# Consistency 7

## Rule: a / an agreement 5

Prefix: \b

### Pattern: a [aeiou]2

In a one-shot scenario, the simple serial dictatorship is an appealing
iff a unique series of moves down the tree occurs, $m_1,m_2,\ldots ### Pattern: an [bcdfghjklmnpqrstvwxz]3 thus the usefulness of the platform as a whole. An historic example using an LP. By satisfying the equity criterion, we trivially are infeasible: we cannot solve an LP every time an individual's budget ## Rule: can not 0 Prefix: \b Suffix: \b ### Pattern: can not0 No matches ## Rule: double words 2 Prefix: \b Suffix: \b ### Pattern: (\w+)\s+\12 quickly exhaust the capacity of those on top. In short order, the the We show that when a collection of budgets and and values permits a ## Rule: Improper Latinate Express 0 Use e.g. to mean "for example", and i.e. to mean "that is". Prefix: \b Suffix: \b ### Pattern: i\.e\.0 No matches ### Pattern: e\.g\.0 No matches ### Pattern: etc\.0 No matches # Formal rules 113 ## Rule: That / Which 98 Prefix: \b Suffix: \b ### Pattern: \w+ \w+ which3 buy'' probability shares, giving a bistochastic matrix which can be propose a substitute which we call the reverse tontine'' (RT) the$m+1$case in which we add a new position$v'$and a new ### Pattern: \w+ \w+ that95 several attractive properties that make it suitable for inefficient, though there are other works that take alternate views, visibility, and even those that do use prices do not rely solely on paid'' results and organic'' search results that are not There are several possible answers, but the general explanation is that individual platform member preferences and willingness to pay do example is that selling visibility might undermine the credibility and companies paying DJ's to play certain songs. The platform might worry that those choosing to pay for position might be adversely selected, economic allocation problem, in that the platform has a number of slots that differ in their value and a number of items---people, platforms---has properties that make the standard economic approaches unattractive. One difference is that the economic literature is problem. Let us assume that we have$n$individuals that we need to stochastic assignment mechanisms, i.e., a mechanisms that will place conditional, i.e., we are assuming that all individuals have identical matrix is identical to that generated by the serial dictatorship.} scenario and our setting is that the economic literature was motivated important. A final difference that is common to many (but not all) two-sided platforms is that the limited capacity of the platform participants suggests that frequent changes in merit'' should be creator is likely to generate strong emotions. Whether they perceive it that way or not, the platform creator is implicitely assigning shares. Further, this is plain from the fact that awards should be according to merit''; for all men agree that what is just in allocate by a lottery that treats individuals exactly the same (a exhausted. This is actually the only mechanism that is fair and efficient, in the sense that a higher-ranked individual would never same set of individuals, it can easily run afoul of Aristotle's theory that equals should be treated equally, as well as his argument that that equals should be treated equally, as well as his argument that$b_1 = 0.8$and$b_2 = 0.2$that must be assigned to two positions that offer value (i.e., visibility) worth$v_1 = 80$and$v_2 =
course means that sometimes the strictly lower-merit individual gets
of fairness in expectation. We want the mechanism to capture the notion that likes should be treated alike, which in this case means
notion that likes should be treated alike, which in this case means that expected value is continuous in merit. It practically goes
effectively zero, which means that every visitor can be shown the
best'' page without that page being consumed. While this
non-rivalrousness is true of web pages, it is not true for many other things that might be returned from search, such as individual buyers
dates, knit so many custom sweaters, etc.). The issue is that
contacts. This characteristic makes it critical for applied purposes that budgets can be updated in nearly real-time.
We show that when a collection of budgets and and values permits a
ensured continuity and monotonicity. However, we demonstrate that this
^{2}\,\] To simplify notation, we suppose without loss of generality that $\sum_{i}b_{i}=\sum_{j}v_{j}=1$ and therefore our equity
condition is merely that $b_{i}=\sum_{j}p_{ij}v_{j}$ for all $i$. It
is obvious that the matrix of marginal probabilities, $P$, should be
of Birkhoff \cite{marcus}. The main idea is that if we are given
\STATE Let $P^{*}$ be the probability matrix that minimizes $\left\Vert Pv-b\right\Vert _{2}^{2}$.
\STATE \COMMENT{By \cite{marcus} it must be the case that $\pi_{ij}=1$
\STATE \COMMENT{By \cite{marcus} it must be the case that this reduces the
\STATE \COMMENT{At this point we now know that $P^{*}$ is a convex combination
algorithm shows that meeting the equity criterion is possible in some
gives us continuity in merit. However, in the platform applications that motivated the problem, this approach is likely to prove
assignments as they are needed, in the order that they are likely to
individuals in that step. The name was chosen because in this
Given the description of the Algorithm~\ref{alg:tontine}, one might worry that it would require $n-k$ re-normalizations of the budgets at
unattractive. Fortunately, there is a simple and fast algorithm that
assigned to that position. We complete the process by moving back up
the tree, repairing'' the values for each tree to reflect that the
continuous. Now, we assume that $x_m(.)$ is continuous and consider
immediately and receives $v^* = \max v \cup v'$. Let us denote that
get selected in that round, some other individual $k$ is selected,
We can see that $x_{m+1}()$ is a finite linear combination of
k < n$. We are interested in the probability that the individual Because$b_h > b_l$and by the assumption that both the high and low S_m(b)^{-1})$, which means that every term of $1-F(k; b_h)$ is less
In this section we show that conventional approaches to ordering
in search results reflects the merit of the individuals that
participate in the results. oDesk is an online labor marketplace that
\item Employers can leverage a search interface that allows them to
retrieve contractor profiles that are relevant to some
contractor visibility and merit estimates. In case of search, the way that oDesk ranks contractors makes them more or less visible to the
employers. Thus, the profile views that the contractors receive
through clicks on the search results are indicative of the visibility that oDesk provides them. In case of job applications, the email
(around 10 applicants) makes it exceeedingly likely that the employer
for consecutive positions in search results. In the next paragraph we show that such distribution is not justified by the contractor merit
rate, defined as the ratio of his job applications that evolved into
the search ranking imposes an artificial boost on visibility that is
distrcibutions shows that there are many contractors who have almost
In this section we compare the algorithms that we have presented in
condition. For our comparison we use synthetic data that capture
population. In all of the scenarios we assume that the visibility in
different rank positions follow the Zipfian distribution that we
section, since our experiments showed that the distribution types have
better the algorithm. We observe that tree-based can rank up to $1000$
convinced us that it cannot be used in practice. To summarize, the
Although we showed that the RT algorithm is fast enough to be applied
merit, two properties that are missing from the vanilla assortative
ranking yields significantly greater RMSE that the other ranking
approaches. Finally, note that the RT ranking yields RMSE that is very
for small values of $\alpha$ that yield distributions that were not
observed in the oDesk dataset. Thus, we can conclude that the RT
algorithm yields visibility that is close to the equitable allocation
number of additional criteria for evaluating assignment mechanisms that
are relevant in the platform/repeated assignment setting that we are
focused on. We show that it is possible to create an assignment that
algorithm, but also show that the standard solution is infeasible for
web-scale applications. As a substitute, we proposed an algorithm RT that has some nice properties that make it well-suited for
that has some nice properties that make it well-suited for
criterion. However, we hope that future work might demonstrate how
past allocations to yield assignments that are equitable in

## Rule: Between / Among 5

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### Pattern: (between|among)5

them. For example, search engines draw a sharp distinction between
difference between this setting the common economic allocation
Another important difference between the classic economic allocation
the two individuals become in merit, gap between their expected payoff
not justified from an individual's merit. The gap between the two

## Rule: Fewer / Less 5

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### Pattern: (fewer|less)5

less. As such, the serial dictatorship and the probabilistic
the matrix $(1-t)^{-1}(P-t\Pi)$ has one fewer nonzero entry than
S_m(b)^{-1})$, which means that every term of$1-F(k; b_h)$is less that oDesk ranks contractors makes them more or less visible to the individuals in much less than a second. The tree-based RT achieves ## Rule: Sentence End in a Preposition 5 Prefix: \b Suffix: \. ### Pattern: about0 No matches ### Pattern: above0 No matches ### Pattern: across0 No matches ### Pattern: after0 No matches ### Pattern: against0 No matches ### Pattern: along0 No matches ### Pattern: alongside0 No matches ### Pattern: amid0 No matches ### Pattern: amidst0 No matches ### Pattern: among0 No matches ### Pattern: amongst0 No matches ### Pattern: around0 No matches ### Pattern: as0 No matches ### Pattern: aside0 No matches ### Pattern: at0 No matches ### Pattern: athwart0 No matches ### Pattern: atop0 No matches ### Pattern: barring0 No matches ### Pattern: beford0 No matches ### Pattern: below0 No matches ### Pattern: beneath0 No matches ### Pattern: beside0 No matches ### Pattern: besides0 No matches ### Pattern: between0 No matches ### Pattern: beyond0 No matches ### Pattern: but0 No matches ### Pattern: by0 No matches ### Pattern: circa0 No matches ### Pattern: concerning0 No matches ### Pattern: despite0 No matches ### Pattern: down0 No matches ### Pattern: during0 No matches ### Pattern: except0 No matches ### Pattern: failing0 No matches ### Pattern: following1 The comparison setting is the following. For different sizes$n$of ### Pattern: for0 No matches ### Pattern: from0 No matches ### Pattern: in0 No matches ### Pattern: insi0 No matches ### Pattern: like0 No matches ### Pattern: minus0 No matches ### Pattern: near0 No matches ### Pattern: next0 No matches ### Pattern: notwithstanding0 No matches ### Pattern: of0 No matches ### Pattern: off0 No matches ### Pattern: on3 where they would eat$1/n\$, and so on. The resultant bistochastic
position and the second position before the third, and so on.
focused on. We show that it is possible to create an assignment that

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option until time runs out. \cite{budish2009implementing} gives a

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