Designing Heaven’s Will:
The job assignment in the Chinese imperial civil service

It has long been recognized that effective state bureaucracies require meritocratic recruitment, competitive compensation, and systematic rules for appointments and promotions that shield away from personal influences (Weber, 1964, 1978). Despite the importance of assignment rules in the bureaucratic system, economic analyses of the civil service have primarily focused on the selection and incentives of civil servants (Dal Bó et al., 2013; Bai and Jia, 2016), while relatively little has been done to understand the rules that determine how selected personnel were appointed to specific jobs. Indeed, economists tend to presume sufficient institutions exist to deploy civil servants in posts and thus enabling them to execute administration tasks. In contrast, historians tend to focus more on why institutions exist and how institutions evolve. In the context of the Chinese civil service, Will (2002) is the first historical study that reviews the origin and evolution of the lots-drawing procedure in imperial China since the late sixteenth century. Watt (1972) offers a panoramic view of the career path for county magistrates—from first-time appointment, promotion, and re-appointment to demotion—during the period between the late eighteenth and the early nineteenth century. Similarly, Gong (1997), Pan (2005) and Zhang (2010) review the civil service systems for Song^$\mathbb{1}$ , Ming^$\mathbb{2}$ and Qing^$\mathbb{3}$ respectively. These historical studies mainly analyze the societal and political background of the appointment systems, but have not evaluated the properties of the appointment procedures. A few recent economics papers evaluate empirically how the appointment methods, and in particular patronage—by means of discretionary appointments through connections, contrary to our focus of rule-based assignment—affect public service, and find that patronage generally leads to worse economic performance or the selection of less competent officials (Xu, 2018; Colonnelli et al., 2020). Thakur (2020) studies a formal assignment procedure currently used in the Indian Administrative Service, focusing its distributional issues and the effect on bureaucratic and economic performance.

By formalizing the civil service assignment as a matching problem, our paper connects to the literature in matching and market design, which has a long tradition of applying formal economic theory to study properties of allocation mechanisms used in real-life applications. Examples include the entry labor market for medical residents (Roth, 1984), the military career in the U.S. (Sönmez, 2013; Sönmez and Switzer, 2013), lawyers to courts in Germany (Dimakopoulos and Heller, 2019), school choice (Abdulkadiroğlu and Sönmez, 2003), and kidney exchange (Roth et al., 2004). Some of the issues that we identify regarding the way in which the matchings are produced sequentially are indirectly related to, for example, the topic of reserve design. Dur et al. (2018) show how the processing order of reserves in school choice design matters and how some processing orders can cause unintended consequences. Other papers include Sönmez et al. (2021) on affirmative action policies in Indian civil service, Pathak et al. (2020a) on H-1B visa allocation rules, and Pathak et al. (2020b) on medical resources allocations. In addition, the search for procedures that guarantee minimize unfilled jobs relates to maximum matchings in random environments. Bogomolnaia and Moulin (2004) look at maximum matchings when randomizing over dichotomous preference. Boczoń and Wilson (2018) study the UEFA Champions League group drawing method, a problem in which matchings are also determined by drawing lots and where there is a concern about maximality.

More broadly speaking, our paper also relates to a burgeoning literature that applies economic theory to analyze important economic and societal institutions in history. Greif (1993) is perhaps the first paper in the field, applying contract theory to analyze contractual relations in eleventh-century Mediterranean trade. Other studies include Greif et al. (1994) on the contract enforcement problem faced by merchants in late medieval Europe, Börner and Hatfield (2017) on the debt-clearing financial mechanisms in preindustrial Europe, and Mackenzie (2019) on the succession rules of Popes. Our paper follows this approach in that we use the toolbox from matching theory to analyze the civil service assignment procedures—another important historical institution—in order to shed light on its properties.

The remainder of the paper is as follows. The next section provides background information on the selection and appointment of civil servants in the historical context. Section 3 describes in detail the assignment procedures used throughout history. Section 4 presents our theoretical framework and formalizes the previously described assignment procedures. Section 5 develops and evaluates comparative static results of the assignments procedures. In Section 6, we conclude and indicate potential theoretical and empirical future work.³³3We also provide some additional background information related to the selection and appointment of civil servants and a documentation of original sources and data used in our historical research in an online supplementary material.

Qualified candidates, depending on their degree, were eligible for different sets of civil service jobs. In general, candidates with higher degrees were appointed to jobs with higher ranks. That is, advanced scholars were more likely to be assigned to more important jobs than tribute scholars or recommended men. Early debates often centered around whether academic achievement should be the sole consideration for measuring one’s competence. However, over time, academic achievement was gradually accepted as the main desideratum, as it provided clear incentives to follow the career system and gave legitimacy to the government rule. Since the precise eligibility requirement varied with time, we will elaborate on the details in Section 3 when discussing the assignment procedures.

This additional criterion could prevent one candidate from being assigned to a particular job even if he was eligible. The rule of avoidance, in fact much older than the civil service appointment, dates back to the second century. Despite the various forms it took over time, avoidance of localities is the most fundamental one for the entry-level jobs (Wei, 1992). It stated that a candidate was prohibited from being appointed to his native province.⁸⁸8The other types of avoidance can overlap with avoidance of localities or involve smaller sets of people. For instance, the avoidance of family, which prevented a candidate from being assigned to a job where he and an incumbent official were direct or indirect family members. Since most people’s families came from the same region, avoidance of localities was often sufficient. The other type was avoidance of teacher and student, which prevented a candidate from serving in a job where he was the student or the teacher of an incumbent officer. Bureaucrats originating from a particular region naturally had an information advantage about their home regions, and therefore assigning them there could be beneficial to the local management. However, as they were also more connected to local elites, this also posed a threat of local capture, especially in the days when regions were more isolated due to communication and transportation constraints. Rule of avoidance was thus intended to prevent the formation of local powers, which was regarded as a major impediment and challenge to unification throughout the Chinese history.

Starting from the late Ming^$\mathbb{2}$ dynasty, candidates were appointed randomly to compatible jobs by drawing them from tubes. Though it might seem odd to randomly assign civil servants to posts, it is a way by which the emperor could essentially eliminate the possibility that local clans would extend their power by influencing these assignments, without taking on the decisions himself. On the other hand, the lack of criteria when deciding individual assignments reduces the ability of matching highly qualified candidates to higher-level jobs. It seems, however, that randomness was gradually accepted as a desirable property from the late sixteenth century onward.⁹⁹9Random assignments have a surprising history in the political sphere. As early as the democratic period of ancient Athens, random assignments were used to select citizens to serve in the Boule (a council appointed to run the daily affairs of the city) and various state offices, through a randomization device known as Kleroterion (Headlam, 1891). In recent years, political scientists have also advocated for a lot-drawing procedure for assigning EU commissioners (Buchstein and Hein, 2009).

With the expansion of the selection of candidates by exams, a more formal assignment procedure started to emerge. From the official documents, we know that selected candidates were eligible for entry-level jobs, including clerks in various ministries in the capital and county magistrates in the provinces, though we do not observe clear rules that define eligible jobs by the categories of degrees candidates obtained.

Selected candidates were appointed four times a year. The assignment procedure was priority-based, and in most cases priorities were determined first by the candidates’ degrees—starting with the advanced scholars then other degree categories—and then by their exam results within each degree category.¹¹¹¹11A hybrid system that combines both merit and patronage was used by the Song^$\mathbb{1}$ dynasty. In addition to exam results, candidates were given higher priority if they were endorsed by incumbent officials. This practice was not continued in the following procedures. All vacancies were announced before the assignment, so that candidates could consider their preferred jobs. During the initial appointment round, officials from the Ministry of Personnel chanted out a candidate’s name following the priority order, who then needed to indicate a job of interest that was still available. The officials in charge would then announce “approve” or “not approve.” An extra round of appointment was organized afterwards for the unassigned candidates and unassigned jobs, using a similar mechanism. If there were still unassigned candidates or jobs, they would be assigned in the next appointment phase three months later.

A priority mechanism continued to be used in the Ming^$\mathbb{2}$ dynasty. However, candidates were no longer asked to express their preferred jobs. Instead, the Ministry of Personnel would assess the candidate, and then decide a suitable job for him. The frequency of the assignment was increased to monthly, with first-time appointments being organized in even months, and promotions or transfers organized in odd months. This organization of monthly appointments continued in the following centuries.

As the civil service system grew in size and importance, the rules on eligibility were clarified. Table 3.1 presents the details. In addition to the general distinction between advanced scholars and tribute scholars together with recommended men, advanced scholars were further differentiated into three grades. The top grade, consisting of three people, were immediately assigned to editor jobs in the Hanlin academy. The rest of the candidates were only appointed upon completion of internships in different ministries and departments in the capital that usually lasted for half a year. Among them, those in the second grade were considered for secretaries in various ministries in the capital, or sub-prefecture magistrates outside the capital. The third-grade advanced scholars were eligible for, among metropolitan jobs, secretaries to the emperor, and officers at various departments and courts. Furthermore, they were eligible for, among jobs outside the metropolitan area, prefecture judges and county magistrates. Finally, tribute scholars and recommended men were eligible only for non-metropolitan jobs, which included prefecture judges, county magistrates, and various study officers.¹²¹²12For candidates qualified through non-examination paths, similar guidelines also existed. It is clear from the table that some jobs such as ministry secretaries only considered advanced scholars, but not candidates with lower degrees, and some jobs such as study officers only considered tribute scholars and recommended men. Other jobs, such as county magistrates and prefecture judges, could be assigned to candidates with all three degrees.

Like the Song^$\mathbb{1}$ procedure, the First Ming^$\mathbb{2}$ procedure followed an order over the candidates when determining their jobs. Advanced scholars had the highest priority, followed by tribute scholars and recommended men, and they were ranked by exam results within each degree category. Following this strict priority order, the Ministry of Personnel looked to place each candidate at a job that he was entitled to, while respecting the rule of avoidance. In general, the ministry tried to look for a job matching the candidate’s skill as measured by exam result. Therefore, higher-ranked jobs were more likely to be assigned to those with higher exam results.

The First Ming^$\mathbb{2}$ procedure could, in principle, handle both eligibility and rule of avoidance well; however, it provided enough leeway for officials or political clans to interfere with the final assignment. It was evident that, by the late sixteenth century, such interference was ubiquitous, to the point that appointment decisions were no longer focusing on one’s competence, but rather which clan he belonged to. This unavoidably undermined the effectiveness of selecting individuals by merit, and as a consequence, reduced competence of governance and fragmented central control (Will, 2002).

The lots-drawing procedure was introduced in 1594. Instead of being evaluated case by case, appointments began to be determined by drawing lots. This was the first procedure that systematically assigned selected candidates to entry-level civil service jobs via lotteries. Information on which candidates were eligible for which jobs was publicly announced before the assignment. Job titles were written on bamboo sticks and were then put into tubes. The procedure then proceeded as follows.

• •

  Advanced scholars drew first, in descending order of their examination results. Each candidate drew a job from a tube filled with jobs that they were eligible for.

• •

  Only after all advanced scholars had a chance to draw, the tribute scholars could proceed to draw. They drew jobs from a tube that filled with jobs that they were eligible for, also following the order determined by their exam results.

• •

  Finally, after all tribute scholars had a chance to draw, the recommended men drew. They drew jobs from a tube filled with jobs that they were eligible for, again following the order determined by their exam results.¹³¹³13Details about the order in which the candidates draw their jobs were vague in the official documents around this time. But from secondary sources, we could infer this drawing order. For example, a report in 1602 written by Li Dai, minister of the Ministry of Personnel at the time, mentions that lots left by the advanced scholars are kept for tribute students and recommended men, while lots left by them are kept for other types candidates (Wu (1609), vol.59).

The Ming^$\mathbb{2}$ lots-drawing procedure respects the eligibility criteria due to the fact that the tubes from which a candidate would draw their job only contained jobs for which they were eligible. So, for example, when matching advanced scholars, the tube with jobs would only contain those that satisfied their eligibility. The official documents around this time mention that the rule of avoidance had to be respected. The documentation is not precise about how that restriction was implemented. From the context, however, and from the descriptions documenting how this restriction was implemented later in the Qing^$\mathbb{3}$ dynasty, it is most plausible that if one drew an incompatible job due to rule of avoidance, he would be allowed to continue drawing until a compatible job was found.

Compared to the First Ming^$\mathbb{2}$ procedure, this lots-drawing procedure limited personal decisions and interference. Moreover, the randomness introduced by lotteries balanced demands from different political clans to place their own people in their home regions. Besides, the drawing of lots has been part of culture in China. Indeed, drawing lots was already used in the assignment of some military posts, as well as the assignment of interns to offices in the capital in earlier times (Will, 2002).

Nevertheless, the new procedure received fierce criticisms from the outset. It was attacked for lack of control in finding the right match (Gu, 1670). One primary issue is that, depending on the realization of chance, higher-priority candidates could draw medium-ranked jobs (i.e. jobs that are also compatible with lower-ranked candidates) while there were still higher-ranked jobs to be filled. Furthermore, it did not take long before the new procedure fell victim to corruption. It was reported that officials selected the best jobs and placed them in a way that, when it was the turn for candidates who bribed them to draw, they could draw the most desirable jobs (Shen, 1619).

Despite the change of dynasty, the lots-drawing procedure survived, as using random assignment to diffuse regional powers was seen as a desirable method to reinforce the unification of the country by the newly founded administration, led by the Manchu ethnic minority.

Besides slight modification on eligibility rules (see Table 3.2), there were a few further developments in the procedure. To begin, additional security measures were introduced to ensure impartiality. First, it was stressed that in addition to executing the lots-drawing procedure publicly, officials from the Censorate, a supervisory agency, were present to oversee the appointments. Second, names of candidates and jobs were sealed before putting them into the tubes. Additionally, to cope with corruption and rigging in the drawing process, candidates no longer followed the order determined by their results when drawing jobs. Instead, they followed a random order also determined by lotteries.

A more fundamental change was the use of a system of partitioned assignments, which reduced the chance of being matched to a lower-ranked job when a high-ranked job was still available, a main concern in the Second Ming^$\mathbb{2}$ procedure. The appointment for newly selected candidates was organized in every even month, as in the Ming^$\mathbb{2}$ dynasty. The Ministry of Personnel first registered all vacancies reported by different administrative units.¹⁴¹⁴14We focus on the Han jobs. Civil service jobs were reserved for different ethnic groups in Qing^$\mathbb{3}$ dynasty: the Manchu jobs (Manchus were the ruling ethnic minority), the Mongol jobs (Mongols were the crucial ally of Manchus), and the Han jobs. The majority of the jobs, especially the entry-level jobs, were Han jobs. Assignment procedures to the Manchu and Mongol jobs were similar to the Han jobs procedure. Vacancies were categorized by their types—for instance, secretaries in various ministries counted as one type, and county magistrates in various locations counted as another. After preparing the list of vacant jobs, the ministry then drew up, for each type of job, a list of candidates—matching the number of jobs— who were eligible. Whenever there were multiple categories of eligible candidates, quotas for each category were specified.¹⁵¹⁵15We provide an illustration for county magistrates taken from Regulations of Ministry of Personnel (QDLBQXZL, 1886). Candidates qualified through other non-examination channels were also considered for the appointment of county magistrates in the Qing^$\mathbb{3}$ dynasty. The quotas for both candidates selected through exams and other channels were as follows: 5 advanced scholars, 5 recommend men, 4 candidates who paid financial tribute (if none, then substitute with recommended men), 3 candidates who were promoted from clerks. These 17 people formed one “class.” Next, we describe, for a given type of job and a given list of candidates, how the First Qing^$\mathbb{3}$ procedure was applied.

• •

  A supervisor in charge of the appointment first drew a stick from the tube of candidates, then drew a stick from the tube of jobs.

• •

  If the pair of candidate and job did not violate the rule of avoidance, then the supervisor would declare the match.

• •

  If the pair did violate the rule of avoidance, then the job stick was put aside, and the official would keep drawing a new job until the candidate did not need to avoid it. After the match, the ineligible job(s) would be put back into the tube.

• •

  The appointment resumed until all candidates were drawn and matched to a job, or all remaining jobs and candidates are mutually incompatible.

Finally, unassigned candidates would be returned to their queues, waiting for the next appointment. Unassigned jobs would also be added to the vacancy lists in the next appointment.

This system of partitioned assignments increased the chance that a higher-priority candidate would end up with a higher-ranked job. Indeed, given that now each partitioned assignment consisted of only one type of job and eligible candidates, and given the way in which the jobs were processed as well as the way in which the contents inside each tube were determined, candidates with higher degrees had a higher chance to draw a job with a higher rank. The lottery only randomized which exact job one eventually got within the same type (and same rank) of job. The Qing^$\mathbb{3}$ administration seemed willing to sacrifice some randomness in the matches to improve the matching quality.

Yet, the partitioned assignments led to another problem. Unlike the Second Ming^$\mathbb{2}$ procedure, where a candidate who first drew an incompatible job had more jobs available next, now the candidate could only draw from a smaller pool of jobs. This reduced the chance of finding a compatible job. Similarly, it also increased the chance that a job was left unassigned. A situation with increasing numbers of unassigned candidates and unserved jobs was not ideal for the functioning of bureaucracy. Of course, this problem was exacerbated by the rule that unassigned candidates could not be considered for other type of jobs in the current assignment. This rule, on the other hand, can be seen as a means to guarantee that candidates would not be assigned to lower-ranked jobs when they could be considered for higher-ranked jobs in the next appointment.

To address problems related to unfilled jobs introduced by the partitioned assignments, the procedure was changed in 1824 such that instead of all candidates being matched at the same time, for a given type of job, those with incompatible jobs among those in their corresponding tube were required to draw first. Only after the prioritized candidates finished drawing could the rest of the candidates proceed to draw jobs. The other aspects of the procedure remained the same as the previous procedure. The following quote taken from the Collected Statutes of Qing^$\mathbb{3}$ (DQHD, 1886) recorded this change:

“1824, it was approved after discussions, for the people who draw lots in the monthly appointment, those who have home provinces to avoid draw first. If they still draw a job that needs to be avoided, remove this job and ask [the candidates] to draw another job. Until a [compatible] lot is drawn, let those who do not need to avoid home provinces draw.”

By prioritizing the candidates who had regions to avoid, their chance of finding a compatible job was increased, and more jobs were matched, as the only factor driving incompatibility in a partitioned assignment is the rule of avoidance.

While we do not have data that would allow us to estimate whether this change had measurable impact on the cardinality of the matchings, there are indications that there was a reduction in the time it took candidates to be matched to a job after the change was introduced. Based on the calculations by Wang (2016) using the archived curricula vitae of civil servants in the Qing^$\mathbb{3}$ dynasty, the average waiting time for advanced scholars to become county magistrates was reduced from 8 years between 1796 and 1820 to 5.5 years between 1821 to 1850. This provides some suggestive evidence of the possible impact of the procedure change.

To sum up before analysing these procedures, the historical account presented in this section elucidates that the assignment procedures gradually moved away from personal influences and resorted to more systematic rules in deciding who should get which job. Despite the fact that candidates’ preferences were considered in the initial Song^$\mathbb{1}$ procedure, they were no longer taken into account by the later procedures, mainly because individual preferences could come into conflict with the prioritization of high-levels. From the late sixteenth century onward, drawing lots was accepted and routinized in the civil service appointments, and developments were therefore made to balance the prioritization of high-levels and the number of total matches, while respecting eligibility, rule of avoidance, and to some extent, randomness.

It is worth mentioning that there are a few common features present in the design of all procedures. First, it was emphasized in all procedures that the assignments should be done publicly, as a means of transparency and, consequently, legitimacy for the assignment outcomes. Second, and perhaps more importantly, all procedures were carried out sequentially, and appointments were declared once a match was found. This provided a simple and intuitive way to produce desirable matchings. When using lots-drawing, for instance, even if there are other ways in theory to find randomized compatible matchings, appointments in the historical practice were made sequentially one at a time using specific priority orders.

A set of workers $W$ is to be matched to a set of jobs $J$. A matching is a function $\mu:W\cup J\to W\cup J\cup\{\emptyset\}$ such that each worker (job) is assigned either to one job (worker) or is left unassigned, and a worker is matched to a job if and only if the job is also matched to him, that is, for any $w\in W$ and $j\in J$, $\mu(w)=j\iff\mu(j)=w$. To account for the restrictions on what matches are acceptable, let $\mathcal{C}$ be a compatibility correspondence $\mathcal{C}:W\twoheadrightarrow J$ that defines which jobs are compatible with each worker. In our application, compatibility could be determined by eligibility or the rule of avoidance. In addition, we say that a matching $\mu$ is feasible if workers only receive jobs that they are compatible with, that is, for any $w\in W$ and $j\in J$, $\mu(w)=j\implies j\in\mathcal{C}(w)$. The set of all feasible matchings is denoted by $\mathcal{M}$. We refer to a triple $\langle W,J,\mathcal{C}\rangle$ as a market.

For any matching $\mu\in\mathcal{M}$ and any subset of jobs $J^{\prime}\subseteq J$, let $\mu(J^{\prime})$ be the set of workers matched to some job in $J^{\prime}$ under matching $\mu$. That is, $\mu(J^{\prime})\equiv\left\{w\in W:\mu(w)\in J^{\prime}\right\}$. Denote by $|\mu|$ the cardinality or the size of $\mu$, that is, $|\mu|=|\mu(J)|$.

This definition says that a matching minimizes unfilled jobs if there is no other feasible matching that matches more workers to jobs. Notice that since a worker can only be matched to a job, minimizing unfilled jobs is equivalent to minimizing unmatched candidates. For our application, it is important to assign as many qualified candidates as possible to jobs, since both unassigned candidates and unfilled jobs could leave important resources idle until the next appointment, typically executed two months later.

An assignment plan is a pair $\left\langle\succ^{W},\left(\succ_{w}^{J}\right)_{w\in W}\right\rangle$, where $\succ^{W}$ is a strict total order over the set of workers, and each element of $\left(\succ_{w}^{J}\right)_{w\in W}$ is a strict total order over the set of jobs. Assignment plans are a key concept that unifies all of the five procedures that we evaluate into a common framework. Depending on the specific application, they can represent a design decision or a realization of chance when drawing lots. The first element of the assignment plan, the order $\succ^{W}$, tells for each pair of workers $w$,$w^{\prime}$, who will be considered for a matching first. In the Song^$\mathbb{1}$ , First, and Second Ming^$\mathbb{2}$ procedures, for instance, an assignment plan in which $w\succ^{W}w^{\prime}$ represents the situation in which worker $w$ obtained a higher exam grade than $w^{\prime}$, and therefore if both obtained the same title (for example, advanced scholar), both procedures will match $w$ to a job (if any) before $w^{\prime}$.¹⁸¹⁸18Note, however, that $w\succ^{W}w^{\prime}$ does not say that $w^{\prime}$ will be matched immediately after $w$, but that $w^{\prime}$ will not be considered for a match before $w$. In the Qing^$\mathbb{3}$ procedures, $\succ^{W}$ indicates the order in which workers in the same tube are drawn. So if, for example, a tube contains workers $w,w^{\prime},w^{\prime\prime}$, then$w\succ^{W}w^{\prime}\succ^{W}w^{\prime\prime}$ represents the realization of chance in which these workers are drawn from that tube in that order. All possible orderings in which workers can be drawn from that tube are represented by all the permutations of $\succ^{W}$. Importantly, a uniform distribution over all of these permutations represents the distribution in which these orders take place in the real-life procedure.

The second element of the assignment plan, the orders over jobs $\left(\succ_{w}^{J}\right)_{w\in W}$, represents the order in which jobs are considered, for each worker. Given a worker $w$ and two jobs $j,j^{\prime}$, $j\succ_{w}^{J}j^{\prime}$ says that, when considering a match for worker $w$, if both jobs are being considered and the worker is matched to $j^{\prime}$, it must be that $j$ is not available anymore or is incompatible with $w$. In the Song^$\mathbb{1}$ procedure, $\succ_{w}^{J}$ represents worker $w$’s preference over the jobs: if both $j$ and $j^{\prime}$ are still available when his turn comes, and $j$ is compatible with him, he will choose $j$ instead of $j^{\prime}$. In the First Ming^$\mathbb{2}$ procedure, it represents whatever criterion was used to choose among eligible jobs. Finally, in the Second Ming^$\mathbb{2}$ and the Qing^$\mathbb{3}$ procedures, orders over jobs also represent a realization of chance of the order of drawing of jobs from the corresponding tube. So if worker $w$ is such that $j\succ_{w}^{J}j^{\prime}\succ_{w}^{J}j^{\prime\prime}$, and these three jobs are in the tube from which he will draw, he will first draw $j$. If $j$ is an eligible job that does not violate the rule of avoidance, he will be matched to that job. If, say, $j$ is incompatible with $w$, then he will next draw $j^{\prime}$, and so on. As in the order over workers, drawing uniformly from the possible permutations over a set of jobs results in the same distribution of outcomes that result from the real-life procedure when $w$ is facing a tube with these contents. Note that, as opposed to the order over workers, which is unique, here each worker might be associated with a different order over jobs. This represents the fact that the drawings of jobs from a tube are independent draws—conditional, of course, on the contents of the tube.

The other key concept when modeling the procedures is an assignment arrangement. An assignment arrangement is a list $\Phi=\left\langle\varphi^{1},\ldots,\varphi^{\ell}\right\rangle$ that partitions a market into $\ell$ independent sub-markets. Each element of the list $\varphi^{i}$ is a sequence of tubes, consisting of a list of subsets of workers $\left\langle W_{1}^{i},W_{2}^{i},\ldots,W_{n_{i}}^{i}\right\rangle$ and a list of subsets of jobs $\left\langle J_{1}^{i},J_{2}^{i},\ldots,J_{m_{i}}^{i}\right\rangle$. These sets of workers and jobs, which we denote by tubes, are such that:

and for every $a\neq c$ or $b\neq d$, $W_{a}^{b}\cap W_{c}^{d}=\emptyset$ and $J_{a}^{b}\cap J_{c}^{d}=\emptyset$. In other words, the sets listed in the sequences of tubes partition the sets of workers and jobs.

An assignment arrangement represents two aspects of the assignment procedure. First, it allows for the problem to be partitioned into independent sub-markets. This is what is done in the Qing^$\mathbb{3}$ procedures: candidates and jobs are split into separate and independent matching sub-problems for each type of job. Second, it represents the order in which different subsets of candidates and jobs are considered. A sequence of tubes $\left\{\left\langle W_{1},W_{2}\right\rangle,\left\langle J_{1},J_{2}\right\rangle\right\}$, for example, indicates that first the workers in $W_{1}$ will be considered and matched to jobs. Only after all the workers in $W_{1}$ are matched to a job (or are left without any remaining compatible job), the workers in $W_{2}$ will be considered. When it comes to jobs, take any worker (regardless of whether it is a worker in $W_{1}$ or $W_{2}$), jobs in $J_{2}$ will only be considered if there are no jobs compatible with that worker among those remaining in $J_{1}$. This representation not only allows the modeling of all the procedures, but also the extension to a new procedure that improves upon the Second Qing^$\mathbb{3}$ procedure in Section 5.2.

Given a market, an assignment plan together with an assignment arrangement can be combined to represent an execution of the matching procedures that we model. Consider a market $\langle W,J,\mathcal{C}\rangle$ and an assignment plan $\left\langle\succ^{W},\left(\succ_{w}^{J}\right)_{w\in W}\right\rangle$. We next describe how the assignment arrangement produces a matching of workers to jobs. To do so, it is helpful to use the following notation. Given a set $I\subseteq W\cup J$ and an order $\succ$, we denote $top^{\succ}(I)$ as the top element of $I$ in the order $\succ$, that is, $top^{\succ}(I)=\left\{i\in I|\forall i^{\prime}\in I\backslash\{i\}:i\succ i^{\prime}\right\}$.

For each element $\varphi^{i}$ in the assignment arrangement, the steps below are followed to produce a matching $\mu$:

1. (1)

   Let $J^{*}=\cup_{k=1}^{m}J_{k}^{i}$.

2. (2)

   For each $t=1,\ldots,n$, let $A^{t}=W_{t}^{i}$ and repeat the following procedure until $A^{t}=\emptyset$:

   • •

     Let $w=top^{\succ^{W}}(A^{t})$ and remove $w$ from $A^{t}$. There are two cases:

     • –

       $\mathcal{C}(w)\cap J^{*}=\emptyset$: let $\mu(w)=\emptyset$.

     • –

       $\mathcal{C}(w)\cap J^{*}\neq\emptyset$: let $a^{*}$ be the lowest value of $a$ such that $\mathcal{C}(w)\cap J^{*}\cap J^{t}_{a}\neq\emptyset$, and $j=top^{\succ_{w}^{J}}\left(\mathcal{C}(w)\cap J^{*}\cap J^{t}_{a^{*}}\right)$. Let $\mu(w)=j$, and remove $j$ from $J^{*}$.

In other words, given $\varphi^{i}$ in the assignment arrangement, we follow the order of tubes of workers in $\varphi^{i}$, matching the workers in $W^{i}_{1}$ first, and only after considering all of them do we move to the next subset of workers $W^{i}_{2}$, and so on. Within each tube, the order in which workers are chosen is determined by $\succ^{W}$. Each worker $w$ is matched to a job in the first tube of jobs that still contains a compatible job. If that tube contains more than one compatible job, then he is matched to the job with highest priority in that tube with respect to $\succ^{J}_{w}$. If, however, there is no compatible job left, then he is left unmatched. We repeat these steps for each $\varphi^{i}$ in the assignment arrangement, which produces a matching of workers to jobs. We refer to the combination of an assignment arrangement with a method of producing an assignment plan as a procedure.

The key advantage of this model, for our purposes, is that it allows us to perform reliable comparative statics between procedures. By fixing an assignment plan and evaluating the matchings produced by two different assignment arrangements, we are able to compare, for example, two random procedures without picking different realizations of chance. Moreover, we are even able to compare deterministic procedures, such as the Song^$\mathbb{1}$ , to a random one, such as the Second Ming^$\mathbb{2}$ . If we are able to analyze the outcomes produced by these two procedures for any fixed assignment plan, then the fact that one is deterministic and the other is random is inconsequential for the analysis.

To facilitate our analysis of the procedures used in history, we consider in what follows a simplified setting where there are two categories of workers, $W^{A}$ (representing advanced scholars), and $W^{B}$ (representing tribute scholars and recommended men).¹⁹¹⁹19While there was an extra division within the category of advanced scholars during the Song^$\mathbb{1}$ and Ming^$\mathbb{2}$ dynasties, our simplified version is equivalent, for our analysis, to the one used during the Qing^$\mathbb{3}$ dynasty. On the other side, jobs are partitioned according to eligibility: jobs in $J^{A}$ and $J^{B}$ are eligible for workers in $W^{A}$ and $W^{B}$ respectively, and jobs in $J^{AB}$ are eligible to all workers. The partition reflects the fact that, in our application, some jobs were assigned only to advanced scholars, and some only to tribute scholars and recommended men, while some others could be matched to candidates from both categories. This simplification allows us to capture the main features of the civil service assignment without unnecessary complications that are not crucial to our results.

To capture the objective of assigning candidates with different degrees to jobs with matching levels, we formulate the following definition when evaluating a matching with respect to our simplified setting.

In other words, a matching $\mu$ prioritizes more high-levels than another matching $\mu^{\prime}$ if (i) $\mu$ does not match fewer workers than $\mu^{\prime}$, (ii) $\mu$ does not match fewer workers from $W^{A}$ than $\mu^{\prime}$, (iii) $\mu$ does not match less workers from $W^{A}$ to jobs in $J^{A}$ than $\mu^{\prime}$, and (iv) it does not match less workers from $W^{B}$ to jobs in $J^{AB}$, except when compensated by an increase in workers from $W^{A}$ matched to these jobs. This definition captures, with relatively simple conditions, the objective of prioritizing the matching of higher-level jobs over lower-level ones, and of higher-level candidates over lower-level ones. Given that eligibility constraints are always satisfied, it guarantees the trade-offs that remain between which candidates and jobs to match are resolved toward matching more advanced jobs to more advanced candidates.

Finally, for our historical analysis, it will be helpful to consider two different compatibility correspondences. The first one, denoted by $\mathcal{C}^{-}$, considers only the constraint on eligibility. That is, for any $w\in W$ and $j\in J$, $j\not\in\mathcal{C}^{-}(w)$ if and only if $w\in W^{A}$ and $j\in J^{B}$ or $w\in W^{B}$ and $j\in J^{A}$. The second one, denoted by $\mathcal{C}^{+}$, considers both the constraints of eligibility and rule of avoidance. That is, for any $w\in W$ and $j\in J$, $j\in\mathcal{C}^{+}(w)$ if and only if $j\in\mathcal{C}^{-}(w)$ and $j$ is not in worker $w$’s native region.

We will use our baseline model of markets, assignment plans, and assignment arrangements to describe the five procedures presented in Section 3 as instances of the model. It should go without saying that, given the nature of the sources that we used, some simplifications and additional assumptions are unavoidable. We are, however, explicit whenever they are consequential, and made a conscious effort to make them parsimonious. If there is a gap in the description of some procedure but is present in a previous or later procedure, then we assumed continuity of that element. Finally, whenever possible we opted for assumptions that make the procedures more similar to each other, as opposed to different, to reduce the reliance of the differences that we obtained on those assumptions. That being said, the vast majority of the elements in the descriptions below can be traced back to the descriptions in the historical documents we used.

All the procedures we evaluate were implemented by matching workers to jobs sequentially. Despite its transparency and simplicity, this type of mechanism may suffer from some shortcomings due to the “greedy” way in which they process the matchings.

More specifically, both the minimization of unfilled jobs and the prioritization of high-levels are properties that, in general, are incompatible with matching each worker with an arbitrary compatible job, one at a time. To see this, consider the example below.²²²²22While the example considers eligibility constraints, one can easily construct an analogous one using only the rule of avoidance.

The issue in the example resides in the fact that if worker $w_{a}$ is matched before $w_{b}$, ignoring the relationship between the sets of compatible jobs of both workers, might not minimize unfilled jobs. At first sight, this problem could be solved (or at least mitigated) if some of the matchings could be revised. If at worker $w_{b}$’s turn only job $j_{a}$ is available, both workers could “exchange” their draws in a mutually compatible way. In fact, this possibility was discussed in the early years of the Second Ming^$\mathbb{2}$ procedure. In 1602, in a correspondence to the court by Li Dai—the minister of the Ministry of Personnel at the time, it was suggested that a candidate who either draws an incompatible job or ends up with no compatible jobs left would be able to exchange his assignment with some candidate matched with a compatible job in a mutually acceptable way (Wu, 1609). However, there is no indication that such exchanges were carried out. One natural interpretation for the reason why these exchanges were not implemented is that making exchanges after a matching was determined reduces at least the perception of transparency. Additionally, revising matchings that were determined randomly goes in the opposite direction of the purpose of the introduction of randomness, bringing back arguably arbitrary changes to the outcome.

Notice that the discussion on the exchange method also provides one more indication that minimizing the number of unfilled jobs were an actual concern at the time. One additional indication that unfilled jobs were an issue affecting the assignments was that from the mid-Qing, one or two extra candidates were added when constructing the list, as a means to reduce the number of unfilled jobs (QDLBQXZL, 1886).

The challenges faced by the designers of these procedures were, in fact, extremely complex even for today’s standards. The first feasible algorithm for finding a maximum matching—the “Hungarian method”— was published in the 1950s (Kuhn, 1955). The literature on “online matching,” which covers methods for determining matchings “greedily” one by one in an attempt to produce approximately maximum matchings, is still evolving (Feng, 2014). Even if one considers only the rule of avoidance, the determination of which workers should be matched to which jobs is a relatively complex combinatorial problem that depends on the numbers of jobs and workers from each region. Given these facts, therefore, it should not come as a surprise that the procedures in general are not optimal. We will show, however, that the designers were relatively close to that in the end.

Our first result shows that, when considering only eligibility constraints, the changes in the procedures were consistent with the objective of minimizing unfilled jobs.

The proposition below shows that when both eligibility and rule of avoidance are considered, the story changes.

Next, we consider the extent to which the matchings produced by all the procedures prioritize high-levels. As before, we first discuss the case where we only take eligibility into account.

As for the minimization of unfilled jobs, rule of avoidance also affects the prioritization of high-levels. The following shows that when we consider both eligibility and rule of avoidance as compatibility constraints, all procedures can result in matchings that do not prioritize high-levels. This result follows immediately from markets constructed from the proof of Proposition 2, and therefore the proof is omitted.

Notice that if we consider only the results involving compatibility in terms of eligibility, we would conclude that both the changes from the Song^$\mathbb{1}$ procedure to the First Ming^$\mathbb{2}$ procedure and from the Second Ming^$\mathbb{2}$ procedure to the First Qing^$\mathbb{3}$ procedure would unambiguously improve the matchings that were produced, both in terms of the minimization of unfilled jobs and prioritization of high-levels. However, when we compare with respect to compatibility in terms of eligibility and rule of avoidance, the same conclusion can no longer be made. In fact, we show in the theorem below that the changes might have resulted in the opposite effect, even while considering the same market and assignment plan.

Even though the First Ming^$\mathbb{2}$ and First Qing^$\mathbb{3}$ procedures were motivated by the need to improve the prioritization of high-levels, our result suggests that there are situations where both procedures lead to worse outcomes, due to its interaction with the rule of avoidance. Moreover, the magnitude of the problem can be substantial, as large as half of the workers.

A striking consequence of the results we produced so far is that the changes of mechanisms might have opposite effects on the minimization of unfilled jobs and prioritization of high-levels, depending on which constraints are being considered, for the same market and assignment plan. That is, this difference is not driven by different orders or realizations of chance, but by whether the rule of avoidance is being considered or not.

Finally, we show that, the change from the First to the Second Qing^$\mathbb{3}$ procedure nonetheless resulted in an improvement in the cardinality of the matchings produced. The proposition below describes this result.

Figure 1 summarizes the above results. Note that we know from Proposition 2, despite the improvement yielded by the Second Qing^$\mathbb{3}$ procedure, in general the Second Qing^$\mathbb{3}$ procedure may still not match as many candidates as possible. In the section below, however, we show that in certain markets the addition of a single tube of jobs could result in a procedure that produces matchings that minimizes unfilled jobs and prioritize high-levels in every realization of chance.

We will now show that, under a relatively simple assumption on the numbers of workers and jobs per region, there is a modification of the Second Qing^$\mathbb{3}$ procedure that always yields matchings that minimize unfilled jobs in each sub-market.

The problems regarding the minimization of unfilled jobs in the Second Qing^$\mathbb{3}$ procedure is driven solely by one of the compatibility constraints—namely, rule of avoidance. Therefore, for ease of illustration, we will consider just one category of workers and jobs that are mutually compatible in terms of degrees. Let $\{r_{1},\ldots,r_{k}\}$ be the set of regions to which workers and jobs might belong to, and $W_{i}$ and $J_{i}$ denote the set of workers and jobs from region $r_{i}$, respectively. In addition, let $W_{-i}$ be the set of all workers except $W_{i}$. Similarly, denote the set of all jobs except $J_{i}$ by $J_{-i}$.

This condition requires, therefore, that for each region that has both workers and jobs in the market, there are no more jobs than workers.²⁵²⁵25Notice that this is a condition involving mutually incompatible sets, and therefore are independent of the conditions for the existence of a matching that covers all workers or all jobs (Hall, 1935).

Without loss of generality, let $W_{1}$ be the set with the largest number of workers (that is, for all $i>1$, $|W_{1}|\geq|W_{i}|$.), and consider the following sequence of tubes:

where $J_{1}$ contains the jobs from the region with the largest number of workers, and $J_{-1}$ contains the jobs from the other regions. In this ordering, workers from the region that has the largest number of workers are given priority to draw a job over the rest of the workers, and the jobs from this region are put into the second tube of jobs.²⁶²⁶26If we do include workers from regions that do not have jobs, then we include them in the second tube on the workers’ side, together with $W_{-1}$. If we include jobs from regions that do not have workers, then we also include them in the first tube of jobs, together with $J_{-1}$.