Another joint work with Ruijie Li, built on our previous research of ridesharing, including A-PASS and Pricing carpool.
In this study, we consider a general problem called the Allocation Problem for the Platform of Platforms – dubbed AP3. Such a problem might arise in a two-sided service market, where a third-party integrator tries to allocate customers to workers separately controlled by a set of online platforms in a manner that satisfies all stakeholders. The integrator, as a leader, influences the outcome of the game by pricing the service, whereas the platforms (followers) are given the freedom to accept or reject customers to maximize their own profit, given the prices set by the integrator (see the plot below for an illustration). A set of nonlinear constraints are imposed on the leader’s problem to eliminate artificial scarcity, orignated from the integrator’s monopoly power. We formulate AP3 as a Stackelberg bipartite matching problem, which is known to be NP-hard in general. Our main result concerns the proof that AP3 can be reduced to a polynomially solvable problem by taking advantage of, somewhat paradoxically, the hard requirement of ruling out artificial scarcity.

A preprint can be downloaded here.
Abstract: We study the Allocation Problem for the Platform of Platforms (abbreviated as AP3) in a two-sided service market, where a third-party integrator tries to allocate customers to workers separately controlled by a set of online platforms in a manner that satisfies all stakeholders. AP3 is a natural Stackelberg game. The integrator, as a leader, influences the outcome of the game by pricing the service, whereas the platforms (followers) are given the freedom to accept or reject customers to maximize their own profit, given the prices set by the integrator. A set of nonlinear constraints are imposed on the leader’s problem to eliminate artificial scarcity, derived from the integrator’s monopoly power. We formulate AP3 as a Stackelberg bipartite matching problem, which is known to be NP-hard in general. Our main result concerns the proof that AP3 can be reduced to a polynomially solvable problem by taking advantage of, somewhat paradoxically, the “hard” requirement of ruling out artificial scarcity. Numerical experiments are conducted using the ride-hail service market as a case study. We find artificial scarcity negatively affects the number of customers served, although the magnitude of the effect varies with market conditions. In most cases, the integrator takes the lion’s share of the profit, but the need to eliminate artificial scarcity sometimes forces them to concede the benefits of collaboration to the platforms. The tighter the supply relative to the demand, the more the platforms benefit from removing artificial scarcity. In an over-supplied market, however, the integrator has a consistent and overwhelming advantage bestowed by its monopoly position.

钱穆的《国史大纲》和许倬云的《万古江河》都是架构宏大的中国通史,两本书覆盖范围差不多,均为上古至民国:国史大纲止于抗战时期,而万古江河以49年作结;风格也类似,都从重构历史出发,探究中华文明特质及精神,解释兴衰变迁,在当今世界中为其定位。就像钱穆说的,“治国史之第一任务,在能于国家民族之内部自身,求得其独特精神之所在。”
对天朝文化感召力的自信,虽然没写圣人出而天下归心,四海定而万国宾服,但意思差相仿佛。 大概因为成书在抗战时期,《国史大纲》民族情结强烈,夷夏之防溢于言表,这一点从他扬明抑清的倾向中表露无疑。比如他说,“明清之际的转变,大部分是明代内部自身的政治问题,说不上民族的衰老”;又说,“中国则因有二百年来满洲部族政权之横梗作病,使之虽欲急起直追而不可得”,恨不得要把中国近代落后的总账一股脑都算在满清这个异族征服者的头上。相比之下,许倬云学贯中西,视野要开阔许多,坚持把中国放在世界地理坐标中来定位,把中国作为东亚文明圈农耕文明的翘楚,但也把北方草原文明—从秦汉的匈奴,到南北朝的五胡,再到辽金元满,乃至西夏吐蕃—放在跟它平等的地位上来描述。《万古江河》从上古中华文明圈的形成发展,到中古时代东亚文明圈内部的重整融合,再到近古时代东亚文明圈跟其他文明圈的交流冲突,最后落到全球化的愿景,鼓吹“真正天下为公的大同境界”,虽然略嫌一厢情愿 (尤其放到出版后十六年的今天,更似乎与浩浩汤汤的反全球化浪潮格格不入),但脉络清晰,逻辑自然,跟我从小读的历史教课书比,至少是一个更客观真实的角度,如果说以史为镜,那《万古江河》应该是想要反躬自省的人更需要的那面镜子。