Designing Approximately Optimal Search on Matching Platforms
Nicole Immorlica, Brendan Lucier, Vahideh Manshadi, Alexander Wei
DDesigning Approximately Optimal Search onMatching Platforms
Nicole Immorlica
Microsoft Research, New York, NY, [email protected]
Brendan Lucier
Microsoft Research, Cambridge, MA, [email protected]
Vahideh Manshadi
Yale School of Management, New Haven, CT, [email protected]
Alexander Wei
UC Berkeley, Berkeley, CA, [email protected]
We study the design of a decentralized two-sided matching market in which agents’ search is guided bythe platform. There are finitely many agent types, each with (potentially random) preferences drawn fromknown type-specific distributions. Equipped with such distributional knowledge, the platform guides thesearch process by determining the meeting rate between each pair of types from the two sides. Focusing onsymmetric pairwise preferences in a continuum model, we first characterize the unique stationary equilibriumthat arises given a feasible set of meeting rates. We then introduce the platform’s optimal directed searchproblem, which involves optimizing meeting rates to maximize equilibrium social welfare. We first showthat incentive issues arising from congestion and cannibalization makes the design problem fairly intricate.Nonetheless, we develop an efficiently computable solution whose corresponding equilibrium achieves atleast 1 / NP -hard to approximate beyond a certain constant factor.
1. Introduction
In recent years, matching platforms have played an increasingly important role in facilitating socialand economic connections: Thirty percent of U.S. adults have used dating platforms to find a partner[22], and 35% of U.S. workers have partaken in some form of freelance labor in the past year, in partthanks to online platforms that have made finding freelance work easier [21]. Platforms, such ase-Harmony, try to improve search by learning users’ characteristics and building match compatibilitymodels which are then used to make matching recommendations. While carefully designed modelshave predictive power, a pair’s compatibility often involves an idiosyncratic component that canonly be discovered upon meeting. Further, since matching requires “coincidence of wants”, the a r X i v : . [ c s . G T ] F e b Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search platform cannot dictate a match even with perfect knowledge of preferences. In such environments,how can matching platforms improve the search process while respecting users’ incentives?Toward answering the above question, we construct a dynamic matching market between twoheterogeneous populations, each composed of finitely many types. To be concrete, we take a datingmarket for heterosexual couples as our base example and thus refer to agents of the two sides aswomen and men. An agent’s type captures their common features observable to the platform. However, in addition to the type-specific component, each agent’s preference for any opposite-sideagent has an idiosyncratic component that is unknown a priori and is only revealed when thepair meets. In our model, agents have cardinal preferences, and each woman-man pair shares asymmetric valuation that is drawn from a distribution corresponding to their types. Symmetricpreferences capture the notion of mutual compatibility in a relationship, i.e., any relationship iseither win-win or lose-lose. Upon meeting, both agents observe their shared valuation; each thendecides whether to match or to wait for another candidate. We consider a continuum model whereagents of different types arrive at exogenous rates and leave the market either upon meeting asatisfactory match or unmatched due to a life event which occurs at a given rate (see Section 2.1and Figure 1). Such an exogenous departure rate captures search friction in the sense that an agentcan only meet a limited number of candidates before leaving unmatched due to a life event.With knowledge of preference distributions and exogenous arrival/unmatched departure rates, theplatform guides the search process by designing meeting rates between pairs of woman-man types.The set of meeting rates for a given type can be viewed as an assortment of opposite-side typesoffered to them over time. Faced with such an assortment, an agent decides on which candidatesto accept or reject in order to maximize their long-run utility. We focus on the stationary andsymmetric equilibria of the underlying dynamic game: We show for any set of “feasible” meetingrates that respect natural physical constraints (see eqs. (2) and (3)), there exists a unique suchequilibrium in which each agent type plays a threshold strategy (see Proposition 3.1). Establishingthe uniqueness of the stationary equilibrium relies on the symmetric structure of preferences. Infact, in our constructive proof, we show that iterating the best response map converges after finitelymany rounds (see Proposition A.4). Uniqueness of equilibrium along with its simple structureenables us to sidestep issues of instability and equilibrium selection which may make the platform’sdesign problem ill-defined. As such, our structural results may be of independent interest in futurework on the design of dynamic matching markets. We remark that platforms such as e-Harmony extract these features using surveys; however, we abstract away fromsuch details and assume that these features are directly observable. Such preference structures have been studied previously in matching literature; see e.g., Ashlagi et al. [6] and Kanoriaand Saban [17]. We highlight that several previous papers considered symmetric preference structures in static settings under othernames such as correlated two-sided or acyclic markets [2, 1]. mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search We remark that by taking such a design approach to search, our work departs from the prevailingsearch environment studied in the literature [8, 24, 3, 18], which assumes agents meet uniformly atrandom. In the presence of sufficient differentiation across types, such a “hands-off” approach tosearch can be arbitrarily sub-optimal: If the platform already knows that sports fans only marrysports fans and outdoor enthusiasts only marry outdoor enthusiasts, it should not waste time byletting outdoor enthusiasts meet sports fans. (For a quantitative example, see Example B.1 inSection B.) At the same time, the optimal meeting design can be far more nuanced than simplyletting preferred pairs meet each other. This is due to disparities in arrival rates combined withstrategic behavior of agents, which can lead to issues of cannibalization and congestion. To alleviatethese issues, sometimes the platform may wish to restrict the choices of a type in order to induce amatching outcome with globally higher welfare. To illustrate these behaviors, in Section 3.3, wepresent an instance of an assortative market with deterministic preferences for which the optimalmeeting design is indeed non-assortative.The aforementioned behaviors suggest that optimal design can be fairly subtle. In fact, weshow that the platform’s optimal directed search problem (formally introduced in Section 3.2)is
APX -hard, i.e., NP -hard to approximate beyond a fixed constant factor (see Theorem 4.2).Nonetheless, we develop a solution (i.e., a set of feasible meeting rates) whose equilibrium welfareis at least 1 / Our work relates to and contributes to several streams of literature on matching markets.
Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Search and Matching.
There is a rich literature in economics that studies decentralized matchingmodels with search frictions and transferable/nontransferable utility. For an informative review, werefer the interested reader to Chade et al. [9]. By and large, this literature focuses on uniformlyrandom meeting among agents and studies the properties of the resulting stationary equilibria.Under a “cloning” assumption—which keeps the distribution of agents unchanged by assertingthat each time two agents match and leave, two single agents identical to them join—Adachi[3] shows that as search friction vanishes, the set of stationary equilibria converges to the set ofstable matching of the corresponding centralized market. Relaxing such a cloning assumption, thusallowing for the agent distribution to be endogenously determined, Lauermann and N¨oldeke [18]show that stationary equilibria converge to stable matchings if and only if there exists a uniquestable matching in a corresponding discrete market. We complement this literature by takinga design approach—motivated by online matching platforms—to optimize directed search (i.e.,non-uniform and type-specific random meetings) in a model with cardinal and symmetric preferences.We highlight that like Lauermann and N¨oldeke [18], in our model, the stationary distribution ofagents is determined endogenously.
Directed Search and Platform Interventions.
Moving beyond random meeting, a series of recentpapers has studied different forms of platform design and intervention to facilitate search. A recentwork of Kanoria and Saban [17] shows limiting the action of agents, e.g., allowing for only one side topropose, and hiding information about the quality of some agents can lead to welfare improvement.In another direction, Halaburda et al. [14] focus on the impact of limiting choice and shows thatwhen agents’ outside options are heterogeneous, a platform which offers limited choice and chargesa higher price can still compete with ones without choice restriction. Compared to these papers,our model and our intervention is more “fine grained,” in that both sides have multiple types, andwe design type-specific meeting rates. The approach of Banerjee et al. [7] in designing visibilitygraphs in a two-sided market with different types of buyers and sellers bears some similarity withours. However, there are several key differences: We consider a matching market without transfersin which the platform directed search design impacts agents’ acceptance thresholds on both sides,while Banerjee et al. [7] focus on a network of buyers and sellers exchanging a single undifferentiatedgood in which the platform impacts clearing prices by choosing a visibility subgraph.Motivated by the role of matching platforms in shaping agent’s choice, a sequence of recentpapers [5, 23, 4] consider the problem of assortment planning in two-sided matching in both staticand dynamic settings. In this novel line of work, agents are non-strategic in that their behavior is We remark that the set of actions of an agent in Kanoria and Saban [17] is richer in that in addition to deciding onwhether to match upon meeting, an agent decides on whether to request meeting a candidate and whether to inspectthat candidate (as a cost). mmorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search captured by a choice model such as the Multinomial Logit model. Consequently, agents are obliviousto the action of those on the other side and the platform design. While sharing a similar motivationwith this emerging literature, our work complements it by capturing agents’ strategic behavior inresponse to the platform design in guiding the search process. In our work, the set of meeting ratesfor a type can be viewed as an assortment of options that the platform (sequentially) offers to anagent type. The equilibrium response to the collection of such assortments determines the matchingand welfare outcome. Matching with Incomplete Information.
Outside the framework of search, several papers explorethe problem of finding a stable matching where preferences are a priori unknown. Taking acommunication complexity approach, Gonczarowski et al. [13] and Ashlagi et al. [6] establish boundson the the amount of communication, measured by the number of bits, needed to find a stablematch in markets with private preferences. The recent work of Immorlica et al. [16] focuses on asetting where learning preferences is costly and show how costly information acquisition impacts anagent’s preference. Further, a few recent papers, such as Liu et al. [20], use the multi-armed banditframework to model the process of learning preferences as an online learning problem and developefficient learning algorithms. Finally, in another direction, Emamjomeh-Zadeh et al. [12] analyze aniterative query process for learning a stable matching under general preferences.
2. Model
In this section, we introduce a search model in a dynamic, two-sided matching market, where thesearch process is facilitated by recommendations from a centralized platform. A graphical depictionof our model, focused on the dynamics around a single type, is given in Figure 1.
We model our matching market as a flow economy, where the market consists of a continuum ofagents of infinitesimal mass. Agents on the two sides are differentiated into finitely many typesbelonging to the sets M and W , respectively. Agents’ types determine prior distributions over theirpreferences (see Section 2.2). We use Θ = M ∪ W to denote the combined type space of all agents.For each type θ ∈ Θ, we denote the population mass of type θ agents present on the platform by η θ .Throughout, we will use the notational convetion that θ (cid:48) refers to a generic type on the oppositeside of the market from type θ .Our matching market takes place in continuous time. To track change in the population masses η θ over time, we use flow rates —the mass of agents arriving or departing per unit time. Weassume entry into the market is fixed exogenously, with type θ agents entering at a constant flow Although η θ can in general vary over time, we suppress the time index on η θ for notational compactness and becauseour eventual focus will be on stationary equilibria, in which case η θ will be constant over time. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Figure 1 This schematic depicts the dynamics for a single type θ . Each dark gray box represents a type, withthe size corresponding to the type’s population mass. The light gray connections between boxes correspond tohow agents meet under directed search. The colored lines depict inflow and outflow for type θ , with widthcorresponding to flow rate: Blue for arrivals, green for matching, and red for leaving unmatched. In stationaryequilibrium, the inflow should balance the outflow (as is depicted). η θ η θ (cid:48) η θ (cid:48) α θ ξ θ · η θ δ · η θ rate of α θ (i.e., an α θ mass of type θ agents enters per unit time). This corresponds to the blueinflow in Figure 1. On the other hand, we let departure from the market be partially determinedendogenously, with all agents eventually either leaving with a match or experiencing an exogenous“life event” that causes them to leave unmatched. More specifically, the two ways an agent candepart are as follows: Matching.
Agents leave the market when they enter into a mutually agreed upon match withanother agent (see the green outflow in Figure 1). We define ξ θ so that ξ θ η θ is the flow rate atwhich type θ agents do so. That is, each individual of type θ (assuming symmetry betweenagents of type θ ) leaves matched with probability ξ θ dt during the infinitesimally small timeinterval [ t, t + dt ). The value of ξ θ is determined both by how the platform recommends potentialmatches and who the agents themselves choose to match with (see Lemma 3.2). Life event.
Agents also leave unmatched when they experience a “life event” (see the red outflowin Figure 1). We assume life events occur randomly for each individual at a constant rate of δ dt .That is, each individual experiences a life event with probability δ dt during each infinitesimallysmall time interval [ t, t + dt ). (Equivalently, the time an agent spends in the market is drawn(unknown to them) from an exponential distribution of rate δ .) At the individual level, thepossibility of leaving unmatched acts as a discount factor and ensures that agents do not searchforever. At the platform level, individuals leave at a total flow rate of δη θ , which ensures thatthe mass of unmatched agents does not grow unboundedly. Note that ξ θ can also vary over time; we suppress the time index as we did for η θ . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search Our focus will be on the stationary equilibria of this market, where the inflow of arriving agentsbalances the outflow of departing agents. We say that stationarity holds if α θ = ( δ + ξ θ ) η θ (1)for all θ ∈ Θ. Note that δ + ξ θ is the total rate at which individual type θ agents exit the market. We assume that agents have cardinal preferences (i.e., a utility associated to matching with eachpotential partner) and normalize the value of leaving unmatched to 0. In our model, the utility valuesof specific potential partners are unknown ex ante. Instead, pairs of agents learn their valuations foreach other when they meet while searching (see Section 2.3), with their utilities being drawn froma prior distribution given by their types. These priors represent the fact that agents may haveidiosyncratic preferences—beauty being in the eye of the beholder—that are not captured by theirtype. We assume that these priors are common knowledge among the platform and the agents.More formally, the utilities that two agents of types m ∈ M and w ∈ W derive if they matchare drawn from some joint distribution D mw . Because we use D mw to capture idiosyncrasies inpreferences, we assume that utilities are drawn independently from D mw for each such pair of agents.To model mutual compatibility, we also assume agents have symmetric valuations , meaning that iftwo agents match, then each agent derives the same utility u ∈ R from the match.Under the symmetric valuations assumption, each joint distribution D mw can be thought of asa distribution over R with cumulative distribution function F mw . To avoid a technical discussionof tiebreaking we make the mild assumption that D mw is a continuous distribution for all ( m, w ).Then F mw is absolutely continuous and F mw ( τ ) = Pr( u ≤ τ ) for u ∼ D mw . Finally, to make ournotation for distributions symmetric, we define D wm := D mw and F wm := F mw for all ( m, w ). A key feature of our model is that search is mediated by a platform. At a high level, the platformdirects search by picking for each agent type θ the rates at which they “meet” other agent typeswhile searching. For each agent, search consists of a sequence of meetings with potential partnersuntil either a match is formed or they leave unmatched. When two agents meet, each must decidewhether to accept or reject the match. If both parties accept (i.e., there is a double coincidence of We assume that agents do not meet twice, since for any agent, there are uncountably many agents of each type thatthey could meet. While all of our arguments work with discrete distributions, one has to take care to define tiebreaking properly.On the other hand, note that any non-continuous distribution can be approximated by a continuous distribution byadding a small amount of (bounded) noise.
Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search wants), then they match with each other and leave the market. Otherwise, both agents remain inthe market and continue their respective searches.At the individual level, the frequency at which an agent encounters other agents of any giventype follows a Poisson process with rate given by the platform’s directed search policy. Upon twoagents of types θ and θ (cid:48) meeting, their valuation u for the match is drawn from D θθ (cid:48) and revealedto them. Each agent then decides whether to accept or reject the match, with the match formingif and only if both accept. If the agents match, each leaves the market with utility u ; otherwise,each continues their search. We assume agents are rate-limited and normalize time so that agentscan meet at most one candidate per unit time on average. That is, we require the total frequencyof meetings for any agent to follow a Poisson process with at most unit rate—this is our capacityconstraint .At the platform level, directed search can be thought of as some mass f mw of type m and type w agents, for each pair of types ( m, w ), meeting each other per unit time. In particular, equal masses of type m and type w agents should meet each other over any time interval. Note that thisproperty must be satisfied if we were to explicitly pair agents in a discrete model. While our modelis continuous, we can still imagine the platform “pairing” equal masses of randomly sampled agentsfor meetings at each instant of time. By requiring that agent-level Poisson processes be consistentwith this fact, we obtain our flow balancing constraint.We now state the directed search model sketched above in more formal terms. For any θ ∈ Θ,define an assortment to be a collection of rates λ θ ( θ (cid:48) ) ≥ θ (cid:48) on the opposite side suchthat type θ agents meet type θ (cid:48) agents following a Poisson process with rate λ θ ( θ (cid:48) ). The platform,as part of its design, chooses an assortment λ θ for each type θ ∈ Θ. (Note that we implicitly restrictthe platform to designs that are anonymous and independent of agents’ histories.)A set of assortments { λ θ } θ ∈ Θ is feasible if it satisfies the following two sets of constraints: Capacity.
The total rate of meetings for type θ agents is bounded by 1 for all types θ ∈ Θ: (cid:88) θ (cid:48) λ θ ( θ (cid:48) ) ≤ . (2)(Note that (cid:80) θ (cid:48) λ θ ( θ (cid:48) ) is the total rate because merging Poisson processes sums their rates.) Flow balance.
For all pairs ( m, w ) ∈ M × W , the mass of type w agents met by type m agentsper unit time equals the mass of type m agents met by type w agents per unit time. By anexact law of large numbers [25, 11] , we can write this condition as f mw := η m λ m ( w ) = η w λ w ( m ) . (3) As in the related literature on search and matching, we formally require a continuous-time exact law of large numbersfor random matching; such a result has only recently been developed rigorously for discrete-time settings by Duffieet al. [11]. mmorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search That is, the mass f mw of type m and w agents that meet per unit time is well-defined. We assume the platform can direct search according to any feasible set of assortments.
3. The Platform’s Design Problem
In this section, we introduce the platform’s design problem, namely:
How should the platformsuggest meetings in directed search to maximize social welfare in equilibrium?
In order to formallydefine the problem, we first describe the agents’ strategic decision-making problem and brieflycharacterize the equilibria that arise.
Any set { λ θ } θ ∈ Θ of assortments defines a game for the agents, in which agents strategically decidewhich potential partners to accept and reject in their meetings. As our focus will be on stationaryequilibria, we assume agents play time- and history-independent strategies. We further assume thatstrategy profiles are symmetric within each type. We can thus write the strategy of type θ agentsas a function σ θ ( θ (cid:48) , u ) taking values in [0 , θ agentaccepts when meeting a type θ (cid:48) agent whom they value at utility u .A property of our setup is that no individual agent can directly influence the behavior of otheragents, as agents are only affected by the action profiles of types in aggregate . Thus, rather thanmodeling agents as playing a full-fledged dynamic game, we can think of each agent as facing aMarkov decision process (MDP) derived from their assortment λ θ and strategies σ θ (cid:48) of the agentson the opposite side. Then, a strategy σ θ is a best response for type θ agents if and only if it is anoptimal policy for this MDP. It follows that a strategy profile { σ θ } θ ∈ Θ is a Nash equilibrium if andonly if the strategy of each type is an optimal policy for the MDP given by their assortment andopposite side’s strategies.In Section A.1, we formally describe the agents’ MDPs and show that any Nash equilibrium in“non-dominated” strategies is equivalent to one in threshold strategies , i.e., strategies σ θ for whichthere exists a threshold τ θ such that σ θ ( θ (cid:48) , u ) = 1 if u ≥ τ θ and σ θ ( θ (cid:48) , u ) = 0 if u < τ θ . In fact,each threshold τ θ in this equilibrium equals the expected utility u θ of type θ agents upon entry.(Note that u θ depends on type θ ’s strategy, the other agents’ strategies, as well as the platform’sassortments—see Lemma 3.3.) Furthermore, we show in Section A.3 that the equilibrium prediction One way to implement meetings would be to have the platform randomly sample a subset of each type to meet someother type at each instant of time. With uniform sampling, individual agents will experience meetings at a Poissonrate. Note that agents are not directly affected by the actions of other agents on the same side. In our analysis, we eliminate dominated strategies that reject matches worth more than one’s expected continuationutility. Performing such a pruning is necessary to rule out degenerate equilibria (e.g., the one where agents reject all oftheir potential matches because they do not expect anyone to ever accept). Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search of our model is unique in the following sense: For any set of assortments, our model makes a uniqueprediction of the distribution of matches that are realized. We record these observations in thefollowing proposition:
Proposition 3.1 (Structure of Nash Equilibria) . Given any fixed set of assortments { λ θ } θ ∈ Θ : There exists a unique strategy profile where each type plays a threshold strategy with thresholdequal to that type’s expected utility. Moreover, this strategy profile is a Nash equilibrium. Any Nash equilibrium in non-dominated strategies produces the same distribution (in both typesand utilities) of realized matches as the preceding Nash equilibrium in threshold strategies.
In light of Proposition 3.1, it suffices to focus our attention on equilibria in threshold strategies.To facilitate our discussion of such equilibria, we state expressions for the rate ξ θ at which agentsmatch and their expected utility u θ when all agents play threshold strategies. Lemma 3.2 (Matching Rate) . Suppose each type θ ∈ Θ plays a threshold strategy with threshold τ θ .Then the rate ξ θ at which individual type θ agents match is ξ θ = (cid:88) θ (cid:48) (cid:32) λ θ ( θ (cid:48) ) (cid:90) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) (cid:33) . (4) Proof.
The rate at which an agent of type θ matches is the sum of the rates at which they meeteach type θ (cid:48) weighted by the probability such a meeting results in a match. The latter probabilityis 1 − F θθ (cid:48) (max( τ θ , τ θ (cid:48) )) = (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) in terms of the agents’ thresholds τ θ and τ θ (cid:48) . Lemma 3.3 (Expected Utility) . Suppose each type θ plays a threshold strategy with threshold τ θ .Then the expected payoff u θ of type θ agents is u θ = (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) u dF θθ (cid:48) (cid:17) δ + (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) (cid:17) . (5) (Note that by Lemma 3.2, the denominator can also be written as δ + ξ θ .) While the proof of Lemma 3.3 depends on the formal definition of the MDP, the intuition for theformula is simple: For any type θ agent, the memorylessness of the MDP implies their expectedutility equals their expected utility conditioned on leaving at any particular instant. Conditioningon such an event, we see that with probability proportional to δ , they left due to a life event,and with probability proportional to λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) F θθ (cid:48) , they left because they matched with anagent of type θ (cid:48) . In equation (5), we simply express the agent’s expected utility as the sum of theirexpected utilities in each of these scenarios weighted by probability of occurrence. mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search The platform’s design problem is to design the search process via assortments in order to induce astationary equilibrium that is (approximately) optimal in terms of social welfare. Thus, in additionto designing the via assortments, the platform should ensure that those assortments are sustainablein equilibrium. When analyzing the design problem, we assume that the platform has access tothe discount rate δ , the arrival rates α θ for all θ ∈ Θ, and the preference distributions D mw for all( m, w ) ∈ M × W .More formally, we say that a set of assortments { λ θ } θ ∈ Θ induces a stationary equilibrium if, forthe equilibrium strategy profile { σ θ } θ ∈ Θ , there exist population masses { η θ } θ ∈ Θ such that:1. Stationarity (as defined in eq. (1)) holds for all types θ ∈ Θ.2. The assortments are feasible with respect to the population masses (i.e., (2) and (3) hold).By Proposition 3.1, a set of assortments { λ θ } θ ∈ Θ implies a unique Nash equilibrium in thresholdstrategies. Furthermore, these threshold strategies determine ξ θ for each θ by Lemma 3.2. Given { ξ θ } θ ∈ Θ , the stationarity condition then determines η θ for each θ . Therefore, any choice of assortments { λ θ } θ ∈ Θ induces at most one stationary equilibrium, making the platform’s optimization problemover the assortments well-defined. Recall that the platform’s optimization objective is social welfare. For stationary equilibria, socialwelfare is defined as the total flow rate of utility realized by agents matching with each other. Notethat the symmetric valuations assumption implies each of the two sides contributes exactly half ofthe total welfare. The platform’s optimization problem is thus to find the assortments { λ θ } θ ∈ Θ thatinduce a stationary equilibrium with maximum welfare.Our setup naturally lets us cast the platform’s design problem of finding a welfare-maximizingpolicy for directed search as a computational one: Given inputs δ , { α θ } θ ∈ Θ , and {D mw } ( m,w ) ∈M×W , compute the assortments { λ θ } θ ∈ Θ that induce a stationary equilibrium with maximum welfare. Wename this problem OptimalDirectedSearch . We remark that we do not dwell on dynamical issues beyond steady state (e.g., convergence toequilibrium from “cold starts”). Instead, we assume that the platform has the power to manipulateentry into the market at time 0 and can ensure that the requisite populations of agents are presentfor the desired equilibrium to sustain indefinitely. This motivates our objective for the platform offinding a welfare-maximizing stationary equilibrium. Since our model uniquely predicts play given assortments, stationary equilibria are also self-sustaining onceestablished. To formalize our computational problem while avoiding details relating numerical computation, we assume thatthe platform’s knowledge of the preference distributions D mw comes in the form of oracle access to the CDFs F mw and the tail expectations (cid:82) ∞ τ u dF mw . We further assume that the platform can solve for each agent’s best responseoptimally. This is a mild assumption because the corresponding optimization problem boils down to a ternary searchon a unimodal function of the aforementioned oracle’s outputs (see Lemma 4.4 and its generalization Lemma C.1 inSection A). Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Stationarity, feasibility,and agents playing best responses can all be encoded (albeit indirectly) as constraints on theassortments. We can thus write
OptimalDirectedSearch as the following constrained optimizationproblem: max λ θ ,τ θ ,ξ θ ,η θ (cid:88) m ∈M (cid:88) w ∈W (cid:18) η m λ m ( w ) (cid:90) max( τ m ,τ w ) u dF mw (cid:19) + (cid:88) w ∈W (cid:88) m ∈M (cid:18) η w λ w ( m ) (cid:90) max( τ m ,τ w ) u dF mw (cid:19) (6)such that α θ = ( δ + ξ θ ) η θ ∀ θ ∈ Θ (6a) η m λ m ( w ) = η w λ w ( m ) ∀ ( m, w ) ∈ M × W (6b)1 ≥ (cid:88) θ (cid:48) λ θ ( θ (cid:48) ) ∀ θ ∈ Θ (6c) ξ θ = (cid:88) θ (cid:48) (cid:32) λ θ ( θ (cid:48) ) (cid:90) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) (cid:33) ∀ θ ∈ Θ (6d) τ θ = 1 δ + ξ θ (cid:88) θ (cid:48) (cid:32) λ θ ( θ (cid:48) ) (cid:90) ∞ max( τ θ ,τ θ (cid:48) ) u dF θθ (cid:48) (cid:33) ∀ θ ∈ Θ (6e) λ m ( w ) , λ w ( m ) ≥ ∀ ( m, w ) ∈ M × W . (6f)Here, constraint (6a) is the stationarity constraint (1); constraints (6b) and (6c) are the feasibilityconstraints of flow balance (3) and capacity (2); constraint (6d) is formula (4) for individuals’ rateof matching; constraint (6e) states that each type’s threshold is a fixed point of their MDP payofffunction (5); and constraint (6f) simply requires that the meetings rates be non-negative.The validity of constraint (6e) follows from Proposition 3.1: Our model’s unique prediction forwhich matches are realized occurs when agents all play threshold strategies with threshold equalto their expected utility. Moreover, the profile of thresholds satisfying this property is uniquelydetermined by the assortments. Hence the social welfare of an assortment is given by the flow ofutility realized when all agents play such threshold strategies.We note that while the optimization program (6) is written as an optimization problem over foursets of variables { λ θ } θ ∈ Θ , { τ θ } θ ∈ Θ , { ξ θ } θ ∈ Θ , and { η θ } θ ∈ Θ , it is really an optimization problem overthe assortments { λ θ } θ ∈ Θ : Agents’ thresholds { τ θ } θ ∈ Θ and matching rates { ξ θ } θ ∈ Θ are determinedentirely by their assortments, and the stationary masses { η θ } θ ∈ Θ are in turn determined entirely bythe matching rates. With our setup of the model and the platform’s design problem now complete, we present as anexample a market where agents have deterministic and strictly vertical preferences. Through thisexample, we illustrate some subtleties involving congestion and cannibalization that arise (even inthis very restricted setting) when solving
OptimalDirectedSearch . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search In our vertically structured market, agents on each side of the market have a common preferenceordering over the types on the opposite side. In such a market, one might expect that the optimaldirected search scheme results in a positively assortative matching when agents’ valuations aresupermodular. However, this turns out not to always be the case. Because agents arrive at disparaterates, matching assortatively can lead to inefficiency due to congestion at the top of the market,and due to “high type” agents cannibalizing demand for “low type” agents. While such phenomenamake finding the exact optimal assortments challenging, we will show in Section 4 that we cannonetheless find a set of assortments that are approximately optimal.
Example 3.4 (A Vertical Market) . Consider a market with strictly vertical preferences, where theagents on each side belong to either a high type H or a low type L . That is, we let M = { m H , m L } and W = { w H , w L } . Suppose the utility distributions F m H w H , F m L w H , F m H w L , and F m L w L are pointmasses at u (1 + 2 ε ), u , 1 + ε , and 1, respectively, for u (cid:29) ε (cid:28) Thus, agents always prefermatching with high-type agents over low-type agents. Suppose further that the arrival rates aresuch that α m H = α w L = 1 and α w H = α m L = 1 /u . Finally, we consider this example in the regimewhere δ is very small (i.e., suppose we are in a nearly frictionless market).To analyze the possible stationary equilibria of this market, we first consider what happens in theequilibria where high-types only match with high-types and low-types only match with with low-types.The maximum possible welfare in such a equilibrium is bounded above by 2((1 + 2 ε ) + α m L ) / (1 + δ ) ≈
2, since the welfare generated by the high-types is at most 2 · α w H · u (1 + 2 ε ) / (1 + δ ) and the utilitygenerated by the low-types is nearly negligible at 2 · α m L · / (1 + δ ). In particular, matching forboth high- and low-types is bottlenecked by the low arrival rate of the less common type. One canthink of these equilibria as being congested at both levels of the market: Due to the disparities inarrival rates, there are many type m H agents hoping to match with only a few type w H agents andmany type w L agents hoping to match with only a few type m L agents. As a result, large numbersof type m H agents and type w L agents go unmatched.Next we observe that such equilibria—where high-types only match with high-types and low-typesonly match with low-types—arise if the expected utility u m H of type m H is greater than 1 + ε .Notably, type m H agents are not willing to match with type w L agents due to the possibility ofmatching with a type w H agent: If u m H > ε , then Proposition 3.1 tells us that type m H agentswill not match with type w L agents, since type m H agents would only obtain 1 + ε utility from doingso. From the platform’s perspective, the presence of w H agents in type m H ’s assortment cannibalizesdemand for type w L agents. Likewise, the possibility of matching with type m H makes type w H An astute reader may notice that, technically, our preference distributions do not satisfy our continuity assumption.We present our example this way for simplicity’s sake—the same example can be made to work with continuousdistributions by adding some small amount of noise to each utility value. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search reject type m L agents—the expected utility of type w H agents is at least (1 + ε ) · α m H /α w H > u from matching with type m H .On the other hand, we can obtain a more efficient equilibrium if the expected utility of type m H isslightly lower, e.g., if they only match with type w L . Consider the assortment where type m H and w L agents only meet each other and type w H agents and m L agents only meet each other. All agents meetat maximal rates; furthermore, all meetings result in matches because no type has other options. As aresult, the social welfare in equilibrium is 2( α m H · (1 + ε ) + α m L · u ) / (1 + δ ) = 2((1 + ε ) + 1) / (1 + δ ) ≈ m H agents only meet type w L agents and another subset of type m H agents only meet type w H agents. However, such treatmentmay be unpalatable to users from a fairness perspective; moreover, if agents are able to rejoin theplatform under a different identity, then such separation is also difficult to enforce. We also show inSection 4 that such an assumption is not restrictive—our solution is constant-factor competitiveagainst the first-best platform policy.
4. Approximation Algorithms
In this section, we show how the platform can construct in polynomial time a set of assortments { λ θ } θ ∈ Θ such that its induced equilibrium obtains a 4-approximation to the optimal social welfare.Furthermore, the equilibrium resulting from our computed assortments has a simple and appealingstructure: It consists of disjoint submarkets such that each submarket has only a single type on oneof the sides. We also establish hardness of approximation for the platform’s optimization problem,meaning that we cannot do better than a constant-factor approximation unless P = NP .We actually show a slightly stronger result: The resulting equilibrium obtains a 4-approximationto the optimal welfare that could be obtained in the first-best matching economy, where ratherthan agents choosing to accept or reject, the platform gets to decide on behalf of the agents. (Inthis first-best economy, the platform gets to choose the assortments as well.) Our theorem can thusalso be interpreted as a price-of-anarchy-style result, that agents’ self-interested behavior causeswelfare to degrade by at most a factor of 4 relative to the first-best optimum. Theorem 4.1 (Computationally Efficient Approximation) . There is a polynomial-time -approximation algorithm for OptimalDirectedSearch . (In fact, this algorithm gets a -approximationto the platform’s first-best optimal welfare.) mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search Our construction of these assortments and the resulting equilibrium proceeds as follows:
Solving for the platform’s first-best.
We start by considering the platform’s optimizationproblem for the first-best matching economy, where the platform gets to plan both the assort-ments and which matches the agents accept. That is, we relax the agents’ incentive constraintand allow the platform to optimize over thresholds τ mw between each pair ( m, w ) of types. Thisrelaxation removes the main barrier—namely, incentives—to solving the optimization problemexactly. Accordingly, we show that this relaxed problem can be reduced to a generalization ofthe classical assignment problem and thus be efficiently solved. Approximating via “star-shaped” markets.
We then show that an optimal solution to this“generalized” assignment problem can be converted into a stationary equilibrium (where wenow take agents’ incentives back into account) with at most a factor 4 loss in welfare. The keyideas for this step are: • Showing that the optimal first-best solution can be 2-approximated by another consistingof “star-shaped” submarkets (which have a side where all the agents are of the sametype). For this step, we draw on some classical observations about the structure of optimalsolutions from previous works on generalized assignment problems [19, 10, 7]. • Showing that for markets where all the agents are of the same type on one side, theplatform can obtain a 2-approximation to the optimal welfare of the first-best matchingeconomy. This step relies on the observation that in such markets, there is an alignmentof incentives between individual agents and the platform.A constant-factor approximation algorithm is likely the best we can hope for, computationally:We show in Section D that approximating the platform’s optimization problem to a factor betterthan is NP -hard. Theorem 4.2 (Hardness of Approximation) . Approximating
OptimalDirectedSearch up to a (cid:0) − ε (cid:1) -factor is NP -hard for any ε > . We prove the above hardness result in Section D. From a technical perspective, our approach veryclosely mirrors that of Chakrabarty and Goel [10], who show hardness of approximation for relatedproblems (e.g., maximum budgeted allocation) by reducing from Hastad’s 3-bit PCP [15]. Whiletheir technique applies to our setting, it is not clear that their hardness results apply directly—achallenge specific to our setting is that our optimization problem is continuous rather than discrete.To make this approach work, we must show that our complicated feasibility set introduces a discreteelement to the optimal allocation. But as a consequence, we obtain a slightly worse constant c .In the remainder of this section, we develop the proof of Theorem 4.1. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
In this section, we study the platform’s first-best optimization problem, where the platform has thepower to plan the entire matching economy. That is, in addition to designing assortments, we allowthe platform to ignore agents’ incentives and dictate each type’s strategy. The first-best relaxation isobtained by modifying the optimization problem (6) as follows: Relax the best response constraint(6e). Then, replace the per-type threshold variables τ θ with pairwise threshold variables τ mw thattake the place of max( τ m , τ w ) in all expressions for all ( m, w ) ∈ M × W in (6).To see why this relaxation corresponds to the platform’s first-best problem, note that rather thanoptimizing over all agent strategies, the platform can restrict its attention to strategies σ θ that havea threshold for each type on the opposite side. This is because whenever two agents of types m and w match at a certain rate, it is optimal for the matches realized to be the highest-valued ones. Inparticular, the platform should never hope that agents of types m and w reject each other at utility u but match at some utility u (cid:48) < u . Rather, there should exist a threshold τ mw such that agents ofthese types match if and only if u ≥ τ mw .While even the relaxed optimization problem looks difficult to work with, our goal in this sectionis to show that a careful reparametrization reduces the optimization problem to a linear program.The resulting linear program is related to (and in fact generalizes) the linear programming relaxationof the assignment problem; the main difference is that instead of having unit capacities on the flowto each node, we allow for real-valued capacities. Formally, the reduction can be stated as follows: Proposition 4.3 (First-Best “Assignment Problem”) . Consider the linear program max β mw · (cid:88) m ∈M (cid:88) w ∈W ρ mw · β mw (7) such that (cid:88) w ∈W β mw ≤ α m ∀ m ∈ M (7a) (cid:88) m ∈M β mw ≤ α w ∀ w ∈ W , (7b) where ρ mw := max τ ≥ (cid:82) ∞ τ u dF mw δ + (cid:82) ∞ τ dF mw . The platform’s first-best optimization problem is equivalent to (7) in the following sense: The two optimization problems have the same optimal objective value. Any feasible choice of { β mw } m ∈M ,w ∈W corresponds to a feasible choice of { λ θ } θ ∈ Θ , { η θ } θ ∈ Θ , { ξ θ } θ ∈ Θ , and { τ mw } m ∈M ,w ∈W that achieves the same objective value. Proposition 4.3 implies that to solve the first-best optimization problem, it suffices to solve thelinear program (7) and convert the resulting solution to an optimal choice of variables for the mmorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search first-best optimization problem. The proof of the Proposition 4.3 proceeds by first reparameterizingthe optimization program (6) using the flow balance constraint: We introduce the variable f mw := η m λ m ( w ) = η w λ w ( m )for the total rate of meetings between each pair ( m, w ) ∈ M × W . We can then encode the flowbalance, capacity, and stationary constraints into the single combined constraint α θ ≥ (cid:88) θ (cid:48) (cid:32) f θθ (cid:48) (cid:32) δ + (cid:90) ∞ τ θθ (cid:48) dF θθ (cid:48) (cid:33)(cid:33) . (8)That is, there exist corresponding values for λ θ ( θ (cid:48) ), ξ θ , and η θ that satisfy the constraints of (6) ifand only if the f θθ (cid:48) variables satisfy (8). To finish, we introduce the variables β mw := f mw (cid:18) δ + (cid:90) ∞ τ mw dF mw (cid:19) . (9)We then substitute into (8) and the objective of (6). The latter becomes2 (cid:88) m ∈M (cid:88) w ∈W (cid:32) β mw · (cid:82) ∞ τ mw u dF mw δ + (cid:82) ∞ τ mw dF mw (cid:33) . (10)While the thresholds τ mw are not yet fixed, these substitutions make the optimization effectivelyunconstrained in τ mw . Optimizing over τ mw in the objective lets us write the objective in terms of ρ mw as in (7), since the only dependence on τ mw is through the expression (cid:82) ∞ τ mw u dF mw δ + (cid:82) ∞ τ mw dF mw (11)We defer full details of this analysis to Section C.The final step in the above is setting τ mw so that expression (11) is maximized. It turns out thatwe can do so by setting τ mw = ρ mw . (That is, the expression achieves its maximum value at a fixedpoint.) We record this observation here: Lemma 4.4 (Fixed Point Structure of Optimal Threshold) . Define A ( τ ) = (cid:82) ∞ τ u dFδ + (cid:82) ∞ τ dF , where δ > and F is a continuous distribution, and let ρ = max τ ≥ A ( τ ) . Then A ( ρ ) = ρ . Moreover, A is monotonically increasing for τ ≤ ρ and monotonically decreasing for τ ≥ ρ . Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Figure 2 This figure depicts the process for converting an optimal first-best solution to a stationary equilibrium,while preserving welfare up to a factor of . We start with a first-best solution whose graph of positive weightvariables forms a forest (i.e., step (1)). The first arrow represents step (2), where we approximate the forest withstars, each corresponding to a star-shaped submarket. The second arrow represents solving the first-best for eachstar-shaped submarket (see Lemma 4.8). The third pair of arrows corresponds to the cases we handle in the proofof Proposition 4.7—in particular, whether the new first-best solution induces a stationary equilibrium (Case 1) ornot (Case 2). -approx re-solve each submarket 2-approx Case 1Case 2 ORfirst-best solution “star-shaped” submarkets first-best solution In the previous section, we described how the platform can solve for a first-best solution, whereagents’ incentives are ignored. In this section, we now bring back agents’ incentives and showhow to convert the first-best solution into a stationary equilibrium. We show that our conversionprocess retains at least of the welfare of the first-best solution, thereby giving us an efficient4-approximation algorithm for OptimalDirectedSearch .At a high-level, our conversion process works as follows (see Figure 2 for a visual depiction):
Step (1).
Find a solution { β ∗ mw } m ∈M ,w ∈W to the linear program (7) such that the graph on Θthat has an edge ( m, w ) for each β ∗ mw > Step (2).
Approximate this edge-weighted forest with a union of vertex-disjoint star graphs suchthat the star graphs contain at least half the total edge weight of the forest.
Step (3).
Construct a set of assortments for each star-shaped submarket (where one side hasonly one type of agent) that induces a stationary equilibrium whose social welfare is at leasthalf the first-best social welfare of the submarket.The first two of the above steps have appeared in other works featuring generalized assignmentproblems [19, 10, 7], where they were similarly used to simplify the problem structure. Our maininsight for this stage of the proof lies in the third step, where we leverage structural properties ofthe agents’ decision problem to show that the optimal stationary equilibrium in each submarketproduces at least half the first-best welfare of that submarket. These steps, together with Proposition 4.3, prove our approximation result Theorem 4.1. Note that this last step is distinct from simply computing the optimal stationary equilibrium. Computing theequilibrium is not enough to show that it actually obtains at least half the first-best welfare. mmorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search We now state the lemmas that give the results needed for steps(1) and (2). The first lemma, having appeared in various forms since Lenstra et al. [19], showsthat there exists an optimal solution for our generalized assignment problem such that the positivevariables β mw > Lemma 4.5 ([19, 10, 7]) . For a generalized assignment problem max (cid:88) m ∈M (cid:88) w ∈W ρ mw · β mw such that (cid:88) w ∈W β mw ≤ α m ∀ m ∈ M (cid:88) m ∈M β mw ≤ α w ∀ w ∈ W , with real-valued capacities α θ , there exists an optimal solution { β ∗ mw } m ∈M ,w ∈W such that the graphon Θ with edge set { ( m, w ) : β ∗ mw > } is a forest. The next lemma, which is the main ingredient for step (2), shows that we can take any edge-weighted forest and approximate it up to a factor of 2 in terms of edge weight with a disjoint unionof star graphs. The precise formulation we cite is due to Banerjee et al. [7], though simillar ideashave also appeared in the contexts of budgeted allocations [19, 10]. This lemma is simple to prove:Root each tree in the forest arbitrarily, and color each edge red or blue based on its distance modulo2 from its root. The colors partition the forest into two subgraphs each made up of star-shapedgraphs; one of these subgraphs must capture at least half the edge weight of the forest. Lemma 4.6 ([7]) . Given an edge-weighted forest, there exists a subgraph that is a union of vertex-disjoint star graphs (i.e., trees of radius ) such that the total edge weight of the subgraph is at leasthalf that of the original tree. In context, Lemma 4.5 tells us that we can find a solution { β ∗ mw } m ∈M ,w ∈W to the linear program(7) for the platform’s first-best optimization problem such that the positive weight variables form aforest on Θ. We weight the edges of this forest by their contribution β mw · ρ mw to social welfarein the first-best optimization problem. Next, Lemma 4.6 tells us that we can 2-approximate thisedge-weighted forest with a union of vertex-disjoint star graphs. Each of these star graphs defines a“star-shaped” submarket; note that each such submarket has a side where all agents are of the sametype. This completes steps (1) and (2). We remark that the constant factor on this lemma is tight: Consider the tree with 2 n + 1 vertices, where the rootvertex has n children and each of these children has another child. Then, any subgraph that is a vertex-disjoint unionof star graphs can only obtain edge weight n + 1. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
We now show that for any star-shaped submarket (i.e., one where one side ofthe market consists entirely of agents of the same type), the platform can design a set of assortmentsthat induce a stationary equilibrium with welfare at least half that of the platform’s first-best solution. Formally, we prove the following proposition:
Proposition 4.7 (Star-shaped Markets) . For any star-shaped market, the platform can chooseassortments that induce a stationary equilibrium whose welfare is a -approximation to the platform’sfirst-best welfare. For the proof of this proposition, a key observation is that the platform’s first-best thresholdcoincides with an agent’s optimal threshold if their assortment is saturated with agents of a singletype. In particular, such agents will not accept more than the platform would want them to in thefirst-best equilibrium. We obtain from this observation a condition for when the first-best solutioninduces a stationary equilibrium. We then show that the induced equilibrium achieves at leasthalf the first-best welfare. When the condition fails to hold, it turns out that we can modify thefirst-best solution appropriately, by only matching agents whose capacity constraint is tight, andstill obtain an equilibrium where at least half the first-best welfare is preserved.For the remainder of this section, we assume without loss of generality that M = { m } , i.e., ourstar-shaped market is such that the M side of the market consists only of type m agents. Structural Observations for Star-Shaped Markets.
Before giving the proof of Proposition 4.7, wedevelop some structural observations about the optimal first-best solution and agents’ thresholds.We defer the proofs of all lemmas in this section to Section C.3.In our first lemma, we characterize the platform’s first-best solution in star-shaped markets:
Lemma 4.8 (First-best Solution of a Star-Shaped Market) . There exists a first-best solution { β ∗ mw } w ∈W and a subset w , w , . . . , w n ⊆ W such that ρ mw ≥ ρ mw ≥ · · · ≥ ρ mw n , β ∗ mw i = α w i for all i < n , and β ∗ mw = 0 if w (cid:54)∈ { w , . . . , w n } . Now that we understand the platform’s first-best optimal solution, we can start to incorporateincentives through the incentive constraint (6e). By Lemma 4.4, each first-best threshold τ mw i inthe first-best solution described in Lemma 4.8 can be taken to be ρ mw i , since ρ mw i maximizes (cid:82) ∞ τ mwi u dF mw i δ + (cid:82) ∞ τ mwi dF mw i . Then, by the definition β mw i = f mw i (cid:16) δ + (cid:82) ∞ τ mwi dF mw i (cid:17) in (9), this first-best solution corresponds tothe choice of f mw i given by f mw i = f ∗ mw i := β ∗ mw i δ + (cid:82) ∞ ρ mwi dF mw i . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search Our next step is to consider incentives and characterize equilibrium when f mw i = f ∗ mw i . Fixing the f mw i values gives us information about equilibrium thresholds τ θ for all types because the incentiveconstraint (6e) is equivalent to τ θ = 1 α θ (cid:88) θ (cid:48) (cid:32) f θθ (cid:48) (cid:90) ∞ max( τ θ ,τ θ (cid:48) ) u dF θθ (cid:48) (cid:33) (12)after scaling both the numerator and the denominator of the right-hand side by η θ . (Recall thatthis fixed point definition of equilibrium play comes from Proposition 3.1, which lets us restrict ourattention to equilibria where each type has threshold equal to their expected utility.)As a point of comparison when analyzing agents’ equilibrium play, we consider the “idealizedthresholds” ˆ τ w i that arise when type w i has absolute market power. These idealized thresholds,which are given by the fixed point equationˆ τ w i = 1 α w i f ∗ mw i (cid:90) ∞ ˆ τ wi u dF mw i , can also be thought of as the threshold that type w i agents would set if τ m = 0. (Note that the left-hand side is monotonically increasing in ˆ τ w i while the right-hand side is monotonically decreasing,so the fixed point exists and is well-defined.) The reason we care about these idealized thresholds isbecause in equilibria where f mw i = f ∗ mw i , the threshold chosen by type m will be a best response tothese idealized thresholds. Namely, type m will play the threshold ˆ τ m which satisfiesˆ τ m = 1 α m n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max(ˆ τ m , ˆ τ wi ) u dF mw i (cid:33) . (13)To prove this claim, we will proceed through the following series of lemmas. These lemmas and thedefinition of ˆ τ m will also be useful for our later analysis.For the idealized thresholds ˆ τ w i , it is not difficult to establish using the definition of ρ mw i andthe monotonicity property in Lemma 4.4 that: Lemma 4.9 (Comparison to First-best Thresholds) . If i < n , then ˆ τ w i = ρ mw i (where ρ mw i isdefined in Proposition 4.3). And at i = n , it holds that ˆ τ w n ≤ ρ mw n . A similar argument shows that these ˆ τ w i upper bound the equilibrium thresholds τ w i for all i : Lemma 4.10 (Comparison to Equilibrium Thresholds) . In any equilibrium where f mw i = f ∗ mw i , τ w i ≤ ˆ τ w i for all i . While the thresholds of type w i are always upper bounded by ˆ τ w i , the quantity max( τ m , τ w i ) isalways at least ˆ τ w i : Intuitively, if τ m < ˆ τ w i , then type w i would set their threshold to be ˆ τ w i , sincethat would be optimal for them. This property, together with Lemma 4.9, guarantees that no type w i for i < n will match more than they would have under the platform’s first-best solution. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Lemma 4.11 (Lower Bound on Equilibrium Matching Thresholds) . In any equilibrium where f mw i = f ∗ mw i , max( τ m , τ w i ) ≥ ˆ τ w i for all i . From Lemmas 4.10 and 4.11, we deduce the threshold τ m played in equilibria where f mw i = f ∗ mw i for all i : These lemmas together imply that max( τ m , τ w i ) = max( τ m , ˆ τ w i ) in such equilibria. Recallthat (by Proposition 3.1) τ m must also satisfy τ m = 1 α m n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max( τ m ,τ wi ) u dF mw i (cid:33) = 1 α m n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max( τ m , ˆ τ wi ) u dF mw i (cid:33) . (14)This is exactly the definition of ˆ τ m as given in (13), so we may conclude τ m = ˆ τ m . Proving the Proposition.
With our structural lemmas, we are now ready to prove Proposition 4.7.We split our analysis into two cases based on the value of ˆ τ m . We show that if ˆ τ m > ρ mw n , thenthere exists a stationary equilibrium (with incentives) corresponding to the first-best f ∗ mw i values.Moreover, this equilibrium obtains at least half the welfare of the first-best optimal. Otherwise,if τ m ≤ ρ mw n , then we consider two further possibilities: If (cid:80) n − i =1 ρ mw i · α mw i is at least half thefirst-best optimal, then we only let type m agents meet type w i agents for i < n . Otherwise, weonly let type m agents meet type w n agents. In each of these two remaining cases, we will showthat we still capture at least half of the first-best optimal welfare. Proof of Proposition 4.7.
As mentioned above, we split our analysis into two cases based on thevalue of ˆ τ m . For both, we use OPT to refer to the welfare value attained by the platform’s first-bestoptimal solution, i.e.,
OPT = 2 (cid:80) ni =1 ρ mw i · β ∗ mw i . We now discuss the two cases: Case 1: ˆ τ m > ρ mw n . If ˆ τ m > ρ mw n , we show that the first-best choice of f ∗ mw i induces a stationaryequilibrium obtaining at least half the first-best optimal welfare.To check feasibility, we need to check that the { ξ θ } θ ∈ Θ and { η θ } θ ∈ Θ given by the thresholdsin the preceding lemmas satisfy the constraints of (6). By our analysis in Proposition 4.3, itis equivalent to check that the combined constraint (8) holds. By Lemmas 4.9 and 4.11, inequilibrium, we have max( τ m , τ w i ) ≥ ˆ τ w i = ρ mw i for all i < n and τ m = ˆ τ m , so max( τ m , τ w n ) ≥ ˆ τ m > ρ mw n . This implies all the thresholds in equilibrium are higher than they were in theplatform’s first-best solution. In other words, the corresponding matching rates ξ θ are no largerthan they were in the first-best solution. It follows that the combined constraint (8) can onlyhave more slack, meaning that a stationary equilibrium is indeed induced by the choice of f ∗ mw i and the thresholds discussed above.Our remaining task is to check that the realized welfare is sufficient by lower bounding ˆ τ m .The intuition for the following calculation is that if type m is rejecting potential matches ofvalue u , then the expected payoff of type m must be at least u . Indeed, we have:12 OPT = (cid:88) w ∈W ρ mw · β ∗ mw mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search = (cid:88) w ∈W (cid:18) f ∗ mw (cid:90) ∞ ρ mw u dF mw (cid:19) = n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max(ˆ τ m ,ρ mwi ) u dF mw i (cid:33) + n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) max(ˆ τ m ,ρ mwi ) ρ mwi u dF mw i (cid:33) ≤ n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max(ˆ τ m , ˆ τ wi ) u dF mw i (cid:33) + n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ ρ mwi ˆ τ m dF mw i (cid:33) ≤ α m · α m n (cid:88) i =1 (cid:32) f ∗ mw i (cid:90) ∞ max(ˆ τ m , ˆ τ wi ) u dF mw i (cid:33) + ˆ τ m n (cid:88) i =1 β ∗ mw i ≤ α m ˆ τ m . The last inequality follows from (13) and the constraint on the β ∗ mw from (7). Since 2 α m ˆ τ m isthe social welfare under the current allocation (where the factor of 2 comes from accountingfor the utilities of both sides), we have the desired lower bound on welfare relative to OPT . Case 2: ˆ τ m ≤ ρ mw n . When ˆ τ m ≤ ρ mw n , it is possible that the first-best choice of f ∗ mw i may notinduce a feasible equilibrium outcome, since type w n might accept more than the platformintended in equilibrium, resulting in the combined constraint (8) being violated for type m .To resolve this, we modify the first-best optimal solution in one of two ways by consideringtwo subcases: In the first subcase, we simply ignore type w n and match type m to types w , w , . . . , w n − . In the second subcase, we only match type m to type w n . At least one ofthese two subcases will obtain half the first-best optimal welfare. Subcase 2(a): (cid:80) n − i =1 ρ mw i · β ∗ mw i > OPT . In this subcase, we consider a modified versionof the first-best optimal solution from before, where β mw n is set to 0. We will show that theresulting equilibrium coincides with the first-best equilibrium for the submarket involvingonly type m and types w , . . . , w n − .First, note that such a modification incurs at most a OPT loss in first-best welfareby assumption. Next, recall that type m ’s threshold in equilibrium must satisfy the fixedpoint equation (13) for the f ∗ mw i values for i < n . It follows that the resulting τ m is at mostˆ τ m (e.g., via the argument for Lemma 4.10). Since ˆ τ m ≤ ρ mw n ≤ ρ mw i = ˆ τ w i for all i < n by Lemmas 4.8 and 4.9, we must have max( τ m , τ w i ) = ˆ τ w i by Lemmas 4.10 and 4.11. Thisshows that the resulting equilibrium is identical to the corresponding first-best solution. (Inparticular, it is feasible.) Hence the social welfare obtained is (cid:80) n − i =1 ρ mw i · β ∗ mw i > OPT . Subcase 2(b): ρ mw n · β ∗ mw n ≥ OPT . In this subcase, we consider the first-best solutionwith β mw = 0 for all w (cid:54) = w n , and set β mw = min( α m , α w n ). In this new “single-edged”market, where we only match types m and w n , the first-best social welfare will be at least Here, we clarify that ˆ τ m is as defined in (13) for the f ∗ mw i values for the unmodified first-best market. Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search OPT by assumption. In equilibrium, the thresholds must satisfy max( τ m , τ w n ) = ρ mw n by appropriate analogs of Lemmas 4.9 to 4.11. Thus, like in Subcase 2(a), the resultingequilibrium is both feasible and identical to that of the corresponding first-best solutionand its social welfare is at least OPT . Wrapping Up.
Theorem 4.1 now follows straightforwardly from our work up to this point. Sincethe total of the first-best welfares for the each of the “star-shaped” submarkets is at least half theplatform’s first-best welfare for the overall market and since Proposition 4.7 shows that at leasthalf of the first-best welfare for each submarket can be realized in stationary equilibrium, we obtainthe desired 4-approximation for Theorem 4.1.
5. Conclusion
Similar to shopping platforms, matching platforms also rely on recommendation systems to facilitatesearch by offering personalized assortments. However, the two-sided and decentralized nature ofthese markets makes the design of their recommendation systems fundamentally different from thoseused for product recommendation. Congestion and misaligned incentives often necessitates makingrecommendations that are sub-optimal for certain agents but improve the overall social welfare. Inthis work, we take a first step toward understanding the intricacies of designing recommendationsystems based on imperfect knowledge about preferences while taking agents’ strategic behaviorinto consideration. Somewhat surprisingly, we show that for general symmetric preferences, carefullydesigned assortments with very limited choices can achieve approximately optimal welfare.
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Designing Approximately Optimal Search
Appendix
A. Formal Definitions of MDP, Strategies, and Equilibrium
This section develops the mathematical formalization for the agents’ decision problem introduced in Section 3.1.We also prove Lemma 3.3 and Proposition 3.1 using the formalization developed here.
A.1. Solving for the Agent’s Best Response
Recall from Section 3.1 that any fixed set of assortments defines a game for the agents, in which agentsdecide who to accept and reject in their meetings. Recall also that agents play symmetric time- and history-independent strategies. We thus write the strategy of type θ agents as a function σ θ ( θ (cid:48) , u ) taking values in[0 ,
1] which specifies the probability that a type θ agent accepts when meeting a type θ (cid:48) agent whom theyvalue at utility u . Then σ θ ( θ (cid:48) , u ) · σ θ (cid:48) ( θ, u ) is the probability that two agents of types θ and θ (cid:48) who value eachother at utility u mutually accept. For measure-theoretic purposes, we restrict our attention to strategies σ θ that are measurable as a function in u for all θ (cid:48) .Rather than modeling a full-fledged dynamic game, we model each type θ agent as facing a continuous-timeMDP defined in terms of their (time-invariant) assortment λ θ and opposing types’ (time-invariant) strategyprofiles σ θ (cid:48) . We may do this because, as noted in Section 3.1, agents are only affected by the actions of othertypes in aggregate; thus any agent’s own strategic decision will not affect the (aggregate) action of others.In this MDP, the agent starts in a “waiting” state. From here, they either transition to an “exited” statewhen they leave unmatched due to a life event or to a “meeting” state when they meet another agent. Atthis “meeting” state, the agent makes a decision to either “accept” or “reject”. Then, if they match, theytransition to “exited” with a payoff; otherwise, they transition back to “waiting.”That life events and meetings occur in a memoryless manner actually means we can rid ourselves of thecontinuous-time aspect of this MDP and get an equivalent discrete-time MDP. Formally, we set up thisdiscrete-time MDP as follows: The agent’s initial state is Waiting . State
Waiting transitions to
Exited withprobability δ/ ( δ + (cid:80) θ (cid:48) λ θ ( θ (cid:48) )) and to PreMeeting θ (cid:48) with probability λ θ ( θ (cid:48) ) / ( δ + (cid:80) θ (cid:48) λ θ ( θ (cid:48) )). State PreMeeting θ (cid:48) transitions to Meeting θ (cid:48) ,u with u drawn from D θθ (cid:48) . State Meeting θ (cid:48) ,u is a decision point where the agent caneither Accept or Reject . If
Accept is chosen, with probability σ θ (cid:48) ( θ, u ) they transition to Exited and receivepayoff u . In all other cases, they return to state Waiting . Finally,
Exited is a terminal state.The optimal policy for this MDP admits a simple informal analysis: If u θ is the expected payoff of anagent in state Waiting , then u θ is also the expected continuation payoff if an agent chooses Reject in state
Meeting θ (cid:48) ,u . Hence the agent should choose Accept if u > u θ and Reject if u < u θ . If u = u θ , the agent isindifferent between the two options; however, since the distributions D mw are continuous, this occurs withprobability 0. We now state this observation more formally: Lemma A.1.
For any best response σ θ , let u θ be its expected payoff. Then σ θ ( θ (cid:48) , u ) = (cid:40) if u > u θ if u < u θ for all θ (cid:48) such that λ θ ( θ (cid:48) ) > and almost all u in the support of σ θ (cid:48) ( θ, u ) dF θθ (cid:48) . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search σ θ such that σ θ ( θ (cid:48) , u ) < θ (cid:48) and u > u θ ). In the subsequent analysis, we rule out such dominated strategies to eliminate “bad” equilibria, e.g.,when all agents reject all potential matches (knowing that no one else ever accepts).Ruling out dominated strategies as above in fact lets us focus on equilibria in threshold strategies , whereeach type has a threshold τ θ and accepts if and only if u ≥ τ θ . Our next lemma shows that any equilibrium innon-dominated strategies is equivalent, in a certain sense, to an equilibrium in threshold strategies, whereeach type’s threshold is also their expected utility: Lemma A.2.
Suppose all agents are playing best responses where they accept all potential matches worth atleast their expected utility. If type θ agents all switch to playing the threshold strategy with threshold τ θ = u θ ,where u θ is the expected utility of type θ agents, then each type’s strategy is still a best response. Moreover,the distribution of matches and the utilities at which they are realized remains the same (up to measure ).Proof. Since type θ agents already accept all potential matches worth at least u θ , by switching to the thresholdstrategy with threshold u θ , the only change is that they now reject all potential matches worth less than u θ .By Lemma A.1, accepting such potential matches was already a probability 0 event to begin with. Hence thedistribution of realized matches remains the same.To see that all strategies are still best responses, notice that the expected utility of type θ agents does notchange. Furthermore, type θ agents rejecting more matches only restricts the choice sets of agents on theopposite side. Hence their strategies σ θ (cid:48) remain best responses also.Applying Lemma A.2 in succession to all types θ ∈ Θ allows us to convert any equilibrium to an equivalentequilibrium in threshold strategies, such that each agent is thresholding at their expected utility. Thus,without loss of generality, we may restrict our attention to such equilibria.We can also show a converse of sorts to Lemma A.2, in which we characterize the optimal threshold fortype θ as a solution to a fixed point equation: Lemma A.3.
There exists a unique fixed point satisfying τ θ = u θ (viewing u θ as a function of τ θ ). Inparticular, if τ θ satisfies the fixed point equation, then τ θ is the unique threshold best response for type θ agents such that τ θ = u θ .Proof. Let τ θ be a best response threshold satisfying the fixed point equation. (By Lemma A.2, such a τ θ exists.) Note that no threshold τ (cid:48) θ > τ θ can satisfy the fixed point equation because τ θ is a best response. Sosuppose τ (cid:48) θ < τ θ . Then, an agent thresholding at τ (cid:48) θ would leave the market no later than an agent thresholdingat τ θ . If they leave strictly earlier, then they must have left with payoff at least τ (cid:48) θ . And conditioned on leavingat the same time, their expected payoff is τ θ . The latter occurs with positive probability, so their expectedpayoff when thresholding at τ (cid:48) θ exceeds τ (cid:48) θ . It follows that τ (cid:48) θ is not a fixed point, making τ θ the unique fixedpoint.8 Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
A.2. Proof of Lemma 3.3
Proof of Lemma 3.3.
To compute the agent’s expected payoff at state
Waiting , we can condition on the eventthat the agent does not return to
Waiting . Then, the conditional probability of matching with an agent oftype θ (cid:48) is λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) δ + (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) (cid:17) and the expected payoff conditioned on matching with an agent of type θ (cid:48) is (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) u dF θθ (cid:48) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) . Summing over all θ (cid:48) the product of the above two expression gives us the first equality. An application ofLemma 3.2 along with the definition ζ θ = ξ θ + δ gives us the second equality. A.3. Proof of Proposition 3.1: Equilibrium Play for Fixed Assortments
In this section, we complete the proof Proposition 3.1, which states that our model makes a unique predictionof equilibrium play given any fixed set of assortments. We show that there exists a strategy profile whereeach strategy is a best response and that this strategy profile is unique. Uniqueness implies that stationaryequilibria are robustly self-sustaining (e.g., a market in stationary equilibrium cannot be disrupted by agentsswitching to another strategy profile of best responses) and that the social welfare of an assortment iswell-defined (see Section 3.2). Our proof of the existence of a profile of best responses is constructive and canbe implemented as an algorithm to compute equilibrium play. Consequently, our model’s unique prediction ofequilibrium play can also be efficiently computed by the platform.To prove that a profile of best responses exists, we show that iterating the best response map convergesafter finitely many iterations. (In a sense, this iteration behaves like a Gale-Shapley operator.) Our uniquenessresult derives from the symmetric valuations assumption that we make (see Section 2.2): Symmetric valuationsinduce a linear ordering over all possible matches by their valuations. This rules out “cycles” in the preferencesand thus the possibility of multiple equilibria.More formally, let the best response map take as input a strategy profile { σ θ } θ ∈ Θ and output a new strategyprofile { σ (cid:48) θ } θ ∈ Θ such that each σ (cid:48) θ is the threshold strategy that thresholds at the expected utility of theagent under the input strategy profile. (This map is a best response by Lemma A.3.) To show existence ofequilibrium, we will show that iterating this map on the strategy profile converges. Proposition A.4.
Given assortments { λ θ } θ ∈ Θ , there is a unique profile { σ θ } θ ∈ Θ in non-dominated strategiessuch that each σ θ is a best response (up to the equivalence given in Lemma A.2). This strategy profile can befound by iterating the best-response map O ( | Θ | ) times. Before we proceed with the proof of Proposition A.4, we note that the claims of Proposition 3.1 followfrom combining Lemmas A.2 and A.3 with Proposition A.4.
Proof.
By applying Lemma A.2, we may restrict our attention to agents playing threshold strategies whereeach type thresholds at their expected utility u θ . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search O ( | Θ | ) steps. In the sequel, when we refer to thethresholds of the i -th iteration, we mean the thresholds after applying the best response map i times.Our first step is to show that each type’s threshold has the following pattern: The sequence of thresholdsafter an odd number of iterations is (weakly) monotonically decreasing and that the sequence of thresholdsafter an even number of iterations is (weakly) monotonically increasing. We prove this claim by induction.For the base case, note that the claim is trivially true when comparing the zeroth and second iterations, sinceall thresholds are initialized to 0. Next, suppose the claim is true for iterations i − i −
1. Then, notethat the thresholds after iterations i − i are each given by the expected utilities of agents. Now, if i isodd, then we know that thresholds after iteration i − i −
3; hence, theexpected utilities, and therefore the new thresholds, are uniformly lower. The analogous argument works foreven i .A similar argument shows that for each type, the infimum of that type’s thresholds after odd-numberediterations is at least the supremum of that type’s thresholds after even-numbered iterations: The claim clearlyholds for the initialization of all thresholds at 0. Next, suppose the claim holds for iteration i −
1. If i is odd,then let j > i be any even number. (Note that the claim follows from the monotonicity property above for j < i .) The thresholds after the j -th iteration are best responses to thresholds after the ( j − j − i − j − i − j -th iteration are uniformly than the thresholdsafter the i -th iteration. The analogous argument works for even i .The final step for showing the convergence of this best response iteration is to show that after eachodd-numbered iteration, at least one additional type will have their threshold “frozen,” meaning that it willnot change in any future iterations. Indeed, after any odd-numbered iteration, consider the agent type θ withthe highest threshold τ θ who has not yet been shown to be frozen. Then, observe that in all future iterations,no unfrozen agent will have a higher threshold by the two claims above—the first claim handles odd-numberediterations and the second claim handles even-numbered iterations. It follows that type θ would not want toalter its best response in all following iterations: It will not be affected by the strategy of any unfrozen type,because their thresholds will be at most τ θ ; the strategies of all frozen types will remain the same. Thus, type θ will be frozen after this iteration. Since at least one additional type has their threshold frozen after eachodd-numbered iteration, iterating the best response map will converge after O ( | Θ | ) iterations.Next, we show the uniqueness of this equilibrium. Suppose for the sake of contradiction that there aretwo distinct strategy profiles { σ θ } θ ∈ Θ and { σ (cid:48) θ } θ ∈ Θ such that each strategy is both a best response and athreshold strategy. Let the two sets of thresholds for these two strategy profiles be { τ θ } θ ∈ Θ and { τ (cid:48) θ } θ ∈ Θ .Furthermore, suppose each threshold is its corresponding type’s expected utility. Since the two strategyprofiles are distinct, there exists a type θ such that τ θ (cid:54) = τ (cid:48) θ and max( τ θ , τ (cid:48) θ ) is maximal. Without loss ofgenerality, we may assume that τ θ > τ (cid:48) θ . It follows that if τ θ (cid:48) > τ θ , then τ θ (cid:48) = τ (cid:48) θ (cid:48) , and if τ θ ≥ τ θ (cid:48) , then τ θ ≥ τ (cid:48) θ (cid:48) .0 Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Consequently, max( τ θ , τ θ (cid:48) ) = max( τ θ , τ (cid:48) θ (cid:48) ) for all θ (cid:48) . But this gives us the following contradiction: Threshold τ θ is type θ ’s expected utility, so Lemma 3.3 tells us τ (cid:48) θ < τ θ = (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) u dF θθ (cid:48) (cid:17) δ + (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ θ (cid:48) ) dF θθ (cid:48) (cid:17) = (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ (cid:48) θ (cid:48) ) u dF θθ (cid:48) (cid:17) δ + (cid:80) θ (cid:48) (cid:16) λ θ ( θ (cid:48) ) (cid:82) ∞ max( τ θ ,τ (cid:48) θ (cid:48) ) dF θθ (cid:48) (cid:17) . That the right-hand side exceeds the left-hand side means thresholding at τ (cid:48) θ is not actually a best response,since we could do better by choosing our threshold to be τ θ . Therefore, two such distinct strategy profilescannot exist, proving the proposition. B. Further Examples
B.1. Example: The Utility of Directed Search in a Horizontal Market
To motivate the utility of platform-directed search, we describe a market involving highly heterogeneouspreferences (i.e., horizontal differentiation) where directed search can lead to significantly greater efficiencythan random meeting.
Example B.1.
Suppose the types M = { m , . . . , m n } and W = { w , . . . , w n } are such that type m i onlyachieves positive utility if matched with type w i . Suppose also that the types are symmetric, with each typehaving the same arrival rate α θ = α for some α > m i , w i ) having thesame preference distribution F m i w i = F . Finally, let δ > m i agents only encounter type w i agents and vice versa. By random meeting, we mean that eachtype θ agent meets type θ (cid:48) agents at at a rate proportional to η θ (cid:48) . Thus, under random meeting, the symmetryof the market implies only a 1 /n fraction of each agent’s meetings are with mutually compatible agents. Onthe other hand, with directed search, one can set assortments so that λ m i ( w i ) = 1 and λ w i ( m i ) = 1 for all i .Then, agents meet other agents of the same type with probability 1. To compare the resulting equilibria, notethat the former can be thought of as identical to the latter, except with the friction δ being n times larger.This is because under random meeting, each agent’s “lifetime” in the market is effectively reduced by a factor n . It is not hard to see that this can lead a welfare loss of up to a factor of n .This example demonstrates that in a market with horizontal preferences, there can be a significant amountof wasteful meeting if agents proceed with random search: In our example, only one in n meetings involvespairs that are compatible with each other. On the other hand, with directed search, the platform couldeliminate such wasteful meeting entirely. While admittedly stylized, this example nonetheless highlights theusefulness of directed search in horizontally differentiated markets. C. Omitted Proofs from Section 4
C.1. Proof of Proposition 4.3
Proof of Proposition 4.3.
We first reparameterize the flow balance constraint by introducing new variables f mw := η m λ m ( w ) = η w λ w ( m ) mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search λ m ( w ) and λ w ( m ). (For symmetry in notation, we will use f mw and f wm interchangeably.) Withthese new variables, we may rewrite the objective symmetrically as2 (cid:88) m ∈M (cid:88) w ∈W (cid:18) f mw (cid:90) ∞ τ mw u dF mw (cid:19) . Furthermore, note that η θ > η θ and replace alloccurrences of η θ λ θ ( θ (cid:48) ) with f θθ (cid:48) to obtain the equivalent constraint η θ ≥ (cid:88) θ (cid:48) f θθ (cid:48) (15)for all θ ∈ Θ. We similarly replace the matching rate equation (6d) with ξ θ η θ = (cid:88) θ (cid:48) (cid:32) f θθ (cid:48) (cid:90) ∞ τ θθ (cid:48) dF θθ (cid:48) (cid:33) . (16)Our next step is to absolve ourselves of the variables ξ θ and η θ . For the former, we can simply mergeconstraints (6a) and (16), as both are equalities involving ξ θ η θ , to get α θ = δη θ + (cid:88) θ (cid:48) (cid:32) f θθ (cid:48) (cid:90) ∞ τ θθ (cid:48) dF θθ (cid:48) (cid:33) . (17)Now, from (17), we can get a definition of η θ , which we can substitute into the only remaining constraint (15)on η θ to get a combined flow balance, capacity, and stationarity constraint α θ ≥ (cid:88) θ (cid:48) (cid:32) f θθ (cid:48) (cid:32) δ + (cid:90) ∞ τ θθ (cid:48) dF θθ (cid:48) (cid:33)(cid:33) (18)for each θ ∈ Θ.To obtain the linear program constraints, we make one final substitution of β mw := f mw (cid:18) δ + (cid:90) ∞ τ mw dF mw (cid:19) , (19)into (18), which gives us constraints (7a) and (7b) of the linear program. We also substitute β mw into theobjective, which yields the equivalent objective2 (cid:88) m ∈M (cid:88) w ∈W (cid:32) β mw · (cid:82) ∞ τ mw u dF mw δ + (cid:82) ∞ τ mw dF mw (cid:33) . (20)The only remaining variables that we haven’t taken into account are the thresholds τ mw . However, noticethat the optimization problem is effectively unconstrained in τ mw —given any choice of β mw and τ mw , we maychoose a f mw so that the definition (19) is satisfied. Thus, we may simply set τ mw so that its contribution (cid:82) ∞ τ mw u dF mw δ + (cid:82) ∞ τ mw dF mw (21)to the objective is maximized. As this is exactly the definition of ρ mw , we conclude that the linear program(7) has the same objective value as the platform’s first-best optimization problem.It remains to prove the second claim in the equivalence between the linear program (7) and the platform’sfirst-best optimization problem. For this, we note that given any feasible choice of { β mw } m ∈M ,w ∈W , we canreverse all of the substitutions made and obtain corresponding values for { λ θ } θ ∈ Θ , { η θ } θ ∈ Θ , { ξ θ } θ ∈ Θ , wherethe variables { τ mw } m ∈M ,w ∈W are set as to maximize (21).2 Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
C.2. Proof of Lemma 4.4
Proof of Lemma 4.4.
The logarithmic derivative of the A ( τ ) is ddτ log( A ( τ )) = F (cid:48) ( τ ) δ + (cid:82) ∞ τ dF − τ F (cid:48) ( τ ) (cid:82) ∞ τ u dF = (cid:18)(cid:90) ∞ τ ( u − τ ) dF − τ δ (cid:19) F (cid:48) ( τ ) (cid:0) δ + (cid:82) ∞ τ dF (cid:1)(cid:0)(cid:82) ∞ τ u dF (cid:1) . The sign of this derivative is given by the first term on the right-hand side, since the fraction on the right-handside is always non-negative. This first term is monotonically decreasing: ddτ (cid:18)(cid:90) ∞ τ ( u − τ ) dF − τ δ (cid:19) = − (cid:18) δ + (cid:90) ∞ τ dF (cid:19) < . Therefore, the expression is maximized when this term vanishes, i.e., when τ = (cid:82) ∞ τ u dFδ + (cid:82) ∞ τ dF = ρ. Moreover, since the derivative is non-negative when τ ≤ ρ and is non-positive when τ ≥ ρ , we have the desiredmonotonicities when τ ≤ ρ and τ ≥ ρ as well.We also state and prove a generalization of Lemma 4.4 to non-zero thresholds and multiple types on theopposite side. That is, the fixed point and monotonicity properties hold generally: Lemma C.1.
Define B ( τ ) = (cid:80) ni =1 (cid:16) λ i (cid:82) ∞ max( τ,τ i ) u dF i (cid:17) δ + (cid:80) ni =1 (cid:16) λ i (cid:82) ∞ max( τ,τ i ) dF i (cid:17) , where δ > , λ i > for all i , τ i ≥ for all i , and F i is a continuous distribution for each i . Let ρ satisfies thefixed point equation ρ = max τ ≥ B ( τ ) . Then B ( ρ ) = ρ . Moreover, B is monotonically increasing for τ ≤ ρ andmonotonically decreasing for τ ≥ ρ .Proof. Note that we can write B ( τ ) as B ( τ ) = (cid:82) ∞ τ u dGδ + (cid:82) ∞ τ dG , where dG = (cid:80) ni =1 ( λ i · ≥ τ i · dF i ) is a new measure defined on R in terms of λ i , τ i , and F i . Here, ≥ τ i denotesthe indicator function that is 1 on inputs at least τ i and 0 otherwise. After appropriately normalizing thenumerator and denominator (by (cid:82) R dG ), the claim follows from Lemma 4.4. C.3. Proofs of Lemmas from Section 4.2.2
Proof of Lemma 4.8.
Recall the platform’s first-best optimization problem (7). In a star-shaped market, thislinear program reduces to a fractional knapsack problem where the platform has a knapsack of size α m and wishes to fill it with items for each w ∈ W of weight α w and value ρ mw . Thus, there exists a first-bestsolution { β ∗ mw } w ∈W that matches type m agents exclusively with some subset { w , . . . , w n } ⊆ W such that ρ mw ≥ ρ mw ≥ · · · ≥ ρ mw n and β ∗ mw i = α w for all i < n . Proof of Lemma 4.9. If i < n , then β ∗ mw i = α w i . Therefore,ˆ τ w i = 1 α w i f ∗ mw i (cid:90) ∞ ˆ τ wi u dF mw i = (cid:82) ∞ ˆ τ wi u dF mw i δ + (cid:82) ρ mwi dF mw i . mmorlica, Lucier, Manshadi, Wei: Designing Approximately Optimal Search τ w I = ρ mw i solves the equation. For i = n , since β ∗ mw n ≤ α w n , we haveˆ τ w n = 1 α w n f ∗ mw n (cid:90) ∞ ˆ τ wn u dF mw n ≤ (cid:82) ∞ ˆ τ wn u dF mw n δ + (cid:82) ρ mwn dF mw n . If ˆ τ w n > ρ mw n , then by Lemma 4.4, we would have that the right-hand side is at most ρ mw n , since theexpression (cid:82) ∞ τ u dF mw n is monotonically decreasing in τ . This contradicts the assumption that ˆ τ w n > ρ mw n .Hence it must be the case that ˆ τ w n ≤ ρ mw n . Proof of Lemma 4.10.
This follows from the fact that for any threshold τ m , we have that τ w i = 1 α θ f ∗ mw i (cid:90) ∞ max( τ m ,τ wi ) u dF mw i ≤ α w i f ∗ mw i (cid:90) ∞ τ wi u dF mw i . The argument used in Lemma 4.9 when i = n then lets us conclude that τ w i ≤ ˆ τ w i for all i . Proof of Lemma 4.11.
The lemma is clearly true if τ m ≥ ˆ τ w i . Now, suppose τ m < ˆ τ w i . Then ˆ τ w i would satisfythe fixed point equation (12), meaning ˆ τ w i would be the equilibrium threshold of type w i . D. Hardness of Approximation
In this section, we show that the platform’s computational problem of finding a c -approximate welfare-maximizing stationary equilibrium is NP -hard for some constant c >
1. This shows that, conditioned on P (cid:54) = NP , one cannot hope to obtain better than a constant factor approximation to the optimal in polynomialtime.From a technical perspective, our approach very closely mirrors that of Chakrabarty and Goel [10], whoshow hardness of approximation for related problems (e.g., maximum budgeted allocation). While theirtechnique applies to our setting, it is not clear that their hardness results apply directly—a challenge specificto our setting is that our optimization problem is continuous rather than discrete. To make this approachwork, we must show that our complicated feasibility set introduces a discrete element to the optimal allocation.But as a consequence, we obtain a slightly worse constant c .At a high level, we show hardness of approximation by reducing the problem of approximating MAX3LIN2 to finding a sufficiently good approximation for the platform’s welfare maximization problem. (Recall that
MAX3LIN2 is the optimization problem where, given m linear equations in n variables over F such that eachequation involves exactly three variables, the objective is to maximize the number of equations simultaneouslysatisfied by an assignment to the variables.) That MAX3LIN2 is hard to approximate is due to H˚astad [15]and is a direct corollary of his celebrated 3-bit PCP. We state this hardness result in terms of a promiseproblem version of
MAX3LIN2 : Theorem D.1 (H˚astad [15]) . It is NP -hard to distinguish between MAX3LIN2 instances where all but εm equations are simultaneously satisfiable and where at most (cid:0) + ε (cid:1) m equations are simultaneously satisfiablefor any ε > . From this classical starting point, we derive hardness of approximation for the platform’s optimizationproblem:4
Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
Theorem D.2.
Computing a (cid:0) − ε (cid:1) -approximate solution to the platform’s welfare maximization problemis NP -hard for any ε > .Proof. As stated above, we reduce instances of
MAX3LIN2 to instances of the platform’s welfare maximizationproblem. Specifically, we reduce to instances where the discount rate δ is small and the distributions D mw arepoint masses. Consider a
MAX3LIN2 instance in n variables x , x , . . . , x n and m equations x i (cid:96) ⊕ x j (cid:96) ⊕ x k (cid:96) = b (cid:96) for (cid:96) ∈ [ m ].We construct an instance of our matching market by defining types, arrival rates, and payoffs based on thesevariables and equations: Types.
Define variable types x i, , x i, ∈ M and switch type s i ∈ W for each variable x i . Define equationtypes e (cid:96), (0 , , , e (cid:96), (0 , , , e (cid:96), (1 , , , e (cid:96), (1 , , ∈ W for each equation x i (cid:96) ⊕ x j (cid:96) ⊕ x k (cid:96) = b (cid:96) , one type for eachsatisfying assignment to equation (cid:96) . That is, type e (cid:96), ( b i ,b j ,b k ) corresponds to the satisfying assignment( x i (cid:96) , x j (cid:96) , x k (cid:96) ) = ( b i ⊕ b (cid:96) , b j ⊕ b (cid:96) , b k ⊕ b (cid:96) ). Arrival rates.
Let the variable and switch types for x i each arrive at rate α i = 4 m i , where m i is thenumber of equations that involve x i . Let each equation type arrive at rate 3. Payoffs.
Let matching type x i,b and type s i yield payoff 2(1 + 2 δ ) for all b . Let matching type x i,b with anyequation type whose corresponding satisfying assignment has the variable x i with parity b yield payoff 1.All other matches provide no utility.With this setup, no agent on side W will ever reject positive utility matches in equilibrium, since all positiveutility matches yield the same payoff for them. Moreover, no switch type agent will ever be rejected inequilibrium, since they yield maximal payoff for variable type agents. And if in equilibrium an agent of type x i,b has expected continuation utility greater than 1, then they must only accept agents of type s i . Theseobservations set the stage for our key lemma: Lemma D.3.
In any welfare-maximizing equilibrium, the expected payoff of all switch type agents will be δ ) / (1 + δ ) . Furthermore, such an equilibrium can be achieved by only showing type s i and type x i,b agents to each other for some b ∈ { , } for each i .Proof. First, we show that some variable type x i,b must have expected payoff greater than 1 for each i ∈ [ n ].Suppose otherwise, that there exists a welfare-maximizing stationary equilibrium such that the expectedpayoffs of both variable types x i,b are at most 1 for some i . Then we could stop matching both types entirely,by setting their assortments to 0. This comes at a loss in welfare of at most 2 · m i . Then, we could matchtype x i, entirely to s i , and obtain an increase in welfare of 4 m i · δ ) / (1 + δ ) > · m i . Finally, note thatthese new assortments also produce a stationary equilibrium: Unmatching the variable types x i,b only addsslack to the assortment feasibility constraints. Unmatching also does not affect any equation type’s threshold, Although point masses are not continuous distributions, which we required in Section 2.2, we can make point massescontinuous by adding a small amount of (bounded) noise. Doing so does not change any of the conclusions. To simplifyour exposition, we focus on the case of point masses only. When δ >
0, these observations still hold for distributions that are not point masses, so long as the variation inpayoff is sufficiently small. mmorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search x i,b whose expected utility is greater than 1 for each i . Notice that this type must exclusively match with type s i agents, since equation type agents only yield payoff 1 and thus cannot be accepted in equilibrium. Toprove the second part of this lemma, we show that we can obtain another welfare-maximizing equilibrium bymatching the entire supply of type s i agents to this variable type x i,b .Suppose we have a welfare-maximizing equilibrium, and we modify it by matching the entire supply oftype s i agents to a variable type x i,b whose expected utility exceeds 1. To restore a stationary equilibrium,note that making this switch may violate two sets of constraints: First, it might violate type x i,b ’s feasibilityconstraint; but this is not a problem, since we can simply stop showing x i,b type agents any types but type s i , since these agents were only accepting type s i agents in the first place. Second, it might cause type x i,b ⊕ to want to accept more equation type agents because they stop matching with switch type agents entirely. Toresolve this, we can simply reduce the supply of equation type agents to type x i,b ⊕ so that by accepting allsuch agents, x i,b ⊕ accepts the same quantity of agents as in the original equilibrium. Finally, to see that thisnew equilibrium has the same welfare as the original equilibrium, note that the flow rates of matches for bothswitch and equation types are at least as high as before.This lemma tells us that there exists a welfare-maximizing equilibrium where, for every i , one of variabletypes x i, and x i, is fully matched to the switch type s i . Moreover, it suffices to optimize over such equilibria.Notice that each such equilibrium also corresponds to an assignment ( x , . . . , x n ) ∈ { , } n , where x i = b iftype x i,b ⊕ is fully matched to the switch type s i .We are now ready to state our analogs of Lemmas 4.6 and 4.7 of Chakrabarty and Goel [10]: Lemma D.4.
Given a
MAX3LIN2 instance with m (cid:48) ≥ (1 − ε ) m simultaneously satisfiable equations, theplatform’s problem it maps to has OPT ≥ (36 + 48 δ − ε ) m/ (1 + δ ) .Proof. Given an assignment ( x , x , . . . , x n ) ∈ { , } n that satisfies m (cid:48) equations, we can match the switchtypes to the variable types as above. We can then match the remaining variable types to the equation typesso that each variable type matches with the equation types of each satisfied equation it is part of at a flowrate of 4 / (1 + δ ). We thus have payoff 2(1 + 2 δ ) / (1 + δ ) · (cid:80) i m i from switch types and payoff 12 / (1 + δ ) persatisfied equation. Summing these quantities gives us the lemma. Lemma D.5.
Given a
MAX3LIN2 instance with m (cid:48) ≤ (cid:0) + ε (cid:1) m simultaneously satisfiable equations, theplatform’s problem it maps to has OPT ≤ (34 . δ + 3 ε ) m/ (1 + δ ) .Proof. Take any assignment ( x , x , . . . , x n ) ∈ { , } n and match switch types to variable types as describedabove. Now, for any equation that is not satisfied, notice that there must be some equation type that cannotbe matched at all to any of the remaining variable types. So the payoff from the equation types for unsatisfiedequations is at most 9 / (1 + δ ). Upper bounding the payoffs of the equation types of the satisfied equationsby 12 / (1 + δ ) per satisfied equation, we get 12 m · δ ) / (1 + δ ) from switch types and at most payoff(( − ε ) m · + ε ) m · / (1 + δ ) from equation types. Summing these quantities proves the lemma.6 Immorlica, Lucier, Manshadi, Wei:
Designing Approximately Optimal Search
To finish, observe that Lemmas D.4 and D.5 show that if we can approximate the platform’s welfare-maximization problem with a factor of 36 + 48 δ − ε . δ + 3 ε , then we would be able to distinguish the two classes of MAX3LIN2 inputs in Theorem D.1. Thus, by takingsufficiently small δ and ε , we see that it is NP -hard to approximate the platform’s welfare-maximizationproblem within any constant factor less than . =2423