The Combinatorial World (of Auctions) According to GARP
aa r X i v : . [ c s . G T ] J u l The Combinatorial World (of Auctions) Accordingto GARP
Shant Boodaghians and Adrian Vetta Department of Mathematics and Statistics, McGill University [email protected] Department of Mathematics and Statistics, and School of Computer Science, McGillUniversity [email protected]
Abstract.
Revealed preference techniques are used to test whether a dataset is compatible with rational behaviour. They are also incorporated as con-straints in mechanism design to encourage truthful behaviour in applicationssuch as combinatorial auctions. In the auction setting, we present an efficientcombinatorial algorithm to find a virtual valuation function with the opti-mal (additive) rationality guarantee. Moreover, we show that there existssuch a valuation function that both is individually rational and is minimum(that is, it is component-wise dominated by any other individually ratio-nal, virtual valuation function that approximately fits the data). Similarly,given upper bound constraints on the valuation function, we show how tofit the maximum virtual valuation function with the optimal additive ra-tionality guarantee. In practice, revealed preference bidding constraints arevery demanding. We explain how approximate rationality can be used tocreate relaxed revealed preference constraints in an auction. We then showhow combinatorial methods can be used to implement these relaxed con-straints. Worst/best-case welfare guarantees that result from the use of suchmechanisms can be quantified via the minimum/maximum virtual valuationfunction.
Underlying the theory of consumer demand is a standard rationality as-sumption: given a set of items with price vector p , a consumer will demandthe bundle x of maximum utility whose cost is at most her budget B . Offundamental import, therefore, is whether or not the decision making be-haviour of a real consumer is consistent with the maximization of a utilityfunction. Samuelson [18,19] introduced revealed preference to provide a the-oretical framework within which to analyse this question. Furthermore, thisconcept now lies at the heart of current empirical work in the field; see, forexample, Gross [11] and Varian [23]. Specifically, Samuelson [18] conjecturedthat the weak axiom of revealed preference ( warp ) was a necessary and suf-ficient condition for integrability – the ability to construct a utility functionwhich fits observed behaviour.owever, Houtthakker [14] proved that the weak axiom was insufficient.Instead, he presented a strong axiom of revealed preference ( sarp ) andshowed non-constructively that it was necessary and sufficient in the casewhere behaviour is determined via a single-valued demand function. Afriat [1]provided an extension to multi-valued demand functions – where ties are al-lowed – by showing that the generalized axiom of revealed preference ( garp )is necessary and sufficient for integrability. Furthermore, Afriat’s approachwas constructive (producing monotonic, concave, piecewise-linear utility func-tions) and applied to the setting of a finite collection of observational data.This rendered his method more suitable for practical use.In addition to its prominence in testing for rational behaviour, revealedpreference has become an important tool in mechanism design. A notablearea of application is auction design. For combinatorial auctions, Ausubel,Cramton and Milgrom [4] proposed bidding activity rules based upon warp .These rules are now standard in the combinatorial clock auction, one ofthe two prominent auction mechanisms used to sell bandwidth. In part,the warp bidding rules have proved successful because they are extremelydifficult to game [6]. Harsha et al. [13] examine garp -based bidding rules,and Ausubel and Baranov [3] advocate incorporating such constraints intobandwidth auctions. Based upon Afriat’s theorem, these garp -based rulesimply that there always exists a utility function that is compatible withthe bidding history. This gives the desirable property that a bidder in anauction will always have at least one feasible bid – a property that cannotbe guaranteed under warp .Revealed preference also plays a key role in motivating the generalisedsecond price mechanism used in adword auctions. Indeed, these position auc-tions have welfare maximizing solutions with respect to a revealed preferenceequilibrium concept; see [23] and [8].
Multiple methods have been proposed to approximately measure how con-sistent a data set is with rational behaviour; see Gross [11] for a comparisonof a sample of these approaches. In this paper, we show how a graphicalviewpoint of revealed preference can be used to obtain a virtual valuationfunction that best fits the data set. Specifically, we show in Section 3 thatan individually rational virtual valuation function can be obtained such thatits additive deviation from rationality is exactly the minimum mean length Afriat [1] gave several equivalent necessary and sufficient conditions for integrability.One of these, cyclical consistency , is equivalent to garp as shown by Varian [21].
2f a cycle in a bidding graph. This additive guarantee cannot be improvedupon. Furthermore, we show there exists a unique minimum valuation func-tion from amongst all individually rational virtual valuation functions thatoptimally fit the data. Similarly, given a set of upper bound constraints, weshow how to find the unique maximum virtual valuation that optimally fitsthe data, if it exists.Imposing revealed preference bidding rules can be harsh. Indeed, Cram-ton [6] states that “there are good reasons to simplify and somewhat weakenthe revealed preference rule”. These reasons include complexity issues, com-mon value uncertainty, the complication of budget constraints, and the factthat a bidder’s assessment of her valuation function often changes as theauction progresses! The concept of approximate rationality, however, natu-rally induces a relaxed form of revealed preference rules. We examine suchrelaxed bidding rules in Section 4, show how they can be implemented com-binatorially, and show how to construct the minimal and maximal valuationfunctions which fit the data, which may be useful for quantifying worst/best-case welfare guarantees.
In this section, we first review revealed preference. We then examine its usein auction design and describe how to formulate it in terms of a biddinggraph.
The standard revealed preference model instigated by Samuelson [18] is asfollows. We are given a set of observations { ( B , p , x ) , ( B , p , x ) , . . . , ( B T , p T , x T ) } . At time t , 1 ≤ t ≤ T , the set of items has a price vector p t and the consumerchooses to spend her budget B t on the bundle x t . We say that bundle x t is (directly) revealed preferred to bundle y , denoted x t (cid:23) y , if y wasaffordable when x t was purchased. We say that bundle x t is strictly revealedpreferred to bundle y , denoted x t ≻ y , if y was (strictly) cheaper than x t when x t was purchased. This gives revealed preference (1) and strict revealed It is not necessary to present the model in terms of “time”. We do so because this bestaccords with the combinatorial auction application. reference (2): p t · y ≤ p t · x t ⇒ x t (cid:23) y (1) p t · y < p t · x t ⇒ x t ≻ y (2)Furthermore, a basic assumption is that the consumer optimises a locallynon-satiated utility function. Consequently, at time t she will spend her en-tire budget, i.e. , p t · x t = B t . In the absence of ties, preference orderings giverelations that are anti-symmetric and transitive. This leads to an axiomaticapproach to revealed preference formulated in terms of warp and sarp byHouthakker [14]. The weak axiom of revealed preference ( warp ) states thatthe relation should be asymmetric, i.e. x (cid:23) y ⇒ y x . Its transitive clo-sure, the strong axiom of revealed preference ( sarp ) states that the relationshould be acyclic. Our interest lies in the general case where ties are allowed.This produces what we dub the k -th Axiom of Revealed Preference ( karp ):Given a fixed integer k and any κ ≤ k x t = x t (cid:23) x t (cid:23) · · · (cid:23) x t κ − (cid:23) x t κ = y ⇒ y ⊁ x t . (3)There are two very important special cases of karp . For k = 1, this issimply warp , i.e. x t (cid:23) y ⇒ y x t . This is just the basic property thatfor a preference ordering, we cannot have that y is strictly preferred to x t if x t is preferred to y . On the other hand, suppose we take k to be arbitrarilylarge (or simply larger than the total number of observed bundles). Then wehave the Generalized Axiom of Revealed Preference ( garp ), the simultaneousapplication of karp for each value of k . In particular, garp encodes theproperty of transitivity of preference relations. Specifically, for any k , if x t = x t (cid:23) x t (cid:23) · · · (cid:23) x t k − (cid:23) x t k = y , then, by transitivity, x t (cid:23) y . The first axiom of revealed preference thenimplies that y ⊁ x t .The underlying importance of garp follows from a classical result ofAfriat [1]: there exists a nonsatiated, monotone, concave utility functionthat rationalizes the data if and only if the data satisfy garp . Brown andEchenique [5] examine the setting of indivisible goods and Echenique et al.[7] consider the consequent computational implications. Local non-satiation states that for any bundle x there is a more preferred bundlearbitrarily close to x . A monotonic utility function is locally non-satiated, but theconverse need not hold. .2 Revealed Preference in Combinatorial Auctions As discussed, a major application of revealed preference in mechanism designconcerns combinatorial auctions. Here, there are some important distinctionsfrom the standard revealed preference model presented in Section 2.1. First,consumers are assumed to have quasilinear utility functions that are linear inmoney. Thus, they seek to maximise profit. Second, the standard assumptionis that bidders have no budgetary constraints. For example, if profitableopportunities arise that require large investments then these can be obtainedfrom perfect capital markets. (This assumption is slightly unrealistic; Harshaet al. [13] show how to implement a budgeted revealed preference model forcombinatorial auctions; see also Section 4.4).Third, the observations ( p t , x t ), for each 1 ≤ t ≤ T , are typically not pur-chases but are bids made over a collection of auction rounds. When offereda set of prices at time t the consumer bids for bundle x t .So what would a model of revealed preference be in this combinatorialauction setting? Suppose that at time t we select bundle x t and that at anearlier time s we selected bundle x s . Assuming a quasi-linear utility functionand no budget constraint, we have revealed: v ( x t ) − p t · x t ≥ v ( x s ) − p t · x s (4) v ( x s ) − p s · x s ≥ v ( x t ) − p s · x t (5)Summing Inequalities (4) and (5) and rearranging gives( p t − p s ) · x s ≥ ( p t − p s ) · x t (6)This is the revealed preference condition for combinatorial auctions proposedas a bidding activity rule by Ausubel, Crampton and Milgrom [4]. The ac-tivity rule simply states that, between time s and time t , the price of bundle x s must have risen by at least as much as the price of x s . If condition (6) isnot satisfied then the auction mechanism will not allow the later bid to bemade.Observe that the bidding rule (6) was derived directly from the assump-tion of utility maximisation. This unbudgeted revealed preference auctionmodel can, though, also be viewed within the framework of the standardbudgeted model of revealed preference. To do this, we assume the bidderhas an arbitrarily large budget B . In particular, prices will never be sohigh that she cannot afford to buy every item. Second, to model quasi-linear utility functions, we treat money as a good. Specifically, given a bun-dle of items x = ( x , . . . , x n ) and an amount x of money we denote by5 x = ( x , x , . . . , x n ) the concatenation of x and x . If p = ( p , . . . , p n ) isthe price vector for the the non-monetary items, then ˆ p = (1 , p , . . . , p n )gives the prices of all items including money.In this n + 1 dimensional setting, let us select bundle ˆ x t at time t . As thebudget B is arbitrarily large, we can certainly afford the bundle x s at thistime. But we may not be able to afford bundle ˆ x s , as then we must also payfor the monetary component at a cost of B − p s · x s . However, we can affordthe bundle x s plus an amount B − p t · x s of money. Applying revealed prefer-ence to { ˆ x , ˆ p } , we have revealed that ˆ x t = ( B − p t · x t , x t ) (cid:23) ( B − p t · x s , x s ).Hence, by quasilinearity, subtracting the monetary component from bothsides, we have,(0 , x t ) (cid:23) (( B − p t · x s ) − ( B − p t · x t ) , x s ) = ( p t · x t − p t · x s , x s ) . Equivalently, v ( x t ) ≥ v ( x s ) + p t · x t − p t · x s . (7)But Inequality (7) is equivalent to Inequality (4). Inequality (5) follows sym-metrically, and together these give the revealed preference bidding rule (6).Note that this bidding rule is derived via the direct comparison of two bun-dles.We can now extend this bidding rule to incorporate indirect comparisonsin a similar fashion to the extension from warp to sarp via transitivity.This produces a garp -based bidding rule. Namely, suppose we bid for themoney-less bundle x i at time t i , for all 0 ≤ i ≤ k , where 1 ≤ t i ≤ T . Thuswe have revealed that(0 , x i ) (cid:23) (( B − p i · x i +1 ) − ( B − p i · x i ) , x i +1 )= ( p i · x i − p i · x i +1 , x i +1 )This induces the inequality v ( x i ) − p i · x i ≥ v ( x i +1 ) − p i · x i +1 . (8)Summing (8) over all i , we obtain k X i =0 ( v ( x i ) − p i · x i ) ≥ k X i =0 ( v ( x i +1 ) − p i · x i +1 ) , where the sum in the subscripts are taken modulo k . Rearranging now givesthe combinatorial auction karp -based bidding activity rule:( p k − p ) · x ≥ k X i =1 ( p i − p i − ) · x i . (9)6or k arbitrarily large, this gives the garp -based bidding rule. In orderto qualitatively analyze the consequences of imposing karp -based activityrules, it is informative to now provide a graphical interpretation of the theserules. Given the set of price-bid pairings { ( p t , x t ) : 1 ≤ t ≤ T } , we create adirected graph G = ( V, A ), called the bidding graph , to which we will assignarc lengths ℓ . There is a vertex in V for each possible bundle – that is, thereare 2 n bundles in an n -item auction. For each observed bid x t , 1 ≤ t ≤ T ,there is an arc ( x t , y ) for each bundle y ∈ V . In order to define the length ℓ x t , y of an arc ( x t , y ), note that Inequality (4) applied to x s = y gives v ( y ) ≤ v ( x t ) + p t · ( y − x t ) , otherwise we would prefer bundle y at time t . For the arc length, we wouldlike to simply set ℓ x t , y = p t · ( y − x t ). Observe, however, that the bundle x t may be chosen in more than one time period. That is, possibly x t = x t ′ for some t = t ′ . Therefore the bidding graph is, in fact, a multigraph. Itsuffices, though, to represent only the most stringent constraints imposed bythe bidding behaviour. Thus, we obtain a simple graph by setting ℓ x t , y = min t ′ { p t ′ · ( y − x t ) : x t ′ = x t } . Now the warp -based bidding rule (6) of Ausubel et al. [4] is equivalent to( p t − p s ) · x s − ( p t − p s ) · x t ≥ . However, ℓ x s , x t + ℓ x t , x s = min s ′ { p s ′ · ( x t − x s ) : x s ′ = x s } + min t ′ { p t ′ · ( x s − x t ) : x t ′ = x t }≤ p s · ( x t − x s ) + p t · ( x s − x t )= ( p t − p s ) · x s − ( p t − p s ) · x t . It is then easy to see that the bidding constraint (6) is violated if and onlyif the bidding graph contains no negative digons (cycles of length two). Fur-thermore, we can interpret karp and garp is a similar fashion. Hence, the k -th axiom of revealed preference is equivalent to requiring that the biddinggraph not contain any negative cycles of cardinality at most k + 1, and garp is equivalent to requiring no negative cycles at all. Thus, we can formalize7he preference axioms in terms of the lengths of negative cycles in a directedgraph. We remark that a cyclic view of revealed preference is briefly outlinedby Vohra [25]. For us, this cyclic formulation has important consequences intesting for the extent of bidding deviations from the axioms. We will quan-tify this exactly in Section 3. Before doing so, though, we remark that thefocus on cycles also has important computational consequences.First, recall that the bidding graph G contains an exponential numberof vertices, one for every subset of the items. Of course, it is not practical towork with such a graph. Observe, however, that a bundle y / ∈ { x , x . . . , x T } has zero out-degree in G . Consequently, y cannot be contained in any cy-cle. Thus, it will suffice to consider only the subgraph induced by the bids { x , x . . . , x T } . In a combinatorial auction there is typically one bid pertime period and the number of periods is quite small. Hence, the inducedsubgraph of the bidding graph that we actually need is of a very manageablesize.Second, one way to implement a bidding rule is via a mathematical pro-gram; see, for example, Harsha et al. [13]. The cyclic interpretation of abidding rule has two major advantages: we can test the rule very quickly bysearching for negative cycles in a graph. For example, we can test for negativecycles of length at most k + 1 either by fast matrix multiplication or directlyby looking for shortest paths of length k using the Bellman-Ford algorithmin O ( T ) time. Another major advantage is that a bidder can interpret theconsequence of a prospective new bid dynamically by consideration of thebidding graph. This is extremely important in practice. In contrast, biddingrules that require using an optimization solver as a black-box are very opaqueto bidders. For combinatorial auctions, Afriat’s result that garp is necessary and suffi-cient for rationalisability can be reformulated as:
Theorem 3.1.
A valuation function which rationalises bidding behaviourexists if and only if the bidding graph has no negative cycle.
This is a simple corollary of Theorem 3.2 below; see also [25]. From an eco-nomic perspective, however, what is most important is not whether agentsare perfectly rational but “whether optimization is a reasonable way to de- For example, in a bandwidth auction there are at most a few hundred rounds. It is then important to study the consequencesof approximately rational behaviour, see, for example, Akerlof and Yellen [2].First, though, is it possible to quantify the degree to which agents are ratio-nal? Gross [11] examines assorted methods to test the degree of rationality.Notable amongst them is the
Afriat Efficiency Index [1,22]. Here the condi-tion required to imply a preference is strengthened multiplicatively. Specifi-cally, x t (cid:23) y only if p t · y ≤ λ · p t · x t where λ <
1. We examine this index withrespect to the bidding graph in Section 4.4. For combinatorial auctions, avariant of this constraint was examined experimentally by Harsha et al. [13].Here we show how to quantify exactly the degree of rationality presentin the data via a parameter of the bidding graph. Moreover, we are able togo beyond multiplicative guarantees and obtain stronger additive bounds.To wit, we say that ˆ v is an ǫ -approximate virtual valuation function if, forall t and for any bundle y ,ˆ v ( x t ) − p t · x t ≥ ˆ v ( y ) − p t · y − ǫ . Note that if ǫ = 0, then the bidder is optimizing with respect to a virtualvaluation function, i.e. is rational. We remark that the term virtual reflectsthe fact that ˆ v need not be the real valuation function (if one exists) of thebidder, but if it is then the bidding is termed truthful . We now examine exactly when a bidding strategy is approximately rational.It turns out that the key to understanding approximate deviations fromrationality is the minimum mean cycle in the bidding graph. Given a cycle C in G , its mean length is µ ( C ) = P a ∈ C ℓ a | C | . We denote by µ ( G ) = min C µ ( C ) the minimum mean length of a cycle in G ,and we say that C ∗ is a minimum mean cycle if C ∗ ∈ argmin C µ ( C ). We canfind a minimum mean cycle in polynomial time using the classical techniquesof Karp [15]. Theorem 3.2. An ǫ -approximate valuation function which (approximately)rationalises bidding behaviour exists if and only if the bidding graph hasminimum mean cycle µ ( G ) ≥ − ǫ . Indeed, several schools of thought in the field of bounded rationality argue that peopleutilize simple (but often effective) heuristics rather than attempt to optimize; see, forexample, [10]. roof. From the bidding graph G we create an auxiliary directed graphˆ G = ( ˆ V , ˆ A ) with vertex set ˆ V = { x , x , . . . , x T } . The arc set is completewith arc lengths ˆ ℓ x s , x t = ℓ x s , x t − µ ( G ) . Observe that, by construction, every cycle in ˆ G is of non-negative length.It follows that we may obtain shortest path distances ˆ d from any arbitraryroot vertex r . Thus, for any arc ( x t , y ), we haveˆ d ( y ) ≤ ˆ d ( x t ) + ˆ ℓ x t , y = ˆ d ( x t ) + ℓ x t , y − µ ( G ) ≤ ˆ d ( x t ) + p t · ( y − x t ) − µ ( G ) . So, if we set ˆ v ( x ) = ˆ d ( x ), for each x , thenˆ v ( x t ) − p t · x t ≥ ˆ v ( y ) − p t · y + µ ( G ) . for all t . Therefore, by definition of ǫ -approximate bidding, we have that ˆ v is a ( − µ )-approximate virtual valuation function.Conversely, let ˆ v be an ǫ -approximate virtual valuation function whichrationalises the graph, and take some cycle C of minimum mean lengthin the bidding graph. Suppose for a contradiction that µ ( C ) < − ǫ . By ǫ -approximability, we haveˆ v ( x s ) − p s · x s ≥ ˆ v ( x t ) − p s · x t − ǫ . But ℓ x s , x t ≥ p s · ( x t − x s ). Therefore ℓ x s x t ≥ ˆ v ( x t ) − ˆ v ( x s ) − ǫ . Summingover every arc in the cycle we obtain ℓ ( C ) = X ( x , y ) ∈ C ℓ xy ≥ X ( x , y ) ∈ C (ˆ v ( y ) − ˆ v ( x ) − ǫ ) = −| C | · ǫ . Thus µ ( C ) ≥ − ǫ , giving the desired contradiction. ⊓⊔ Recall that, the bidding behaviour is irrational only if µ ( G ) is strictlynegative. We emphasize that Theorem 3.2 applies even when µ ( G ) is positive,but in this case, we have an ǫ -approximate virtual valuation function where ǫ is negative! What does this mean? Well, setting δ = − ǫ , we then have, forall t and for any bundle y , that ˆ v ( x t ) − p t · x t ≥ ˆ v ( y ) − p t · y + δ . Thus, x t is not just the best choice, but it provides at least an extra δ units ofutility over any other bundle. Thus, the larger δ is, the greater our degree ofconfidence in the revealed preference-ordering and valuation.10 .2 Individually Rational Virtual Valuation Functions Theorem 3.2 shows how to obtain a virtual valuation function with the bestpossible additive approximation guarantee: any valuation rationalising thebidding graph G must allow for an additive approximation of at least − µ ( G ).However, there is a problem. Such a valuation function may not actually becompatible with the data; specifically, it may not be individually rational. For individual rationality , we require, for each time t , that ˆ v ( x t ) − p t · x t ≥ r since we have ˆ v ( x r ) = 0.It is possible to obtain an individually rational, approximate, virtualvaluation function simply by taking the ˆ v from Theorem 3.2 and adding ahuge constant to value of each package. This operation, of course, is entirelyunnatural and the resulting valuation function is of little practical value. The Minimum Individually Rational Virtual Valuation Function.
We say that v () is the minimum individually rational, ǫ -approximate virtualvaluation function if v ( x t ) ≤ ω ( x t ) for each 1 ≤ t ≤ T , for any other indi-vidually rational, ǫ -approximate virtual valuation function ω (). This leadsto the questions: (i) Does such a valuation function exist? and (ii) Can it beobtained efficiently? The answer to both these questions is yes . Theorem 3.3.
The minimum individually rational, µ -approximate virtualvaluation function exists and can be found in polynomial time.Proof. We create an auxiliary directed graph H from ˆ G by adding a sinkvertex z . We add an arc ( x t , z ) of length − p t · x t , for each 1 ≤ t ≤ T ,allowing for repeated arcs. Because ˆ G contains no negative cycle, neitherdoes H . Therefore, there exist shortest path distances in H . Denote by ˆ d ()the shortest path distance from vertex x t to z in H . We claim that setting v ( x t ) = − ˆ d ( x t ) gives the minimum individually rational, µ -approximatevirtual valuation function.To begin, let’s verify that v () is an individually rational, µ -approximatevirtual valuation function. First, we require that v () is individually rational.Now the direct path consisting of the arc ( x t , z ) is at least as long as theshortest path from x t to z . Thus, − p t · x t ≥ ˆ d ( x t ). Individual rationalitythen follows as v ( x t ) = − ˆ d ( x t ) ≥ p t · x t .Second we need to show that v () is µ -approximate. Consider a pair { x s , x t } . The shortest path conditions imply that − v ( x s ) = ˆ d ( x s ) ≤ ˆ ℓ st + ˆ d ( x t ) = ( ℓ st − µ ) + ˆ d ( x t ) = ( ℓ st − µ ) − v ( x t ) . d (). There-fore, by definition of ℓ st , v ( x t ) ≤ v ( x s ) + ℓ st − µ = v ( x s ) + min s ′ { p s ′ · ( x t − x s ) : x s ′ = x s } − µ ≤ v ( x s ) + p s · ( x t − x s ) − µ . Hence, v () is µ -approximate as desired.Finally we require that v () is minimum individually rational. So, take anyother individually rational, µ -approximate virtual valuation ω (). We mustshow that v ( x t ) ≤ ω ( x t ) for every bundle x t . Now consider the shortest pathtree T in H corresponding to ˆ d (). If ( x t , z ) is an arc in T (and at least onesuch arc exists) then − p t · x t = ˆ d ( x t ). Thus v ( x t ) − p t · x t = ( − p t · x t ) − ˆ d ( x t ) = 0 ≤ ω ( x t ) − p t · x t . Here the inequality follows by the individual rationality of ω (). Thus v ( x t ) ≤ ω ( x t ).Now suppose that v ( x s ) > ω ( x s ) for some x s . We may take x s to be theclosest vertex to the root z in T with this property. We have seen that x s cannot be a child of z . So let ( x s , x t ) be an arc in T . As x t is closer to theroot than x s , we know v ( x t ) ≤ ω ( x t ). Then, as T is a shortest path tree, wehave ˆ d ( x s ) = ˆ ℓ st + ˆ d ( x t ). Consequently − v ( x s ) = ˆ ℓ st − v ( x t ), and so ω ( x t ) ≥ v ( x t ) = ˆ ℓ st + v ( x s ) > ˆ ℓ st + ω ( x s ) . But then ω ( x t ) > ω ( x s ) + ℓ st − µ = ω ( x s ) + min s ′ { p s ′ · ( x t − x s ) : x s ′ = x s } − µ . It follows that there is at least one time period when x s was selectedin violation of the µ -optimality of ω (). So v () is a minimum individuallyrational, µ -approximate virtual valuation function. ⊓⊔ The Maximum (Individually Rational) Virtual Valuation Function.
The minimum individually rational virtual valuation function allows us toobtain worst-case social welfare guarantees when revealed preference is usedin mechanism design, see Section 4. For the best-case welfare guarantees, weare interested in finding the maximum virtual valuation function. In general,this need not exist as we may add an arbitrary constant to each bundle’svaluation given by the minimum individually rational virtual valuation func-tion. But, it does exist provided we have an upper bound on the valuation12f at least one bundle. This is often the case. For example in a combinatorialauction if a bidder drops out of the auction at time t + 1, then p t +1 · x t isan upper bound on the value of bundle x t . Furthermore, in practice, bidders(and the auctioneer) often have (over)-estimates of the maximum possiblevalue of some bundles.So suppose we are given a set I and constraints of the form v ( x i ) ≤ β i foreach i ∈ I . Then there is a unique maximum µ -approximate virtual valuationfunction. Theorem 3.4.
Given a set of constraints, the maximum µ -approximate vir-tual valuation function exists and can be found in polynomial time.Proof. Let v ( x i ) ≤ β i for each i ∈ I . We construct a graph H from ˆ G byadding a source vertex z with arcs of length β i from z to x i , for each i ∈ I .Since z has in-degree zero, H has no negative cycles because ˆ G does not.Denote by ˆ d () the shortest distance of every vertex from z . We claim thatsetting v ( x ) = ˆ d ( x ) gives us the desired maximum µ -approximate valuationfunction.To prove this, we first begin by checking that it satisfies the upper-boundconstraints. This is trivial, because for each i ∈ I there is a path consistingof one arc of length β i from z to x i . Thus the shortest path to x i has lengthat most β i . Second, the valuation function v () = ˆ d () is µ -approximate bythe choice of arc length in ˆ G . Third, we show that this valuation functionis maximum. So, take any other µ -approximate virtual valuation ω () thatsatisfies the upper bound constraints I . We must show that v ( x t ) ≥ ω ( x t ) forevery bundle x t . For a contradiction, suppose that P = { z , y , y , . . . , y r } isthe shortest path from z to y r in H and that v ( y r ) < ω ( y r ). Observe thatthe node adjacent to z on P must be y = x i for some i ∈ I . Now because ω () is a µ -approximate valuation function, we have r − X j =1 ω ( y j +1 ) ≤ r − X j =1 (cid:0) ω ( y j ) + ℓ y j , y j +1 − µ (cid:1) = r − X j =1 (cid:16) ω ( y j ) + ˆ ℓ y j , y j +1 (cid:17) . Cancelling terms produces ω ( y r ) ≤ ω ( y ) + r − X j =1 ˆ ℓ y j , y j +1 ≤ β j + r − X j =1 ˆ ℓ y j , y j +1 = ˆ d ( y r ) = v ( y r ) . Here the second inequality follows by the facts that y = x i , for some i ∈ I ,and ω () satisfies the upper bound constraint ω ( x i ) ≤ β i . This contradictsthe assumption that v ( y r ) < ω ( y r ). ⊓⊔ β t = p t · x t ,for all 1 ≤ t ≤ T . Individual rationality then implies that v ( x t ) must equal p t · x t for every bundle. In general, however, such a valuation function is not µ -approximate. In such cases no individually rational µ -approximate virtualvaluation functions may exist that satisfy the upper bound constraints. Onthe other hand, suppose such a virtual valuation function does exist. Thenthe maximum µ -approximate virtual valuation function in Theorem 3.4 mustbe individually rational by maximality. So far, we have focused upon how to test the degree of rationality reflectedin a data set. Specifically, we saw in Theorem 3.2 that the minimum meanlength of a cycle, µ ( G ), gives an exact and optimal goodness of fit measurefor rationality. Furthermore, Theorem 3.3 explained how to quickly obtainthe minimum individually rational valuation function that best fits the data.Recall, however, that revealed preference is also used as tool in mecha-nism design. In particular, we saw in Section 2.2 how revealed preference isused to impose bidding constraints in combinatorial auctions. We will nowshow how to apply the combinatorial arguments we have developed to createother relaxed revealed preference constraints. Consider a combinatorial auction at time (round) t where our prior price-bundle bidding pairs are { ( p , x ) , ( p , x ) , . . . , ( p t − , x t − ) } . By Inequality(6) in section 2.2, rational bidding at time t implies that v ( x t ) − p t · x t ≥ v ( x s ) − p t · x s , for all s < t .Moreover, a necessary condition is then that ( p t − p s ) · x s ≥ ( p t − p s ) · x t and this can easily be checked by searching for negative length digons in thebidding graph induced by the first t bids. If such a cycle is found then thebid ( p t , x t ) is not permitted by the auction mechanism.The non-permittal of bids is clearly an extreme measure, and one thatcan lead to the exclusion of bidders from the auction even when they stillhave bids they wish to make. In this respect, it may be desirable for themechanism to use a relaxed set of revealed preference bidding rules. Thenatural approach is to insist not upon strictly rational bidders but rather just14pon approximately rational bidders. Specifically, the auction mechanismmay (dynamically) select a desired degree ǫ of rationality. This requires thatat time t , v ( x t ) − p t · x t ≥ v ( x s ) − p t · x s − ǫ, for all s < t .A necessary condition then is ( p t − p s ) · x s ≥ ( p t − p s ) · x t − ǫ , and we cantest this relaxed warp -based bidding rule by insisting that every digon hasmean length at least − ǫ . Similarly, the relaxed karp -based bidding rule is( p k − p ) · x ≥ k X i =1 ( p i − p i − ) · x i − ( k + 1) · ǫ (10)The relaxed garp -based bidding rule applies the relaxed karp -basedbidding rule for every choice of k . The imposition of the relaxed garp -basedbidding rule ensures approximate rationality. Theorem 4.1.
A set of price-bid pairings { ( p t , x t ) : 1 ≤ t ≤ T } has acorresponding ǫ -approximate individually rational virtual valuation functionif and only if it satisfies the relaxed garp -based bidding rule.Proof. Suppose the relaxed garp -based bidding rule is satisfied. By The-orem 3.2, it suffices to show that the minimum mean cycle in the biddinggraph with arc lengths ℓ is at least − ǫ . So take any collection { x i } ki =1 ofbundles. Let t i be the time when ℓ x i , x i +1 was minimized, and let p i := p t i .Then we have − ( k + 1) · ǫ ≤ ( p k − p ) · x − X ki =1 ( p i − p i − ) · x i = X ki =0 p i · ( x i +1 − x i )= X ki =0 ℓ x i , x i +1 Here, the inequality follows because the relaxed garp -based bidding rule issatisfied. (Again the subscripts are taken modulo k + 1.) Since, the corre-sponding cycle contains k + 1 arcs, we see that the length of the minimummean cycle is at least − ǫ .Conversely, if the bidding data has a corresponding ǫ -approximate indi-vidually rational virtual valuation function then the relaxed bidding rulesare satisfied. ⊓⊔ .2 Relaxed KARP-Based Bidding Rules Theorem 4.1 tells us that imposing the relaxed garp -based bidding ruleensures approximate rationality. But, in practice, even warp -based biddingrules are often confusing to real bidders. There is likely therefore to be someresistance to the idea of imposing the whole gamut of garp -based biddingrules. We believe that this combinatorial view of revealed preference, wherethe bidding rules can be tested via cycle examination, will eradicate some ofthe confusion. However, for simplicity, there is some worth in quantitativelyexamining the consequences of imposing a weaker relaxed karp -based bid-ding rule rather than the garp -based bidding rule. To test for the relaxed karp -based bidding rules, we simply have to examine cycles of length atmost k + 1. Now suppose the karp -based bidding rules are satisfied. Byfinding the µ ( G ) in the bidding graph we can still obtain the best-fit ad-ditive approximation guarantee, but we no longer have that this guaranteeis ǫ . We can still, though, prove a strong additive approximation guaranteeeven for small values of k . To do this we need the following result. Theorem 4.2.
Given a complete directed graph G with arc lengths ℓ . If ev-ery cycle of cardinality at most k + 1 has non-negative length then the mini-mum mean length of a cycle is at least − ℓ max k , where ℓ max = max e ∈ E ( G ) | ℓ e | .Proof. Take any cycle C with cardinality | C | > k + 1. Let the arcs of C be { e , e , . . . , e | C | } in order. Then | C | X i =1 i + k − X j = i ℓ e j = k · | C | X i =1 ℓ e i = k · ℓ ( C ) = k · | C | · ℓ ( C ) | C | . (11)Above, the inner summation is taken modulo | C | . On the other hand takeany path segment P = { e i , e i +1 , . . . , e i + k − } , where again the subscript sum-mation is modulo | C | . Because the graph is complete and the maximum arclength is ℓ max , the length of P is at least − ℓ max . Otherwise, we have a neg-ative length cycle of cardinality k + 1 by adding to P the arc from the headvertex of e i + k − to the tail vertex of e i . Thus, | C | X i =1 i + k − X j = i ℓ e j ≥ −| C | · ℓ max . (12)Combining Equalities (11) and Inequality (12) gives that ℓ ( C ) | C | ≥ − ℓ max k . Asevery cycle of cardinality at most k + 1 has non-negative mean length, thisimplies that the minimum mean length of any cycle in G is at least − ℓ max k . ⊓⊔ karp -based bidding rule. Corollary 4.1.
Given a set of price-bid pairings { ( p t , x t ) : 1 ≤ t ≤ T } thatsatisfy the relaxed karp -based bidding rule, there is a ( b max k + ǫ ) -approximateindividually rational virtual valuation function, where b max is the maximumbid made by the bidder during the auction.Proof. The relaxed karp -based bidding rule (10) implies that every cycle ofcardinality at most k + 1 in the bidding graph G has mean length at least − ǫ . Let G ′ be the modified graph with arc lengths ℓ ′ x s , x t := ℓ x s , x t + ǫ . Thenevery cycle in G ′ of cardinality at most k + 1 has non-negative length. ByTheorem 4.2, the minimum mean length of a cycle in G ′ is then at most ( ℓ ′ ) max k . Furthermore, ( ℓ ′ ) max = ℓ max + ǫ ≤ b max + ǫ . Theorems 3.2 and 3.3then guarantee the existence of a ( b max k + ǫ )-approximate individually rationalvirtual valuation function. ⊓⊔ One may ask whether the additive approximation guarantee in Corol-lary 4.1 can be improved. The answer is no ; Theorem 4.2 is tight. Lemma 4.1.
There is a graph G where each cycle of cardinality at most k + 1 has non-negative length and the minimum mean length of a cycle is − ℓ max /k .Proof. Let G be a complete directed graph with vertex set V = { v , v , . . . , v n } .We will define arc lengths ℓ such that all ( k +1)-cycles in G have non-negativelength, but the minimum mean length of a cycle is − ℓ max k . First consider thecycle C = { v , v , . . . , v k +2 , v } . Give each arc in C a length − ℓ max k . Thus C has cardinality k + 2 and mean length − ℓ max k . Now let every other arc e have length ℓ max . It immediately follows that the only cycle in G with nega-tive length is C . Thus, all cycles of length at most k + 1 have non-negativelength, but the minimum mean length of a cycle is − ℓ max k , as desired. Our results from Section 3 give approximate rationality guarantees on indi-vidual bidders. We briefly outline this here. By applying the above relaxedrevealed preference bidding rules to each bidder, we can now obtain guaran-tees on the overall social welfare of the entire auction. For example, considerimposing the relaxed garp bidding rules. Now suppose each bidder usesa minimum, individually rational, ǫ -approximate virtual valuation function17hat satisfies the gross substitutes property. It is known that if bidder valu-ation functions satisfy the gross substitutes property then the simultaneousmulti-round auction (SMRA) will converge to a Walrasian equilibrium andmaximize social welfare [17,12,16]. Consequently, the output allocation nowmaximizes virtual welfare to within an additive factor per bidder. One mayexpect there is some maximum discrepancy (say, in the L ∞ norm) betweenthe true valuation function and some virtual approximate virtual valuation.If so, because the implemented virtual valuations are minimum, we can thenlower-bound the true social welfare. Similarly, best-case bounds follow usingthe maximum approximate virtual valuation function. Interestingly other bidding rules used in practice or proposed in the literaturecan be viewed in the graphical framework. For example, bid withdrawalscorrespond to vertex deletion in the bidding graph, whilst budget constraintsand the Afriat Efficiency Index can be formulated in terms of arc-deletion.We briefly describe these applications here.
Revealed Preference with Budgets.
Recall that, in Section 2.2, we haveassumed that, in the quasilinear model, bidders have no budgetary con-straints. This is not a natural assumption. Harsha et al. [13] explain how toimplement budgeted revealed preference in a combinatorial auction. Theirmethod applies to the case when the fixed budget B is unknown to the auc-tion mechanism. To do this, upper and lower bounds on feasible budgets aremaintained dynamically via a linear program. It is also straightforward todo this combinatorially using edge-deletion in the bidding graph; we omitthe details as the process resembles that of the following subsection. The Afriat Efficiency Index.
Recall that to determine the Afriat Effi-ciency Index we reveal x t (cid:23) y only if p t · y ≤ λ · p t · x t where λ <
1. Thisis equivalent, in Afriat’s original setting, to removing from the graph anyarc ( x t , x s ) for which p t · x s > λ · p t · x t . Of course, for the applicationof combinatorial auctions, we assume quasi-linear utilities. Therefore, theappropriate implementation is to remove any arc ( x t , x s ) for which v ( x s ) − p t x s > λ · ( v ( x t ) − p t x t ) . How, though, can we implement this rule as v () is unknown? We can simplyapply the techniques of Section 3 and use for v the minimum individually18ational virtual valuation function. We can now determine the best choiceof λ that gives a predetermined, ǫ additive approximation guarantee ǫ . Thiscan easily be computed exactly by bisection search over the set of arcs, aseach arc a has its own critical value λ a at which it will be removed. Theoptimal choice arises at the point where the minimum mean cycle in thebidding graph rises above − ǫ . When ǫ = 0, the corresponding choice of λ isthe anolog of the Afriat Efficiency Index. Revealed Preference with Bid Withdrawals.
Some iterative multi-itemauctions allow for bid withdrawals, most notably the simultaneous multi-round auction (SMRA). Bid withdrawals may easily be implemented alongwith revealed preference bidding rules. At time t , a bid withdrawal corre-sponds to the removal of (a copy of) a vertex x s , where s < t . This may beimportant strategically. To see this, suppose the bid x t is invalid under the karp -based bidding rules because it would induce a negative cycle of cardi-nality at most k + 1 in the bidding graph on { x , x , . . . , x t } . If x s lies onall such negative cycles then x t becomes a valid bid after the withdrawal of x s . Because auctions typically restrict the total number of bid withdrawalsallowed, the optimal application of bid withdrawals correspond to the prob-lem of finding small hitting sets for the negative length cycles of cardinalityat most k + 1. References
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