An Efficient Auction for Non Concave Valuations - CiteSeerX

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a non concave valuation. It is based on Ausubel's clinching auction. Our main result is stated below. Theorem 1: When all agents but one have concave utility.
An Efficient Auction for Non Concave Valuations Junjik Bae∗ , Eyal Beigman† , Randall Berry∗ , Michael L. Honig∗ , and Rakesh Vohra† ∗ EECS

Department Northwestern University, Evanston, IL 60202 [email protected], {rberry, mh}@ece.northwestern.edu † CMS-EMS, Kellogg School of Management Northwestern University, Evanston, IL 60208 {e-beigman, r-vohra}@northwestern.edu

Abstract—We study an implementation problem for settings where some of the participants have non concave valuations. These valuations are common in the wireless industry in cases where primary users with property rights on a spectrum band would like to lease some of it to low power users. The non concavity void the efficiency results for standard designs of dynamic auctions. Moreover, policy concerns in such settings is often to prevent collusion and fraudulent bidding, therefore static Vickrey mechanisms do not provide the right incentives. We present an alternative mechanism that selects a core outcome that minimizes seller revenue. Such an allocation is efficient in equilibrium, limits the incentives to use shills, maximizes incentives for truthful bidding, and gives a Vickrey outcome whenever the latter is in the core.

I. I NTRODUCTION The prime objective of auction design is to construct a mechanism through which one or more goods are allocated between several agents when there is complete information asymmetry, namely, no agent has information on the preferences of any agent apart from herself. VCG theory shows that in this setting an efficient outcome can be obtained through a truthful mechanism. [1] show that this mechanism is essentially unique. The revenue equivalence theorem guarantees that these merits do not come at the expense of the seller, as her revenue in a second price auction is equal to the expected revenue in a first price auction or in any other reasonable mechanism for allocating a single indivisible good. However, in multiple unit and combinatorial auctions the merits of VCG mechanisms may certainly come at the expense of the seller. By selling through an auction, a seller is committing to both the allocation and the payments dictated by the mechanism, regardless of the share of wealth (namely, the payments) that are transferred to the seller from the overall value that is created by this exchange. Indeed, in some cases the payments to the seller may be zero. This is further aggravated by the vulnerability of VCG mechanisms to various types of misrepresentation and collusion which all are at the expense of the seller. These create an incentive for the seller to deviate from the alleged dominant strategy equilibrium. Such This research was supported in part by NSF under grant CNS-0519935.

deviations would typically increase the welfare of the seller, but decrease the overall efficiency. In the formal construction of the auction, this is, of course, impossible as the seller does not influence the outcome, but in practice the seller has ”outside” options to limit participation and nullify unfavorable outcomes. This is in particular true for wireless markets, where even in the most favorable circumstances there is a positive probability that technical issues may come up, and there is informational asymmetry that favors the seller as to the circumstances in which this happens. Moreover, given that a strategic seller may choose to play these unofficial strategies, other agents have an incentive to deviate from their official strategy as well, hence truthful bidding may no longer be a dominant strategy equilibrium even if the seller does not eventually deviate. One attempt to address these reservations would be to design auctions that would maximize revenue rather than social welfare. The revenue equivalence theory implies that for a single indivisible unit auction this would conflict with both efficiency and truthfulness, [2], [3] show that the same is true in the wider settings. Moreover, it would still leave in place the seller’s outside options through which she could obtain even higher revenues. Our objective is therefore to incorporate the options the seller has de facto into the mechanism and to find optimal allocation and payment schemes under these constraints. In this mechanism, each agent would make a bid which would allegedly be her valuation. The seller would then choose a subset of agents that she would like to service and price them at a level that is below their bid. In this mechanism, the seller has the option to turn down all agents while the agents are committed to any price below their bid. The solution concept in this case is the core, the set of all outcomes, namely, allocations and payments, for which there is no deviation by a coalition of agents that would improve the outcome of all agents in the coalition. In our setting, any coalition to which goods are allocated must include the seller. Thus the core consists of all the outcomes for which there is no coalition of bidders that could create more value than is suggested by the allocation. This in particular implies that the allocation is efficient, the bidders cannot benefit from shill

bids or collusions, and the seller cannot benefit from excluding bidders (though she can of course allocate them nothing). The core in most cases is a large set and the strategic behavior of the agents depends on the way we choose a core allocation. If the bidders bid truthfully, then obviously any allocation that maximizes the collective surplus and charges each agent a notch below her bid is a core allocation that transfers the surplus to the seller. In this case, the agents would bid below their true valuations and the allocation may be inefficient. A. Wireless Markets A central issue for wireless networks is the allocation of scarce radio resources. The traditional approach to this resource allocation problem was to optimize a control parameter, such as total power or throughput. More recent approaches to resource allocation in wireless networks are based on the utility of the agents in the network defined over transmission rates, levels of interference and potential delays. The objective is to implement algorithms for maximizing a target function which can represent welfare (Kelly Maulloo and Tan [4],Liu Berry and Honig [5]), revenue (Acemoglu Ozdaglar and Srikant [6]) or a notion of fairness (Kelly [7]). The resource being allocated is usually bandwidth taken to be a divisible good. While this approach has advantages in networks with few agents, it puts severe restriction on frequency reuse. In wireless settings there are often a very large number of agents that compete for very restricted bandwidth resources, and in these cases frequency reuse is essential. The CDMA technology taps in the need to optimize frequency utilization by spreading transmission on the whole bandwidth. In this case the resource allocation is power and not bandwidth. It is well known that transmission creates externalities within the network as well as to the environment, for this reason these markets tend to be highly regulated. In the usual setting the wireless operator obtains the transmission rights for a certain bandwidth and enables agents to transmit in this frequency within a certain cell. However with the surge in congestion in wireless systems in recent years it has often been the case that primary users have an incentive to lease some of their transmission rights in a secondary market. The primary users are typically broadcasting operators or government agencies that have a bandwidth allocation for emergencies which is latent most of the time. In these cases there are two types of agents, primary agents with a constant transmission level required for its operations and secondary agents which purchase power transmission rights from the primary agent. We elaborate in appendix II on the derivation of the agents utility function, at this point it suffices to say that the secondary agents have a concave utility in power while the primary agent has a non concave utility in power not allocated to secondary use. It should be noted that primary agents could always free ride on each other, and therefore efficiency can only be obtained through proper contracting. For this reason the network is

parceled into cells which contain at most one primary agent. It is reasonable to assume in many cases that the primary agents are static, but even roaming primary agents can be delt with through a breathing cellular network (Bahl Hajiaghayi Jain Mirrokni Qiu and Saberi [8]). B. Main Results The non concavity of the utility of the primary agent or equivalently the complimentarity of the commodity is a problem that has so far been largely ignored by the literature. Ausebel [9] conditions his auction on concave utility and therefore cannot apply, moreover his auction uses one trajectory to discover market clearing price. We construct a new auction, termed the Fallback auction, for multiple identical goods where one designated agent may have a non concave valuation. It is based on Ausubel’s clinching auction. Our main result is stated below. Theorem 1: When all agents but one have concave utility functions, truthful bidding gives a core allocation that minimizes the payment to the seller and is therefore a subgame perfect equilibrium of the Fallback auction. C. Paper Structure In section II we describe the specific setting which motivates our main result. Section III introduces the Fallback auction, section IV gives some examples and section V describes the main result. The result holds in both discrete and continuous time settings. The former requires additional tie breaking rules which add unnecessary difficulties to the analysis. We therefore do the analysis for a continuous time auction. II. U TILITY D ERIVATION IN A W IRELESS S ETTING Our model is motivated by the wireless communication scenario illustrated in Figure 1. There are two transmitterreceiver pairs. The two transmitters spread their power over the same bandwidth, so that the received discrete-time signal at receiver i is √ √ ri = q1 hi1 x1 + q2 hi2 x2 + ni where xj is the transmitted symbol from transmitter j, assumed to have zero mean and unit variance, qj is the associated transmitted power, hij is the channel gain from transmitter j to receiver i, and ni is additive noise with zero mean and variance N0 . For i = 1, the first term on the right-hand side is the desired signal, and the second term is interference. We assume that the utility derived by user i is a function of the received Signal-to-Interference and Noise Ratio (SINR) γi (q) =

N0 +

qh Pi ii

j6=i qj hji

for i = 1, 2

Fig. 1.

A. Ausubel’s Clinching Auction

Wireless Network with Roaming Transmitters

where q = {q1 , q2 } is transmission power. Letting ui (γi ) denote this utility, an efficient allocation of power q maximizes u1 (γ1 ) + u2 (γ2 ). We assume that the transmitters are not cooperative, so that in the absence of any information exchange between the transmitter-receiver pairs, each transmitter would increase its power to a maximum limit. This is in contrast to previous work in which distributed power allocation is considered with cooperative transmitters [10]. There a distributed power allocation algorithm is proposed, which can achieve an efficient allocation; however, that algorithm is not incentive compatible. To address the incentive compatibility issue, we consider an auction mechanism for allocating power. This approach is also discussed in [11] in the context of dynamic spectrum sharing with an exclusive use spectrum sharing model. However, a key difference is that here a particular user’s power is auctioned (allocated) across all participating users. Namely, for the scenario in Figure 1, we assume that the power q2 > 0 is fixed and both users bid for power q1 . That is, user 1 bids for power q11 to increase her own utility, whereas user 2 bids power q12 to decrease the power assigned to user 1, and thereby reduce interference. We assume that 0 ≤ q1 ≤ q1max , i.e. there is a fixed supply of maximum power designated for user 1, so that q11 + q12 ≤ q1max . The utility function u1 (q1 ) is assumed to be a monotonic increasing concave function of q1 . On the other hand, u2 (q1 ) 2 is a decreasing function of q1 (i.e. du dq1 < 0), and it can be a convex function of q1 if the following relation holds, d2 u2 (q1 ) h22 q2 h212 n d2 u2 du2 o > 0, = γ + 2 2 2 2 dq1 (N0 + h12 q1 )3 dγ2 dγ2 or the coefficient of relative risk aversion is − for all γ2 .

γ2 u00 2 (γ2 ) u02 (γ2 )

< 2

If p is the per unit price of the good, strict concavity of Ui implies that arg max Ui (x) − px 0≤x≤1

is unique. Let xi (p) = argmax0≤x≤1 Ui (x), be agent i’s demand at price p. Notice that xi (0) = 1, xi (p) is continuous, strictly decreasing and xi (p) → 0 as p goes to infinity. Initially the price is set at 0 and increased continuously. At each P price p, each bidder i is asked to report xi (p). As n long as i=1 xi (p) > 1, the price is increased. The instant Pn x (p) = 1 the auction terminates. The terminal price is i=1 i called p∗ . Notice that this is the market clearing price. That such a price exists follows from the continuity of the xi (p)’s and the fact that they are decreasing. At each price p ≤ p∗ , agentPi clinches, i.e., is allocated the quantity Ci (p) = [0, 1 − j6=i xj (p)]+ and charged p. Notice that Ci (p) ≤ xi (p). If p−i is the market clearing price when agent i is excluded, it is easy to see that Ci (p) = 0 for all p ≤ p−i . Hence, the total payment of agent i upto price p ≥ p−i will be P Z p Z p d( j6=i xj ) dCi (ρ) dρ = − dρ ρ ρ dρ dρ p−i 0 Notice also that p−i ≤ p∗ and xj (p−i ) ≥ xj (p∗ ) for all j 6= i. The VCG payment of agent i is X X Pi = Uj (xj (p−i )) − Uj (xj (p∗ )) j6=i

=

X

j6=i

[Uj (xj (p )) − Uj (xj (p∗ ))].

j6=i

The trick is to show that these payments can be derived from information revealed during the auction. The first order conditions for optimality imply that uj (xj (p)) = p. Therefore, Pi

=

X

−i

A seller wishes to allocate one unit of a divisible good, in our case transmission power, among n agents. Agent i obtains utility Ui (x) from consuming x units of the good. Utilities are assumed to be quasi-linear with derivative ui . When all Ui are concave, Ausubel’s [9] clinching auction can be used to allocate the good. Sincere bidding is an equilibrium of the auction. At this equilibrium the VCG outcome obtains. We describe the the clinching auction below. For expositional convenience we assume the Ui ’s to be strictly concave.



Uj (xj (p )) − Uj (xj (p )) =

j6=i

III. T HE M ODEL

−i

XZ j6=i

xj (p−i )

uj (x)dx

xj (p∗ )

p∗

X Z p∗ dxj dxj dρ = − ρ dρ dρ dρ −i −i j6=i p j6=i p P Z p∗ X Z p∗ d( j6=i xj ) dxj ρ ( )dρ = − ρ dρ = − dρ dρ p−i p−i

= −

XZ

uj (xj (ρ))

j6=i

This shows that Pi coincides with payments imposed by the auction. Therefore algorithm 1 with concave utilities terminates at p∗ with the VCG outcome. The key insight here is that to compute VCG payments one does not need to know the utilities themselves but the

differences in utilities. These differences can be inferred from how agents reduce their demands as prices in the auction rise.

Pn t and i=1 xi (p) < 1 for all p > p . This means there is excess supply when the auction terminates.

We now modify the description of the auction. We emphasize that the modification does not change the auction, only its description. The change in description will be useful when we adapt the auction to the case when one of the bidders has a non-concave utility function.

When p > pt , x1 (p) = 0. Hence agent 1’s demand at a price that exceeds pt can be less than what he has clinched. In the clinching auction an agent is not allowed to relinquish his clinch. If agent 1’s utility were convex, his surplus is forced to be zero or negative creating an incentive to deviate from sincere bidding.

In the auction, at each price p we ask agent i to report argmax0≤x≤1 [Ui (x) − px] which ignores the fact that agent i may already have clinched something by the time price in the auction reaches p. For example, if agent i has clinched q units by the time price has reached p, it would be more natural to ask the agent to report the quantity argmax0≤x≤1 [U (x+q)−px]. In words, the additional amount above the amount clinched that would maximize her surplus. Concavity of the utility functions means that the two reports would differ by q, so making no difference to the outcome of the auction. More generally, the surplus that agent i enjoys from x units when the price at auction is p is Z p dCi (ρ) + dρ. πi (x, p) = Ui (x) − p · [x − Ci (p), 0] − ρ dρ 0 Now we redefine xi (p) to be arg max0≤x≤1 S(x, p) and at each price p during the auction we ask each agent i to report xi (p). Concavity of the utility function means that throughout the auction xi (p) ≥ Ci (p). Algorithm 1 Clinching Auction Initialization: p ← 0 ; Ui , Uji ← 0 ; xj , x− j ← 1 for i, j = 1, . . . , n Dynamic: Pn while i=1 xj > 1 do p ← p + ∆p Ask each agent her demand xi for price p. P − xi ← max{xi , 1 − j6=i x− j } ; xi ← xi for i = 1, . . . , n UiP ← Ui + p · ∆xj if j6=i xj = 1 then Uji ← Uj for j 6= i end if end while P Pi ← j6=i (Uj − Uji ) for i = 1, . . . , n Return (x1 , . . . , xn ) and (P1 , . . . , Pn )

B. The Fallback Auction The Fallback auction modifies Ausubel’s clinching auction to account for the presence of a single agent with a non-concave utility function, henceforth called agent 1. All other agents have strictly concave Pn utilities. In this case there may be no price p such that i=1 xi (p) = 1. The difficulty arises because x1 (p) while non-increasing, may have jumpPdiscontinuities. n Therefore, there is a pt such that limp%pt i=1 xi (p) ≥ 1

The Fallback auction surmounts this difficulty by allowing agent 1 and only agent 1, to relinquish some of the units he has already clinched. If a fallback price is reached, we allow agent 1 to choose some smaller quantity clinched earlier in the auction. The remaining units must then be reallocated to the other agents. t Definition is called a fallback price Pn1: p limp%pt i=1 xi (p) ≥ 1 and x1 (p) = 0 for all p > pt .

if

To understand when a fallback occurs, suppose that agent 1 decides to fallback at some price p. At this point agent 1 is free to fallback to an earlier clinched quantity, i.e., C1 (p0 ) where p0 < p. Agent 1’s surplus from choosing C1 (p0 ) R p0 1 (ρ) dρ. Clearly agent 1 would would be U1 (C1 (p0 ))− p−1 ρ dCdρ 0 choose the value of p that would maximize this quantity. This motivates the following definition. Definition 2: The price p∗ that solves Z p dC1 (ρ) dρ max U1 (C1 (p)) − ρ dρ pt ≥p≥p−1 −1 p is called the security price. The security price is the price that agent 1 chooses to maximize the payoff when it falls back. Therefore, the following equality hols at the fallback price. Z p∗ dC1 (ρ) t t ∗ π1 (x1 (p ), p ) = U1 (C1 (p )) − ρ dρ dρ p−1 = π1 (x1 (p∗ ), p∗ ) Note that in the definition of security price, the maximization could be taken over the interval [p−1P , ∞). If the fallback n auction terminates in a price p where i=1 xi (p) = 1, the resulting allocation and payments coincide with those of the clinching auction. In particular, the VCG outcome is obtained. If the fallback auction terminates in a fallback price, pt , then we allocate to each agentPi ≥ 2 the quantity xi (p∗ ) and to agent 1 the amount 1 − i≥2 xi (p∗ ). The payment agent 1 makes is Z p∗ dC1 (ρ) dρ. P1 = ρ dρ 0 To describe the payment that agent i ≥ 2 makes we must first modify the definition of p−i . If the fallback auction applied to agents in N \ i terminates in a market clearing price than

1 2 3 0 0 7 4 2 0 4 1 0 TABLE I AGENT A CONVEX , B AND C

p−i is that price otherwise it is the relevant security price. The payment that agent i ≥ 2 makes is Z pt dxi t dρ Pi = p · limt xi (p) − ρ p%p dρ ∗ p

A B C

= pt · limt xi (p) + Ui (xi (p∗ )) − Ui (xi (pt )) p%p

0≤p 2 agents, the truthful bidding outcome of the fallback auction reaches core allocation with the seller’s minimum payoff. From the argument of Whinston [14], it follows that the bidding strategies of the agents according to the truncation profile is a full information equilibrium. Lemma 2: Assuming the truthful bidding among the agents, the allocation of the fallback auction is efficient. Lemma 3: Suppose n = 2 and agent 1 has an increasing convex utility function. Then the truthful bidding is an ex post perfect equilibrium of the fallback auction that charges its VCG payment to each agent. Proof : When the fallback flag is up, the payment of agent 1 is Z p∗ Z p∗ dC1 (ρ) dC1 (ρ) P1 = ρ dρ = ρ dρ dρ dρ 0 p−1 = U2 (x2 (p−1 )) − U2 (x2 (p∗ )), which is by definition the VCG payment of agent 1. In addition, the payment of agent 2 is given by equation (1): P2 (p∗ ) = pt · limt x2 (p) + U2 (x2 (p∗ )) − U2 (x2 (pt )). p%p

From the definition of the fallback price, π1 (x1 (pt ), pt ) = π1 (x1 (p∗ ), p∗ ). Namely, Z

pt

dC1 (ρ) dρ dρ 0 p∗ dC1 (ρ) ρ dρ, dρ −1 p

U1 (1) − pt · (1 − C1 (pt )) − Z ∗ = U1 (C1 (p )) −

ρ

or t

t



Z

pt

U1 (1) − p · x2 (p ) = U1 (x1 (p )) +

ρ p∗

dC1 (ρ) dρ dρ

= U1 (x1 (p∗ )) + U2 (x2 (p∗ )) − U2 (x2 (pt )). Here, we use the following facts in the auction: x1 (pt ) = 1 and C1 (pt ) = 1 − x2 (pt ). Therefore, the payment of agent 2 becomes P2 = U1 (1) − U1 (x1 (p∗ )) = U1 (x1 (p−2 )) − U1 (x1 (p∗ )), and this is exactly the VCG payment of agent 2. For n = 2, the fallback auction with truthful bidding reaches the efficient allocation with VCG payment. From [15], any incentive compatible, individually rational and efficient mechanism must charge VCG payments. Therefore, for n = 2, the truthfull bidding is an ex post perfect equilibrium of the fallback auction that returns VCG outcome 2 In [16], Day and Milgrom argue that core-selecting auctions that minimize the seller’s payoff maximize incentives for truthful reporting and they produce the Vickery outcome when it lies in the core. Theorem 3 in [17], especially, show that truncation report is a full information equilibrium. From the following Lemma, we show that the fallback auction minimizes the seller’s payoff with the core allocation for all agents and the seller. Lemma 4: For n > 2, if all agents bid truthfully, the fallback auction finds an imputation in the core with minimum payoff of the seller. Therefore, the bidding strategies of the agents according to the profile of πi truncation of Ui (xi ) is a full information equilibrium in the fallback auction and this leads to the efficient outcome. Corollary 5: The bidding strategies of the agents according to the profile of πi truncation of Ui (xi ) is a full information equilibrium in the fallback auction.

VI. C ONCLUSIONS AND O PEN P ROBLEMS 1) Fallback Auction for more than one non concave valuations. 2) Minimal Information elicitation required by iterated auctions for obtaining efficient outcomes.

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