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Serial dictatorship mechanisms with reservation prices

Abstract

We propose a new set of mechanisms, which we call serial dictatorship mechanisms with individual reservation prices for the allocation of homogeneous indivisible objects, e.g., specialist clinic appointments. We show that a mechanism \(\varphi \) satisfies minimal tradability, individual rationality, strategy-proofness, consistency, independence of unallocated objects, and non-wasteful tie-breaking if and only if there exists a reservation price vector r and a priority ordering \(\succ \) such that \(\varphi \) is a serial dictatorship mechanism with reservation prices based on r and \(\succ \). We obtain a second characterization by replacing individual rationality with non-imposition. In both our characterizations r, \(\succ \), and \(\varphi \) are all found simultaneously and endogenously from the properties. Finally, we illustrate how our model, mechanism, and results capture the normative requirements governing the functioning of some real-life markets and the mechanisms that these markets use.

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Notes

  1. 1.

    Strategy-proofness is a key property that is “obviously” satisfied—in the sense of Li (2017)—by all the classical serial dictatorship mechanisms and by our own serial dictatorship with reservation prices.

  2. 2.

    For instance, restricted endowment inheritance mechanisms introduced by Pápai (2000) and characterized by Pápai (2000) and Ehlers et al. (2002) are essentially serial dictatorships where in each iteration, we might have either single or twin dictators.

  3. 3.

    A finite set of potential agents would not change any of our results.

  4. 4.

    Setting the set of payment vectors equal to the Cartesian product of a discrete or finite price set would not change any of our results. As we discuss in detail in Sect. 5, ruling out negative payments or transfers is natural in certain contexts of rationing, e.g., when allocating appointments for certain medical services.

  5. 5.

    Requiring continuity of \(u_i\) would be a less general assumption that guarantees the existence of valuation \(v_i\).

  6. 6.

    Thomson (2015) provides an extensive survey of consistency in various applications.

  7. 7.

    Private insurance is compulsory for anyone who is not an Australian citizen or a permanent resident.

  8. 8.

    There are many private insurers, each offering many policies that differ in coverage, “embargo” waiting periods imposed, prices, discounts available, levels of excess or co-payments required, and so on.

  9. 9.

    See the “Access Policy” white paper by the Health Service Programs Branch (2013).

  10. 10.

    While our description above applies in many situations, there are several exceptions and limitations. For instance, for organ transplants appointments are made using a different dynamic matching procedure (Akbarpour et al. 2019). Emergency room rules for dealing with life-threatening situations are also different. More generally, people in very serious conditions are unlikely to pass their turn. Our description is best suited for procedures that are less severe, but nevertheless serious enough to require a specialist-led appointment, including for instance most “watchful waiting” scenarios in which the condition does require a specialist-led appointment that the patients then decide whether or not to take.

  11. 11.

    Clinical need is most of the time established based on external referrals, the arrival time is random, and hospitals use consistent procedures to determine the priorities. Thus, although the priority order is created within the hospital, it can be interpreted as being essentially exogenously given.

  12. 12.

    Non-imposition allows patients who are not interested in an appointment to withdraw from consideration at no cost.

  13. 13.

    As we described in Sect. 5 when considering the allocation of next-available consultant-led medical appointments, the mechanisms are first required to ensure that patients are prioritized based on clinical need and that there is “equality of access.” The patients’ payments, while important, come second.

  14. 14.

    Other related characterizations of second-price auctions are obtained by Saitoh and Serizawa (2008) and Ohseto (2006).

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Author information

Correspondence to Bettina Klaus.

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Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We thank the editor (Nicholas Yannelis), an anonymous referee, David Delacrétaz, Fuhito Kojima, and Steven Williams for helpful discussions and suggestions.

Bettina Klaus gratefully acknowledges financial support from the Swiss National Science Foundation (SNFS), Project 100018_162606.

Alexandru Nichifor gratefully acknowledges financial support via the Australian Research Council’s Discovery Early Career Researcher Award (DECRA), Project DE170101183.

Appendices

Appendix

Proof of Lemmas 1 and 2

Proof of Lemma 1

(a) Assume that mechanism \(\varphi \) satisfies individual rationality. Let \(N \in \mathcal {N}\), \(k\in \mathbb {Z}_+\), and \((N,u,k) \in \Gamma (N,k)\) with associated valuation vector \(v\in \mathcal {V}(N)\). Let \(i\in N\) such that \(v_i=0\). If \(\alpha _i(N,u,k)=0\), then (IR1) implies \(\pi _i(N,u,k)=0\). If \(\alpha _i(N,u,k)=1\), then (IR2) implies \(\pi _i(N,u,k)\le v_i=0\), and thus, \(\pi _i(N,u,k)=0\). Hence, \(\varphi \) satisfies non-imposition.

(b) Assume that mechanism \(\varphi \) satisfies non-imposition and strategy-proofness. Let \(N \in \mathcal {N}\), \(k\in \mathbb {Z}_+\), and \((N,u,k) \in \Gamma (N,k)\) with associated valuation vector \(v\in \mathcal {V}(N)\). Let \(i\in N\) and \(u'=(u'_i,u_{-i}) \in \mathcal {U}(N)\) with associated valuation vector \(v'=(0,v_{-i}) \in \mathcal {V}(N)\).

(IR1) Suppose that \(\alpha _i(N,u,k)=0\) and, in contradiction to (IR1), \(\pi _i(N,u,k)>0\). By property (i) of utility function \(u_i\), \(u_i(\varphi _i(N,u,k))=u_i(0,\pi _i(N,u,k))\overset{\text {(i)}}{<}u_i(0,0)\). By property (ii) of utility function \(u_i\), \(u_i(0,0) \overset{\text {(ii)}}{\le } u_i(1,0)\).

By non-imposition, we have \(\pi _i(N,u',k)=0\). Hence, \(\varphi _i(N,u',k)\in \{(0,0),(1,0)\}\) and \(u_i(\varphi _i(N,u,k))<u_i(\varphi _i(N,u',k))\), contradicting strategy-proofness. Thus, \(\alpha _i(N,u,k)=0\) implies \(\pi _i(N,u,k) = 0\).

(IR2) Suppose that \(\alpha _i(N,u,k)=1\) and, in contradiction to (IR2), \(\pi _i(N,u,k) >v_i\ (\ge 0)\). By property (iii) of utility function \(u_i\), \(v_i\ne \infty \) and \(u_i(1,v_i)=u_i(0,0)\). By property (i) of utility function \(u_i\), \(u_i(\varphi _i(N,u,k))=u_i(1,\pi _i(N,u,k))\overset{\text {(i)}}{<}u_i(1,v_i)=u_i(0,0)\). By property (ii) of utility function \(u_i\), \(u_i(0,0) \overset{\text {(ii)}}{\le } u_i(1,0)\).

By non-imposition, we have \(\pi _i(N,u',k)=0\). Hence, \(\varphi _i(N,u',k)\in \{(0,0),(1,0)\}\) and \(u_i(\varphi _i(N,u,k))<u_i(\varphi _i(N,u',k))\), contradicting strategy-proofness. Thus, \(\alpha _i(N,u,k)=1\) implies \(\pi _i(N,u,k)\le v_i\)\(\square \)

Proof of Lemma 2

Assume that mechanism \(\varphi \) satisfies individual-rationality (IR1), strategy-proofness, and non-wasteful tie-breaking. Let \(N \in \mathcal {N}\), \(k\in \mathbb {Z}_+\), and \((N,u,k) \in \Gamma (N,k)\) with associated valuation vector \(v\in \mathcal {V}(N)\). Let \(i\in N\) such that \(\alpha _i(N,u,k)=1\) and suppose that, in contradiction to (IR2), \(\pi _i(N,u,k) >v_i\ (\ge 0)\).

Consider \(u'=(u'_i,u_{-i}) \in \mathcal {U}(N)\) with associated valuation vector \(v'=(\pi _i(N,u,k),v_{-i}) \in \mathcal {V}(N)\), hence agent i now values the object at exactly his previous payment \(\pi _i(N,u,k)\). If \(\varphi _i(N,u,k)=(1,\pi _i(N,u,k))=\varphi _i(N,u',k)\), then non-wasteful tie-breaking is violated. Hence, \(\alpha _i(N,u,k)=1\ne \alpha _i(N,u',k)\) or \(\pi _i(N,u,k)\ne \pi _i(N,u',k)\).

If \(\alpha _i(N,u,k)=1=\alpha _i(N,u',k)\) and \(\pi _i(N,u,k)\ne \pi _i(N,u',k)\), then strategy-proofness is violated because either at u or \(u'\) agent i could misreport his utility function to be charged a lower price while still receiving the object and by property (i) of utility function \(u_i\), he would be better off. Thus, \(\alpha _i(N,u',k)=0\) and by (IR1), \(\pi _i(N,u',k)=0\). By property (i) of utility function \(u_i\), \(u_i(\varphi _i(N,u,k))=u_i(1,\pi _i(N,u,k))\overset{\text {(i)}}{<}u_i(0,0) =\varphi _i(N,u',k)\), contradicting strategy-proofness.\(\square \)

Proof of Theorem 1

It is easy to see that any serial dictatorship mechanism with reservation prices induced by some reservation price vector \(r\in \mathcal {F}\) and some priority ordering \(\succ \in \mathcal {P}\) satisfies all the properties in the theorem.

For the uniqueness proof, we assume that \(\varphi \) satisfies all the properties in the theorem; we split the proof into four parts: First, we construct the individual reservation price vector \(r\in \mathcal {F}\); second, we construct the priority ordering \(\succ \in \mathcal {P}\) over \(\mathbb {N}\); third, we prove that \(\varphi =\psi ^{(r,\succ )}\) for single-object problems, i.e., for \(k=1\); fourth, we extend the result that \(\varphi =\psi ^{(r,\succ )}\) to any \(k\in \mathbb {Z}_+\) via an induction argument.

Part 1: individual reservation prices

We first establish the existence of an individual reservation price vector.

Lemma 3

Assume that mechanism \(\varphi \) satisfies minimal tradability, individual rationality, and strategy-proofness. Then, for each agent \(i\in \mathbb {N}\), there exists an individual reservation price \(r_i\ge 0\) such that for each utility function \(u_i\in \mathcal {U}(\{i\})\) with associated valuation \(v_i\in \mathcal {V}(\{i\})\):

(i):

\(v_i> r_i\) implies \(\varphi _i(\{i\},u_i,1)=(1,r_i)\),

(ii):

\(v_i= r_i\) implies \(\varphi _i(\{i\},u_i,1)\in \{(0,0),(1,r_i)\}\), and

(iii):

\(v_i<r_i\) implies \(\varphi _i(\{i\},u_i,1)=(0,0)\).

Proof

Assume that mechanism \(\varphi \) satisfies all the properties in the lemma. For each \(i\in \mathbb {N}\), we define an individual reservation price\(r_i\ge 0\) as follows. Let \(N=\{i\}\) and \(k=1\). Define the price range of mechanism \(\varphi \) for agent i with preferences \(u_i\) as the set of all possible prices at which he could obtain the object, i.e.,

$$\begin{aligned} P_i^{\varphi } =\left\{ p_i\in \mathbb {R}_+ : \varphi _i(\{i\},u_i,1)=(1,p_i)\text { for some }u_i\in \mathcal {U}(\{i\})\right\} . \end{aligned}$$

By minimal tradability, \(|P_i^{\varphi }| \ge 1\).

Suppose that \(|P_i^{\varphi }|>1\). Then, there exist \(p_i,p'_i\in P_i^{\varphi }\) and, without loss of generality, assume \(p_i>p'_i\). Hence, there exist utility functions \(u_i,u'_i\in \mathcal {U}(\{i\})\) such that \(\varphi _i(\{i\},u_i,1)=(1,p_i)\) and \(\varphi _i(\{i\},u'_i,1)=(1,p'_i)\). Then, agent i with preferences represented by \(u_i\) can receive the object at the lower price \(p'_i\) if he pretends his preferences are represented by \(u'_i\). Thus, in contradiction to strategy-proofness, by property (i) of utility function \(u_i\), \(u_i(\varphi _i(\{i\},u'_i,1))=u_i(1,p'_i)\overset{\text {(i)}}{>}u_i(1,p_i)= u_i(\varphi _i(\{i\},u_i,1))\).

Thus, we have \(|P_i^{\varphi }|=1\) and \(r_i\) is defined via \(P_i^{\varphi }=\{r_i\}\). Hence, if \(\alpha _i(\{i\},u_i,1)=1\), then \(\pi _i(\{i\},u_i,1)=r_i\) and by individual rationality (IR2), \(v_i\ge r_i\). By individual rationality (IR1), if \(\alpha _i(\{i\},u_i,1)=0\), then \(\pi _i(\{i\},u_i,1)=0\). We now have the following implications for agent i’s allotment:

(i):

if \(v_i> r_i\), then \(u_i(1,r_i)>u_i(0,0)\) and by strategy-proofness, \(\varphi _i(\{i\},u_i,1)=(1,r_i)\);

(ii):

if \(v_i= r_i\), then \(u_i(0,0)=u_i(1,r_i)\) and \(\varphi _i(\{i\},u_i,1)\in \{(0,0),(1,r_i)\}\); and

(iii):

if \(v_i<r_i\), then \(u_i(0,0)>u_i(1,r_i)\) and by strategy-proofness, \(\varphi _i(\{i\},u_i,1)=(0,0)\).

\(\square \)

By our next lemma, for any problem, if an agent receives an object, then his valuation has to be weakly larger than his individual reservation price (which also equals his payment); otherwise, his payment is necessarily null.

Lemma 4

Assume that mechanism \(\varphi \) satisfies minimal tradability, individual rationality, strategy-proofness, consistency, and independence of unallocated objects. Then, for each \(N \in \mathcal {N}\), each \(k\in \mathbb {Z}_+\), each \(\gamma \in \Gamma (N,k)\) with associated valuation vector \(v\in \mathcal {V}(N)\), and each \(i\in N\), if \(\alpha _i(\gamma )=1\), then \(\pi _i(\gamma )=r_i\le v_i\) (with \(r_i\) as in Lemma 3). Furthermore, if \(\gamma =(N,u,1)\), i.e., \(k=1\), then independence of unallocated objects is not necessary.

Proof

Assume that mechanism \(\varphi \) satisfies all the properties in the lemma. Let \(N \in \mathcal {N}\), \(k\in \mathbb {Z}_+\), and \(\gamma =(N,u,k)\in \Gamma (N,k)\) with associated valuation vector \(v\in \mathcal {V}(N)\). Let \(i\in N\) and \(\alpha _i(\gamma )=1\). If all agents but agent i leave with their allotments, then the reduced problem is \(\gamma _{\{i\}}=(\{i\},u_i,k_{\{i\}})\), where \(k_{\{i\}}=k-\sum _{j\in N\setminus \{i\}}\alpha _j(\gamma )\ge 1\). By consistency, \(\varphi _i(\gamma _{\{i\}})=\varphi _i(\gamma )\) and \(\alpha _i(\gamma _{\{i\}})=\alpha _i(\gamma )=1\). If \(k_{\{i\}}=1\), then \(\gamma _{\{i\}}=(\{i\},u_i,1)\). If \(k_{\{i\}}>1\), then using independence of unallocated objects, we obtain \(\varphi _i(\{i\},u_i,1)=\varphi _i(\gamma _{\{i\}})\).

Thus, \(\varphi _i(\{i\},u_i,1)=\varphi _i(\gamma )\) and \(\alpha _i(\{i\},u_i,1)=\alpha _i(\gamma )=1\). By Lemma 3, \(v_i\ge r_i\) and \(\varphi _i(\{i\},u_i,1)=(1,r_i)=\varphi _i(\gamma )\). In particular, \(\pi _i(\gamma )=r_i\le v_i\).

If \(k=1\), then in the proof above, \(k_{\{i\}}=1\) and independence of unallocated objects is not necessary. \(\square \)

Part 2: priority ordering

In this part, we consider single-object problems, i.e., \(k=1\).

Let \(i,j\in \mathbb {N}\), \(i\ne j\). By minimal tradability, there exists \(u=(u_i,u_j)\in \mathcal {U}(\{i,j\})\) with associated valuation vector \(v=(v_x,v_y)\in \mathcal {V}(\{x,y\})\) such that for an agent \(x\in \{i,j\}\equiv \{x,y\}\), \(\alpha _x(\{x,y\},u,1)=1\). By consistency and Lemma 3 (i) and (ii), \(\varphi _x(\{x,y\},u,1)=\varphi _x(\{x\},u_x,1)=(1,r_x)\).

Let \(u'=(\bar{u}_x,u_y)\in \mathcal {U}(\{x,y\})\) with associated valuation vector \(v'=(r_x+1,v_y)\in \mathcal {V}(\{x,y\})\). Then, by strategy-proofness, \(\alpha _x(\{x,y\},u',1)=1\) (in fact, we even have \(\varphi _x(\{x,y\},u',1)=(1,r_x)\)).

Let \((\bar{u}_x,\bar{u}_y)\in \mathcal {U}(\{x,y\})\) with associated valuation vector \((r_x+1,r_y+1)\in \mathcal {V}(\{x,y\})\). By consistency, the object continues to remain allocated at problem \((\{x,y\},(\bar{u}_x,\bar{u}_y),1)\). To see this, observe that otherwise, if the object is not allocated anymore, starting from \((\{x,y\},(\bar{u}_x,\bar{u}_y),1)\) and removing agent y, by consistency we would have \(\alpha _x(\{x\},\bar{u}_x,1)=0\), which would contradict that \(\varphi _x(\{x\},\bar{u}_x,1)=(1,r_x)\) (by Lemma 3 (i)). Thus, one of the agents in \(\{i,j\}\equiv \{x,y\}\) receives the object. If \(\alpha _i(\{i,j\},(\bar{u}_i,\bar{u}_j),1)=1\), then set \(i\succ j\). Otherwise, if \(\alpha _j(\{i,j\},(\bar{u}_i,\bar{u}_j),1)=1\), then set \(j\succ i\).

We now prove the transitivity of \(\succ \). Assume, by contradiction, that there exist distinct agents \(i,j,l\in \mathbb {N}\) such that \(i\succ j\), \(j\succ l\), and \(l\succ i\). Assume that for any of these agents \(a\in \{i,j,l\}\), \(\bar{u}_a\) is the utility function used to determine \(\succ \) with associated valuation \(r_a+1\). Hence, \(\alpha _i(\{i,j\},(\bar{u}_i,\bar{u}_j),1)=1\), \(\alpha _j(\{j,l\},(\bar{u}_j,\bar{u}_l),1)=1\), and \(\alpha _l(\{i,l\},(\bar{u}_i,\bar{u}_l),1)=1\).

By minimal tradability, there exists \(u=(u_i,u_j,u_l)\in \mathcal {U}(\{i,j,l\})\) with associated valuation vector \(v=(v_x,v_y,v_z)\in \mathcal {V}(\{x,y,z\})\) such that for an agent \(x\in \{i,j,l\}\equiv \{x,y,z\}\), \(\alpha _x(\{x,y,z\},u,1)=1\). By consistency and Lemma 3 (i) and (ii), \(\varphi _x(\{x,y,z\},u,1)=\varphi _x(\{x\},u_x,1)=(1,r_x)\).

Let \(u'=(\bar{u}_x,u_y,u_z)\in \mathcal {U}(\{x,y,z\})\) with associated valuation vector \(v'=(r_x+1,v_y,v_z)\in \mathcal {V}(\{x,y,z\})\). Then, by strategy-proofness, \(\alpha _x(\{x,y,z\},u',1)=1\) (in fact, we even have \(\varphi _x(\{x,y,z\},u',1)=(1,r_x)\)).

Let \(u''=(\bar{u}_x,\bar{u}_y,u_z)\in \mathcal {U}(\{x,y,z\})\) with associated valuation vector \(v''=(r_x+1,r_y+1,v_z)\in \mathcal {V}(\{x,y,z\})\). By consistency, the object continues to remain allocated at problem \((\{x,y,z\},u'',1)\). To see this, observe that otherwise, if the object is not allocated anymore, starting from \((\{x,y,z\},u'',1)\) and removing agent z, by consistency we would have \(\alpha _x(\{x,y\},(\bar{u}_x,\bar{u}_y),1)=0\) and \(\alpha _y(\{x,y\},(\bar{u}_x,\bar{u}_y,1))=0\), which would contradict that either \(x\succ y\) or \(y\succ x\). Thus, one of the agents in \(\{i,j,l\}\equiv \{x,y,z\}\) receives the object.

Let \((\bar{u}_x,\bar{u}_y,\bar{u}_z)\in \mathcal {U}(\{x,y,z\})\) with associated valuation vector \((r_x+1,r_y+1,r_z+1)\in \mathcal {V}(\{x,y,z\})\). By consistency (if agent z did not receive the object before) or by strategy-proofness (if agent z did receive the object before), one of the agents in \(\{i,j,l\}\equiv \{x,y,z\}\) receives the object, without loss of generality, agent i, i.e., \(\alpha _i(\{i,j,l\},(\bar{u}_i,\bar{u}_j,\bar{u}_l),1)=1\). By consistency, \(\alpha _i(\{i,l\},(\bar{u}_i,\bar{u}_l),1)=1\), contradicting \(l\succ i\) (and hence, \(\alpha _l(\{i,l\},(\bar{u}_i,\bar{u}_l),1)=1\)).

Part 3: single-object problems

We show for single-object problems, i.e., \(k=1\), that \(\varphi \) always assigns the object and payments as if it is a serial dictatorship mechanism based on \(r\in \mathcal {F}\) (from Part 1) and \(\succ \in \mathcal {P}\) (from Part 2). That is, we show that for each \(N \in \mathcal {N}\), each \(\gamma =(N,u,1) \in \Gamma (N,1)\) with associated valuation vector \(v\in \mathcal {V}(N)\), and \(U^{r}(\gamma )\equiv \{j\in N: v_j > r_j\}\), \(\varphi \) assigns the uniquely determined outcome such that for each \(i\in N\),

(a):

if \(i=\arg \max _{\succ } U^{r}(\gamma )\), then \(\psi ^{(r,\succ )}_i(\gamma )=(1,r_i)\) and

(b):

if \(i\ne \arg \max _{\succ } U^{r}(\gamma )\), then \(\psi ^{(r,\succ )}_i(\gamma )=(0,0)\).

Recall that by individual rationality (IR1), if \(i\in N\) and \(\alpha _i(\gamma )=0\), then \(\pi _i(\gamma )=0\). Furthermore, by Lemma 4, if \(i\in N\) and \(\alpha _i(\gamma )=1\), then \(\pi _i(\gamma )=r_i\). Hence, we only need to prove that the allocation rule \(\alpha =\alpha ^{(r,\succ )}\). We proceed by contradiction, considering a different object allocation in each of the Cases (a) and (b); to simplify the proof, we start with Case (b).

Case (b): There exists \(i\ne \arg \max _{\succ } U^{r}(\gamma )\) such that \(\alpha _i(\gamma )=1\).

Case (b.1): \(i\not \in U^{r}(\gamma )\)

By Lemma 4, \(\pi _i(\gamma )=r_i\) and by \(i\not \in U^{r}(\gamma )\), we have \(v_i \le r_i\). If \(v_i<\pi _i(\gamma )\), then individual rationality (IR2) is violated. If \(v_i=\pi _i(\gamma )\), then non-wasteful tie-breaking is violated.

Case (b.2): \(i\in U^{r}(\gamma )\) but there exists an agent \(j\in U^{r}(\gamma )\) such that \(j\succ i\) and \(\alpha _i(\gamma )=1\).

Assume that \((\bar{u}_i,\bar{u}_j)\) is the utility profile used to determine \(j\succ i\) with associated valuation vector \((r_i+1,r_j+1)\). Hence, \(\varphi _j(\{i,j\},(\bar{u}_i,\bar{u}_j),1)=(1,r_j)\).

Starting from problem (Nu, 1), by consistency and Lemma 3, \(\varphi _i(\{i,j\},(u_i,u_j),1)=(1,r_i)\). By strategy-proofness, \(\alpha _i(\{i,j\},(\bar{u}_i,u_j),1)=1\). Hence, \(\alpha _j(\{i,j\},(\bar{u}_i,u_j),1)=0\) and by individual rationality (IR1), \(\varphi _j(\{i,j\},(\bar{u}_i,u_j),1)=(0,0)\).

Since \(j \in U^{r}(\gamma )\), we have \(v_j > r_j\). Then, in contradiction to strategy-proofness, we have that \(u_j(\varphi _j(\{i,j\},(\bar{u}_i,\bar{u}_j),1))= u_j(1,r_j)>u_j(0,0)= u_j(\varphi _j(\{i,j\},(\bar{u}_i,u_j),1))\) (agent j with utility function \(u_j\) and valuation \(v_j\) will beneficially misreport utility function \(\bar{u}_j\) with valuation \(r_j+1\)).

Case (a): for \(j=\arg \max _{\succ } U^{r}(\gamma )\), we have \(\alpha _j(\gamma )=0\).

The contradiction obtained for Case (b) above implies that for each \(i\in N \setminus \{j\}\), \(\alpha _i(\gamma )=0\). If now also \(\alpha _j(\gamma )=0\), then the object is not allocated. By consistency, starting from problem \(\gamma =(N,u,1)\) and removing all agents but j, we obtain \(\alpha _j(\{j\},u_j,1)=0\). However, since \(j\in U^{r}(\gamma )\), we have \(v_j > r_j\), which by the definition of \(r_j\) in Lemma 3 (i) implies that \(\alpha _j(\{j\},u_j,1)=1\), a contradiction. \(\square \)

Part 4: an arbitrary number of objects

We now show by induction on the number of objects that \(\varphi =\psi ^{(r,\succ )}\) for the general domain of all problems, i.e., \(k\in \mathbb {Z}_+\).

Induction Basis\(\varvec{k=0,1}\): Let \(N \in \mathcal {N}\), \(k\in \{0,1\}\), and \(\gamma =(N,u,k) \in \Gamma (N,k)\). Then, \(\varphi (\gamma )=\psi ^{(r,\succ )}(\gamma )\) follows for \(k=0\) by individual rationality (IR1) and for \(k=1\) by Part 3.

Induction Hypothesis\(\varvec{k'\le k}\): On the subdomain of problems where at most \(k\ge 1\) objects are available, we assume \(\varphi =\psi ^{(r,\succ )}\).

Induction Step\(\varvec{k+1}\): We show that for problems where \(k+1\) objects are available, we have \(\varphi =\psi ^{(r,\succ )}\). Let \(\varphi =(\alpha ,\pi )\) and \(\psi ^{(r,\succ )}=(\alpha ',\pi ')\)

Consider a set of agents \(N\in \mathcal {N}\) and a utility profile \(u\in \mathcal {U}(N)\). If no agent in N would like to receive an object at problem \((N,u,k+1)\), i.e., if for all \(i\in N\), \(v_{i}\le r_i\), then by Lemma 4 and non-wasteful tie-breaking, for all \(i\in N\), \(\varphi _i(N,u,k+1)=(0,0)=\psi _i^{(r,\succ )}(N,u,k+1)\). Hence, assume that for some agent \(i\in N\), \(v_{i}> r_i\).

Without loss of generality, assume that agent 1 is the highest priority agent in N according to \(\succ \) such that \(v_{1}> r_1\). Assume, by contradiction, that \(\alpha _1(N,u,k+1)=0\).

Case 1. There exists an agent \(i\in N\setminus \{1\}\) such that \(\alpha _i(N,u,k+1)=1\).

Recall that \(\alpha _1(N,u,k+1)=0\). Hence, when all agents except agents 1 and i leave with their allotments, we obtain the reduced problem \((\{1,i\},u_{\{1,i\}},k')\) where \(1\le k'\le k+1\). By consistency, we then have

$$\begin{aligned} \alpha _1(N,u,k+1)=\alpha _1(\{1,i\},u_{\{1,i\}},k')=0 \end{aligned}$$

and

$$\begin{aligned} \alpha _i(N,u,k+1)=\alpha _i(\{1,i\},u_{\{1,i\}},k')=1. \end{aligned}$$

Note that only one object is allocated (to agent i). Hence, when removing all unallocated objects from reduced problem \((\{1,i\},u_{\{1,i\}},k')\), we obtain the problem \((\{1,i\},u_{\{1,i\}},1)\). By independence of unallocated objects, we then have

$$\begin{aligned} \alpha _1(\{1,i\},u_{\{1,i\}},k')=\alpha _1(\{1,i\},u_{\{1,i\}},1\})=0, \end{aligned}$$

contradicting the Induction Basis (since for problems with one object, agent 1 as the highest priority agent who wants an object should receive it).

Case 2. For all agents \(i\in N\setminus \{1\}\), \(\alpha _i(N,u,k+1)=0\).

Recall that \(\alpha _1(N,u,k+1)=0\). Hence, when all agents except agent 1 leave with their allotments, we obtain the reduced problem \((\{1\},u_{1},k+1)\). By consistency, we then have

$$\begin{aligned} \alpha _1(N,u,k+1)=\alpha _1(\{1\},u_{1},k+1)=0. \end{aligned}$$

By minimal tradability, there exists a utility function \(\hat{u}_1\) such that \(\alpha _1(\{1\},\hat{u}_{1},k+1)=1\). By Lemma 4, \(\varphi _1(\{1\},\hat{u}_{1},k+1)=(1,r_1)\) and \(\varphi _1(\{1\},u_{1},k+1)=(0,0)\). Since \(v_1>r_1\),

$$\begin{aligned} u_1(\varphi _1(\{1\},\hat{u}_{1},k+1))>u_1(\varphi _1(\{1\},u_{1},k+1), \end{aligned}$$

contradicting strategy-proofness.

Cases 1 and 2 now imply that \(\alpha _1(N,u,k+1)=1\). Recall that \(\alpha '_1(N,u,k+1)=1\). Hence, when agent 1 leaves problem \((N,u,k+1)\) with his allotment under both mechanisms, \(\varphi \) as well as \(\psi ^{(r,\succ )}\), we obtain the reduced problem \((N\setminus \{1\},u_{N\setminus \{1\}},k)\). By consistency, for all \(i\in N\setminus \{1\}\), we then have

$$\begin{aligned} \varphi _i(N,u,k+1) =\varphi _i(N\setminus \{1\},u_{N\setminus \{1\}},k) \end{aligned}$$

and

$$\begin{aligned} \psi ^{(r,\succ )}_i(N,u,k+1) =\psi ^{(r,\succ )}_i(N\setminus \{1\},u_{N\setminus \{1\}},k). \end{aligned}$$

By the Induction Hypothesis, for all \(i\in N\setminus \{1\}\), we have

$$\begin{aligned} \varphi _i(N\setminus \{1\},u_{N\setminus \{1\}},k) =\psi ^{(r,\succ )}_i(N\setminus \{1\},u_{N\setminus \{1\}},k). \end{aligned}$$

Together with \(\varphi _1(N,u,k+1)=\psi ^{(r,\succ )}_1(N,u,k+1)=(1,r_1)\) (by Lemma 4), this completes the proof that

$$\begin{aligned} \varphi (N,u,k+1)=\psi ^{(r,\succ )}(N,u,k+1). \end{aligned}$$

\(\square \)

Independence of axioms

The following examples present mechanisms that satisfy all the properties in Theorem 1 and Corollary 2, except for the one(s) in the title of the example.

Example 1

(Minimal Tradability) The no-trade mechanism never allocates any object and no payments are made.

Example 2

(Individual Rationality, Non-Imposition) Note that by Lemma 2, we can only show independence of (IR1) (since (IR2) is implied by (IR1), strategy-proofness, and non-wasteful tie-breaking)

Fix a positive price \(P>0\) and assign objects sequentially at price \(P>0\) to the agents with the lowest indices within the set of agents who have a valuation larger than P, until we run out of objects or agents, all remaining agents, except agent 1, pay nothing; if agent 1 is present, even if his valuation is not larger than P, then he pays price P.

This mechanism, \(\varphi ^1\), does neither satisfy individual rationality (IR1) nor non-imposition, e.g., for problem \(\gamma =(N,u,1)\) with \(1\in N\) and \(u_1\) such that \(v_1=0\), we have \(\alpha ^1_1(\gamma )=0\) and \(\pi ^1_1(\gamma )=P>0\). Note that \(\varphi ^1\) satisfies individual rationality (IR2).

Example 3

(Strategy-Proofness) We assign objects sequentially to the agents with the lowest indices within the set of agents who have a positive valuation, until we run out of objects or agents, agents who obtain an object pay half their valuation, all remaining agents pay nothing.

Example 4

(Consistency) Let \(f\in \mathcal {F}\) be a vector of reservation prices and \(\vartriangleright ,\vartriangleright '\in \mathcal {P}\) be two distinct priority orderings. We apply \(\psi ^{(f,\vartriangleright )}\) to problems \(\gamma \in \Gamma (N,k)\) where the set of agents N has cardinality 2 and \(\psi ^{(f,\vartriangleright ')}\) otherwise.

Example 5

(Independence of Unallocated Objects) Mechanism \(\varphi '\) is defined as follows. Let \(f\in \mathcal {F}\) be a vector of reservation prices and \(\vartriangleright \in \mathcal {P}\) be a priority ordering. Then, if fewer agents than there are objects want an object (i.e., their valuation is higher than their reservation price), no object is allocated and no payment is made, i.e., \(\varphi '\) coincides with the no-trade mechanism. Otherwise, \(\varphi '=\psi ^{(f,\vartriangleright )}\).

Note that this is an adjustment of the no-trade mechanism in such a way that if at least as many agents as there are objects want an object, all objects are allocated, and hence, minimal tradability is satisfied. Furthermore, for single-object problems, we have \(\varphi '=\psi ^{(f,\vartriangleright )}\). For problems with \(k>1\), the cases (i) “fewer agents than there are objects want an object” and (ii) “at least as many agents than there are objects want an object” are unchanged when agents leave with their allotments, and hence, consistency is satisfied.

Example 6

(Non-Wasteful Tie-Breaking) Consider a modification of our serial dictatorship mechanism with reservation prices in which also agents who are indifferent between [not receiving the object and not paying anything] and [receiving the object and paying his reservation price], as long as objects are still available, receive an object.

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Klaus, B., Nichifor, A. Serial dictatorship mechanisms with reservation prices. Econ Theory (2019). https://doi.org/10.1007/s00199-019-01223-6

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Keywords

  • Serial dictatorship
  • Individual reservation prices
  • Priority ordering
  • Axiomatic characterization
  • Strategy-proofness
  • Consistency

JEL Classification

  • C78
  • D47
  • D71