Eliciting Information from participants with Competing Incentives and Dependent Beliefs

TL;DR

This paper investigates belief elicitation mechanisms in scenarios with competing incentives and dependent beliefs, revealing fundamental tradeoffs, optimal strategies, and methods to ensure truthful reporting under various conditions.

Contribution

It introduces new insights into incentive-compatible belief elicitation, analyzing the impact of competing incentives and dependent beliefs, and proposes mechanisms to mitigate manipulation.

Findings

Quadratic Scoring Rule is worst-case optimal against manipulation.

Robust truthful mechanisms exist when incentives stem from rational bribers.

Decoupling methods can address dependence issues in belief elicitation.

Abstract

In this paper, we study belief elicitation about an uncertain future event, where the reports will affect a principal's decision. We study two problems that can arise in this setting: (1) Agents may have an interest in the outcome of the principal's decision. We show that with intrinsic competing incentives (an interest in a decision that is internal to an agent) truthfulness cannot be guaranteed and there is a fundamental tradeoff between how much the principal allows reports to influence the decision, how much budget the principal has, and the degree to which a mechanism can be manipulated. Furthermore, we show that the Quadratic Scoring Rule is worst-case optimal in minimizing the degree of manipulation. In contrast, we obtain positive results and truthful mechanisms in a setting where the competing incentives stem instead from a rational briber who wants to promote a particular decision. We show that the budget required to achieve this robustness scales with the sum of squares of the degree to which agent reports can influence the decision. (2) We study the setting where the future event is only observed conditionally on the decision taken. We give a category of mechanisms that are truthful when agent beliefs are independent but fails with dependent beliefs, and show how to resolve this through a decoupling method.

Equations

$$\text{textsc}pact(r)=\min (\max [L\cdot \left(\frac{1}{n}\sum _{i\in [n]}{r}_i-t\right),\epsilon ],1),$$

$$\text{textsc}{epay}_i(r,o):={\mathbb{E}}_{X\sim \text{textsc}rand}[\text{textsc}{pay}_i(r,o,X)],\text{textsc}pact(r):={\mathbb{E}}_{X\sim \text{textsc}rand}[\text{textsc}act(r,X)]$$

$$P_i(o,q_{\neg i},c_{\neg i},r_{\neg i}|\text{textsc}mech,c_i)$$

$$f_i^q\in {\mathcal{D}}^q:=$$

$$f_i^c\in {\mathcal{D}}^c:=$$

$$f_i^r\in {\mathcal{D}}^r:=$$

$$\times$$

$$\text{textsc}{util}_i\left(r,o,c_i,\text{textsc}mech\right):=\text{textsc}{epay}_i\left(r,o\right)+c_i\cdot \text{textsc}pact\left(r\right).$$

$$\text{textsc}{se-util}_i\left(r_i,t_i,\text{textsc}mech\right):$$

$$r_i^{\ast }\left(t_i,\text{textsc}mech\right)$$

$$\text{textsc}se-util\left(r,t,\text{textsc}mech\right)$$

$$={\mathbb{E}}_{O\sim q}\left[\text{textsc}epay\left(r,O\right)\right]+c\cdot \text{textsc}pact\left(r\right)$$

$$\text{textsc}se-util\left(r,t,\text{textsc}mech\right)$$

$$\text{textsc}{util}_b(c,d,q,f,\text{textsc}mech)$$

$$\left(d-\sum _{i\in [n]}{c}_i\right)\cdot$$

$$\text{textsc}{se-util}_b(c,(d,\varphi ),\text{textsc}mech)$$

$$c^{\ast }\left((d,\varphi ),\text{textsc}mech\right):=\arg \underset{c\ge 0}{\max }\text{textsc}{se-util}_b(c,(d,\varphi ),\text{textsc}mech).$$

$$\text{textsc}{se-util}_i\left(q_i,(q_i,f_i,c_i),\text{textsc}mech\right)$$

$$\forall r_i\ne q_i\in [0,1],f_i\in \mathcal{F},c\in \mathcal{C}.$$

$$\text{textsc}{se-util}_b(0,(d,\varphi ),\text{textsc}mech)>\text{textsc}{se-util}_b(c,(d,\varphi ),\text{textsc}mech)\forall \varphi \in \Phi ,c\in {\mathbb{R}}_{>0}^n$$

$$\text{textsc}{util}_b(0,d,q,f,\text{textsc}mech)>\text{textsc}{util}_b(c,d,q,f,\text{textsc}mech)\forall q\in [0,1{]}^n,f\in {\mathcal{D}}^n,c\in {\mathbb{R}}_{>0}^n.$$

$$\text{textsc}manip\left(n,\mathcal{F},\mathcal{C},\text{textsc}mech\right):=$$

$$\underset{q\in [0,1{]}^n,f\in {\mathcal{F}}^n,c\in \mathcal{C}}{\max }\left|\text{textsc}pact\left(r_1^{\ast }\left((q_1,f_1,c_1),\text{textsc}mech\right),..,r_n^{\ast }\left((q_n,f_n,c_n),\text{textsc}mech\right)\right)-\text{textsc}pact\left(q_1,..,q_n\right)\right|.$$

$${\Delta }_i(r_{\neg i})=\underset{r_i,r_i^{\prime }\in [0,1]}{\max }\left|\text{textsc}pact(r_i,r_{\neg i})-\text{textsc}pact(r_i^{\prime },r_{\neg i})\right|\forall i\in [n],\forall r_{\neg i}\in {[0,1]}^{n-1}.$$

$$L$$

$$L_i(r_{\neg i})$$

$$\text{textsc}cost(\epsilon ):=\underset{q\in [0,1]:q+\epsilon \in [0,1]}{\min }\left(S(q,q)-S(q+\epsilon ,q)\right).$$

$$\text{textsc}cost(\epsilon )$$

$$\text{textsc}epay(r,o)=\beta \left((2r-r^2)o+(1-r^2)(1-o)\right).$$

$$S(r,q)=\beta \left((2r-r^2)q+(1-r^2)(1-q)\right).$$

$$\text{textsc}cost(\epsilon )={\epsilon }^2\beta .$$

$$\textsc{util}_{i}\left(r_{i}\big{|}r_{\neg i},o,c_{i},\textsc{mech}\right):=\textsc{epay}_{i}\left(r,o\right)+c_{i}\cdot\textsc{pact}\left(r\right).$$

$$\text{textsc}manip\left(\{c\},\text{textsc}mech\right)\ge \frac{\Delta {}^2}{8\beta /c+2\Delta }.$$

$$\beta =c\cdot {\Delta }^2\cdot \Omega \left(\frac{1}{\epsilon }\right)\text{as }\epsilon \to 0.$$

Full text

1 Introduction

There are many settings where it is helpful to elicit information from a population in order to support a decision, and in particular in elicting beliefs about the probability of an uncertain event. For example: Will expanding the port facility in Seattle lead to an increase of 5M tons of freight/year in 2025? Will a start-up achieve revenue growth of more than 20% in its target markets in 2023? Will a high speed trainline, if built from Manchester to the center of London, meet a target of 2M trips a year in 2027?

In many ways, this may look like a well-studied problem. This is a setting of information elicitation with verification, in that there is a downstream uncertain event whose future value can be verified and then scored against, and solutions such as scoring rules and their multi-user extensions to market-scoring rules and prediction markets may come to mind (Brier et al., 1950; Gneiting and Raftery, 2007; Hanson, 2007; Wolfers and Zitzewitz, 2004). And yet, there are a number of aspects of this problem that make it non-standard and unsolved. In particular, the presence of a decision that will be made in a way that depends on the aggregation of reports brings about new challenges.

First, the outcome of the uncertain future event may only be observed conditioned on the decision being made affirmatively. This means that payments depend in turn on the decision and in turn on reports, and this changes incentives. Elicitation together with decisions has been studied before, notably in the context of decision markets (Chen et al., 2011) and VCG-based scoring mechanisms (York et al., 2021). But to our knowledge, this has not been studied together with participants who also have competing incentives. In the above examples, perhaps a participant is a shipper who would personally benefit from the port expansion. Perhaps a participant is subject to a bribe from the CEO of the start-up, or a bribe from a politician in the North of England. This raises the following question:

Can robust elicitation mechanisms be designed when there is a decision to be made and participants may face competing incentives?

Moreover, the coupling of a decision-contingent observation of the outcome together with participants whose beliefs about the future event can depend on the beliefs of others brings about novel challenges. This introduces a subtle interaction, whereby a participant who conditions on the payments that arise when a decision is made should also condition on what this implies about others’ reports (and thus beliefs), and in turn what this should imply in regard to updates about their own belief.111Although in a different context, this style of reasoning pattern is likely familiar in the context of the “winner’s curse” in common-value auctions, wherein a bidder should reason about what inference they would make in the event that their bid is the highest (Krishna, 2009). In the above examples, perhaps a participant believes that others who report beliefs about the impact on freight in Seattle are doing this with their own independent data and analysis, and that they may be more skilled in forecasting. This raises the following question:

Can elicitation mechanisms be designed when the observation of the uncertain future event is decision-contingent and participants have dependent beliefs about this event?

More formally, we consider in this paper a principal who wants to elicit information from $n$ recommenders about a hidden binary variable, $O$, and then use this information to take a binary decision, $A$. We design mechanisms that make use of a payment function, determining the payments to each recommender, and a decision function, choosing an action $A$ based on the elicited information. In the simpler set-up, we assume the realized outcome is always observed. This would be the case in the setting of forecasting revenue growth at the start-up, or whether it will rain on a May weekend in London (this information to be used to decide whether or not to throw a coronation party). More generally, and as would be the case in the other motivating examples, the realized outcome $O$ is only observed contingent on an affirmative decision, and for this reason the payment function can only depend on $O$ when action $A=1$.

Contributions. We characterize the space of truthful mechanisms in the face of these difficulties of competing incentives, observing the outcome conditionally on the decision taken, and belief dependencies between recommenders. We offer a number of answers, both positive and negative. We work in a model where each recommender can form a belief about the other recommenders’ beliefs, competing incentives, and in turn the reports of others, and we can consider the worst-case manipulation over recommender beliefs.

The style of negative result is to show that there can always be a participant who will prefer to misreport their belief, to some degree, when a decision rule is sensitive in even a small way to their input and there is an intrinsic competing incentive, i.e., some kind of interest in the decision that does not rely on a bribe from an interested party. We prove in Theorem 1 that the maximum cost of misreporting that can be imposed by a scoring rule scales quadratically in the size of the misreport (i.e., the loss in payment scales with ${(r_{i}-q_{i})}^{2}$ uniformly across all beliefs). As a result, no rule can do better than the Quadratic Scoring Rule (see Lemma 1). This leads to a fundamental tradeoff between the degree to which a rule can be manipulated in the presence of competing incentives and its sensitivity to reports. If recommenders can have a large influence on the decision they also have a larger incentive to misreport. However, if we design a decision rule to be insensitive to reports, then the rule has no utility. We show that this conflict cannot be avoided by any mechanism, even if we are free to design any decision function and scoring rule. In particular, we give a lower bound on the extent to which the decision can be manipulated, this depending on the influence that recommenders can exert and the budget that is available for payments (Theorem 2).

We also study the setting where the competing incentive is not intrinsic but comes about from a rational briber, who cares about trying to promote a particular decision. This opens up new possibilities for positive results. In particular, the briber will choose not to bribe in equilibrium with the rational best response of recommenders when the effect of the bribe on the decision is too small relative to the cost of the bribe. Truthful mechanisms exist in this setting, and we provide conditions on the sensitivity to agent reports and the budget of the mechanism. The budget required for truthfulness scales with the sum of squares of the sensitivity of the rule to reports of agents (Theorem 6). As a result, the total amount of influence that recommenders can have can grow with $\sqrt{n}$, where $n$ is the number of recommenders, while maintaining truthfulness or fixed low manipulation.

We also address the additional challenge when the outcome is only observed conditionally on the decision. For this, we propose a decoupling construction (Def. 19), akin to importance sampling, that can be used to disentangle the payment and decision rule. It can be combined with any mechanism that is truthful without this censoring, allowing to avoid the conditional observation problem while preserving expected payments to recommenders and maintaining the truthfulness of the underlying mechanism (Lemma 5).

Outline. Section 2 defines the problem, recommender utility, competing incentives, rational briber, and truthfulness. Section 3 provides upper and lower bounds to the cost to the recommender of lying in the presence of a proper scoring rule. Section 4 gives bounds on the manipulability of any mechanism in the face of competing incentives, gives an impossibility result for zero-collusion when recommenders have instrinsic outside incentives, and a positive no-collusion result in the case of rational bribers. Section 5 gives a category of mechanisms that cannot be truthful when recommender beliefs are dependent, and proposes the decoupling construction to handle this. Finally, Section 6 concludes.

1.1 Related work

We delineate between our work and the prior literature along the axes of single vs. multi-agent elicitation, whether agent beliefs are independent or dependent, whether or not there is a decision to make (perhaps with an outcome observed conditionally on this decision), and whether or not agents have preferences on the decision (“competing incentives”).

A first connection is with scoring rules, which elicit subjective information from a single agent about an uncertain future event and align incentives with truthful reports (Brier et al., 1950; Winkler, 1994; Gneiting and Raftery, 2007). Unlike our setting, scoring rules are single agent and do not model settings in which there is a decision to be made (including settings with competing incentives). While Wagering mechanisms (Freeman and Pennock, 2018) and prediction markets (Wolfers and Zitzewitz, 2004) extend this setting to multiple agents and support belief aggregation, they do not model a setting in which there is a decision to be made (and consequently, do not handle competing incentives). These mechanisms allow for agents with dependent beliefs, at least implicitly through the use of sequential elicitation, which allows an agent to incorporate relevant information from earlier in making their own report. Zhang et al. (2011) study the use of multi-agent scoring rules for the routing of prediction tasks and with dependent beliefs (agents have beliefs about each others’ prediction ability), but without a decision to make (and thus, without competing incentives).

Chen et al. (2011) study decision markets, where there is a principal who uses the aggregation of beliefs in a prediction market to make a decision, this leading to a decision-contingent observation. They prove that incentive alignment requires randomized decision rules with full support in a setting with sequential elicitation. For this reason, all of our truthful mechanisms also require full support on the space of possible decisions. In contrast with our model, their agents do not have competing incentives, and incentives are aligned for the myopic beliefs of an agent at the time they make a report (and thus, the subtle interaction between dependent beliefs and inference in regard to beliefs of others in the event of a decision is unmodeled). York et al. (2021) also study a model with multiple agents and with a decision to make based on the aggregation of their reports. In contrast to Chen et al. (2011), they consider agents who make simultaneous reports about a future event, and achieve interim incentive alignment without appeal to a randomized decision rule by appealing to uncertainty about reports of others. In contrast to our setting, they do not handle dependent beliefs or participants with competing incentives.

The peer prediction literature (Miller et al., 2005; Jurca and Faltings, 2009; Prelec, 2004; Witkowski and Parkes, 2012, e.g.) studies an elicitation setting with multiple agents and dependent beliefs, but without a decision to make, and thus without competing incentives. Another major difference is that the problem studied in peer prediction is that of information elicitation without verification—eliciting information in the absence of an uncertain future event against which to score.

In the different setting of social choice, Alon et al. (2011) study a setting with a decision to make and competing incentives, considering the problem of selecting a committee from a set of voters where each voter would prefer to be selected. They suggest a mechanism that is able to align incentives through the use of randomization to introduce suitable independence between an agent’s own report and whether or not it is selected; see also Kurokawa et al. (2015). These approaches are not applicable in the present setting. We also make brief mention of transitive trust, a setting that is again distinct from that studied here but that does include participants with dependent beliefs and competing incentives, in that participants care about their own trust ranking. In particular, Hopcroft and Sheldon (2007) develop a variation on PageRank (Page et al., 1999) that is robust to misreports. Bribery has also been studied in voting systems (Keller et al., 2018; Elkind et al., 2009; Faliszewski and Rothe, 2016; Parkes et al., 2017), for example in regard to how many votes a briber must flip to change a discrete decision.

In summary, this paper is the first we are aware of to develop truthful mechanisms for the aggregation of beliefs where there is a decision to make (and an outcome is observed contingent on this decision), there are competing incentives, and where agents’ beliefs may be dependent.

2 Problem Definition

Before we define the problem in full generality, we start with an example (inspired by the setting studied in York et al. (2021)) to illustrate the problems we will study.

Example 2.1** (Loan Allocation).**

A lender wants to decide whether to make a loan to a potential borrower ($A=1$) or not ($A=0$) and does not know if the borrower will return the loan ($O=1$) or not ($O=0$). The lender wants to elicit information regarding trustworthiness of the borrower from a group of $n$ recommenders. Each recommender $i$ holds their own belief, i.e., their own estimate $q_{i}\in{[0,1]}$ of the probability of the loan being returned $O=1$ if granted. Recommenders make reports, $r=(r_{1},..,r_{n})$ and the lender determines the probability of allocating the loan using a (possibly randomized) decision rule pact, i.e., $P(A=1)=\textsc{pact}(r)$.

To ensure truthful reporting, i.e. $r=q$, the lender pays each recommender according to some payment rule $\textsc{pay}_{i}(r,O)$, that rewards accurate reports. The payment can only depend on $O$ in the case that the loan is made, and we need $\textsc{pay}_{i}(r,o)\geq 0~{}\forall r,o$. Further, the lender does not want the sum of payments to exceed a budget, $\beta>0$. By way of example, the lender may decide whether to allocate the loan using the decision rule,

[TABLE]

which is parametrized by a minimum allocation probability, $\varepsilon\in{[0,1]}$, a threshold $t\in{[0,1]}$, and a slope $L\geq 0$.

We will see that the magnitude of $L$ is one of the key tradeoffs that principals must make when defending against recommender manipulation due to outside incentives. The principal would like to fully use reported information, ideally using a deterministic decision function with $\varepsilon=0$ and $L\to\infty$. Not doing so could lead to concerns around perceived fairness (“Why did he get a loan and I didn’t, even though my rating was higher?”). However, we will show that the more decision influence we give to recommenders, the more incentive they have to misreport, with deterministic decision functions always being subject to some manipulation.

2.1 Notation

We use upper-case variables to denote random variables (RVs). We use $P(E)$ to denote the probability of an event $E$ and $\mathbb{E}[Y|E]$ to denote a conditional expectation of the RV $Y$ given event $E$. When clear in the context, we abbreviate the event $Z=z$ by simply $z$; i.e., we write $P(Y=y)$ as $P(y)$ and $\mathbb{E}[Y|Z=z]$ as $\mathbb{E}[Y|z]$. The expectation always only applies to the RVs inside the expectation; i.e., to the upper-case variables. For instance, $\mathbb{E}[f(Y,z)]=\sum_{y\in\mathcal{Y}}f(y,z)P(y)$. With a slight abuse of notation, for a continuous RV $Y$, we will use $P(y)$ to denote the probability density function of $Y$. We will use $\mathcal{PDF}(\mathcal{Y})$ to denote the set of all possible probability density functions of a random variables that take values in $\mathcal{Y}\subseteq\mathbb{R}^{m}$, for some $m$.

Importantly, we use a subscript $i$, i.e., $P_{i}(Y=y)$ and $\mathbb{E}_{i}[Y]$, when the distribution of an RV $Y$ is subjective and hence specific to recommender $i$. Whenever there is no subscript, for example $P(Y=y)$ and $\mathbb{E}[Y]$, this means that the distribution of $Y$ is objective, fully defined by the mechanism and known to everyone. Further, for sequences $z=(z_{1},..,z_{n})$ we will use $z_{\neg i}$ to denote $(z_{1},..,z_{i-1},z_{i+1},..,z_{n})$. With some abuse of notation, whenever clear from the context, we will use $(y_{i},z_{\neg i})$ to denote $(z_{1},..,z_{i-i},y_{i},z_{i+1},..,z_{n})$. We will use $[n]$ to denote the set $\{1,..,n\}$.

2.2 Basic Set-up

There is a principal, whose goal is to elicit from the recommenders (also referred as agents when this causes no confusion) information about an unknown outcome, $O\in\{0,1\}$, in order to take an action $A\in\{0,1\}$ of interest to the principal. There are $n$ recommenders, and possibly a briber (unless we mention the briber explicitly, we assume that there is no briber present). Each recommender $i$ holds a private, subjective belief, $Q_{i}\in{[0,1]}$, regarding the probability of $O=1$. The recommender is asked to reveal $Q_{i}$ to an elicitation mechanism and submits a report $R_{i}\in{[0,1]}$, which, if truthful, equals $Q_{i}$. The recommender may also have a private incentive that yields utility $C_{i}\geq 0$ if $A=1$, and [math] otherwise. This utility is distinct from any utility they will receive as a result of the payments in the mechanism, and may compete with the incentives for truthfulness. We study two settings: (i) The recommender may have an intrinsic preference regarding the outcome. In this setting, we don’t make any assumption about the underlying process that generates $C$, other than that it lies in some domain $\mathcal{C}$. (ii) In the second setting, we assume that the competing incentive is a bribe offered by a self-interested briber, paid conditional on the decision being $A=1$.

2.3 Game and Mechanism

To determine payments to recommenders and the action to be taken, the principal makes use of a mechanism, which is known to all agents and defined as follows:

Definition 1** (Mechanism).**

A mechanism is defined by the tuple $\textsc{mech}=(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$, with

rand* being a probability distribution on a domain $\mathcal{X}$,* 2. 2.

$\textsc{act}:{[0,1]}^{n}\times\mathcal{X}\to\{0,1\}$* being the function that produces a decision based on the reports, and* 3. 3.

the payments $\textsc{pay}_{i}:{[0,1]}^{n}\times\{0,1\}\times\mathcal{X}\to\mathbb{R}_{+}$.

For convenience, we also define

[TABLE]

and we call $\beta:=\max_{r,o}\sum_{i\in[n]}\textsc{epay}_{i}(r,o)$ the budget of the mechanism.

We study two different settings, one where $O$ is always observed, regardless of the action taken, and one in which the outcome $O$ is only observed in the case of decision $A=1$. Note that the second case imposes constraints on the payment functions $\textsc{pay}_{i}$, as they may only depend on $o$ whenever action $A=1$ is taken. Either way, the rules of the mechanism are common knowledge to the recommenders and the briber (if any). A mechanism induces the following reporting game, which is the focus of our analysis:

In the first step, the competing incentives $C=(C_{1},..,C_{n})$ are observed by the respective recommenders (stemming either from intrinsic preferences or from a briber). Note that $C_{i}$ is unknown to the principal and all recommenders other than $i$. 2. 2.

Each recommender, $i\in[n]$, then submits their report, $R_{i}\in{[0,1]}$ to the mechanism. 3. 3.

The mechanism then takes a decision $A\in\{0,1\}$ based on the recommenders’ reports, according $A:=\textsc{act}(R,X)$, where $X\in\mathcal{X}$ is a random variable internal to the mechanism. 4. 4.

If a briber is present, the briber pays $C_{i}$ to each recommender $i\in[n]$ when the desired action $A=1$ was taken by the mechanism. 5. 5.

Either outcome $O$ is unconditionally observed, or (depending on the setting), outcome $O$ is only observed for decision $A=1$. 6. 6.

Each recommender $i\in[n]$ receives a payment, $\textsc{pay}_{i}(R,O,X)$, from the mechanism, based on how accurate their prediction was, where $X$ can be used for randomization.

Since we would like the mechanism to be truthful, at least in the absence of competing incentives, we will mostly focus on proper mechanisms, defined as follows:

Definition 2** (Proper Mechanism).**

We say that a mechanism $\textsc{mech}=(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$ is proper if for all recommenders $i$, $\textsc{epay}_{i}(r,o)$ is a strictly proper scoring rule with respect to $r_{i}$ (regardless of others’ reports $r_{\neg i}$).

In the setting where the outcome $O$ is observed, regardless of the decision taken, it is straightforward to ensure that a mechanism be proper, so we will mostly focus on proper mechanisms. When the outcome is only observed conditionally on the decision being $A=1$, ensuring properness is not as straightforward, we will discuss this problem in Section 5.

2.4 Recommenders

We define the recommender type such that it contains the information required to model recommenders’ behavior. This information consists of recommenders’ beliefs and their competing incentives. To recommender $i$, the variables $O,Q_{\neg i},C_{\neg i}$, and $R_{\neg i}$ are unknown, hence $i$ holds a private, subjective belief $P_{i}(o,q_{\neg i},c_{\neg i},r_{\neg i}|\textsc{mech},c_{i})$ regarding these variables (given the $c_{i}$ and mech, which are revealed to $i$; note that we don’t view $q_{i}$ as an observation, but instead as part of recommender $i$’s belief, hence we don’t explicitly condition on it). Each recommender’s beliefs may be different, but we assume that all of them respect the independences implied by the Bayesnet in Figure 1, which can be interpreted as the the causal process giving rise to the random variables.

Given the independences expressed in Figure 1, recommender $i$’s belief factorizes as follows:

[TABLE]

Hence, a recommender’s belief can be defined by defining each of these distributions. This belief, along with the recommender’s competing incentive $c_{i}$ forms the recommender’s type, leading to the following definition:

Definition 3** (Recommender Types and Profile).**

A recommender profile is a tuple $(n,t)$, where $n\in\mathbb{N}_{\geq 1}$ is the number of recommenders and $t=(t_{1},..,t_{n})$ is the list of the recommenders’ types. Each type $t_{i}$ is a tuple $t_{i}=\left(q_{i},f_{i},c_{i}\right)$, where

$q_{i}\in[0,1]$* defines $i$’s belief $P_{i}(O=1)=q_{i}$, which is the information we want to elicit,* 2. 2.

$c_{i}\in\mathbb{R}$* is the competing incentive, i.e., the utility $i$ gains if $A=1$, and* 3. 3.

$f_{i}=(f_{i}^{q},f_{i}^{c},f_{i}^{r})\in\mathcal{D}:=\mathcal{D}^{q}\times\mathcal{D}^{c}\times\mathcal{D}^{r}$* defines $i$’s belief regarding the other recommenders’ beliefs $Q_{\neg i}$, incentives $C_{\neg i}$, and reports $R_{\neg i}$:*

[TABLE]

where $\mathcal{M}$ is the set of all mechanisms.

For a single recommender, we simplify the type as $(q,c)$, where $q\in[0,1]$ is this recommender’s belief and $c\in\mathbb{R}$ is the competing incentive. Without competing incentive, the type is just belief $q\in[0,1]$, and this reduces to the standard setting in information elicitation. Similarly, if we only consider mechanisms that make payments independently, i.e. $\textsc{epay}_{i}(r,o)=\textsc{epay}_{i}(r^{\prime},o)~{}\forall r,r^{\prime}:r_{i}=r_{i}^{\prime},o$, and assume that there are no competing incentives (i.e., $c=0$), then each recommenders’ utility is independent of others’ reports. In this case, the types simplify to $(q_{i})$, and we have the standard setting of eliciting information from $n$ independent recommenders.

2.4.1 Utility

Each recommender has the following utility:

Definition 4** (Recommender Utility).**

Given a mechanism $\textsc{mech}=(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$, recommender $i\in[n]$ with competing incentive $c_{i}$, has utility (for a realization $R_{\neg i}=r_{\neg i},O=o$, and in expectation with respect to $X$) of

[TABLE]

Hence, the subjective expected utility of a recommender (Def. 3) of type $t_{i}$ is

[TABLE]

where, with slight abuse of notation, we used $R_{\neg i},O\sim(t_{i},\textsc{mech})$ to denote that $i$’s subjective distribution of $R_{\neg i},O$ is fully specified by $t_{i},\textsc{mech}$ (see Def. 3).

Recommenders will submit the report that maximizes their subjective expected utility:

[TABLE]

In the single-recommender setting, these definitions simplify as follows:

Definition 5** (Single-Recommender Setting).**

In the case of a single recommender, the type profile simply consists of $t=(q,c)$, with $q\in[0,1]$ and $c\in\mathbb{R}$, and the subjective expected utility Def. 4 simplifies to

[TABLE]

Defining S(r,q):=$$\mathbb{E}_{O\sim{q}}\left[\textsc{epay}\left(r,O\right)\right], for simplicity, we have

[TABLE]

2.4.2 Recommender Domains

Throughout this article, we will consider different domains of recommenders, defined as follows:

Definition 6** (Recommender Domains).**

We say that a recommender profile $\left(n,(q_{i},f_{i},c_{i})_{i\in[n]}\right)$ lies in a belief domain $\mathcal{F}\subseteq\mathcal{D}$ (with $\mathcal{D}$ being the full belief domain, defined in Def. 3) if $f_{i}\in\mathcal{F}~{}\forall i$. Further, we say that a recommender profile lies in the incentive domain $\mathcal{C}\subset\mathbb{R}^{n}$ if $(c_{1},..,c_{n})\in\mathcal{C}$. We call the tuple $\mathcal{T}=(\mathcal{F},\mathcal{C})$ the type domain, and we say a recommender profile lies in the type domain $\mathcal{T}$ if it lies both in the belief domain $\mathcal{F}$ and in the incentive domain $\mathcal{C}$.

We will consider the following belief domains:

Definition 7** (Belief domains).**

We will consider the following sets of beliefs:

$\mathcal{F}_{all}:=\mathcal{D}$* is the set of all beliefs , with $\mathcal{D}$ as defined in (Def. 3),* 2. 2.

\mathcal{F}_{indep}:=\left\{\text{f\in}\mathcal{D}:f^{q}(q_{\neg}|0)=f^{q}(q_{\neg}|1)\quad\forall q_{\neg}\in[0,1]^{n-1}\right\}, i.e., each recommender $i$ believes that, given their own estimate $q_{i}$, other recommenders’ estimates $Q_{\neg i}$ do not carry any additional information about $O$. This implies, through the independences in Figure 1, that $R_{\neg i}$ and $O$ are independent as well, according to $i$’s subjective belief.

2.5 Bribers

We define a briber, who may attempt to manipulate the action by bribing recommenders, as follows:

Definition 8** ($d$-Rational Briber).**

A $d$-rational briber gains utility $d>0$ in the case of action $A=1$, and utility [math] for action $A=0$. The briber holds their own belief $\phi\in\Phi$, where $\Phi$ is the set of all possible beliefs about the recommenders’ beliefs, $Q_{1},..,Q_{n}$ and $F_{1},..,F_{n}$ (Def. 3), hence the briber’s type is $(\phi,d)$.

The briber may offer a bribe to each recommender, which is only paid if the desired action $A=1$ is taken. We model this as the briber determining the competing incentives $c_{1},..,c_{n}\geq 0$ in the recommenders’ types (Def. 3).

The briber knows the mechanism, and that recommenders will maximize their subjective expected utility, i.e. the briber knows that recommender $i$ will pick their report according to $r_{i}^{*}\left((Q_{i},F_{i},c_{i}),\textsc{mech}\right)$ 3. However, $Q_{i}$ and $F_{i}$ are unknown to the briber. Hence, given a mechanism $\textsc{mech}=(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$, the briber’s utility (for a realization of recommenders’ beliefs $Q=q,F=f$, and in expectation with respect to $X$) is

[TABLE]

Therefore, the subjective expected utility of the briber is,

[TABLE]

and they will hence pick the bribe

[TABLE]

Unless we state otherwise, we will assume that there is no briber present.

2.6 Incentive Compatibility

We will study two properties of mechanisms. The first is truthfulness, i.e., whether it is a dominant strategy for each recommender to reveal their true $q_{i}$. Whenever this is not the case, we will study to what extent the decision is affected by misreporting (i.e., the manipulability of a mechanism).

2.6.1 Truthfulness

For a mechanism to be strictly truthful, we require that all recommenders’ subjective expected utilities are strictly larger for reporting their truthful beliefs, than for any other report, regarless of other recommenders’ reports:

Definition 9** (Strict truthfulness).**

A mechanism mech (Def. 1) is strictly truthful, with respect to a type domain (Def. 6) $(\mathcal{F},\mathcal{C})$, iff for all $i\in[n]$ we have

[TABLE]

In this case, truthful reporting is a dominant strategy of each recommender. In the presence of a briber, two more notions will be useful:

Definition 10** (Strict truthfulness with briber).**

A mechanism mech (Def. 1) is strictly truthful in the presence of a d-rational briber* *(Def. 8) with respect to a belief domain $\mathcal{F}$, iff the mechanism is strictly truthful (Def. 9) with respect to the type domain $\mathcal{T}=(\mathcal{F},\mathcal{C})$, where $\mathcal{C}$ is the set of competing incentives that could possibly be induced by a briber type $\phi$ (7), i.e. $\mathcal{C}=\left\{c^{*}\left((d,\phi),\textsc{mech}\right):\phi\in\Phi\right\}$.

Definition 11** (Bribe-freeness).**

*We say that mechanism mech (Def. 1) is bribe-free given a d-rational-briber *(Def. 8) if it is irrational for a briber of any type to bribe, i.e. a bribe would reduce the subjective expected utility 6:

[TABLE]

which is equivalent to

[TABLE]

2.6.2 Manipulability

It is often impossible to achieve strict truthfulness in the presence of competing incentives. Therefore, we also study the extent to which misreports affect the decision:

Definition 12** (Manipulability).**

We define the manipulability of a mechanism mech (Def. 1), with respect to a type domain (Def. 6) $\mathcal{T}=(\mathcal{F},\mathcal{C})$, as the worst-case (in terms of types) manipulation:

[TABLE]

For a single recommender, we use the notation $\textsc{manip}\left(\mathcal{C},\textsc{mech}\right)$ instead of $\textsc{manip}\left(1,\emptyset,\mathcal{C},\textsc{mech}\right)$.

We will characterize the assumptions on the mechanism and the recommender types for which strict truthfulness or bounds on manipulability can or cannot be achieved.

2.7 Sensitivity to Reports

Intuitively, if agents have a larger influence on the decision, then their incentive for manipulation, in consideration of a competing incentive, becomes larger. However, if the decision probability, $\textsc{pact}(r)$, is completely insensitive to reports, then the mechanism has no utility. To characterize the trade-off between budget, sensitivity to reports, and truthfulness (or the amount of manipulation), we will use the following definitions:

Definition 13** (Sensitivity).**

For a single recommender, a decision function, $\textsc{pact}(r)$, has sensitivity, $\Delta$, if the recommender can affect the decision probability by $\Delta$, i.e., $\Delta=\max_{r,r^{\prime}\in{[0,1]}}\left|\textsc{pact}(r^{\prime})-\textsc{pact}(r)\right|$.

For multiple recommenders, the sensitivity may be different with respect to each recommender, and it may be a function of others’ reports. Hence, in the multi-recommender-setting, $\Delta=(\Delta_{1},..,\Delta_{n})$ represents a series of functions $\Delta_{i}:{[0,1]}^{n-1}\to[0,1]$:

[TABLE]

Definition 14** (Max/Min-Uniform-Sensitivity).**

For a single recommender, we say that a decision rule, pact, has max-uniform-sensitivity, $L$, on an interval $a\subseteq[0,1]$, if

[TABLE]

and min-uniform-sensitivity as defined as above, except with a $\min$ instead of a $\max$.

For multiple recommenders, the sensitivity may be different with respect to each recommender, and it may be a function of others’ reports. Hence, in the multi-recommender-setting, $L=(L_{1},..,L_{n})$, represents a series of functions $L_{i}:{[0,1]}^{n-1}\to[0,1]$, and we have a profile of intervals, $a=(a_{1},..,a_{n})$, with $a_{i}\subseteq[0,1]$. We say that pact has max-uniform-sensitivity $L$ on interval profile, $a$ if

[TABLE]

and min-uniform-sensitivity on interval profile $a$ as defined as above, except with a $\min$ instead of a $\max$. Whenever we refer to max/min-uniform sensitivity without specifying an interval, we mean the interval is the full domain $a=[0,1]$.

If pact is linear on an interval $a$, then min-uniform-sensitivity and max-uniform-sensitivity are identical on that interval. Further, if act has min-uniform-sensitivity $L$ on an interval $a$, then it has sensitivity of at least $L|a|$.

3 Lying a Little is Cheap

In this section, we consider the single-recommender setting (Def. 5) and we study the properties of the expected payment function $S$, which determines the incentive for truthfulness, in isolation. In later sections, we will tie these results to the setting where we explicitly model agents’ competing incentives in regard to the action chosen.

In particular, we are interested in how much cost a recommender incurs for deviations from truthfulness, and we make the following definition:

Definition 15** (Cost of $\varepsilon$-Lying).**

For a single recommender, we define the cost of $\varepsilon$-lying of a scoring rule (payment function) with expected payment $S$, for a given $\varepsilon\in(-1,1)$, as the minimal (in terms of true belief) expected cost that is associated with an $\varepsilon$ deviation, i.e.,

[TABLE]

Intuitively, this is the best case expected cost to a recommender, for a given scoring rule, for changing their report by a small amount. We will give upper bounds on the cost of $\varepsilon$-lying that could possibly be achieved by any payment function with a fixed budget $\beta$. Further, we will show that the quadratic scoring rule is optimal in terms of cost of $\varepsilon$-lying.

3.1 Upper Bound on the Cost of Lying

Theorem 1** (Upper bound on the cost of lying).**

Any expected-payment function $S(r,q)\in[0,\beta]\ \forall r,q\in{[0,1]}$ has a cost of $\varepsilon$-lying (Def. 15) of at most,

[TABLE]

for any deviation $\varepsilon\in(-1,1)$.

This result shows that the cost of lying scales quadratically in the size of the misreport $\varepsilon$, meaning that small deviations are cheap for any scoring rule.

3.2 Lower Bound on the Cost of Lying

We now show that the quadratic scoring rule is essentially optimal in terms of the cost of $\varepsilon$-lying (Def. 15). The following is the quadratic scoring rule (Brier et al., 1950), normalized so that payments lie in $[0,\beta]$.

Definition 16** ($\beta$-Quadratic Scoring Rule).**

The payment made by the $\beta$-Quadratic Scoring Rule rule is,

[TABLE]

This yields an expected payment, given a true belief $q$, of

[TABLE]

We have $\max_{r,o}[\textsc{epay}(r,o)]=\beta$, and $\min_{r,o}[\textsc{epay}(r,o)]=0$.

Lemma 1** (Lower bound on the cost of lying).**

The $\beta$-Quadratic Scoring Rule has a cost of $\varepsilon$-lying (Def. 15) of

[TABLE]

Proof.

The result follows straightforwardly by plugging the definition of the $\beta$-Quadratic Scoring Rule into the definition of $\varepsilon$-lying (Def. 15). ∎

Comparing this result to Theorem 1, we see that Quadratic Scoring is essentially optimal in terms of the cost of $\varepsilon$-lying.

4 Competing Incentives and Bribery

We have seen that small deviations from truthfulness are cheap for any payment scheme. However, there may still be hope for strict truthfulness (Def. 9) in the setting where the competing incentives (called $c$ in Def. 3) stem from the utility that recommenders gain from the mechanism picking $A=1$ through the function pact. To see this, recall the definition of recommenders’ utilities 2:

[TABLE]

For example, if we design act (and hence pact) to be independent of the reports $r$, then recommenders cannot manipulate the action and have hence no incentive for misreporting. Although this extreme mechanism would have no utility, one may hope to achieve truthfulness, or at least a bound on the manipulation of the decision (Def. 12), through the careful co-design of act and pay, so that wherever pact has a large sensitivity, the incentives from epay are strong enough to overcome the temptation of manipulation.

In fact, we show that as soon as there is even one recommender (Def. 3) with a fixed competing incentive $c_{i}>0$, no mechanism can achieve strict truthfulness (Def. 9) unless it completely ignores all reports (i.e., it has sensitivity (Def. 13) equal to [math]). We then quantify to what extent misreporting affects the decision taken, i.e., we provide lower bounds on the manipulability (Def. 12) as a function the sensitivity $\Delta$ (Def. 13), the budget $\beta$ (Def. 1), and the competing incentives $(c_{1},..,c_{n})$. For the single-recommender case, we again provide a matching upper bound, achieved by a mechanism making payments according to the Quadratic Scoring rule.

On the other hand, if the competing incentives stem from a $d$-rational briber (Def. 8), we give a positive result—it is possible to attain truthfulness. We provide necessary and sufficient conditions for truthfulness as a function of the sensitivity $\Delta$, the budget $\beta$, and the incentive of the briber $d$.

At a high level, our results establish that the budget $\beta$ of a mechanism (Def. 1) must grow with the sum of squares of the sensitivities, i.e., $\sum_{i}{\Delta_{i}}^{2}$, for the mechanism to guarantee small manipulation or, whenever possible, truthfulness. Of interest, is that this allows the total influence of recommenders on the decision (i.e., the sensitivity of the decision rule) to grow with $\sqrt{n}$, while keeping the budget constant. Hence, with a sufficient number of recommenders, we can attain a large aggregate sensitivity, while maintaining truthfulness, or at least low manipulability.

4.1 Single Recommender

We first consider the single-recommender setting (Def. 5), and then generalize the results to the multi-recommender setting. We start by showing that no mechanism is strictly truthful when the recommender has a fixed competing incentive, $c>0$. Thereafter, we consider a setting where the incentive for misreporting is not intrinsic, but comes from a $d$-rational briber, and we show that in this setting truthfulness can be achieved.

4.1.1 Recommender with an Intrinsic, Competing Incentive

The following theorem gives a lower-bound on manipulability in the presence of a recommender with an intrinsic (i.e., fixed) competing incentive $c$:

Theorem 2** **(Lower-bound on single-recommender

manipulability).

In the single-recommender setting (Def. 5), any proper mechanism mech (Def. 2) with a budget $\beta>0$ and sensitivity $\Delta\geq 0$ (Def. 13) has manipulability (Def. 12) of at least

[TABLE]

It follows that for any $c>0$, there is no strictly truthful mechanism, unless it ignores the recommender’s report, i.e., $\Delta=0$. Furthermore, this result implies that to guarantee that $\textsc{manip}\left(\{c\},\textsc{mech}\right)\leq\varepsilon$, we need a budget of at least

[TABLE]

This result has far-reaching implications. It states that as soon as the recommender may have any (arbitrarily-small) competing incentive $c>0$, there exists no proper mechanism that can guarantee that the decision will not be manipulated by the recommender. Even if we co-design the decision act and payment pay function in possibly intricate ways, allowing for discontinuities and randomness, it is impossible to guarantee that there will be no manipulation. Before we discuss this lower bound quantitatively, we provide a complementary upper bound on the achievable manipulability.

Theorem 3** **(Upper-bound

on single-recommender manipulability).

In the single-recommender setting (Def. 5), any proper mechanism mech (Def. 2) with max-uniform-sensitivity $L$ (Def. 14), that makes payments according to the $\beta$-quadratic scoring rule (Def. 16), has manipulability (Def. 12) of at most

[TABLE]

This result implies that to guarantee that $\textsc{manip}\left(\{c\},\textsc{mech}\right)\leq\varepsilon$, we need a budget of at most

[TABLE]

To illustrate this result, suppose that we pick the decision function $\textsc{pact}(r)=r\cdot L$, for some $0\leq L\leq 1$, (along with the $\beta$-quadratic payment rule). This mechanism has sensitivity $\Delta=L$, and we can compare Theorem 3 with Theorem 2. Together, they imply that the budget $\beta$ required to guarantee a maximum manpulation $\varepsilon>0$, while exhibiting senstivity of at least $\Delta$, satsifies

[TABLE]

where $c>0$ is the competing incentive. The budget must be proportional to the competing incentive $c$, and inversely proportional to the admissible manipulation $\varepsilon$. Interestingly, it scales quadratically with the sensitivity. We will discuss the implications of these dependences after discussing the setting where a briber is present.

4.1.2 Self-Interested, Rational Briber

As we have seen in the previous section, whenever the competing incentive of the recommender is non-zero ($c>0$) and the decision rule has sensitivity $\Delta>0$, then no mechanism is truthful. However, if the competing incentive comes from a briber, we shall see that it is possible to design a mechanism in which it is not in the interest of the briber (Def. 8) to make a bribe. This means that strict truthfulness (Def. 10) can be achieved and is equivalent to bribe-freeness (Def. 11). In the following, we characterize under what conditions truthfulness can be achieved in this setting, by providing a necessary and then a sufficient condition.

Theorem 4** **(Single Recommender with Briber: Necessary Condition

for Truthfulness).

In the single-recommender setting, for a proper mechanism mech (Def. 2) to be strictly truthful in the presence of a $d$-rational briber (Def. 10), it is necessary that

[TABLE]

with $\beta>0$ being the budget and $\Delta\geq 0$ the sensitivity (Def. 13) of the mechanism.

We now turn to a positive result, showing the possibility of strict truthfulness in the presence of competing incentives stemming from a rational briber:

Theorem 5** **(Single Recommender with Briber: Sufficient Condition

for Truthfulness).

Consider the single-recommender setting with a $d$-rational briber and a proper mechanism (Def. 2) with max-uniform-sensitivity, $L$ (Def. 14), and making payments according to the $\beta$-Quadratic Scoring Rule (Def. 16). In this case, a sufficient condition for strict truthfulness is

[TABLE]

As in the previous section, suppose that we pick the decision function $\textsc{pact}(r)=r\cdot L$, for some $0\leq L\leq 1$, (along with the $\beta$-quadratic payment rule). This mechanism has sensitivity $\Delta=L$, we can hence combine Theorem 5 with Theorem 4, and obtain

[TABLE]

As in the case of a fixed, intrinsic incentive, the budget grows quadratically with the sensitivity $\Delta$. As we shall see, in the multi-recommender setting, this property will allow for truthful mechanisms with a low budget, yet large aggregate sensitivity to recommenders’ reports.

4.2 Multiple Recommenders

When eliciting information form multiple recommenders, several of them may have competing incentives. We will consider the incentive domain $\mathcal{C}(\bar{c})$ (Def. 6) that contains all incentive profiles such that the sum of incentives is bounded by $\bar{c}$, i.e. $\mathcal{C}(\bar{c}):=\left\{c\in\mathbb{R}_{\geq 0}^{n}:\sum_{i\in[n]}c_{i}\leq\bar{c}\right\}$.

4.2.1 Recommenders with Intrinsic, Competing Incentives

The lower bound on the manipulability from the single-recommender case translates straighforwardly to the multi-recommender setting:

Corollary 1** (Lower-bound on multi-recommender manipulability).**

Any proper mechanism mech (Def. 2) with budget $\beta>0$ and sensitivity $\Delta$ (Def. 13), has manipulability (Def. 12) (with respect to the full belief domain (Def. 6) $\mathcal{F}_{all}$) of at least

[TABLE]

This implies that if there is any competing incentive $|\bar{c}|>0$, there is no strictly truthful mechanism, unless it ignores all recommenders’ reports, i.e., $\Delta_{i}(r_{\neg i})=0\forall i\in[n],r_{\neg i}\in[0,1]^{n-1}$.

Proof.

Consider recommender $i$. Suppose this is the only recommender with a conflicting incentive, i.e., $c_{i}=\bar{c}$, so that the others report truthfully, with $R_{\neg i}=q_{\neg i}$. Further, suppose that recommender $i$ happens to reason correctly about others’ reports, i.e. $i$ knows $R_{\neg i}$. In the best-case scenario, the mechanism happens to allocate the entire budget $\beta$ to recommender $i$. From Theorem 2, we know this recommender will manipulate the action probability by at least

[TABLE]

Taking the worst-case recommender $i$ and beliefs $q_{\neg i}$, we obtain the result.∎

Hence, as in the single-recommender setting, it is impossible to design a mechanism that guarantees strict truthfulness in the presence of a fixed, intrinsic competing incentive.

4.2.2 Self-Interested, Rational Briber

We now turn to the setting where the competing incentives in this multi-recommender setting come from a rational briber who attempts to manipulate the decision. The necessary condition for a single recommender translates straightforwardly to the multi-recommender setting:

Corollary 2** (Multiple Recommenders with Briber: Necessary Condition for Truthfulness).**

For a proper mechanism mech (Def. 2) to be strictly truthful in the presence of a $d$-rational briber (Def. 10), it is necessary that

[TABLE]

with $\beta>0$ being the budget and $\Delta$ the sensitivity (Def. 13) of the mechanism.

Proof.

Suppose the briber (Def. 8) only targets a single recommender. In the best case, the mechanism happens to allocate the entire budget $\beta>0$ to this recommender. The result then follows from Theorem 4 by taking the worst-case in terms of who the briber targets and the the bribers’ beliefs $\phi$. ∎

We now present a sufficient condition for truthfulness in the briber setting, and we shall see in the following that this insight allows for truthful mechanisms with large aggregate sensitivity.

Theorem 6** **(Multiple recommenders with briber: sufficient

condition for truthfulness).

Consider the single-recommender setting with a $d$-rational briber and a proper mechanism (Def. 2) with max-uniform-sensitivities $L_{1},..,L_{n}$ (independently of others’ reports) and budget $\beta$. Suppose the mechanism makes payments to each recommender $i$ according to the $\beta_{i}$-Quadratic Scoring Rule (Def. 16), with $\beta_{i}=\beta\frac{L_{i}^{2}}{\bar{L}}$ and $\bar{L}:=\sum_{j\in[n]}L_{j}^{2}$. In this case, a sufficient condition for strict truthfulness is

[TABLE]

Comparing this result to Theorem 5, we see that we now have the sum of the squares of the max-uniform-sensitivities on the left-hand-side. Hence, to maintain truthfulness when adding more recommenders (suppose we give an equal amount of influence, that is the sensitivity of the decision rule to reports, to everyone), each recommender’s influence has to scale with $L_{i}=\frac{1}{\sqrt{n}}$, and hence the sum of max-uniform-sensitivities constants can grow with $\sqrt{n}$, i.e., the total influence of recommenders can grow as more recommenders are added.

5 Conditional Observations and Dependent Recommenders

It is often the case that we observe the outcome only conditionally on the action taken. For instance, we will only observe the quality of a product if we decide to buy it or the repayment or not of a loan if we decide to make it. Hence, in this section we address the challenge that $O$ is only observed if $A=1$, which imposes the following structure on the payment function.

Definition 17** (Conditional Payment Function (CPF)).**

We say that a payment function is a conditional payment function (CPF) if it can be expressed as,

[TABLE]

This decision-conditional outcome structure does not allow for $\textsc{pay}_{i}$ to be a standard scoring rule. Chen et al. (2011) study this setting in the context of decision markets and York et al. (2021) in the context of a VCG-based scoring rule for information elicitation. However, neither model is able to handle the presence of competing incentives and moreover, we show that these styles of mechanism are not truthful for dependent recommenders and develop a novel mechanism that addresses this problem.

Before we proceed to study this problem, we derive a condition for truthfulness that is simpler than Def. 9 but equivalent.

5.1 Equivalent Conditions for Truthfulness

The following result reformulates the condition for strict truthfulness in a manner that will be convenient in this section.

Lemma 2** (Strict truthfulness).**

A mechanism mech (Def. 1) is strictly truthful (Def. 9), with respect to the type domain $(\mathcal{F}_{all},\{\mathbf{0}\})$ (Def. 6) of recommenders with any belief and no competing incentives, iff for all $i\in[n]$ we have

[TABLE]

Further, a mechanism mech (Def. 1) is strictly truthful (Def. 9), with respect to the type domain $(\mathcal{F}_{indep},\{\mathbf{0}\})$ (Def. 6) of recommenders with any belief and no competing incentives, iff for all $i\in[n]$ we have

[TABLE]

5.2 Independent vs Dependent Recommenders

We first present a slight generalization of the VCG-Scoring Mechanism (York et al., 2021) that is truthful for independent recommenders and then show the difficulties that arise when recommenders are dependent.

Definition 18** (Critical-Payment Mechanism (CPM)).**

In the critical-payment mechanism, the payment and decision functions are deterministic, and $\textsc{act}(r)$ is strictly increasing in each $r_{i}$ individually. We define the critical value, at which the decision changes from 1 to 0, as

[TABLE]

Given this, the payment is defined as,

[TABLE]

Lemma 3** (Based on York et al. (2021)).**

CPM is weakly truthful for independent recommenders in the absence of competing incentives.

Proof.

Recall the condition for strict truthfulness for independent recommenders (Lemma 2) which, for weak truthfulness takes the form

[TABLE]

in the absence of competing interests, i.e. with $c_{i}=0$. The expected payment of CPM is

[TABLE]

and satisfies the weak truthfulness condition. Further, the optimal report satisfies

[TABLE]

∎

The following lemma shows, however, that the CPM mechanism is not truthful for dependent recommenders and illustrates the following intuition: Suppose recommender $i$ believes that recommender $j$’s report is informative, i.e., $i$ assumes that $j$’s report is larger when the hidden outcome is $O=1$ than when $O=0$ (i.e., $r_{j}^{1}>r_{j}^{0}$). Then, if recommender $i$ knew $r_{j}$’s report, they would adjust their own report $r_{i}$ towards $r_{j}$. Since our mechanism can only evaluate recommenders’ predictions in the case $A=1$, the mere fact of scoring on $O$ must itself reveal information about others’ reports (assuming that the decision depends in some way on others’ reports).

Lemma 4**.**

CPM is not weakly truthful for dependent recommenders.

Proof.

Recall the condition for strict truthfulness for dependent recommenders (Lemma 2) which, for weak truthfulness and in the absence of competing interests, i.e., $c_{i}=0$, takes the form

[TABLE]

The expected payment of CPM is now

[TABLE]

with $x:=\textsc{act}_{i}^{-1}(r_{\neg i}^{1})$ and $y:=\textsc{act}_{i}^{-1}(r_{\neg i}^{0})$ (which implies that $\textsc{act}(r_{i},r_{\neg i}^{1})=1$ iff $r_{i}\geq x$ and $\textsc{act}(r_{i},r_{\neg i}^{0})=1$ iff $r_{i}\geq y$). Suppose that $x<y$, which is the case if $i$ believes that other recommenders’ beliefs are positively correlated with the outcome. We then have

[TABLE]

This implies that the recommender will always report $x\leq r_{i}<y$, regardless of their true belief, $q_{i}$. ∎

5.3 Truthful Mechanisms for Dependent Recommenders

We now propose a method for decoupling payments and the action taken, thereby preventing the leakage of information through the action. This decoupling can be applied to make the payment functions of any mechanism take the form of CPFs (Def. 17), and hence admissible in the conditional-observation setting.

Definition 19** ($\alpha$-decoupling).**

An $\alpha$-decoupling takes any mechanism, $(\textsc{act}^{\prime},\textsc{pay}^{\prime}_{1:n},\textsc{rand}^{\prime})$, as input and produces a modified mechanism, $(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$, where the new payment functions $\textsc{pay}_{1:n}$ are CPFs. First, we draw a Bernoulli RV, $X_{1}\sim\mathcal{B}(1-\alpha$), and, if the original mechanism is randomized, we also draw from its distribution $X_{2}\sim\textsc{rand}^{\prime}$. Let $X=(X_{1},X_{2})$. The new decision and payment functions are defined as

[TABLE]

Lemma 5** (Decoupling maintains properties).**

The mechanism $(\textsc{act},\textsc{pay}_{1:n},\textsc{rand})$ obtained by applying $\alpha$-decoupling to mechanism $(\textsc{act}^{\prime},\textsc{pay}^{\prime}_{1:n},\textsc{rand}^{\prime})$ satisfies $\textsc{epay}_{i}(r,o)=\textsc{epay}_{i}^{\prime}(r,o)$ and $\textsc{pact}(r)=\alpha+(1-\alpha)\textsc{pact}^{\prime}(r)$. Further, if the original mechanism is strictly truthful, so is its decoupled version.

Proof.

We have

[TABLE]

and

[TABLE]

Strict truthfulness of the decoupled mechanism follows because the expected payments are identical, and the influence of recommenders on the action probability is smaller, so incentives for misreporting are smaller everywhere. ∎

Hence, the results from the previous section can easily be carried over to the conditional-observation setting.

6 Conclusion

We have demonstrated that no decision scoring mechanism is completely robust to intrinsic competing recommender incentives, and that the best performance can be achieved through the use of the Quadratic Scoring Rule. However, when a rational briber is the cause of the competing incentives, we can get a no-manipulation result, as long as the mechanism has sufficient budget relative to the maximum influence that recommenders can have over the decision. For multiple recommenders, we can allow the total recommender influence to grow as $\frac{1}{\sqrt{n}}$ while preserving the same incentives. We also show that dependent recommender beliefs can cause an additional violation of strict-truthfulness, but that this can be resolved with a general, decoupling construction.

To our knowledge, this problem of competing incentives in the context of elicitation and decision making, also with dependent recommender beliefs, has not been formally studied before and we believe it is a rich area for future work. One interesting direction is to develop optimal scoring rules for different types of allocation functions, in particular aligning the magnitude of incentive with the magnitude of decision impact of a recommender. Another interesting direction is to allow recommenders to explicitly rate the quality of other recommenders, and be rewarded for their posterior beliefs. This can open up improved avenues for informative belief aggregation.

Appendix A Appendix

A.1 Proof of Theorem 1

In order to prove Theorem 1, we first derive the following result:

Lemma 6**.**

For any proper scoring rule with $S(x,y)\geq 0,\ \forall x,y\in{[0,1]}$, and any monotone sequence $x_{1},..,x_{K}\in{[0,1]}$, such that $|x_{k+1}-x_{k}|\leq\varepsilon\ \forall k$, we have

[TABLE]

The first line holds with equality if $|x_{k+1}-x_{k}|=\varepsilon\quad\forall k$.

Proof.

Any scoring rule for dichotomous RVs can be written as

[TABLE]

for some $g(x)$ and $h(x)$. Using the identity

[TABLE]

we have

[TABLE]

With $r=0$, we have

[TABLE]

and with $r=1$, we have

[TABLE]

Subtracting the second equality from the first, we obtain

[TABLE]

Now, for an increasing sequence $x_{1},..,x_{K}$ with $|x_{k+1}-x_{k}|\leq\varepsilon\quad\forall k$, we have

[TABLE]

By symmetry, the same inequality holds for decreasing sequences, and the result follows. ∎

We are now ready to prove Theorem 1, which we restate here for convenience: See 1

Proof.

For a given $1>\varepsilon>0$, let $K=\left\lfloor\frac{1}{\varepsilon}\right\rfloor+1$ and let the sequence $x_{1},..,x_{K}\in{[0,1]}$ be such that $x_{k+1}-x_{k}=\varepsilon$. Then Lemma 6 states that,

[TABLE]

[TABLE]

Since both terms are $\geq 0$ the result follows. ∎

A.2 Proof of Theorem 2

We will use the following result to prove Theorem 2:

Lemma 7**.**

Let $S(\cdot,\cdot)$ be the expected payment function of a proper scoring rule, and suppose an agent gains additional utility, $T(r)$ from its report, with $[\max_{r}T(r)-\min_{r}T(r)]\geq\delta$, for some $\delta>0$, such that the total expected utility of the agent is,

[TABLE]

For any scoring rule with expected payment $S(x,y)\in[0,\beta],\forall x,y\in{[0,1]}$ and some $\beta>0$, there exists a true belief $q\in{[0,1]}$, such that the optimal report, $r\in\arg\max_{x}u(x,q)$, satisfies the following inequality:

[TABLE]

Proof.

For a given $\varepsilon$, satisfying $1>\varepsilon>0$, let $K=\left\lceil\frac{1}{\varepsilon}\right\rceil+1$, and let $x_{1},..,x_{K}\in{[0,1]}$ be a monotone sequence such that $x_{1}\in\arg\min_{r}T(r)$, $x_{K}\in\arg\max_{r}T(r)$, and $|x_{k+1}-x_{k}|\leq\varepsilon$. Then Lemma 6 says that

[TABLE]

Further, we have

[TABLE]

Summing the two inequalities, we have

[TABLE]

and hence

[TABLE]

With $\varepsilon=\delta/(4\beta)$, we have

[TABLE]

By the pigeonhole principle, it follows that there is an $i$, such that

[TABLE]

Hence, if the true belief is $q=x_{i}$, then the optimal report $r$ satisfies

[TABLE]

Finally, we have

[TABLE]

from which the result follows. ∎

We are now ready to prove Theorem 2, which we restate, for convenience:

See 2

Proof.

Recall that the subjective expected utility of a single recommender 4 with type $t=(q,c)$ is

[TABLE]

Applying Lemma 7 with $T(r)=c\cdot\textsc{pact}\left(r\right)$ and hend $\delta=c\cdot\Delta$, we obtain

[TABLE]

from which the result follows.

∎

A.3 Proof of Theorem 3

We restate Theorem 3 for convenience: See 3

Proof.

Recall that the subjective expected utility of a single recommender 4 with type $t=(q,c)$ is

[TABLE]

Clearly, the agent will report $r$ such that,

[TABLE]

Combining this with the (Def. 14) condition

[TABLE]

we obtain

[TABLE]

∎

A.4 Proof of Theorem 4

We restate Theorem 4 for convenience: See 4

Proof.

In the presence of a d-rational briber (Def. 8), strict truthfulness (Def. 10) is equivalent to bribe-freeness (Def. 11):

[TABLE]

In the sinlge-recommender setting, the utility of the briber 5 is

[TABLE]

Hence, we can write the condition for bribe-freeness as

[TABLE]

From Theorem 2, we know that

[TABLE]

hence we obtain the necessary condition for truthfulness

[TABLE]

It is easy to verify that with

[TABLE]

the right-hand side is equal to [math], and hence strict truthfulness is not satisfied. Hence, whenever $c^{*}>0$, strict truthfulness is violated, which is the case if

[TABLE]

∎

A.5 Proof of Theorem 5

We restate Theorem 5 for convenience: See 5

Proof.

In the presence of a d-rational briber (Def. 8), strict truthfulness (Def. 10) is equivalent to bribe-freeness (Def. 11):

[TABLE]

In the sinlge-recommender setting, the utility of the briber 5 is

[TABLE]

Hence, we can write the condition for bribe-freeness as

[TABLE]

From Theorem 3, we know that

[TABLE]

Hence, we obtain the sufficient condition for truthfulness

[TABLE]

∎

A.6 Proof of Theorem 6

See 6

Proof.

In the presence of a d-rational briber (Def. 8), strict truthfulness (Def. 10) is equivalent to bribe-freeness (Def. 11):

[TABLE]

From Theorem 3, we know that given a competing incentive, $c_{i}$, each recommender will manipulate the action by at most

[TABLE]

Hence, the briber’s utility 5 for a given set of bribes is at most

[TABLE]

where we have used the definition of $\beta_{i}$, and defined $\bar{c}:=\sum_{i}c_{i}$. Hence, a sufficient condition for bribe-freeness is

[TABLE]

from which the result follows. ∎

A.7 Proof of Lemma 2

See 2

Proof.

Without competing incentives, the condition for strict truthfulness (Def. 9) becomes

[TABLE]

By the law of total expectation, we can write the expected payment from recommender $i$’s perspective as

[TABLE]

Defining

[TABLE]

we can write the condition for strict IC as

[TABLE]

Note that the recommenders’ beliefs may be such that $p^{0},p^{1}$ are concentrated in one point each $x^{0},x^{1}$, which means that the condition above implies

[TABLE]

To see that the reverse implication also holds, note that a pointwise inequality implies an average inequality. We can hence integrate on both sides in $x^{0}$ and then in $x^{1}$ (since both $p^{0},p^{1}$ integrate to 1)

[TABLE]

Now take for instance the first term in the sum (with $o=0$) on the right-hand side:

[TABLE]

Following a similar reasoning for the other terms, we obtain

[TABLE]

which is identical to 34 and hence closes the circle of implications, meaning that 34 is equivalent to 35. This concludes the proof for the first part of the result (i.e. with $\mathcal{F}_{all}$). The second part (with $\mathcal{F}_{indep}$) can easily be derived along a similar line of reasoning, noting that in that case $p^{0}$ and $p^{1}$ must be identical. ∎