Merging and splitting endowments in object assignment problems

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1 Merging and splitting endowments in oject assignment prolems Nanyang Bu, Siwei Chen, and William Thomson April 26, Introduction We consider a group of agents, each endowed with a set of indivisile goods, called ojects, and equipped with a preference relation over susets of the union of their endowments. An allocation rule associates with each economy of this type a redistriution among them of their endowments. We are concerned aout two kinds of opportunities to manipulate a rule to their advantage that agents may have. First, imagine an agent giving control of his endowment to some other agent, and withdrawing; the rule is applied; the agent with the augmented endowment may now e assigned a undle that can e partitioned into two undles, one for himself and one for the agent who withdrew, that make oth of them at least as well off as they would have een otherwise and at least one of them etter off. Alternatively, an agent may invite some outsider, someone who has no endowment, and give him control over part of his endowment. In the enlarged economy, the two of them may e assigned undles whose union the agent who invited the outsider prefers to what he would have een assigned otherwise. We are interested in selections from the correspondence that selects for each economy its efficient allocations that each agent finds his component of it at least as desirale as his endowment. We inquire aout the existence of University of Rochester. Filename: mergingsplittingapril30.tex 1

2 such selections that are immune to one or the other of these ehaviors. Our main results are negative: even if preferences are restricted in natural ways, there is no such selection. Vulneraility of rules to such ehaviors is examined in the context of classical economies y Thomson (2006), who reports negative results for rules that are central to the literature. The current note suggests that the prolem may e quite pervasive. 2 Model and axioms The model is standard. It pertains to the prolem of reassignment among a group of agents of the undles of resources that they privately own. Here, the resources are indivisile; we call them ojects. Since the form of manipulation that we consider involve variations in the set of active agents, we need to cast our analysis in a variale-population framework. Formally, there is an infinite set of potential agents, N. To define an economy, we draw a finite suset of them from this infinite population. Let N denote the class of finite susets of N, with generic elements N and N. Endowments consist of indivisile resources, or ojects. Ojects are denoted a,,.... For each i N, let ω i e agent i s endowment. Preferences are defined over sets of ojects. An economy with agent set N is otained y specifying, for each agent, an endowment, and a preference relation defined over the susets of the union of everyone s endowment. Formally, it is a pair (R, ω), where for each i N, R i designates agent i s complete, transitive, and anti-symmetric preference relation, defined over susets of ω i. Let P i e the strict preference associated with R i and I i the indifference relation. Let E N e the set of all economies with agent set N. An allocation for e (R, ω) E N is a list of undles x (x i ) i N such that x i = ω i. Let X(e) denote the set of allocations of e. A solution is a correspondence defined over E N that associates with each N N and each e (R, ω) E N a non-empty suset of X(e). A rule is a single-valued solution. We will consider several specifications of preferences. First, a preference relation is size monotone if given two sets of ojects, the one with more items is at least as desirale as the one with fewer items. It is additive if numers can e assigned to ojects so that two sets can e compared y comparing the sums of the numers assigned to the ojects in each set. It respects the desiraility of individual ojects if for each set and each 2

3 pair of ojects that are not in the set, adding to the set the more desirale oject yields an augmented set that is at least as desirale as the one that results y adding the less desirale oject. Two important solutions are defined next. We will e interested in rules that select from their intersection. An allocation is efficient if there is no other allocation that each agent finds at least as desirale to it and one agent prefers. Our first solution is the solution that associates with each economy its set of efficient allocations. The notation we choose for it is a reference to Pareto: Efficiency solution, P : For each N N, each (R, ω) E N, x P (R, ω) if and only if there is no x X(e) such that for each i N, x i R i x i and for at least one i N, x i P i ω i. Next is the solution that associates with each economy its set of allocations that each agent finds at least as desirale as his endowment: Endowments lower ound solution, B: For each N N and each (R, ω) E N, x B(R, ω) if and only if for each i N, x i R i ω i. Next, we turn to the axioms imposed on rules. We formulate two kinds of manipulation that agents may engage in and require immunity to the manipulation. Let ϕ e a rule. Consider a pair of agents; imagine that one of them entrusts his endowment to the other and withdraws; the rule is applied without him; at the allocation that results, the second agent s assignment may e a undle that can e partitioned etween the two of them in such a way that each of them is at least as well off as he would have een if they had not merged their endowments, and at least one of them is etter off. We require immunity to this sort of ehavior: Endowments-merging proofness: For each N N, each (R, ω) E N with x ϕ(r, ω), each {i, j} N, with x ϕ(r N\{j}, ω i, ω N\{i,j} ), where ω i ω i ω j, and each pair (ˆx i, ˆx j ) of undles such that ˆx i ˆx j = x i, it is not the case that for each k {i, j}, ˆx k R k x k, and for at least one k {i, j}, ˆx k P k x k. 3

4 A weaker requirement is that the rule should not e such that oth agents end up etter off. Symmetrically, an agent may transfer some of his endowment to some agent who was not initially present and whose endowment is the empty set; the rule is applied, and the union of what the two of them are assigned is a undle that the first agent prefers to his initial assignment. We require immunity to this sort of ehavior: 1 Endowments-splitting proofness: For each N N, each (R, ω) E N, with x ϕ(r, ω), for each i N, each j / N, each R j R, and each pair (ω i, ω j) of undles such that ω i ω j = ω i, with x ϕ(r, R j, ω i, ω N\{i}, ω j), we have x i R i (x i x j). Here, a weaker requirement is that the rule should not e such that agent i can partition his assignment into two undles, one for himself and one for the person he invited in, so that oth of them end up etter off. Related strategic moves have een defined in other models. In the context of the adjudication of conflicting claims, it is their claims that agents can merge or split (O Neill, 1982). In the context of queueing, agents can split or merge jos (Moulin, 2007). In each case, one may require of a rule that it not e vulnerale to these actions. Counterparts of these properties for classical economies are studied y Thomson (2006). 3 The results We estalish two impossiility results. In the proofs, we slightly ause notation and omit rackets around sets of ojects. For instance, instead of {a, }, we write a. We present preferences in tale form. We place endowments in oxes. 1 Alternatively, we could allow the second agent to have his own endowment and require that the union of their assignments e such that this agent e ale to retrieve his endowment, the first agent ending up etter off with the remaining resources. Finally, we could declare the manipulation successful if there is a partitioning of the union of their assignments that make oth of them at least as well off as they would have een without the maneuver and at least one of them etter off. 4

5 Proposition 1. On the size-monotone domain, no selection from the endowments lower ound and Pareto solution is weakly endowments-merging-proof. Proof. Let ϕ B P. Let N {1, 2, 3, 4}, ω 1 a, ω 2, ω 3 cd, and ω 4 ef. Preferences (R i ) i N are descried in the following tale. R 1 R 2 R 3 R 4 S s.t S 2 S s.t S 3 S s.t S 3 S s.t S 5 d df ad adf c a ac cdef a S s.t S = 2 cd S s.t 4 S 3 S s.t S = 1 f S s.t S 2 f e e ef S s.t S = 1 S s.t S 2 Let e (R, ω) E N e the economy so defined. Let x X(e) e defined y x 1 c, x 2 e, x 3 ad, and x 4 f. Let y X(e) e defined y y 1 d, y 2 f, y 3 ac, and y 4 e. We omit the proof that B(e) P (e) = {x, y}. Since ϕ BP, there are two cases to consider. Case 1: ϕ(e) = x. Agents 1 and 2 merge their endowments and agent 1 withdraws. Let e (R 1, (a, cd, ef)) E N\{1} e the resulting economy. Let x X(e ) e defined y x 2 df, x 3 ac, and x 4 e. We omit the proof that B(e ) P (e ) = {x }. Thus, ϕ(e ) = x. Let ˆx 1 d and ˆx 2 f. Then, ˆx 1 ˆx 2 = x 2. Since ˆx 1 P 1 x 1 and ˆx 2 P 2 x 2, weak endowments-mergingproofness is violated. Case 2: ϕ(e) = y. Agents 3 and 4 merge their endowments and agent 3 withdraws. Let e (R 3, (a,, cdef)) E N\{3} e the resulting economy. Let y X(e ) e defined y y 1 c, y 2 e, and y 4 adf. We omit the proof that B(e ) P (e ) = {y }. Thus, ϕ(e ) = y. Let ŷ 3 ad and ŷ 4 f. Then, ŷ 3 ŷ 4 = y 4. Since ŷ 3 P 3 y 3 and ŷ 4 P 4 y 4, weak endowments-merging-proofness is violated. 5

6 The next proposition pertains to a narrower domain, ut the violation that we estalish is for endowments-merging proofness itself, not for its weak version. Proposition 2. On the intersection of the size-monotone and additive domains, no selection from the endowments lower ound and Pareto solution is endowments-merging-proof. Proof. Let ϕ B P. Let N {1, 2, 3}, ω 1 a, ω 2, and ω 3 c. Preferences (R i ) i N are descried in the following tale. R 1 R 2 R 3 ac ac S s.t S 2 ac ac a c a a c c c a a c Let e (R, ω) E N e the economy so defined. Let x X(e) e defined y x 1 c, x 2, and x 3 a. Let y X(e) e defined y y 1 c, y 2 a, and y 3. We omit the proof that B(e) P (e) = {x, y}. Since ϕ BP, there are two cases to consider. Case 1: ϕ(e) = x. Agents 1 and 2 merge their endowments and agent 1 withdraws. Let e (R 1, (a, c)) e the resulting economy. Let x X(e ) e defined y x 2 ac, and x 3. We omit the proof B(e ) P (e ) = {x }. Thus, ϕ(e ) = x. Let ˆx 1 c and ˆx 2 a. Then, ˆx 1 ˆx 2 = x 2. Since ˆx 1 = x 1 and ˆx 2 P 2 x 2, endowments-merging-proofness is violated. Case 2: ϕ(e) = y. Agents 1 and 3 merge their endowments and agent 3 withdraws. Let e (R 3, (ac, )) e the resulting economy. Let y X(e ) e defined y y 1 ac, and y 2. We omit the proof B(e ) P (e ) = {y }. Thus, ϕ(e ) = y. Let ŷ 1 c, and ŷ 3 a. Then, ŷ 1 ŷ 3 = y 1. Since ŷ 1 = y 1 and ŷ 3 P 3 y 3, endowments-merging-proofness is violated. 6

7 Next, we turn to our second property and for it too, estalish a negative result. Proposition 3. On the size-monotone domain, no selection from the endowments lower ound and Pareto solution is endowments-splitting-proof. Proof. Let ϕ B P. Let N {2, 3}, ω 2 a, and ω 3 cd. Preferences (R i ) i N are as descried in the following tale. R 2 R 3 S s.t S 3 S s.t S 3 c d ac ad a cd S s.t S = 2 S s.t S = 2 d c a c S s.t S = 1 S s.t S = 1 Let e (R, ω) E N e the economy so defined. Let x X(e) e defined y x 2 ac and x 2 d. Let y X(e) e defined y y 1 c and y 2 ad. We omit the proof that B(e) P (e) = {x, y}. Since ϕ BP, there are two cases to consider. Case 1: ϕ(e) = x. Agent 2 invites agent 1, with preference R 1 = R 2 and no endowment, and provides him with as endowment. Let e ((R 1, R), (, a, cd)). Let x X(e ) e defined y x 1, x 2 c, and x 3 ad. We omit the proof that B(e ) P (e ) = {x }. Thus, ϕ(e ) = x. Since c = x 1 x 2 P 2 x 2 = ac, endowments-splitting-proofness is violated. Case 2: ϕ(r, ω) = y. Agent 3 invites agent 4, with preference R 4 = R 3 and no endowment, and provides him with d as endowment. Let e ((R, R 3 ), (a, c, d)) E N. Let y X(e ) e defined y y 2 ac, y 3, and y 4 d. We omit the proof that B(e ) P (e ) = {y }. Thus, ϕ(e ) = y. Since d = y 3 y 4 P 3 y 3 = ad, endowments-splitting-proofness is violated. 7

8 We oserve that in the proof of Proposition??, the preferences of the new agent are the same as that of the agent who invited him. So, we need not invoke exotic preferences to prove it. To estalish the independence of the axioms in our propositions, consider the rule that, for each prolem, assigns to each agent his endowment. This rule is oviously endowments-merging proof as well as endowmentssplitting proof, ut it is not efficient. Other rules can e defined with these properties that do etter from the efficiency viewpoint: for many economies, they achieve a Pareto improvement over the endowment profile. Any selection from the Pareto solution is efficient and violates oth strategic requirements. 4 Related literature Other manipulation opportunities in economies of the kind considered here have een studied in earlier literature. An agent may enefit y withholding some of his endowment or y destroying some of it (Atlamaz and Klaus, 2007). An agent may enefit y artificially augmenting it y orrowing from the outside, or from one of his fellow traders (Atlamaz and Thomson, 2006). An agent may enefit y grouping the ojects he owns into undles (Klaus, Dimitrov and Haake, 2006). Most results concerning immunity of rules to these ehaviors are negative. It is unfortunate that we had to add to this list of impossiilities. Note that they these other types of manipulation do not involve changes in the set of active agents. Such changes are the focus of the present note. In the context of claims prolems, the property of immunity to the merging or splitting of claims studied in the literature cited in the introduction is akin to the properties we analyzed here. In that context, it can e met however. 8

9 References Atlamaz, M. and B. Klaus, Manipulation via endowments in exchange markets with indivisile goods, Social Choice and Welfare 28 (2007), Atlamaz, M. and W. Thomson, Borrowing-proofness in economies with indivisile goods, mimeo, Klaus, B., Dimitrov, D., and C.-J. Haake, Bundling in exchange markets with indivisile goods, Economics Letters 93 (2006), Moulin, H., On scheduling fees to prevent merging, splitting and transferring of jos, Mathematics of Operations Research 32 (2007), , Proportional scheduling, split-proofness, and merge-proofness, Games and Economic Behavior 63 (2008), O Neill, B., A prolem of rights aritration from the Talmud, Mathematical Social Sciences 2 (1982), Postlewaite, A., Manipulation via endowments, Review of Economic Studies 46 (1979), Thomson, W., New variale-population paradoxes for resource allocation, mimeo, 2006; revised