Counterbalancing Learning and Strategic Incentives in Allocation Markets

1 Introduction . This paper contributes to the growing literature in the intersection of learning and mechanism design , by considering a fundamental allocation problem in the presence of learning and strategic agents . In our setting , a planner needs to allocate a single indivisible object to at most one among several agents waiting in a queue . The object ’ s true quality is either high ( “ good ” ) or low ( “ bad ” ) , but is ex-ante unknown to both the planner and the agents . Agents are privately informed , in the sense that each receives a private signal that is informative about the true quality of the object , and strategic since they choose to accept or reject the object based on the action maximizing their expected utility . In this framework , agents thus engage in a social learning process as they observe the actions of their predecessors but not their private signals . The planner ’ s goal is to design an incentive-compatible mechanism that optimizes correctness , i.e. , maximizes the ex-ante accuracy of allocation . This problem finds a natural application in the allocation of donated organs to patients on a national waitlist . The median waiting time for a first kidney transplant in the U.S. is around 3.6 years [ 25 ] , whereas the rate at which deceased donor kidneys are offered to patients but eventually discarded has been growing steadily [ 23 ] , exceeding 21.2 % in 2020 [ 30 ] . The reasons leading to high discard rates – even of seemingly high-quality organs – remain unclear . Clinical studies attribute this increase 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . to herding behavior and inefficiency in allocation mechanisms [ 11 , 29 ] . Improving the allocation outcome of these markets is undoubtedly crucial as it could save thousands of future lives . Towards understanding the source of high discard rates , our framework highlights the often overlooked role of social learning . In most cases , when offered an organ , patients not only have access to publicly available data about the organ ( e.g. , donor ’ s age , organ size ) but also consult with their own physicians for private opinions . Each physician may assess the same organ differently based on their prior experience and/or medical expertise . Empirical works [ 13 , 31 ] have recently proposed an explanation that patients rationally ignore their own private information ( e.g. , a physician ’ s medical assessment ) about the quality of an organ and follow the preceding patients ’ rejections . Our proposed model captures this phenomenon and serves as a theoretical sandbox for the design and assessment of alternative allocation policies . We theoretically show that sequentially offering the object to agents – which is a policy commonly used in practice – can result in poor correctness , since the rejections by the few initial agents can cause a cascade of rejections by the subsequent agents . To address this issue , we introduce a novel class of incentive-compatible allocation mechanisms characterized by a batch-by-batch , dynamic voting process under a simple majority rule . Our proposed greedy mechanism dynamically partitions the list into batches of varying sizes . Then , it sequentially offers the object to each batch until the object is allocated to some agent . The planner privately asks each agent in the current batch to either opt in or opt out . If the majority of the current batch vote to opt in , then the planner randomly allocates the object to one of the current agents who opted in . If the mechanism exhausts the list without success , the object is discarded . Interestingly , several Organ Procurement Organizations ( OPOs ) already offer organs in batches , although no voting is used ; 1 the reasons are both to speed up the allocation process and to avoid herding . In our main result , we show that if private signals are at least as informative as the prior , there always exists an incentive-compatible voting mechanism that strictly improves correctness and thus helps avoid unwanted discards . Furthermore , we suggest a simple , greedy algorithm to implement such a mechanism . For all other values of private signal precision , we establish that there is no incentive-compatible voting mechanism . Moreover , every voting mechanism results in the same correctness as the sequential offering mechanism . Incentive-compatible voting mechanisms have several interesting properties . First , the batch size increases after each failed allocation attempt . The intuition is simple . Failed allocations make the current belief about the object quality more pessimistic . Therefore , the planner increases the batch size to ensure that the object is allocated only if a sufficient number of agents receive positive signals while each agent ’ s decision remains pivotal . Second , offering to another batch after a failed attempt strictly improves correctness ( compared to sequential offering ) . As our simulations showcase , this improvement is significant and generally becomes larger as the prior and signal precision grow ; in fact , a voting mechanism with just two batches significantly outperforms sequential offering . Finally , we find that the prior and signal precision have opposite effects on correctness : the correctness tends to decrease as the prior increases but the marginal effect of signal precision is positive . An intuitive explanation is as follows . Since the optimal batch size decreases as the prior increases , the probability to incorrectly allocate a bad object increases . However , as the signal precision improves , the magnitude of the error decreases since agents ’ signals become more reliable . We highlight the tension between agents ’ strategic incentives and the planner ’ s learning goal . From the planner ’ s perspective , agents ’ private signals contain valuable information about the unknown quality of the object . By increasing the number of collected data points , the planner can improve her confidence about the true quality of the object . Agents ’ strategic incentives , however , impose a constraint on the maximum size of the batch that allows the truthful elicitation of agents ’ signals : as the batch size increases , the chance to be pivotal decreases which in turn reduces the incentive for an agent with a negative signal to be truthful . Our proposed voting mechanisms enable the planner to crowdsource agents ’ private information in a simple , truthful , and effective manner . Related literature . Our model draws upon seminal social learning papers [ 7 , 8 , 28 ] and their extensions in various game structures [ 1 , 2 , 14 , 24 ] . Several empirical studies also examine social learning in organ allocation settings [ 13 , 31 ] . However , none of these works consider the existence of a planner that controls the flow of information . Furthermore , our proposed mechanism is inspired by the voting literature ( e.g. , [ 6 , 12 ] ) ; our notion of correctness is adopted from [ 3 ] . Finally , our paper 1Based on authors ’ personal communication . See , e.g. , [ 21 ] for allocation in batches . is loosely connected to information design ( e.g. , [ 10 , 18 , 26 , 27 ] ) and Bayesian exploration ( e.g. , [ 15–17 , 20 , 22 ] ) . In contrast to our model , this literature does generally not consider an underlying social learning process . We include an extended overview of the related literature in Appendix A . 2 Model . We consider a model in which a social planner wishes to allocate a single indivisible object of unknown quality to privately informed agents in a queue . Object and agents . The object is characterized by a fixed quality ! 2 { G , B } , where G and B denote a good and bad quality , respectively . The true quality ! of the object is ex-ante unknown to both the planner and agents ; however , both share a common prior belief µ = P ( ! = G ) 2 ( 0 , 1 ) . Agents are waiting in a queue ; we denote by i the agent in position i . Each each agent knows his own position . For simplicity , we assume that the queue consists of an arbitrarily large number I of agents . Each agent i has a private binary signal si 2 S , { g , b } that is informative about the true quality of the object ! . Each si is identically and independently distributed conditional on ! . Furthermore , each si is aligned with ! with probability q = P ( si = g | ! = G ) = P ( si = b | ! = B ) , where the signal precision q 2 ( 0.5 , 1 ) is commonly known.2 Agents are risk-neutral . The utility of the agent who receives the object is 1 if ! = G and 1 if ! = B ( This symmetry simplifies the exposition and analysis but does not qualitatively change the results. ) . Any agent who does not receive the object , because he either declines or is never offered , receives a utility of 0 . As a tie-breaking rule we assume that indifferent agents always decline . Voting mechanisms . The planner designs a mechanism which potentially asks ( a batch of ) agents to report their private signals , and based on their report , decides whether and how to allocate the object . We consider a class of voting mechanisms denoted by V. A voting mechanism V 2 V offers the object sequentially to odd-numbered batches of agents , where each batch size may be set dynamically based on the information from previous batches . In any given batch , if the majority of agents vote to opt in , the object is allocated uniformly at random to one of the current agents who opted in . Formally , a voting mechanism V { ⇡j } 1j=1 is defined by a sequence of functions ⇡1 , ⇡2 , ... , such that for each batch j , the corresponding function ⇡j : [ 0 , 1 ] ! N maps a current belief µj 1 about the object quality ! to a batch size Kj ( we let µ0 = µ ) . The mechanism begins with offering to the first batch of agents , who are in positions 1 , . . . , ⇡1 ( µ ) . If the object is not allocated to batch j 1 , it is offered to the agents in the next ⇡j ( µj 1 ) positions . When batch j is offered the object , each agent i in batch j chooses an action ( a vote ) ↵i 2 A , { y , n } , where y and n correspond to opting in ( “ yes ” ) and opting out ( “ no ” ) , respectively . If Yj , the number of opt in votes from batch j , constitutes the majority of batch j , i.e. , Yj d⇡j ( µj 1 ) 2 e , the object is allocated uniformly at random among agents in batch j who opt in . To simplify exposition , we sometimes denote by Kj the ex-post size of batch j ( while omitting the dependency on the current belief µj 1 ) . We assume that the agents in batch j as well as the planner observe all the votes of agents in the previous batches j0 < j , and this is commonly known . After any batch j and given a current prior µj 1 , the belief µj shared by the planner and agents is updated recursively in terms of the realized batch size Kj and voting outcome Yj as µj , P ( ! | µj 1 , Kj , Yj ) = µj 1qYj ( 1 q ) Kj Yj µj 1qYj ( 1 q ) Kj Yj + ( 1 µj 1 ) qKj Yj ( 1 q ) Yj . ( 1 ) A commonly applied voting mechanism is a sequential offering mechanism , which offers the object to one agent at a time . Another type of voting mechanism is single-batch voting mechanisms , which offer the object only once ; thus , the object is discarded if the majority in the batch choose to opt out . Let ui ( ↵i ; si ) denote the expected utility of an agent who receives signal si 2 S and takes action ↵i 2 A . A voting mechanism V 2 V is incentive-compatible ( IC ) if ui ( y ; g ) ui ( n ; g ) and ui ( n ; b ) ui ( y ; b ) . Thus , agent i gets higher utility from opting in than opting out when he has received signal si = g , but prefers to opt out when si = b . 2The condition q 2 ( 0.5 , 1 ) is without loss of generality . The common prior belief µ and signal precision q are standard in the social learning literature [ 7 , 8 , 28 ] . Correctness . The planner ’ s goal is to maximize the probability of a correct allocation outcome , or correctness ; the allocation outcome is correct if it allocates a good object or does not allocate a bad object . Formally , under a voting mechanism V 2 V , the planner ’ s decision whether to allocate the object or not is denoted by the random variable Z = { 0 , 1 } . A mechanism V 2 V achieves correctness c ( V ) if c ( V ) , EV [ I ( ! = G \ Z = 1 ) + I ( ! = B \ Z = 0 ) ] . Since we allocate a single object , a cascade can only form on the opt out action . Therefore , maximizing correctness is equivalent to maximizing social welfare . Upper bound on correctness . A natural benchmark on correctness would be the optimal solution in a setting where agents are not strategic and thus are willing to truthfully reveal their private signal to the planner . In the absence of strategic incentives , the planner ’ s optimal solution would be to ( i ) ask all agents in the queue to reveal their private signals , ( ii ) compute her posterior belief based on the gathered information , and ( iii ) allocate the object to a random agent if and only if her posterior exceeds 0.5 . Equivalently , as we show in Lemma A.1 in Appendix A , the planner allocates the object if and only if the number of positive signals is at least y , 0.5 log ⇣ 1 µ µ ⌘⇣ log ⇣ q 1 q ⌘⌘ 1 + 0.5 I , where I is the number of agents in the queue . In that case , we can show that correctness equals P ( XI y ) , ( 2 ) where XI is a Binomial ( I , q ) random variable ( see also Lemma 3 ) . As I grows , correctness approaches 1 . Importantly , this upper bound is not achievable by an incentive-compatible mechanism . Discussion of modeling assumptions . The model is based on several modeling choices that we discuss next . First , the assumption that voting outcomes are fully revealed to subsequent batches is made for simplicity . Dropping this knowledge complicates the analysis substantially : one should compute a posterior belief of an agent by taking expectations of possible outcomes that led to them being offered an object . Second , assuming that batches are always odd-numbered is of technical nature . To attain incentive compatibility , our analysis relies on the monotonic behavior of the upper and lower bounds on the prior . This result depends on a probabilistic argument about binomial distributions ( Lemma D.2 ) , which fails to hold if batches can be both of odd and even sizes.3 Finally , we consider the allocation of just a single object rather than a setting in which objects arrive and are allocated sequentially . In particular we abstract away from dynamic incentives and restrict attention to the learning aspect . We leave the interaction between the two for future work .

Summary: The authors study a setting in which a central planner allocates a single indivisible good to agents showing up sequentially. The true quality of the item (good or bad) is unknown to both agents and the mechanism designer. The agents receive a signal about the quality of the item, and the planner aims to leverage those signals to learn the quality of the item and how to allocate it efficiently.