On the Interplay between Incentive Compatibility and Envy Freeness
Edith Cohen, Michal Feldman, Amos Fiat, Haim Kaplan, Svetlana Olonetsky
aa r X i v : . [ c s . G T ] M a r On the Interplay between Incentive Compatibility and EnvyFreeness
Edith Cohen ∗ Michal Feldman † Amos Fiat ‡ Haim Kaplan § Svetlana Olonetsky ¶ Abstract
We study mechanisms for an allocation of goods among agents, where agents have no incentive tolie about their true values (incentive compatible) and for which no agent will seek to exchange outcomeswith another (envy-free). Mechanisms satisfying each requirement separately have been studied exten-sively, but there are few results on mechanisms achieving both. We are interested in those allocations forwhich there exist payments such that the resulting mechanism is simultaneously incentive compatibleand envy-free.Cyclic monotonicity is a characterization of incentive compatible allocations, local efficiency is acharacterization for envy-free allocations. We combine the above to give a characterization for allo-cations which are both incentive compatible and envy free. We show that even for allocations thatallow payments leading to incentive compatible mechanisms, and other payments leading to envy freemechanisms, there may not exist any payments for which the mechanism is simultaneously incentivecompatible and envy-free. The characterization that we give lets us compute the set of Pareto-optimalmechanisms that trade off envy freeness for incentive compatibility. ∗ AT & T Labs-Research, 180 Park Avenue, Florham Park, NJ. † School of Business Administration and Center for the Study of Rationality, The Hebrew University of Jerusalem. ‡ The Blavatnik School of Computer Science, Tel Aviv University. § The Blavatnik School of Computer Science, Tel Aviv University. ¶ The Blavatnik School of Computer Science, Tel Aviv University.
Introduction
We consider allocation problems, where a set U of objects should be allocated among m agents, each havinga valuation function v i assigning a value to every bundle. We wish to find a partition of the objects among theagents so as to achieve some goal. Typically, this goal is to maximize (or approximate) the social welfare,i.e., the sum of the agents’ valuations for their bundles. A mechanism M = h a, p i is a protocol that receivesthe set of the agents’ valuations as input and returns a tuple consisting of an allocation a and payments p forthe agents. The utility of an agent is the sum of valuations of the items minus any payments (quasi-linearutility).Two natural desired properties of any mechanism are incentive compatibility and envy-freeness. Amechanism is incentive compatible if it is a dominant strategy for every agent to report her private infor-mation truthfully [5]. In 1987, Rochet [9] defined the notion of a cycle-monotonic allocation, and provedthat this was a necessary and sufficient condition of an IC-implementable allocation – an allocation havingassociated payments that jointly form an incentive compatible mechanism. A mechanism is envy-free if noagent wishes to switch her outcome with that of another [2, 3, 10, 6, 7, 11]. The notion of locally efficient al-location has been defined by Haake et. al. [4] who showed that this was a necessary and sufficient conditionof an EF-implementable allocation – allocation having associated payments that jointly form an envy-freemechanism.While much work has been done on each of these properties independently, not much is yet understoodabout the interrelation between incentive compatibility and envy freeness The interaction between these twonotions is the focus of our paper.Our contribution is a characterization of allocations that are both incentive compatible and envy-free. Weuse this to derive various Pareto-optimal mechanisms that trade off envy freeness for incentive compatibility,we also use this to obtain negative results and to show separation results for different types of allocationsand problems. Motivated by the cyclic monotonicity characterization of [9] and the locally efficient characterization of [4]we now consider two new categories of allocation functions: EF ∪ IC -implementable: An allocation function a is called incentive compatible or envy-free imple-mentable ( EF ∪ IC -implementable) if there exists a payment function p such that M = h a, p i isincentive compatible and a (possibly different) payment function p ′ such that M = h a, p ′ i is envy-free. EF ∩ IC -implementable: An allocation function a is incentive compatible and envy-free implementable( EF ∩ IC -implementable) if there exists a payment function p such that the mechanism M = h a, p i is incentive compatible and envy-free. Clearly, every function which is EF ∩ IC -implementable isalso EF ∪ IC -implementable.Many natural questions arise regarding the properties above: Are these classes identical? Empty? Whatinteresting problems fall into each of these classes? Figure 1 gives different payment functions for the socialwelfare maximizing allocation of one indivisible item. It is easy to verify the properties claimed for thevarious payment functions. The social welfare maximizing allocation is indeed EF ∩ IC , Clarke pivotspayments give a mechanism that is both incentive compatible and envy free (the third entry).1ayments Properties p i = min j = i v j p j = min k = j v k − v i , for all j = i Incentive compatible, not envy-free(VCG, not Clarke pivot payments) p i = 0 p j = − max k = i v k Envy-free, not incentive compatible(not VCG) p i = max k = i v k p j = 0 , for all j = i Envy-free and incentive compatible(VCG with Clarke pivot payments)Figure 1: Allocating a single item to the agent of highest valuation, (agent i ), the utility of agent i is v i − p i ,the utility of agents j = i is − p j .In Section 3 we provide a characterization for allocations that are EF ∩ IC -implementable. We then usethe obtained characterization to derive some useful observations regarding the spectrum of “best possible”tradeoffs between (approximate) envy-freeness and (approximate) incentive compatibility.Finally, in Section 5 we explore the relationships between the different classes. In particular, we showthat not every IC -implementable allocation is also EF -implementable, and vice versa. Additionally, usingthe characterization provided in Section 3, we show that not every IC ∪ EF -implementable allocation isalso IC ∩ EF -implementable. Let U be a set of objects, and associated with agent i , ≤ i ≤ m , is a valuation function v i ∈ V i that mapssets of objects into ℜ . Let v = < v , v , . . . , v m > be a sequence of valuation functions, and let ( v ′ i , v − i ) bethe sequence of valuation functions arrived by substituting v i by v ′ i ( v ′ i ∈ V i ) , i.e., ( v ′ i , v − i ) = < v , . . . , v i − , v ′ i , v i +1 , . . . , v m > . An allocation function a maps a sequence of valuation functions v = < v , v , . . . , v m > , into a parti-tion of U consisting of m parts, one for each agent. I.e., a ( v ) = < a ( v ) , a ( v ) , . . . , a m ( v ) >, where S i a i ( v ) = U and a i ( v ) ∩ a j ( v ) = ∅ for i = j . Let A denote the set of possible allocations. Apayment function is a mapping from v to ℜ m , p ( v ) = < p ( v ) , p ( v ) , . . . , p m ( v ) > , p i ( v ) ∈ ℜ . Paymentsare from the agent to the mechanism (if the payment is negative then this means that the transfer is from themechanism to the agent).A mechanism is a pair of functions, M = h a, p i , where a is an allocation function, and p is a paymentfunction. For a sequence of valuation functions v = h v , v , . . . , v m i , the utility to agent i is defined as v i ( a i ( v )) − p i ( v ) . Such a utility function is known as quasi-linear. Here we deal with indivisible allocations, although our results also extend to divisible allocations with appropriate modifica-tions. In this paper we consider only deterministic mechanisms and can therefore omit the allocation as an argument to the paymentfunction. ickrey-Clarke-Groves (VCG) mechanism: A celebrated result in mechanism design is the family of
Vickrey-Clarke-Groves (VCG) mechanisms. A mechanism M = h a, p i is called a VCG mechanism if: • a ( v ) ∈ argmax a ∈ A P mi =1 v i ( a i ( v )) , and • p i ( v ) = h i ( v − i ) − P j = i v j ( a j ( v )) , where h i does not depend on v i , i = 1 , . . . , m .For all { h i } mi =1 , the VCG mechanism is incentive compatible (See, e.g., [8]). The Clarke pivot payment for a VCG mechanism takes h i ( v − i ) = max a ′ ∈ A X j = i v j ( a ′ ) . We next define mechanisms that are IC, EF, or both IC and EF. • A mechanism is incentive compatible if it is a dominant strategy for every agent to reveal her truevaluation function to the mechanism. I.e., if for all i , v , and v ′ i : v i ( a i ( v )) − p i ( v ) ≥ v i ( a i ( v ′ i , v − i )) − p i ( v ′ i , v − i ); this is equivalent to: p i ( v ) ≤ p i ( v ′ , v − i ) + (cid:16) v i ( a i ( v )) − v i ( a i ( v ′ i , v − i )) (cid:17) . (1) • A mechanism is envy-free if no agent seeks to switch her allocation and payment with another. I.e., iffor all ≤ i, j ≤ m and all v : v i ( a i ( v )) − p i ( v ) ≥ v i ( a j ( v )) − p j ( v ); this is equivalent to: p i ( v ) ≤ p j ( v ) + (cid:16) v i ( a i ( v )) − v i ( a j ( v )) (cid:17) . (2) • A mechanism ( a, p ) is incentive compatible and envy-free if ( a, p ) is both incentive compatible andenvy-free. EF ∩ I C -implementable
Before presenting our characterization for allocations that are EF ∩ IC -implementable, we present severalknown characterizations. Definition 3.1. (Locally Efficient([4])) An allocation function a is said to be locally efficient if for all v , andall permutations π of , . . . , m , m X j =1 v i ( a i ( v )) ≥ m X j =1 v i ( a π ( i ) ( v )) . Theorem 3.2. ([4]) A necessary and sufficient condition for an allocation function a to be EF -implementableis that a is locally efficient. efinition 3.3. (Cycle monotonicity [9]) We require Rochet’s notion of cyclic monotonicity : an allocationfunction satisfies cycle monotonicity if for every player i , every integer K , and every v i , v i , . . . , v Ki ∈ V i ,we have K X k =1 h v ki (cid:16) a i ( v ki , v − i ) (cid:17) − v ki (cid:16) a i ( v k +1 i , v − i ) (cid:17)i ≥ (3)where v K +1 i = v i . Note that the summand is the same as the expression in Equation (1). Theorem 3.4. ([9]) A necessary and sufficient condition that an allocation function is IC -implementableis that it is cycle monotonic. G a For an allocation function a , let G a be a weighted digraph with vertices ( i, v ) , where ≤ i ≤ m , and v = < v , . . . , v m > is a sequence of valuation functions.The arcs of the graph are classified as either EF arcs or IC arcs as follows: • EF arcs: For all v , all ≤ i, j ≤ m , there is an arc from ( j, v ) to ( i, v ) of weight v i ( a i ( v )) − v i ( a j ( v )) . • IC arcs: For all ≤ i ≤ m , all v − i , all v i , and all v ′ i , there is an arc from ( i, ( v ′ i , v − i )) to ( i, ( v i , v − i )) of weight v i ( a i ( v i , v − i )) − v i ( a i ( v ′ i , v − i )) .If we consider only EF arcs, then G a consists of vertex disjoint complete digraphs, each of whichcorresponds to a different v . An allocation is EF -implementable if and only if there are no negative cyclesof EF arcs in G a . This is equivalent to the locally efficient characterization of Theorem 3.2.If we consider only IC arcs, then G a consists of vertex disjoint complete digraphs, each of whichcorresponds to a different pair i and v − i . An allocation is IC -implementable if and only if there are nonegative cycles of IC arcs in G a . This is equivalent to the cycle monotone characterization of incentivecompatible mechanisms of Theorem 3.4.This suggests the following characterization of allocations that are EF ∩ IC -implementable: Theorem 3.5.
Allocation function a is incentive compatible and envy-free implementable ( EF ∩ IC -implementable) if and only if there are no negative cycles in G a .Proof. A mechanism ( a, p ) is EF and IC if and only if there exists a payment function p ( v ) = < p ( v ) , p ( v ) , . . . p m ( v ) > such that the following two conditions hold: • For any IC arc e = (( i, v ′ ) , ( i, v )) , it holds that v i ( a i ( v )) − p i ( v ) ≥ v i ( a i ( v ′ )) − p i ( v ′ ) ;which is equivalent to v i ( a i ( v )) − v i ( a i ( v ′ )) ≥ p i ( v ) − p i ( v ′ ) . • For any EF arc e = (( j, v ) , ( i, v )) , it holds that v i ( a i ( v )) − p i ( v ) ≥ v i ( a j ( v )) − p j ( v ) ;which is equivalent to v i ( a i ( v )) − v i ( a j ( v )) ≥ p i ( v ) − p j ( v ) . G a , w ( e ) = v i ( a i ( v )) − v i ( a i ( v ′ )) for an IC arc e = (( i, v ′ ) , ( i, v )) , therefore w ( e ) ≥ p i ( v ) − p j ( v ) .Similarly, w ( e ) = v i ( a i ( v )) − v i ( a j ( v )) for an EF arc e = (( j, v ) , ( i, v )) , therefore w ( e ) ≥ p i ( v ) − p i ( v ′ ) .If we sum up these inequalities over the set of arcs forming a cycle (consisting of alternate IC and EF arcs), we get that the sum of arc weights must be non-negative as the right hand sides cancel out.If G a does not contain a negative cycle, we can compute shortest paths from any arbitrary start vertex ( i, v ) , and interpret the length of the shortest path from ( i, v ) to a vertex ( j, v ′ ) as p j ( v ′ ) . Shortest pathsobey the required condition.It follows that an allocation is EF ∪ IC -implementable (potentially different payments for incentivecompatibility and for envy-freeness) if and only if all negative cycles in G a include at least one EF arcand at least one IC arc. Figure 2 (in Appendix 5) helps visualize G a by illustrating a subgraph of a G a thatcontains a cycle of EF and IC arcs that is not a union of complete IC and EF cycles (this 8-node subgraphis the smallest subgraph with this property.) A mechanism ( a, p ) has ∆ -approximate envy-freeness if for all v , all ≤ i, j ≤ m , v i ( a i ( v )) − p i ( v ) ≥ v i ( a j ( v )) − p j ( v ) − ∆ (no agent envies another by more than ∆ ). Similarly, ( a, p ) has ∆ -approximateincentive compatibility if no agent has incentive to lie that is larger than ∆ , i.e. for all ≤ i ≤ m , for all v and v ′ i , v i ( a i ( v )) − p i ( v ) ≥ v i ( a i ( v ′ i , v − i )) − p j ( v − i , v ′ i ) − ∆ .For a graph G a as defined in Section 3.1 and a pair ( c ef , c ic ) of nonnegative values, we define G +( c ef ,c ic ) a to be such that c ef is added to the weight of all EF arcs and c ic is added to the weight of all IC arcs in G a .A pair ( c ef , c ic ) is cycle-correcting for G a if G +( c ef ,c ic ) a does not contain negative cycles. By comput-ing shortest paths from an arbitrary node in G +( c ef ,c ic ) a we obtain payments (and a mechanism) with c ef -approximate envy-freeness and c ic -approximate incentive compatibility. A cycle-correcting pair ( c ef , c ic ) is minimal if there is no cycle-correcting pair ( x, y ) = ( c ef , c ic ) such that x ≤ c ef and y ≤ c ic .The set T a of minimal cycle-correcting pairs defines a tradeoff between approximate envy-freeness andapproximate incentive compatibility of the allocation function a . If a is EF ∩ IC -implementable, then T a = { (0 , } . If a is EF -implementable (but not necessarily IC -implementable), there is a point ofthe form (0 , c ic ) ∈ T a . The corresponding mechanism is envy-free and has the best possible (that is, c ic -approximate) incentive compatibility subject to envy-freeness. Similarly, if a is IC -implementable, thereis a point ( c ef , ∈ T a with a corresponding incentive compatible mechanism that has the best possible(that is, c ef -approximate) envy freeness subject to incentive compatibility. These “best” tradeoffs can becomputed in time polynomial in the size of G f (minimum ratio cycles [1]). Assume that one can find either envy free or truthful prices, but not both. Then, one can find prices thatenforce envy freeness for some of the agents, and truthfulness for the complement, and can choose how toclassify the agents.Consider the EF and IC arcs graph such that there are no negative cycles of only IC or only EF arcs, butthere might be a mixed cycle. In particular, any negative cycle must include a vertex where the arc enteringthe vertex is an EF arc and the arc exiting the vertex is an IC arc.5e do as follows: We remove all IC arcs entering ( i, X ) where i is trusted, and remove all EF arcsentering ( i, X ) where i is untrusted. Any negative cycle must include both trusted and untrusted agents(otherwise it is limited to edges of one type). There are no EF edges from trusted to untrusted vertices, andIC edges only connect vertices associated with the same agent. Thus, there is no cycle including both trustedand untrusted vertices.It follows that no trusted agent will be envious, since envy is prevented by the incoming EF edges.Likewise, no untrusted agent has incentive to lie, this is guaranteed by the incoming IC edges . IC arcEF arc , v v , v v , v v , v v , v v , v v , v v , v v Figure 2: A cycle in G a that includes two agents (1 and 2), and two valuation functions for each, { v , v } for agent 1, { v , v } for agent 2. The valuation functions for all other agents are fixed in this cycle.In this section we prove that not every allocation that is IC -implementable is also EF -implementable,and vice versa. Additionally, we show that not every allocation that is IC ∪ EF -implementable is also IC ∩ EF -implementable. Claim 5.1.
Not every allocation that is IC -implementable is also EF -implementable. Proof.
Consider a single item auction and give it to agent . If the mechanism pays nothing to any agent thisis incentive compatible. However, there is no payment function that makes this envy-free, this allocation isnot locally efficient (and thus not EF -implementable) unless agent has the highest valuation. In fact, we could try to add more EF and IC edges, subject to there being no negative cycle, and get a wider set of envy/incentive-compatibility constraints. We could combine this idea with that of increasing the remaining edges by some ∆ , so as to get someguarantees on the envy of the untrusted agents and the incentive to lie of the trusted agents. laim 5.2. Not every allocation that is EF -implementable is also IC -implementable. Proof.
Consider a single divisible good. Let a i ( v ) = α i denote what fraction of the good is given to agent i , where P mi =1 α i = 1 . For the examples below, we take the valuation function to be proportional, i.e., forall i , v i ( a i ( v )) = α i z i , z i is agent i ’s value of the entire object.Consider the following assignment function: a ( v , v ) = (1 / , / z = z ( − z z + z ) , + z z + z ) ) z < z ( + z z + z ) , − z z + z ) ) z > z This assignment is locally efficient, but the fraction assigned to an agent does not monotonically increasewith the agent valuation. This contradicts cyclic monotonicity (on a cycle of length 2).
Claim 5.3.
Not every allocation that is IC ∪ EF -implementable is also IC ∩ EF -implementable. Proof.
Consider a single divisible good and three agents { , , } where each of the agents, 1 and 2, canchoose one of two specific valuation functions: Agent i = 1 , , has valuation function v i ∈ { v i , v i } , andagent 3 has only one valuation function, v .As before, let z ji = v ji (1) (the valuation for the entire good). Consider a setting where z ≥ z > z ≥ z ≥ z . (For concreteness, take z = 0 , z = z = 1 , z = z = 2 for a divisible good and z = − , z = z = − , z = z = − for a divisible task.)We denote a b b i ≡ a i ( v b , v b , v ) and consider the allocation a = a = 0 . , a = 0 . and for b b ∈ { , , } , a b b = a b b = 0 . , a b b = 0 .We show that a is IC -implementable (all IC cycles in G a are nonnegative). Any IC cycle must involveone of the agents { , } (agent can not change valuations) and the two valuations of this agent. There arefour IC cycles (2-cycles) with weights, for b b ∈ { , } , : ( a b − a b ) · ( z − z ) ≥ a b − a b ) · ( z − z ) ≥ (Using the property that for any agent i ∈ { , } , fixing the valuation of other agents, the fraction allocatedis nondecreasing with valuation (weak monotonicity).)One could argue directly that all EF cycles in G a are nonnegative and therefore a is EF -implementable.It is easier to see that a is locally efficient. The two agents 1 and 2 always receive the same fraction andagent 3, whose valuation is smaller, receives a smaller fraction.We now show that G a has a negative cycle, and therefore, using Theorem 3.5, a is not EF ∩ IC -implementable. Consider the 8-cycle C (see Figure 2) over players , and valuations ( v b , v b , v ) for b b ∈ { , , , } . The weight of C is w ( C ) = ( a − a ) · z + % EF arc (2 , v v ) → (1 , v v )( a − a ) · z + % IC arc (2 , v v ) → (2 , v v )( a − a ) · z + % EF arc (1 , v v ) → (2 , v v )( a − a ) · z + % IC arc (1 , v v ) → (1 , v v )( a − a ) · z + % EF arc (2 , v v ) → (1 , v v )( a − a ) · z + % IC arc (2 , v v ) → (2 , v v )( a − a ) · z + % EF arc (1 , v v ) → (2 , v v )( a − a ) · z % IC arc (1 , v v ) → (1 , v v ) w ( C ) = 0 . z − z ) < . References [1] R. K. Ahuja, T. L. Magnanti, and J. B. Orlin.
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