matching supply and demand with mismatch-sensitive players*people.duke.edu/~mc468/matching.pdf ·...
TRANSCRIPT
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.
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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
4 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 5
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
6 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 7
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.
8 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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),
Author: Matching Supply and Demand with Mismatch-Sensitive Players 9
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)
10 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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.
Author: Matching Supply and Demand with Mismatch-Sensitive Players 11
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.
12 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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)
Author: Matching Supply and Demand with Mismatch-Sensitive Players 13
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)
14 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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.
Author: Matching Supply and Demand with Mismatch-Sensitive Players 15
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)
16 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 17
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.
18 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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) +
Author: Matching Supply and Demand with Mismatch-Sensitive Players 19
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.
20 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 21
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
22 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 23
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)
24 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
Author: Matching Supply and Demand with Mismatch-Sensitive Players 25
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.
References
Adlakha, S., Johari, R., and Weintraub, G. (2015). Equilibria of dynamic games with many players: Existence,
approximation,and market structure. Journal of Economic Theory, 156:269–316.
Afeche, P., Diamant, A., and Milner, J. (2014). Double-sided batch queues with abandonment: Modeling
crossing networks. Operations Research, 62(5):1179–1201.
Akan, M., Alagoz, O., Ata, B., and Erenay, F. (2012). A broader view of designing the liver allocation
system. Operations Research, 60(4):757–770.
Akbarpour, M., Li, S., and Gharan, S. (2019). Thickness and information in dynamic matching markets.
Working Paper.
Anderson, R., Ashlagi, I., Y., K., and Gamarnik, D. (2017). Efficient dynamic barter exchange. Operations
Research, 65(6):1446–1459.
Ashlagi, I., Burq, M., Jaillet, P., and Manshadi, V. (2019). On matching and thickness in heterogeneous
dynamic markets. Operations Research, 67(4):927–949.
26 Author: Matching Supply and Demand with Mismatch-Sensitive Players
Buke, B. and Chen, H. (2017). Fluid and diffusion approximations of probabilistic matching system. Queueing
System, 86:1–33.
Chen, Z., Hu, M., and Zhou, Y. (2019). Optimal dynamic matching under impatient demand and patient
supply. Working Paper.
Gurvich, I. and Ward, A. (2014). On the dynamic control of matching queues. Stochastic Systems, 4(2):479–
523.
Hu, M. and Zhou, Y. (2019). Dynamic type matching. Working Paper.
Iglehart, D. (1965). Limiting diffusion approximations for the many server queue and repairement problem.
Journal of Applied Probability, 2(2):429–411.
Kanoria, Y. and Saban, D. (2019). Facilitating the search for partners on matching platforms. working paper.
Ozkan, E. and Ward, A. (2016). Dynamic matching for real-time ridesharing. Working Paper.
Pavan, A. and Gomes, R. (2019). Price customization and targeting in matching markets. Working paper.
Salop, S. C. (1979). Monopolistic competition with outside goods. The Bell Journal of Economics, 10(1):141–
156.
Su, X. and Zenios, S. (2006). Recipient choice can address the efficiency-equity and trade-off in kidney
transplantation: A mechanism design model. Management Science, 52(11):1647–1660.
Unver, M. (2010). Dynamic kidney exchange. review of economic studies, 77(1):372–414.
Wolff, R. W. (1982). Poisson arrivals see time averages. Operations Research, 30(2):223–231.
Zenios, S., Chertow, G., and Wein, L. (2000). Dynamic allocation of kidneys to candidates on the transplant
waiting list. Operations Research, 48(4):549–569.
Author: Matching Supply and Demand with Mismatch-Sensitive Players 27
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)
28 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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
].
Author: Matching Supply and Demand with Mismatch-Sensitive Players 29
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)
30 Author: Matching Supply and Demand with Mismatch-Sensitive Players
+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),
Author: Matching Supply and Demand with Mismatch-Sensitive Players 31
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∗.
32 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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)
Author: Matching Supply and Demand with Mismatch-Sensitive Players 33
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.
34 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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}
Author: Matching Supply and Demand with Mismatch-Sensitive Players 35
+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)
36 Author: Matching Supply and Demand with Mismatch-Sensitive Players
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.