text:
\section{Introduction}
Distributed estimation theory to solve the problem of fusing data from a group of sensors to estimate a parameter or a random variable is a well-developed field. More recently, the emerging areas of social computing and crowdsourcing have enabled many large scale sensing and estimation tasks that leverage many humans (or human owned and operated devices) to collect data about phenomena of interest (see works such as~\cite{guo2014participatory,ma2014opportunities} for an overview). An example is that of aggregating information and opinions of a `crowd' recruited using Amazon Mechanical Turk to perform tasks that are time consuming and difficult to scale such as image labeling. Similar applications have been proposed or demonstrated in the fields ranging from environmental monitoring \cite{boulos2011crowdsourcing}, health data collection \cite{pryss2015mobile}, traffic monitoring \cite{pan2013crowd}, and so on.

Beyond already existing challenges in traditional distributed estimation or detection~\cite{berthet2016resource,cattivelli2010diffusion,varshney2012distributed}, new challenges arise in the design of such a crowdsensing system since data sources may not have any incentive to provide the data aggregator with the quality of data that it desires~\cite{allahbakhsh2013quality}. This might be due to the fact that sensors may have to exert resources (e.g., time, power, or bandwidth) to produce an accurate measurement~\cite{gao2015survey}. Further, even though the sensors may have accurate data, they may still wish to corrupt data before transmission either to gain privacy or for some other selfish reason~\cite{gong2016optimal}. Early work in this field (e.g., \cite{mohan2008nericell,thiagarajan2009vtrack,rana2010ear}) ignored these issues and assumed that participants were voluntary recruits who would collect and provide high quality data. More recently, it has been recognized that without a suitable incentive being present, such voluntary providers of data may not be enough to generate an estimate of desired quality. As an illustrative example, \cite{hu2006can} studied product reviews on Amazon.com and concluded that users with a moderate outlook are unlikely to report; thus, while controlled experiments on the same items reveal normally distributed opinions, voluntarily reported ratings often follow bi-modal, U-shaped distributions where most of the ratings are either very good or very bad.

Accordingly, there has been recent work on designing mechanisms that incentivize data sources (i) to participate and generate measurements of sufficient quality (i.e. effort exertion), and (ii) to report these measurements and their quality accurately (i.e. truthful elicitation). Incentivization may be through monetary or non-monetary rewards for the sensors. A review of various incentive mechanisms, including both monetary and non-monetary incentives, is provided in \cite{gao2015survey,jaimes2015survey,restuccia2016incentive}. As an illustration, if the agents cannot falsify data and the problem is solely to incentivize effort exertion, mechanisms such as those in~\cite{cai2015optimum,farokhi2015budget, witkowski2013dwelling, radanovic2016incentives, lee2010sell,cao2015target, chen2016auction} have been proposed.
Similarly, for the problem of truthful elicitation, the class of mechanisms called peer prediction mechanisms has been developed with different information structures (see e.g., \cite{prelec2004bayesian, miller2005eliciting, jurca2009mechanisms, witkowski2012robust,radanovic2013robust}) to incentivize the agents to report truthfully in a game-theoretic equilibrium.

However, most of the works in the literature focused on either  truthfulness elicitation or effort elicitation, without considering how much total reward is to be paid, or the trade-off between the total payment and the estimate accuracy at the estimator. A systematic theory that addresses the challenges of incentive mechanism design with the objective of optimizing the overall cost function at the estimator is not well studied.
In this paper, we address the mechanism design problem at the estimator of minimizing expected total compensation to be made to the strategic agents while guaranteeing a specified quality of global estimate, with both the measurements and their accuracy from the strategic agents unknown to the estimator.

The works that are closest seem to be \cite{luo2018parametric, liu2016learning,liu2017sequential,dobakh}.
\cite{luo2018parametric} considered a model that determines the compensation to strategic agents by verifying their reports with the ground truth of the phenomenon of interest, which was assumed available to the estimator after the estimation process. However, this is not applicable in many cases where the reports to be collected are subjective, such as collecting ratings for a product from consumers. In this paper, we do not require the availability of ground truth.
\cite{liu2016learning} considered the mechanism design problem from the perspective of an estimator who cares about both the estimate accuracy and the total payment with the assumption that the actual effort costs of the agents are drawn from a known distribution. Although this assumption can be relaxed by learning the distribution in a sequential setting \cite{liu2017sequential}, their model is limited to a binary-answer task (e.g., an image contains a certain object or not) and a binary effort model (e.g., either exert a fixed level of effort at a fixed cost or no effort and no cost at all).
Different from solving the discrete value tasks (i.e., detection problem or classification problem) such as image labeling and spectrum occupancy sensing, we focus on estimation tasks with continuous measurements and continuous effort models.
Finally, unlike \cite{dobakh} where the model is limited to a specific effort cost function, we do not restrict the format of effort cost function.

The rest of the paper is organized as follows.
In Section II, the problem statement is presented. In Section III, we solve the optimization problem at the estimator to minimize total compensation while guaranteeing a certain estimation accuracy. Next, in Section IV, we present an optimal mechanism that achieves the desired behavior from the strategic agents in a Nash equilibrium when the cost functions of the strategic agents satisfy a certain property. In Section V, we provide a feasibility-guaranteed sub-optimal mechanism when the cost functions of the strategic agents do not satisfy that property. Section VI provides some simulation experiments and Section VII concludes the paper.

\paragraph*{Notation}
$\E_X [f]$ denotes the expectation of function $f$ taken with respect
to the random variable $X$; when $X$ is explained from the context, the notation is abbreviated as $\E[f]$.
A Gaussian distribution is denoted by $\mathcal{N} (m, \sigma^{2})$ where $m$ is the mean and $\sigma$ is the standard deviation.
A tuple of $n$ elements is denoted with parentheses by $(e_1, e_2, \cdots, e_n)$.

\section{Problem Statement}   \label{sec_ch2_problem statement}
\paragraph*{Estimation Setup} Consider a scalar-valued random variable $X$ that is distributed according to a prior distribution $X \sim\mathcal{N} (0, \sigma^{2}_{x})$ and takes an unknown value $x$ in an experiment. An estimator (also called an aggregator) seeks to estimate the value $x$ using observations from $N$ sensors (also called agents). The $i$-th sensor generates an observation $y_{i}\in\mathbb{R}$ according to the relation
\begin{equation}
\begin{split}
y_{i} &= x + v_{i}, \\
\end{split}
\end{equation}
where $v_{i}$ is the measurement noise that is distributed according to a Gaussian distribution with mean 0 and variance $\sigma^{2}_{i}.$ We assume that the measurement noises for the $N$ sensors are mutually independent and further independent with the variable $X$. For notational ease, we denote by $\xi$ the reciprocal of the variance, i.e., $\xi_{x}=\sigma_{x}^{-2}$ and $\xi_{i}=\sigma_{i}^{-2}$.

For each sensor $i$, given the measurement $y_i$, the minimum mean square error (MMSE) estimate $\hat{x}_{i}$ and the corresponding local mean squared error (MSE) $\Sigma_{i}$ can be computed as
\begin{equation}  \label{ch2_eq_local_estimate}
\hat{x}_{i}=\frac{\sigma^{2}_{x}}{\sigma^{2}_{x}+\sigma^{2}_{i}}y_{i},
\end{equation}
\begin{equation} \label{ch2_eq_local_MSE}
\Sigma_{i}=\frac{1}{\sigma^{-2}_{x}+\sigma^{-2}_{i}}=\frac{1}{\xi_{x}+\xi_{i}}.
\end{equation}
We denote $\hat{x}_{i}$ as the local estimate and $\Sigma_{i}$ as the local MSE at the $i$-th sensor since these quantities are obtained based on the information at each sensor.
These local estimates can be fused to obtain the global MMSE estimate $\hat{x}_{g}$
using the relation~\cite{xu2014distributed}
\begin{equation}  \label{eq:fuse}
\Sigma_{g}^{-1}\hat{x}_{g} = \sum_{i=1}^{N}\Sigma_{i}^{-1}\hat{x}_{i},
\end{equation}
where $\Sigma_{g}$ is the global MSE corresponding to $\hat{x}_{g}$ and can be calculated as
\begin{equation}  \label{global MSE}
\Sigma_{g}^{-1} = \sum_{i=1}^{N}\Sigma_{i}^{-1} - (N-1)\sigma_{x}^{-2}=
\xi_{x}+\sum_{i=1}^{N}\xi_{i}.
\end{equation}

\paragraph*{Effort Cost} The variance $\sigma_{i}^{2}$ affects the quality of the measurement at sensor $i$ and is assumed to be a parameter that is under the control of the sensor. In other words, the sensor can put in more {\em effort} and decrease the variance $\sigma_{i}^{2}$ while incurring a higher effort cost. The effort cost may represent usage of battery, time, or some other resource. For simplicity and without loss of generality, we assume that $\xi_{i}$ is the effort level of agent $i$ that incurs an effort cost $c_{i}(\xi_{i})$. We make some weak assumptions on the cost function that describes the effort cost.
\begin{assumption}
The cost function of each sensor $c_{i}(\xi_{i})$ satisfies the following properties:
\begin{itemize}
    \item $c_{i}(\xi_{i}) \geq 0$, i.e., effort cost is non-negative;

    \item ${\dfrac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}}} > 0$, i.e., more effort cost is incurred to obtain a measurement with higher accuracy;
    \item $\xi_{i} \in [0, \xi_{iu}]$ and $c_{i}(\xi_{i}) \in [0, c_{iu}]$.
\end{itemize}
\end{assumption}
Note that when sensor $i$ does not put in any effort, i.e., $\xi_i=0$ and $\sigma^2_i = \infty$, then the effort cost is zero, i.e., $c_i(0)=0$ and its local MSE is equal to the variance of the prior distribution of $X$, i.e., $\Sigma_i = \sigma^2_x$.

\paragraph*{Formulation as a Mechanism Design Problem} We are interested in a formulation in which the estimator and the sensors are all self-interested. The estimator is interested in generating a global estimate with a specified accuracy as measured by the global MSE. To do so, it must incentivize sensors to generate and transmit measurements with sufficiently low local MSE. On the other hand, the sensors do not gain directly from the estimator being able to generate an accurate global estimate. Since they incur effort costs to generate measurements with low local MSE, the estimator must compensate the sensors using a payment mechanism of some sort, for simplicity, we assume the payment is monetary, although money may be thought of as a proxy of some other resource such as battery charging. The problem we consider in this paper is to minimize the payment from the estimator to incentivize self-interested sensors to obtain and report measurements with sufficient accuracy that allow the global MSE to be below a specified level.

We now formulate this interaction as a mechanism design problem. The timeline of the interaction is as shown in Fig~\ref{fig/ch2_communication topology}.
\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_communication_topology.pdf}}
    \caption{Timeline and communication topology of the incentive mechanism design problem.}
    \label{fig/ch2_communication topology}
  \end{center}
\end{figure}
The estimator asks each sensor to report its measurement and local estimate. Note that reporting this pair is equivalent to reporting the local estimate and the local MSE. The strategy sets and the utility functions of each player are given as below.
\begin{itemize}
\item {\em Strategy sets:} Each sensor can choose the level of effort to exert and the values of its measurement and local estimate that it reports. For each sensor $i$, we define its strategy as choosing each element in the following tuple
\begin{equation*}
    s_{i} = (\xi_i, \hat{x}_{ri}, y_{ri}),
\end{equation*}
where $\hat{x}_{ri}$ is the reported local estimate and $y_{ri}$ is the reported measurement.
Denote the set of all feasible $s_i$'s by $S_i$. With a slight abuse of standard notation in game theory, when sensor $i$ adopts strategy $s_{i}$, denote by $s_{-i}=(s_1, s_2, \cdots, s_{i-1}, s_{i+1}, \cdots, s_N)$ the strategy profile of all the other sensors except for sensor $i$. The estimator decides how much payment each sensor $i$ will obtain and how to fuse the reports from the sensors. Since the sensors may misreport their local estimates,  (\ref{eq:fuse}) may not be the optimal way to fuse local reported estimates from the sensors.
Thus, the strategy of the estimator includes the payment functions that map each strategy profile of the sensors to their payments and the fusion rule, i.e.,
\begin{equation*}
    s_{e} = (p_{i}(s_{1},\cdots,s_{N}),\ell(s_{1},\cdots,s_{N})),
\end{equation*}
where $p_{i}(s_{1},\cdots,s_{N})$ denotes the payment made to sensor $i$ which is in general a function of the strategies of all the sensors, and $\ell(s_{1},\cdots,s_{N})$ is the fusion rule used to obtain the global estimate. Note that the payment $p_i(s_1, s_2, \cdots, s_N)$ can also be expressed as $p_{i}(s_{i}, s_{-i})$. Denote the set of all feasible strategies $s_e$'s by $S_e$.
\item {\em Utility Functions:} The expected utility of each sensor $i$ is given by
\begin{equation}
\operatorname{\E}\expectarg{U_{i}} = \operatorname{\E}\expectarg{p_{i}(s_{i}, s_{-i}) - c_{i}(\xi_{i})},
\end{equation}
where the expectation is taken over the uncertainties of the random variable $X$ and measurement noises.
Thus each sensor $i$ optimizes over the effort level and reports to maximize its expected utility,
\begin{equation}
\max_{s_{i} \in S_i} \\ \operatorname{\E}\expectarg{U_i}.
\end{equation}
On the other hand, the estimator is interested in minimizing the expected total payment while obtaining a global estimate with MSE less than a certain threshold. Formally, the optimization problem at the estimator is given as follows
\begin{equation}   \label{optimization of center}
\begin{split}
\min_{s_e \in S_e} \quad &\operatorname{\E}\expectarg*{\sum_{i=1}^{N} p_{i}(s_{i}, s_{-i})}\\
s.t. \quad &\Sigma_{g} \leq \Sigma_{t},\\
& \operatorname{\E}\expectarg{p_{i}(s_{i}, s_{-i})-c_{i}(\xi_{i})}\geq 0, \forall i,\\
& s_i = \argmaxA\operatorname{\E}\expectarg{p_{i}(s_{i}, s_{-i})- c_{i}(\xi_{i})} , \forall i,
\end{split}
\end{equation}
where $\Sigma_{t}$ is the specified threshold on the global MSE. The second constraint above ensures individual rationality, which is necessary for the sensors to participate.
\end{itemize}
In the sequel, we solve problem~(\ref{optimization of center}). Note that problem (\ref{optimization of center}) specifies a game among the sensors since their utilities depend on actions taken by all of them. We will consider the solution of the optimization problem when the behavior of the sensors is specified according to a Nash equilibrium.

\section{Optimization Problem at the Estimator}
\label{sec_ch2:optimization problem at the estimator}
To understand why the problem~(\ref{optimization of center}) is difficult to solve, we note why some intuitive incentive mechanisms may not work.
\begin{itemize}
    \item A payment scheme $p_{i} = c$ for a constant $c$ that does not depend on the reports will lead to each sensor not making any effort and reporting some arbitrary value to the estimator. In economics, this is termed as the problem of \textit{moral hazard}.
    \item A payment scheme that specifies $p_{i}$ as a decreasing function of $\Sigma_{i}$ or $\xi_{i}$ can be considered to incentivize the sensors to exert effort and take accurate measurements. However, it will lead sensors reporting very low local MSE irrespective of the actual effort made. This is termed as the problem of \textit{adverse selection}.
\end{itemize}
In either case, note that the actual measurements $y_i$, local estimates $\hat{x}_{i}$ and the local MSE $\Sigma_{i}$ are all unknown to the estimator, fusing reported local estimates to obtain a global estimate that satisfies the constraint $\Sigma_{g} \leq \Sigma_{t}$ is also a nontrivial problem. The overall optimization problem (\ref{optimization of center}) is even more difficult.

\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_flow_chart.pdf}}
    \caption{The procedure of designing the incentive mechanism with strategic data sources.}
    \label{fig_ch2_flow_chart}
  \end{center}
\end{figure}
Our results are organized as shown in Fig. \ref{fig_ch2_flow_chart}. Specifically, we show that the following technical condition on the effort cost functions plays an important role in the  simplification of the problem:
\begin{equation} \label{constraint of realization}
    -2 \frac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}}
    -
    \frac{\partial^{2} c_{i}(\xi_{i})}{\partial \xi_{i}^{2}}
    (\xi_{x}+{\xi}_{i})
    <0, \enskip \forall \xi_i \enskip \text{and} \enskip i.
\end{equation}

\begin{remark} \label{Remark:quadratic satisfy constraint}
If $c_{i}(\xi_{i})$ is convex over $\xi_{i}$, the constraint (\ref{constraint of realization}) holds for any $\xi_{i}$.
\end{remark}

Depending on whether~(\ref{constraint of realization}) is satisfied or not, we have the following result.
\begin{proposition} \label{proposition of using M1}
Consider the setup of problem~(\ref{optimization of center}). If condition~(\ref{constraint of realization}) is satisfied, the estimator can  specify a payment design such that
\begin{enumerate}
    \item the selected sensors exert the effort levels specified by the estimator;
    \item the selected sensors report truthfully about their measurements and local estimates;
    \item the expected payment to each selected sensor is the effort cost of the sensor for the specified effort level.
\end{enumerate}
\end{proposition}
Note that under these three conditions, the estimator can choose the optimal effort levels from agents that yield minimum payment while meeting the constraints in problem (\ref{optimization of center}) with the fusion rule as shown in (\ref{eq:fuse}). Thus, in this case, the optimization problem (\ref{optimization of center}) reduces to
\begin{equation} \label{eq:P3}
    \begin{split}
    \min_{\phi, \xi}  \quad & \sum_{i=1}^{N}\phi_{i}c_{i}(\xi_{i})\\
    s.t.  \quad &  \frac{1}{\xi_{x}+\sum_{i=1}^{N}\phi_{i}\xi_{i}}
    \leq \Sigma_{t},\\
    &  \phi_{i} \in (0,1),\\
    &  \xi_{i} \in [0,\xi_{iu}],
    \end{split}
\end{equation}
where $\xi = (\xi_1, \xi_2, \cdots, \xi_N)$ and $\phi = (\phi_{1}, \phi_{2}, \cdots, \phi_{N})$. $\phi_{i}$ is an indicator about whether or not the estimator selects agent $i$: $\phi_{i}=1$ represents the case where the estimator selects agent $i$ and $\phi_{i}=0$
represents the case where the estimator does not select agent $i$, which can be implemented by, for instance, setting $p_{i}=0$.

In Section \ref{sec: M1}, we show that if the cost functions satisfy constraint (\ref{constraint of realization}), a mechanism $\mathcal{M}_1$ can be designed that specifies a payment design according to Proposition \ref{proposition of using M1}. Thus, problem (\ref{optimization of center}) can be solved optimally.
Otherwise if the cost functions does not satisfy constraint (\ref{constraint of realization}), we solve the problem in a sub-optimal way through the following proposition. This will be proved through the design of a mechanism $\mathcal{M}_2$ presented in Section \ref{sub-optimal}.
\begin{proposition}
Consider the setup of problem~(\ref{optimization of center}). If the condition~(\ref{constraint of realization}) is not satisfied, the estimator can  specify a payment design such that
\begin{enumerate}
    \item the selected sensors would exert their maximum effort levels;
    \item the selected sensors report truthfully about their measurements and local estimates;
    \item the expected payment to each selected sensor $i$ is the effort cost of the sensor for its maximum effort level.
\end{enumerate}
\end{proposition}
Note that under these three conditions, the estimator cannot choose the optimal effort levels from sensors as in Proposition \ref{proposition of using M1}.
But the estimator can still select a subset of all agents given that the selected agents will exert their maximum efforts and the estimator will pay the corresponding costs incurred at maximum effort levels.
Formally, problem (\ref{optimization of center}) is transformed to the following well-defined binary knapsack problem (KP):
\begin{equation} \label{eq:P7}
\begin{split}
\min_{{\phi}}  \quad &  \sum_{i=1}^{N}\phi_{i}c_{i}(\xi_{iu})\\
s.t. \quad &  \frac{1}{\xi_{x}+\sum_{i=1}^{N}\phi_{i}\xi_{iu}}
    \leq \Sigma_{t},\\
& \phi_{i} \in (0,1).
\end{split}
\end{equation}
This problem is NP-hard but can be solved exactly in pseudo-polynomial time through dynamic programming algorithms \cite{kellerer2003knapsack}.

\section{Optimal Mechanism When (\ref{constraint of realization}) Holds}       \label{sec: M1}

In this section, we address the cases where (\ref{constraint of realization}) holds. We first simplify the optimization problem (\ref{eq:P3}) and present two interesting special cases. Then we present an optimal mechanism to prove Proposition \ref{proposition of using M1}.

\subsection{Solving Problem (\ref{eq:P3})}
(\ref{eq:P3}) can be rewritten as
\begin{equation} \label{eq:rewrite P3}
    \begin{split}
    \min_{{\phi}, {\xi}} \quad & \sum_{i=1}^{N}\phi_{i}c_{i}(\xi_{i})\\
     s.t. \quad &
    \sum_{i=1}^{N}\phi_{i}\xi_{i}
    \geq \Sigma_{t}^{-1}-\xi_{x},\\
    &  \phi_{i} \in (0,1), \\
    &  \xi_{i} \in [0,\xi_{iu}].
    \end{split}
\end{equation}
This is a mixed-integer nonlinear programming problem and specifically known as the general knapsack problem (GKP) with variable coefficients \cite{suzuki1978generalized} \cite{holmberg1996solving}, which is difficult to solve in general. However, since $c_i(0)=0, ~ \forall i$, we can transform problem (\ref{eq:rewrite P3}) to problem (\ref{eq:P4}) according to the following result.

\begin{lemma}
Problem
(\ref{eq:rewrite P3}) can be solved by constructing solution of the following optimization problem
\begin{equation} \label{eq:P4}
    \begin{split}
    \min_{{\xi}}  \quad &  \sum_{i=1}^{N}c_{i}(\xi_{i})\\
    s.t.  \quad &
    \sum_{i=1}^{N}\xi_{i}
    \geq \Sigma_{t}^{-1}-\xi_{x},\\
    & \xi_{i} \in [0,\xi_{iu}].
    \end{split}
\end{equation}
\end{lemma}

\begin{IEEEproof}
Suppose that the minimum of ($\ref{eq:rewrite P3}$), denoted by $O_{1}$, is achieved at $({\xi}^{O1}, {\phi}^{O1})$ and the minimum of ($\ref{eq:P4}$), denoted by $O_{2}$, is achieved at ${\xi}^{O2}$.  We have $O_{1}\leq O_{2}$ because ($\ref{eq:P4}$) is a special case of ($\ref{eq:rewrite P3}$) by fixing all $\phi_{i}=1$. On the other hand, $O_{1}\geq O_{2}$, because any value achieved in ($\ref{eq:rewrite P3}$) can be achieved in ($\ref{eq:P4}$) by constructing ${\xi}^{O2}$ from $({\xi}^{O1}, {\phi}^{O1})$ as
\begin{equation}
    \xi^{O2}_{i} =
    \begin{cases}
      \xi^{O1}_{i}, & \text{for } \phi^{O1}_{i}=1, \\
      0, & \text{for } \phi^{O1}_{i}=0. \\
    \end{cases}
\end{equation}
Thus, $O_{1}=O_{2}$. In general, it is easier to solve ($\ref{eq:P4}$) first and then construct $({\xi}^{O1}, {\phi}^{O1})$ from ${\xi}^{O2}$ by setting
\begin{equation}
    (\xi^{O1}_{i}, \phi^{O1}_{i}) =
    \begin{cases}
      (\xi^{O2}_{i}, 1), & \text{for } \xi^{O2}_{i}\neq 0, \\
      (r, 0), & \text{for } \xi^{O2}_{i} = 0, \\
    \end{cases}
\end{equation}
where $r$ can be any number since $\phi^{O1}_{i}=0$.
\end{IEEEproof}

We now present two interesting special cases.
\subsubsection{Special Case: Continuous Quadratic Cost Function}
A quadratic effort cost $c_{i}(\xi_{i}) = l\xi_{i}^{2}$ is quite popular, e.g., in control theory. In this case,
then the optimization problem (\ref{eq:P4}) becomes,
\begin{equation} \label{eq:P6}
    \begin{split}
    \min_{{\xi}}  \quad & \sum_{i=1}^{N}l\xi^{2}_{i}\\
    s.t. \quad  &
    \sum_{i=1}^{N}\xi_{i}
    \geq \Sigma_{t}^{-1}-\xi_{x},\\
    &  \xi_{i} \in [0,\xi_{iu}],
    \end{split}
\end{equation}
which is a standard Quadratic Programming (QP) problem.
According to Cauchy-Schwarz inequality, the optimal solution of (\ref{eq:P6}) is given by
\begin{equation} \label{eq:P6 solution}
    \tilde{\xi}_{1} = \tilde{\xi}_{2} = ... = \tilde{\xi}_{N} = \frac{\Sigma_{t}^{-1}-\xi_{x}}{N},
\end{equation}
assuming for simplicity that $\frac{\Sigma_{t}^{-1}-\xi_{x}}{N}\leq\xi_{iu}$.

As stated in Remark \ref{Remark:quadratic satisfy constraint}, since the cost function is convex, the constraint (\ref{constraint of realization}) holds for any possible $\xi_{i}$. Under our optimal mechanism (presented in Section \ref{subsection: optimal mechanism}), there is a Nash equilibrium where all agents select the effort level as $\frac{\Sigma_{t}^{-1}-\xi_{x}}{N}$
and
report truthfully about their local estimates and their measurements. Meanwhile, the minimum expected total payment that can be achieved to ensure global MSE to be no greater than $\Sigma_{t}$ is $\dfrac{l(\Sigma_{t}^{-1}-\xi_{x})^{2}}{N}$.

\subsubsection{Special Case: Discrete Linear Cost Function} \label{special_case_discrete}
A natural model is that each agent can increase the accuracy of its local estimate by taking more measurements and estimating based upon the sample mean. For instance, if agent $i$ takes $\eta_{i}$ number of measurements of the following
\begin{equation}
\begin{split}
&y_{i}(1) = x + v_{i}(1), \\
&y_{i}(2) = x + v_{i}(2), \\
&\vdots \\
&y_{i}(\eta_{i}) = x + v_{i}(\eta_{i}),
\end{split}
\end{equation}
where $v_{i}(k)$ follows i.i.d. Gaussian distribution $\mathcal{N} (0,\sigma^{2}_{io})$. Denote the effort cost of taking each measurement by a cost of $c_{io}$.
Then the noise level, effort level and effort cost of the sample mean $\bar{y}_{i} =  x + \bar{v}_{i}$ averaged from taking $\eta_{i}$ measurements are respectively given by
\begin{equation}
\begin{split}
    &\sigma^{2}_{i} = \frac{\sigma^{2}_{io}}{\eta_{i}}, \\
    &\xi_{i}=\eta_{i}\sigma_{io}^{-2},\\
    &c_{i}(\xi_{i}) = \eta_{i}c_{io}=\sigma^{2}_{io}c_{io}\xi_{i}.
\end{split}
\end{equation}
Therefore in this case, the effort cost function a linear function and the effort level depends on the number of measurements taken.
We denote the corresponding maximum number of measurements by $\eta_{i}^{m}$.
The optimization problem (\ref{eq:P4}) becomes,
\begin{equation} \label{eq:P5}
    \begin{split}
    \min_{{\eta}}  \quad &  \sum_{i=1}^{N}\eta_{i}c_{io}\\
    s.t.  \quad  &
    \sum_{i=1}^{N}\eta_{i}\sigma^{-2}_{io}
    \geq \Sigma_{t}^{-1}-\xi_{x},\\
    & \eta_{i} \in (0, 1, ..., \eta^{m}_{i}),
    \end{split}
\end{equation}
which is known as the Bounded Knapsack Problem (BKP). It is NP-hard but it can be solved exactly in pseudo-polynomial time through dynamic programming algorithms \cite{kellerer2003knapsack}\cite{pisinger2000minimal}. Denote by $\tilde{{\eta}} = (\tilde{\eta}_{1}, \tilde{\eta}_{2}, ..., \tilde{\eta}_{N})$ the optimal solution of (\ref{eq:P5}).

Under our optimal mechanism, there is a Nash equilibrium where each agent takes $\tilde{\eta}_{i}$ number of measurements and report truthfully about the local estimate and measurement. Meanwhile, the minimum expected total payment that can be achieved to ensure global MSE to be no greater than $\Sigma_{t}$ is  $\sum_{i=1}^{N}\tilde{\eta}_{i}c_{io}$.

\subsection{Optimal Mechanism} \label{subsection: optimal mechanism}
Denoting the optimal solution of problem (\ref{eq:P4}) by $\tilde{{\xi}} = (\tilde{\xi}_1, \tilde{\xi}_2, ..., \tilde{\xi}_N)$, we now present mechanism $\mathcal{M}_{1}$ that fulfills Proposition \ref{proposition of using M1}, i.e., under Mechanism $\mathcal{M}_{1}$, all the agents exerting the desired effort levels and reporting their measurements and local estimates truthfully is a Nash equilibrium. In addition, the expected payment to each agent $i$ is the effort cost of the agent for the specified effort level.

In our proposed incentive mechanism $\mathcal{M}_{1}$, agents are asked to report two items $(\hat{x}_{ri}, y_{ri})$, where $\hat{x}_{ri}$ is the reported local estimate and $y_{ri}$ is the reported measurement. Note that $\hat{x}_{ri} \neq \hat{x}_{i}$ and $y_{ri} \neq y_{i}$ in general since agents may falsify their reports to maximize their utilities.
The payment function is given by
\begin{equation} \label{p2}
p_i(\hat{x}_{ri},y_{rj})=
  \gamma_i-\beta_i(\hat{x}_{ri}-y_{rj})^2,
\end{equation}
where $y_{rj}$ is the reported measurement from another agent $j \neq i$.
As before, agents are interested in maximizing their expected utilities,
\begin{equation}
s^{*}_{i} = \argmaxA_{s_{i} \in S_i}\E\left[U_{i}\right]=\argmaxA_{s_{i} \in S_i}\E\left[p_i(\hat{x}_{ri},y_{rj})-c_{i}(\xi_{i})\right].
\end{equation}

Now, we state our results about the optimal mechanism $\mathcal{M}_{1}$.
\begin{theorem} \label{Theorem 1}
Consider the problem (\ref{optimization of center}) when (\ref{constraint of realization}) holds. Let (\ref{p2}) be the payment function to each sensor $i$ with
\begin{equation} \label{beta}
\beta_{i}=\evalat[\bigg]{\frac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}}}{\xi_{i}=\tilde{\xi}_i}(\xi_{x}+\tilde{\xi}_{i})^2,
\end{equation}
and
\begin{equation} \label{gamma}
    \gamma_{i}=\beta_{i}\left(\frac{1}{\xi_{x}+\tilde{\xi}_{i}}+\tilde{\xi}_{j}^{-1}\right)+c_{i}(\tilde{\xi}_{i}).
\end{equation}
The strategy profile
$s^{*}=(s_{1}^{*},s_{2}^{*},...,s_{N}^{*})$ with
\begin{equation}
s^{*}_{i} =
  (\xi_{i}=\tilde{\xi}_{i}, \hat{x}_{ri}=\hat{x}_{i}, y_{ri}=y_{i})
\end{equation}
is a Nash equilibrium of the mechanism design problem (\ref{optimization of center}).
In addition, the expected payment to each agent is the effort cost, i.e., $\operatorname{\E}\expectarg{p_{i}}=c_{i}(\tilde{\xi}_{i})$.
\end{theorem}

\begin{IEEEproof}
See Appendix \ref{apdx:ch2_th1}.
\end{IEEEproof}

Intuitively, the payment is designed as a function of the difference between the reports from the agents, which motivates each agent to estimate the information of another agent based on her own information. The accuracy of the agent's estimate, and the corresponding expected payment, will depend on how much effort is exerted when the agent obtains her own information. Therefore, $\beta_{i}$ can be designed as in (\ref{beta}) so that the effort level that the estimator wishes each agent to exert (i.e., $\tilde{\xi_{i}}$) will turn out to be exactly the optimal choice for agent $i$ when all the other agents exert the effort levels desired by the estimator, i.e., $\xi_{j} = \tilde{\xi}_j, \forall j\neq i$. Further, $\gamma_{i}$ can be designed as in (\ref{gamma}) so that the expected payment is small but enough to cover the effort cost.

It is worth remarking that the estimation of the reports among the agents is in a \textit{game} setting, which means the desired strategy profile $s^{*}$ from all the agents is obtained in a Nash equilibrium sense.
Further, if there exists an \textit{`honest'} agent who reports its measurement and effort level truthfully, it is no longer needed to ask the strategic agents to report their measurements. In this case, the desired strategy profile $s^{*}$ is an {\it equilibrium in strictly dominant strategies} in which every $s_i^*$ is the strictly dominant strategy for agent $i$. We present the result in the following corollary.

\begin{corollary} \label{corollary:unique NE}
Consider the setting of Theorem \ref{Theorem 1} with an honest agent $h$ who reports its measurement and effort level truthfully, i.e., $y_{rh}=y_{h}=x+v_h$, where $v_h\sim\mathcal{N} (0, \xi_{h}^{-1})$. Let the payment function to each sensor $i$ be specified by (\ref{p2}), (\ref{beta}) and (\ref{gamma}) after replacing $y_{rj}$ and $\tilde{\xi}_j$ with $y_{rh}$ and ${\xi}_h$ respectively.
The strategy profile $s^{*'}=(s_{1}^{*'},s_{2}^{*'},...,s_{N}^{*'})$ with
\begin{equation}
s^{*'}_{i} =
  (\xi_{i}=\tilde{\xi}_{i}, \hat{x}_{ri}=\hat{x}_{i})
\end{equation}
is the unique equilibrium in strictly dominant strategies.
\end{corollary}
\begin{IEEEproof}
The proof is similar to the proof of Theorem \ref{Theorem 1}. The only difference in this case is that the strategic agents now estimate the measurement from the honest agent instead of estimating the measurement from another strategic agent. The utility of each strategic agent will no longer depend on other strategic agents, thus, the strategy in the Nash equilibrium is the strictly dominant strategy for each agent and the Nash equilibrium is the unique equilibrium.
\end{IEEEproof}

\section{A Sub-optimal Mechanism When (\ref{constraint of realization}) Does Not Hold} \label{sub-optimal}
In this section, we provide a feasibility-guaranteed sub-optimal mechanism $\mathcal{M}_{2}$ for the cases where the constraint (\ref{constraint of realization}) can not be satisfied. $\mathcal{M}_{2}$ achieves truthful reporting and elicits maximum effort from the selected agents with expected payment to selected agent $i$ being the effort cost $c_{i}(\xi_{iu})$ in a Nash equilibrium sense.

Denote the optimal solution to (\ref{eq:P7}) as $\tilde{{\phi}} = (\tilde{\phi}_{1}, \tilde{\phi}_{2}, ..., \tilde{\phi}_{N})$. We present the incentive mechanism $\mathcal{M}_{2}$ that only selects the agents for which $\tilde{\phi}_{i}=1$ and elicits their maximum efforts.

\begin{theorem} \label{Theorem 2}
Consider the problem (\ref{optimization of center}) when (\ref{constraint of realization}) does not hold. Let the payment to each sensor $i$ with $\tilde{\phi}_{i}=1$ be determined by comparing its reported local estimate with the reported measurement from another agent $j$ with $\tilde{\phi}_{j}=1$, i.e.,
\begin{equation} \label{payment theorem 2}
p_i(\hat{x}_{ri},y_{rj})=
\begin{cases}
  \gamma_i-\beta_i(\hat{x}_{ri}-y_{rj})^2,
  & \text{for } \tilde{\phi}_{i}=1 \\
  0,
  & \text{for } \tilde{\phi}_{i}=0
\end{cases}
\end{equation}
with
\begin{equation} \label{beta theorem 2}
\beta_{i}>\max_{\xi_{i}\in [0,\xi_{iu}]}{\frac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}}}(\xi_{x}+\xi_{i})^{2}
\end{equation}
and
\begin{equation} \label{gamma theorem 2}
    \gamma_{i}=\beta_{i}(\frac{1}{\xi_{x}+\xi_{iu}}+\xi_{ju}^{-1})-c_{i}(\xi_{iu}).
\end{equation}
The strategy profile ${s}^{*}=(s_{1}^{*}, s_{2}^{*}, ..., s_{N}^{*})$ with
\begin{equation}
s^{*}_{i} =
\begin{cases}
  (\xi_{i}=\xi_{iu}, \hat{x}_{ri}=\hat{x}_{i}, y_{ri}=y_{i}),
  & \text{for } \tilde{\phi}_{i}=1 \\
  (\xi_{i}=0, \hat{x}_{ri}=\hat{x}_{i}, y_{ri}=y_{i}),
  & \text{for }  \tilde{\phi}_{i}=0
\end{cases}
\end{equation}
is a Nash equilibrium of the mechanism design problem (\ref{optimization of center}).
In addition, the expected payment to each agent is the effort cost for its maximum effort level, i.e., $\operatorname{\E}\expectarg{p_{i}}=c_{i}(\xi_{iu})$.
\end{theorem}

\begin{IEEEproof}
See Appendix \ref{apdx:ch2_th2}.
\end{IEEEproof}

Similarly, if there exists an honest agent who reports its measurement and effort level truthfully, it is no longer needed to ask the strategic agents to report their measurements. In this case, the desired strategy profile is the unique equilibrium with strictly dominant strategies.
\begin{corollary}
Consider the setting in Theorem \ref{Theorem 2} with an honest agent $h$ who reports its measurement and effort level truthfully, i.e., $y_{rh}=y_{h}=x+v_h$, where $v_h\sim\mathcal{N} (0, \xi_{h}^{-1})$. Let the payment function be specified by (\ref{payment theorem 2}), (\ref{beta theorem 2}) and (\ref{gamma theorem 2}) after replacing $y_{rj}$ and $\tilde{\xi}_j$ with $y_{rh}$ and ${\xi}_h$ respectively.
The strategy profile ${s}^{*'}=(s_{1}^{*'},s_{2}^{*'},...,s_{N}^{*'})$ with
\begin{equation}
s^{*'}_{i} =
\begin{cases}
  (\xi_{i}=\xi_{iu}, \hat{x}_{ri}=\hat{x}_{i}),
  & \text{for } \tilde{\phi}_{i}=1 \\
  (\xi_{i}=0, \hat{x}_{ri}=\hat{x}_{i}),
  & \text{for }  \tilde{\phi}_{i}=0
\end{cases}
\end{equation}
is the unique equilibrium in strictly dominant strategies.
\end{corollary}

The proof is omitted since it is similar to that of Corollary \ref{corollary:unique NE}.

\section{Simulation Experiments}

\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_distribution_of_parameters.pdf}}
    \caption{Scatter plot and histograms of $\sigma^{2}_{il}$ and $c_{iu}$.}
    \label{fig:distribution of parameters}
  \end{center}
\end{figure}

\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_effect_of_eta_m.pdf}}
    \caption{Minimum payments with $N=100$, and $\eta^{m}=2$, $\eta^{m}=4$, $\eta^{m}=100$ respectively.}
    \label{fig:different eta_m}
  \end{center}
\end{figure}

\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_compare_N.pdf}}
    \caption{Minimum payments with $\eta^{m}=2$, and $N=100$, $N=500$, $N=1000$ respectively.}
    \label{fig: comparison_N}
  \end{center}
\end{figure}

\begin{figure}[tpb]
  \begin{center}
    \centerline{\includegraphics[width=\linewidth]{fig/fig_compare_suboptimal.pdf}}
    \caption{Comparison of minimum payments in the sub-optimal case and the optimal case.} \label{fig:compare_suboptimal}
  \end{center}
\end{figure}

In this section, we demonstrate our mechanisms with simulation experiments. We first consider the setting of the problem in Section \ref{special_case_discrete} and investigate the minimum payment at different threshold $\Sigma_{t}$. $N=100$ agents are simulated and the variance of the prior distribution is selected as $\sigma^{2}_{x}=1000$. Further, fixing the minimum variance of each agent $\sigma^{2}_{il}$ and its corresponding maximum effort $c_{iu}$ allows us to study the effect $\eta^{m}_{i}$, which can be interpreted as the quantization level of the effort cost of each agent.
Without loss of generosity, we set $\eta^{m}_{i}=\eta^{m}$ for all $i$.
To make the agents heterogeneous on their highest accuracies, we randomly generate $\sigma^{-2}_{il}\sim\mathcal{U}[0.0001, 0.01]$, which is selected such that roughly a half of $\sigma^{2}_{il}$ fall in the range $[100, 200]$ and the other half of $\sigma^{2}_{il}$ fall in the range $[200, 10000]$. $c_{iu}$ is randomly generated from a mixture Gaussian distribution $c_{iu}\sim.5\mathcal{N}(50, 100)+.5\mathcal{N}(100, 100)$.
The scatter plot and histograms of these two parameters are shown in Fig. \ref{fig:distribution of parameters}.

The minimum payments with $\eta^{m}=2$, $\eta^{m}=4$, and $\eta^{m}=100$ at different threshold $\Sigma_{t}$ are shown in Fig. \ref{fig:different eta_m}. In general, greater $\eta^{m}$ yields smaller payment.
On the other hand, we also study the effect of $N$. We use the same distributions to generate $\sigma^{2}_{il}$ and $c_{iu}$. $\eta^{m}$ is fixed as $\eta^{m}=2$. As shown in Fig. \ref{fig: comparison_N}, more agents being available generally yields smaller payments.
Lastly, we compare our sub-optimal case with the optimal case considered in Section \ref{special_case_discrete} under the same setting. Recall that in the sub-optimal solution, our mechanism $\mathcal{M}_{2}$ yields all selected agents exerting maximum effort.
Using the same simulated parameters, the optimization problem (\ref{eq:P7}) in the sub-optimal case can be viewed a problem similar to (\ref{eq:P5}), but the decision variables are limited to be either $0$ or $\eta^{m}$.
In Fig. \ref{fig:compare_suboptimal}, we show the comparison of minimum payments between the sub-optimal case and the optimal case at different $\Sigma_{t}$ with $\eta^{m}=2$ and $N=100$.

\section{Summary}
In this paper, we designed incentive mechanisms for a static estimation problem where the data sources are strategic agents whose measurements and accuracies are both unknown to the estimator. The objective of the incentive mechanism is to minimize the expected total payment made to the agents with a guaranteed quality of global estimate. We formulate the problem in a very general setting without assuming any specific format of the agents' cost functions. Instead, we designed an optimal incentive mechanism for the cases where the cost functions satisfy certain property and provided a sub-optimal incentive mechanism for the other cases. We also demonstrated our mechanisms by two special cases with continuous quadratic cost function and discrete linear cost function. Both in the special case with the discrete linear cost function and in the sub-optimal case, the optimization problem were transformed to knapsack problems, which can be solved in pseudo-polynomial time by dynamic programming.
Future work will include extending the results to dynamic estimation problems.

\appendices
\section{Proof of Theorem \ref{Theorem 1}} \label{apdx:ch2_th1}

It suffices to prove that if the strategy profile of all the other agents follow the stated equilibrium, denoted as ${s}_{-i}={s}_{-i}^{*}$, agent $i$ does not have another strategy which yields greater expected utility than ${s}_{i}^{*}$. Mathematically, when ${s}_{-i}={s}_{-i}^{*}$, the optimal strategy for agent $i$ is given by
\begin{equation}
\begin{split}
  (\xi^{*}_{i}, \hat{x}^{*}_{ri}, y^{*}_{ri})
  &=  \argmaxA \operatorname{\E}\expectarg{U_{i}} \\
  &=  \argmaxA \operatorname{\E}\expectarg*{\gamma_i-\beta_i(\hat{x}_{ri}-y_{j})^2 - c_{i}(\xi_{i})|\hat{x}_{i}}. \\
\end{split}
\end{equation}

First notice that the estimator can not verify the reports $\hat{x}^{*}_{ri}$ and $y^{*}_{ri}$ jointly, since
\begin{equation}
    \hat{x}_{i} = \frac{\xi_{x}^{-1}}{\xi_{x}^{-1}+\xi_{i}^{-1}}y_{i},
\end{equation}
and $\xi_{i}$ is unknown to the estimator. Therefore, the agent can optimize $\hat{x}^{*}_{ri}$ and $y^{*}_{ri}$ independently. However, the expected utility is indifferent to $y_{ri}$, hence no other value can yield greater utility than $y^{*}_{ri}=y_{i}$.

Next, we prove that for any given $\xi_{i}$, the optimal $\hat{x}^{*}_{ri}=\hat{x}_{i}$. Since $\beta_{i}$ and $\gamma_{i}$ are positive constants and $\xi_{i}$ is fixed,
\begin{equation}
\begin{split}
    \hat{x}^{*}_{ri} &= \argminA \operatorname{\E}\expectarg*{(\hat{x}_{ri}-y_{j})^{2}|\hat{x}_{i}} \\
    &= \operatorname{\E}\expectarg*{y_{j}|\hat{x}_{i}} \\
    &= C_{y_{j}\hat{x}_{i}}C_{\hat{x}_{i}\hat{x}_{i}}^{-1}\hat{x}_{i}, \\
\end{split}
\end{equation}
where $C_{y_{j}\hat{x}_{i}}$ and $C_{\hat{x}_{i}\hat{x}_{i}}$ are computed as
\begin{equation}
     C_{y_{j}\hat{x}_{i}}
     = \operatorname{\E}\expectarg*{(x+v_{j})\frac{\xi_{x}^{-1}}{\xi_{x}^{-1}+\xi_{i}^{-1}}(x+v_{i})}
     = \frac{\xi_{x}^{-2}}{\xi_{x}^{-1}+\xi_{i}^{-1}},
\end{equation}
and
\begin{equation}
     C_{\hat{x}_{i}\hat{x}_{i}}
     = \operatorname{\E}\expectarg*{\left(\frac{\xi_{x}^{-1}}{\xi_{x}^{-1}+\xi_{i}^{-1}}(x+v_{i})\right)^{2}}
     = \frac{\xi_{x}^{-2}}{\xi_{x}^{-1}+\xi_{i}^{-1}},
\end{equation}
since the noises and $x$ are all independent to each other.

Now, we show that with $\beta_{i}$ given by (\ref{beta}) and if the constraint (\ref{constraint of realization}) is satisfied at $\tilde{\xi}_{i}$, then the optimal $\xi^{*}_{i}=\tilde{\xi}_{i}$. For any given $\xi_{i}$, the expected utility with the optimal choice of $\hat{x}_{ri}$ is given by
\begin{equation}
    \operatorname{\E}\expectarg{U_{i}(\xi_{i})} = \gamma_{i} - \beta_{i}\operatorname{\E}\expectarg{(\hat{x}_{i}-y_{j})^{2}} - c_{i}(\xi_{i}),
\end{equation}
where
\begin{equation}
\begin{split}
    \operatorname{\E}\expectarg{(\hat{x}_{i}-y_{j})^{2}}
    &= \operatorname{\E}\expectarg*{\left(\frac{\xi_{x}^{-1}}{\xi_{x}^{-1}+\xi_{i}^{-1}}(x+v_{i})-(x+v_{j})\right)^{2}} \\
    &= \frac{\xi_{i}^{-1}\xi_{x}^{-1}}{\xi_{i}^{-1}+\xi_{x}^{-1}} + \xi_{j}^{-1} \\
    & = \frac{1}{\xi_{i}+\xi_{x}} + \tilde{\xi}_{j}^{-1}.
\end{split}
\end{equation}
Thus, setting the first derivative of $\operatorname{\E}\expectarg{U_{i}(\xi_{i})}$ over $\xi_{i}$ to zero yields the unique maximum $\xi^{*}_{i}$ if $\operatorname{\E}\expectarg{U_{i}(\xi_{i})}$ is concave:
\begin{equation} \label{eq:derivative over y_i}
    \frac{\partial\operatorname{\E}\expectarg{U_{i}(\xi_{i})}}{\partial \xi_{i}}
    = \frac{\beta_{i}}{(\xi_{i}+\xi_{x})^{2}}
    -
    \frac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}} = 0,
\end{equation}
where the solution is $\tilde{\xi}_{i}$ if $\beta_{i}$ is given by
\begin{equation}
\beta_{i}=\evalat[\bigg]{\frac{\partial c_{i}(\xi_{i})}{\partial \xi_{i}}}{\xi_{i}=\tilde{\xi}_{i}}(\tilde{\xi}_{i}+\xi_{x})^2.
\end{equation}
To guarantee the concavity of $\operatorname{\E}\expectarg{U_{i}(\xi_{i})}$,
\begin{equation}
    \frac{\partial^{2}\operatorname{\E}\expectarg{U_{i}(\xi_{i})}}{\partial \xi_{i}^{2}}
    = \frac{-2\beta_{i}}{(\xi_{i}+\xi_{x})^{3}}
    -
    \frac{\partial^{2} c_{i}(\xi_{i})}{\partial \xi_{i}^{2}}<0,
\end{equation}
which implies the constraint (\ref{constraint of realization}) should be satisfied for any $\xi_{i}$.

Lastly, the maximum expected utility of agent $i$ is given by
\begin{equation}
    \operatorname{\E}\expectarg{\tilde{U}_{i}} = \gamma_{i} - \beta_{i}\left(\frac{1}{\xi_{x}+\tilde{\xi}_{i}}+\tilde{\xi}_{j}^{-1}\right) - c_{i}(\tilde{\xi}_{i}).
\end{equation}
Therefore, $\gamma_{i}$ given by (\ref{gamma}) is designed to satisfy individual rationality. Meanwhile, the expected payment is as small as the effort cost $c_{i}(\tilde{\xi}_{i})$.

\section{Proof of Theorem \ref{Theorem 2}} \label{apdx:ch2_th2}
The proof is similar to the proof of Theorem \ref{Theorem 1}, except for that $\beta_{i}$ is designed to ensure that the derivative over $\xi_{i}$ as shown in (\ref{eq:derivative over y_i}) is always positive so that the selected agents would prefer to exert maximum effort.

\bibliographystyle{IEEEtran}

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Krystal Joy Brown
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Home News Renfrewshire Council gets to grips with fuel poverty in IoT pilot
Renfrewshire Council gets to grips with fuel poverty in IoT pilot
Renfrewshire Council has seen a 600 percent return on investment from a project that uses IoT sensors to tackle health and heating problems experienced by tenants in social housing.
A pilot project carried out by local authority Renfrewshire Council in Scotland has helped to tackled fuel poverty in social housing using IoT sensors, according to a report from Scottish Housing News.
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An opening paragraph that hooks 'em in so they read to the end.
How important your font is. Size AND colour.
What you can do to make your Blog easier to read.
More tips to keep eyeballs on your Blog.
How to end your Blog that inspires action from your reader.
Prefer to read. Here's the transcript.
And today we're going be talking about something very sad.
Your unloved dusty blogs that are sitting on your website that haven't seen the light of day in a long time. No one's finding them, no one's reading them, no one's leaving comments on them. It's just too sad to bear.
It's time to dust them down and you know give them a little bit of loving so that you can get them out there.
The first thing I'd like you to do is go to your website and find a blog that hasn't had much loving for a very very long time.
Open it up and read it. Then what I'd like you to do is to read it again.
This time, you're going to step into the shoes of your ideal client.
Read it as if you are them.
And get more eyeballs on it.
Is it what your ideal client, your audience, are will type into a search engine like Google?
Is it something that they actually want to know about?
Is it something that they're actually struggling with and need help with?
Does it tell them that they're in the right place?
Does it give them hope that, yes, I have found what it is I am looking for.
Do you make it clear what your Blog is about and the benefits they'll get by reading it?
There's a few things here that come under the 'easy to read' category.
Let's start with the font size.
What works well is a white or a pale background with black type. Put your type in black.
And also as well, the spacing of it.
Are the sentences close together?
Can they easily keep their head in one place and read your fabulous content? (your reader doesn't want to risk a neck injury if your Blog takes up the whole width of their PC screen).
Are you using lists, bullet points, images, subtitles and titles to break up your Blog.
Looking for what we want.
Break up your wall of text with images?
Images can be the most helpful things that you can put in a blog for your reader.
Keep it easy for them to read – think about the words.
Are you writing for someone who doesn't know very much compared to yourself, or are you writing for your peers?
Now what I mean by this is keep it really simple for them.
They want to get an answer to what is bothering them, like that.
They don't want to have to spend ages working out what the heck it is you're talking about.
Use simple language so that they can go away and go, yeah that was easy to understand.
At the end of your blog have a Call to Action?
Now it doesn't have to be complicated or convoluted. It can be simple. We like simple.
Enjoy this blog? Got any questions? I'd love to hear your comments below.
Include a link to one of your freebies or a report, something that they can download.
It could be to arrange a call with you.
Or a link that takes them onto another blog that's on a similar topic.
A quick summary on how you can dust down those dusty old blogs of yours, give them a bit of fresh air, a bit of loving, and get them back out into the world in a way that's gonna be more easy for your audience to find them and to read them and enjoy them and in a way that gets you more leads and clients.
And if you've got any questions, pop them below. I'm always happy to answer them for you.
So off you pop, get dusting down those blogs of yours.


Kawasaki Vulcan 750 Forum : Kawasaki VN750 Forums > Forums Administration > Suggestions / Tech Support / Site Help > Upcoming Changes to the Motorcycle.com App.
We're here to announce some changes to the Motorcycle.com app. Firstly, we regret to say that Tapatalk is pulling its support of our branded Motorcycle.com apps, both the free and paid versions. As a result no further updates for them will be released and they will no longer be available for download as of December 31st 2015. Sites that request it can be added to the main version of Tapatalk, but all app support will need to go through Tapatalk itself.
However, we can now announce that we have developed a brand new Motorcycle App developed with a different company that we will be working very closely with. This will allow for issues to be addressed in a much timelier manner. The new app can be found here!


Southport Community Building, 223 E. Bay Street, Southport.
The special meeting is being called by the Chair of the Board pursuant to Article III, Section 2 and Article V, Section 1 of the Bylaws of the Authority.
Regular Meetings. From time to time, the Board shall establish the time and place for its regular meetings. Such regular meetings shall be held no less frequently than once per quarter.
meeting as is possible so that your request can be considered.
The brick walkway in front of the Fort Johnston Garrison House, heading toward the public bathrooms and the parking area on Davis Street, has been rerouted and repaired. Memorial bricks sold by the Southport Historical Society line the walkway. The concrete sidewalk now continues from the parking area to the bathrooms. Work was done in partnership with the Public Works Department and the Southport Historical Society.
A special thank you to Mark, David, Addison, George and Jody for a wonderful job.
The last brick is in!
The Board of Aldermen voted for their top five (5) priorities and members volunteered to champion specific items and to provide monthly updates at Board meetings.


text:
WP1 - Labour Market barriers and enablers
WP2 - Legal Barriers and Enablers
WP3 - Policy Barriers and Enablers
WP4 - Civil Society Barriers and Enablers
WP5 - Social Partners Barriers and Enablers
WP6 - Individual Barriers and Enablers
WP reports and results
Policy Learning Platform
Priority Action Networks
Legacy Networks
Charles University in Prague, the Czech Republic, was founded in 1348, making it one of the oldest universities in the world. It is the largest and most renowned Czech university, and is also the best-rated Czech university according to international rankings.
The university contains 17 faculties, one of which is the Faculty of Social Science, where the Prague research team is based. The Faculty of Social Sciences was founded in 1990 and its main goal is to develop teaching and research in those aspects of the social sciences, which relate to the public administration, or to the economic and cultural life of society. The Faculty of Social Science (FSV) comprises of five Institutes, relevant to the present project is the Institute of Sociological Studies (ISS).
Since September 2016, the ISS hosts Jean Monnet Chair project "European Political Sociology" and since 2017, the ISS is involved in the project "Countering Islamophobia Through the Development of Best Practice in the use of Counter-Narratives in EU Member States" (JUST/2015/ACTIONGRANTS). Moreover, it hosts 8 research projects funded by the Czech national funding agency.
https://www.cuni.cz/UKEN-1.html
Dino Numerato
Dr Dino Numerato is Assistant Professor and Head of Department of Sociology at the Faculty of Social Sciences, Charles University (Prague, Czech Republic). He holds a PhD in Sociology from Masaryk University (Brno, Czech Republic). Prior to joining the Faculty of Social Sciences at Charles University, he worked as a research fellow at Loughborough University (UK) and Bocconi University (Milan, Italy). As part of the SIRIUS project, he will be responsible for leading the WP4 "Civil society barriers and enablers". He will take advantage of his research experience and focus on various forms of civic engagement, notably in the areas sport and health care. He is currently the principal investigator of the project "Civic engagement and politics of health care" funded by the Czech Science Foundation. Previously, he carried out Marie Curie Fellowship (EU FP7) research project FANSREF on mobilization of football fans in the European context. As a team member, he has also been involved in several other international research projects, in particular EUROHOPE (EU FP7), SC-SPORT-EX (EU FP6) and DIOSCURI (EU FP6).
Karel Čada
Dr Karel Čada is a researcher at of Department of Sociology at the Faculty of Social Sciences, Charles University in Prague. He completed his doctoral studies at the same department (2008 to 2015), which included years 2013-2014 which he spent as a visiting scholar at the Australian National University. He is currently the principal investigator of the project "Dynamics of Poverty and Social Exclusion in the Czech Republic" funded by the Czech Science Foundation. With Dr Jakub Grygar, he received the ERSTE Foundation Fellowship for the project on Vietnamese cuisine and socio-material proximities in the making of cosmopolitan city (2015-2016). Between 2014 and 2015, he was a leader of the research team which mapped Czech socially excluded localities (The Ministry for Labour and Social Affair). He has published in the area of social exclusion, poverty, education inequalities, non-governmental organisations and social and health policies.
Petr Soukup
Dr Petr Soukup is Assistant Professor at the Faculty of Social Sciences, Charles University (Prague, Czech Republic). He holds a PhD in Sociology from Charles University. Prior to joining the Faculty of Social Sciences at Charles University, he worked as a research in market research company. As part of the SIRIUS project, he will be responsible for data collection and analysis in the WP1. He is mainly interested in quantitative data analysis, teaches courses of basic and advanced data analysis.
Karina Hoření
Karina joined the team of Charles University and she is responsible for qualitative research and coordination of the team. She holds M.A. in sociology and history from Masaryk University, Brno and during her studies she obtained scholarships from University of Salford, University of Bergen and University of Vienna. She worked as a researcher at the Institute for the Study of Totalitarian Regimes/Prague and was main editor of online educational portal SocialismRealised. She collaborated on number of projects about migration nad history such as Leperiben- collecting oral histories of Czechoslovak Roma. She teaches courses and lectures about Czech social and cultural history and and her interests are qualitative research, ethnic relations in the Czech republic and cross- country skiing.
Migrants and Refugees in Europe Work Integration in Comparative Perspective
SIRIUS Latest Open Access Book is Now Out
Junior Workshop on The Economics of Migration
FundingGrant awarded: 2.5 million euros
Awarding body: European Union, Horizon 2020 Research programme H2020-SC6-REVINEQUAL-2017
Project reference: 770515
Project coordinationSimone Baglioni
Department of Economics and Management
simone.baglioni@unipr.it
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SIRIUSSkills and Integration of Migrants, Refugees and Asylum Applicants in European Labour Markets


HSAs are more important than ever with a strong record of growth in both number of accounts and assets. HSAs give financial institutions a new deposit source in the huge and growing healthcare marketplace. Better yet, banks and credit unions are in the best position to win in this emerging yet competitive field.
The key HSA rules can be allocated to a few simple categories that will be covered in the webinar.
HSA Eligibility. Although more Americans than ever are eligible to open an HSA, the question of "am I eligible for an HSA?" remains one of the most troublesome questions to answer. Learn how to answer that question confidently, quickly and without liability.
Opening an HSA. The rules for opening an HSA are fairly straight-forward, but this is an area that is the custodian's responsibility. So, you have to do it correctly. The webinar will cover issues on the "establishment date," authorized signers, beneficiaries, family/single HSAs, and other key account opening issues.
Contributing to an HSA. A lot of HSA owners wisely want to maximize their HSA contribution. The answer to how to do that is more difficult than just referencing the current HSA limits. Gain the knowledge to guide a customer through the process of calculating his or her individual HSA contribution limit.
Distributions from HSAs. You'll probably get more questions on what is an eligible medical expense than any other area. The good news is that there are a lot of good answers to this question. Learn the basic distribution rules, the most frequently asked questions and how to handle the questions you have never heard before.
Custodial Issues. The webinar will also give an overview of the responsibilities of the HSA custodian. Most of these rules fall into the other categories, but include IRS reporting, fixing mistakes, and tracking HSA balances.