# Eliciting Social Knowledge for Creditworthiness Assessment

Mark York
Harvard University
<EMAIL_ADDRESS>Munther Dahleh
MIT
<EMAIL_ADDRESS>David C. Parkes
Harvard University
<EMAIL_ADDRESS>
###### Abstract

Access to capital is a major constraint for economic growth in the developing
world. Yet those attempting to lend in this space face high defaults due to
their inability to distinguish creditworthy borrowers from the rest. In this
paper, we propose two novel scoring mechanisms that incentivize community
members to truthfully report their signal on the creditworthiness of others in
their community. We first design a truncated asymmetric scoring-rule for a
setting where the lender has no liquidity constraints. We then derive a novel,
strictly-proper VCG scoring mechanism for the liquidity-constrained setting.
Whereas Chen et al. (2011) give an impossibility result for an analogous
setting in which sequential reports are made in the context of decision
markets, we achieve a positive result through appeal to interim beliefs about
the reports of others in a setting with simultaneous reports. Moreover, the
use of VCG methods allows for the integration of linear belief aggregation
methods.

##  1 Introduction

Access to capital has become the primary anti-poverty tool in development.
Global microfinance grew from 13 million borrowers and $7 billion in loans in
1995 (Kassim and Rahman, 2018) to 140 million borrowers and $129 billion in
loans in 2019 (Porter, 2020). A particular challenge with microfinance is that
the unbanked have minimal credit history, creating an information asymmetry
problem between lenders and borrowers.

Muhammad Yunus launched microfinance in 1976 with the Grameen bank. They lend
to groups of people who are jointly-liable to repay the loan. This creates
self-selection based on community information (Grameen, 2020), but it also
imposes significant cost on lenders and borrowers through bi-weekly meetings,
the risk of default by fellow group members, and administration.

Another solution is the advent of data-analytics based lenders. These lenders
typically give loans to individuals, and they leverage demographic or other
information to select borrowers. Branch, operating in Kenya, requires users to
own a smartphone with their app installed and runs analytics on the calls,
text messages, emails, and other usage data from the phone. Based on the
performance of past borrowers, these companies determine how likely a new
potential borrower is to repay. Loans are as small as five USD, and interest
rates start at 18% monthly (199% APR) (Branch.co, 2021). While this expands
credit access, it excludes people who do not have smartphones and the interest
rates are high. Another issue is that un-creditworthy borrowers learn which
factors the algorithm considers, and they can modify their behavior to receive
loans (e.g. (Björkegren and Grissen, 2020)).

Fortunately, research shows that community members are knowledgeable about the
creditworthiness of people in their community. Maitra et al. (2013) deployed
an agent-intermediated lending scheme in West Bengal, India through which they
appointed agents to select borrowers and administer loans. These agents were
compensated based on repayment rates, and the repayment rates were higher than
those for group lending schemes in the same region. Hussam et al. (2021) went
one step further and deployed a commmunity-recommendation scheme employing the
Robust Bayesian Truth Serum (RBTS) to reward recommenders for giving reports
that conform closely to those of their peers (Witkowski and Parkes, 2012a).
RBTS was found to partially nullify the incentives of recommenders to lie on
behalf of family members. Of note, RBTS does not reward recommenders based on
repayment outcomes.

In this paper, we propose a new information elicitation system that
incentivizes community members to report their true beliefs about the
likelihood that others will repay a loan.111An initial deployment of the
scheme, conducted under Harvard University’s IRB, is underway in Uganda with
100 agricultural borrowers, thanks to a partnership with Makere University and
generous support from the Global Challenges in Economics and Computation. We
handle lenders with a minimum profit threshold on the belief that a borrower
will repay a loan and handle lenders with and without liquidity constraints
that limit how many loans they can make.

The goal is to support a lender who wants to make loans to the best borrowers,
where this is defined according to the belief aggregation rule of a lender and
its liquidity constraints and profit thresholds. We achieve this through
incentive alignment so that recommenders will have strict incentives to prefer
to report their true beliefs over other possible reports. This is the concept
of strict properness from the scoring rule literature. Here, we seek strict
properness in the interim, so that it holds for any information of a
recommender (any belief about the likelihood of repayment of borrowers) and in
expectation with respect to a prior on the beliefs of other recommenders about
the likelihood of repayment of borrowers.

The main results are the following:

  1. 1.

A mechanism, the truncated Winkler mechanism, that is strictly proper for a
reasonable (grain of no veto) technical condition for a lender without
liquidity constraints and multiple recommenders and multiple borrowers, and
for a monotone non-decreasing belief aggregation rule. We show that the
mechanism is not incentive compatible for a liquidity-constrained borrower.

  2. 2.

A mechanism, the Vickrey-Clarke-Groves (VCG) scoring mechanism, that is
strictly proper under reasonable conditions (reasonably uniform aggregation
weights across recommenders), for a lender with or without liquidity
constraints and multiple recommenders and multiple borrowers, and for a
weighted linear belief aggregation rule.

  3. 3.

The VCG scoring mechanism also aligns incentives with recommenders wanting to
receive larger weights in the aggregator, and thus higher quality predictions
when this leads to higher weights over time. Moreover, the VCG scoring
mechanism can be configured to ensure that all recommenders have non-negative
utility from participation, whatever the outcome from making loans.

In regard to the first result, which makes use of the Winkler scoring rule, it
is important that we use asymmetric scoring rules such that the minimum
expected score from the scoring rule is associated with the lender’s threshold
on minimum probability of repayment at which making a loan is profitable. We
develop the grain of no veto condition, which provides strict incentives, by
reasoning about the interim utility and uncertainty faced by any given
recommender.

While making use of the well known VCG mechanism, the application of the VCG
scoring mechanism to this context is novel and nonstandard. In particular, we
use payments both in the normal sense of VCG and also to construct the
valuation functions of recommenders for different outcomes. We make use of
outcome-contingent payments to, in effect, give a recommender a valuation
functions for making a loan to a borrower that is proportional to the belief
the recommender has as to the repayment probability of the borrower. By
folding the typical VCG style payments on top, we in effect take a constant
(trivially proper but not strictly proper) scoring rule and generate an
elicitation mechanism that is strictly proper. Moreover, the valuation
functions are defined so that the allocation rule of the mechanism corresponds
to a belief aggregation model, and can embed weights assigned by a belief
aggregator to recommenders in an incentive-compatible way. This use of linear
weighted belief aggregators corresponds naturally to well-studied belief
aggregation systems (Soule et al., 2020).

While our work is inspired by lending, these two mechanisms are broadly
applicable in decision settings where interim uncertainty creates full support
on the decision space from the agent perspective, and where outcomes are
observed. Examples include employee screening, tenant screening, insurance
underwriting, and service provider ratings.

The remainder of this paper is structured as follows. Section 1.1 contains a
brief literature review of prediction markets, decision markets, scoring rules
and report aggregation. Section 2 describes our recommendation gathering
system and defines notation and key terms. Section 3 discusses the
unconstrained liquidity setting and the truncated Winkler mechanism. Section 4
introduces the VCG scoring mechanism for the liquidity-constrained setting and
gives an analysis of the incentive properties of the mechanism. Section 5
concludes and points to directions for future work.

###  1.1 Related Work

One way to formulate the problem of gathering community lending
recommendations is as a peer prediction problem, i.e., as a problem of
information elicitation without verification. The approach in peer prediction
is to leverage correlation and mutual-information structure between reports to
promote incentive alignment around true reports. A number of peer prediction
mechanisms have been proposed, each requiring varying levels of knowledge on
the part of the designer, on the kinds of reports, and on the task e.g.
(Miller et al., 2005; Witkowski and Parkes, 2012a, b; Shnayder et al., 2016;
Agarwal et al., 2017; Jurca and Faltings, 2009; Kong and Schoenebeck, 2019;
Radanovic et al., 2016; Wang et al., 2019; Waggoner and Chen, 2014). Indeed,
this was the approach taken to belief elicitation for microfinance in Hussam
et al. (2021).

Our framing of the microfinance problem is that of information elicitation
with verification, where a lender will observe whether or not a borrower makes
a repayment or defaults in the future, making this setting well-suited for the
methods of scoring rules, prediction markets, and decision markets. Scoring
rules are methods to elicit beliefs about uncertain future events where the
outcome will be later observed (Gneiting and Raftery, 2007). The basic
framework is that a participant reports her belief, $\hat{p}$ of the
probability of an event. Her true belief is denoted as $p$. After the event is
observed, participants are paid a reward (potentially negative) of
$s(\hat{p},o)$, where $o\in\mathcal{O}$ is the outcome of the event,
$\mathcal{O}$ is a finite, exhaustive set of mutually-exclusive outcomes, and
$s$ is a scoring rule mapping the report $\hat{p}$ and outcome $o$ to reward.
A proper scoring rule is one in which the expected score from a truthful
report is at least as great as the expected score from any non-truthful
report, i.e.,

$\mathbb{E}_{o\sim p}[s(p,o)]\geq\operatorname{\mathbb{E}}_{o\sim
p}[s(\hat{p},o)]\;;\;\forall\hat{p}\neq p$ (1)

A strictly proper scoring rule replaces this inequality with a strict
inequality. Common strictly proper scoring rules for the binary outcome case
include the logarithmic scoring rule and the Brier or quadratic scoring rule.
These are symmetric in the sense that the expected score when truthfully
reporting is minimized at $p=0.5$ and symmetric about that point. A useful
modification is the class of Winkler scoring rules (Winkler, 1994) which allow
designers to set the minimum score point at any arbitrary location
$c\in(0,1)$.

Prediction markets can be used in a way that combines scoring rules with
sequential elicitation from multiple participants, with the current market
price reflecting the aggregate belief of the population about the outcome of
an uncertain event (Chen and Pennock, 2010). The market starts with an initial
prediction $p_{0}\in\Delta(\mathcal{O})$, where $\Delta(\mathcal{O})$ is the
set of distributions on outcomes. In a prediction market with an automated
market maker, for example the logarithmic market-scoring rule, agent $i$’s
report $\hat{p}_{i}$ is in effect scored relative to the preceding report
$\hat{p}_{i-1}$ (Hanson, 2007).

Scoring rules and prediction markets are analyzed in a setting in which agents
have no influence on whether information about whether an event is realized.
This is not the case in the present paper, where agents’ reports will
determine who gets a loan. This brings us close to the decision market
framework where a principal makes a decision based on market prices (Chen et
al., 2011). This creates new incentive challenges. For example, suppose there
is one loan to allocate and two borrowers, with current market price 0.8 and
0.6 for each borrower, and an agent with corresponding beliefs 0.9 and 0.89.
In a market-scoring rule context, the agent would have a higher expected
payoff than truthful reporting by leaving the price on borrower 1 unchanged
and buying borrower 2 to a price of 0.89, so that the second borrower gets the
loan.

Chen et al. (2011) provide a characterization of strict properness that
requires randomization over decisions and full support on the set of possible
decisions. This can present a challenge to many applications, for example to
decisions about construction projects or the present context of loan
decisions, and a hurdle to the real-world implementation of decision markets.
As we discuss in Sections 3 and 4, we provide a counterpoint to this
requirement of randomization with full support. The key difference is that the
incentive analysis is conducted in the interim when an agent knows its own
belief report but is uncertain about the belief reports of others, this
uncertainty maintained through simultaneous reports. This interim uncertainty
enables strict incentive alignment for a deterministic decision rule that
lends to the set of borrowers that are most likely not to default.

VCG concepts have been used together with scoring rules but not in the way
described here. Papakonstantinou et al. (2011) use a two-stage elicitation
mechanism whereby agents participate in a second price procurement auction for
information with the winner paid using a scoring rule based on the cost of the
second-lowest-cost agent. Other work has considered the problem of selection
of agents from within the same group as those who provide yes/no approval
information on others (Alon et al., 2011). The goal is to select agents with
the maximum number of approvals from others and there is no downstream
observation of an uncertain event. Whereas we assume the recommenders is
disjoint from the set of potential borrowers, Alon et al. (2011) focus on the
incentive issues that arise when this is not the case.

This work also relates to belief aggregation, which we briefly review in the
Appendix.

##  2 Preliminaries

A lender has a set of candidate borrowers $M=\\{1,\ldots,m\\}$ and recruits a
set $N=\\{1,\ldots,n\\}$ of recommenders who know the candidate borrowers
personally and provide reports on the $m$ borrowers. Each recommender $i\in N$
has a subjective belief $p_{iq}\in[0,1]$ of the likelihood with which
candidate borrower $q$ will repay a loan. We write
$p_{i}=(p_{i1},\ldots,p_{im})$, $p_{q}=(p_{1q},\ldots,p_{nq})$, and
$p=(p_{1},\ldots,p_{n})$. We also refer to belief $p_{i}$ as the type of the
recommender. We let ${\mathcal{D}}$ denote a prior on beliefs, such that
$p\sim{\mathcal{D}}$. We write
$p_{-i}=(p_{1},\ldots,p_{i-1},p_{i+1},\ldots,p_{n})$, and write
$p_{-i}\sim{\mathcal{D}}_{-i}$, marginalizing out over recommender $i$. We
assume ${\mathcal{D}}$ and ${\mathcal{D}}_{-i}$ are common knowledge.

Recommender $i$ makes a report $\hat{p}_{iq}\in[0,1]$ to the lender
(principal), and we allow $\hat{p}_{iq}\neq p_{iq}$. We use $\hat{p}_{i}$ to
denote the profile of all belief reports of recommender $i$. In the mechanisms
that we design, recommenders make reports independently with no knowledge of
other recommenders’ reports. The repayment outcome for a borrower $q$ who
receives a loan is a binary variable, $o_{q}\in\\{0,1\\}$, with $1$
representing repayment and $0$ representing default.

The lender makes a decision about which borrowers will receive a loan. We
assume that the lender has a profit threshold $c\in[0,1]$, such that the
lender makes profit when making a loan where the repayment probability is $c$
or higher. The lender forms a belief about the repayment probability of a
borrower $q\in M$ with an aggregation function $B_{q}(p_{q})$, which
represents the lender’s belief where $p_{q}=(p_{1q},\ldots,p_{nq})$. We assume
this is weakly monotone increasing in $p_{i}$, for each $i$. We also sometimes
work with a linear aggregator, $B_{q}(p_{q})=\sum_{i\in N}w_{i}p_{iq}$, with
weight $w_{i}>0$ on recommender $i$ and $\sum_{i\in N}w_{i}=1$. The lender
also has a liquidity constraint $K\leq m$, and if $K<m$ then can only make
loans to a limited number of borrowers.

######  Definition 1 (Elicitation mechanism).

We design an elicitation mechanism $\mathcal{M}=(x,t,s)$:

  1. 1.

Elicit belief reports $\hat{p}=(\hat{p}_{1},\ldots,\hat{p}_{n})$ from
recommenders

  2. 2.

Determine the set of borrowers, $x(\hat{p})\in\\{0,1\\}^{m}$, that will
receive a loan, such that $\sum_{q\in M}x_{q}(\hat{p})\leq K$, and define two-
part payments:

    1. (a)

An immediate payment  $t_{i}(\hat{p})\in\mathbb{R}$ made by each recommender
$i\in N$

    2. (b)

An outcome-contingent payment $s_{iq}(\hat{p}_{iq},o_{q})\in\mathbb{R}$ made
to each recommender $i$ for each borrower $q\in a=x(\hat{p})$, i.e., for each
borrower for which $a_{q}=1$.

Given reports $\hat{p}$ and outcome profile $o=(o_{1},\ldots,o_{m})$, the
realized utility to recommender $i$ is

$\displaystyle u_{i}(\hat{p}_{i},\hat{p}_{-i},o)=\sum_{q\in
x(\hat{p})}s_{iq}(\hat{p}_{iq},o_{q})-t_{i}(\hat{p}).$ (2)

Here,
$\hat{p}_{-i}=(\hat{p}_{1},\ldots,\hat{p}_{i-1},\hat{p}_{i+1},\ldots,\hat{p}_{n})$.
The utility for recommender $i$ with belief $p_{i}$ is

$\displaystyle U_{i}(p_{i},\hat{p}_{i},\hat{p}_{-i})$
$\displaystyle=\sum_{q\in
x(\hat{p})}\left(p_{iq}s_{iq}(\hat{p}_{iq},1)+(1-p_{iq})s_{iq}(\hat{p}_{iq},0)\right)-t_{i}(\hat{p})$
(3) $\displaystyle=\sum_{q\in x(\hat{p})}\mathbb{E}_{o_{q}\sim
p_{iq}}[s_{iq}(\hat{p}_{iq},o_{q})]-t_{i}(\hat{p})$

This quantity is ex post with respect to the reports of others, and takes an
expectation over borrower outcomes with respect to the beliefs of recommender
$i$. It is useful to interpret the outcome-contingent payment to the
recommender as inducing a term that plays a similar role as an agent’s
valuation in mechanism design, where $\mathbb{E}_{o_{q}\sim
p_{iq}}[s_{iq}(\hat{p}_{iq},o_{q})]$ is the recommender’s “value” for the
lender’s decision to lend to borrower $q$.

There are a number of possible desiderata for an elicitation mechanism in this
setting.

  * •

Allocative efficiency means that the mechanism allocates to the borrowers with
the maximum probability of repayment amongst those better than the profit
threshold $c$. For lender belief $B_{q}(\hat{p}_{q})$, this requires that

$\displaystyle x(p)\in$
$\displaystyle\operatorname*{arg\,max}_{a\in\\{0,1\\}^{m}}\sum_{q\in
M:B_{q}(p_{q})>c}B_{q}(p_{q})\times a_{q}$ (4) s.t. $\displaystyle\sum_{q\in
M}a_{q}\leq K$

  * •

Weak ex post incentive compatibility (weak EPIC) means that each recommender’s
utility is weakly maximized by reporting truthfully, regardless of the reports
of other recommenders (or ex post proper in the language of the scoring rule
literature, adapted here to also consider the effect of reports of others).
This is

$\displaystyle U_{i}(p_{i},p_{i},\hat{p}_{-i})\geq
U_{i}(p_{i},\hat{p}_{i},\hat{p}_{-i})\;;\forall i,\ \forall p_{i},\
\forall\hat{p}_{i},\ \forall\hat{p}_{-i}$ (5)

  * •

Strict ex post incentive compatibility (strict EPIC) means that each
recommender’s utility is strictly maximized by reporting truthfully,
regardless of the reports of other recommenders (or ex post strict proper in
the language of the scoring rule literature, adapted here to also consider the
effect of reports of others). This is

$\displaystyle
U_{i}(p_{i},p_{i},\hat{p}_{-i})>U_{i}(p_{i},\hat{p}_{i},\hat{p}_{-i})\;;\forall
i,\ \forall p_{i},\ \forall\hat{p}_{i}\neq p_{i},\ \forall\hat{p}_{-i}$ (6)

These incentive concepts are ex post with respect to the reports of others and
interim with respect to a recommender’s own belief on the outcome of making
loans to borrowers, and defined before repayment outcomes are observed.

  * •

Strict interim incentive compatibility (strict IIC) means that each
recommender’s interim utility, considering beliefs of others, is strictly
maximized by reporting truthfully (or strict properness in the language of the
scoring rule literature). This is

$\displaystyle\mathbb{E}_{p_{-i}\sim{\mathcal{D}}_{-i}}[U_{i}(p_{i},p_{i},p_{-i})]>\mathbb{E}_{p_{-i}\sim{\mathcal{D}}_{-i}}[U_{i}(p_{i},\hat{p}_{i},p_{-i})]\;;\forall
i,\ \forall p_{i},\ \forall\hat{p}_{i}\neq p_{i}$ (7)

  * •

Ex post Individually Rational (IR) means that all recommenders that make a
truthful report have a non-negative expected utility once loans are allocated,
but before repayment outcomes are observed. This is

$\displaystyle U_{i}(p_{i},p_{i},\hat{p}_{-i})\geq 0\;;\forall i,\ \forall
p_{i},\ \forall\hat{p}_{-i}.$ (8)

  * •

Strong ex post IR means that all recommenders that make a truthful report have
a non-negative realized utility even after repayment outcomes are observed.
This is

$\displaystyle u_{i}(p_{i},\hat{p}_{-i},o)\geq 0\;;\forall i,\ \forall p_{i},\
\forall\hat{p}_{-i},\ \forall o.$ (9)

The first concept of IR is ex post with respect to the reports of other
recommenders. For this reason, we adopt the phrasing strongly ex post IR for
the second notion, which holds once outcomes are observed.

##  3 Unconstrained-Liquidity Setting: Truncated Winkler Mechanism

In this section, we use strictly proper scoring rules to design elicitation
mechanisms for the unconstrained liquidity setting, where the problem
decomposes to make a separate decision for each borrower. We make use of the
Winkler scoring rule. Relative to the otherwise more general VCG scoring
mechanism, this approach can accommodate non-linear belief aggregation rules.

###  3.1 One Recommender, One Borrower

We first consider the case of a single recommender and a single borrower, and
drop subscripts for notational simplicity. Without care, this setup can be
prone to the problem outlined in Chen et al. (2011) for decision markets, and
a recommender may be incentivized to misreport to ensure that the borrower
receives a loan and they have a chance of being compensated. To illustrate
this, consider setting the immediate payment $t$ to zero, and using making use
of a truncated quadratic scoring rule for the outcome-contingent payment:

$\IEEEeqnarraybox[][c]{rcl}s^{TQ}(\hat{p},o)=\begin{cases}2\hat{p}-\hat{p}^{2}-(1-\hat{p})^{2}&\mbox{if
$\hat{p}>c,o=1$, and}\\\ 2(1-\hat{p})-\hat{p}^{2}-(1-\hat{p})^{2}&\mbox{if
$\hat{p}>c,o=0$.}\\\ \end{cases}\IEEEstrut$ (10)

The rule is “truncated” because it only defines a score when report
$\hat{p}>c$. The recommender receives no outcome-contingent payment when
$\hat{p}\leq c$ and a loan is not made.

Figure 1: Expected utility for a truncated quadratic scoring rule under
truthful reporting vs. expected utility when reporting $\hat{p}=.6+\epsilon$,
for small $\epsilon>0$. The threshold is set to $c=0.6$.

Let $U(p)$ denote the expected utility under truthful reporting. For $p>c$
this is $\mathbb{E}_{o\sim p}[s^{TQ}(p,o)]$ and otherwise this is zero. See
Figure 1, for the case of $c=0.6$. A recommender with true belief $p<0.6$ can
achieve a higher utility by misreporting $\hat{p}=0.6+\epsilon$ for small
$\epsilon>0$. This is akin to the problem with incentive alignment in decision
markets.

One idea to address this is to add an immediate payment $t(\hat{p})$, set to
compensate for the forfeited payment in the case that the report $\hat{p}\leq
c$. For the quadratic scoring rule, we would set

$\displaystyle t^{TQ}(\hat{p})$
$\displaystyle=\left\\{\begin{array}[]{ll}-[c^{2}+(1-c)^{2}]&\mbox{if
$\hat{p}\leq c$,}\\\ 0&\mbox{otherwise.}\end{array}\right.$ (13)

We negate the quantity here, because the convention is that this is the
payment made by a recommender to the mechanism. In this way, there is
continuity in the expected payment across this report threshold. However, this
elicitation mechanism aligns incentives as long as threshold $c\geq 0.5$, but
not otherwise.

Figure 2 illustrates the combined effect of the one-time and outcome-
contingent payments, plotting $U(p)$. For $c=0.6$, $U(p)$ is convex, which is
a sufficient condition for the rule to be proper (Gneiting and Raftery, 2007).
In particular, it is strictly convex and strictly proper for $p>c$ and weakly
proper for $p\leq c$. The strictness for beliefs above $c$ ensures the lender
will make the correct decision for any optimal report of the recommender.
However, for $c<0.5$, $U(p)$ is non-convex and a recommender with belief
$p\in(c,1-c)$ can obtain a higher utility by reporting $\hat{p}\leq c$. This
is because the non-truncated quadratic scoring rule $s^{Q}$ is symmetric with
$G^{Q}(p)=G^{Q}(1-p)$, for $G^{Q}(p)=\mathbb{E}_{o\sim p}[s^{Q}(p,o)]$, and
with a minimum $G^{Q}(p)$ at $p=.5$.

Figure 2: Expected utility for a truncated quadratic scoring rule together
with immediate payment $t^{TQ}$ for threshold values $c=.6$ and $c=.3$. Note
the properness when $c\geq 5$ and the lack of properness when $c<.5$

To avoid this, we use an asymmetric scoring rule whose minimum expected score
under truthful reporting is at $p=c$. Let $s$ be any symmetric proper scoring
rule, and consider some $c\in(0,1)$. The Winkler scoring rule $s^{W}$
(Winkler, 1994) is

$s^{W}(\hat{p},o)=\frac{s(\hat{p},o)-s(c,o)}{T(c,\hat{p})},\quad\mbox{where}\quad
T(c,\hat{p})=\begin{cases}s(0,0)-s(c,0)&\mbox{if $\hat{p}\leq c$}\\\
s(1,1)-s(c,1)&\mbox{otherwise.}\end{cases}$ (14)

The Winkler rule $s^{W}$ is (strictly) proper when $s$ is (strictly) proper.
We choose to build $s^{W}$ from the logarithmic scoring rule, obtaining:

$\IEEEeqnarraybox[][c]{rcl}s^{W}(\hat{p},o)&\ =\
\begin{cases}\frac{\ln(\hat{p})-\ln(c)}{-\ln(c)}&\mbox{if $\hat{p}>c,o=1$}\\\
\frac{\ln(1-\hat{p})-\ln(1-c)}{-\ln(c)}&\mbox{if
$\hat{p}>c,o=0$.}\end{cases}\IEEEstrut$ (15)

Figure 3: Left: Expected score $G^{W}(p)$ with truthful reporting under the
Winkler Scoring Rule based on the Logarithmic Scoring Rule. Right: Expected
utility $U(p)$ with truthful reporting for the truncated Winkler elicitation
mechanism and lender threshold $c=0.3$.

Figure 3 (left) shows the expected score $G^{W}(p)=\mathbb{E}_{o\sim
p}[s^{W}(p,o)]$ from the Winkler rule. $G^{W}(p)=0$ for $p=c$. This leads to
the following elicitation mechanism.

######  Definition 2 (Truncated Winkler elicitation mechanism (1 recommender,
1 borrower)).

The Truncated Winkler elicitation mechanism for 1 recommender and 1 borrower
and lender profit threshold $c$ is defined as following:

  * •

Allocation: $x(\hat{p})=1$ if $\hat{p}>c$ and $x(\hat{p})=0$ otherwise

  * •

Payment

    * –

Immediate payment: zero

    * –

Outcome-contingent payment: $s(\hat{p},o)=s^{W}(\hat{p},o)$ as per Eq. (15).

Figure 3 (right) shows the expected utility $U(p)$ from truthful reporting in
a truncated Winkler mechanism with $c=0.3$. The utility is convex and given by

$\IEEEeqnarraybox[][c]{rcl}U(p)&\ =\ \begin{cases}0&\mbox{, if $p\leq c$}\\\
p\frac{\ln(p)-\ln(c)}{-\ln(c)}+(1-p)\frac{\ln(1-p)-\ln(1-c)}{-\ln(c)}&\mbox{otherwise.}\end{cases}\IEEEstrut$
(16)

######  Theorem 3.1.

For the one-borrower, one-recommender case, the truncated Winkler elicitation
mechanism with lender profit threshold $c$ is ex post IR, and strict proper
for beliefs $p>c$ and proper for all beliefs.

###### Proof.

For a single recommender setting, strict ex post proper and ex post proper are
equivalent to simply strict proper and proper, since there are no reports of
others. The expected score $G^{W}(p)$ from the Winkler rule under truthful
reporting is strictly positive when $p>c$ (Winkler, 1994). Since reports of
$\hat{p}\leq c$ yield payment of $0$, risk-neutral agents will always prefer
$\hat{p}>c$ when $p>c$. Given that $s^{W}$ is strictly proper, the full
mechanism is also strictly proper when $p>c$. When $p\leq c$, the convexity of
$U(p)$ and the fact that $U(c)=0$ guarantees that a recommender weakly
maximizes its expected payment from truthful reporting when $p\leq c$. This
gives properness for all beliefs. Individual rationality is immediate, since
$U(p)\geq 0$. $\blacksquare$ ∎

###  3.2 Multiple Recommmenders, Multiple Borrowers

We now consider the multiple recommenders, multiple borrowers case. Because
liquidity is unconstrained, the lending decision decomposes easily across
borrowers.

A new aspect of the analysis is that with multiple recommenders we can obtain
strict interim IC (strict properness) by reasoning about the interim
uncertainty a recommender has about the reports of other recommenders. We
bring back subscripts, and recommender $i$’s report on borrower $q$ is denoted
$\hat{p}_{iq}$ and borrower $q$’s repayment outcome is $o_{q}$.

For borrower $q$, a loan is made if and only if $B_{q}(\hat{p}_{q})>c$. In
defining the outcome-contingent payment, we let $c_{iq}$ denote the minimum
report by recommender $i$ on borrower $q$ such a loan is made to $q$, i.e.,

$c_{iq}\ \triangleq\
\underset{p^{\prime}\in[0,1]}{\mathrm{inf}}p^{\prime}\;\mathrm{s.t.}\;B_{q}(\hat{p}_{iq}=p^{\prime},\hat{p}_{-iq})>c,$
(17)

where $p_{-iq}=(p_{1q},\ldots,p_{i-1,q},p_{i+1,q},\ldots,p_{nq})$. $c_{iq}$ is
a function of the reports of others but we leave this implicit. By weak
monotonicity of the belief aggregator, for any $\hat{p}_{iq}>c_{iq}$ we have
that borrower $q$ receives a loan. For example with a linear aggregator
$B_{q}(\hat{p}_{q})=\sum_{i\in N}w_{i}\hat{p}_{iq}$, we have

$c_{iq}=\mathrm{min}(1,\mathrm{max}(0,\frac{1}{w_{i}}(c-\sum_{j\neq
i}w_{j}\hat{p}_{jq}))).$ (18)

Each recommender’s outcome-contingent payment follows the Winkler scoring rule
(15), with Winkler parameter $c$ replaced by $c_{iq}$. We let
$s^{W}_{iq}(\cdot,\cdot)$ denote this modified Winkler scoring rule.

######  Definition 3 (Truncated Winkler elicitation mechanism ($n$
recommenders, $m$ borrowers)).

The Truncated Winkler elicitation mechanism with unconstrained lender
liquidity, lender profit threshold $c$, and monotone belief aggregation
$B_{q}$, is defined as following:

  * •

Allocation: for each borrower $q$, $x_{q}(\hat{p})=1$ if
$B_{q}(\hat{p}_{q})>c$ and $x_{q}(\hat{p})=0$ otherwise

  * •

Payment:

    * –

Immediate payment: zero

    * –

Outcome-contingent payment: for each borrower that receives a loan,
$s_{iq}(\hat{p}_{iq},o_{q})=s^{W}_{iq}(\hat{p}_{iq},o_{q})$ as per Eq. (15),
and with parameter $c$ in the Winkler rule set to $c_{iq}$ as per Eq. (17).

Lemma 1 follows immediately from the proof of Theorem 3.1, recognizing that
the thresholds $c_{iq}$ play the role of $c$ (and allowing for the incentive
and IR properties to be stated ex post, i.e., for any reports of others).

######  Lemma 1.

For unconstrained liquidity, multiple borrowers and multiple recommenders, and
a weak monotone-increasing aggregation function $B_{q}$, the truncated Winkler
mechanism is ex post individually rational, and ex post proper for a report of
recommender $i$ on borrower $q$ when $p_{iq}\leq c_{iq}$ and strictly ex post
proper when $p_{iq}>c_{iq}$.

We now state our first main theorem. For this, define no veto for $i$ and $q$
at threshold $c$ to mean $B_{q}(0,p_{-i,q})>c$, i.e., beliefs of others such
that even with $p_{iq}=0$ from recommender $i$ the borrower will receive a
loan under truthful reports, and thus also with any belief of recommender $i$
(by the monotonicity of $B_{q}$).

######  Definition 4 (Grain of no veto).

The distribution on beliefs satisfies a grain of no veto at $c$ when
$\mathbb{P}[p_{-i,q}\sim{\mathcal{D}}_{-i}:B_{q}(0,p_{-i,q})>c]>0$, for all
recommenders $i$, all borrowers $q$, i.e., no veto is satisfied with non-zero
measure of the type distribution.

######  Theorem 3.2.

For unconstrained liquidity, multiple borrowers and multiple recommenders, and
a weak monotone-increasing aggregation function $B_{q}$, the truncated Winkler
mechanism is ex post IR, and also strictly interim IC (strict proper) when the
distribution on beliefs satisfies grain of no veto.

###### Proof.

Per Lemma 1, when $p_{iq}>c_{iq}$, the Winkler scoring mechanism is strictly
ex post proper with regard to $i$’s report on $q$. Otherwise, it is ex post
proper. By the grain of no veto property, for any belief of $i$ on $q$
$p_{iq}$, there is non-zero probability that $p_{iq}>c_{iq}$. This implies
$\mathbb{E}_{p_{-i}\sim{\mathcal{D}}_{-i}}[U_{i}(p_{i},p_{i},p_{-i})]>\mathbb{E}_{p_{-i}\sim{\mathcal{D}}_{-i}}[U_{i}(p_{i},\hat{p}_{i},p_{-i})]$,
for all $\hat{p}_{-i}\neq p_{i}$. IR follows immediately from Lemma 1.
$\blacksquare$ ∎

For a weighted linear aggregator, the grain of no veto condition requires (for
all $i$, all $q$) that $\mathbb{P}[p_{-i,q}\sim{\mathcal{D}}_{-i}:\sum_{j\neq
i}w_{j}p_{jq}>c]>0$. That is, there is some non-zero probability that the
weighted sum over reports of all but one recommender is large enough. As a
special case, we can state a corollary for the case that beliefs have full
support on $[0,1]$.

######  Corollary 1.

For unconstrained liquidity, multiple borrowers and multiple recommenders, and
a belief distribution with full support, the truncated Winkler mechanism with
the weighted linear aggregator is ex post IR, and also strictly interim IC
(strict proper) for lender profit threshold $c$ when $\max_{i}w_{i}<1-c$,
which is only possible when $c<(n-1)/n$.

###### Proof.

Since $\max_{i}w_{i}<1-c$, then $\sum_{j\neq i}w_{j}>c$, for all $j\in N$.
From this, we have $\mathbb{P}[p_{-i,q}\sim{\mathcal{D}}_{-i}:\sum_{j\neq
i}w_{j}p_{jq}>c]>0$, by full support, and thus the grain of no veto condition.
Moreover, since $\max_{i}w_{i}\geq 1/n$, we need $1/n<1-c$ and thus
$c<(n-1)/n$. ∎

As the threshold increases the system needs more recommenders to provide
strict properness and the lender becomes less able to put a very large weight
on any single recommender.

##  4 Constrained-Liquidity Setting: The VCG Scoring Mechanism

In this section, we give a mechanism that provides strict properness for the
possibly constrained-liquidity case, i.e., handling $K<m$ in addition to
$K=m$. We first illustrate the failure of the truncated Winkler mechanism, and
then introduce the VCG scoring mechanism that achieves strict properness
together with a linear belief aggregator.

###  4.1 Failure of the Truncated Winkler Mechanism

The following theorem also implies that the Truncated Winkler mechanism fails
to be strict interim IC under the grain of no veto property.

######  Theorem 4.1.

In the constrained-liquidity setting with more than one recommender, the
Truncated Winkler mechanism is not weakly ex post IC (ex post proper).

###### Proof.

Consider three recommenders, two borrowers, budget $K=1$, lending threshold
$c=.5$, and $s^{W}$ based on the logarithmic scoring rule. Suppose an
aggregator that is a simple average of reports. The beliefs are as in Table 1.
If all recommenders report truthfully, lender’s belief will be $~{}.57$ and
$.55$, for borrowers 1 and 2 respectively, and borrower 1 will be allocated.
The expected utility (payment) of recommender 2 will be
$p_{2,1}s^{W}(p_{2,1},1)+(1-p_{2,1})s^{W}(p_{2,1},0)=.4\frac{ln(.4)-ln(.2)}{-ln(.2)}+(1-.4)\frac{ln(1-.4)-ln(1-.2)}{-ln(.2)}=.4*.43+.6*(-.18)=.07$.
If, recommender 2 misreports $\hat{p}_{21}=0$, then the lender beliefs will be
$.43$ and $.55$ for borrowers 1 and 2, and borrower 2 will be allocated. In
this case, recommender 2’s expected utility will be
$p_{2,2}s^{W}(p_{2,2},1)+(1-p_{2,2})s^{W}(p_{2,2},0)=.85\frac{ln(.85)-ln(.7)}{-ln(.7)}+(1-.85)\frac{ln(1-.85)-ln(1-.7)}{-ln(.7)}=.85*.54+.15*(-1.94)=.17$.
Similar examples can be constructed for any number of recommenders greater
than one. $\blacksquare$ ∎

| Recomender 1 | Recommender 2 | Recommender 3
---|---|---|---
Belief on Borrower 1 | .7 | .4 | .6
Belief on Borrower 2 | .4 | .85 | .4
Borrower 1 threshold $c_{i1}$ | .5 | .2 | .4
Borrower 2 threshold $c_{i2}$ | .25 | .7 | .25
Expected utility, Honest | .12 | .07 | .09
Expected utility, Recommender 2 Misreport | .04 | .17 | .04
Table 1:  Perverse incentives with the Truncated Winkler mechanism in the
liquidity-constrained setting.

###  4.2 The VCG Scoring Mechanism

For the VCG scoring mechanism, we define the outcome-contingent payment to be
that of a constant scoring rule, and

$s^{VCG}_{iq}(\hat{p}_{iq},o_{q})=\begin{cases}w_{i},\mbox{if $o_{q}=1$}\\\
0,\mbox{otherwise.}\end{cases}$ (19)

This scoring rule is trivially proper, but not strictly proper (it doesn’t
depend on the report). Interestingly, this will provide strictly proper
incentives when embedded within the framework of the VCG mechanism. It is
useful to define the value function of a recommender for loan decisions
$a\in\\{0,1\\}^{m}$ in the context of the VCG scoring mechanism as

$\displaystyle v_{i}(a)$ $\displaystyle\triangleq\sum_{q\in
M}a_{q}\times\operatorname{\mathbb{E}}_{o_{q}\sim
p_{iq}}[s^{VCG}_{iq}(p_{iq},o_{q})]=\sum_{q\in M}a_{q}w_{i}p_{iq}.$ (20)

Similarly, we define the reported value function in the context of the VCG
scoring mechanism as

$\displaystyle\hat{v}_{i}(a)$ $\displaystyle\triangleq\sum_{q\in
M}a_{q}\times\operatorname{\mathbb{E}}_{o_{q}\sim\hat{p}_{iq}}[s^{VCG}_{iq}(\hat{p}_{iq},o_{q})]=\sum_{q\in
M}a_{q}w_{i}\hat{p}_{iq}.$ (21)

These play the typical role of valuations and reported valuations in the
analysis of the incentive properties of a VCG mechanism. A non-standard aspect
is that the weight $w_{i}$ is under control of the mechanism designer because
it reflects the payment made in the event that a borrower makes a repayment
(the outcome-contingent payments). As the designer has direct control over
this aspect of the value function, we can use these weights without using a
weighted VCG mechanism.

This aligns the allocation of loans that maximizes the total reported value
with the allocation that lends to the borrowers who are best in the sense of
having maximum total weighted reported beliefs from recommenders, and thus are
most likely to repay according to the lender’s linear weighted belief
aggregator. That is, the value-maximizing allocation is just:

$\displaystyle\max_{a\in\\{0,1\\}^{m}}$ $\displaystyle\sum_{q\in
M}a_{q}\left(\sum_{i\in N}w_{i}\hat{p}_{iq}\right)$ (22) s.t.
$\displaystyle\sum_{q\in M}a_{q}\leq K.$

To introduce a lending threshold $c>0$, we can also add $K$ imaginary reserve
borrowers to the system and a reserve recommender with weight 1 who reports
$c$ for a loan decision to each of these borrowers and 0 for other borrowers.
We leave the weights to other recommenders unchanged. Adopting $R$ to
represent the set of reserve borrowers, the modified allocation rule is

$\displaystyle x^{VCG}(\hat{p})$
$\displaystyle\in\operatorname*{arg\,max}_{a\in\\{0,1\\}^{m}}\left[\sum_{q\in
M}a_{q}\left(\sum_{i\in N}w_{i}\hat{p}_{iq}\right)+\sum_{q\in R}a_{q}c\right]$
(23) s.t. $\displaystyle\sum_{q\in M\cup R}a_{q}\leq K.$

Going forward we will incorporate the reserve recommender into the set of $N$
recommenders and the $K$ reserve borrowers into the set $M$ of borrowers.

######  Definition 5 (VCG scoring mechanism ($n$ recommenders, $m$
borrowers)).

The VCG scoring mechanism with possibly constrained lender liquidity, lender
profit threshold $c$, and linear weighted belief aggregation $B_{q}$ with
weights $w=(w_{1},\ldots,w_{n})$, is defined as following:

  * •

Allocation: adopt $x^{VCG}(\hat{p})$

  * •

Payment (no payments are made by or collected from the reserve recommender):

    * –

Immediate payment:

$t^{VCG}_{i}(\hat{p}_{i},\hat{p}_{-i})=\sum_{j\neq
i}\hat{v}_{j}(x^{-i}(\hat{p}_{-i}))-\sum_{j\neq
i}\hat{v}_{j}(x(\hat{p}_{i},\hat{p}_{-i})),$ (24)

where $x^{-i}$ is the allocation decision that would be made without $i$
present, i.e., ignoring the reports from recommender $i$.

    * –

Outcome-contingent payment: for each borrower that receives a loan,
$s^{VCG}_{iq}(\hat{p}_{iq},o_{q})=w_{i}$ if $o_{q}=1$ and $0$ otherwise.

The realized utility of recommender $i$ after repayment outcomes are known is

$u_{i}(\hat{p}_{i},\hat{p}_{-i},o)\triangleq\sum_{q\in
x^{VCG}(\hat{p})}w_{i}o_{q}-t^{VCG}_{i}(\hat{p}_{i},\hat{p}_{-i}).$ (25)

###  4.3 Strict Properness of the VCG Scoring Mechanism

######  Theorem 4.2.

The VCG Scoring Mechanism is efficient, satisfies weak ex post IC (ex post
proper), and is ex post individually rational.

Once valuation functions are set-up to correspond to aggregate belief reports
this proof follows the standard recipe for the IC and IR properties of a VCG
mechanism ; see the Appendix.

We also want strict interim IC (strict properness), so that it is a unique
best response of a recommender to report its true beliefs.

We define an equal-shift misreport as a misreport $\hat{p}_{i}\neq p_{i}$ for
which $p_{iq}-p_{iq^{\prime}}=\hat{p}_{iq}-\hat{p}_{iq^{\prime}}$ for every
$q$, every $q^{\prime}$. We say a mechanism is strictly proper up to equal-
shift misreports if truthful reporting is a unique best response, maximizing
interim utility except for possible tie-breaking amongst equal-shift
misreports.

We now state the first of our two main theorems.

######  Theorem 4.3.

For constrained liquidity $(K<m)$, two or more borrowers, three or more
recommenders, and a belief distribution with full support, the VCG Scoring
mechanism without a reserve (i.e., $c=0$) is strict interim IC (strictly
proper) up to equal-shift misreports when $\max_{i}[w_{i}]<1/2$.

###### Proof.

Three or more recommenders are required for
$\max_{i^{\prime}}[w_{i^{\prime}}]<1/2$. Two or more borrowers allows for
constrained liquidity. We consider recommender $i$, belief $p_{i}$, any
$\hat{p}_{i}\neq p_{i}$ that is not an equal-shift misreport, and establish a
non-zero measure on the beliefs $p_{-i}$ of others such that the allocation
changes in a way that reduces the total value (i.e., not selecting the
borrowers with the top $K$ aggregate belief of repayment). Since VCG is weakly
ex post IC (Theorem 4.2), this establishes strict interim IC up to equal-shift
misreports.

For any $q$, let $B(q)$ denote the aggregate belief on $q$ at $p_{i}$ and
$\hat{B}(q)$ at report $\hat{p}_{i}$. If $p_{-i}$ satisfies
$p_{jq}=\frac{1/2-w_{i}p_{iq}}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}=p^{*}_{q}$, $\forall j\neq i$, then $B(q)=1/2$. This belief
$p_{-i}$ is feasible by full support, and since for $p_{iq}=0$ we have
$p^{*}_{q}=(1/2)/\sum_{j^{\prime}\neq i}w_{j^{\prime}}<1$ since
$\sum_{j^{\prime}\neq i}>1/2$ from $w_{i}<1/2$. For $p_{iq}=1$ we have
$p^{*}_{q}=(1/2-w_{i})/\sum_{j^{\prime}\neq i}w_{j^{\prime}}>0$ since
$w_{i}<1/2$.

For a non equal-shift misreport, there are borrowers $q$ and $q^{\prime}$,
such that
$p_{iq}-p_{iq^{\prime}}=\hat{p}_{iq}-\hat{p}_{iq^{\prime}}+\epsilon$, for
$\epsilon>0$; i.e., with the relatively disadvantaged borrower labeled $q$.
Consider a profile $p_{-i}$ that satisfies the following properties:

  1. 1.

$K$ borrowers, including borrower $q$, are allocated:

     * •

For $q^{\prime\prime}\neq q$, set
$p_{jq^{\prime\prime}}\in(p^{*}_{q^{\prime\prime}},1]$, for $j\neq i$, such
that $B(q^{\prime\prime})>1/2$, where this belief of others is feasible since
$p^{*}_{q^{\prime\prime}}<1$.

     * •

For borrower $q$, set
$p_{jq}\in(p^{*}_{q},\min(1,p^{*}_{q}+\frac{1}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}\frac{w_{i}\epsilon}{2}))$, so that
$B(q)\in(\frac{1}{2},\frac{1}{2}+\frac{w_{i}\epsilon}{2})$, where this belief
of others is feasible since $p^{*}_{q}<1$.

  2. 2.

$K-m$ borrowers, including borrower $q^{\prime}$, are not allocated.

     * •

For $q^{\prime\prime}\neq q^{\prime}$, set
$p_{jq^{\prime\prime}}\in[0,p^{*}_{q^{\prime\prime}})$, for $j\neq i$, such
that $B(q^{\prime\prime})<1/2$, where this belief of others is feasible since
$p^{*}_{q^{\prime\prime}}>0$.

     * •

For borrower $q^{\prime}$, set
$p_{jq^{\prime}}\in(\max(0,p^{*}_{q^{\prime}}-\frac{1}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}\frac{w_{i}\epsilon}{2})),p^{*}_{q^{\prime}})$, so that
$B(q^{\prime})\in(\frac{1}{2}-\frac{w_{i}\epsilon}{2},\frac{1}{2})$, where
this belief of others is feasible since $p^{*}_{q^{\prime}}>0$.

There is a non-zero measure on beliefs $p_{-i}$ satisfying these properties by
the full support assumption. For any such $p_{-i}$, at misreport $\hat{p}_{i}$
we have $\hat{B}(q^{\prime})>\hat{B}(q)$, since
$\hat{B}(q^{\prime})-B(q^{\prime})=\hat{B}(q)-B(q)+w_{i}\epsilon$ and
$B(q)-B(q^{\prime})<w_{i}\epsilon$. By the monotonicity of the VCG allocation
rule, this implies one of the following at this misreport:

  1. 1.

Borrower $q^{\prime}$ but not $q$ is allocated, which is an outcome with lower
total value since $B(q^{\prime})<B(q)$.

  2. 2.

Neither $q$ nor $q^{\prime}$ are allocated, which is an outcome with lower
total value since $B(q)>1/2$ and only $K-1$ other borrowers $q^{\prime\prime}$
have true aggregate belief $B(q^{\prime\prime})>1/2$.

  3. 3.

Both $q$ and $q^{\prime}$ are allocated, which is an outcome with lower total
value since $B(q^{\prime})<1/2$ while $K$ borrowers $q^{\prime\prime}$
(including $q$) have aggregate belief $B(q^{\prime\prime})>1/2$.

This completes the proof. $\blacksquare$ ∎

We now state the second main theorem.

######  Theorem 4.4.

For possibly constrained liquidity $(K\leq m)$, one or more borrowers, three
or more recommenders, and a belief distribution with full support, the VCG
Scoring mechanism with a lender profit threshold $c$, with $0<c<1$, is strict
interim IC (strictly proper) when $\max_{i}[w_{i}]<\min(1-c,c)$ (which
requires $n>1/\min(1-c,c)$ recommenders).

###### Proof.

We need three or more recommenders because $\min(1-c,c)\leq 1/2$, and thus
$n>1/(1/2)=2$. We consider recommender $i$, belief $p_{i}$, any
$\hat{p}_{i}\neq p_{i}$, and establish a non-zero measure on the beliefs
$p_{-i}$ of others such that the allocation changes in a way that reduces the
total value (i.e., not selecting the top borrowers amongst those with
aggregate belief at least $c$). Since VCG is weakly ex post IC (Theorem 4.2),
this establishes strict interim IC up to equal-shift misreports.

For any $q$, let $B(q)$ denote the aggregate belief on $q$ at $p_{i}$ and
$\hat{B}(q)$ at report $\hat{p}_{i}$. If $p_{-i}$ satisfies
$p_{jq}=\frac{c-w_{i}p_{iq}}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}=p^{*}_{q}$, $\forall j\neq i$, then $B(q)=c$. This belief
$p_{-i}$ is feasible by full support, and since for $p_{iq}=0$ we have
$p^{*}_{q}=c/\sum_{j^{\prime}\neq i}w_{j^{\prime}}<1$ since
$\sum_{j^{\prime}\neq i}w_{j^{\prime}}>c$ from $w_{i}<1-c$. For $p_{iq}=1$,
$p^{*}_{q}=(c-w_{i})/\sum_{j^{\prime}\neq i}w_{j^{\prime}}>0$ since $w_{i}<c$.

For misreport $\hat{p}_{i}$, consider borrower $q$ with $\hat{p}_{iq}\neq
p_{iq}$.

(Case 1: $\hat{p}_{iq}<p_{iq}$) Let $\hat{p}_{iq}=p_{iq}-\epsilon$, some
$\epsilon>0$. Consider a profile $p_{-i}$ that satisfies the following
properties:

  1. 1.

$B(q)\in(c,c+w_{i}\epsilon)$, by setting
$p_{jq}\in(p^{*}_{q},\min(1,p^{*}_{q}+\frac{1}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}w_{i}\epsilon))$, all $j\neq i$, where this belief of others
is feasible since $p^{*}_{q}<1$ and $c<1$.

  2. 2.

At least $m-K\ (\geq 0)$ other borrowers $q^{\prime}\neq q$ have
$B(q^{\prime})<c$, by setting $p_{jq^{\prime}}\in[0,p^{*}_{q^{\prime}})$, all
$j\neq i$, where this belief of others is feasible since
$p^{*}_{q^{\prime}}>0$.

There is a non-zero measure on beliefs $p_{-i}$ satisfying these properties by
the full support assumption. Given (1) and (2), at true beliefs we have
borrower $q$ allocated since $B(q)>c$ and at least $m-K$ others cannot be
allocated, so $q$ is in the top $K$ of those with aggregate belief above the
threshold $c$. For any such $p_{-i}$, at misreport $\hat{p}_{i}$ we have
$\hat{B}(q)=B(q)-w_{i}\epsilon<c$, since $B(q)\in(c,c+w_{i}\epsilon)$ and
$\hat{p}_{iq}=p_{iq}-\epsilon$. This implies that $q$ is not allocated,
resulting in an outcome with lower total value since $q$ was in the top $K$
and with true aggregate belief above the threshold.

(Case 2: $\hat{p}_{iq}>p_{iq}$) Let $\hat{p}_{iq}=p_{iq}+\epsilon$, some
$\epsilon>0$.

Consider a profile $p_{-i}$ that satisfies the following properties:

  1. 1.

$B(q)\in(c-w_{i}\epsilon,c)$, by setting
$p_{jq}\in(\max(0,p^{*}_{q}-\frac{1}{\sum_{j^{\prime}\neq
i}w_{j^{\prime}}}w_{i}\epsilon),p^{*}_{q})$, all $j\neq i$, where this belief
of others is feasible since $p^{*}_{q}>0$ and $c>0$.

  2. 2.

At least $m-K\ (\geq 0)$ other borrowers $q^{\prime}\neq q$ have
$B(q^{\prime})<c$, by setting $p_{jq^{\prime}}\in[0,p^{*}_{q^{\prime}})$, all
$j\neq i$, where this belief of others is feasible since
$p^{*}_{q^{\prime}}>0$.

Given (1), at true beliefs borrower $q$ is not allocated. At misreport
$\hat{p}_{i}$, we have $\hat{B}(q)=B(q)+w_{i}\epsilon>c$, since
$B(q)\in(c-w_{i}\epsilon,c)$ and $\hat{p}_{iq}=p_{iq}+\epsilon$. This implies
one of the following at this misreport:

  1. 1.

Borrower $q$ is allocated, resulting in an outcome with lower total value
since the true aggregate belief on $q$ is below the threshold (that is, by
causing $q$ to be allocated, $i$ displaces a reserve borrower
$q^{\prime\prime}$ with $B(q^{\prime\prime})=c$, and $i$ must pay this
difference to the system).

  2. 2.

If $q$ is not allocated, then since $\hat{B}(q)>c$ there must be $K$ others
allocated, by the definition of the VCG outcome rule. At least $m-K$ others
have $B(q^{\prime})<c$, and thus at most $(m-1)-(m-K)=K-1$ others have
$B(q^{\prime})\geq c$. This means that at least one other borrower with
$B(q^{\prime})<c$ is allocated, and the outcome has lower total value.

This completes the proof. $\blacksquare$ ∎

###  4.4 Strong Ex post IR

We can also achieve strong ex post IR by ensuring that the immediate payment
by each agent is weakly negative and noting that the outcome-contingent
payments to each recommender are weakly-positive. For this, define
$\mathit{tcomp}_{i}(\hat{p}_{-i})$ as the worst-case immediate payment in VCG
given reports of others. This quantity is independent of the recommender’s own
report. At the same time, we introduce a multiplier $\alpha>0$ to the outcome-
contingent payments, so that
$s_{iq}^{aVCG}(\hat{p}_{iq},o_{q})=\alpha_{i}w_{i}$ if $o_{q}=1$, and 0
otherwise. Neither change affects the incentive analysis. Modifying the
definition of reported valuations accordingly, for example with
$\hat{v}_{i}(a)=\sum_{q\in M}a_{q}\alpha w_{i}\hat{p}_{iq}$, we have

$\mathit{tcomp}_{i}(\hat{p}_{-i})=\mathrm{max}_{\hat{p}_{i}}\left(\sum_{j\neq
i}\hat{v}_{j}(x_{-i}(\hat{p}_{-i}))-\sum_{j\neq
i}\hat{v}_{j}(x^{*}(\hat{p}_{i},\hat{p}_{-i}))\right).$ (26)

We refer to this as the rescaled VCG scoring mechanism. By worst-case deficit
we mean the worst-case, total payment made by the mechanism to the agents,
considering both the immediate payments and the outcome-contingent payments.

######  Theorem 4.5.

In the possibly constrained liquidity setting, and with multiple recommenders
and multiple borrowers, there is some value of $\alpha_{0}>0$ such that for
any $\alpha<\alpha_{0}$ the rescaled VCG scoring mechanism is strong ex-post
IR, has worst-case deficit at most $\epsilon>0$, and strict proper incentives
as stated in Theorems 4.3 and 4.4.

###### Proof.

For strong ex post IR, this follows from the definition of
$\mathit{tcomp}_{i}(\hat{p}_{-i})$ and outcome-contingent payments being non-
negative. For the strict properness, this follows from the invariance of
incentive analysis to scaling payments by any $\alpha>0$ and that
$\mathit{tcomp}_{i}(\hat{p}_{-i})$ is independent of recommender $i$’s
reports. The claim of deficit smaller than $\epsilon$ for any
$\alpha<\alpha_{0}$, for some $\alpha_{0}>0$ follows from linearity,
recognizing that $\alpha$ scales all payments. ∎

###  4.5 Incentive Alignment with Better Reporting Quality

We show in this section that a recommender in the VCG scoring mechanism
strictly improves its utility by having a higher weight. This nicely aligns
incentives in the broader ecosystem, similar to the way in which advertisers
with higher quality make smaller payments in the generalized second pricing of
internet advertising, and provides an incentive for a recommender to improve
its reporting quality and thus attain a higher weight over time in the
aggregation rule.

######  Theorem 4.6.

Whatever the reports of others, for any recommender $i$, increasing the weight
$w_{i}$ to $w_{i}^{\prime}>w_{i}$, fixing the weights of others, increases the
utility $U_{i}(p_{i},p_{i},\hat{p}_{-i})$ to the recommender from truthful
participation in the VCG scoring mechanism.

###### Proof.

The utility to recommender $i$ is

$U_{i}(p_{i},p_{i},\hat{p}_{-i})=v_{i}(x^{VCG}(p))+\sum_{j\neq
i}\hat{v}_{j}(x^{VCG}(p_{i},\hat{p}_{-i}))-\sum_{j\neq
i}\hat{v}_{j}(x^{-i}(\hat{p}_{-i})),$

where $x^{-i}$ is the allocation decision that would be made in the VCG
scoring mechanism without $i$. Let $v_{i}$ and $v^{\prime}_{i}$ denote the
recommender’s valuation for weight $w_{i}$ and $w^{\prime}_{i}$, respectively.
The third term does not depend on its weight. Consider the first two terms,
and let $a$ and $a^{\prime}$ denote the allocation for $w_{i}$ and
$w^{\prime}_{i}>w_{i}$, respectively. We have
$v^{\prime}_{i}(a^{\prime})+\sum_{j\neq i}\hat{v}_{j}(a^{\prime})\geq
v^{\prime}_{i}(a)+\sum_{j\neq i}\hat{v}_{j}(a)>v_{i}(a)+\sum_{j\neq
i}\hat{v}_{j}(a)$. The first inequality holds trivially when $a^{\prime}=a$,
and if $a^{\prime}\neq a$ then by the optimizing property of the VCG
allocation rule. The second inequality holds since
$v^{\prime}_{i}(a)=\sum_{q}a_{q}w^{\prime}_{i}p_{iq}>\sum_{q}a_{q}w_{i}p_{iq}=v_{i}(a)$,
and since reported values of others are unchanged. $\blacksquare$ ∎

###  4.6 Relation to Chen et al.’s Impossibility Result

Theorem 2 of Chen et al. (2011) states that a decision market, which uses
belief reports as reflected in market prices to make a decision, is strictly
proper if and only if the decision is randomized and the distribution has full
support. That is, for strict properness every decision must be taken with non-
zero probability. The key difference between our model and that of Chen et al.
(2011) is that our agents report their beliefs simultaneously without
awareness of the reports of others against whom they will be judged. This
creates interim uncertainty about the allocation given the common prior
$\mathcal{D}$ and the technical conditions stated in Theorems 4.3 and 4.4.
Whereas Chen et al. (2011) have certainty about belief aggregation and thus
need the decision rule to be randomized will full support, we have interim
uncertainty about belief aggregation and the decision rule can be
deterministic as a function of reports (still providing what is in effect full
interim support) and thus strict properness.

###  4.7 Linear Belief Aggregation

As shown in theorems 4.3 and 4.4 above, the VCG scoring mechanism is Interim
IC when $max_{i\in N}(w_{i})$ is sufficiently small. There are many linear
aggregators in the literature which can be adjusted to satisfy this condition.
In the Appendix, we describe the linear aggregator by (Budescu and Chen,
2015), which is based on reports from previous rounds and therefore doesn’t
affect current-round incentives.

##  5 Conclusion

Creditworthiness detection remains an unsolved problem, and our forumlation of
the problem as an elicitation mechanism gathering community beliefs brings a
novel source of information and a rich body of mechanism design literature to
the task. In so doing, we developed a class of truncated, asymmetric scoring
rules that are strict ex-post proper in the sufficient-liquidity case. We also
connected the scoring rule and VCG literatures in a novel way, creating the
VCG scoring mechanism through which we construct agents’ values via scoring
rules. These values then feed into a VCG allocation and payment mechanism that
is strictly interim IIC (and thus strictly proper) in both liquidity-
constrained and liquidity-unconstrained settings, and with or without a lender
profit threshold. Given impossibility results in the analagous case of
decision scoring rules (Chen et al., 2011), these results expand the range of
settings in which information can be elicited and paid for based on outcomes
by carefully leveraging agents’ interim uncertainty. We also connected these
mechanisms with the belief aggregation literature, allowing us to retain
incentive compatibility properties with linear aggregation techniques.

Much future work remains. Lenders have suggested we incorporate considerations
of collusion, where recommenders may misreport to help friends. Further work
remains to ensure optimal aggregation. Bayesian and maximum likelihood
estimator techniques could optimize the weighting of unknown agents and
potential borrowers, incorporating factors such as demographics or other
background information. Online-learning models could be built for specific use
cases, such as optimizing the exploration and exploitation tradeoff for
lenders building client bases in new communities. Also considering the
required payments to motivate recommenders to invest appropriately in
providing good information is an area that we have seen as important from an
ongoing field study in Uganda. Finally, additional field work should continue
in parallel to ensure that the theory focuses on the most critical problems
which crop up in practice.

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##  6 Appendix

###  6.1 Brief Aggregation Literature Review

Linear belief aggregators have been shown to outperform individual experts in
a phenomena known as “wisdom of the crowds” [Mannes et al., 2014]. Moreover,
in many contexts expert quality will vary significantly, and aggregating
expert reports based on quality improves prediction accuracy [Soule et al.,
2020, Satopää et al., 2017, Lee and Danileiko, 2014, Wang et al., 2019]. We
also make use of belief aggregation methods, which can be divided into linear
[Budescu and Chen, 2015, Soule et al., 2020, Winkler, 1981] and non-linear
[Raykar et al., 2010, Zhou et al., 2014, Venanzi et al., 2014, Welinder and
Perona, 2010, Kim et al., 2020, Karger et al., 2013] approaches. They can be
also be divided into approaches that weight based on previous prediction
quality, e.g. Budescu and Chen [2015] and ones that exploit structure in
participants’ reports to provide ratings even before any outcomes have been
realized, e.g. Raykar et al. [2010]. Many of these latter approaches build on
the expectation-maximization work of Dawid and Skene [1979]. Any aggregation
scheme that is a monotonically-increasing function of the reports will work
for the truncated Winkler rule of Section 3. However, the aggregation must be
a linear combination to work with the VCG Scoring mechanism of Section 4. As
such, we focus here on linear aggregators that weight based on prediction
quality for realized outcomes, where quality measures are derived from past
performance.

###  6.2 Proof of Theorem 4.2

###### Proof.

Suppose by way of contradiction that recommender $i$ has a useful misreport
$\hat{p}_{i}\neq p_{i}$ such that
$\operatorname{\mathbb{E}}[u_{i}(\hat{p}_{i},\hat{p}_{-i},o)]>\operatorname{\mathbb{E}}[u_{i}(p_{i},\hat{p}_{-i},o)]$.
Let $x^{\prime}$ be the allocation under $i$’s misreport. Since $i$’s expected
utility is strictly greater under $x^{\prime}$ than $x$, we have

$v_{i}(x^{\prime})-\sum_{j\neq i}\hat{v}_{j}(x^{-i})+\sum_{j\neq
i}\hat{v}_{j}(x^{\prime})>v_{i}(x)-\sum_{j\neq
i}\hat{v}_{j}(x^{-i})+\sum_{j\neq i}\hat{v}_{j}(x)$ (27)

$\Rightarrow v_{i}(x^{\prime})+\sum_{j\neq
i}\hat{v}_{j}(x^{\prime})>v_{i}(x^{*})+\sum_{j\neq i}\hat{v}_{j}(x^{*})$ (28)

But this is a contradiction, as the social choice function should then have
allocated $x^{\prime}$ to maximize recommender value. Note that this does not
depend on the reports of others, whether they are being truthful, or even
whether their reports are known. Thus, there can be no useful misreport, and
the mechanism satisfies weak IC.

Efficiency, which is the property of a mechanism choosing the allocation that
maximizes sum of value for all agents, follows from the fact that the
allocation function explicitly maximizes the value of the aggregation
function.

For IR, consider recommender $i$, consider the borrowers $x_{new}$ who are
allocated when $i$ is present but not otherwise and $x_{old}$ who are not
allocated when $i$ is present but allocated otherwise. For other borrowers,
they contribute a non-negative utility to $i$, as they do not affect $t^{VCG}$
and the outcome-contingent payments are non-negative. For the borrowers in
$x^{new}$, expected utility is $\operatorname{\mathbb{E}}_{o\sim
p_{i}}[u_{i}(p_{i},\hat{p}_{-i},o)]=v_{i}(x^{new})-\sum_{j\neq
i}v_{j}(x^{old})+\sum_{j\neq i}v_{j}(x^{new})$. This term is also non-
negative, because if it were negative, then the allocation function would have
allocated to $x^{old}$ instead of $x^{new}$. Therefore, each recommender’s
expected utility is non-negative, and the mechanism is individually-rational.
$\blacksquare$ ∎

###  6.3 Linear Belief Aggregation Example

Following the lead of Budescu and Chen [2015], we calculate the weights to
associate with a recommender based on the reports and outcomes from previous
rounds. First, we calculate a score $Q$ when considering the system of all
recommenders and the set $M^{*}$ of all borrowers from previous rounds that
have received loans on which outcomes are available.

$Q=a+b\sum_{q\in M^{*}}\left(\sum_{r\in\\{0,1\\}}(o_{qr}-m_{qr})^{2}\right),$
(29)

where $m_{q1}$ is the average (unweighted) reported belief that borrower $q$
will repay the loan across all recommenders and $m_{q0}=1-m_{q1}$. It is
convenient to set $a=100$ and $b=-50$, leading to $0\leq Q\leq 100$. Define
$Q_{-i}$ to be the analogous quantity, but where the average reported belief
is calculated over those reports from recommenders $j\neq i$. Given this, we
define the accuracy contribution $C_{i}=(Q-Q_{-i})/|M^{*}|$. The weight
$w_{i}$ is defined to be zero for a recommender with $C_{i}\leq 0$. For the
set of recommenders $N_{+}=\\{i\in N:C_{i}>0\\}$, the weight $w_{i}$ for $i\in
N_{+}$ is set in proportion to $C_{i}$ and normalized such that
$\sum_{i}w_{i}=1$.