mixed equilibria which is approximately 1.185. By following previous study on the procurement via contest, we further study two other standard objectives, revenue and maximum bid. We evaluate the performance of all-pay auctions in the prior-free setting, i.e. no distribution over bidders' valuation is assumed. We show that both the revenue and the maximum bid of any mixed Nash equilibrium are at least as high as v 2 /2, where v 2 is the second highest valuation. In contrast, the revenue and the maximum bid in some mixed Nash equilibrium may be less than v 2 /2 when using reward structure other than allocating the entire reward to the highest bidder. This result coincides with the optimal crowdsourcing contest developed in [6] for the setting with prior distributions. We also show that in conventional procurements (modeled by first-price auctions), v 2 is exactly the revenue and maximum bid in the worst equilibrium. So procurement via all-pay contests is a 2-approximation to the conventional procurement in the context of worst-case equilibria.

Related work
The inefficiency of Nash equilibria in auctions has been a well-known fact (see e.g. [17]). Existence of efficient equilibria of simultaneous sealed bid auctions in full information settings was first studied by Bikhchandani [3]. Christodoulou, Kovács and Schapira [7] initiated the study of the (Bayesian) price of anarchy of simultaneous auctions with item-bidding. Several variants have been studied since then [2,13,12], as well as multi-unit auctions [9,18].
Syrgkanis and Tardos [23] proposed a general smoothness framework for several types of mechanisms and applied it to settings with fractionally subadditive bidders obtaining several upper bounds (e.g., first price auction, all-pay auction, and multi-unit auction). Christodoulou et al. [8] constructed tight lower bounds for first-price auctions and showed a tight price of anarchy bound of 2 for all-pay auctions with subadditive valuations. Roughgarden [20] presented an elegant methodology to provide price of anarchy lower bounds via a reduction from the hardness of the underlying optimization problems.
All-pay auctions and contests have been studied extensively in economic theory. Baye, Kovenock and de Vries [1], fully characterized the Nash equilibria in single-item all-pay auction with complete information. The connection between all-pay auctions and crowdsourcing contests was proposed in [10]. Chawla et al. [6] studied the design of optimal crowdsourcing contest to optimize the maximum bid in all-pay auctions when agents' value are drawn from a specific distribution independently.

Preliminaries
In a combinatorial auction, n players compete on m items. Every player (or bidder) i ∈ [n] has a valuation function v i : {0, 1} m → R + which is monotone and normalized, that is, The outcome of the auction is represented by a tuple of (X, p) where X = (X 1 , . . . , X n ) specifies the allocation of items (X i is the set of items allocated to player i) and p = (p 1 , . . . , p n ) specifies the buyers' payments (p i is the payment of player i for the allocation X). In the simultaneous item-bidding auction, every player i ∈ [n] submits a non-negative bid b ij for each item j ∈ [m]. The items are then allocated by independent auctions, i.e. the allocation and payment rule for item j only depend on the players' bids on item j. In a simultaneous all-pay auction the allocation and payment for each player is determined as follows: each item j ∈ [m] is allocated to the bidder i * with the highest bid for that item, i.e. i * = arg max i b ij , and each bidder i is charged an amount equal to p i = j∈[m] b ij . It is worth mentioning that, for any bidder profile, there always exists a tie-breaking rule such that mixed equilibria exist [22].
The classes of the above valuations are in increasing order of inclusion.
Multi-unit Auction. In a multi-unit auction, m copies of an item are sold to n bidders. Here, bidder i 's valuation is a function that depends on the number of copies he gets. That is v i : {0, 1, . . . , m} → R + and it is non-decreasing and normalized, with v i (0) = 0. We say a valuation v i is submodular, if it has non-increasing marginal values, i.
Nash equilibrium and price of anarchy. We use b i to denote a pure strategy of player i which might be a single value or a vector, depending on the auction. So, for the case of m simultaneous auctions, . . , b n ) the strategies of all players except for i. Any mixed strategy B i of player i is a probability distribution over pure strategies.
For any profile of strategies, b = (b 1 , . . . , b n ), X(b) denotes the allocation under the strategy profile b. The valuation of player i for the allocation The utility u i of player i is defined as the difference between her valuation and payment: Clearly, any pure Nash equilibrium is also a mixed Nash equilibrium.
Our global objective is to maximize the sum of the valuations of the players for their received allocations, i.e., to maximize the social welfare SW (X) = i∈ [n] is an optimal allocation if SW (O) = max X SW (X). In Sect. 5, we also study two other objectives: the revenue, which equals the sum of the payments, i p i , and the maximum payment, max i b i . We also refer to the maximum payment as the maximum bid. , where E(I) is the class of mixed Nash equilibria for the instance I ∈ I. The pure PoA is defined as above but restricted in the class of pure Nash equilibria.
Let B = (B 1 , . . . , B n ) be a profile of mixed strategies. Given the profile B, we fix the notation for the following cumulative distribution functions (CDF): G ij is the CDF of the bid of player i for item j; F j is the CDF of the highest bid for item j and F ij is the CDF of the highest bid for item j if we exclude the bid of player i. Observe that F j = k G kj and F ij = k =i G kj . We also use ϕ ij (x) to denote the probability that player i gets item j by bidding x. Then, ϕ ij (x) ≤ F ij (x). When we refer to a single item, we may drop the index j. Whenever it is clear from the context, we will use shorter notation for expectations, e.g. we use

Combinatorial Auctions
In this section we prove an upper bound of 1.82 for the mixed price of anarchy of simultaneous all-pay auctions when bidders' valuations are fractionally subadditive (XOS). This result improves over the previously known bound of 2 due to [23]. We first state our main theorem and present the key ingredients. Then we prove these ingredients in the following subsections.  . . , O n ) be a fixed optimal solution, that maximizes the social welfare. We can safely assume that O is a partition of the items. Since v i is an XOS valuation, let ξ O i i be a maximizing additive function with respect to O i . For every item j we denote by o j item j's contribution to the optimal social welfare, that is, In order to bound the price of anarchy, we consider only items with o j > 0, as it is without loss of generality to omit items with o j = 0.
For a fixed mixed Nash equilibrium B, recall that by F j and F ij we denote the CDFs of the maximum bid on item j among all bidders, with and without the bid of bidder i, respectively. For As a key part of the proof we use the following two inequalities that bound from below the social welfare in any mixed Nash equilibrium B.
Inequality (1) suffices to provide a weaker upper bound of 2 (see [8]). The proof of (2) is much more involved, and requires a deeper understanding of the equilibria properties of the induced game. We postpone their proofs in Sect. 3 (1) and (2), for every λ ≥ 0. It suffices to bound from below the right-hand side of (3) with respect to the optimal social welfare. For any cumulative distribution function F , and any positive real number v, let (3) can then be rewritten as SW (B) ≥ 1 1+λ j R(F j , o j ). Finally, we show a lower bound of R(F, v) that holds for any CDF F and any positive real v.
The proof of (4) is given in Sect. 3.3 (Lemma 3.19). Finally, we obtain that for any λ > 0, By taking λ = 0.56, we conclude that the price of anarchy is at most 1.82.

Proof of Inequality (1)
This section is devoted to the proof of the following lower bound. Recall that the definition o j is from the definition of XOS functions.

Proof. Recall that
We can bound bidder i's utility in the Nash equilibrium B by u i (B) ≥ j∈O i A j . To see this, consider the deviation for bidder i, where he bids only for items in O i , namely, for each item j, he bids the value x j that maximizes the expression and the bids x j must be paid in any case, the expected utility with these bids is at least By summing up over all bidders,

The first equality holds because
The second inequality follows because i b ij ≥ max i b ij and the last one is implied by the definition of the expected value of any positive random variable.

Proof of Inequality (2)
Here, we prove the following lemma for any mixed Nash equilibrium B.
First we show a useful lemma that holds for XOS valuations. We will further use the technical Proposition 3.5, whose proof is deferred to Appendix A.
Proposition 3. 5. For any integer n ≥ 2, any positive reals G i ≤ 1 and positive reals g i , for We are now ready to prove Lemma 3. 3. We first state a proof sketch here to illustrate the main ideas.
Sketch of Lemma 3.3. Recall that G ij is the CDF of the bid of player i for item j. For simplicity, we assume G ij (x) is continuous and differentiable, with g ij (x) being the PDF of player i's bid for item j. The general case will be considered later. First, we define the expected marginal valuation Given the above definition and a careful characterization of mixed Nash equilibria, we are able to for any x in the support of G ij . Let g ij (x) be the derivative of G ij (x). Using Lemma 3.4, we have where the second inequality follows by the law of total probability. By using the facts that dx , for any x > 0 such that g ij (x) > 0 (x is in the support of player i) and F j (x) > 0, we obtain .
For every x > 0, we use Proposition 3.5 only over the set S of players with g ij (x) > 0. After summing over all bidders we get, The above inequality also holds for F j (x) = 0. Finally, by merging the above inequalities, we Now we show the complete proof for Lemma 3.3. Recall that o j is the contribution of item j to the optimum social welfare. If player i is the one receiving item j in the optimum allocation, then The proof of Lemma 3.3 needs a careful technical preparation that we divided into a couple of lemmas.
First of all, we define the expected marginal valuation of item j for player i. For given mixed strategy B i , the distribution of bids on items in [m]\{j} depends on the bid b ij , so one can consider the given conditional expectation: For a given B, let ϕ ij (x) denote the probability that bidder i gets item j when she bids x on item j. It is clear that ϕ ij is non-decreasing and ϕ ij (x) ≤ F ij (x) (they are equal when no ties occur). Lemma 3.7. For a given B, for any bidder i, item j and bids x ≥ 0 and y ≥ 0, The second equality is due to ; the third one holds because b ij = y, and that other players' bids have distribution × k =i B k . The fourth one is obvious, since is independent of the condition j ∈ X i (b ) and of the player i's bid on item j. Definition 3. 8. Given a Nash equilibrium B, we say a bid x is good for bidder i and item j ( 9. Given a Nash equilibrium B, for any bidder i and any item j, Pr[b ij is bad ] = 0. Proof. The lemma follows from the definition of Nash equilibrium; otherwise we can replace the bad bids with good bids and improve the bidder's utility.
Lemma 3. 10. Given a Nash equilibrium B, for any bidder i, item j, good bid x and any bid y ≥ 0, Moreover, for a good bid x > 0, ϕ ij (x) > 0 holds.
Now we consider the difference between the above two terms:

The second equality holds since
; the third equality holds by Lemma 3.7. Finally, ϕ ij (x) > 0 for positive good bids follows by taking y = 0, since with ϕ ij (x) = 0 the left hand side of the inequality would be negative.
Next, by using the above lemma, we are able to show several structural results for Nash equilibria.
Definition 3. 11. Given a mixed strategy profile B, we say that a positive bid 12. Given a mixed strategy profile B, if a positive bid x is in bidder i's support on item j, then for every ε > 0, there exists x − ε < x ≤ x such that x is good. Proof. Suppose on the contrary that there is an ε > 0 such that for all x , such that Lemma 3. 13. Given a Nash equilibrium B, if x > 0 is in bidder i's support on item j, then there must exist another bidder k = i such that x is also in the bidder k's support on item j, i.e. for all Proof. Assume on the contrary that for each player Lemma 3.12, there exists x − ε < x ≤ x such that x is good for player i. Since ϕ ij is a non-decreasing function and ϕ ij ( Lemma 3.14. Given a Nash equilibrium B, for bidder i and item j, there are no x > 0 such that there are no mass points in the bidding strategy, except for possibly 0. Proof. Assume on the contrary that there exists a bid x > 0 such that Pr[b ij = x] > 0 for some bidder i and item j. By Lemma 3.9, x is good for bidder i and item j, and ϕ ij (x) > 0 by Lemma 3.10. According to Lemma 3.13, there must exist a bidder k such that x is in her support on item j. We can pick a sufficiently small ε such that ε < This can be done since (x − ε) increases when ε decreases. Due to Lemma 3.12 there exists x − ε < x ≤ x such that x is good for bidder k and item j. Now we consider the following two cases for x . 10. The first inequality holds by the case assumption. The second holds because player k cannot get item j with bid x whenever player i gets it by bidding x. The last inequality holds because both ϕ ij (x) > 0 and Pr which contradicts Lemma 3. 10. Here the first inequality holds because the probability that player k gets the item with bid x + ε is at least the probablity that he gets it by bidding x plus the probability that i bids x and gets the item (these two events for b −k are disjoint). The second inequality holds by case assumption, and the rest hold by our assumptions on ε and ε . Lemma 3.15. Given a Nash equilibrium B, for any bidder i and item j, ϕ ij (x) = F ij (x) for all x > 0. Proof. The lemma follows immediately from Lemma 3.14. The probablity that some player k = i bids exactly x is zero. Thus F ij (x) equals the probability that the highest bid of players other than i is strictly smaller than x, and 1 − F ij (x) is the probability that it is strictly higher. Therefore Lemma 3. 16. Given a Nash equilibrium B, for any bidder i, item j and good bids Lemma 3. 17. Given a Nash equilibrium B and item j, let T = sup{x|x is in some bidder's support on item j}. For any bid x < T , x is in some bidder's support on item j. Proof. Assume on the contrary that there exist a bid x < T such that x is not in any bidder's support. Then there exists δ > 0 such that G ij (x) = G ij (x − δ) for all bidder i. Let y = sup{z|∀i, G ij (x) = G ij (z)}. By Lemma 3.14, G ij is continuous. So we have G ij (y) = G ij (x) = G ij (x − δ) for any bidder i. That is F ij (y) = F ij (x − δ) for any bidder i. By the definition of supremum, there exits a bidder k such that for any ε > 0, G kj (y + ε) > G kj (x) = G kj (y). By Lemma 3.9, there exists a good bid y + ∈ (y, y + ε] for bidder k and item j. We pick a sufficient small ε such that (F kj (y + ) − F kj (y)) · v kj (y + ) < δ. This can be done since F kj is continuous by Lemma 3.14 and v kj is non-decreasing by Lemma 3. 16.
which contradicts Lemma 3.10 and Lemma 3.15.
Lemma 3. 18. Given a Nash equilibrium B, if x > 0 is a good bid for bidder i and item j, and F ij is differentiable in x, then Proof. Notice that v ij (x) = 0 by Lemma 3. 10. By Lemma 3.10 and 3.15, we have That is, The lemma follows by taking the limit when ε goes to 0.
of Lemma 3. 3. Since G ij (x) is non-decreasing, continuous (Lemma 3.14) and bounded by 1, G ij (x) is differentiable on almost all points. That is, the set of all non-differentiable points has Lebesgue measure 0. So it will not change the value of integration if we remove these points. Therefore it is without loss of generality to assume G ij (x) is differentiable for all x. Let g ij (x) be the derivative of G ij (x), i.e. probability density function for bidder i's bidding on item j. Using Lemma 3.4, we have The second inequality follows by the law of total probability, and the third is due to Lemmas 3.7 and 3. 15. By Lemma 3.18

and the fact that
.
By concentrating on a specific item j, let S x be the set of bidders so that x is in their support. We next show that |S Let h ij = min{x|F ij = 1} (we use minimum instead of infimum, since, by Lemma 3.14, F ij is continuous). By definition h ij should be in some bidder's support. Lemma 3.17, for all x ∈ (0, o j − A j ], x is in some bidder's support and by Lemma 3.13, there are at least 2 bidders such that x is in their supports. By the definition of derivative, for all i ∈ S x , g ij (x) = 0. Similarly, we have g ij (x) > 0 and G ij (x) > 0 for all i ∈ S x by definition 3. 11. Moreover, for every i ∈ S x , x is good for bidder i and item j, since x is in their support. So, for any fixed , and according to Proposition 3.5, Merging all these inequalities,

Proof of Inequality (4)
In this section we prove the following technical lemma.

Lemma 3.19. For any CDF F and any real
In order to obtain a lower bound for R(F, v) as stated in the lemma, we show first that we can restrict attention to cumulative distribution functions of a simple special form, since these constitute worst cases for R(F, v). In the next lemma, for an arbitrary CDF F we will define a simple piecewise linear functionF that satisfies the following two properties: Once we establish this, it is convenient to lower bound R(F , v) for the given type of piecewise linear functionsF . Lemma 3.20. For any CDF F and real v > 0, there always exists another CDFF such that Proof. For any CDF F and real v > 0, there always exists another CDFF such that First notice that max x≥0 {F (x) · v − x} = A. By the definition of Riemann integration, we can represent the integration as the limit of Riemann sums. For any positive integer l, let R l be the Riemann sum if we partition the interval [0 . For any given l, let i * be the index such that i>i * ( We defineF l as follows: It is straight-forward to check thatF (x) = lim l→∞Fl (x), as described in the statement of the lemma. We will show that for any l, R l (F, v) ≥ R l (F l , v). Then the lemma follows by taking the limit, since R l (F, v) → R(F, v), and R l (F , v) → R(F , v). Figure 1(a) illustratesF (x) (when we take the limit of l to infinity).
By the construction ofF l , it is easy to check that l−1 i=0 F (x i ) = l−1 i=0F l (x i ) and max x {F l (x)·v− x} = A. Then in order to prove R l (F, v) ≥ R l (F l , v), it is sufficient to prove that l−1 show thatF l (x) has the minimum value for the expression l−1 i=0 F l (x i ) within Q.
Assume on the contrary that some other function Q ∈ Q has the minimum value for l−1 i=0 Q(x i ) within Q and Q(x j ) =F l (x j ) for some x j . Let i 1 be the smallest index such that Q(x i 1 ) > 0 and i 2 be the largest index such that Q(x i 2 ) < (x i 2 + A)/v. By the monotonicity of Q, we have i 1 ≤ i 2 .
Due to the assumption that Q(x j ) =F l (x j ) for some x j and l−1 Figure 1(b) shows how we modify Q to Q . It is easy to check Q ∈ Q and l−1 Q (x i ) which contradicts the optimality of Q. The inequality holds because of of Lemma 3. 19. By Lemma 3.20, for any fixed v > 0, we only need to consider the CDF's in the following form: for any positive A and x 0 such that By optimizing over t, the above formula is minimized when t = λ 2 ≤ 1. That is,

Multi-unit Auctions
In this section, we propose a randomized all-pay mechanism for the multi-unit setting, where m identical items are to be allocated to n bidders. Markakis and Telelis [18] and de Keijzer et al. [9] have studied the price of anarchy for several multi-unit auction formats. The current best upper bound obtained was 1.58 for both pure and mixed Nash equilibria.
We propose a randomized all-pay mechanism that induces a unique pure Nash equilibrium, with an improved price of anarchy bound of 4/3. We call the mechanism Random proportional-share allocation mechanism (PSAM), as it is a randomized version of Kelly's celebrated proportional-share by using Birkhoff-von Neumann decomposition theorem such that x i ≤ X ≤ x i and the expectation of sampling X i is x i ; else Set X = 0 and p = 0; Return X i and p i for all i ∈ [n]; allocation mechanism for divisible resources [15]. The mechanism works as follows (illustrated as Mechanism 1).
Each bidder submits a non-negative real b i to the auctioneer. After soliciting all the bids from the bidders, the auctioneer associates a real number x i with bidder i that is equal to x i = m·b i i∈[n] b i . Each player pays their bid, p i = b i . In the degenerate case, where i b i = 0, then x i = 0 and p i = 0 for all i.
We turn the x i 's to a random allocation as follows. Each bidder i secures x i items and gets one more item with probability x i − x i . An application of the Birkhoff-von Neumann decomposition theorem guarantees that given an allocation vector (x 1 , x 2 , . . . , x n ) with i x i = m, one can always find a randomized allocation 1 with random variables X 1 , X 2 , . . . , X n such that E[X i ] = x i and Pr[ x i ≤ X i ≤ x i ] = 1 (see for example [11,4]) . We next show that the game induced by the Random PSAM when the bidders have submodular valuations is isomorphic to the game induced by Kelly's mechanism for a single divisible resource when bidders have piece-wise linear concave valuations. For convenience, we review the definition of isomorphism between games as appears in Monderer and Shapley [19]. be the strategy sets in Γ k , and let (u i k ) i∈[n] be the utility functions in Γ k . We say that Γ 1 and Γ 2 are isomorphic if there exists bijections φ i : a i 1 → a i 2 , i ∈ [n] such that for every i ∈ [n] and every (a 1 , a 2 , . . . , a n ) ∈ × i∈[n] A i 1 , u i 1 (a 1 , a 2 , . . . , a n ) = u i 2 (φ 1 (a 1 ), φ 2 (a 2 ), . . . , φ n (a n )) .

Theorem 4.2.
Any game induced by the Random PSAM applied to the multi-unit setting with submodular bidders is isomorphic to a game induced from Kelly's mechanism applied to a single divisible resource with piece-wise linear concave functions. Proof. For each bidder i's submodular valuation function f i : {0, 1, . . . , m} → R + , we associate a concave function g i : [0, 1] → R + such that, Essentially, g i is the piecewise linear function that comprises the line segments that connect f i (k) with f i (k + 1), for all nonnegative integers k. It is easy to see that g i is concave if f i is submodular (see also Fig. 4 for an illustration). We use identity functions as the bijections φ i of Definition 4.1. Therefore, it suffices to show that, for any pure strategy profile b, u i (b) = u i (b), where u i and u i are the bidder i's utility functions in the first and second game, respectively. Let We next show an equivalence between the optimal welfares. Lemma 4. 3. The optimum social welfare in the multi-unit setting, with submodular valuations f = (f 1 , . . . , f n ), is equal to the optimal social welfare in the divisible resource allocation setting with concave valuations g = (g 1 , .

Proof. For any valuation profile v and (randomized) allocation
. Note that for any fractional allocation x, such that j x j = m, Theorem 4.2 and Lemma 4.3, allow us to obtain the existence and uniqueness of the pure Nash equilibrium, as well as the price of anarchy bounds of Random PSAM by the corresponing results on Kelly's mechanism for a single divisible resource [14]. Moreover, it can be shown that there are no other mixed equilibria by adopting the arguments of [5] for Kelly's mechanism. The main conclusion of this section is summarized in the following Corollary.
Corollary 4. 4. Random PSAM induces a unique pure Nash equilibrium when applied to the multiunit setting with submodular bidders. Moreover, the price of anarchy of the mechanism is exactly 4/3.

Single item auctions
In this section, we study mixed Nash equilibria in the single item all-pay auction. First, we measure the inefficiency of mixed Nash equilibria, showing tight results for the price of anarchy. En route, we also show that the price of anarchy is 8/7 for two players. Then we analyze the quality of two other important criteria, the expected revenue (the sum of bids) and the quality of the expected highest submission (the maximum bid), which is a standard objective in crowdsourcing contests [6]. For these objectives, we show a tight lower bound of v 2 /2, where v 2 is the second highest value among all bidders' valuations. In the following, we drop the word expected while referring to the revenue or to the maximum bid.
We quantify the loss of revenue and the highest submission in the worst-case equilibria. We show that the all-pay auction achieves a 2-approximation comparing to the conventional procurement (modeled as the first price auction), when considering worst-case mixed Nash equilibria; we show in Appendix B that the revenue and the maximum bid of the conventional procurement equals v 2 in the worst case. We also consider other structures of rewards allocation and conclude that allocating the entire reward to the highest bidder is the only way to guarantee the approximation factor of 2. Roughly speaking, allocating all the reward to the top prize is the optimal way to maximize the maximum bid and revenue among all the prior-free all-pay mechanisms where the designer has no prior information about the participants. Throughout this section we assume that the players are ordered based on decreasing order of their valuations, i.e. v 1 ≥ v 2 ≥ . . . ≥ v n . We also drop the word expected when referring to the revenue or to the maximum bid.
, where players k + 1 through n bid zero with probability 1. W.l.o.g., we assume that v 1 = 1 and v i = v > 0, for 2 ≤ i ≤ k. Let P 1 be the probability that bidder 1 gets the item in any such mixed Nash equilibrium denoted by B. Then the expected utility of bidder 1 in b ∼ B can be expressed by Based on the characterization in [1], no player would bid above v in any Nash equilibrium and nobody bids exactly v with positive probability. Therefore, if player 1 deviates to v, she will gets the item with probability 1. By the definition of Nash equilibrium, we It has been shown in the proof of Theorem 2C in [1], that E[b 1 ] is minimized when players 2 through k play symmetric strategies. Following their results, we can extract the following equations (for a specific player i): Recall that G i (x) is the CDF according to which player i bids in B. Since players 2 through k play symmetric strategies, G i (x) should be identical for i = 1. Then, for some i = 1, Note that 1 − v + x ≤ 1, and so we get is minimized for v ≈ 0.5694 and therefore, the price of anarchy is at most T (0.5694) ≈ 1.185. Particularly, for two players, E[SW (b)] ≥ 1 − v/2 + v 2 /2, which is minimized for v = 1/2 and therefore the price of anarchy for two players is at most 8/7.
Lower bound: Consider n players, with valuations v 1 = 1 and v i = v > 0, for 2 ≤ i ≤ n. Let B be the Nash equilibrium, where bidders bid according to the following CDFs, is the probability of bidder i getting the item when she bids x, for every bidder i.
If player 1 bids any value x ∈ [0, v], her utility is Bidding greater than v is dominated by bidding v. If any player i = 1 bids any value x ∈ [0, v], her utility is Bidding greater than v results in negative utility. Hence, B is a Nash equilibrium. Let P 1 be the probability that bidder 1 gets the item in B, then When n goes to infinity, , which for v = 1/2 results in price of anarchy at least 8/7. Theorem 5. 2. In any mixed Nash equilibrium of the single-item all-pay auction, the revenue and the maximum bid are at least half of the second highest valuation. Proof. Let k be any integer greater or equal to 2, such that be the CDF of the maximum bid h. By the characterization of [1], in any mixed Nash equilibrium, players with valuation less than v 2 do not participate (always bid zero) and there exist two players 1, i bidding continuously in the interval [0, v 2 ]. Then, by [1], for any x ∈ (0, v 2 ]. Therefore, we get In the proof of Theorem 2C in [1], it is argued that G i 1 (x) is maximized (and therefore the maximum bid is minimized) when all the k players play symmetrically (except for the first player, Finally we get The same lower bound also holds for the revenue, which is at least as high as the maximum bid. This lower bound is tight for the maximum bid, as indicated by our analysis, when k goes to infinity and for the symmetric mixed Nash equilibrium. In the next lemma, we show that this lower bound is also tight for the revenue. Lemma 5.3. For any > 0, there exists a valuation vector v = (v 1 , . . . , v n ), such that in a mixed Nash equilibrium of the induced single-item all-pay auction, the revenue and the maximum bid is at most v 2 /2 + . Proof. In [1], the authors provide results for the revenue in all possible equilibria. For the case that v 1 = v 2 , the revenue is always equal to v 2 . To show a tight lower bound, we consider the case where v 1 > v 2 and there exist k players with valuation v 2 playing symmetrically in the equilibrium, by letting k go to infinity. For this case, based on [1], the revenue is equal to 2 when k goes to infinity. By substituting we get, By taking limits, we finally derive that lim v→0 The same tightness result also holds for the maximum bid, which is at most the same as the revenue.
Finally, the next theorem indicates that allocating the entire reward to the highest bidder is the best choice. In particular a prior-free all-pay mechanism is presented by a probability vector q = (q i ) i∈[n] , with i∈[n] q i = 1, where q i is the probability that the i th highest bidder is allocated the item, for every i ≤ n.
Theorem 5. 4. For any prior-free all-pay mechanism that assigns the item to the highest bidder with probability strictly less than 1, i.e. q 1 < 1, there exists a valuation profile and mixed Nash equilibrium such that the revenue and the maximum bid are strictly less than v 2 /2. Proof. We will assert the statement of the theorem for the valuation profile (1, v, 0, 0, . . . , 0), where v ∈ (0, 1) is the second highest value. It is safe to assume that q 2 ∈ [0, q 1 ) 3 . We show that the following bidding profile is a mixed Nash equilibrium. The first two bidders bid on the interval [0, v(q 1 − q 2 )] and the other bidders bid 0. The CDF of bidder 1's bid is G 1 (x) = x v(q 1 −q 2 ) and the CDF of bidder 2's bid is G 2 (x) = x/(q 1 − q 2 ) + 1 − v. It can be checked that this is a mixed Nash equilibrium by the following calculations. For every bid x ∈ [0, v(q 1 − q 2 )], When v goes to 0, the revenue go to v(q 1 − q 2 )/2 < v/2 since q 1 − q 2 < 1. Obviously, the same happens with the maximum bid, which is at most the same as the revenue.
In the above inequalities we used that r > 1 and r 2 ≤ H i (G,g) H j (G,g) . The claim contradicts the assumption that H(G, g) is the minimum, so the lemma holds.
Lemma A. 3. Under constraints (6), if G and g minimize H(·, ·), then for every G i = G j = 1, g i = g j . Proof. For the sake of contradiction, suppose that there exist G i = G j = 1 such that g i = g j . We will prove that for g = ( g i +g j 2 , g i +g j 2 , g −ij ) (i.e. for every k = i, j, g k = g k , and g i = g j = g i +g j 2 ), H(G, g) > H(G, g ).
Notice that for every k = i, j, H k (G, g ) = H k (G, g), since g i + g j = g i + g j and G i = G j = 1. Hence it is sufficient to show that H i (G, g) + H j (G, g) ≥ H i (G, g ) + H j (G, g ). Let A ij = t =j,t =i gt Gt .
H i (G, g) + H j (G, g) − H i (G, g ) − H j (G, g ) + g j (g j + A ij )((g i + g j + 2A ij ) − 2(g i + A ij )) (g j + A ij )(g i + A ij )(g i + g j + 2A ij ) = g i (g i + A ij )(g i − g j ) + g j (g j + A ij )(g j − g i ) (g j + A ij )(g i + A ij )(g i + g j + 2A ij ) = (g i − g j )(g 2 i − g 2 j + A ij (g i − g j )) (g j + A ij )(g i + A ij )(g i + g j + 2A ij ) = (g i − g j ) 2 (g i + g j + A ij ) (g j + A ij )(g i + A ij )(g i + g j + 2A ij ) > 0 , which contradicts the assumption that G and g minimize H(·, ·).
Lemma A. 4. If H i = H j , then: (g i = rg j > 0 and r ≥ 1) ⇒ G i ≥ r 2 G j . Proof. Let A ij = t =j,t =i gt Gt ; then H i = g i g j G j +A ij . By assumption: If g i = g j then 1 G j − 1 G i = 0, so G i = G j . If G i = G j then (g i − g j )(g i + g j + A ij G i ) = 0 . Under constraints (6), A ij G i > 0 and g i , g j > 0, so g i − g j = 0 which results in g i = g j . If g i = rg j , with r ≥ 1 then (g i − g j )A ij ≥ 0 and so 1 G j − r 2 G i ≥ 0, which implies G i ≥ r 2 G j .
which is true by the case assumption. Therefore, L is non-increasing and so it is minimized for g = ∞. Hence, L ≥ k k−1 a ≥ a. Let g i = r i g j , for every i ∈ S. Since j = arg min i∈S g i , then for every i ∈ S, r i ≥ 1. By using Lemma A.4:

B Conventional Procurement
In this section we give bounds on the expected revenue and maximum bid of the single-item firstprice auction. In the following, we drop the word expected when referring to the revenue or to the maximum bid.
Theorem B. 1. In any mixed Nash equilibrium, the revenue and the maximum bid lie between the two highest valuations. There further exists a tie-breaking rule, such that in the worst-case, these quantities match the second highest valuation (This can also be achieved, under the no-overbidding assumption).