Clearing complex exchange markets are one of the successful applications of algorithmic economics and multi-agent systems (see e.g., [Sönmez and Ünver, 2011]). Key domains for which algorithms have been designed include housing markets and kidney exchange markets. In recent years, market designers are turning their attention to liver and lung exchange markets. There are proposals to explore the efficiency gains via multi-organ exchange markets (see e.g., [Dickerson and Sandholm, 2017]). Apart from exchanging organs or housing, several new digital platforms have come up that facilitate bartering of goods. Exchange markets are also useful to model time-bank scenarios in which people exchange services rather than items.
In most of the organ exchange markets, it is supposed that agents have single-minded dichotomous preferences over outcomes: they are either satisfied with an allocation or they are not. We consider a general model of item or organ exchange that captures all such organ markets. Each agent is endowed with a set of items and each agent has a set of target sets of items. An agent is satisfied if she gets any one of those target sets of items.
The primary focus both in theory and practice is on the optimisation problem of satisfying the maximum number of agents. The problem is computationally complex even for single-unit allocations if there is a bound on the size of the exchange cycles [Abraham et al., 2007]. It is also computationally hard if there are agents who have multi-unit demands as is the case of needing two liver or lung components [Luo and Tang, 2015]. In this paper, we do not pursue computational issues but focus on the issue of strategyproofness of mechanisms.
In exchange settings, strategyproofness is an important property to ensure the stability and efficiency of a market. Manipulable mechanisms may give incentives for agents to hide their resources which can affect the efficiency of the market. They can also give incentives to agents to misreport their preferences in which case the mechanisms may be optimising on the wrong input. In contrast to traditional voting settings, in which strategyproofness and Pareto optimality be achieved by implementing some variant of serial dictatorship, the same argument need not work for exchange settings in which agents have the ability to ‘veto’ certain outcomes because individual rationality is imposed.
Our primary contribution is formalizing a general property of mechanisms called weak consistency and proving that any mechanism satisfying weak consistency is strategyproof. Within the class of weakly consistent mechanisms, we highlight two subclasses of mechanisms which have proved useful in restricted domains. The first one is called constrained priority (CP) and the second one is called constrained utilitarian priority (CUP). Both mechanisms run a serial dictatorship type priority mechanism on the set of feasible and individually rational allocations. We show that the mechanisms satisfy strategyproofness and constrained Pareto optimality. We then show how a subtle difference in imposing a different version of individual rationality results in impossibility results even for single-minded agents.
2 Related Work
The design of individually rational, Pareto efficient and strategyproof algorithms for exchange problems is one of the most intensely studied topics in market design. Under strict preferences, single-unit demands and single-unit allocations, the well-known Gale’s Top Trading Cycles algorithm satisfies all the three ‘gold standard’ properties. If the preferences are allowed to be weak, there exist extensions of the TTC algorithm that satisfy the three properties (see e.g., [Jaramillo and Manjunath, 2012, Plaxton, 2013, Saban and Sethuraman, 2013]).
There has also been work on strategyproof exchange with multiple endowments. When preferences are not dichotomous and agents own more than one item, then strategyproofness, Pareto efficiency, and individual rationality are generally impossible to achieve (see e.g., [Papai, 2007, Konishi et al., 2001, Todo et al., 2014]). Some positive results depend on the fact that agents have lexicographic preferences over items or items types. Under these assumptions, it is possible to construct extensions of TTC that achieve strategyproofness, individual rationality, and Pareto optimality (see e.g., [Fujita et al., 2015, Sikdar et al., 2017]). The preferences we consider do not assume any of these structural restrictions that are based on lexicographic or conditional lexicographic comparisons. Biró et al.  examined a discrete exchange setting in which agents have strict preferences over objects but there can be multiple copies of objects. They study the conditions under which strategyproofness can be achieved.
There has also been work on multi-item exchanges when preferences are dichotomous. Roth et al.  show that for pairwise kidney exchanges, a priority mechanism is strategyproof. In a highly insightful paper, Hatfield  extended the argument to any kidney exchange framework with any kind of constraints on the feasible allocation including bounds on the exchange cycles. More recently, Luo and Tang  considered lung exchange and again showed that a priority mechanism applied to the set of allocations satisfying the maximum number of agents is strategyproof. Abbassi et al.  considered general exchange problems in which agents partition the items into liked and disliked. Agents’ utilities are not single-minded because they are interested in maximizing the number of liked items. For their model, Abbassi et al.  presented an algorithm that is strategyproof and Pareto optimal. However, the algorithm relies on the fact that there are no constraints such as on the size of exchange cycles.
When preferences are single-minded, computing an allocation that satisfies the maximum number of agents is an NP-hard problem even if each agent has two items [Luo and Tang, 2015]. Even for kidney exchange in which each agent gets at most one item, computing an allocation that satisfies the maximum number of agents is an NP-hard problem if there is an upper bound on the size of the exchange cycles [Abraham et al., 2007]. In this paper, we do not focus on computational problems.
Dichotomous preferences are well-studied in other context as well such as two-sided matching [Bogomolnaia and Moulin, 2004], auctions Mishra and Roy , non-cooperative games [Harrenstein et al., 2001], single-winner voting [Brams and Fishburn, 2007], committee voting [Aziz et al., 2017] and probabilistic voting [Bogomolnaia et al., 2005].
3 Model and Concepts
An exchange market is a tuple where be a set of agents and
be the set of items. The vectorspecifies the endowment of each agent . We suppose that and . Each agent has a demand set . Each element of is a bundle of items that is acceptable to agent and meets her goal of being in the exchange market.
Just like the endowment, any allocation specifies the allocated bundle of each agent . In any allocation , . Where the context is clear, we refer to the allocation bundle as the allocation of agent . We say that allocation satisfies agent is for some .
For any two allocations , one can define the preference relation of agent where if and only if satisfies implies that satisfies . The weak preference relation gives rise to the indifference relation which holds if and . It also gives rise to the strict part of the relation where if and .
Consider an exchange market in which there are 3 agents. Agent has 4 cars . Agent 2 has a valuable painting . Agent has a sports bike and a helmet . Agent 1 will only be satisfied if she gets a painting and keeps one of the cars. Agent will only be satisfied if she gets car or sports bike and helmet . Agent will be satisfied if she gets car . The information can be captured in the form of an exchange market where
; ; .
A possible allocation that satisfies all the agents is where .
An allocation is strongly individually rational (S-IR) if or for some . Informally, if an agent’s goal is not met, she is not interested in using up any of her endowed resources. This is a standard assumption is settings such as kidney exchange. Except for the last section, we will restrict our attention to S-IR allocations. Enforcing the S-IR requirement can also be seen as enforcing the standard individual rationality requirement for agents that have a special type of trichotomous preferences. In such special trichotomous preferences, agents ideally want to get an acceptable allocation, then they next prefer getting their endowment, and they least prefer a bundle which does not satisfy them and which is not their endowment. Enforcing the S-IR requirement can also be viewed as a special type of feasibility constraint.
Our model can also model altruistic donors who are happy with any allocation. We are interested in maximizing the number of agents who are satisfied. We say that an allocation is constrained Pareto optimal within the set of allocations satisfying set of properties if there exists no allocation satisfying such that for all and for some .
The Generality of the Model
Our model captures any kind of exchange market in which agents are interested in getting one of the acceptable bundles. If for each and for each , the market can model a kidney exchange market [Abraham et al., 2007]. If for each and for each , the market can model a lung exchange market [Luo and Tang, 2015]. It can also model multi-organ exchange market [Dickerson and Sandholm, 2017]. The model also allows for altruistic donors who have real items to give but in return, they are happy to get null items. We allow to contain dummy or null items so as allow imbalanced exchanges in which an agent may get different numbers of items that she gives away. In that case, any allocation for has a one-to-one correspondence with a set of cycles in which each item points to her agent and each agent points to an item she gets in the allocation. The cycles are referred to as exchange cycles.
A mechanism is a function that maps each problem instance into an allocation. We say that a mechanism is S-IR if it returns an S-IR allocation for each instance.
Let denote the set of allocations that are feasible with respect to some feasibility constraints denoted by . Depending on what the feasibility constraints capture, the term could denote the set of allocations based on pairwise exchanges or exchanges cycles of size at most 3, or satisfying some distribution or diversity constraint. In principle, could also capture the S-IR requirement. Whatever feasibility conditions we assume on the model, we will assume that they allow the endowment allocation to be a feasible allocation. That is the only minimal assumption we make on the feasibility constraints. It ensures that there always exists at least one allocation satisfying the feasibility constraints.
Since we are interested in mechanisms that satisfy any given feasibility constraints represented by , we can view a mechanism as a function that maps -feasible allocations to some allocation while taking into account the information in .
Without loss of generality, we can think of mechanisms as choice functions that choose one of the ‘best’ allocations from the set of feasible allocations.
For any mechanism and restriction , one can define the mechanism that only returns some allocation from the set of allocations . More generally, we will refer to as applying to the problem instance but only allowing to return an allocation from the set of allocations .
Whereas each agent’s private information is , she can report any and . We will assume can only declare an endowment that she actually has, therefore any reported endowment , we assume that . We say that a mechanism is strategyproof if there exists no instance under which some agent has an incentive to misreport her private information to get a more preferred outcome for herself. For strategyproof mechanisms, truth-telling is a dominant strategy for each agent.
4 Weak consistency implies strategyproofness
We first present a general definition of weak consistency that applies to any social choice setting. Recall that in our setting, for any agent with demand set , and for any two allocation and , it is the case that if both and satisfy with respect to or neither satisfy her.
Consider a set of feasible allocations defined with respect to some feasibility constraints . A mechanism is weakly consistent if
for any such that , and for any and such that for all implies that where for all .
Informally, weak consistency requires that if the set of feasible outcomes is contracted but it is still possible for each agent to get the same satisfaction level, then each agent does get the same satisfaction level from the returned outcome.
Note that weak consistency is weaker than the consistency condition considered by Hatfield . Weak consistency condition is a weaker form of Sen’s condition [Sen, 1971] or Maskin monotonicity [Maskin, 1999].
Any weakly consistent and S-IR mechanism is strategyproof.
Consider a weakly consistent mechanism . Suppose that an agent reports and gets an allocation . If for some , then agent has no incentive to misreport. Therefore, we suppose for contradiction that for each .
Suppose that reports such that the resultant allocation is and gets an acceptable bundle according to . Since we assumed in our setup that can only declare an endowment that she actually has, therefore . We denote the set of resources given by to the market by . Since is weakly consistent, if only reports bundle as acceptable and only reports , then the mechanism should still choose the same allocation for agent . Formally, if reports and , then the allocation for is .
Since is weakly consistent and S-IR, we can view as a choice mechanism choosing from the set of feasible and S-IR allocations. It chose an allocation when reported . Now consider expanding her reported endowment to and her reported demand set to . When reports , some additional feasible and S-IR allocations may emerge as well. When reports , the only new feasible S-IR allocations are those in which gets one of her acceptable bundles according to . By weak consistency, one of the three cases hold:
the same allocation is returned in which case is satisfied with respect to
some other allocation is returned such that all agents are indifferent with . In this case, is satisfied with respect to .
some allocation is returned such that some agent is not indifferent between and . In this case the allocation must be one of the new allocation resulted from expanding her reported endowment to and her reported demand set to . Such a new allocation cannot be one in which does not trade any item because in that case, it would even be feasible, when did not expand her reported sets and . Therefore, it must be the case that did trade some items. Since we assume the S-IR requirement, if did trade, then gets some allocation that satisfies her with respect to one her demands in .
Since is satisfied with respect to in all the cases, it contradicts the assumption that benefited from misreporting . ∎
Our proof technique is along the same lines as Theorem 2 of Hatfield . In contrast to the result by Hatfield , (1) our result applies to any exchange market where agents can have any number of items (2) it considers a property which is weaker than the consistency property used by him and (3) it explicitly uses the S-IR requirement which was implicitly used in the proof by Hatfield. As a corollary, we recover the central result from the paper by Hatfield. Since we focus on the S-IR requirement explicitly, it leads to a better understanding when the more permissive requirement of IR is used.
Corollary 1 (Theorem 2 of Hatfield ).
For exchange markets with dichotomous preferences and single-unit demands and endowments, any consistent mechanism is strategyproof.
5 Constrained Priority Mechanisms
In this section, we focus on two natural weakly consistent mechanisms. The mechanisms are adaptations of the idea of applying serial dictatorship and priority over the set of all feasible outcomes (see e.g. [Aziz et al., 2013]). The mechanisms are parametrized with respect to (a set of feasibility constraints) and which is a priority ordering over the agent set in which the agent in -th turn is denoted by . Both rules are based on lexicographic comparisons.
For any permutation of N, the CP mechanism is defined as follows.
For any permutation of N, the CUP mechanism is defined as follows.
CP starts from the set of feasible allocations and then refines this set by using a priority ordering over the agents. CUP starts from the set of feasible allocations satisfying the maximum number of agents and then refines this set by using a priority ordering over the agents. Both CP and CUP are flexible enough to enforce or not enforce S-IR as part of the feasibility constraints.
The following remark points out that CUP is a general mechanism that has been used in restricted domains when S-IR is enforced.
When applied to pairwise kidney allocation, CP and CUP are equivalent to the priority mechanism considered by Roth et al. . When applied to kidney exchange, CUP is equivalent to the priority mechanism considered by Hatfield . When applied to kidney exchange, CUP is equivalent to the lexicographic mechanism considered by Luo and Tang . When applied to housing allocation with existing tenants, CUP is equivalent to the MIR algorithm of Aziz .
Next, it is shown that CUP and CP are weakly consistent.
CUP and CP are weakly consistent.
For any set of feasible allocations, CP and CUP can be seen as choice functions that choose the ‘best’ allocation from the set. When that allocation is removed the set, the next choices of CP and CUP can be selected. Hence both CP and CUP can be seen as inducing a weak order on the set of feasible allocations. In such a weak order, any two feasible allocations in which each agent gets the same utility, are in the same indifference class. If an allocation is removed from the set of feasible allocations, then note that the relative ordering of the other allocations does not change. Hence CP and CUP both satisfy weak consistency. ∎
For any feasibility restriction on the set of S-IR allocations, CUP is strategyproof.
For any feasibility restriction on the set of S-IR allocations, CP is strategyproof.
As corollaries of the above theorem, we obtain several results where the mechanisms referred to in the corollaries are essentially applying CP to a restricted domain of exchange problems where S-IR is enforced.
Corollary 2 (Theorem 1, Roth et al. ).
For pairwise kidney exchange markets, the priority mechanism is strategyproof.
Corollary 3 (Corollary 3, Hatfield ).
For kidney exchange markets, the priority mechanism is strategyproof even if we impose any constraint on the exchange cycle sizes.
Corollary 4 (Theorem 2, Luo and Tang ).
For lung exchange markets where no cycle constraints are imposed, the lexicographic mechanism [Luo and Tang, 2015] is strategyproof.
Corollary 5 (Proposition 5, Aziz ).
For house allocation with existing tenants, the MIR algorithm [Aziz, 2019] is strategyproof.
If we include S-IR in the set of feasibility requirements , then note that both CP and CUP return an allocation that is Pareto optimal allocation within the set of allocations satisfying S-IR and . Hence, we can rephrase our result in the form of the following theorem.
Under dichotomous preferences, for any restriction on allocations , there exists a strategyproof mechanism that returns an allocation that satisfies constrained Pareto optimality among the set of all allocations that satisfy and S-IR.
Since imposing S-IR is similar to imposing IR when agents have trichotomous preferences, we can rephrase the theorem above as follows.
Under trichotomous preferences where each agent only has her own endowment as the second preferred outcome, for any restriction on allocations , there exists a strategyproof mechanism that returns an allocation that satisfies constrained Pareto optimality among the set of all allocations that satisfy and IR.
6 Impossibility Results
In the previous sections, we limited our attention to allocations that satisfy S-IR (strong individual rationality). We now explore the consequence of dropping the S-IR requirement. An allocation is individually rational (IR) if for some implies that for some .
Whereas IR is a less stringent requirement than S-IR, constrained Pareto optimality with respect to allocations satisfying IR is a stronger property than constrained Pareto optimality with respect to allocations satisfying S-IR. In fact, the property is stronger enough that our central results in the previous sections collapse. In particular Theorem 5.4 changes into an impossibility result as we expand the set of feasible allocations.
The proof is based on an adaptation of an impossibility result for strategyproof mechanisms maximizing total utility of agents in which agents want to get as many desirable items as possible and their utilities are not single-minded (Theorem 1, [Abbassi et al., 2015]). We adapt the argument to only consider agents having single-minded utilities: they are only satisfied if they get one of their targets sets of items. The subtle difference from the previous section is that we do not enforce S-IR which expands the set of feasible of allocations and leads to an impossibility result.
We say that an allocation is a result of pairwise exchanges if it is a result of pairs of agents make one-for-one exchange for items and with each item changing ownership at most once. Considering the exchange cycles view of allocations as mentioned in Section 3, an allocation as a result of pairwise exchange is characterized by exchange cycles in which each cycle has at most two agents in it.
Consider as the restriction of allowing allocations that are a result of pairwise item exchanges in which agents get desirable items. For , there exists no strategyproof mechanism that returns an allocation that is constrained Pareto optimal among the set of all allocations that satisfy .
Consider a set of three agents .
Each agent likes some items :
In any allocation that satisfies and is Pareto optimal, two agents can be satisfied. Since only allocations as a result of pairwise exchanges are allowed, the maximum sized matching for 9 item nodes is at most 8. However at least one agent can get most two acceptable items so it is not satisfied.
Suppose agent 1 is not satisfied. Then agent 1 can report and . In that case, agent 1 is not willing to get item in any exchange and can force this because of the requirement we have in . Agent 2 has one item which is useless to other agents so in any -feasible allocation it cannot be satisfied because it does not get three items from . Therefore the only constrained Pareto optimal allocations are those in which agent 1 exchanges with , and and with two the items among . Therefore by changing her preference relation, agent gets satisfied.
Suppose agent 2 is not satisfied. Then agent 1 can report and . In that case, agent 2 is not willing to get item in any exchange and can force this because of the requirement we have in . Agent 1 has one item which is useless to other agents so it in any -feasible allocation it cannot be satisfied because it does not get three items from . Therefore the only constrained Pareto optimal allocations are those in which agent 2 exchanges with , and and with two the items among . Therefore by changing her preference relation, agent gets satisfied.
Suppose agent 3 is not satisfied. Then agent 3 can report and . In that case, agent 3 is not willing to get item in any exchange and can force this because of the requirement we have in . Agent 1 has one item which is useless to other agents so it in any -feasible allocation it cannot be satisfied because it does not get three items from . Therefore the only constrained Pareto optimal allocations are those in which agent 3 exchanges all her items with the items in the set . Therefore by changing her preference relation, agent gets satisfied. ∎
The impossibility result is tight in the sense that for , the properties can be easily satisfied by the CP mechanism with the constraint of individual rationality. Next we note that Theorem 6.1 can be rephrased as follows.
Consider as the restriction of allowing allocations that are a result of pairwise item exchanges in which agents gets desirable items. For , there exists no strategyproof mechanism that returns an allocation that is constrained Pareto optimal among the set of all allocations that satisfy and IR.
Consider the proof of the previous theorem. Since none of the agents is satisfied by their endowment, the IR requirement has no bite. In particular, the set of allocations satisfying IR coincides with the set of all allocations. Hence, constrained Pareto optimality among the set of all allocations that satisfy is equivalent to constrained Pareto optimal among the set of all allocations that satisfy and IR. ∎
We have provided a clearer understanding of strategyproof multi-item exchange markets in which agents have dichotomous preferences. We focussed on scenarios where agents have single-minded preferences: they are either satisfied if they get a target bundle or they are dissatisfied if they do not get a target bundle. Under these preferences, we showed that strategyproofness can be achieved even under arbitrary restrictions on feasible allocations. In fact, there exist mechanisms that satisfy constrained Pareto optimality. We complement these results with a couple of impossibility results when we strengthen constrained Pareto optimality by enforcing a weaker notion of individual rationality. We envisage further algorithmic and game-theoretic research on the general multi-item exchange model discussed in the paper.
Haris Aziz is supported by a Scientia Fellowship. He thanks Barton Lee and Bahar Rastegari for useful feedback.
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