Truth and Envy in Capacitated Allocation Games
Edith Cohen, Michal Feldman, Amos Fiat, Haim Kaplan, Svetlana Olonetsky
aa r X i v : . [ c s . G T ] F e b Truth and Envy with Capacitated Valuations
Edith Cohen ∗ Michal Feldman † Amos Fiat ‡ Haim Kaplan § Svetlana Olonetsky ¶ October 23, 2018
Abstract
We study auctions with additive valuations where agents have a limit on the number ofgoods they may receive. We refer to such valuations as capacitated and seek mechanisms thatmaximize social welfare and are simultaneously incentive compatible, envy-free, individuallyrational, and have no positive transfers.If capacities are infinite, then sequentially repeating the 2nd price Vickrey auction meetsthese requirements. In 1983, Leonard showed that for unit capacities, VCG with Clarke Pivotpayments is also envy free. For capacities that are all unit or all infinite, the mechanism producesa Walrasian pricing (subject to capacity constraints).Here, we consider general capacities. For homogeneous capacities (all capacities equal) weshow that VCG with Clarke Pivot payments is envy free (VCG with Clarke Pivot payments isalways incentive compatible, individually rational, and has no positive transfers). Contrariwise,there is no incentive compatible Walrasian pricing.For heterogeneous capacities, we show that there is no mechanism with all 4 properties, butat least in some cases, one can achieve both incentive compatibility and envy freeness.
We consider settings where a set [ s ] = { , . . . , s } of s goods should be allocated amongst n agentswith private valuations. An agent’s valuation function is a mapping from every subset of thegoods into the non negative reals. A mechanism receives the valuations of the agents as input,and determines an allocation a i and a payment p i for every agent. We assume that agents havequasi-linear utilities; that is, the utility of agent i is the difference between her valuation for thebundle allocated to her and her payment.We seek mechanisms that are1. Efficient — the mechanism maximizes the sum of the valuations of the agent. Alternately,efficient mechanisms are said to maximize social welfare. ∗ AT&T Labs-Research, 180 Park Avenue, Florham Park, NJ. † School of Business Administration and Rationality Center, 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.
1. Incentive compatible (truthful) — it is a dominant strategy for agents to report their privateinformation [8].3. Envy free - no agent wishes to exchange her outcome with that of another [5, 6, 16, 10, 11, 18].4. Make no positive transfers — the payments of all agents are non-negative.5. Individually rational for agents — no agent gets negative utility.We use the acronyms IC, EF, NPT, and IR to denote incentive compatible, envy-free, no positivetransfers, and individually rational, respectively.Our main results concerns the class of capacitated valuations: every agent i has an associatedcapacity c i , and her value is additive up to the capacity, i.e. , for every set S ⊆ [ s ], v i ( S ) = max X j ∈ T v i ( j ) (cid:12)(cid:12)(cid:12) T ⊆ S, | T | = c i , where v i ( j ) denotes the agent i ’s valuation for good j .Consider the following classes of valuation functions:1. Gross substitutes: good x is said to be a gross substitute of good y if the demand for x ismonotonically non-decreasing with the price of y , i.e. , ∂ (demand x ) /∂ (price y ) ≥ . A valuation function is said to obey the gross substitutes condition if for every pair of goods x and y , good x is a gross substitute of good y .2. Subadditive valuations: A valuation v : 2 [ s ] → R ≥ is said to be subadditive if for every twodisjoint subsets S, T ⊆ [ s ], v ( S ) + v ( T ) ≥ v ( S ∪ T ).3. Superadditive valuations: A valuation v : 2 [ s ] → R ≥ is said to be superadditive if for everytwo disjoint subsets S, T ⊆ [ s ], v ( S ) + v ( T ) ≤ v ( S ∪ T ).Capacitated valuations are a subset of gross substitutes, which are themselves a subset ofsubadditive valuations.In a Walrasian equilibrium (See [7]), prices are item prices , that is, prices are assigned to individual goods so that every agent chooses a bundle that maximizes her utility and the marketclears. Thus, Walrasian prices automatically lead to an envy free allocation. Every Walrasianpricing gives a mechanism that is efficient and envy free, has no positive transfers, and is individuallyrational [2].We remark that while Walrasian pricing ⇒ EF, NPT, IR, the converse is not true. Evena mechanism that is EF, NPT, IR, and
IC does not imply Walrasian prices. Note that envyfree prices may be assigned to bundles of goods which cannot necessarily be interpreted as itemprices. It is well known that in many economic settings, bundle prices are more powerful than itemprices [1, 14, 3].Gul and Stacchetti [7] showed that every allocation problem with valuations satisfying grosssubstitutes admits a Walrasian equilibrium. For the superset of gross substitutes, subadditivevaluations, a Walrasian equilibrium may not exist.2s capacitated valuations are also gross substitutes (see Theorem 2.4 in Section 2.2), it followsthat capacitated valuations always have a Walrasian equilibrium. Walrasian prices, however, maynot be incentive compatible. In fact, we show (Proposition 3.1) that even with 2 agents withcapacities 2 and 3 goods, there is no IC mechanism that produces a Walrasian equilibrium.For superadditive valuations it is known that Walrasian equilibrium may not exist. P´apai [13]has characterized the family of mechanisms that are simultaneously EF and IC under superadditivevaluations. In particular, VCG with Clarke pivot payments satisfies these conditions. However,P´apai’s result for superadditive valuations does not hold for subadditive valuations. Moreover,Clarke pivot payments do not satisfy envy freeness even for the more restricted family of capacitatedvaluations, as demonstrated in the following example:
Example 1.1.
Consider an allocation problem with two agents, { , } , and two goods, { a, b } .Agent 1 has capacity c = 1 and valuation v ( a ) = v ( b ) = 2, and agent 2 has capacity c = 2 andvaluation v ( a ) = 1, v ( b ) = 2. According to VCG with Clarke pivot payments, agent 1 is given a and pays 1, while agent 2 is given b and pays nothing (as he imposes no externality on agent1). Agent 1 would rather switch with agent 2’s allocation and payment (in which case, her utilitygrows by 1), therefore, the mechanism is not envy free.Two extremal cases of capacitated valuations are “no capacity constraints”, or, all capacitiesare equal to one. If capacities are infinite, running a Vickrey 2nd price auction [17] for every good,independently, meets all requirements (IC + Walrasian ⇒ efficient, IC, EF, NPT, IR). If all agentcapacities are one, [9] shows that VCG with Clarke pivot payments is envy free, and it is easy to seethat it also meets the stronger notion of an incentive compatible Walrasian equilibrium. For arbi-trary capacities (not only all ∞ or all ones), we distinguish between homogeneous capacities, whereall agent capacities are equal, and heterogeneous capacities, where agent capacities are arbitrary.When considering incentive compatible and heterogeneous capacities, we distinguish betweencapacitated valuations with public or private capacities: being incentive compatible with respect toprivate capacities and valuation is a more difficult task than incentive compatible with respect tovaluation, where capacities are public. In this paper, we primarily consider public capacities.The main results of this paper (which are also summarized in Figure 1) are as follows: • For arbitrary homogeneous capacities c , such that( c ≡ c = c = · · · = c n ): – VCG with Clarke pivot payments is efficient, IC, NPT, IR, and EF. (Section 3). – However, there is no incentive compatible mechanism that produces Walrasian prices,even for c = 2. (Section 3). • For arbitrary heterogeneous capacities c = ( c , c , . . . , c n ): – Under the VCG mechanism with Clarke Pivot payments (public capacities), a highercapacity agent will never envy a lower capacity agent. (Section 3). – There is no mechanism that is IC, NPT, and EF (for public and hence also for privatecapacities). (Section 4). – We also deal with some special cases: 3 ubadditive Gross substitutes capacitated -heterogeneous capacitated -homogeneousWalras. NO[7] YES[7] ( → ) YES ( → ) YESWalras.+IC NO ( ← ) NO ( ← ) NO ( ← ) NO (Proposition 3.1)EF + IC ? YES for m = 2 , n = 2(Corollary 6.2) ?( → ) YES for m = 2 , n = 2 ? YES for m = 2(Prop. 5.1) YES( ↑ )EF + IC + NPT NO ( ← ) NO ( ← ) NO (Theorem 4.1) YES (Corollary 3.10)
Figure 1: This table specifies the existence of a particular type of mechanism (rows) for variousfamilies of valuation functions (columns). Efficiency is required in all entries. The valuation familiessatisfy capacitated homogeneous ⊂ capacitated heterogeneous ⊂ gross substitutes ⊂ subadditive.Wherever results are implied from other table entries, this is specified with corresponding arrows.We note that for the family of additive valuations (no capacities), all entries are positive, as theClarke pivot mechanism satisfies all properties. ∗ ∗ Let [ s ] = { , . . . , s } be a set of goods to be allocated to a set [ n ] = { , . . . , n } of agents.An allocation a = ( a , a , . . . , a n ) assigns agent i the bundle a i ⊆ [ s ] and is such that S i a i ⊆ [ s ]and a i ∩ a j = ∅ for i = j . We use L to denote the set of all possible allocations.For S ⊆ [ s ], let v i ( S ) be the valuation of agent i for set S . Let v = ( v , v , . . . , v n ), where v i isthe valuation function for agent i .Let V i be the domain of all valuation functions for agent i ∈ [ n ], and let V = V × V × · · · × V n .An allocation function a : V maps v ∈ V into an allocation a ( v ) = ( a ( v ) , a ( v ) , . . . , a n ( v )) . A payment function p : V maps v ∈ V to R n ≥ : p ( v ) = ( p ( v ) , p ( v ) , . . . , p n ( v )), where p i ( v ) ∈ R ≥ is the payment of agent i . Payments are from the agent to the mechanism (if the payment isnegative then this means that the transfer is from the mechanism to the agent).A mechanism is a pair of functions, M = h a, p i , where a is an allocation function, and p isa payment function. For a valuation v , the utility to agent i in a mechanism h a, p i is defined as v i ( a i ( v )) − p i ( v ). Such a utility function is known as quasi-linear.For a valuation v , we define ( v ′ i , v − i ) to be the valuation obtained by substituting v i by v ′ i , i.e.,( v ′ i , v − i ) = ( v , . . . , v i − , v ′ i , v i +1 , . . . , v n ) . Here we deal with indivisible goods, although our results also extend to divisible goods with appropriate modifi-cations. 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 holds if and only if 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 (EF) if for all i, j ∈ [ n ] and all v : v i ( a i ( v )) − p i ( v ) ≥ v i ( a j ( v )) − p j ( v );this holds if and only if p i ( v ) ≤ p j ( v ) + (cid:16) v i ( a i ( v )) − v i ( a j ( v )) (cid:17) . (2)Given valuation functions v = ( v , v , . . . , v n ), a social optimum Opt is an allocation thatmaximizes the sum of valuations Opt ∈ arg max a ∈L n X i =1 v i ( a i ) . Likewise, the social optimum when agent i is missing, Opt − i , is the allocationOpt − i ∈ arg max a ∈L X j ∈ [ n ] \{ i } v j ( a j ) . A mechanism M = h a, p i is called a VCG mechanism [17, 4] if: • a ( v ) = Opt, 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 ∈ [ n ].For connected domains, the only efficient incentive compatible mechanism is VCG (See Theo-rem 9.37 in [12]). Since capacitated valuations induce a connected domain, we get the followingproposition. Proposition 2.1.
With capacitated valuations, a mechanism is efficient and IC if and only if it isVCG.VCG with
Clarke pivot payments has h i ( v − i ) = max a ∈L X j = i v j ( a ) (= X j = i v j (Opt − ij )) . Agent valuations for bundles of goods are non negative. The only mechanism that is efficient,incentive compatible, individually rational, and with no positive transfers is VCG with Clarke pivotpayments.The following proposition, which appears in [13], provides a criterion for the envy freeness of aVCG mechanism.
Proposition 2.2. [13] Given a VCG mechanism, specified by functions { h i } i ∈ [ n ] , agent i does notenvy agent j iff for every v , h i ( v − i ) − h j ( v − j ) ≤ v j (Opt j ) − v i (Opt j ) . .2 Gross substitutes and capacitated valuations We define the notion of gross substitute valuations and show that every capacitated valuation(i.e., additive up to the capacity) has the gross substitutes property. As this discussion refers to avaluation function of a single agent, we omit the index of the agent.Fix an agent and let D ( p ) be the collection of all sets of goods that maximize utility for theagent under price vector p , D ( p ) = arg max S ⊆ [ s ] { v ( S ) − P j ∈ S p j } . Definition 2.3. [7] A valuation function v : 2 [ s ] → R ≥ satisfies the gross substitutes conditionif the following holds: Let p = ( p , . . . , p s ) and q = ( q , . . . , q s ) be two price vectors such that theprice for good j is no less under q than under p : i.e. , q j ≥ p j , for all j . Consider the set of all itemswhose price is the same under p and q , E ( p, q ) = { ≤ j ≤ s | p j = q j } , then for any S p ∈ D ( p ) thereexists some S q ∈ D ( q ) such that S p ∩ E ( p, q ) ⊆ S q ∩ E ( p, q ). Theorem 2.4.
Every capacitated valuation function (additive up to the capacity) obeys the grosssubstitutes condition.
Proof.
Fix a capacitated valuation v , and prices p, q such that p i ≤ q i for every good i . Fix alsosome set S p ∈ D ( p ). Let S pq = { i ∈ S p : p i = q i } (= S p ∩ E ( p, q )). We show that there exists a setin D ( q ) that contains the set S pq .Let S q be an arbitrary set in D ( q ). Consider the following case analysis:1. S pq ⊆ S q : we’re done.2. S q ⊂ S p but S pq S q , this means that S q is smaller than the capacity of the agent. Let˜ S q = S q ∪ ( S pq \ S q ), it follows from the optimality of S p that for all goods j ∈ S pq ( ⊆ S p ), v ( j ) − p j ≥
0. Thus,the utility from ˜ S q is at least equal to the utility from S q , ergo, ˜ S q ∈ D ( q ).3. S q S p and S pq S q , : Let x = min {| S q \ S p | , | S pq \ S q |} . Replace x arbitrary goods in( S q \ S p ) by x arbitrary goods in ( S pq \ S q ), and let ˜ S q denote the result.From the optimality of S p , for every i ∈ ( S pq \ S q ) and j ∈ ( S q \ S p ) we have v ( j ) − p j ≤ v ( i ) − p i .Substituting p j ≤ q j and p i = q i , we get v ( j ) − q j ≤ v ( i ) − q i , and therefore the utility under q , u q ( S q ∪ { i } \ { j } ) ≥ u q ( S q ).Inductively, it holds that u q ( ˜ S q ) ≥ u q ( S q ), and thus ˜ S q ∈ D ( q ). If S pq ⊆ ˜ S q then we’re doneas in case 1 above. Otherwise, we can apply case 2 above.As a corollary, we get that capacitated valuations admit a Walrasian equilibrium. However, notnecessarily within an IC mechanism. The main result of this section is that Clarke pivot payments are envy free when capacities arehomogeneous. This follows from a stronger result, which we establish for heterogeneous capacities,showing that with Clarke pivot payments, no agent envies a lower-capacity agent.We first observe that one cannot aim for an incentive compatible mechanism with Walrasianprices (if this was possible then envy freeness would follow immediately).6 b cAgent 1 1 + ǫ ǫ − ǫ Agent 2 1 − ǫ/ ǫ (a) Matrix v a b cAgent 1 1 − ǫ − ǫ/ ǫ (b) Matrix v ′ Figure 2: No IC mechanism with Walrasian pricing for these inputs.
Proposition 3.1.
Capacitated valuations with homogeneous capacities c ≥ Proof.
Consider the valuations v given in Figure 2(a), which represents valuations for three goodsand two agents, each with capacity 2. Assume that Walrasian prices exist. I.e. , for every valuationmatrix v there exist prices p a ( v ), p b ( v ), and p c ( v ) such that every agent chooses a bundle of maximalutility under these prices, and this allocation maximizes social welfare. The social optimum hasOpt ( v ) = { a, b } Opt ( v ) = { c } . It follows that the price paid by agent 1 for { a, b } is p ( v ) = p a ( v ) + p b ( v ) . (3)However, it also follows from Proposition 2.1 that the only mechanism that is efficient andincentive compatible is the VCG mechanism. Therefore, the price paid by agent 1 is also of thefollowing form: p ( v ) = h ( v ) − v (Opt ( v ))= h ( v ) − v ( { c } ) = h ( v ) − (1 + ǫ ) , for some function h (that does not depend on v ).Combining Equations (3) and (4) we get that p a ( v ) + p b ( v ) = h ( v ) − (1 + ǫ ) . (4)If p a ( v ) < − ǫ/ a (as agents choose bundles of maximal utilityunder Walrasian pricing). As agent 2 receives Opt = { c } it follows that p a ( v ) ≥ − ǫ/
2. Similarly, p b ( v ) ≥
1. Substituting in (4) gives h ( v ) ≥ ǫ/ . (5)Now consider the valuations v ′ given in Figure 2(b). The social optimum here is Opt ( v ′ ) = { a } and Opt ( v ′ ) = { b, c } . As the mechanism is VCG, the payment for agent 1 must be of the form p ( v ′ ) = h ( v ′ ) − v ′ (Opt ( v ′ )) for h that does not depend on v ′ . As v ′ = v , we have p ( v ′ ) = h ( v ′ ) − v ′ (Opt ( v ′ ))= h ( v ) − v ( { b, c } )= h ( v ) − − ǫ ≥ − ǫ/ . v ′ agent 1 gets the bundle Opt ( v ′ ) = { a } and hence p a ( v ′ ) = p ( v ′ ) ≥ − ǫ/ a is 1 − ǫ . Agent 1 receives Opt ( v ′ ) = { a } . The utility to agent 1 is v ′ ( { a } ) − p a ( v ′ ) ≤ (1 − ǫ ) − (1 − ǫ/ < The following theorem establishes a general result for capacitated valuations: in a VCG mechanismwith Clarke-pivot payments, no agent will ever envy a lower-capacity agent.
Theorem 3.2.
If we apply the VCG mechanism with Clarke-pivot payments on the assignmentproblem with capacitated valuations, then • The mechanism is incentive compatible, individually rational, and makes no positive transfers(follows from VCG with CPP). • No agent of higher capacity envies an agent of lower or equal capacity.The input to the VCG mechanism consists of capacities and valuations. The agent capacity, c i ≥ i ), is publicly known. The number of units of good j , q j ≥ v i ( j ) — the value to agent i of a unit of good j , are private. b -Matching Graph Given capacities c i , q j , and a valuation matrix v , we construct an edge-weighted bipartite graph G as follows: • We associate a vertex with every agent i ∈ [ n ] on the left, let A be the set of these vertices. • We associate a vertex with every good j ∈ [ s ] on the right, let I be the set of these vertices. • Edge ( i, j ), i ∈ A , j ∈ I , has weight v i ( j ). • Vertex i ∈ A (associated with agent i ) has degree constraint c i . • Vertex j ∈ I (associated with good j ) has degree constraint q j .We seek an allocation a (= a ( v )) where a ij is the number of units of good j allocated to agent i . The value of the allocation is v ( a ) = P ij a ij v i ( j ). We seek an allocation of maximal value thatmeets the degree constraints: P j a ij ≤ c i , P i a ij ≤ q j , this is known as a b -matching problem andhas an integral solution if all constraints are integral, see [15]. Let a i = ( a i , a i , . . . , a in ) denotethe i ’th row of a , which corresponds to the bundle allocated to agent i .Let v k ( a i ) = P j ∈ [ s ] a ij v k ( j ) denote the value to agent k of bundle a i . Let M denote someallocation that attains the maximal social value, M ∈ arg max a v ( a ). Finally, let G − i be the graphderived from G by removing the vertex associated with agent i and all its incident edges, and let M − i be a matching of maximal social value with agent i removed.Specializing the Clarke-pivot rule to our setting, the payment of agent k is p k = v ( M − k ) − v ( M ) + v k ( M k ) . (6)8n the special case of permutation games (the number of agents and goods is equal, and everyagent can receive at most one good), the social optimum corresponds to a maximum weightedmatching in G . Permutation games were first studied by [9] who showed that Clarke-pivot paymentsare envy free. However, the shadow variables technique used in this proof does not seem to generalizefor larger capacities. Remark:
Our proof is given in terms of fractional allocations (where a ij ≥ P j a ij ≤ c i , P i a ij ≤ q j ) but also holds for integral allocations (where a ij ∈ Z ≥ , P j a ij ≤ c i , P i a ij ≤ q j ).This is because when capacities and quantities are integral, there is always an integral socialoptimum. Proof.
Let agent 1 and agent 2 be two arbitrary agents such that c ≥ c . Agent 1 does not envyagent 2 if and only if v ( M ) − p ≥ v ( M ) − p By substituting the Clarke pivot payments (6) and rearranging, this is true if and only if v ( M − ) ≥ v ( M − ) + v ( M ) − v ( M ) . (7)Thus in order to prove the theorem we need to establish (7).We construct a new allocation D − on G − (from the allocations M and M − ) such that v ( D − ) ≥ v ( M − ) + v ( M ) − v ( M ) . (8)From the optimality of M − , it must hold that v ( M − ) ≥ v ( D − ). Combining this with (8) shallestablish (7), as required.In what follows we make several preparations for the construction of the allocation D − . Given M and M − , we construct a directed bipartite graph G f on A ∪ I coupled with a flow f as follows.For every pair of vertices i ∈ A and j ∈ I , • If M ij − M − ij >
0, then G f includes arc i → j with flow f i → j = M ij − M − ij . • If M ij − M − ij <
0, then G f includes arc j → i with flow f j → i = M − ij − M ij . • If M ij = M − ij , then G f contains neither arc i → j nor arc j → i .We define the excess of a vertex in G f to be the difference between the amount of flow flowingout of the vertex and the amount of flow flowing into the vertex. I.e., the excess χ i of a vertex i ∈ A in G f is χ i = X ( i → j ) ∈ G f f i → j − X ( j → i ) ∈ G f f j → i = X j (cid:0) M ij − M − ij (cid:1) , and the excess χ j of a vertex j ∈ I in G f is χ j = X ( j → i ) ∈ G f f j → i − X ( i → j ) ∈ G f f i → j = X i (cid:0) M − ij − M ij (cid:1) . Clearly the sum of all excesses is zero.A vertex is said to be a source if its excess is positive, and said to be a target if its excess isnegative. The aforementioned definitions imply the following observation.9 bservation 3.3.
To summarize,0 ≤ X j M − ij + | χ i | = X j M ij ≤ c i ∀ source i ∈ A . (9)0 ≤ X j M ij + | χ i | = X j M − ij ≤ c i ∀ target i ∈ A . (10)0 ≤ X i M ij + | χ j | = X i M − ij ≤ q j ∀ source j ∈ I . (11)0 ≤ X i M − ij + | χ j | = X i M ij ≤ q j ∀ target j ∈ I . (12)Using the flow decomposition theorem, we can decompose the flow f into simple paths andcycles, where each path connects a source to a target. Associated with each path and cycle T is apositive flow value f ( T ) >
0. Given an arc x → y , f x → y is obtained by summing up the values f ( T )of all paths and cycles T that contain x → y . Notice that M − j = 0 for all j and therefore f → j ≥ j . It follows that there are no arcs of the form j → G f . The following observation canbe easily verified. Observation 3.4.
For each path P = u , u , . . . , u t in a flow decomposition of G f , where u is asource and u t is a target, it holds that f ( P ) ≤ min { χ u , | χ u t |} .We define the value of a path or a cycle T = u , u , . . . , u t in G f , to be v ( P ) = X u i ∈ A ,u i +1 ∈ I v u i ( u i +1 ) − X u i ∈ I ,u i +1 ∈ A v u i +1 ( u i ) . It is easy to verify that P T f ( T ) · v ( T ) = v ( M ) − v ( M − ), where we sum over all paths and cycles T in our decomposition.We will repeatedly do the following procedure: Let M , M − , f and G f be as above. Lemma 3.5.
Let T = u , u , . . . , u t be a cycle in G f or a path in the flow decomposition of G f ,and let ǫ be the minimal flow along any arc of T . We construct an allocation c M (= c M ( T )) from M by canceling the flow along T , start with c M = M and then for each ( u i , u i +1 ) ∈ T set: c M u i u i +1 = M u i u i +1 − ǫ u i ∈ A , u i +1 ∈ I c M u i +1 u i = M u i +1 u i + ǫ u i ∈ I , u i +1 ∈ A . Alternatively, we construct c M − (= c M − ( T )) from M − , starting from c M − = M − and then foreach ( u i , u i +1 ) ∈ T set c M − u i u i +1 = M − u i u i +1 + ǫ u i ∈ A , u i +1 ∈ I , c M − u i +1 u i = M − u i +1 u i − ǫ u i ∈ I , u i +1 ∈ A . The allocations c M , c M − are valid (do not violate capacity constraints).10 roof. If T is a cycle then our manipulations do not affect the total quantity allocated to an agent,nor the total demand for a good.If T is a path, then our manipulations do not affect the quantities/demands for all internalvertices, i.e. , it is sufficient to show that the capacities/demands of u and u t are not exceeded.It follows from Observation 3.4 that the flow along a path T , f ( T ) ≤ min { χ u , | χ u t |} .Consider the first vertex along T , u , if u is an agent, then by Observation 3.3 it holds that X j M − u j ≤ c u − | χ u | ≤ c u − ǫ. Thus we can increase the allocation of M − u u by ǫ , while not exceeding the capacity of agent u ( c u ). If u is a good, agent u can release ǫ units of good u without violating any capacityconstraints. For vertex u t , we can follow a similar argument and use Observation 3.3 to show thatthe capacity constraint of u t is not violated either.The remainder of the proof requires several preparations that are cast in the following lemmata. Lemma 3.6.
It is without loss of generality to assume that M − is such that1. There are no cycles of zero value in G f .2. There is no path P = u , u , . . . , u t of zero value such that u = 1 is a source and u t is atarget. Proof.
Assume that there is a cycle or a path T in the flow decomposition of G f such that v ( T ) = 0.Let ǫ be the smallest flow along an arc e of T . Let c M − = c M − ( T ) as in Lemma 3.5, it followsfrom the Lemma that c M − is a valid assignment.Furthermore, v ( c M − ) = v ( M − ) − xv ( T ) = v ( M − ) and if we replace M − by c M − then thenew G f (for M and c M − ) is derived from the old G f (for M and M − ) by decreasing the flowalong every arc of T by ǫ , and removing arcs whose flow is zero. In particular, at least one arc willbe removed and no new arcs added. We repeat this process until G f does not contain any cycle orpath of zero value, as required.Thus, in the sequel we assume that M − satisfies conditions (1) and (2) of Lemma 3.6 . Lemma 3.7.
The graph G f does not contain a cycle. Proof.
Assume that G f contains a cycle C which carries ǫ > v ( C ) <
0, let c M = c M ( C ). According to Lemma 3.5, c M is a valid assignment. The value of c M is v ( c M ) = v ( M ) − ǫv ( C ) > v ( M ) , which contradicts the maximality of M .If v ( C ) >
0, let c M − = c M − ( C ). According to Lemma 3.5, c M − is a valid assignment. Wenow show that c M − allocates nothing to agent 1. For there to be an edge j → G f , it must bethat M j − M − j <
0, but M j ≥ M − does not contain agent 1 at all so M − j = 0. It follows Since Inequality (8) depends only on the value of M − it does not matter which M − we work with. G f , and thus cannot be part of any cycle. The value of c M − is v ( c M − ) = v ( M − ) + ǫv ( C ) > v ( M − ) , which contradicts the maximality of M − .Also, by Lemma 3.6, there are no cycles of value zero in G f and this concludes the proof.In particular, Lemma 3.7 implies that there are no cycles in our flow decomposition. We nextshow that the only source vertex in G f is the vertex corresponding to agent 1. Lemma 3.8.
The vertex that corresponds to agent 1 is the unique source vertex.
Proof.
Proof via reduco ad absurdum. Consider the graph G f , let u = 1 be a source vertex, u t atarget vertex, and, by assumption, let P = u , u , . . . u t , be some path in G f with flow ǫ >
0. Sincethe vertex corresponding to agent 1 has no incoming arcs, P does not contain vertex 1. Accordingto Lemma 3.6, such a path P cannot have value zero.According to Lemma 3.5, the allocations c M − = c M − ( P ) and c M ( P ) are valid (preserve capacityconstraints).Consider the following two cases. case a: v ( P ) >
0. It follows that v ( c M − ) = v ( M − ) + ǫv ( P ) > v ( M − ) . case b: v ( P ) <
0. It follow that v ( c M ) = v ( M ) − ǫv ( P ) > v ( M ) . In both cases we’ve reached contradiction, either to the optimality of M − (case a) or to theoptimality of M (case b).Lemma 3.8 implies that all the paths in our flow decomposition originate at agent 1. We arenow ready to describe the construction of the allocation D − :1. Stage I: initially, D − := M − .2. Stage II: for every good j , let x = min { M j , M − j } , and set D − j := M − j − x and D − j := x .3. Stage III: for every flow path P in the flow decomposition of G f that contains agent 2, let ˆ P bethe prefix of P up to agent 2. For every agent to good arc ( i → j ) ∈ ˆ P set D − ij := D − ij + f ( P ),and for every good to agent arc ( j → i ) ∈ ˆ P set D − ij := D − ij − f ( P ).It is easy to verify that D − indeed does not allocate any good to agent 2. Also, the allocation toagent 1 in D − is of the same size as the allocation to agent 2 in M − . Since c ≥ c , D − is avalid allocation.To conclude the proof of Theorem 3.2 we now show that: Lemma 3.9.
Allocation D − satisfies (8). 12 roof. Rearranging (8), we obtain v ( D − ) ≥ v ( M − ) (13)+ s X j =1 ( v ( j ) − v ( j )) · min( M j , M − j ) (14)+ X j : M j >M − j ( v ( j ) − v ( j )) ( M j − M − j ) . (15)At the end of stage I, we have D − = M − and so the inequality above at line (13) (Excludingexpressions (14) and (15)) holds trivially. It is also easy to verify that at the end of stage II, theinequality above that spans expressions (13) and (14) (and excludes expression (15)) holds. Whatwe show next is that at the end of stage III, the full inequality above will hold.Consider a good j such that M j > M − j . In G f we have an arc 2 → j such that f → j = M j − M − j , therefore in the flow decomposition we must have paths P , . . . , P ℓ , all containing thearc 2 → j , such that ℓ X k =1 f ( P k ) = f → j = M j − M − j . (16)For every k = 1 , . . . , ℓ , let b P k denote the prefix of P k up to agent 2. Consider the cycle C consisting of b P k followed by arcs 2 → j and j →
1. We claim that the value of this cycle isnon-negative.Consider the allocation c M ( C ) which is a valid allocation from Lemma 3.5. Observe that v ( c M ) = v ( M ) − ǫv ( C ) > v ( M ). This now contradicts the assumption that M maximizes v over allallocations. We obtain v ( b P k ) + v ( j ) − v ( j ) ≥ . Rearranging and multiplying by f ( P k ), it follows that f ( P k ) v ( b P k ) ≥ f ( P k ) ( v ( j ) − v ( j )) . Summing over all paths k = 1 , . . . , ℓ , we get ℓ X k =1 (cid:16) f ( P k ) v ( b P k ) (cid:17) ≥ ( v ( j ) − v ( j )) ℓ X k =1 f ( P k ) . Substituting (16) in the last inequality establishes the following inequality: ℓ X k =1 (cid:0) f ( P k ) v ( b P k ) (cid:1) ≥ (cid:0) v ( j ) − v ( j ) (cid:1)(cid:0) M j − M − j (cid:1) . (17)The left hand side of (17) is exactly the gain in value of the allocation when applying stageIII to the paths b P , . . . , b P ℓ during the construction of D − above. The right hand side is the termwhich we add in (15).To conclude the proof of Lemma 3.9, we note that stage III may also deal with other pathsthat start at agent 1 and terminate at agent 2. Such paths must have non-negative value and13hus can only increase the value of D − . Otherwise we can construct an allocation ˜ M , such that v ( ˜ M ) > v ( M ) by decreasing M ij by ǫ for every arc ( i → j ) ∈ P and increasing M ij by ǫ forevery arc ( j → i ) ∈ P as we did constructing c M in Lemma 3.5. The allocation ˜ M is valid sinceit preserves capacities of vertices that are internal on the path and decreases only arcs with flowon them, M ij > ǫ . Finally, the capacity of a source agent vertex can be increased according toObservation 3.3.This concludes the proof of Theorem 3.2.The following is a direct corollary of Theorem 3.2. Corollary 3.10.
If all agent capacities are equal, then the VCG allocation with Clarke-pivotpayments is EF.
Do Clarke-pivot payments work also under heterogeneous capacities? The answer is no, as demon-strated in Example 1.1. In this section we prove a stronger result, showing that any mechanismthat is both incentive compatible and envy-free for heterogeneous capacities must have positivetransfers. We remark that IC, NPT, and IR ⇔ Clarke-pivot payments, which, along with Ex-ample 1.1 implies that one cannot have an efficient mechanism that is IC, NPT, IR, and EF forheterogeneous capacities, here we prove that even without the individual rationality requirement,this is impossible.
Theorem 4.1.
Consider capacitated valuations with heterogeneous capacities such that the num-ber of goods exceeds the smallest agent capacity. There is no mechanism that is simultaneouslyefficient, IC, EF, and has no positive transfers. That is, any IC and EF efficient mechanism hassome valuations v for which the mechanism pays an agent. Remark:
Note that the conditions on the capacities of the agents and the number of goodsare necessary. If capacities are homogeneous or the total supply of goods is at most the minimumagent capacity, then Clarke-pivot payments, that are known to be incentive compatible, individuallyrational, and have no positive transfers, are also envy-free.
Proof.
We start with a warm-up of capacitated valuations with two agents and two goods whereagent i = 1 , i . We then generalize the proof to arbitrary heterogeneous settings. Toease the notation we abbreviate v i ( j ) to v ij . Two agents and two goods:
One can easily verify that the social optimum is as follows. (we omit cases with ties). • If v > v and v > v , then Opt = { , } and Opt = ∅ . We refer to this class ofvaluations as class A. The social optimum is unique when there are no ties. Valuations v ’s with ties form a lower dimensional measure0 set. It suffices to consider valuations without ties for both existence or non-existence claims of IC or EF payments.This is clear for non-existence, for existence, the payments for a v with ties is defined as the limit when we approachthis point through v ’s without ties that result in the same allocation. Clearly IC and EF properties carry over, alsoIR and NPT. If v − v > max { , v − v } , then Opt = { } and Opt = { } . We refer to this class ofvaluations as class B . • If v − v > max { , v − v } , then Opt = { } and Opt = { } . We refer to this class ofvaluations as class B . x + 3 ǫ x + ǫ (a) x + 3 ǫ x + ǫx + ǫ x (b) x + ǫ x (c) Figure 3:
These matrices correspond to three valuation profiles, where in each matrix the rows correspondto the agents and the columns correspond to goods. The valuations in matrices (a) and (b) belong to class B , and the valuation in matrix (c) belongs to class A . Substituting the above in Proposition 2.2 we obtain that for v ∈ B , agent 1 does not envyagent 2 if and only if h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt ) = v − v , (18)and agent 2 does not envy agent 1 if and only if h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt ) = v − v . (19)Fix an ǫ >
0, and some x >
0, and consider the valuation v where v = x + 3 ǫ , v = x + ǫ and v = v = 0 (see Figure 4(a)). This valuation is clearly in B . Substituting in (19), agent 2 doesnot envy agent 1 in v if and only if − ( x + 3 ǫ ) ≤ h (0 , − h ( x + 3 ǫ, x + ǫ ) . (20)Next consider the valuation v where v = x + 3 ǫ , v = x + ǫ , v = x + ǫ , and v = x (seeFigure 4(b)). This valuation is clearly in B as well. Substituting in (18), agent 1 does not envyagent 2 in v if and only if h ( x + ǫ, x ) − h ( x + 3 ǫ, x + ǫ ) ≤ x − ( x + ǫ ) = − ǫ. (21)Combining (20) and (21), it follows that h ( x + ǫ, x ) ≤ h ( x + 3 ǫ, x + ǫ ) − ǫ ≤ h (0 ,
0) + x + 2 ǫ (22)The no positive transfers requirement states that for any v , p i ≥ i ; in particular, p = h ( v ) − v (Opt ) ≥
0; i.e., h ( v ) ≥ v (Opt ) . (23)Finally, consider the valuations v where v = v = 0, v = x + ǫ , and v = x (see Figure 4(c)).Clearly, the optimal allocation is one in which agent 2 gets both goods, thus v ∈ A and v (Opt ) =2 x + ǫ . From (22) it follows that h ( x + ǫ, x ) ≤ h (0 ,
0) + x + 2 ǫ . From (23) it follows that h ( x + ǫ, x ) ≥ v (Opt ) = 2 x + ǫ . Combining we obtain h (0 , ≥ x − ǫ . However, h (0 ,
0) cannotbe a function of x ; in particular if h (0 , < x − ǫ , then we obtain a contradiction.15his simple case gives us essentially all the intuition and structure that is required for solvingthe general case. Heterogeneous capacities, multiple agents and goods:
Let c be the smallest agent capacity and rename the agents such that c is the capacity of agent1, and the capacity of agent 2 is strictly greater than c . Consider an instance with at least c + 1goods, and valuation functions satisfying v ij = 0 if i > j > c + 1, and v ij = v ij ′ for i = 1 , ≤ j, j ′ ≤ c + 1.It is easy to verify that an optimal allocation Opt is obtained as follows (where we omit caseswith ties and only define the allocation of goods j = 1 , . . . , c + 1). • If v > v and v > v , then Opt = { , . . . , c + 1 } and Opt = ∅ . We refer to this classof valuations as class A. • If v > v and v < v , then Opt = { } and Opt = { , . . . , c + 1 } . We refer to this classof valuations as class B . • If v − v > v − v and v > v , then Opt = { , . . . , c } and Opt = { c + 1 } . We referto this class of valuations as class B +1 . • If v − v > max { , v − v } , then Opt = { } and Opt = { , . . . , c + 1 } . We refer to thisclass of valuations as class B .Substituting the above in proposition 2.2 we obtain that for v ∈ B +1 , agent 1 does not envyagent 2 if and only if h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt ) = v − v , (24)and agent 2 does not envy agent 1 if and only if h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt )= v + ( c − v − v − ( c − v . (25)Fix an ǫ > x >
0, and consider the valuation v where v = x + 3 ǫ , v j = x + ǫ for j = 2 , . . . , c + 1, and v j = 0 for j = 2 , . . . , c + 1. This valuation is clearly in B +1 . Substituting thecorresponding values in (25) we obtain that agent 2 does not envy agent 1 if and only if − cx − ( c + 2) ǫ ≤ h (0 , − h ( x + 3 ǫ, x + ǫ ) . (26)Next consider the valuation v where v = x + 3 ǫ , v j = x + ǫ for j = 2 , . . . , c + 1, v = x + ǫ , and v j = x for j = 2 , . . . , c + 1. This valuation is clearly in B +1 as well. Substituting the correspondingvalues in (24) we obtain that agent 1 does not envy agent 2 if and only if h ( x + ǫ, x ) − h ( x + 3 ǫ, x + ǫ ) ≤ − ǫ . (27)Combining (26) and (27) we obtain, h ( x + ǫ, x ) ≤ h ( x + 3 ǫ, x + ǫ ) − ǫ ≤ h (0 ,
0) + cx + ( c + 2) ǫ − ǫ = h (0 ,
0) + cx + ( c + 1) ǫ. (28)16inally, consider the valuations v where v = v j = 0 for j = 2 , . . . , c + 1, v = x + ǫ , and v j = x for j = 2 , . . . , c + 1. Clearly, the optimal allocation is one in which agent 2 gets all c + 1 goods, thus v ∈ A and v (Opt ) = ( c + 1) x + ǫ . From (28) it follows that h ( x + ǫ, x ) ≤ h (0 ,
0) + cx + ( c + 1) ǫ . In order to satisfy no positive transfers, according to (23), it holds that h ( x + ǫ, x ) ≥ v (Opt ) = ( c + 1) x + ǫ . Combining we obtain h (0 , ≥ x − cǫ . However, h (0 , x and c ; in particular if h (0 , < x − cǫ , then we obtain a contradiction. In the previous section we showed that it is impossible to satisfy EF, IC and NPT simultaneously.Here we show that if we forego the NPT requirement, then capacitated valuations with two agents(and arbitrary capacities and number of goods) admits a mechanism which satisfies the other twoproperties as well as IR.
Proposition 5.1.
Proof.
Let c i be the capacity of agent i and assume without loss of generality that c ≤ c . Givena vector ( x , x . . . ) let top b { x } be the set of the b largest entries in x .We show that h ( v ) = X j ∈ top c { v } v j and h ( v ) = X j ∈ top c { v } v j (29)give VCG payments which are envy-free.By Proposition 2.2, it is sufficient to show that for i = 1 , j = 2 and for i = 2 , j = 1 it holds that h i ( v j ) − h j ( v i ) ≤ v j (Opt j ) − v i (Opt j ) . By substituting h and h from (29), this is equivalent to X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ v (Opt ) − v (Opt ) (30)and X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ v (Opt ) − v (Opt ) . (31)Assume first that the number of goods is exactly c + c . Clearly, in the optimal solution, agent1 will get the c goods that maximize v j − v j and agent 2 will get the c goods that minimize thisdifference.We first establish (31): X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ X j ∈ top c { v } ( v j − v j ) ≤ X j ∈ top c { v − v } ( v j − v j )= X j ∈ Opt ( v j − v j )= v (Opt ) − v (Opt )17here the inequalities follow by the fact that for every v i ∈ R s ≥ and S ⊂ [ s ], it holds that X j ∈ top | S | { v i } v ij ≥ X j ∈ S v ij . (32)In what follows we establish (30). We use the following additional notation: for a subset Y of entries, let top b { x | Y } denote the set of b largest entries in x projected on Y . In addition, let S = top c { v } ∩ (Opt \ top c { v | Opt } ); i.e., S is the (possibly empty) set of goods that are amongthe top c goods for v , are also in Opt , but are not among the top c goods for v in Opt . v (Opt ) − v (Opt ) = X j ∈ Opt v j − X j ∈ top c { v | Opt } v j = X j ∈ Opt \ top c { v | Opt } v j + X j ∈ top c { v | Opt } ( v j − v j ) ≥ X j ∈ S v j + X j ∈ top c { v | Opt } ( v j − v j ) ≡ X j ∈ S v j + X j ∈ top c { v | Opt } ( v j − v j ) . (33)Let S ′ be a set of | S | goods from Opt ∪ top c { v | Opt } which are not contained in top c { v } .Such a set always exists because there are 2 c goods in Opt ∪ top c { v | Opt } , and exactly c − | S | of them are in top c { v } , and therefore we have c + | S | goods to choose S ′ from. Therefore, inorder to establish (30), it suffices to show that X j ∈ S v j + X j ∈ top c { v | Opt } ( v j − v j ) ≥ X j ∈ top c { v } v j − X j ∈ top c { v } v j . This is established in what follows. X j ∈ top c { v } v j − X j ∈ top c { v } v j = X j ∈ S v j + X j ∈ top c { v }\ S v j − X j ∈ top c { v } v j ≤ X j ∈ S v j + X j ∈ S ′ ∪ top c { v }\ S v j − X j ∈ top c { v } v j ≤ X j ∈ S v j + X j ∈ S ′ ∪ top c { v }\ S ( v j − v j ) (34) ≤ X j ∈ S v j + X j ∈ top c { v | Opt } ( v j − v j ) (35)Inequality (34) follows from Equation (32) since S ′ ∪ top c { v } \ S by definition contains exactly c goods. Finally, to establish Inequality (35), observe that all the goods in top c { v | Opt } belongto Opt , while all the goods that belong to S ′ ∪ top c { v } \ S but not to top c { v | Opt } belong toOpt . Recalling that v j − v j ≥ v j ′ − v j ′ for every j ∈ Opt , j ′ ∈ Opt concludes the derivationof the inequality.It remains to analyze the cases where the number of goods is different than c + c . If thenumber of goods is less than c + c , then consider a set D of “dummy” goods that are added to18he set of “real” goods, such that v j = v j = 0 for every j ∈ D . These dummy goods do notchange the optimal allocation projected on the real goods, and for every agent and every bundle,the valuation of the agent to the bundle is equal to her valuation for the set of real goods in thebundle. In addition, the values of h and h are also equal to their values as defined with respectto the real goods alone. Therefore, the aforementioned argument (for the case of c + c goods) canbe applied here as well.We next consider the case in which there are more than c + c goods. Observe that all thegoods involved in Equations (30) and (31) participate in the optimal solution (as top c { v } and top c { v } must both be included in the optimal solution). Therefore, it is sufficient to consider theset of c + c goods that participate in the optimal solution. In previous sections we restricted attention to capacitated valuations. Here, we turn to the moregeneral family of subadditive valuations, but restrict attention to the case of two agents and twogoods. For this case, we construct a mechanism that is simultaneously IC, EF and IR. This issummarized in the following proposition.
Proposition 6.1.
For any subadditive allocation setting with two agents and two goods, a VCGmechanism with the following h , h functions is envy free and individually rational: h ( v ) = max( v ( { } ) , v ( { } )) h ( v ) = max( v ( { } ) , v ( { } )) Proof.
By Proposition 2.2, a VCG mechanism is envy free if and only if h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt ) , and (36) h ( v ) − h ( v ) ≤ v (Opt ) − v (Opt ) . (37)The only possible allocations are both goods allocated to same agent or each agent gets onegood. Wlog., we can assume that good 2 is allocated to agent 2. Case 1:
Opt = { , } , Opt = ∅ . Case 2:
Opt = { } , Opt = { } .We establish via case analysis that (36) and (37) hold in these two cases. To simplify presen-tation we use v i (1) , v i (2) , v i (1 ,
2) when we refer to v i ( { } ) , v i ( { } ), and v i ( { , } ) respectively. Inaddition, we use max v i and min v i to denote max { v i (1) , v i (2) } and min { v i (1) , v i (2) } for every agent i . Establishing (36) for Case 1:
From subadditivity, v (1 , ≤ v (1) + v (2) = max v + min v . From optimality, v (1 , ≥ max { v (1) + v (2) , v (2) + v (1) } ≥ max v + min v . v (Opt ) − v (Opt ) = v (1 , − v (1 , ≥ max v + min v − max v − min v = max v − max v = h ( v ) − h ( v ) Establishing (37) for Case 1:
From subadditivity, v (1 , ≤ v (1) + v (2) = max v + min v . From optimality, v (1 , ≥ max { v (1) + v (2) , v (2) + v (1) } ≥ max v + min v . Combining together, we get max v ≥ max v and therefore, h ( v ) − h ( v ) = max v − max v ≤ v (Opt ) − v (Opt ) . Establishing (36) for Case 2:
We need to show, that v (Opt ) − v (Opt ) − ( h ( v ) − h ( v )) = v (2) − v (2) − max v + max v ≥ . If max v = v (2), then the above inequality trivially holds. If max v = v (1), then the aboveinequality follows from optimality of allocation, v (2) + v (1) ≥ v (2) + v (1).We omit the proof of (37) for Case 2 since it is similar to (36) for Case 2.This establishes the assertion of the proposition.Recall that the valuation of an agent i for bundle B is defined as v i ( B ) ≡ P j ∈ top ci { v i | B } v ij ,which is a special case of subadditive valuations. The following is, therefore, a direct corollary ofProposition 6.1. Corollary 6.2.
For capacitated valuations (public or private) with 2-agent and 2-goods, the VCGmechanism with the following h , h functions is EF and IC: h ( v , c ) = ( max( v , v ) c ∈ { , } c = 0 h ( v , c ) = ( max( v , v ) c ∈ { , } c = 0 Remark:
Note that with two goods, all c i ≥ ∈ { , , } . This work initiates the study of efficient, incentive compatible, and envy free mechanisms forcapacitated valuations.Our work suggests a host of problems for future research on heterogeneous capacitated valua-tions and generalizations thereof. 20e know that, generally, there may be no mechanism that is both IC and EF even if we allowpositive transfers .First, is there a mechanism for games with more than two agents that is efficient, IC, and EF ?We conjecture that such mechanisms do exist and believe this is also the case for any combinatorialauction with subadditive valuations (which generalizes capacitated valuations with private or publiccapacities). We provided such mechanisms for capacitated valuations with two agents (publiccapacities) and for subadditive valuations with two agents and two goods.Second, our work focused on efficient mechanisms; i.e., ones that maximize social welfare. Anatural question is how well the optimal social welfare can be approximated by a mechanism thatis IC, EF, and NPT. Michal Feldman is partially supported by the Israel Science Foundation (grant number 1219/09)and by the Leon Recanati Fund of the Jerusalem school of business administration. Amos Fiat,Haim Kaplan, and Svetlana Olonetsky are partially supported by the Israel Science Foundation(grant number 975/06).
References [1] Milgrom P. R. Ausubel, L. M. Ascending auctions with package bidding.
Frontiers of Theo-retical Economics , 1:1–42, 2002.[2] Liad Blumrosen and Noam Nisan. Combinatorial auctions. In E. Tardos V. Vazirani N. Nisan,T. Roughgarden, editor,
Algorithmic Game Theory . Cambridge University Press, 2007.[3] Liad Blumrosen and Nosam Nisan. Informational limitations of ascending combinatorial auc-tions.
Journal of Economic Theory , 145:1203–1223, 2001.[4] E. Clarke. Multipart Pricing of Public Goods.
Public Choice , 1:17–33, 1971.[5] L.E. Dubins and E.H. Spanier. How to cut a cake fairly.
American Mathematical Monthly ,68:1–17, 1961.[6] D. Foley. Resource allocation and the public sector.
Yale Economic Essays , 7:45–98, 1967.[7] Frank Gul and Ennio Stacchetti. Walrasian equilibrium with gross substitutes.
Journal ofEconomic Theory , 87:95–124, 1999.[8] Leonard Hurwicz. Optimality and informational efficiency in resource allocation processes. InK.J. Arrow, S. Karlin, and P. Suppes, editors,
Mathematical Methods in the Social Sciences ,pages 27–46. Stanford University Press, Stanford, CA, 1960.[9] Herman B. Leonard. Elicitation of honest preferences for the assignment of individuals topositions.
The Journal of Political Economy , 91:3:461–479, 1983. As an example, consider a setting with two goods, a, b and three agents, where v ( a ) = v ( b ) = v ( { a, b } ), v ( a ) = v ( b ) = v ( { a, b } ), and v ( a ) = v ( b ) = 0, while v ( { a, b } ) >
0. One can easily verify that this setting has nomechanism that is simultaneously incentive compatible and envy free. This example is due to Noam Nisan.
G. Feiwel (ed.), Arrow and theFoundations of the Theory of Economic Policy (essays in honor of Kenneth Arrow) , volume59:4, pages 341–349, 1987.[11] Herve Moulin.
Fair Division and Collective Welfare . MIT Press, 2004.[12] Noam Nisan. Introduction to mechanism design. In Noam Nisan, Tim Roughgarden, Eva Tar-dos, and Vijay Vazirani, editors,
Algorithmic Game Theory , chapter 16. Cambridge UniversityPress, 2007.[13] Szilvia p´apai. Groves sealed bid auctions of heterogeneous objects with fair prices.
Socialchoice and Welfare , 20:371–385, 2003.[14] David Parkes. Iterative combinatorial auctions: Achieving economic and compu- tationaleciency.
Ph.D. Thesis, Department of Computer and Information Science, University of Penn-sylvania , 2001.[15] W. Pulleyblank. Dual integrality in b -matching problems. In R. W. et. al. Cottle, editor, Combinatorial Optimization , volume 12 of
Mathematical Programming Studies , pages 176–196.Springer Berlin Heidelberg, 1980. 10.1007/BFb0120895.[16] L.-G. Svensson. On the existence of fair allocations.
Journal of Economics , 43:301–308, 1983.[17] William Vickrey. Counterspeculation, Auctions, and Competitive Sealed Tenders.
Journal ofFinance , 16:8–37, 1961.[18] H. Peyton Young.
Equity: In Theory and Practice . Princeton University Press, 1995.22 r X i v : . [ c s . G T ] F e b Truth and Envy in Capacitated Allocation Games
Edith Cohen ∗ Michal Feldman † Amos Fiat ‡ Haim Kaplan § Svetlana Olonetsky ¶ Abstract
We study auctions with additive valuations where agents have a limit on the number of items theymay receive. We refer to this setting as capacitated allocation games.
We seek truthful and envy freemechanisms that maximize the social welfare. I.e., where agents have no incentive to lie and no agentseeks to exchange outcomes with another.In 1983, Leonard showed that VCG with Clarke Pivot payments (which is known to be truthful,individually rational, and have no positive transfers), is also an envy free mechanism for the special caseof n items and n unit capacity agents. We elaborate upon this problem and show that VCG with ClarkePivot payments is envy free if agent capacities are all equal. When agent capacities are not identical, weshow that there is no truthful and envy free mechanism that maximizes social welfare if one disallowspositive transfers.For the case of two agents (and arbitrary capacities) we show a VCG mechanism that is truthful, envyfree, and individually rational, but has positive transfers. We conclude with a host of open problems thatarise from our work. We consider allocation problems where a set of objects is to be allocated amongst m agents, where everyagent has an additive and non negative valuation function. We study mechanisms that are truthful, envy free,and maximize the social welfare (sum of valuations). The utility of an agent i is the valuation of the bundleassigned to i , v i ( OPT ) , minus any payment, p i .A mechanism is incentive compatible (or truthful) if it is a dominant strategy for every agent to reporther private information truthfully [4]. A mechanism is envy-free if no agent wishes to switch her outcomewith that of another [1, 2, 9, 6, 7, 10].Any allocation that maximizes the social welfare has payments that make it truthful — in particular —any payment of the form p i = h i ( t − i ) − X j = i v j ( OPT ) (1)where OPT is an allocation maximizing the social welfare and t − i are the types of all agents but agent i .Similarly, any allocation that maximizes the social welfare has payments that make it envy free, this followsfrom a characterization of envy free allocations (see [3]). Unfortunately, the set of payments that make themechanism truthful, and the set of payments that make the mechanism envy free, need not intersect. In this ∗ 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. i has maximal valuation, then agent i will receive all items.Leonard [5] considered the problem of assigning people to jobs, n people to n positions, and calledthis problem the permutation game. The Vickrey 2nd price auction is irrelevant in this setting because noperson can be assigned to more than one position. Leonard showed that VCG with Clarke Pivot payments issimultaneously truthful and envy free. Under Clarke Pivot payments, agents internalize their externalities,i.e., h i ( t − i ) = X j = i v j ( OPT − i ) (2)where OPT − i is the optimal allocation if there was no agent i . By substituting P j = i v j ( OPT − i ) for h i ( t − i ) in Equation 1 one can interpret Clarke Pivot payments as though an agent pays for how much others lose byher presence, i.e., the agent internalizes her externalities.Motivated by the permutation game, we consider a more general capacitated allocation problem whereagents have associated capacities. Agent i has capacity U i and cannot be assigned more than U i items.Like Leonard, we seek a mechanism that is simultaneously truthful and envy free. The private types weconsider may include both the valuation and the capacity (private valuations and private capacities) or onlythe valuation (private valuations, public capacity). Leonard’s proof uses LP duality and it is not obvioushow to extend it to more general settings.Before we address this question, one needs to ask what does it mean for one agent to envy another whenthey have different capacities? A lower capacity agent may be unable to switch allocations with a highercapacity agent. To deal with this issue, we allow agent i , with capacity less than that of agent i ′ to choosewhatever items she desires from the i ′ bundle, up to her capacity. I.e., we say that agent i envies agent i ′ if agent i prefers a subset of the allocation to agent i ′ , along with the price set for agent i ′ , over her ownallocation and price.The VCG mechanism (obey Equation 1) is always truthful. In fact, any truthful mechanisms that choosethe socially optimal allocation in capacitated allocation problems must be VCG [8]. We obtain the following:1. For agents with private valuations and either private or public capacities, under the VCG mechanismwith Clarke Pivot payments, a higher capacity agent will never envy a lower capacity agent. Inparticular, if all capacities are equal then the mechanism is envy free. (See Section 3).2. For agents with private valuations, and either private or public capacities, any envy free VCG paymentmust allow positive transfers. (See Section 4).3. For two agents with private valuations and arbitrary public capacities, there exist VCG payments suchthat the mechanism is envy free. It follows that such payments must allow positive transfers. (SeeSection 5).4. For two agents with private valuations and private capacities, and for two items, there exist VCGpayments such that the mechanism is envy free. (See Section 6).2 Preliminaries
Let U be a set of objects, and let v i be a valuation function associated with agent i , ≤ i ≤ m , that mapssets of objects into ℜ . We denote by v a sequence < v , v , . . . , v m > of valuation functions one for eachagent.An allocation function a maps a sequence of valuation functions v = < v , v , . . . , v m > into a partitionof U consisting of m parts, one for each agent. I.e., a ( v ) = < a ( v ) , a ( v ) , . . . , a m ( v ) >, where ∪ i a i ( v ) ⊆ U and a i ( v ) ∩ a j ( v ) = ∅ for i = j . A payment function is a mapping from v to ℜ m , p ( v ) = < p ( v ) , p ( v ) , . . . , p m ( v ) > , p i ( v ) ∈ ℜ . We assume that payments are from the agent to themechanism (if the payment is negative then this means that the transfer is from the mechanism 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.Let v = < v , v , . . . , v m > be a sequence of valuations, we define ( v ′ i , v − i ) to be the sequence ofvaluation functions arrived by substituting v i by v ′ i , i.e., ( v ′ i , v − i ) = < v , . . . , v i − , v ′ i , v i +1 , . . . , v m > . We next define mechanisms that are incentive compatible, envy-free, and both incentive compatible andenvy-free. • A mechanism is incentive compatible ( IC ) if it is a dominant strategy for every agent to reveal hertrue valuation 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 ); ⇔ 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) . (3) • A mechanism is envy-free ( EF ) if no agent seeks to switch her allocation and payment with another.I.e., if for all ≤ i, j ≤ m and all v : v i ( a i ( v )) − p i ( v ) ≥ v i ( a j ( v )) − p j ( v ); ⇔ p i ( v ) ≤ p j ( v ) + (cid:16) v i ( a i ( v )) − v i ( a j ( v )) (cid:17) . (4) • A mechanism ( a, p ) is incentive compatible and envy-free ( IC ∩ EF ) if ( a, p ) is both incentivecompatible and envy-free. Vickrey-Clarke-Groves (VCG) mechanism:
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 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. 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 .It is known that any mechanism whose allocation function a maximizes P mi =1 v i ( a i ( v )) (social welfare)is incentive compatible if and only if it is a VCG mechanism (See, e.g., [8], Theorem 9.37). In the followingwe will denote by opt an allocation a which maximizes P mi =1 v i ( a i ( v )) .The Clarke-pivot payment for a VCG mechanism is defined by h i ( v − i ) = max a ′ ∈ A X j = i v j ( a ′ ) . A capacitated allocation game has m agents and n items that need to be assigned to the agents. Agent i is associated with a capacity U i ≥ , denoting the limit on the number of items she can be assigned, andeach item j is associated with a capacity Q j ≥ , denoting the number of available copies of item j . Thevaluation v i ( j ) denotes how much agent i values item j , and P j ∈ S v i ( j ) is the valuation of agent i to thebundle S .A capacitated allocation game has a corresponding bipartite graph G , where every agent ≤ i ≤ m hasa vertex i associated with it on the left side, and every item ≤ j ≤ n has a vertex j associated with it onthe right side. The weight of the edge ( i, j ) is v i ( j ) . An assignment is a subgraph of G that satisfies thecapacity constraints, i.e. agent i is assigned at most U i items and item j is assigned to at most Q j agents.Recall that we denote by opt an assignment of maximum value. We describe opt by a matrix M where M ij is the number of copies of item j allocated to agent i in opt .For player i , the graph G − i is constructed by removing the vertex associated with agent i and its incidentedges from G . The assignment with maximum value in G − i is defined by a matrix M − i .Let M be an assignment (either in G or in G − i for some i .). We denote by M i r the i ’th row of M , ( M i , M i , . . . , M in ) which gives the bundle that agent i gets. We define v k ( M i ) = P nj =1 M ij v k ( j ) and v ( M ) = P mi =1 v i ( M i ) .The Clarke-pivot payment of agent k is p k = v ( M − k ) − v ( M ) + v k ( M k ) . (5)The main result of this section is that in a VCG mechanism with Clarke-pivot payments, no agent willever envy a lower-capacity agent. In particular, this says that if all agents have the same capacity, the VCGmechanism with Clarke-pivot payments is both incentive compatible and envy-free.The proof of our main result (Theorem 3.1) is given in terms of a factional assignment but also holds forintegral assignments.Special case of capacitated allocation games, in which there are n items and n agents, and each agentcan get at most a single item was first introduced in a paper by Leonard [5], and was called a permutationgame. Leonard proved Theorem 3.1 for this special case only, and its proof technique does not seem togeneralize for larger capacities. Our proof is different.Here is our main theorem. Theorem 3.1.
Consider a VCG mechanism consisting of an optimal allocation M and Clarke-pivot pay-ments (5). Then if U i ≥ U j , agent i does not envy agent j . and agent be arbitrary two agents such that the capacity of agent 1 is ≥ that of agent 2,that is U ≥ U .Let M be an optimal assignment, M − an optimal assignment without agent , and M − some optimalassignment without agent . Agent does not envy agent iff v ( M ) − p ≥ v ( M ) − p Based on Equation 5, this is true when: v ( M ) − ( v ( M − ) − v ( M ) + v ( M )) = v ( M ) − v ( M − ) ≥ v ( M ) − ( v ( M − ) − v ( M ) + v ( M )) = v ( M ) + v ( M ) − v ( M − ) − v ( M ) Rearranging we obtain that agent does not envy agent iff v ( M − ) ≥ v ( M − ) + v ( M ) − v ( M ) . (6)We prove the theorem by establishing (6). We use the assignments M and M − to construct an assign-ment D − on G − such that v ( D − ) ≥ v ( M − ) + v ( M ) − v ( M ) . (7)From the optimality of M − , v ( M − ) ≥ v ( D − ) , which combined with (7) implies (6).Given assignments M and M − , we construct a flow f on an associated bipartite digraph, G f , withvertices for every agent and item. We define arcs and flows on arcs in G f for every agent i and item j : • If M ij − M − ij > then G f includes an arc i → j with flow f i → j = M ij − M − ij . • If M ij − M − ij < then G f includes an arc j → i with flow f j → i = M − ij − M ij . • If M ij = M − ij then G f contains neither i → j not j → i .We define the excess of an agent i in G f , and the excess of an item j in G f , to be ex i = X ( i → j ) ∈ G f f i → j − X ( j → i ) ∈ G f f j → i = X j (cid:0) M ij − M − ij (cid:1) ,ex j = X ( j → i ) ∈ G f f j → i − X ( i → j ) ∈ G f f i → j = X i (cid:0) M − ij − M ij (cid:1) , respectively.In other words the excess is the difference between the amount flowing out of the vertex and the amountflowing into the vertex. Clearly the sum of all excesses is zero. We say that a node is a source if its excessis positive and we say that a node is a target if its excess is negative.5 bservation 3.2. To summarize, i is an agent and a source ⇒ ≤ X j M − ij + | ex i | = X j M ij ≤ U i ; (8) i is an agent and a target ⇒ ≤ X j M ij + | ex i | = X j M − ij ≤ U i ; (9) j is an item and a source ⇒ ≤ X i M ij + | ex j | = X i M − ij ≤ Q j ; (10) j is an item and a target ⇒ ≤ X i M − ij + | ex j | = X i M ij ≤ Q j . (11)By the standard flow decomposition theorem we can decompose f into simple paths and cycles whereeach path connects a source to a target. Each path and cycle T has a positive flow value f ( T ) > associatedwith it. Given an arc x → y , if we sum the values f ( T ) of all paths and cycles T including x → y then weobtain f x → y .Notice that M − j = 0 for all j and therefore f → j ≥ for all j . It follows that there are no arcs of theform j → in G f . Observation 3.3.
For each path P = u , u , . . . , u t in flow decomposition G f , where u is a source and u t is a target, we have f ( P ) ≤ min { ex u , | ex u t |} .We define the value of a path or a cycle T = u , u , . . . , u t in G f , to be v ( P ) = X agent u i , item u i +1 v u i ( u i +1 ) − X item u i , agent u i +1 v u i +1 ( u i ) . It is easy to verify that the P T f ( T ) · v ( T ) over all paths and cycles in our decomposition is v ( M ) − v ( M − ) . Lemma 3.4.
Without loss of generality, we can assume that M − is such that1. There are no cycles of zero value in G f .2. There is no path P = u , u , . . . , u t of zero value such that u = 1 is a source and u t is a target.Proof. Assume that there is a cycle or a path T in the flow decomposition of G f such that v ( T ) = 0 . Let x be the smallest flow along an arc e of T . We modify M − as follows: For every agent to item arc i → j ∈ T we increase M − ij by x and for every item to agent arc j → i ∈ T we decrease M − ij by x . Let the resultingflow be ˜ M − .If T is a cycle then the capacity constraints are clearly preserved. If T is not a cycle, then the capacityconstraints are trivially preserved for all nodes other than u and u t . From Equation (8) we know that X j M − u j ≤ U u − | ex u | ≤ U u − x if u is an agent.6rgo, if u is an agent we can increase the allocation of M − u u by x , while not exceeding the capacity ofagent u ( U u ). If u is an item, agent u can release x units of item u without violating any capacityconstraints.We can similarly see that the capacities constraints of u t are not violated (Equation (11)).Furthermore v ( ˜ M − ) = v ( M − ) − xv ( T ) = v ( M − ) and if we replace M − by ˜ M − then G f changesby decreasing the flow along every arc of T by x , and removing arcs whose flow becomes zero (in particularat least one arc will be removed). This process does not introduce any new edges to G f .We repeat the process until G f does not contain zero cycles or paths as defined.From now on we assume that M − is chosen according to Lemma 3.4 . Lemma 3.5.
The flow f in G f does not contain cycles.Proof. Assume that f contains a cycle C which carries ǫ > flow. Clearly C does not contain agent sincethere is not any arc entering agent in G f .Assume first that v ( C ) < . Create an assignment c M from M by decreasing M ij by ǫ for each agentto item arc i → j ∈ C and increasing M ij by ǫ for each item to agent arc j → i ∈ C . This can be donebecause M − M − has a flow of ǫ along the agent to item arc i → j , so, it must be that M ij ≥ ǫ . Similarly, M − M − has a flow of ǫ along item to agent arcs j → i so it must be the M ij ≤ U i − ǫ . Since C is a cyclethe assignment c M still satisfies the capacity constraints. Furthermore v ( c M ) = v ( M ) − ǫv ( C ) > v ( M ) which contradicts the maximality of M .If v ( C ) > we create assignment c M − from M − as follows. For every item to agent arc j → i ∈ C we decrease M − ij by ǫ and for every agent to item arc i → j ∈ C we increase M − ij by ǫ . This can be donebecause M − − M has a flow of ǫ along the item to agent arc j → i , so, it must be that M − ij ≥ ǫ . Since C is a cycle c M − still satisfies the capacity constraints. Furthermore v ( c M − ) = v ( M − ) + ǫv ( C ) > v ( M − ) which contradicts the maximality of M − .We need to argue that c M − makes no assignment to agent , this follows because agent has noincoming flow in G f and cannot lie on any cycle.By assumption, there no cycles of value zero in G f .In particular Lemma 3.5 implies that there are no cycles in our flow decomposition. Lemma 3.6.
Agent is the only source node.Proof. We give a proof by contradiction, assume some other node, u = 1 , is a source. Then, there is a flowpath P = u , u , . . . u t from that node to a target node u t . Since there are no arcs incoming into vertex ,the path P cannot include agent .Let ǫ be the flow along the path P in the flow decomposition.If v ( P ) > define c M − ij = M − ij + ǫ for each agent to item arc i → j in P and c M − ij = M − ij − ǫ foreach item to agent arc j → i in P . For all other item/agent pairs ( i, j ) , let c M − ij = M − ij . We have that v ( c M − ) = v ( M − ) + ǫv ( P ) > v ( M − ) this would contradict the maximality of M − if c M − is a legal assignment. Since Equation (7) depends only on the value of M − it does not matter which M − we work with v ( P ) < define c M ij = M ij − ǫ for each agent to item arc i → j in P and c M ij = M ij + ǫ for eachitem to agent arc j → i in P . For all other item/agent pairs ( i, j ) , let c M ij = M ij . We have that v ( c M ) = v ( M ) − ǫv ( P ) > v ( M ) which contradicts the maximality of M .We still need to argue that the assignment c M − (if v ( P ) > ) and the assignment c M (if v ( P ) < ) arelegal. Because P has a flow of ǫ , M − ij ≥ ǫ for each item to agent arc j → i along P , and M ij ≥ ǫ for eachagent to item arc i → j along P .We also worry about exceeding capacities at the endpoints of P , since the size of assignments ofagents/items that are internal to the path do not change.We increase the capacity of u while constructing M − only if u is an agent, and increase the capacityof u t while constructing M − only if it is an item. By Observation 3.2 this is legal. A similar argumentshows that in c M the assignment of u and u t is smaller than their capacities.According to the way we choose M − , it cannot be that v ( P ) = 0 and that P carries a flow in G f .In particular Lemma 3.6 implies that all the paths in our flow decomposition start at agent .We construct D − from M − as follows.1. Stage I: Initially, D − := M − .2. Stage II: For each item j let x = min { M j , M − j } . Set D − j := M − j − x and D − j := x .3. Stage III: For each flow path P in the flow decomposition of G f that contains agent we considerthe prefix of the path up to agent . For each agent to item arc i → j in this prefix we set D − ij := D − ij + f ( P ) , and for each item to agent arc j → i in this prefix we set D − ij := D − ij − f ( P ) .It is easy to verify that D − indeed does not assign any item to agent . Also, the assignment to agent in D − is of the same size as the assignment to agent in M − . Since U ≥ U , D − is a legal assignment. Lemma 3.7.
The assignment D − satisfies Equation (7).Proof. Rearranging Equation (7) v ( D − ) ≥ v ( M − ) (12) + n X j =1 ( v ( j ) − v ( j )) · min( M j , M − j ) (13) + X j | M j >M − j ( v ( j ) − v ( j )) · ( M j − M − j ) . (14)At the end of stage I, we have D − = M − and so the inequality above at line (12) (without adding (13)and (14)) holds trivially. It is also easy to verify that at the end of stage II, the inequality above that spans(12) and (13) but without (14) holds. Finally, at the end of stage III, the full inequality in (12), (13) and (14)will hold as we explain next.Consider an item j such that M j > M − j . In G f we have an arc → j such that f → j = M j − M − j .Therefore in the flow decomposition we must have paths P , . . . , P ℓ all containing → j such that ℓ X k =1 f ( P k ) = f → j = M j − M − j (15)8et b P k be the prefix of P k up to agent . Consider the cycle C consisting of b P k followed by → j and j → . It has to be that that value of this cycle is non-negative. (Otherwise, construct c M by decreasing eachagent to item arc i → j on the cycle c M ij = M ij − ǫ and increasing each item to agent arc j → i on thecycle c M ij = M ij + ǫ . It follows, that v ( c M ) = v ( M ) − ǫv ( C ) > v ( M ) in contradiction of maximality ofM. The matching v ( c M ) is legal since it preserves capacities and decreases assignment associated with arcswith flow on them.)Therefore, v ( b P k ) + v ( j ) − v ( j ) ≥ ⇒ v ( b P k ) ≥ ( v ( j ) − v ( j )); ⇒ f ( P k ) v ( b P k ) ≥ f ( P k )( v ( j ) − v ( j )); ⇒ ℓ X k =1 (cid:0) f ( P k ) v ( b P k ) (cid:1) ≥ ( v ( j ) − v ( j )) ℓ X k =1 f ( P k ) . Substituting Equation (15) into the above gives us that ℓ X k =1 (cid:0) f ( P k ) v ( b P k ) (cid:1) ≥ (cid:0) v ( j ) − v ( j ) (cid:1)(cid:0) M j − M − j (cid:1) . (16)The left hand side of equation (16) is exactly the gain in value of the matching when applying stage IIIto the paths b P , . . . , b P ℓ during the construction of D − above. The right hand side is the term which we addin Equation (14).To conclude the proof of Lemma 3.7, we note that stage III may also deal with other paths that start atagent 1 and terminate at agent 2. Such paths must have value ≥ and thus can only increase the value ofthe matching D − . (Otherwise we can build assignment c M , such that v ( c M ) > v ( M ) by decreasing M ij by ǫ for each arc i → j ∈ P and increasing M ij by ǫ for each arc j → i ∈ P as we did before. The matching c M is legal since it preserves capacities on inner nodes of the path, decreases only arcs with flow on them, M ij > ǫ . Capacity of a source agent node can be increased according to Observation 3.2.) Corollary 3.8.
If all agent capacities are equal then the VCG allocation with Clarke-pivot payments isenvy-free.
Do Clarke-pivot payments work also under heterogeneous capacities? The answer is no. This followssince in the next section we show that any mechanism that is both incentive compatible and envy-free musthave positive transfers, and Clarke-pivot payments do not. ∩ EF payments implypositive transfers
Consider an arbitrary VCG mechanism. Let opt = < opt , opt , . . . , opt n > denote the allocation and let p i = h i ( v − i ) − v − i ( opt ) (17)9e the payments, where v − i ( opt ) = X ≤ j ≤ nj = i v j ( opt j ) . Let v ( opt ) = P nj =1 v j ( opt j ) and let opt − i = < opt − , opt − , . . . , opt − ii − , ∅ , opt − ii +1 , . . . , opt n >, be the allocation maximizing v − i ( opt − i ) = X ≤ j ≤ nj = i v j ( opt − ij ) . We substitute the VCG payments (17) into the envy-free conditions (4) and obtain that i does not envy j if and only if v i ( opt j ) − p j ≤ v i ( opt i ) − p i ⇔ p i − p j ≤ v i ( opt i ) − v i ( opt j ) ⇔ h i ( v − i ) − v − i ( opt ) − (cid:0) h j ( v − j ) − v − j ( opt ) (cid:1) ≤ v i ( opt i ) − v i ( opt j ) ⇔ h i ( v − i ) − h j ( v − j ) ≤ v − i ( opt ) − v − j ( opt ) + v i ( opt i ) − v i ( opt j ) ⇔ h i ( v − i ) − h j ( v − j ) ≤ v ( opt ) − ( v ( opt ) − v j ( opt j )) − v i ( opt j ) ⇔ h i ( v − i ) − h j ( v − j ) ≤ v j ( opt j ) − v i ( opt j ) . (18) Theorem 4.1.
Consider a capacitated allocation game with heterogeneous capacities such that the numberof items exceeds the smallest agent capacity. There is no mechanism that simultaneously optimizes the socialwelfare, is IC ∩ EF, and has no positive transfers (the mechanism never pays the agents). That is, any IC ∩ EFmechanism has some valuations v for which the mechanism pays an agent. Note that the conditions on the capacities of the agents and the number of items are necessary – If capac-ities are homogeneous or the total supply of items is at most the minimum agent capacity then Clarke-pivotpayments, that are known to be incentive compatible, individually rational, and have no positive transfers,are also envy-free.In the rest of this section we prove Theorem 4.1. We start with a capacitated allocation game with twoagents and two items where agent i has capacity i ( i = 1 , ). We then generalize the proof to arbitraryheterogeneous games.To ease the notation we abbreviate in the rest of the paper v i ( j ) to v ij .We partition the valuations into three sets A , B , and B as follows (we omit cases with ties). The optimal allocation that maximizes social welfare is uniquely defined when there are no ties. Valuations v ’s with ties forma lower dimensional measure 0 set. It suffices to consider valuations without ties for both existence or non-existence claims of IC or EF payments. This is clear for non-existence, for existence, the payments for a v with ties is defined as the limit when weapproach this point through v ’s without ties that result in the same allocation. Clearly IC and EF properties carry over, also IR andnonnegativity of payments. (A) v > v and v > v . For these valuations in an optimal allocation agent obtains the bundle { , } and agent obtains the empty bundle. • (B ) v − v > max { , v − v } . For these valuations in an optimal allocation item is assignedto agent and item to agent . • (B ) v − v > max { , v − v } . For these valuations in an optimal allocation item is assignedto agent and item to agent .Substituting the above in (18) we obtain that for v ∈ B , agent 1 does not envy agent 2 if and only if h ( v ) − h ( v ) ≤ v ( opt ) − v ( opt ) = v − v . Agent 2 does not envy agent 1 if and only if h ( v ) − h ( v ) ≤ v ( opt ) − v ( opt ) = v − v . Combining we obtain that there is no envy for v ∈ B , if and only if v − v ≤ h ( v ) − h ( v ) ≤ v − v . (19)For a fixed ǫ > , and x > ǫ , the valuation v such that v = x + 3 ǫ , v = x + ǫ , v = v = 0 isclearly in B . Substituting in (19) we obtain − ( x + 3 ǫ ) ≤ h (0 , − h ( x + 3 ǫ, x + ǫ ) ≤ − ( x + ǫ ) (20). The valuation v such that v = x + 3 ǫ , v = x + ǫ , v = x + ǫ , and v = x is also clearly in B andfrom (19) we obtain x + ǫ − ( x + 3 ǫ ) ≤ h ( x + ǫ, x ) − h ( x + 3 ǫ, x + ǫ ) ≤ x − ( x + ǫ ) hence − ǫ ≤ h ( x + ǫ, x ) − h ( x + 3 ǫ, x + ǫ ) ≤ − ǫ . (21)Combining (20) and (21) we obtain h ( x + ǫ, x ) ≤ h ( x + 3 ǫ, x + ǫ ) − ǫ ≤ h (0 ,
0) + x + 3 ǫ (22)The no positive transfers requirement is that for any v , h ( v ) ≥ v ( opt ) . (23)Consider now the valuations v such that v = x + ǫ , v = x , v = v = x − ǫ . Clearly, v ∈ A (agent getsboth items), hence v ( opt ) = 2 x − ǫ . Substituting this and (22) in (23) we obtain x − ǫ ≤ h (0 , x +3 ǫ ,hence h (0 , ≥ x − ǫ . Clearly, for valuations with large enough x we obtain a contradiction, that is, thereexist valuations where the mechanism pays an agent. Heterogeneous capacities, multiple agents and items:
Let c be the smallest agent capacity and assume itis the capacity of agent 1. Let agent be any agent with capacity > c . There are ≥ c + 1 items. It sufficesto consider restricted valuation matrices v where v ij = 0 when i > or when j > c + 1 and v ij ≡ v i for i = 1 , and ≤ j ≤ c + 1 . We partition these valuations into four sets A , B , B +1 , B , as follows (we omitcases with ties and only define the assignment of items , . . . , c + 1 ):11 (A) v > v and v > v . For these valuations in an optimal allocation agent obtains the bundle { , . . . , c + 1 } . • (B ) v > v and v < v . For these valuations in an optimal allocation items is assigned toagent and items , . . . , c + 1 to agent . • (B +1 ) v − v > v − v and v > v . For these valuations in an optimal allocation items , . . . , c are assigned to agent and item c + 1 is assigned to agent . • (B ) v − v > max { , v − v } . For these valuations in an optimal allocation item is assignedto agent and items , . . . , c + 1 to agent .Substituting the above in (18) we obtain that for v ∈ B +1 , agent 1 does not envy agent 2 if and only if h ( v ) − h ( v ) ≤ v ( opt ) − v ( opt ) = v − v . Agent 2 does not envy agent 1 if and only if h ( v ) − h ( v ) ≤ v ( opt ) − v ( opt )= v + ( c − v − v − ( c − v . Combining we obtain that there is no envy for v ∈ B +1 , if and only if v + ( c − v − v − ( c − v ≤ h ( v ) − h ( v ) ≤ v − v . (24)For a fixed ǫ > and for x > ǫ , the valuation v such that v = x + 3 ǫ , v = x + ǫ , v = v = 0 isclearly in B +1 . For such v the left hand side of (24) is v + ( c − v − v − ( c − v = − ( x + 3 ǫ ) − ( c − x + ǫ )= − cx − ( c + 2) ǫ Substituting in (24) we obtain − cx − ( c + 2) ǫ ≤ h (0 , − h ( x + 3 ǫ, x + ǫ ) ≤ − ( x + ǫ ) . (25)The valuation v such that v = x + 3 ǫ , v = x + ǫ , v = x + ǫ , and v = x is also clearly in B +1 .For such v the left hand side of (24) is v + ( c − v − v − ( c − v = x + ǫ + ( c − x − ( x + 3 ǫ ) − ( c − x + ǫ )= − ( c + 1) ǫ From (24) we obtain − ( c + 1) ǫ ≤ h ( x + ǫ, x ) − h ( x + 3 ǫ, x + ǫ ) ≤ − ǫ. (26)Combining (25) and (26) we obtain, h ( x + ǫ, x ) ≤ h ( x + 3 ǫ, x + ǫ ) − ǫ ≤ h (0 ,
0) + cx + ( c + 2) ǫ − ǫ = h (0 ,
0) + cx + ( c + 1) ǫ For valuations v = x + ǫ , v = x , v = v = x − ǫ , we clearly have v ∈ A (agent gets all items),hence v ( opt ) = ( c + 1) x − ( c + 1) ǫ .For a sufficiently large x (relative to ǫ and h (0 , ), h ( v ) = h ( x + ǫ, x ) ≤ h (0 ,
0) + cx + ( c + 1) ǫ < ( c + 1) x − ( c + 1) ǫ = v ( opt ) , which contradicts the no positive transfers requirement (23).12 In this section we assume that capacities are public and derive IC ∩ EF payments for any game with twoplayers. Lemma 5.1.
Any 2-player capacitated allocation game with public capacities has an IC ∩ EF individuallyrational mechanism.Proof. Let c i be the capacity of player i and assume without loss of generality that c ≤ c . For a vector ( x , x . . . ) let top b { x } be the set of the b largest entries in x . We show that h ( v ) = X j ∈ top c { v } v j and h ( v ) = X j ∈ top c { v } v j give VCG payments which are envy-free.It suffices to show that for { i, j } = { , } , h i ( v − i ) − h j ( v − j ) ≤ v j ( opt j ) − v i ( opt j ) . That is, X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ v ( opt ) − v ( opt ) (27)and X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ v ( opt ) − v ( opt ) . (28)Assume first that the number of items is exactly c + c . In the optimal solution, player will get the c items that maximize v j − v j and player will get the c items that minimize this difference.We establish (28) as follows X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ X j ∈ top c { v } ( v j − v j ) ≤ X j ∈ top c { v − v } ( v j − v j )= X j ∈ opt ( v j − v j ) = v ( opt ) − v ( opt ) .
13e establish (27) as follows X j ∈ top c { v } v j − X j ∈ top c { v } v j ≤ X j ∈ top c { v } ( v j − v j ) ≤ X j ∈ top c { v − v } ( v j − v j ) ≤ X j ∈ opt v j − X j ∈ top c { v ( opt ) } v j where v ( opt ) is the vector of the values of player to the items player 2 gets in the optimal solution.If there are fewer than c + c items, we add “dummy” items with valuations v j = v j = 0 and thelemma follows from the previous argument for the case with c + c items.If there are more than c + c items then consider the set of c + c items than participate in the optimalsolution. We now observe that (27) and (28) only involve items that participate in the optimal solution( top c { v } and top c { v } must both be included in the optimal solution). In this section, valuations and capacities are private. We give VCG payments which are envy-free andindividually rational for any game with two agents and two items. We specify the payments by giving thefunctions h ( v , c ) and h ( v , c ) . Note that with two items, all c i ≥ are equivalent, therefore we onlyneed to consider capacities ∈ { , , } .We show that the following give envy-free payments h ( v , c ) = (cid:26) max( v , v ) c ∈ { , } c = 0 h ( v , c ) = (cid:26) max( v , v ) c ∈ { , } c = 0 The payments are envy-free if and only if δ = h ( v , c ) − h ( v , c ) ≤ v ( opt ) − v ( opt ) ,δ = h ( v , c ) − h ( v , c ) ≤ v ( opt ) − v ( opt ) . The conditions when { c , c } = { , } were worked out in the previous section and the correctness for h ( v , and h ( v , carries over (and symmetrically, if we switch capacities of the agents). Consider thefollowing remaining cases. • c = c = 2 : agent 1 does not envy agent 2 if and only if: h ( v , − h ( v , ≤ v + v − v − v if v > v , v > v v − v if v < v , v > v v − v if v > v , v < v if v < v , v < v h ( v , − h ( v , ≤ v + v − v − v if v > v , v > v v − v if v < v , v > v v − v if v > v , v < v if v < v , v < v Combining, we obtain the condition min { v − v , } + min { v − v , }≤ h ( v , − h ( v , ≤ max { v − v , } + max { v − v , } . (29)We now show that our particular h ’s satisfy (29). It suffices to establish one of the inequalities: We have v ≤ max { v , v } + max { v − v , } v ≤ max { v , v } + max { v − v , } Combining, we obtain the desired relation: max { v , v }≤ max { v , v } + max { v − v , } + max { v − v , } . • c = c = 1 : agent 1 does not envy agent 2 if and only if: h ( v , − h ( v , ≤ (cid:26) v − v v + v > v + v v − v v + v < v + v Symmetrically, agent 2 does not envy agent 1 if and only if: h ( v , − h ( v , ≤ (cid:26) v − v v + v > v + v v − v v + v < v + v Combining, we obtain min { v − v , v − v }≤ h ( v , − h ( v , ≤ max { v − v , v − v } (30)We now show that our particular h ’s satisfy (30). It suffices to establish one of the inequalities: We have v ≤ max { v , v } + v − v v ≤ max { v , v } + v − v } max { v , v } ≤ max { v , v } + max { v − v , v − v } . • c = 1 , c = 0 : No agent envies the other if and only if h ( v , − h ( v , ≤ h ( v , − h ( v , ≤ max { v , v } Combining, we obtain − max { v , v } ≤ h ( v , − h ( v , ≤ (31)Symmetrically, when c = 0 , c = 1 : − max { v , v } ≤ h ( v , − h ( v , ≤ (32)Our particular h ’s trivially satisfy (31) and (32). • c = 2 , c = 0 : No agent envies the other if and only if h ( v , − h ( v , ≤ h ( v , − h ( v , ≤ v + v Combining, we obtain − v − v ≤ h ( v , − h ( v , ≤ (33)Symmetrically, when c = 0 , c = 2 : − v − v ≤ h ( v , − h ( v , ≤ (34)Our particular h ’s trivially satisfy (33) and (34). We have begun to study truthful and envy free mechanisms for maximizing social welfare for the capacitatedallocation problem.There is much left open, for example:1. Is there a truthful and envy free mechanism (with positive transfers) for the capacitated allocationproblem (arbitrary capacities):(a) With public capacities and more than two agents.(b) With private capacities for more than 2 agents and 2 items?2. How well can we approximate the social welfare by a mechanism that is incentive-compatible, envy-free, invidually rational, and without positive transfers for capacitated allocations ?3. Noam Nisan has observed that for superadditive valuations, there may be no mechanism that is bothtruthful and envy free. We conjecture that one can obtain mechanisms that are both truthful and envyfree for subadditive valuations. 16 eferences [1] L.E. Dubins and E.H. Spanier. How to cut a cake fairly.
American Mathematical Monthly , 68:1–17,1961.[2] D. Foley. Resource allocation and the public sector.
Yale Economic Essays , 7:45–98, 1967.[3] Claus-Jochen Haake, Matthias G. Raith, and Francis Edward Su. Bidding for envyfreeness: A proce-dural approach to n-player fair-division problems.
Social Choice and Welfare , 19:723–749, 2002.[4] Leonard Hurwicz. Optimality and informational efficiency in resource allocation processes. In K.J.Arrow, S. Karlin, and P. Suppes, editors,
Mathematical Methods in the Social Sciences , pages 27–46.Stanford University Press, Stanford, CA, 1960.[5] Herman B. Leonard. Elicitation of honest preferences for the assignment of individuals to positions.
The Journal of Political Economy , 91:3:461–479, 1983.[6] Eric S. Maskin. On the fair allocation of indivisible goods. In
G. Feiwel (ed.), Arrow and the Foun-dations of the Theory of Economic Policy (essays in honor of Kenneth Arrow) , volume 59:4, pages341–349, 1987.[7] Herve Moulin.
Fair Division and Collective Welfare . MIT Press, 2004.[8] Noam Nisan. Introduction to mechanism design. In Noam Nisan, Tim Roughgarden, Eva Tardos, andVijay Vazirani, editors,
Algorithmic Game Theory , chapter 16. Cambridge University Press, 2007.[9] L.-G. Svensson. On the existence of fair allocations.
Journal of Economics , 43:301–308, 1983.[10] H. Peyton Young.