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 the problem of allocating a single indivisible item to a group of agents, without money. Motivated by kidney exchange, the item can only have two qualities: either good or bad. Every agent (as well as the mechanism designer) agree that the prior probability of the item good is $\mu = Pr[\text{item is good}]$. However, each agent i has their own private signal s_i about the item's quality. It is assumed that these signals are iid and depend on the true quality of the item: $q = Pr[s_i \text{ is good } |\text{ item is good}] = Pr[s_i \text{ is bad }|\text{ item is bad}]$, where this q is the accuracy of their signal. The utility of each agent is +1 upon receiving a good item and -1 for a bad one. The goal is to develop truthful mechanisms that maximize the probability that the item is correctly allocated: i.e., given to an agent if it is good or discarded if it is bad.  The authors study sequential allocation mechanisms, where the decisions of previous agents might reveal information about the item and affect following choices. This is empirically observed in Kidney Exchange, where if a kidney is rejected early, it is very likely that potential recipients might keep rejecting in, further decreasing the probability that someone will accept it.  The first result of the authors is to study the sequential offering mechanism. Here, the agents are ordered and the item is offered to each one, stopping and allocating the item the first time an agent accepts it. They show (roughly) that the item is allocated if $\mu > q$ or if the first two agents receive a good signal and it's never allocated for $\mu < 1-q$. Also, notice that his mechanism is not truthful. The reason the item might not get allocated is because the agents are actively updating their prior $\mu$.  The second (and main result) is the development of a *batched* version of the previous mechanism. In the simplest version, a batch of K agents is selected and asked to report their signals. If the majority considers the item good, then it is allocated at random to one agent reporting a good signal. Depending on $\mu$ and $q$, the authors show that there exists a range of batch sizes such that the mechanism is truthful. The second step is to greedily combine batches together: starting for a truthful batch size for $\mu$, if the item is not allocated then the next batch size would be a function of $\mu'$, the updated prior. Even though the game is sequential, the agents only participate once and only care about their current prior and the size of the batch they participate in.  Putting the two results together, the authors show that for $\mu > q$ there is no truthful mechanism and every mechanism is at most as good as the sequential one. This makes sense, as in this case the prior is more informative than the private signal. For $\mu < q$ however, their batched mechanism is incentive compatible and improves the guarantee. Finally, the authors have experiments on synthetic date showing the benefit of using 1 or 2 batches, compared to the sequential mechanism,