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When can we design efficient and strategy-proof rules in package assignment problems?

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We show that if a domain includes a partially quasi-linear domain, no rule is efficient and strategy-proof. It is known that there exists an efficient and strategy-proof rule in the quasi-linear domain (the set of quasi-linear preferences) (Holmström, 1979). A preference relation is r-partially quasi-linear if it is quasi-linear on the r-relevant consumption set.

For the two-agent case, we show that the r-partially quasi-linear domain (the set of r-partially quasi-linear preferences) is a maximal domain for efficiency and strategy-proofness (Theorem 1 (ii)). Further, we show that in the quasi-linear domain r-partially, the Vickrey rules generalized by r (a generalization of the Vickrey rule in non-quasi-linear domains) are the only efficient and strategy-stable rules (Theorem 1 ( i)) . For example, the monotonic object domain includes the r-partially quasi-linear domain for every r ∈R.

Figure 1: An illustration of the consumption set for M = {1, 2} and ¯ x = (1, 1).
Figure 1: An illustration of the consumption set for M = {1, 2} and ¯ x = (1, 1).

Rules and their properties

The valuation function of a quasi-linear preference relation corresponds to the payment levels at the bundles on the indifference curve passing through (0,0) as shown in Figure 2. Groves and Vickrey rules are defined on the quasi-linear domain because they are defined by of valuation functions. The preferences defined below are non-quasi-linear preferences that preserve quasi-linearity on the r-relevant consumption set.

For each r ∈ R, let RP(r) be a class of r-partial quasi-linear preferences and call it an r-partial quasi-linear domain. R-partial quasi-linearity requires that the preference relation is quasi-linear when the consumption set is restricted to the r-relevant consumption set. It is known that on the quasi-linear domain Groves rules are the only efficient and strategically safe rules.

Figure 3: An illustration of the r-relevant consumption set for R i for ¯ x = (1, 1).
Figure 3: An illustration of the r-relevant consumption set for R i for ¯ x = (1, 1).

Two-agent case

More than two-agent case

However, as we see in Section 5.3, we can determine the class of maximal domains for efficiency, strategy certainty, and some additional properties. The black indifference curves are those for R1, the bold indifference curves are those for R2, and the dashed indifference curve is that for R3. Contrary to the case of n = 2, the following result shows that when n ≥ 3, efficiency and strategy certainty are incompatible even on the partially quasi-linear domain for every r∈R.

We demonstrate this by showing that for every r ∈ R there exists a domain that is "close" to an r-partially quasi-linear domain, and that there exists an efficient and strategically secure rule on the domain. Consider a preference relation Ri that satisfies the following conditions: For each (x, t)∈X(Ri) and each y∈X, ifVRi(y,(x, t))≥0,. The first condition ˆRP is the same as the partial quasi-linearity requirement.

Figure 6: Indifference curves of R i ∈ R ˆ P .
Figure 6: Indifference curves of R i ∈ R ˆ P .

Further results with additional properties

By fact 2, if a rule on Rn with R ⊇ RQ satisfies efficiency, strategy certainty, individual rationality, and no support for losers, then it coincides with a Vickrey rule on (RQ)n. R is a maximal domain of efficiency, strategy certainty, individual rationality, and no subsidy to losers if and only if R=RP. If there is a rule about Rn that satisfies efficiency, strategy certainty, individual rationality, and no subsidies to losers, then R (RP.

From fact 1 it is easy to see that if a rule on Rn with R ⊇ RQ satisfies efficiency, strategy stability, and general payment for losers, it coincides with an r-generalized Vickrey rule for some r∈ R on (RQ)n. There exists maximum domain efficiency, strategy resilience, and a common payment for losers if and only if R=RP(r) for some r∈R. If there exists a rule for Rn that satisfies efficiency, strategy stability, and common payment for losers, then R (RP(r) for somer ∈R.

Impossibility results on various domains

A preference relation has both non-negative and non-positive income effects if and only if it is quasi-linear. It is also clear that RN N I and RN P I contain a preference relation so that for every r ∈ R it is not r-partially quasi-linear. Since individual rationality and no subsidy for losers imply general payoff for losers, Corollary 2 implies that no rule satisfies efficiency, strategy consistency, individual rationality and no subsidy for losers on (RN N I)n or (RN P I )n not.

Another class of preferences studied in the literature is the class of quasi-linear preferences with borrowing costs (Saitoh and Serizawa, 2008). An extreme case is that the income is infinite - in this case the agent does not need to borrow money and therefore her preference relation is quasi-linear. No rule about Rn satisfies efficiency, strategy constancy, individual rationality and no subsidy for losers.

Public goods model

For example, consider a preference relation Ri ∈ RQ,B with a valuation function vi and an income level wi such that for some x∈ X, wi < vi(x). The formal model can be found in the supplementary material (Kazumura introduces the notion of partial quasi-linearity in the public goods model.19 Their main result implies that in the public goods model, if the domain is larger than the partially quasi-linear domain , no rule satisfies efficiency, strategy testability, individual rationality, and no subsidy, where no subsidy requires each agent's payoff to be nonnegative. In the public goods model, if there are at least three agents and six alternatives, no rule does not satisfy the efficiency and provability of the strategy in the partially quasilinear domain.

The key idea for this result is that we can integrate the parcel allocation model into the public goods model. If there are six alternatives in the collective goods model, we can associate each alternative with one of these parcel allocations. Then, for each preference relation in the parcel allocation model, we can find a corresponding preference relation in the public goods model.

Since this set of preferences is contained in the partially quasi-linear domain in the public goods model, we obtain the impossibility result. We have demonstrated that the quasi-linearity of preferences plays an important role for the existence of an efficient and strategy-fixed rule in a package assignment model. An underlying assumption in this paper is that the domain contains all the quasi-linear preferences.

However, in some situations in practice, objects are substitutes or complements, and therefore we cannot justify that agents can have any quasi-linear preference relation in such situations. Although the results in those theorems and propositions hold for every r-partially quasi-linear domain, we only provide the proofs for the (0-)partially quasi-linear domain. However, the proofs can easily be modified so that they work for every r-partially quasi-linear domain.

First, we define several classes of preferences and establish lemmas that guarantee the existence of some preferences that we choose in the proofs.

Preferences

Definition10 requires that the bundles indifferent to (x, tx) must be located to the left of the kinked dashed line. Given a vector t∈R|X| and x∈X, letRM Tt,x be the set of preferences that are monotonic transformations, often at x. In some of the proofs we choose a partially quasi-linear preference relation that is a monotone transformation of two monotone object vectors.

20 The right-hand side of both inequalities in definition 9 can be equal to zero for some preference relation.

Figure 7: An illustration of a monotonic transformation of t ∈ R |X| at x ∈ X.
Figure 7: An illustration of a monotonic transformation of t ∈ R |X| at x ∈ X.

Implications of properties of rules

If R ⊇ RQ, then by Fact 1 an efficient and strategy-proof rule on Rn coincides with a Groves rule on the quasi-linear domain. Let R be such that R ⊇ RQ and f are an efficient and strategy-proof rule for Rn. The next lemma states that in some specific situations efficiency and strategy resilience give a range of payment for an agent who does not receive an object.

Lemma 11(i) states that when n = 2 and the domain is a partially quasilinear domain, the loser's payoff is zero as long as the other agent has a quasilinear preference. The* in Lemma 11(ii) is the maximum of the sum of valuations that can be achieved by agents other than agents i and j under the assumption that no agent receives all objects. However, for every r ∈ R we can prove the strategic impermeability of r-generalized Vickrey rules on (RP(r))n in the same way.

21 We also use this claim in the proof of Theorem 1(i) when we prove the efficiency of the generalized Vickrey rules. As we mentioned at the beginning of the Appendix, we only give the proof for the partially quasi-linear domain. In the proof of Theorem 1, whenever we get an agent i, the other agent is denoted by j.

Proof of Theorem 1 (i)

Using Step 4 and following the proof from Step 2, we can show that for every R∈ R2,.

Figure 8: An illustration of Case 1 in the proof of Step 2
Figure 8: An illustration of Case 1 in the proof of Step 2

Proof of Theorem 1 (ii)

The first property gives an upper bound on the sum of the valuations that agents other than agents 1 and 2 can obtain under the assumption that no agent receives ¯x. This fact will be used in later steps to determine the payoffs for agents 1 and 2 at some preference profiles. Assume, conversely, that there is an efficient and strategy-safe rule f on Rn such that it is a (generalized) Vickrey rule on (RQ)n.

Rn∈ RQ, the option sets of agent 1 under for−1 and R0−1 coincide with those under a Vickrey rule, respectively.

Figure 11: An illustration of R i for i ∈ {1, 2}.
Figure 11: An illustration of R i for i ∈ {1, 2}.

Proof of Lemma 2

Proof of Lemma 3

To show that Ri object is monotonic, we need to prove that for every0 ∈R, the vector (VRi(x0,(0, t0)))x0∈X object is monotonic. Finally, since its object is monotonic and sufficiently close to 0, it is immediate that (VRi(x00,(y, its)))x00∈X object is monotonic.

Figure 23: An illustration of t, s, t ∗ , and R i .
Figure 23: An illustration of t, s, t ∗ , and R i .

Proof of Lemma 4

First note that since t∗ is weakly object monotonic and (x0)x0∈X object is monotonic, (VRi(x0,(x, tx)))x0∈X object is monotonic.

Figure 24: An illustration of t, s, t ∗ , and R i . Let d : X → R be such that for each x 0 ∈ X,
Figure 24: An illustration of t, s, t ∗ , and R i . Let d : X → R be such that for each x 0 ∈ X,

Proof of Lemma 5

Proof of Lemma 6

Proof of Lemma 7

Proof of Lemma 8

Proof of Lemma 9

Proof of Lemma 11

We only focus on the proof of s0ea =tee, because the same argument applies to s00ea =tee. Svensson (2016): "The Transfer of Ownership of Public Housing to Existing Tenants: A Market Design Approach," Journal of Economic Theory. “Supplement to 'When can we design efficient and strategy-resistant rules in packet assignment problems?',” Mimeo.

2020b): “Strategy-safe multi-objective mechanism design: Ex-post revenue maximization with non-quasilinear preferences,” Journal of Economic Theory. Serizawa (2016): "Efficiency and Strategy Safety in Object Allocation Problems with Multi-Demand Preferences," Social Choice and Welfare. May (2012): "A note on the incompatibility of strategy certainty and Pareto optimality in quasi-linear settings with public budgets," Economics Letters.

Serizawa (2015): "Strategy-proofness and efficiency with non-quasi-linear preferences: A characterization of minimum price Walrasian rule",.

Figure 26: An illustration of R 2 , R 0 2 , and R 2 00 .
Figure 26: An illustration of R 2 , R 0 2 , and R 2 00 .

Figure 1: An illustration of the consumption set for M = {1, 2} and ¯ x = (1, 1).
Figure 2: Indifference curves of a quasi-linear preference relation.
Figure 3: An illustration of the r-relevant consumption set for R i for ¯ x = (1, 1).
Figure 4: An illustration of an r-partially quasi-linear preference relation.
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