Matching Supply and Demand with Mismatch-Sensitive Players*people.duke.edu/~mc468/Matching.pdf · Author: Matching Supply and Demand with Mismatch-Sensitive Players 3 angle of our

Matching Supply and Demand withMismatch-Sensitive Players*

Mingliu ChenThe Fuqua School of Business, Duke University. [email protected]

We study matching over time with short and long-lived players who are very sensitive to mismatch, using

a novel method to characterize the mismatch. In particular, players’ preferences are uniformly distributed

on a circle, so the mismatch between two players is characterized by the one-dimensional circular angle

between them. This framework allows us to capture matching processes in applications ranging from ride

sharing to job hunting. Our analytical framework relies on threshold matching policies, and is focused on a

limiting regime where players demonstrate low tolerance towards mismatch. This framework yields closed-

form optimal matching thresholds. If the matching process is controlled by a centralized social planner (e.g. an

online matching platform), the matching threshold reflects the trade-off between matching rate and matching

quality. The corresponding optimal matching threshold is smaller than myopic matching threshold, which

helps building market thickness. We further compare the centralized system with decentralized systems,

where players decide their matching partners. We find that matching controlled by short-lived players hurts

all players’ expected utilities in general. Furthermore, if the matching threshold is designed by long-lived

players, the platform may want to induce them to be more tolerant.

Key words : Matching, Peferences of Players, Queueing Theory.

History : Current version: December 11, 2019.

1. Introduction

The booming development of online matching platforms draw lots of attention from the operations

community, for good reasons. Easily accessible to public, these matching platforms gather millions

of potential clients every day. The sheer size of user base means diverse preferences among users.

Players’ preferences mainly has two dimensions: quality and timing of matches. Take carpooling

in ride-sharing platforms as an example. During morning(evening) rush hours, riders and drivers

depart from the same neighborhood(business district), but post different individual destinations on

the platform. Obviously, players prefer to be matched with others who have similar destinations.

Moreover, riders typically operate on tight time constraints, and may actively look for alternatives

elsewhere if not being matched immediately on the platform. Drivers can be more patient but need

to depart eventually after some time. DiDi, the largest ride-sharing company in China, offers a peer-

to-peer car-sharing platform to serve riders and drivers in the example above, called DiDi Hitch. It

* This is a joint work with Professors J.G. Dai, Peng Sun and Zhixi Wan.

1

2 Author: Matching Supply and Demand with Mismatch-Sensitive Players

is based on a two-sided search processes that involve both riders’ and drivers’ inputs of destinations

and mutual acceptance where drivers are mainly commuters instead of professional drivers. In this

service, each rider submits her final destination to the platform and receives a list of potential

drivers who are going the the similar directions in return. Then the rider proposes to a driver

of her choice. The corresponding driver may accept or reject it. Blablacar in Europe has similar

services as well. Moreover, based on our conversation with DiDi Hitch, the carpooling service is

mostly decentralized at the moment. However, decentralization may introduce many inefficiencies

because individuals may behave myopically or overly strategic. Thus, besides characterizing players’

preferences, mismatch and the corresponding matching process, it is worth studying who should be

the match maker: the platform (centralized) or individual players (decentralized)? What tolerance

level and matching criteria should the match maker use? If some decentralized matching systems

lack efficiency compared with centralized ones, what are the driving forces behind them? Are

individuals too picky or too tolerant in decentralized matching when comparing with centrally

optimal ones?

To answer these questions, we consider a two-sided matching market where a platform serves

as an intermediary who matches players arriving following Poisson processes. Building upon the

observation from DiDi Hitch, one side of players (riders) are short-lived, who leave the platform

immediately if not being matched upon arrival. The other side of players (drivers) are long-lived,

who stay on the platform for an exponentially distributed period of time without a match. When

matching players, the platform needs to consider their individual preferences. Otherwise, players

may reject the match.

A salient feature of this paper is that we provide a novel method to characterize players’ individ-

ual preferences over the quality of matches. In other words, we construct a measure for mismatch

between players that directly affects matching outcomes. We have all incoming players’ preferences

uniformly distributed on the boundary of a circle. The mismatch between two players can be char-

acterized by the one dimensional circular angle between them, which we refer to as the mismatch

angle. Thus, throughout our model and analysis, we can use this one-dimensional scaler to describe

the compatibility between any two players. Our characterization of players’ mismatch is natural

in ride sharing and others scenarios with spatial features. In the example of DiDi Hitch, mismatch

is the difference between destination of a rider’s and a drivers’ destinations. Closer destinations

translate to smaller mismatch angles in our model.

Before going into analytical schematics and results, it is worth noting that our model is not

restrictive to carpooling in ride-sharing. On a gig job hunting platform, candidates want to find

suitable jobs as soon as possible. Talent seekers may be more patient but will not settle with any

candidate whose skills are not compatible with their opening positions. In this context, a mismatch

Author: Matching Supply and Demand with Mismatch-Sensitive Players 3

angle of our model captures the difference between the skill a candidate possessed and the talent

a seeker is looking for. The smaller the angle is, the more compatible two players are.

Since our model captures mismatch between players using a circular angle, we focus on threshold

policies that match two players if their mismatch angle is small enough. We are interested in the

optimal matching threshold under different objectives ranging from maximizing social welfare to

maximizing the utility of a particular group of players. Furthermore, in each match, each individual

may not need to bear the entire mismatch angle since two matching partners can split the mismatch

with each other. Consider the carpooling example. A rider and a driver may agree on a drop

off location in between their destinations. Thus, we also identify the optimal mismatch splitting

decision for players.

Furthermore, we are interested in understanding the relationship between market thickness and

matching policies. An aggressive matching policy that tries to match as many players as possible

may generate good short-term revenue at the moment but leads to a thinner market. Intuitively,

matching in a thin market is difficult and hurts long-term revenue. Therefore, what is a good

matching policy that balances the market thickness and the immediate matching rate? Note that

since short-lived players never stay on the platform, their arrival rate represents their side of market

thickness. However, describing the market thickness for long-lived players is non-trivial. We model

the number of long-lived players on the platform as a Continuous Time Markov Chain (CTMC),

to be more specific, a Birth-Death process. We perform all analysis of the matching system under

the steady state of this CTMC.

Aiming to characterize the matching system with closed-form expressions, we consider the case

where players only accept matching partners who have really small mismatch angle. Under this

limiting regime, we introduce appropriate M/M/∞ queues to bound the original CTMC. The

classic result on the stationary distribution of M/M/∞ queues (see e.g., Iglehart 1965) greatly

simplifies our analysis. Moreover, using Taylor expansions on players’ tolerance towards mismatch

angles helps us to obtain various results in closed-form.

Closed-form expressions reveal clear relationship between optimal matching thresholds and the

market thickness. First, intuitively, larger matching thresholds lead to thinner market on the side

of long-lived players. Second, if the platform decides the matching threshold, it has an inverse

relationship to the thickness of the market. That is, if either side of the market becomes thicker, a

smaller matching threshold is always optimal. Moreover, the platform should not match myopically.

Instead, we provide a closed-form expression of the optimal matching threshold, which is smaller

than the myopic threshold. Here the intuition is that using a smaller threshold improves matching

quality. Although it may decrease overall matching rate, a smaller threshold thickens the market


of long-lived players. Thus, the slight drop in matching rate is compensated by more potential

matching parters and better matching quality.

We also study two decentralized systems where the platform allows short-lived and long-lived

players designing their own matching thresholds. Both scenarios are formulated as games with

potentially infinite number of players. In the first scenario, we consider a game among short-lived

players. In equilibrium, they match myopically. This myopic behavior of short-lived players ignores

the benefit of building a thicker market–improving matching quality by having more matching

partners. Furthermore, the platform can improve not only the social welfare by deviating from this

decentralized system, but also the utilities of all players, including that of short-lived players. We

also provide a simple method that can induce players to be pickier so as to achieve better individual

and social utilities. If long-lived players design their matching thresholds, on the other hand, we

provide a numerical procedure that allows us to compute the equilibrium matching threshold. We

find that long-lived players’ equilibrium matching threshold appears smaller than the one designed

by the platform with an intention to maximize long-lived players’ total utility. That is, under

centralized matching, all long-lived players should be less picky, and use a larger threshold than

the one in equilibrium. In this case, from a focal player’s perspective, larger threshold reduces the

number of other long-lived players on the platform, which leads to less competition and thus better

utility.

Recent literature on matching intermediaries paves the way for our work. A stream of related

papers study matching using fixed matching probability for all players. In this literature, players are

homogeneous in the sense that they are indifferent to whom they are matched with (see e.g., Unver

2010, Anderson et al. 2017, Buke and Chen 2017, Akbarpour et al. 2019). In particular, Akbarpour

et al. (2019) consider a dynamic matching system in a networked market with departure. They

show that market thickness is important if the platform can identify which player is about to leave

and only match these players. Many papers (see e.g., Unver 2010, Anderson et al. 2017, Akbarpour

et al. 2019) show that myopic policy is near optimal. In comparison, our results indicate that

strategically delayed matching can be beneficial.

Other papers consider heterogeneous players where players can only be matched or have higher

matching probability if their (discrete) types are compatible. Ashlagi et al. (2019) study a dynamic

matching market with easy and hard to match players. In their model, hard to match players have

significantly lower matching probability compared with that of easy to match players, and all players

want to be matched as soon as possible. They analyze the performance of myopic matching policies

involving bilateral and chain matching. Ozkan and Ward (2016) consider a matching problem

for ride-sharing. They use players’ origin or destination as types. Riders only accept drivers who

can arrive in a certain time window. They take advantage of large market, where players’ arrival


rate is large, and identify policies that match the most number of players. In other applications,

although any pair of players can be matched, the matching reward depends on players’ types. Hu

and Zhou (2019) consider a two-sided matching problem in a discrete-time finite horizon setting.

Players from both supply and demand side can abandon the system in any period. Chen et al.

(2019) consider a market with short-lived demand side and long-lived supply side. Players from

each side has two types and matching rewards has a supermodular structure according to players’

types. Our work differs from the papers above as we use a novel circular model to connect players’

individual preferences(types) with matching outcomes. This innovation leads us to obtain concise

and informative results.

Another steam of research study two-sided matching via queueing methods. Afeche et al. (2014)

study trading systems using double-sided queues. They also consider short and long-lived players

similar to our model without circular preference. They provide performance measure of queues

under FCFS policy such as expected waiting time, etc. Many other papers analyze matching policies

under fluid limits (see e.g., Zenios et al. 2000, Su and Zenios 2006, Akan et al. 2012, Gurvich and

Ward 2014, Ozkan and Ward 2016, Kanoria and Saban 2019). Our work exploits the similarity

between our matching system with M/M/∞ queues.

Our modeling approach also resembles the Hotelling’s circular city model in economics. The

original model is discussed in Salop (1979), in which they use a circular model as a geographical

representation of a city. They consider that suppliers have fixed locations on a circle and consumers

have preferences over their relative locations to the suppliers. Recently, Pavan and Gomes (2019)

extend this model to a three dimensional space as a cylindrical model with preferences in two

dimensions. However, there is no arrivals or departures of players in either of the two papers above.

The rest of this paper is organized as follows. We introduce the model and formulate the match-

ing system in Section 2. In Section 3, we introduce results when the platform designs matching

thresholds, and compare with results on decentralized systems in Section 4. We conclude our dis-

cussion in Section 5 and highlight some future directions. All proofs and some minor derivations

are presented in the Appendix.

2. Model Setup

Consider two classes {L,S} of players arriving to the matching platform. Type L players are long-

lived (patient). They follow a Poisson arrival process with rate λL. Each arrival has a “life time”

that is exponentially distributed with rate γ; if no match occurs by the end of its life time, the

player disappears from the platform at that time. Type S players are short-lived (impatient) with

exponential arrival rate λS and leave the platform immediately if not matched upon arrival. Two


players are matchable only if they are from different classes. Therefore, matches can only be made

upon arrival of short-lived players.

We use a “circular” model to describe compatibility between players. Upon arrival, each player

from either class uniformly and independently claims a random spot on the edge of a circle. Between

players from two classes, their mismatch is measured by the arc, or, equivalently, the central angel

between their spots on the circle. To simplify notations in future sections, we define φ∈ [0,1] as the

mismatch ratio, which is the ratio between the actual mismatch angle to the maximum possible

mismatch angle π. For example, two players with mismatch angle π/4 (or 45 degrees) have a

mismatch ratio 0.25. Mismatch ratio effectively normalizes mismatch angle into a variable inside

the interval [0,1].

We assume that in each potential match with mismatch ratio φ, a player form either class may

not need to bear the entire mismatch. Instead, short-lived and long-lived players bear mismatch

ratios αS(φ)∈ [0, φ] and αL(φ)∈ [0, φ] such that αS(φ)+αL(φ) = φ. We treat the mismatch splitting

functions αL(·) and αS(·) as part of the matching design in this paper as they should be in reality.

In the carpooling example, the rider may bear very limited portion of the total mismatch as her

penalty to mismatch is typically much higher comparing to that of drivers. It is unlikely that a

rider shall participate at all in a match that requires her to walk for a significant distance to her

final destination.

2.1. Players’ utility and matching policy

Each successful matching generates a fixed revenue to players involved. Therefore, define a class

i∈ {L,S} player’s utility in a match with mismatch ratio φ and splitting function αi as,

Wi(αi(φ)) = ui− ciαi(φ), i∈ {L,S}, (2.1)

where ui is the fixed revenue generated by each match and ci is the mismatch penalty coefficient

for players in class i.

In this paper, we consider mismatch-sensitive players, in the sense that the penalty of a bad

match generates is far greater than the fixed revenue it creates. In order to capture this feature,

we assume uL� cL, uS� cS1, and define

ε :=uLcL� 1, and s :=

uScLcSuL

. (2.2)

The scaling factor ε represents mismatch tolerance for long-lived players. In the limiting regime

considered in later sections, we let ε go to zero. The fixed constant s ∈ [0,∞) represents the ratio


between short and long-lived players’ mismatch tolerances. Following definitions of ε and s, we

have εs= uS/cS. Thus, we can rewrite players’ utility in (2.1) as the following,

WL(αL(φ)) = cL(ε−αL(φ)), and WS(αS(φ)) = cS(εs−αS(φ)), φ∈ [0,1]. (2.3)

Each matching process follows the same sequence of events below. The match maker observes

the newly arrived short-lived player and x≥ 0 number of long-lived players on the platform. Denote

φi, i∈ {1, ..., x} as the mismatch ratio between each of the x long-lived players and the short-lived

player. Moreover, let φ(x) = min{φi | i= 1, ..., x} represents the minimum mismatch ratio between

an short-lived player and the x long-lived players. After observing {φi}xi=1, the match maker decides

whether to match between the short-lived player and a long-lived player. If a long-lived player

k ∈ {1, ..., x} is the one matched, the match maker then informs players their share of mismatch

{αS(φk), αL(φk)}. Players from both sides decide whether to accept or reject the match based on

their utilities WS(αS(φk)) and WL(αL(φk)). Players then claim their utilities if both sides accept,

which leads to a successful match. We assume that the platform eliminates any player who rejects

a match from future matching processes. Therefore, when involving in a match purposed by the

match maker, both players are myopic and accept the match if and only if

WS(αS(φ))≥ 0 and WL(αL(φ))≥ 0, (PC)

when their mismatch ratio is φ. We refer to this condition as players’ participation constraint or

(PC). Note that (PC) is similar to players’ ex-post individual rationality constraints in economics

literatures.

In this paper, we restrict our attention to stationary threshold policies that are only based on the

minimum mismatch ratio φ(x) = min{φi | i= 1,2, ...x} between a short-lived player and x long-lived

players. Therefore, a match is purposed for the arriving short-lived player if and only if φ(x) is no

greater than the threshold εθ design by the match maker. Note that according to (PC), thresholds

designed by the match maker also need to scale with ε.

Formally, a threshold policy {εθ,αS(·), αL(·)} matches the arriving short-lived player with an

existing long-lived player i if and only if φ(x) = φi and φ(x)≤ εθ when there are x long-lived players

on the platform. Functions αS(φ(x)) and αL(φ(x)) specifies the mismatch splits.

Furthermore, we also assume that the match maker focuses on policies {εθ,αS(·), αL(·)} such

that WS(αS(φ))≥ 0 and WL(αL(φ))≥ 0 for all φ≤ εθ. In other words, the match maker ensures

that (PC) is always satisfied for each purposed match under any policy.


2.2. Matching probability and Birth-Death process

As matches can only occur with arrivals of short-lived players, in this paper, we use the term

matching probability to represent the probability that upon the arrival of a short-lived player, she

can be matched with a long-lived player.

When there are x long-lived players available and the matching threshold is εθ, the matching

probability is

pεθ(x) = 1− (1− εθ)x, (2.4)

where inside the parentheses is the probability that the short-lived player cannot be matched with

a long-lived player, whose location is uniformly distributed on the circle. Since long-lived players

are located independently, (1−εθ)x is the probability that all x number of long-lived players cannot

be matched with the short-lived player.

Next we derive the distribution function of minimum mismatch ratio φ(x), which is a random

variable that depends on the number of current long-lived players. Recall φi is the mismatch ratio

between the arriving short-lived player and a long-lived player i ∈ {1,2, ..., x} when there are x

number of long-lived players in total. The probability that the minimum mismatch ratio φ(x) is

no greater than φ∈ [0,1] is

Hx(φ) = 1−Πxi=1P(φi >φ) = 1− (1−φ)x, ∀0≤ φ≤ 1, (2.5)

which is the C.D.F. of the random variable φ(x). By differentiating H(·), we reach its P.D.F.

gx(φ) = x(1−φ)x−1, ∀0≤ φ≤ 1. (2.6)

Note that both the C.D.F. and the P.D.F. of the minimum mismatch ratio is parametrized by the

number of long-lived players on the platform, x. Recall that we focus on policies with thresholds on

the minimum mismatch ratio. The distribution of φ helps us characterize the quality of matching

outcomes.

As we can see, both matching probability and distribution of the minimum mismatch ratio

critically depend on the number x of long-lived players on the platform. Therefore, in order to

characterize the dynamics of the long-lived players, we formulate the arrival and departure of long-

lived players as a Continuous-Time Markov Chain (CTMC). We use a “birth-death” Markov chain,

Xεθ = {Xεθ(t), t≥ 0}, which is parametrized by the matching threshold εθ, to capture the number

of long-lived players. Denote fεθ(x) to represent the density function of the stationary distribution

of process Xεθ, which solves the following system of equations,

λLfεθ(0) = (λspεθ(1) + γ)fεθ(1),


λLfεθ(x− 1) + (λspεθ(x+ 1) + γ(x+ 1))fεθ(x+ 1) = (λL +λspεθ(x) + γx)fεθ(x), x= 1,2, ...

This system of difference equations has the following unique solution2:

fεθ(x) =λL

xfεθ(0)

Πxk=1(λspεθ(k) + γk)

, ∀x≥ 0, (2.7)

and

fεθ(0) =

(1 +

∞∑x=1

λLx

Πxk=1(λspεθ(k) + γk)

)−1

. (2.8)

We denote Xεθ as the random variable according to the steady state distribution fεθ of process

Xεθ. It is worth noting that Xεθ reflects the market thickness of long-lived players. More players

available leads to thicker market. Thickness of short-lived players is characterized by their arrival

rate λS since they never stay on the market.

In the following sections, we conduct analysis under steady state of the matching system where

the number of long-lived players follows the stationary distribution of process Xεθ in every match.

As the result, any utility functions are evaluated by taking expectations over the random variable

Xεθ.

3. Centralized Matching

In this section, the platform is the match maker. We explore the optimal matching threshold and

mismatch splitting functions when the platform designs them. We measure the efficiency of the

matching platform using the social welfare it generates, which is defined as the sum of short and

long-lived players’ utilities. That is, for each successful match made under policy {εθ,αS(·), αL(·)},

the social welfare it generates is simply

WSW (αS(φ), αL(φ)) :=WS(αS(φ)) +WL(αL(φ)), ∀0≤ φ≤ εθ. (3.1)

Therefore, the expected social welfare rate under threshold policy {εθ,αS(·), αL(·)} is defined as

USW (εθ,αL(·), αL(·)) := λSEφ(Xεθ)

[WSW (αS(φ(Xεθ)), αL(φ(Xεθ)))I{φ(Xεθ)≤ εθ}

], (3.2)

where the expression inside the expectation follows (3.1). Since a match can only occur upon the

arrival of a short-lived player, the welfare per match is multiplied by λS.

Acting as a social planner, the platform aims to maximize the social welfare rate in (3.2).

Therefore, the platform’s optimization problem is

maxθ,αS(·),αL(·)

USW (εθ,αS(·), αL(·)) (3.3)


s.t. 0≤ αS(φ)≤ εs, ∀φ≤ εθ0≤ αL(φ)≤ ε, ∀φ≤ εθ,αL(φ) +αS(φ) = φ, ∀φ≤ εθ,θ≥ 0.

Note that the first two constraints represents players’ (PC) and the third constraint follows the

definition of the mismatch splitting function.

3.1. Optimal mismatch splitting functions

Since the platform can focus on matching thresholds and mismatch splitting functions that always

induce players to accept purposed matches, we can decompose the optimization problem in (3.3)

into a two-staged stochastic program. In the first stage, the platform designs a matching threshold

εθ based on the expected social welfare rate. In the second stage, when facing a realized minimum

mismatch ratio that is no greater than εθ, the platform specifies mismatch splitting functions for

players.

We solve the second stage problem first. Suppose the platform faces a mismatch ratio φ between

two potential matching partners. Then the second stage problem is

maxαS(·),αL(·)

cL(ε−αL(φ)) + cS(εs−αS(φ)), (3.4)

s.t. αL(φ) +αS(φ) = φ,0≤ αS(φ)≤ εs,0≤ αL(φ)≤ ε.

We summarize the optimal solution to this linear optimization problem in the next lemma.

Lemma 1. Fixing φ∈ [0,1], the optimization problem in (3.4) has optimal solutions:

(i) if cS ≥ cL,

{α∗S(φ), α∗L(φ)}=

{{0, φ}, if 0≤ φ≤ ε,{φ− ε, ε}, if ε < φ≤ ε(1 + s);

(3.5)

(ii) if cS < cL,

{α∗S(φ), α∗L(φ)}=

{{φ,0}, if 0≤ φ≤ εs,{εs,φ− εs}, if εs < φ≤ ε(1 + s).

(3.6)

Lemma 1 indicates that the optimal mismatch splitting functions are Bang-Bang controls. This

intuitive solution suggests that the platform should let the player with greater mismatch penalty

bear minimum mismatch ratio within the other player’s tolerance. Consider the carpooling example,

where the rider has much higher mismatch penalty compared with the driver. If their mismatch is

very small, the rider should be dropped off very close to her destination. If their mismatch ratio is

bigger than the driver’s tolerance, then the rider should be dropped off at the location that makes

the driver indifferent between accepting or rejecting the match.


From Lemma 1, we also observe that the platform’s centralized problem in (3.3) has a myopic

matching threshold εθM , where

θM := 1 + s. (3.7)

This myopic matching threshold guarantees a non-negative social utility per match. Later in this

section, we shall see the relationship between this myopic threshold and the optimal one.

3.2. Optimal matching threshold

With the optimal solution to the second stage problem, the platform’s matching design problem is

simplified to designing the optimal matching threshold εθ. Furthermore, Lemma 1 suggests that the

platform can restrict attention to θ≤ θM = 1 + s since players may not accept matches otherwise.

For our limiting regime where ε approaches 0, we assume that 1 + s� 1/ε.

Following Lemma 1 and with slight abuse of notation, we can rewrite the objective function in

(3.3) as

USW (εθ) =

λSEφ(Xεθ)

[WSW (0, φ(Xεθ))Iφ(Xεθ)≤min ε{θ,1}+WSW (φ(Xεθ)− ε, ε)Imin ε{θ,1}≤φ(Xεθ)≤εθ

], if cS ≥ cL,

λSEφ(Xεθ)

[WSW (φ(Xεθ),0)Iφ(Xεθ)≤min ε{θ,s}+WSW (εs,φ(Xεθ)− εs)Imin ε{θ,s}≤φ(Xεθ)≤εθ

], if cS < cL,

= λSEXεθhε(Xεθ, εθ), (3.8)

where function hε takes the form of

hε(x, εθ) =

∫ εθ

0

[ε(scS + cL)− cLφ]gx(φ)dφ+

∫ εθ

εθ

cS[ε(1 + s)−φ]gx(φ)dφ, θ= min{1, θ}, if cS ≥ cL,∫ εθ

0

[ε(scS + cL)− cSφ]gx(φ)dφ+

∫ εθ

εθ

cL[ε(1 + s)−φ]gx(φ)dφ, θ= min{s, θ}, if cS < cL.

(3.9)

The integration is over the minimum mismatch ratio φ(Xεθ), which has a distribution function

gx(·) in (2.6). Note that (3.9) has close-form expressions, though complex. Thus, the platform’s

problems can be analyzed if we can evaluate (3.8) effectively.

3.3. Approximations

It is hard to obtain analytical results with closed-form expressions from the matching system

purposed in the previous sections. Even though (3.9) has a closed-form expression, its expectation

w.r.t. Xεθ is hard to compute in closed-form due to the complicated expressions of its steady state

distribution. Therefore, we purpose an approximation schematic in this section when ε approaches

0.


Before going into the limiting regime where ε approaches 0, we first purpose an M/M/∞ queue

related to our original CTMC. Consider an M/M/∞ queue Yεθ = {Yεθ(t), t≥ 0} with arrival rate

λ= λL and service rate µ= γ+λSεθ.

Note if using this M/M/∞ queue to mimic the original birth-death process Xεθ, when there are

x long-lived players on the platform, the system departure rate is x(εθ+ γ). Thus, the “matching

probability” in this system is simply εθx when there are x long-lived players. The next lemma

shows the relationship between matching probabilities under process Xεθ and the M/M/∞ queue

Yεθ.

Lemma 2. Fixing 0≤ εθ≤ 1, we have

εθx≥ pεθ(x) = 1− (1− εθ)x, ∀x≥ 1.

The intuition behind this upper bound of matching probability is simple. Note εθx is the matching

probability when the x long-lived players have no overlap in their preference (tolerance angle).

Thus, their preferences cover the maximum possible angle on the circle, which leads to greater

matching probability. Therefore, it is an upper bound of pεθ(x). As the result, thisM/M/∞ system’s

departure rate is higher or equal to the one in the original CTMC Xεθ.

Following classic results on M/M/∞ queue (see, e.g. Iglehart 1965), the Poisson random variable

Yεθ with load parameter R(εθ) :=λL

λSεθ+ γhas the stationary distribution of process Yεθ. Note that

Yεθ has departure rate no smaller than the one in Xεθ for all states. Similarly, we can consider an

M/M/∞ queue Y0, which has departure rate no greater than the one in Xεθ for all states. The

next proposition formalizes the relationship between Y0, Yεθ and Xεθ.

Proposition 1. Fixing 0≤ εθ≤ 1, we have

Y0 �1 Xεθ �1 Yεθ. (3.10)

Moreover, we have

EYεθhε(Yεθ, εθ)≤EXεθhε(Xεθ, εθ)≤EY0hε(Y0, εθ). (3.11)

We use the notation �1 representing the first order stochastic dominance of cumulative probability

distributions3. Proposition 1 shows that replacing Xεθ by Yεθ and Y0 lead to lower and upper

bounds for the platform’s objective function, respectively.

From this point forwards, we use Yεθ to replace Xεθ and let ε approaches 0 when seeking analytical

results. The next proposition formally establishes this treatment.

Proposition 2. Consider function hε in (3.9), we have

limε→0|EYεθhε(Yεθ, εθ)−EXεθhε(Xεθ, εθ)|= 0. (3.12)


Proposition 2 states that using random variable Yεθ instead of Xεθ in the objective function is a

good approximation when ε approaches 0. It sheds light on searching for analytical results for the

social welfare rate USW in (3.8) as EYεθhε(Yεθ, εθ) has closed-form expressions. From Proposition 2,

we know that Y0 is also a good approximation for Xεθ when ε goes to 0. The reason that we do not

use Y0 in this paper is that players departing only by reneging the corresponding system. Thus, it

has no matching aspect at all.

Furthermore, note that its expression (provided in the appendix) is still complex. Thus, we per-

form Taylor expansions over EYεθhε(Yεθ, εθ). The next proposition provides a simple approximation

of (3.8) when letting ε approach 0.

Proposition 3. Consider the Poisson random variable Yεθ with load parameter R(εθ) := λLλSεθ+γ

,

and denote J(θ, ε) :=EYεθhε(Yεθ, εθ)

λL. There exists a third order polynomial J of θ and ε such that,

limε→0

1

ε3

∣∣∣J(θ, ε)− J(θ, ε)∣∣∣= 0, or J(θ, ε) = J(θ, ε) + o(ε3). (3.13)

We provide the closed-form expression of function J(θ, ε) in the appendix. Proposition 3 greatly

simplifies expressions of utility functions under the limiting regime when ε goes to 0. In particular,

the expected utility can be approximated by λSλLJ(θ, ε), which is a simple polynomial of θ. Thus,

we have set up an approximation method for analytical results. Proposition 2 and 3 are the building

blocks for solving platform’s problems formulated in (3.3).

3.4. Optimal matching threshold

With the help of our approximation framework under limiting regime where ε approaches 0, we

seek analytical result of the optimal matching threshold. Note that using our approximation, social

welfare rate in (3.2) can be approximated as

λSλLJ(θ, ε), (3.14)

which is a simple polynomial polynomial of θ according to Proposition 3. Furthermore, denote

θ∗(ε)∈ [0,1+s] to be a maximum of J(θ, ε) for a given ε. We present the optimal matching threshold

in the next Theorem.

Theorem 1. Consider matching threshold εθ∗(ε), where

θ∗(ε) =

θM −

λS2cSγ

[cLθ2M + (cS − cL)s(2 + s)]ε, if cS ≥ cL,

θM −λS

2cLγ[cSθ

2M + (cL− cS)(1 + 2s)]ε, if cS < cL,

(3.15)

where θM follows (3.7). We have

θ∗(ε) = θ∗(ε) + o(ε), (3.16)


and there exists a ε≥ 0 such that for all ε≤ ε,

J(θ∗(ε), ε)≥ J(θM , ε). (3.17)

It is worth pointing out that function J(θ, ε) is concave in θ if ε is small (we provide a sufficient

condition in the proof of Theorem 1 in appendix). Therefore, solving for θ∗(ε) and θ∗(ε) are very

simple.

In Theorem 1, (3.15) provides an approximate optimal solution that maximizes the approximate

objective function J . The inequity in (3.17) confirms that despite the approximations, solution

θ∗(ε) indeed out-performs the myopic threshold θM for the original objective function J .

The expression of the optimal centralized matching threshold in (3.15) shows the fundamental

connection between market thickness and platform’s matching decisions. We can interpret the ratioλSγ

as the average number of short-lived players that a long-lived player shall encounter during the

time on the platform. Intuitively, the larger the value of λS is, the thicker the market of short-lived

players is. As a result, each long-lived player may encounter more matching partners on average,

which leads to better quality matches. Similarly, as γ decreases, long-lived players stay for longer

periods of time on average, which leads to thicker market as well. Under a thicker market, the

benefit of better matching quality outweighs the slight reduction in matching rate by using a

smaller matching threshold. The thicker the market is, the smaller θ∗(ε) is.

Furthermore, as it turns out, θ∗(ε) is no greater than θM . The reason that the platform is pickier

than matching myopically is to build market thickness. This is consistent with the strategical

delay/waiting in the matching literature (see e.g. Akbarpour et al. 2019). We can see the effect

of this move from looking at the Poisson random variable Yεθ, which represents the number of

long-lived players on the platform. Note Yεθ has meanλL

λSεθ+ γ, which increases as θ decreases (the

same result holds for random variable Xεθ as well). In other words, by building a thicker market of

long-lived players, the platform gives up some marginal welfare from low profit matches but gains

greater future welfare. Furthermore, we observe that the gap between the optimal and myopic

thresholds disappears when γ goes to infinity, representing the case where long-lived customers

become more like short-lived. This effect demonstrates the importance of having long-lived players

on the platform.

Not only the optimal matching threshold reveals the key connection between market density and

matching decision, it also has direct and practical implementations in reality. Consider the DiDi

Hitch example again. The platform provides a list of potential drivers to each requesting rider.

This can be viewed as a centralized decision if the platform can design the list, instead of showing

all potential drivers. The platform may only present the driver to the rider if their mismatch is no

greater than the optimal matching threshold.


4. Decentralized Matching

In this section, we study situations, under which the platform does not design the matching thresh-

old anymore. Instead, the platform only has control over the mismatch splitting functions, which is

announced prior to players’ arrivals. Thus, players from either side report their acceptable thresh-

olds upon arrivals based on the mismatch splitting function. Then both the platform and the

players commit to the thresholds. The platform has to show all matching partners to the focal

player so that she is eligible to match with anyone that leads to a mismatch ratio no greater than

her reported threshold. This setting forms a game among players since each player needs to consider

the best response for other players’ actions.

4.1. Short-lived players setting thresholds

Consider the case where short-lived players choose their thresholds. Since short-lived players never

stay, their game is simple and leads to myopic decisions.

Suppose the platform still follows the same mismatch splitting function defined in (3.5) and (3.6)

as it has the intention to maximize social welfare. The corresponding myopic threshold for short-

lived player is simply εθM . Apparently, this is not optimal if the platform maximizes the social

welfare according to (3.15). Short-lived players’ myopic behaviors ignore the benefit of building

market thickness.

In addition, the myopic behaviors of short-lived players may hurt their own overall utilities under

certain conditions. Define short and long-lived players’ expected utilities under our approximation

scheme as

US(εθ) :=

λSEφ(Yεθ)

[WS(0)Iφ(Yεθ)≤min ε{θ,1}+WS(φ(Yεθ)− ε)Imin ε{θ,1}≤φ(Yεθ)≤εθ

], if cS ≥ cL,

λSEφ(Yεθ)

[WS(φ(Yεθ))Iφ(Yεθ)≤min ε{θ,s}+WS(εs)Imin ε{θ,s}≤φ(Yεθ)≤εθ

], if cS < cL,

(4.1)

UL(εθ) :=

λSλL

Eφ(Yεθ)

[WL(φ(Yεθ))Iφ(Yεθ)≤min ε{θ,1}+WL(ε)Imin ε{θ,1}≤φ(Yεθ)≤εθ

], if cS ≥ cL,

λSλL

Eφ(Yεθ)

[WL(0)Iφ(Yεθ)≤min ε{θ,s}+WL(φ(Yεθ)− εs)Imin ε{θ,s}≤φ(Yεθ)≤εθ

], if cS < cL,

(4.2)

when the platform uses mismatch splitting functions in Lemma 1. We compare players’ expected

utilities between cases where the platform maximizes social welfare and short-lived players match

myopically, in the next Proposition.

Proposition 4. There exists a ε≥ 0, such that for all ε≤ ε,

US(εθM)≥US(εθ∗(ε)), if and only if s≤√cLcS

+ 1− 1, and cS ≥ cL. (4.3)


Furthermore,

UL(εθM)≥UL(εθ∗(ε)), if and only if s≥ cLcS

(1 +

√cS

(1 +

1

cL

)), and cS < cL. (4.4)

As Proposition 4 suggests, matching myopically only benefits short-lived player when s is small

and cS ≥ cL. In this parameter regime, εθM is only slightly bigger than ε. According to the splitting

functions in Lemma 1, short-lived player shares no mismatch if the actual mismatch is less than

ε. Thus, in this case, short-lived players only incur penalty if the mismatch is between ε and εθM ,

which is a very small region as s is small. As short-lived players’ mismatch penalty is low, the

increment in matching rate by using εθM instead of εθ∗(ε) outweighs the slight loss in matching

quality. The same intuitions explains the second statement in Proposition 4 for long-lived players.

In any parameter regime other than the two in Proposition 4, inducing short-lived players to be

pickier benefits all players.

Next, consider the cases where the platform solely maximizes short-lived players’ utilities, instead

of the social welfare. This setting can also be interpreted as all short-lived players form an union

who decides a matching threshold. Obviously, the mismatch splitting function in Lemma 1 is no

longer optimal. Instead, short-lived players always bear as little mismatch as possible and the

mismatch splitting function follows (3.5). Thus, short and long-lived players have utilities

US(εθ) := λSEφ(Yεθ)

[WS(0)Iφ(Yεθ)≤min ε{θ,1}+WS(φ(Yεθ)− ε)Imin ε{θ,1}≤φ(Yεθ)≤εθ

], (4.5)

UL(εθ) :=λSλL

Eφ(Yεθ)

[WL(φ(Yεθ))Iφ(Yεθ)≤min ε{θ,1}+WL(ε)Imin ε{θ,1}≤φ(Yεθ)≤εθ

]. (4.6)

Note that the utility function US is the same as US if cS ≥ cL. Therefore, if the platform aims to

maximize short-lived players’ utility, the optimal matching threshold εθ∗S has

θ∗S(ε) = θM −λS2γ

(2 + s2)ε, (4.7)

according to Theorem 1 when cS ≥ cL = 0.

Proposition 5. There exists a ε≥ 0, such that for all ε≤ ε,

US(εθM)≤US(εθ∗S(ε)), and UL(εθM)≤UL(εθ∗S(ε)). (4.8)

According to Proposition 5, if the mismatch splitting function always favors short-lived players,

the myopic matching threshold εθM always hurts players utilities comparing to εθ∗S. Thus, inducing

short-lived players to be pickier benefits everyone, including long-lived players.

Finally, it is worth noting that as long as the platform has control over the mismatch splitting

functions, it can still induce short-lived players using thresholds desired by the platform. For


example, if cS ≥ cL and the platform aims to maximize social welfare, it can use the following

mismatch splitting functions

{α∗S(φ), α∗L(φ)}=

{0, φ}, if 0≤ φ≤ ε,{φ− ε, ε}, if ε < φ≤ εθ∗(ε),{φ,0}, if φ> εθ∗(ε),

(4.9)

so that short-lived players is pickier and would not accept any match with mismatch ratio greater

than εθ∗(ε) defined in (3.15). If the platform deems to maximize short-lived players’ utilities, it can

substitute θ∗(ε) by θ∗S(ε) in (4.7) and achieves the goal.

4.2. Long-lived players setting thresholds

Suppose the platform allows each long-lived player to commit to a threshold εθ prior to joining the

platform. This is a game with potentially infinite number of players, and each long-lived player’s

utility depends on all players’ thresholds. In this section, we study the equilibrium matching thresh-

olds of long-lived players. Due to the complexity of this setting, we restrict our attention to the

case where long-lived players bear the entire mismatch ratio in each match, i.e., given mismatch

ratio φ, we assume that αL(φ) = φ and αS(φ) = 0. In reality, this mismatch splitting function is

well-suited for ride-sharing situations as each rider (short-lived) needs to be dropped off at her

requested destination while the driver (long-lived) is penalized for the entire mismatch ratio. With

this assumption, long-lived players can ignore short-lived players’ (PC) and report any 0≤ εθ≤ ε.

For the rest of this section, we first derive the value function for each long-lived player and provide

a heuristic method to evaluate it efficiently. We conclude this section with a numerical comparison

of this decentralized system to a centralized systems where the platform designs the matching

threshold. Using the numerical comparison, we identify the relationship between market thickness

and long-lived players’ thresholds, which is more complicated than the one with short-lived players.

Recall that φi represents the mismatch ratio between long-lived player i and a short-lived player.

Moreover, φ(x) = min{φi | i = 1, ..., x} represents the minimum mismatch ratio between a short-

lived player and the x long-lived players. We extend the definition of threshold policy to describe

this setting. When there are x long-lived players on the platform and let {εθi}i=1,...,x represents the

set of their reported thresholds. A threshold policy matches the arriving short-lived player with an

existing long-lived player i if and only if φi = φ(x) and φi ≤ εθi.

Next, we derive long-lived players’ equilibrium threshold so that no individual player would

deviate when all other players commit to it. Consider the situation where all other x long-lived

players commit to threshold εθ and the focal player uses a threshold εθ. We note there are two

immediate outcomes upon an arrival of a short-lived player.


First, the focal player is matched with the short-lived player and claims expected utility. Define

function

A(x, y, z) :=

∫ z

0

∫ φ

0

cL(ε− φ)dφhx(φ)dφ+

∫ 1

z

∫ y

0

cL(ε− φ)dφhx(φ)dφ, ∀0≤ y, z ≤ 1 and x≥ 0,

(4.10)

which represents the expected utility of a long-lived player whose threshold is y while all other x

long-lived players on the market use threshold z. Then there is

Eφ,φ(x)

[WL(φ)Iφ≤min{φ(x),εθ}

]=

{A(x, εθ, εθ), if θ≤ θ,A(x, εθ, εθ), if θ > θ,

(4.11)

where function WL is defined in (2.1); random variable φ represents the mismatch ratio between

the focal player and the short-lived player, follows uniform distribution on [0,1]; random variable

φ(x) represents the minimum mismatch ratio among all other players, follows distribution in (2.6);

and the indicator function inside expectation states the condition that the focal player shall be

matched.

Second, the focal player may be unmatched. Define functions

B1(x, y, z) =

∫ y

0

∫ 1

φ

dφhx(φ)dφ+

∫ z

y

∫ 1

y

dφhx(φ)dφ, (4.12)

B2(x, y, z) =

∫ 1

z

∫ 1

y

dφhx(φ)dφ, ∀0≤ y, z ≤ 1 and x≥ 1. (4.13)

Then we have

P(φ >min{φ(x), εθ} and φ(x)≤ εθ

)=

{B1(x, εθ, εθ), if θ≤ θ,B1(x, εθ, εθ), if θ > θ,

(4.14)

representing the probability that the focal player is unmatched while another long-lived player is

matched. Furthermore, we have

P(φ >min{φ(x), εθ} and φ(x)> εθ

)=B2(x, εθ, εθ), (4.15)

representing the probability that no player is matched.

Now we can define V (x, εθ, εθ) as the expected utility function for a focal player when all other

x long-lived players commit to a threshold with θ and the focal one uses a threshold with εθ.

We use Xεθ,εθ = {Xεθ,εθ(t), t≥ 0} to denote the process describing the total number of other long-

lived players on the platform besides the focal player. Moreover, denote N = {NL,NS,Nγ} as

a three-dimensional counting process describing the total number of arrival of long/short-lived

players and departure of long-lived players, respectively. Furthermore, we split NS(t) =NSm(t) +


NSn(t) representing the total numbers of short-lived arrivals that are matched and not by time t,

respectively. Thus, we have

Xεθ,εθ(t) =Xεθ,εθ(0) +NL(t)−Nγ(t)−NSm(t), ∀ t≥ 0. (4.16)

Therefore, we can write the utility function as an integral over time as

V (x, εθ, εθ) =

EN[∫ ∞

0

e−γtA(X(t), εθ, εθ)dNS(t)∣∣∣X(0) = x

], if θ≤ θ,

EN[∫ ∞

0

e−γtA(X(t), εθ, εθ)dNS(t)∣∣∣X(0) = x

], if θ > θ,

(4.17)

where function A defined in (4.10) represents the expected utility of the focal player upon arrival

of a short-lived player. Intuitively, e−γt comes from the P.D.F. of the exponential distribution for

reneging. It is equivalent to a discount factor.

The utility function V (x) (we drop variables εθ, εθ when there is no confusion) also solves the

following difference equation for x≥ 1:

V (x) =

λSA(x, εθ, εθ) +

[λSB1(x, εθ, εθ) +xγ

]V (x− 1) +λLV (x+ 1)

λL +λS(1−B2(x, εθ, εθ)) + (x+ 1)γ, if θ≤ θ,

λSA(x, εθ, εθ) + [λSB1(x, εθ, εθ) +xγ]V (x− 1) +λLV (x+ 1)

λL +λS(1−B2(x, εθ, εθ)) + (x+ 1)γ, if θ≤ θ,

(4.18)

with boundary condition

λSA(0, εθ, εθ) = (λL +λSB2(0, εθ, εθ) + γ)V (0)−λLV (1), (4.19)

where functions A, B1 and B2 are defined in (4.10), (4.12) and (4.13), respectively. We leave the

detailed derivation of the difference equations in Appendix C.

Next, define a long-lived player’s expected utility upon joining the platform when she reports εθ

while all other long-lived players report εθ as

EXεθ [V (Xεθ, εθ, εθ)], 0≤ εθ, εθ≤ 1, (4.20)

where the expectation follows P.A.S.T.A. That is, the number of long-lived players follows its steady

state distribution from each arriving player’s perspective (see, e.g., Wolff 1982). Furthermore, define

the market equilibrium threshold as εθE such that

εθE ∈ arg max0≤εθ≤1

EXεθE [V (XεθE , εθ, εθE)]. (4.21)

Therefore, in order to evaluate the equilibrium threshold of long-lived players, we need to compute

function V first.


It is very hard to find a closed-form solution to the difference equation in (4.18) even with our

limiting regime. Note that x can go to infinity in (4.18). Thus, solving this system of difference

equations numerally also appears challenging. Here, we provide a good approximation method to

solve for function V heuristically.

Proposition 6. Fix 0< θ, θ≤ 1.

(i) There exists an upper bound B such that V (x, θ, θ)≤B <∞ for all x≥ 0.

(ii) Fixing x∈Z+, denote Vx(x, θ, θ) as the solution to the system of equations that solves (4.19)

and (4.18) for x< x with Vx(x, θ, θ) = 0. We have

0≤ V (x, θ, θ)−Vx(x, θ, θ)≤B(

1 + γλL

)x−x , ∀0≤ x≤ x. (4.22)

Proposition 6(ii) suggests a very efficient heuristic to compute the value function V numerically,

which only involves solving a system of linear equations with x variables. Furthermore, Proposition

6(ii) also indicates the error introduced by this heuristic calculation decreases exponentially with

the choice of x. In fact, as Figure 1 suggests, the improvement on the value function of using

x= 5000 instead of x= 500 is negligible for all x < 500. So we do not need to choose a very large

x. Moreover, according to Proposition 1, we have Xεθ �1 Y0, where Y0 is a Poisson random variable

with loadλLγ

. Thus, the distribution function of Xεθ is also “light-tailed”. Therefore, following

Proposition 6(ii), our heuristic evaluation of function V also has little impact on long-lived players’

expected utility upon arrival, EXεθ [V (Xεθ, εθ, εθ)].

0 1000 2000 3000 4000 50000

0.01

0.02

0.03

0.04

0.05

0.06

Figure 1 Function Vx with x = 500 and x = 5000


Next, we compare the decentralized system with a centralized system. Consider the platform

solely maximizes long-lived players’ expected utility by designing a matching threshold. We define

the optimal centralized threshold as

εθ∗ ∈ arg max0≤εθ≤1

Eφ(Xεθ)

[cL(ε−φ(Xεθ)

)Iφ(Xεθ)≤εθ

]. (4.23)

In the following numerical examples, we focus on value function Vx instead of function V . As a

result, for x≥ 1, the market equilibrium threshold εθE in numerical examples is computed as

εθE ∈ arg max0≤εθ≤1

EXεθE [Vx(XεθE, εθ, εθE)]. (4.24)

20 25 30 35 400.5

0.52

0.54

0.56

0.58

0.6

0.62

0.64optimalequlibrium

(a)

optimalequlibrium

(b)

Figure 2 Equilibrium Thresholds v.s. Optimal Centralized Thresholds

In our numerical study, we find that θE ≤ θ∗. We provide examples in Figure 2. The message here

is quite clear. In this decentralized system, long-lived players are too picky. If all long-lived players

can increase their thresholds simultaneously to the optimal threshold εθ∗, everyone is better off. The

optimal matching threshold provides the perfect balance between mismatch penalty and matching

probability for long-lived players as a group. However, the optimal centralized matching threshold

does not lead to market equilibrium in the decentralized setting. When everyone else commits to

the optimal threshold, the focal player has the incentive to deviate by choosing a smaller threshold

as it leads to better match quality without matching probability scarifies much. If other players’

matching thresholds decrease at the same time, the system-wise matching rate decreases, which

leads to more long-lived players on the platform. Thus, it adds a secondary effect, which increases

market thickness on the side of long-lived players. When other players’ common matching threshold

drops to the equilibrium point, the focal player cannot benefit from being pickier. The reason


is that competition level between players increases as market thickness increases. There exists a

threshold in thickness that decreasing matching threshold leads to significant loss in matching rate

that outweighs the slight gain in matching quality.

The observations above encourage the platform to induce long-lived players increasing their

threshold. Platform can consider monetary tools such as subsidizing long-lived players so they can

be less picky. Operational tools such as setting deadlines on their stay on the market may also

induce players to be more tolerant.

5. Further Discussions, Extensions and Concluding Remarks

In this section, we provide discussions over some of the assumptions we have made in previous

sections. Furthermore, we relax some of them and provide detailed analysis. In particular, we

construct a numerical example relates our model to daily operations of a ride-sharing firm. We

conclude by summarizing the core of this paper and highlighting some extensions that are currently

beyond the scope of this paper.

5.1. Heterogeneous players

One of the assumptions we have made in this paper is that players are homogeneous within each

class. That is, all short-lived players have a same tolerance level and all long-lived players have

another tolerance level. It turns out, if there is heterogeneity among short-lived players, our model

may still be easily applied to. Suppose there are finite observable types of short-lived players who

have different tolerance levels. Then the platform can simply offer a unique threshold for each type.

We provide a study on centralized matching where there are two types of short-lived players in

Appendix D.1. The same solution method can still be applied and the structure of the optimal

matching thresholds greatly resembles that of (3.15). Having two types of short-lived players does

not change the insights we obtained in Section 3.

However, it is currently unclear what are the optimal thresholds if types are unobservable, which

leads to questions on mechanism design.

5.2. Platform’s objectives

In previous sections, we have focused on the case where the platform is a social planner maximizing

social welfare. Naturally, we can consider other objectives. Our analytical framework is able to

handle such extensions with properly defined utility functions of the platform. For example, the

platform may be profit maximizing by extract a portion of revenue generated per match. Consider

a ride-sharing platform, like Didi Hitch, who takes α ∈ [0,1] portion of the revenue generated in

each match and maximizes its own expected profit. We also assume that riders are short-lived


and drivers are long-lived as riders are actively looking for outside options (e.g. taxi, bus, etc.).

Furthermore, we assume drivers do not have utility besides the fare from riders. For each successful

match, the rider pays u to the platform and it takes a portion α ∈ [0,1] of it before distribute

the rest to the driver. In each match, drivers bear the mismatch with coefficient c and riders

are dropped off at requested destinations. We define ε :=u

cas a driver’s maximum tolerance on

mismatch ratio if the platform gives all fare to the driver. In Appendix D.2, we derive some results

based on our analytical framework for this profit maximizing platform.

Furthermore, we conduct a numerical study based on this profit maximizing platform. The

numerical example not only relates our model to a ride sharing platform’s daily operations, it also

shows that our closed-form expression of the optimal matching threshold is able to perform well

in reality.

For this numerical example, we consider a circular city with radius 7.5km and all riders and

drivers depart from the center of the circle. Drivers and riders have the same arrival rate of λS =

λL = 3 people per minute. Drivers are long-lived players who have reneging rate of γ = 0.5 people

per minute. In other words, drivers may stay for 2 min on average without a match.

Driver CharacteristicsTravel speed 20km/hrTotal fare $ 2.170Income $ 1.953Opportunity cost $1.500/kmTravel cost $0.102/kmTotal cost $1.602/kmMax detour $1.180kmMax mismatch ratio 0.057

Table 1 Driver Characteristics

In this example, we use DiDi Hitch’s pricing schematics. The driver earns 0.7 dollar for the first 2

kilometers and earns 0.14 dollar per kilometer afterwards. The platform takes 10 percent of the fare

(α= 0.1) paid to the driver as its profit, which is the current policy at DiDi Hitch. Furthermore,

drivers are commuters who has outside opportunity cost of 30 dollar per hour. Using rush hour

average travel speed of 20 kilometers per hour, we convert drivers’ costs into the ones with units

of dollar per kilometer, reported in Table 1. The travel cost is estimated with a car consumes 10

liter per 100 kilometers in Beijing (gas price $1.02 per liter). Furthermore, we can calculate the

maximum detour length a driver is willing to take in this setting. Since the detour is the length

of the arc between a driver and a rider’s destinations on a circle with 7.5km radius, we have the

maximum mismatch ratio a driver would take as

ε=u

c=

$2.170

$1.602/km×π× 7.5km= 0.057. (5.1)


0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 10.022

0.024

0.026

0.028

0.03

0.032

0.034

0.036

0.038

0.04

taylor exppoissonrealopt

te

opt

Figure 3 Platform’s Profit Function

ResultsOptimal Matching threshold 1.329kmClosed-form Matching threshold 1.310kmOptimal Profit $0.03841/minProfit with Closed-form Matching threshold $0.03838/min

Table 2 Results Summery

In the example, we also consider the platform may subsidize drivers. We leave the detailed

derivations in Appendix D.2. In this section, we only highlight that using our solution schematics,

the optimal matching threshold can be approximated as εθ∗ with

θ∗ = 1− λS(2−α)α

2γε+ o(ε)≈ 0.967. (5.2)

Equivalently, it represents a threshold on the detour with maximum length of 1.310 kilometers.

The results of this numerical example are presented in Figure 3 and Table 2. In Figure 3, we

can observe that using Poisson distribution (dash-dot line) to approximate the distribution of the

original (solid line) Birth-Death process hardly sacrifices accuracy. The main discrepancy is intro-

duced with Taylor Expansion over the objective function (dash line). However, our approximated

optimal matching threshold (indicated by the solid dot in Figure 3) with closed-form expression in

(5.2) still performs very well as it is very close to the optimal threshold (indicated by the star).

5.3. Concluding remarks

We study threshold policies in a matching market with short and long-lived players. Players’

individual preferences are characterized by a circular model, which links preferences of players


directly with different matching probabilities. Under the limiting regime where players are very

mismatch-sensitive, ε approaches 0, we are able to obtain optimal matching thresholds in closed-

form for various scenarios. The closed-form expressions of optimal thresholds provide clear intuition

on the relationship between matching policies and market thickness.

In Section 4 with short-lived players deciding their matching thresholds, we provide a simple

method inducing players to be pickier. However, if the platform loses control over the mismatch

splitting function, it needs other operational tools to achieve this goal. Similarly, we have men-

tioned some tools to induce long-lived players to be more tolerant if they are match makers. It is

interesting to analyze the effectiveness of any given method. Next, when long-lived players design

their matching thresholds, we assume that they bear the entire mismatch ratio in each match.

We can certainly relax this restriction and obtain numerical results. Furthermore, we do not allow

long-lived players to adjust their matching thresholds dynamically due to the complexity of the

game. Recent development on stationary equilibrium concept (Adlakha et al. 2015) may shed light

if we allow dynamic thresholds.

Endnotes

1. Throughout this paper, A�B represents A is much less than B.

2. We show that the stationary distribution always exists in Appendix A.

3. For any two random variables X and Y with cumulative probability distributions FX and FY

respectively, X first-order stochastically dominates X, written X �1 Y , if and only if FX(k)≤ FY (k)

for all k.

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Appendix

A. Existence of the Stationary Distribution

It is well understood that a birth-death process has stationary distribution if and only if

∞∑x=1

Πxk=1

λ(k− 1)

µ(k)≤∞,

where λ(x) is the system birth rate and µ(x) is the system death rate when system state is x. Forprocess Xεθ, we have

∞∑x=1

λxLΠxk=1(λSpεθ(k) + γk)

≤∞∑x=1

λxLΠxk=1γk

=∞∑x=1

(λxLγ

)k1

x!= exp

(λLγ

)− 1<∞, (A.1)

where the last equality follows the p.m.f. of a Poisson random variable with load λL/γ. Thus, thestationary distribution of Xεθ always exists for positive and finite λL and γ.

B. Proofs

Proof of Lemma 2. We prove this result by showing that g1(x) := εθx− [1− (1− εθ)x] is non-negative for all x≥ 1.

By taking the first and second order derivatives of g1(x), we reach

dg1(x)

dx= εθ+ (1− εθ)x ln(1− εθ), and

d2 g1(x)

dx2= (1− εθ)x(ln(1− εθ))2.

Since 0≤ εθ≤ 1, we haved2 g1(x)

dx2≥ 0 and therefore

dg1(x)

dx≥ dg1(x)

dx

∣∣∣∣∣x=1

= εθ+ (1− εθ) ln(1− εθ) =: g2(θ).

We can verify that g2(θ) is increasing in θ by checking

dg2(θ)

dθ=−ε ln(1− εθ)≥ 0,

since 0≤ εθ≤ 1. Therefore, we have

dg1(x)

dx≥ g2(0) = 0.

As the result, we have g1(x) is increasing in x. Since g1(1) = 0, we reach the final result thatg1(x)≥ 0 for all x≥ 1.

We show a more general result in order to prove Proposition 1.

Lemma 3. Consider two Birth-Death Processes X1 = {X1(t), t≥ 0} and X2 = {X2(t), t≥ 0} withthe same arrival rate λ. Process X1 and X2 have departure rates µ1(x) and µ2(x) such that µ1(x)≥µ2(x) for all states x≥ 1. Denote X1 and X2 as the random variables that take stationary distri-butions (suppose they exist) of processes X1 and X2. We have

X2 �1 X1. (B.1)


Proof of Lemma 3. Denote the P.M.F. (C.M.F.) of X1 and X2 as f1(·) and f2(·) (F1(·) andF2(·)), respectively. We show that F1(x)≥ F2(x) for all x≥ 0.

First, note that for Birth-Death processes, one can verify that

f1(x)

f2(x)=f1(0)

f2(0)Πxi=1

µ2(x)

µ1(x). (B.2)

Since we have µ1(x)≥ µ2(x) for all states x≥ 1, there is Πxi=1

µ2(x)

µ1(x)≤ 1, which implies f1(0)≥ f2(0)

since f1(·) and f2(·) are well-defined p.m.f.Next, we prove the desired result by contradiction. Suppose there exists some x≥ 1 such that

F1(x)< F2(x) and let k := min{x |F1(x)< F2(x)}. By the definition of k, we have F1(k)< F2(k)and F1(k− 1)≥ F2(k− 1). These two inequalities imply that f1(k)< f2(k).

Note by (B.2), we have that

f1(x+ 1)

f2(x+ 1)=f1(x)µ2(x+ 1)

f2(x)µ1(x+ 1),

which implies that f1(x) < f2(x) for all x > k sinceµ2(x+ 1)

µ1(x+ 1)≤ 1 for all x ≥ 1. Therefore, since

both f1(·) and f2(·) are well-defined p.m.f., we reach the following inequalities

∞∑i=k+1

f1(i)<∞∑

i=k+1

f2(i), andk∑i=0

f1(i)<k∑i=0

f2(i).

By adding up the two inequalities above, we reach contradiction as 1 < 1. Therefore, we haveF1(x)≥ F2(x) for all x≥ 0.

Proof of Proposition 1. The proof of the first statement follows the result of Lemma 3 directly.In addition, for Poisson random variables Yεθ and Y0, first order stochastic dominance can be showndirectly by comparing the expressions of their C.M.F.

In order to prove the second statement, we show that hε(x, εθ) is a increasing function of x≥ 0.Then the desired result follows by the property of first order stochastic dominance. We prove thestatement above for cS ≥ cL.

First consider θ≤ 1. Let A= scS + cL, and

G(A,x) := εA− cL1 +x

+cL(1 + εθx)− εA(1 +x)

1 +x(1− εθ)x, (B.3)

so one can verify that G(A,x) = hε(x, εθ). Thus, we only need to show that G(A,x) is increasingw.r.t. x. We verify this statement by taking the first derivatives,

dG(A,x)

dx=

1

(1 +x)2

[cL(1− (1− εθ)1+x) + (1 +x)(1− εθ)x[cL(1 + εθx)− εA(1 +x)] ln(1− εθ)

].

By taking the first derivative w.r.t. A, there is

−(1− εθ)x ln(1− εθ)ε > 0,

since εθ < 1. Thus,dG(A,x)

dxis increasing in x and by using A− cLθ≥ 0,

dG(x)

dx>

cL(1 +x)2

[[(1 +x) ln(1− εθ)− 1] (1− εθ)1+x + 1

].


One may verify that the right hand side of the inequality is increasing in θ when εθ < 1, so bysetting θ= 0, we reach

dG(x, εθ)

dx> 0.

Next, consider 1< θ≤ 1 + s. Again, take the first derivatives of hε(x, εθ) w.r.t x and θ, we have

dh2ε(x, εθ)

dxdθ= cSε

2(1 + s+ θ)(1− εθ)x−1(1 +x ln(1− εθ)),

which is first increasing then decreasing w.r.t. x. Thus, we only need to verify for θ= 1 and 1 + s.

dhε(x, εθ)

dx

∣∣∣θ=1≥ 0, and

dhε(x, εθ)

dx

∣∣∣θ=1+s

= 0

Thus, we havedhε(x, εθ)

dx> 0 for all 1< θ≤ 1+s. Therefore, we have function hε(x, εθ) is increasing

w.r.t. x.The proof for the case where cS < cL follows the same step and thus the proof is omitted.

Proof of Proposition 2. First, recall that Yεθ is a Poisson random variable with parameter R(εθ).Therefore, as ε→ 0, Yεθ converges to Y0 in distribution by definition.

Next, note function hε(x, ·) is strictly increasing in x. Recall Proposition 1 and by definition offirst order stochastic dominance, we have

EYεθhε(Yεθ, εθ)≤EXεθhε(Xεθ, εθ)≤EY0hε(Y0, εθ),

which gives

|EYεθhε(Yεθ, εθ)−EXεθhε(Xεθ, εθ)| ≤ |EYεθhε(Yεθ, εθ)−EY0hε(Y0, εθ)|.

Consider another M/M/∞ queue with service rate λSθ. By Lemma 3, we have that barYεθ �1 Yθ

since ε� 1 where Yθ represents the Poisson random variable with loadλL

γ+λSθ.

Next, consider function hε(x, ·) in (3.9), which is an increasing function w.r.t. x according tothe proof of Proposition 1. One can easily verify that limx→∞ hε(x, ·) = εA, where A= scS + cL ifcS ≥ cL and A= scL + cS otherwise. Since ε� 1, we have function hε is upper bounded by A<∞.

Thus we have

limε→0|EYεθhε(Yεθ, εθ)−EY0

hε(Y0, εθ)| ≤ limε→0|EYθhε(Yθ, εθ)−EY0

hε(Y0, εθ)|

= |EYθ limε→0

hε(Yθ, εθ)−EY0limε→0

hε(Y0, εθ)|= 0,

where the first equality follows the fact EY0A<∞, EYθA<∞ together with Dominated Conver-

gence Theorem, and the last equality follows simple algebra.

Proof of Proposition 3. We only derive the result for cS ≥ cL as its counterpart follows the

exact same steps. Since Yεθ is a Poisson random variable with load factor R(εθ) =λL

γ+λSεθ, one

can verify that

J(θ, ε) =EYεθhε(Yεθ, εθ) =e−ε(2+θ)R(εθ)

R(εθ)

[cLe

ε(1+θ)R(εθ) (1 + (εR(εθ)− 1))eεR(εθ)


+cS(−eε(1+θ)R(εθ) + εsR(εθ)eε(2+θ)R(εθ) + (1 + ε(θ− 1− s)R(εθ))e2εR(εθ)

)],

(B.4)

with the help of the probability generating function for Poisson random variables.Next, we perform third order Taylor expansions over (B.4). Denote

J(θ, ε) :=(cL− cS(1− 2(1 + s)θ+ θ2))

2γε2

+cL(λL + 3λSθ)− 3λScSθ(1− 2(1 + s)θ+ θ2)−λLcS(1− 3(1 + s)θ2 + 2θ3)

6γ2ε3. (B.5)

Following the same steps, we can derive function J for cS < cL,

J(θ, ε) :=cSs

2− cL(s− θ)2 + 2cLθ

2γε2

+3(cL− cS)λLs

3 + 3(cL− cS)λSs2θ− 3cL(λL + 2λS)(1 + s)θ2 + cL(2λL + 3λS)θ3

6γ2ε3.

(B.6)

Using the definition of function J(θ, ε), we have,

EYεθhε(Yεθ, εθ) = λLJ(θ, ε) = λLJ(θ, ε) + o(ε3),

which implies the result according to the definition of “little-o” notations.Note that one can also perform Taylor expansion over function h directly before taking the

expectation w.r.t. Yεθ. However, the “higher-order” terms (o(ε3) and higher) contain the randomvariable Yεθ, which can be infinity. Then one need to verify that these “higher-order” terms areindeed going to zero as ε→ 0, which can be shown using the “light-tail” property of Poisson randomvariables. We omit details for this procedure.

Proof of Theorem 1. First, we provide procedures on obtaining the expression of θ∗ for casewhere cS ≥ cL. Fixing ε ≤ ε (we provide the expression of ε towards the end of this part of theproof), solve the maximization problem

max0≤θ≤1+s

J(θ, ε). (B.7)

Suppose 1≤ θ≤ 1 + s. According to (B.5), we have

max0≤θ≤1+s

J(θ, ε) = max0≤θ≤1+s

ε2

[(cL− cS(1− 2(1 + s)θ+ θ2))

2γ

+cL(λL + 3λSθ)− 3λScSθ(1− 2(1 + s)θ+ θ2)−λLcS(1− 3(1 + s)θ2 + 2θ3)

6γ2ε

],

which is concave and gives unique finite optimal solution if ε≤ ε1 :=2cSγs

λS(cL + cSs(2 + s)),

θ∗(ε) =εcS((λL + 2λS)(1 + s)−

√cS(cS(γ+ ε(λL + 2λS)(1 + s))2− ε(2λL +λS)(2cSγ(1 + s) +λSε(cS − cL)))

cS(2λL + 2λS)ε

= θ∗(ε) + o(ε),


where θ∗(ε) is defined in (3.15), and the second equality follows Taylor expansion. Therefore, theoptimal objective function is

J(θ∗(ε), ε) = ε2

[cL + cSs(2 + s)

2γ− cL(λL + 3λS(1 + s)) + cSs(3 +λS(2 + 3s+ s2) +λL(3 + 3s2 + s2))

6γ2ε+ o(ε)

].

= J(θ∗(ε), ε) + o(ε), (B.8)

where the second equality can be verified easily by using the closed-form expression of J(θ, ε) in(B.5).

Second, suppose 0≤ θ < 1. We have

J(θ, ε) = ε2

[θ(2cSs+ cL(2− θ))

2γ− 3cS(λL + 2λS)s+ 3cLλS(2− θ)− cLλL(3− 2θ)

6γ2ε

],

which is strictly increasing as ε≤ ε2 :=2cSγs

cLλS + 2cSs(λL +λS). Thus, we have θ∗ = 1, which gives

J(1, ε) = ε2[cL + 2cSs

2γ− 3cSs(λL + 2λS) + cL(λL + 3λS

6γ2ε

]. (B.9)

By comparing expressions in (B.8) and (B.9), we conclude that the optimal solution is

θ∗(ε) = θ∗(ε) + o(ε), (B.10)

if ε≤ ε3 :=3cSγs

3cLλS + cS(λL + 3λS)(3 + s)s. Thus, we have obtained the expressions of θ∗ for cS ≥ cL

and ε≤ ε := min{ε1, ε2, ε3}.The same procedure can be applied to the case where cS < cL and thus is omitted.For the second statement in (3.17), one one can verify that

limε→0

1

ε4(J(θ∗(ε), ε)−J(1 + s, ε)) =

λ2SλL(cL + cSs(2 + s))2

8cSγ3≥ 0, (B.11)

by plugging in θ= θ∗(ε) and θ= 1 + s into (B.4) respectively, and perform 4th order Taylor expan-sions. The statement in (3.17) follows afterwards.

Note that the method to reach the closed-form expression of ε with function J can be extendedto other proofs in the appendix. Thus, we omit this procedure from now on.

Proof of Proposition 4. Since our approximation method provides closed-form expressions usingTaylor expansions, we can prove this Proposition using simple algebra.

Suppose cS ≥ cL, we have

limε→0

1

ε4(US(εθ∗(ε))−US(εθM)) =

λLλ2S

8cSγ3[c2Ss

2(2 + s)2− c2L]. (B.12)

Note the right-hand-side has non-negative positive critical point of s∗ =

√cLcS

+ 1−1. One may also

verify that the right hand side of (B.12) has first order derivatives w.r.t. s greater than 0 at s= s∗.Therefore, we conclude that there exists a ε such that for all ε≤ ε we have US(εθ∗(ε))−US(εθM)≥ 0if and only if s≤ s∗.


Suppose cS < cL, we have

limε→0

1

ε4(US(εθ∗(ε))−US(εθM)) =

λLλ2Ss

2

4cSγ3[cL + 2cLs+ cSs

2]≥ 0. (B.13)

Thus, we have shown the first statement.Suppose cS ≥ cL, we have

limε→0

1

ε4(UL(εθ∗(ε))−UL(εθM)) =

λLλ2S

4cSγ3[cL + cSs(2 + s)]≥ 0. (B.14)

Finally, suppose cS < cL, we have

limε→0

1

ε4(UL(εθ∗(ε))−UL(εθM)) =

λLλ2S

8cSγ3[(cL + 2cLs)

2− c2Ss

4], (B.15)

where the right hand side has a unique non-negative critical point s∗ =cLcS

(1 +

√cScL

+ cS

). Fur-

thermore, one can easily verify that the right hand side of (B.15) has negative first order derivativew.r.t. s at s= s∗. Therefore, we have shown the second statement as well.

Proof of Proposition 5. This prove follows the same step as the proof of Proposition 4.We have

limε→0

1

ε4(US(εθ∗S(ε))−US(εθM)) =

λLλ2ScS

8γ3(2 + s2)2 ≥ 0, (B.16)

and

limε→0

1

ε4(UL(εθ∗S(ε))−UL(εθM)) =

λLλ2ScL

8γ3(2 + s2)≥ 0. (B.17)

Proof of Proposition 6. (i) We show the result for θ≤ θ. First we show that g(x, θ) :=A(x, θ, θ)is decreasing w.r.t. x. Note that

∂ g(x, θ)

∂ x=

1

(1 +x)2(2 +x)2

{3− 4ε+ 2x− 4εx− εx2− (1− εθ)x+1(3 + 2x+ ε[θ(1 +x)2− (2 +x)2])

+(1− εθ)1+x(1 +x)(2 +x)(1 + ε(θ− 2 +x(θ− 1))) ln(1− εθ)}. (B.18)

Furthermore, we have

∂2 g(x, θ)

∂ x∂ θ=−ε2(θ− 1)(1− εθ)x ln(1− εθ)≤ 0,

since θ≤ 1. Thus, we have

∂ g(x, θ)

∂ x≤ ∂ g(x,0)

∂ x= 0.

Therefore, we reach the result that for any x≥ 1,

A(x, θ, θ)≤A(0, θ, θ). (B.19)


Recall the expression of the value function as an integral in (4.17). We have

V (x, εθ, εθ) = EN[∫ ∞

0

e−γtA(x, θ, θ)dNS(t)

]≤ EN

[∫ ∞0

e−γtA(0, θ, θ)dNS(t)

]≤ A(0, θ, θ)

∫ ∞0

e−γt dt=A(0, θ, θ)

γ=:B, ∀x≥ 0, (B.20)

where the first inequality follows (B.19) and the second inequality follows the definition of NλS .

The proof for the case where θ≤ θ follows the same logic so it is omitted.(ii) Again, we only show the result for θ≤ θ as the counterpart follows the same steps.Consider a pure birth process with X = {X(t), t≥ 0} with arrival rate λL. Define τ = min{t≥

0 |X(t) = x} and τ = min{t≥ 0 | X(t) = x}. Note that τ follows Erlang distribution with parameterλL and x− X(0) since X is a pure Birth (Poisson) process.

We show that if two processes have X(0) = X(0)≤ x, then there is P(tκ < t)≥ P(tκ < t) for allt≥ 0 through coupling. First, note that X is a counting process. Next define left-continuous jumpprocess Y such that

Y (t) = X(t)−Z(Y (t−)), t≥ 0, (B.21)

where Z(Y ) = {Z(Y (t−)) | t≥ 0} is a counting process with arrival rate γy+λSB1(θ, θ, y) if Y (t−) =

y. Denote τ = min{t ≥ 0 |Y (t) = x}. Thus, by construction, Y (t)D= X(t) (equal in distribution),

which implies that

τD= τ. (B.22)

Since γ > 0, we have X(t)≥ Y (t) almost surely for all t≥ 0, which implies that

τ ≤ τ , a.s., (B.23)

which gives

P(τ < t)≥ P(τ < t) = P(τ < t), ∀ t≥ 0, (B.24)

where the inequality follows (B.23) and the equality follows (B.22).By the definition of first order stochastic dominance and the fact that e−γt is strictly decreasing

w.r.t. t, we reach

Eτ[e−γτ

]≤Eτ

[e−γτ

]. (B.25)

Now, we can write the utility functions V (x), Vx(x) as an integrals over time similar to (4.17).For X0 = x, there are

V (x, εθ, εθ) = EN[∫ τ

0

e−γtA(X(t), θ, θ)dNS(t) + e−γτV (x)

], (B.26)

Vx(x, εθ, εθ) = EN[∫ τ

0

e−γtA(X(t), θ, θ)dNS(t) + 0

]. (B.27)

Therefore,

V (x)−Vxδ(x) = Eτ[e−γτ

]V (x)≤Eτ

[e−γτ

]V (x)≤Eτ

[e−γτ

]B =

(1 +

γ

λL

)−(x−x)

B,

where the first inequality follows (B.25), second inequality follows part(i) and last equality followsthe moment generating function of Erlang random variables.


C. Patient Players’ Utility Function

Then we can write out value functions recursively in a heuristic manner. Fix θ≤ θ and consider aninfinitesimal time period [t, t+ δ),

V (x) = (1− γδ){λLδV (x+ 1) +xγδV (x− 1) + (1−λLδ−λSδ−xγδ)V (x)

+λSδ[A(x, εθ, εθ) +B1(x, εθ, εθ)V (x− 1) +B2(x, εθ, εθ)V (x)

]}= (1− γδ)

{λSδA(x, εθ, εθ) +λLδV (x+ 1) +

[1−λLδ−λSδ[1−B2(x, εθ, εθ)]−xγδ

]V (x)

+[λSB1(x, εθ, εθ) +xγ

]δV (x− 1)

}.

By dividing both sides with δ and then take δ→ 0, we reach,

V (x) =λSA(x, εθ, εθ) +

[λSB1(x, εθ, εθ) +xγ

]V (x− 1) +λLV (x+ 1)

λL +λS(1−B2(x, εθ, εθ)) + (x+ 1)γ.

The derivation for θ > θ follows the same steps and thus it is omitted.

D. Extensions

D.1. Two types of short-lived players

Consider heterogeneity within short-lived players. Assume with probability p1 ∈ [0,1], an incomingshort-lived player is type 1 and with probability p2 := 1−p1, she is type 2. Type 1 players are moretolerant comparing to type 2 such that

s1 ≥ s2, where s1 :=u1

cSε, and s2 :=

u2

cSε, (D.1)

where u1 and u2 represent players’ valuations per match for type H and L, respectively.The platform can design two matching thresholds θ1 and θ2 for type 1 and type 2 players,

respectively. Thus, the expected matching probability is

pεθ1,εθ2(x) = 1− [p1(1− εθ1)x + p2(1− εθ2)x], (D.2)

which is upper bounded by (p1θ1 + p2θ2)εx.Using the similar approximation method as in Section 3, we choose an M/M/∞ system with load

factor R(εθ1, εθ2) :=λL

γ+λSε(p1θ1 + p2θ2). We denote Xεθ1,εθ2 (we drop the subscript when there

is no confusion) as the Poisson random variable with load factor R(εθ1, εθ2). With slight abuse ofnotation, we modify the definition of function W in (2.3) to

WS1= cS(εs1−αS1

(φ)), and WS2= cS(εs1−αS2

(φ)),∀φ∈ [0,1],

where αS1(φ) and αS2

(φ) represents the mismatch ratio shared by Type 1 and Type 2 short-livedplayers. Furthermore, we modify the definition of function WSW in (3.1) into

WSW1(αS1

(φ), αL(φ)) := WS1(αS1

(φ)) +WL(αL(φ)), ∀0≤ φ≤ εθ1,

WSW2(αS2

(φ), αL(φ)) := WS2(αS2

(φ)) +WL(αL(φ)), ∀0≤ φ≤ εθ2.

Therefore, the platform’s problem is

maxθ1,θ2,αS1

(·),αS2(·),αL(·)

λSEφ(X) [p1WSW1(αS(φ(X)), αL(φ(X)))I{φ(X)≤ εθ1}


+p2WSW2(αS2

(φ(X)), αL(φ(X)))I{φ(X)≤ εθ2}] (D.3)

s.t.

0≤ αS1(φ)≤ εs1, ∀φ≤ εθ1

0≤ αS2(φ)≤ εs2, ∀φ≤ εθ2

0≤ αL(φ)≤ ε, ∀φ≤ εθ,αL(φ) +αS1

(φ) = φ, ∀φ≤ εθ1,αL(φ) +αS2

(φ) = φ, ∀φ≤ εθ2,θ≥ 0.

We can use the same approximation methods offered in Section 3 to reach optimal matchingthresholds in closed-forms as ε approaches zero. Here, we only highlight some results for the casewhere cS ≥ cL.

First, as an analogy to Lemma 1, we have

{α∗S1(φ), α∗L(φ)}=

{{0, φ}, if 0≤ φ≤ ε,{φ− ε, ε}, if ε < φ≤ ε(1 + s1);

(D.4)

and

{α∗S2(φ), α∗L(φ)}=

{{0, φ}, if 0≤ φ≤ ε,{φ− ε, ε}, if ε < φ≤ ε(1 + s2),

(D.5)

as the optimal mismatch splitting decisions for Type 1 and Type 2 short-lived players, respectively.Moreover, as an analogy to Theorem 1, the optimal matching thresholds in the limiting regime areεθ∗1 and εθ∗2 , such that

θ∗1 = 1 + s1−λS[cL + cSp1s1(2 + s1) + cSp2s2(2 + s2)]

2γcSε+ o(ε), (D.6)

θ∗2 = 1 + s2−λS[cL + cSp1s1(2 + s1) + cSp2s2(2 + s2)]

2γcSε+ o(ε). (D.7)

As the expressions above greatly resemble the thresholds in (3.15), the intuition behind the optimalmatching thresholds is the same as the one in Section 3.

D.2. Platform’s profit

Unlike previous sections, we assume that the platform can subsidize drivers with amount S(·),which is a function of the realized mismatch ratio in each match. The reason we do not considersubsidy for riders is that it may greatly affect riders’ arrival (market entry) rate as they have manyoutside options. On the other hand, drivers are commuters, instead of professional drivers, so theirside of arrival is relatively insensitive towards subsidy.

Driver’s utility in this setting with subsidy is defined as

WD(S(φ)) = c[(1−α)ε−φ] +S(φ), ∀0≤ φ≤ εθ, (D.8)

if the matching threshold is εθ. Therefore, the platform’s profit maximization problem is

maxθ,S(·)

Eφ(Xεθ)

[αεc−S(φ(Xεθ))

]I0≤φ(Xεθ)≤εθ (D.9)

such that {WD(S(φ))≥ 0, ∀φ≤ εθ,θ≥ 0,

where the first constraint enforces long-lived players’ participation. Immediately, we can reach theoptimal subsidy decision S∗(·) as the objective is decreasing w.r.t. S(φ) given any realized mismatchratio 0≤ φ≤ εθ and α∈ [0,1]. Therefore, according to the second constraint, we have

S∗(φ) =

{c[φ− (1−α)ε], if (1−α)ε < φ≤ εθ,0, otherwise.

(D.10)


Therefore, the optimization problem in (D.9) can be simplified to

maxθ≥0

Eφ(Xεθ)αεcI0≤φ(Xεθ)≤(1−α)ε + c(ε−φ(Xεθ)

)I(1−α)ε<φ(Xεθ)≤εθ. (D.11)

Using similar approximation method in Section 3, we can approximate the objective function as

λSc exp(− λLεθ

λSεθ+γ

)λL

[γ

(1− exp

(λLε(θ+ 1−α)

λSεθ+ γ

))+λSεθ

(1− exp

(−λLε(θ+ 1−α)

λSεθ+ γ

))

−λLε(

1−α exp

(λLεθ

λSεθ+ γ

)+ θ

)]

=cλLλSε

2

γ

[1

2

((2−α)α− (θ− 1)2

)− 1

6γ

[3λS((2−α)α− (θ− 1)2)θ+λL((1−α)3− 3θ2 + 2θ3)

]ε+ o(ε)

](D.12)

and have the optimal matching threshold εθ∗S(ε) with

θ∗S(ε) = 1− λS(2−α)α

2γε+ o(ε), (D.13)

when ε is close to zero.

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