text:
James Neelly to Thomas Jefferson, 18 October 1809
From James Neelly
Nashville Tennessee 18th Octr 1809
It is with extreme pain that I have to inform you of the death of His Excellency Meriwether Lewis, Governor of upper Louisiana who died on the Morning of the 11th Instant and I am Sorry to Say by Suicide;
I arrived at the Chickasaw Bluffs on or about the 18th of September, where I found the Governor (who had reached there two days before me from St Louis) in Very bad health—It appears that his first intention was to go around by Water to the City of Washington; but his thinking a War with England probable, & that his Valuable papers might be in dainger of falling into the hands of the British, he was thereby induced to Change his route, and to come through the Chickasaw nation by land; I furnished him with a horse to pack his trunks &c1 on, and a man to attend to them; having recovered his health in Some digree at the Chickasaw Bluffs, we Set out together. And on our arrival at the Chickasaw nation I discovered that he appeared at times deranged in mind, we rested there two days & came on. one days Journey after crossing Tennessee River & where we encamped we lost two of our horses, I remained behind to hunt them & the Governor proceeded on, with a promise to wait for me at the first houses2 he Came to that was inhabited by White people; he reached the house of a Mr Grinder about Sun Set, the man of the house being from home, and no person there but a woman who discovering the governor to be deranged gave him up the house & slept herself in one near it, his Servant and mine Slept in the Stable loft some distance from the other houses,3 the woman reports that about three OClock She heard two pistols fire off in the Governors Room: the Servants being awakined by her, came in but too late to save him. he had shot himself in the head with one pistol & a little below the Breast with the other—when his4 Servant came in he Says; I have done the business my good Servant give me Some water. he gave him water, he Survived but a short time, I came up Some time after, & had him as decently Buried as I could in that place—if there is any thing wished by his friends to be done to his grave I will attend to their Instructions,
I have got in my possession his two trunks of papers (amongst which is said to be his travels to the pacific Ocean) and probably some Vouchers for expenditures of Public Money for a Bill which he Said had been protested by the Secy of war; and of which act to his death, he repeatedly complained. I have also in my Care his Rifle, Silver watch, Brace of Pistols, dirk & tomahawk: one of the Governors horses was lost in the wilderness which I will endeavour to regain, the other I have Sent on by his Servant who expressed5 a desire to go to the governors mothers & to Monticello:6 I have furnished him with fifteen Dollars to Defray his expences to Charlottsville; Some days preveous to the Governors death he requested of me in Case any7 Accident happened to him, to Send his trunks with the papers therein to the President, but I think it Very probable he meant to You—I wish to be informed what arrangements may be considered best in Sending on his trunks &c—I have the honor to be
With Great respect Yr Ob Sert
James Neelly
U.S. agent to the Chickasaw nation
the governor left two of his trunks at the Chickasaw Bluffs in the Care of Capt Gilbert C. Russell. Commanding officer, & was to write to him from Nashville what to do with them.
RC (DLC); in John Brahan's hand, signed by Neelly; at foot of text: "The Honble Thomas Jefferson late President of United States Monticello near Charlottsville Va"; endorsed by TJ as received 21 Nov. 1809 and so recorded in SJL. Dupl (ViW: TC-JP); entirely in a clerk's hand; endorsed by TJ as a "Duplicate." Enclosed in TJ to James Madison, 26 Nov. 1809, and Madison to TJ, 11 Dec. 1809.
James Neelly served as United States agent to the Chickasaw Nation from 8 July 1809 until 4 June 1812. He apparently served in the military during the War of 1812, and in 1817 Chickasaw leader Tishomingo recommended that he be reinstalled as agent (Madison, Papers description begins William T. Hutchinson, Robert A. Rutland, John C. A. Stagg, and others, eds., The Papers of James Madison, 1962– , 31 vols.: Congress. Ser., 17 vols.; Pres. Ser., 5 vols.; Sec. of State Ser., 6 vols description ends , Pres. Ser., 4:592n; Jackson, Papers description begins Sam B. Smith, Harold D. Moser, and others, eds., The Papers of Andrew Jackson, 1980– , 6 vols. description ends , 4:476).
This and John Brahan's letter of the same date are evidently the earliest extant accounts of Meriwether Lewis's death and burial at Grinder's Inn, on the Natchez Trace in Tennessee. TJ subsequently received other accounts (Gilbert C. Russell to TJ, 4, 31 Jan. 1810) and was involved with and kept apprised of the recovery of Lewis's effects (TJ to C. & A. Conrad & Co., 23 Nov., 11 Dec. 1809; TJ to Madison, 26 Nov. 1809; Isaac A. Coles to TJ, 5 Jan. 1810; William Dickson to TJ, 20 Feb. 1810; TJ to Dickson, 20 Apr. 1810; TJ to Benjamin Smith Barton, 6 Oct. 1810; Barton to TJ, 16 Oct. 1810).
Lewis's term as territorial governor had been difficult and contentious. He was coming to washington to defend protested official financial instruments. Even if they were honored, unsuccessful land investments had put him on the brink of financial ruin. With him he carried the journals of the Lewis and Clark Expedition, which TJ counted on him to prepare for publication but on which Lewis had in fact done no significant work. Given this background and these accounts, those who knew Lewis best, including TJ and William Clark, seem to have accepted the death as a suicide without hesitation or subsequent doubt. The counterhypothesis that Lewis was murdered did not emerge in TJ's lifetime but has since attracted its share of supporters (Nashville Democratic Clarion, 20 Oct. 1809; Saint Louis Missouri Gazette, 4 Oct. 1809, 2 Nov. 1809; TJ to Paul Allen, 18 Aug. 1813; James J. Holmberg, ed., Dear Brother: Letters of William Clark to Jonathan Clark [2002], 206–9, 216–23; Jackson, Letters of Lewis and Clark description begins Donald Jackson, ed., Letters of the Lewis and Clark Expedition with Related Documents, 1783–1854, 2d ed., 1978, 2 vols. description ends , 2:470–4, 487–8, 573–5n; Peale, Papers description begins Lillian B. Miller and others, eds., The Selected Papers of Charles Willson Peale and His Family, New Haven, 1983–2000, 5 vols. in 6 description ends , vol. 2, pt. 2, p. 1238).
Lewis's servant was John Pernier.
1. Abbreviation omitted in Dupl.
2. Dupl: "house."
4. Dupl: "my."
5. RC: "expessed." Dupl: "expressed."
6. RC: "Monticllo." Dupl: "Monticello."
7. In Dupl "thing" is canceled here.
Clark (Clarke), William; and journals of Lewis and Clark Expedition search
Eustis, William; as secretary of war search
Grinder, Mrs. Robert search
Grinder, Robert search
Lewis, Meriwether; death of search
Lewis, Meriwether; Lewis and Clark Expedition search
Lewis and Clark Expedition; journals of search
Neelly, James; and M. Lewis's death search
Neelly, James; identified search
Neelly, James; letters from search
Pernier (Purney), John; and M. Lewis's death search
Russell, Gilbert Christian; and M. Lewis's belongings search
suicide; and M. Lewis search
Neelly, James
"James Neelly to Thomas Jefferson, 18 October 1809," Founders Online, National Archives, https://founders.archives.gov/documents/Jefferson/03-01-02-0478. [Original source: The Papers of Thomas Jefferson, Retirement Series, vol. 1, 4 March 1809 to 15 November 1809, ed. J. Jefferson Looney. Princeton: Princeton University Press, 2004, pp. 606–608.]


text:
\section{Introduction}
Mobile video traffic is expected to represent 78\% of the world's mobile data traffic by 2021 \cite{cisco}. This traffic includes video conferencing, video on demand, and streaming videos which are bandwidth-intensive applications that are rapidly straining the capacity of wireless cellular systems. To support all this traffic and ensure the required application-specific quality-of-service (QoS), mobile network operators (MNOs) have started exploiting the possibility of leveraging the somewhat under-used unlicensed spectrum. This is done by enabling the base stations to offload part of their traffic over the WiFi network using the so-called LTE over unlicensed bands (LTE-U) technology \cite{zhanglte,zhang2015coexistence,liu2014small}. However, to successfully deploy LTE-U, one must ensure a fair usage of the unlicensed bands by developing new coexistence frameworks \cite{zhanglte,zhang2015coexistence,liu2014small}. Moreover, over the unlicensed bands, there is no QoS guarantee and the MNOs must incite the users to accept the best effort service based on their traffic type \cite{zhanglte}.

In this regard, several works \cite{GuanINFOCOM16,khairy2017hybrid,bairagi2018multi,bennis2013cellular,liu2014small, chen2016echo,gu2015exploiting,elsherif2015resource,kang2014mobile,chen2016impact,challita2017proactive} have addressed the problem of coexistence over the unlicensed channels so that the LTE traffic served over unlicensed spectrum does not degrade the performance of the WiFi users (WUs).
The authors in \cite{GuanINFOCOM16} proposed a mechanism that jointly determines a dynamic channel selection, carrier aggregation and fractional spectrum access for LTE-U systems. In \cite{bennis2013cellular}, the authors proposed a cross-system learning algorithm that allows the LTE-U small base stations (SBSs) to autonomously select the unlicensed channels over which they transmit and determine their transmit power. The authors in \cite{liu2014small} formulated the load balancing between licensed and unlicensed bands as an optimization problem to maximize the overall users' rate. In the model of \cite{liu2014small}, the optimal time usage of the unlicensed channels by cellular base stations (BSs) is determined in a centralized way based on the global traffic. The work in\cite{chen2016echo} formulated the unlicensed spectrum allocation problem with uplink-downlink decoupling as a noncooperative game in which the BSs are the players that select the unlicensed channels over which they serve their users. The goal of the BSs is to optimize the uplink and downlink sum-rate while balancing the licensed and unlicensed spectra between the users. The authors in \cite{gu2015exploiting} formulated the unlicensed spectrum allocation problem as a student-project matching problem with externalities. In the model of \cite{gu2015exploiting}, a swap-matching algorithm is proposed in which the BSs propose to the users a list of unlicensed bands over which they can be served.  In \cite{elsherif2015resource}, an optimization problem is formulated to jointly allocate licensed and unlicensed bands while maximizing the users' sum-rate and ensuring fairness among these users.

 The work in \cite{kang2014mobile} proposed a distributed traffic offloading scheme for LTE-U scenarios with a single base station. The authors in \cite{chen2016impact} proposed an economic model for the allocation of licensed band to the LTE-U BSs that can serve mobile and fixed users over unlicensed channels for free. In \cite{challita2017proactive}, the authors proposed a dynamic coexistence mechanism based on deep learning enabling multiple small base stations to select the unlicensed channels and the time they use every channel while guaranteeing fairness with WiFi users and other LTE-U operators. The work in \cite{bairagi2018multi} used multi-game theory to propose a novel coexistence framework between LTE and WiFi users. In particular, a cooperative Nash bargaining game (NBG) and a bankruptcy game were formulated to allocate unlicensed resources. The authors in \cite{xiao2018optimizing} introduced the concept of right sharing for mobile network operators allowing the operators to share and trade their spectrum access rights to the unlicensed bands. In \cite{garcia2018orla}, the authors proposed an orthogonal listen-before talk coexistence paradigm, a licensed-assisted access (LAA) scheme, that provides a substantial improvement in performance compared to classical listen-before talk mechanism, as it imposes no penalty on existing WiFi networks. The authors in \cite{khairy2017hybrid} proposed a hybrid MAC protocol to jointly optimizing the sleep period and the contention window size in LTE-U, to maximize the network performance in terms of the total network throughput and throughput fairness of LTE-U and WiFi. The work in \cite{gu2017dynamic} proposed a coexistence mechanism based on matching theory and proposed a mechanism that accounts for network dynamics and wad evaluated under two typical user mobility models. Other works \cite{mozaffari2016unmanned,challita2017network} have explored other alternatives to improve the backhaul connectivity via the deployment of unmanned aerial vehicles (UAVs). In \cite{mozaffari2016unmanned}, the authors studied the coexistence problem of device-to-device (D2D) and cellular networks in the presence of a single UAV that provides downlink transmission to support the users. The work in \cite{challita2017network} proposed a backhaul framework in which the UAVs are utilized to link the core network and SBSs when the ground backhaul is either unavailable or limited in capacity.

Remarkably, none of these works \cite{GuanINFOCOM16,khairy2017hybrid,bairagi2018multi,bennis2013cellular,liu2014small, chen2016echo,gu2015exploiting,elsherif2015resource,kang2014mobile,chen2016impact,challita2017proactive} have addressed the economic aspect of LTE-U and all have strictly focused on the fair coexistence over the unlicensed bands. However, mobile video content that requires high QoS represents most of the traffic that is exchanged by the users over cellular network. Such QoS requirements cannot be guaranteed over the unlicensed bands due the medium access protocol that is used by the WUs and imposed to the BSs. Thus, without a suitable pricing mechanism, introducing LTE-U can create a new concern in which some users with privilege would not be incentivized to use the unlicensed bands while other users might be always offloaded to the unlicensed bands \cite{elec2015fair}. This economic concerns has been also raised in \cite{shih2018unlicensed,yu2017auction,zhang2017multi}. In \cite{yu2017auction}, the authors proposed an auction mechanism, which enables the LTE provider to negotiate with the WiFi access point owners an exclusive or shared access of the unlicensed channels. The authors in \cite{zhang2017multi} formulated the allocation problem of the unlicensed bands as a Stackelberg game in which each operator sets an interference penalty price for each user that causes interference to the WiFi access point (WAP), and the users choose the unlicensed channel and determine the optimal transmit power in the chosen channel. Hence, it is necessary to develop economic models that incite the LTE users to receive their traffic via the unlicensed channels. Moreover, the spectrum allocation mechanism must accounts for the traffic type of the users in terms of QoS requirements since no coexistence framework exists in the literature in which the requirement in terms of QoS of two separate devices is considered \cite{elec2015fair}.

The main contribution of this paper is to introduce a novel incentive mechanism for facilitating the deployment of LTE-U over cellular networks. We consider a model in which an MNO proposes agreements to a number of LTE-U users that are subscribed to its services to incite them to accept being served over the WiFi network. We consider a practical model with \emph{asymmetric and incomplete information} to account for the heterogeneity of the QoS that is required by each user and is unknown to the MNO as well as other users. The proposed approach, based on contract theory \cite{borgers2015introduction,zhang2017survey}, allows an MNO to define a contract for each user in a model with asymmetric and incomplete information, by fixing the price of serving each content and the fraction of the content that is served via the licensed and unlicensed channels. Unlike classical contract-theoretic models \cite{gao2011spectrum,duan2014cooperative,zhang2015contract,hamidouche2016breaking}, we consider a \emph{Bayesian model} with incomplete information in which the private information of a given user as well as its selected contract is not revealed to the other users. The goal of the MNO is to maximize its own reward while ensuring the feasibility of the contract it proposes to the users. For this contract model, we derive the closed-form of the price and prove its optimality and uniqueness. Moreover, we model the licensed and unlicensed spectrum allocation in an LTE-U system as a \emph{priority-based Bayesian matching game}  to determine the amount of traffic to serve over each of the frequency bands. For the formulated game, we propose a distributed algorithm to assign the licensed and unlicensed frequency bands to the users based on their priority and their QoS requirement. Then, we prove that the algorithm is guaranteed to converge to a Bayesian stable matching. Simulation results show that the proposed pricing mechanism can increase the fraction of users that achieve their requirements in terms of QoS with up to 45\% compared to a uniform pricing mechanism. Moreover, the performance of the proposed mechanism for the case of incomplete information is shown to approach the performance of the same mechanism when all the information about users' requirements is available to the MNO.

The rest of the paper is organized as follows. The system model is provided in Section \ref{model}. In Section \ref{contract}, the problem is formulated as a contract theory problem. In Section \ref{matching}, the allocation problem of licensed and unlicensed bands is formulated as a Bayesian matching game. Section \ref{sim} presents the simulation results. Conclusions are drawn in Section \ref{con}.
\section{System Model}
\label{model}
Consider a wireless network composed of a set $\mathcal{S}$ of $S$ LTE-U BSs that can serve the downlink traffic of a set $\mathcal{N}$ of $N$ users over both licensed and unlicensed bands. Since the BSs cannot guarantee any QoS when they serve users over the unlicensed bands, the MNO has to design an economic model that motivates users to accept being served over unlicensed channels depending on their QoS requirement. To define the optimal pricing policy and spectrum allocation mechanism, we use the two frameworks of contract theory and Bayesian matching games. Contract theory is a powerful mathematical tool for creating economic incentives while matching games enable to capture the interconnection between the users that share the same resources and deal with the combinatorial nature of such resource allocation problems. The pricing and spectrum allocation mechanisms are illustrated in Fig. \ref{fig:model}.

\begin{figure}
\centering
\includegraphics[scale=0.58]{lteu_contract-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Pricing and spectrum allocation in an LTE-U system.}\vspace{-0.3cm}
\label{fig:model}
\end{figure}

In the proposed model, each MNO specifies a performance-reward bundle \emph{contract} $(\alpha_i,\beta_i,\pi_i)$, where $\pi_i$ is the monetary reward that is charged by the MNO to user $i\in\mathcal{N}$, $\alpha_i$ is the fraction of user $i$'s traffic that is served over the licensed bands, and $\beta_i$ is the fraction of traffic that is served via the unlicensed bands. Since users have different QoS requirements depending on their traffic, we introduce a type $\theta_i\in\Theta_i=[\ubar{\theta}_i,\bar{\theta}_i]$ for each user $i$. The type represents the willingness of the user to pay for its traffic. In fact, the higher the required download rate, the more the user would be willing to pay. We sort the users' requirements in terms of QoS in an ascending order and classify them into $K$ types denoted by $\theta_1,...,\theta_K$ with $K\leq C$. Each type represents the type of the user's traffic or the willingness of a user to pay for the service offered by the MNO. This information is private to each user and is not available to the other users. The types are ordered as follows:
\begin{equation*}
\theta_1<...<\theta_k<...<\theta_K,  ~~~k\in \mathcal{K}=\{1,...,K\}.
\end{equation*}
Here, we assume that all of the requests generated by the users are launched by different users.

Instead of offering the same contract to all of the users, the MNO must offer different contracts to different users according to their types. This enables the MNO to account for the heterogeneous QoS requirements of the users and their willingness to pay for the service offered by the MNO. Moreover, the MNO must account for the interdependence between the contract designed for each type that appears in the performance achievable by every user when selecting the contract designed for its type. Such interdependence appears in the achievable rate by each user which depends on the offload policy $\{(\alpha_i,\beta_i)\}_{i\in\mathcal{K}}$ that determines the amount of traffic that is served over both licensed and unlicensed bands for all users' types. The users are free to accept or decline any type of contracts. If a user declines to receive any contract, we assume that a user signs a contract $(\alpha_i,\beta_i,\pi_i) =(0,0,0)$. Next, we define the utilities of the MNO and users in the system.

\subsection{Mobile Network Operator's Utility}
The utility of an MNO corresponds to the difference between the price charged to its users and the cost for serving them which is widely used in the literature as it captures practical considerations \cite{shah2017incentive,hajimirsadeghi2017joint,liu2017design}. The utility of the MNO when serving a user of type $\theta_i$ can be given by:
\begin{equation}
u_{\text{o},i}(\boldsymbol{\theta})=  \pi_i(\theta_i,\boldsymbol{\theta}_{-i})-c_i(\alpha_i(\boldsymbol{\theta}),\theta_i,\boldsymbol{\theta}_{-i}),
\end{equation}
where $\pi_i(\theta_i,\boldsymbol{\theta}_{-i})$ is the price charged by the MNO to a given user of type $i$, $\boldsymbol{\theta}$ is a vector that groups the types of all the users, and $\boldsymbol{\theta}_{-i}$ represents the same vector without the type of user $i$.

We define the cost $c_i(\alpha(\boldsymbol{\theta}),\theta_i,\boldsymbol{\theta}_{-i})$ as the transmit power needed to serve a user of type $\theta_i$ with a rate $r_{si}$ from its associated BS $s\in\mathcal{S}$ which depends on the rates required by the other users. Thus, the transmit power cost can be given by:
\begin{equation}
c_i\left(\alpha_i(\boldsymbol{\theta}),\theta_i,\boldsymbol{\theta}_{-i}\right)= p_{si}(\alpha_i(\boldsymbol{\theta}))=\frac{\left(e^{\frac{r^{\text{req}}_{i}}{w}}-1\right)\left(\sigma^2_0+\sum_{j\in\mathcal{S}\setminus s}\sum_{k\in\mathcal{U}}|h_{ji}|^2\delta_{jk}p_{ji}\left(\alpha_k(\boldsymbol{\theta})\right)\right)}{|h_{si}|^2},
\end{equation}
where $r^{\text{req}}_{i}$ is the required rate of user $i$, $p_{si}(.)$ is the transmit power from BS $s$ to user $i$, $h_{si}$ is the channel gain, $w$ is the bandwidth, and $\sigma^2_0$ is the noise variance. $\delta_{jk}$ is a binary variable that is equal to $1$ when SBS $j$ is connected to user $k$ and $0$ otherwise. when We note that the transmit power $p_{ji}(\alpha_k(\boldsymbol{\theta}))$ of the other BSs depends on the type of all users as the served user is affected by the interference from all the other BSs. For notational convenience, we use $\alpha_i$, $\alpha_i(\theta_i,\boldsymbol{\theta}_{-i})$ and $\alpha_i(\boldsymbol{\theta})$ interchangeably.
\subsection{Users' Utility}
The goal of a user $i$ of type $\theta_i$ is to maximize its benefit which can be defined as the difference between the achievable rate and the price charged by the MNO:
\begin{equation}
\label{user_utility}
u_i(\alpha_i,\boldsymbol{\theta})=\theta_i v_i\left(\alpha_i(\theta_i,\boldsymbol{\theta}_{-i}),\beta_i(\theta_i,\boldsymbol{\theta}_{-i}),\theta_i\right)-\pi_i(\theta_i,\boldsymbol{\theta}_{-i}),
\end{equation}
where $v_i$ is the valuation of the user. This function can be defined as the time needed for the MNO to serve the user's traffic or equivalently, the achievable rate by the user. Note that $v_i$ is a decreasing function of the types $\boldsymbol{\theta}_{-i}$. In fact, the larger the type of other users, the higher is the traffic load on the licensed bands, and thus the QoS experienced by the user decreases. The valuation function of a user is given by:
\begin{equation}
\label{utility_sbs}
v_i(\theta_i,\boldsymbol{\theta}_{-i})=\eta (r_{si}(\alpha_i(\boldsymbol{\theta}),\beta_i(\boldsymbol{\theta}),\theta_i )-r_{i}^{\text{req}})^2,
\end{equation}
where $\eta$ is a parameter and $r_{si}(\alpha_i(\boldsymbol{\theta}),\beta_i(\boldsymbol{\theta}),\theta_i )$ represents the achievable data rate by user $i$ when served from BS $i$ over both licensed and unlicensed bands. We choose a quadratic function to model the valuation of users to capture the fact that user $i$ is only satisfied by the rate when it attains its target rate. Thus, the valuation function of a user $i$ is maximized at $r_{i}^{\text{req}}$ and decreases when the rate achieved by the user exceeds or is lower than $r_{i}^{\text{req}}$. The achievable rate by user $i$ is given by:
\begin{equation}
\label{rate}
r_{si}(\alpha_i(\boldsymbol{\theta}),\beta_i(\boldsymbol{\theta}),\theta_i )=\mathbb{E}_t\left[w_{si}\log{\Big(1+\frac{p_{si}(\alpha_i(\boldsymbol{\theta}))|h_{si}|^2}{\sigma^2+I(\alpha_i(\boldsymbol{\theta}),\boldsymbol{\theta})}\Big)}\right]+\mathbb{E}_t\left[\kappa_{si}(\beta_i(\boldsymbol{\theta}),\boldsymbol{\theta})\right],
\end{equation}
where $I=\sum_{l\in\mathcal{S}\setminus s}\sum_{k\in\mathcal{U}}{\delta_{lk}p_{li}(\alpha_k(\boldsymbol{\theta}))|h_{li}|^2}$ is the interference experienced by user $i$ from the BSs over the licensed bands, and $\kappa_{si}(\beta_i(\boldsymbol{\theta})),\boldsymbol{\theta})$ is the rate that user $i$ achieves when served by BS $s$ over the unlicensed band and is given by:
\begin{equation}
    \kappa_{si}(\beta_i(\boldsymbol{\theta}),\boldsymbol{\theta})=
    \begin{cases}
w^{\prime}_{si}\log{\Big(1+\frac{p^{\prime}_{si}(\alpha_i,\theta_i)|h^{\prime}_{si}|^2}{\sigma^{\prime 2}+I^{\prime}(\beta_i(\boldsymbol{\theta}),\boldsymbol{\theta})}\Big)}, & \text{if}\ I^{\prime}\leq I_{\text{th}}, \\
      0, & \text{otherwise},
    \end{cases}
\end{equation}
where $w^{\prime}_{si}$ is the bandwidth of the unlicensed channel, $p^{\prime}_{si}$ is the transmit power from BS $s$ to user $i$ over the unlicensed channel, and $I^{\prime}=\sum_{w\in\mathcal{S}\cup\mathcal{W}\setminus s}{p^{\prime}_{wi}|h_{wi}|^2}$ is the interference experienced by user $i$ from all the transmitting BSs and WiFi users in $\mathcal{W}$ over the same unlicensed channel. For the unlicensed channel, we define an interference threshold $I_{\text{th}}$ that is used by each BS to decide whether to transmit or not over the unlicensed channel. When a given BS senses an interference level that exceeds $I_{\text{th}}$, it does not transmit its content which results in a backoff period. Note that the interference experienced by any user depends on the types of all the other users and the amount of traffic that is served over the licensed and unlicensed bands.

Since users do not have information about other users' types, we assume that the profile type $\boldsymbol{\theta}=[\theta_1,..., \theta_n]$ is drawn from the set $\Theta=\times\Theta_i$ according to a distribution $P(\boldsymbol{\theta})$ with density $p(\boldsymbol{\theta})$ which is common knowledge to all users. The types of users are assumed to be statistically independent such that $\phi(\boldsymbol{\theta})=\prod_{i\in\mathcal{N}}\phi_i(\theta_i)$. Moreover, considering a model with incomplete information, each user's strategy not only depends on its beliefs about $\phi(\boldsymbol{\theta})$, but also on its beliefs about the author users' beliefs about $\phi(\boldsymbol{\theta})$. Given the distribution of the users' types, each user seeks to maximize its expected utility given by:
\begin{equation}
\label{exp_utility_users}
\bar{u}_i(\theta_i)= \int_{\Theta_{-i}} u_i(\alpha_i,\theta_i,\boldsymbol{\theta}_{-i}).
\end{equation}
The expected valuation function is defined as $\bar{v}_i(\theta_i)=\int_{\Theta_{-i}}v_{i}(\alpha(\boldsymbol{\theta}),\beta(\boldsymbol{\theta}),\theta_i)$ and the expected price is $\bar{\pi}_i(\theta_i)=\int_{\Theta_{-i}}\pi_i(\theta_i,\boldsymbol{\theta_{-i}})$.
The optimization problem of the MNO can be defined as follows:
\begin{equation}\label{PF}
\begin{aligned}
& \underset{\{\alpha_i(.),\pi_i(.)\}_{\forall i}}{\max}
&& \int_{\Theta}\sum_{i\in\mathcal{N}}\left[\pi_i(\theta_i,\boldsymbol{\theta}_{-i})-c_i(\theta_i,\boldsymbol{\theta}_{-i})\right]dP(\theta), \\
& \text{s. t.}
&& 0\leq \alpha_i(\boldsymbol{\theta})\leq 1.
\end{aligned}
\end{equation}
In this formulation, we do not make any constraint on the participation of the users. Thus, when proposing the contracts resulting from solving (\ref{PF}), users may prefer not to select any of the contracts or select contracts that are not designed for their types. To analyze this economic incentive problem, we propose a solution based on the Nobel-prize winning framework of contract theory for designing feasible contracts \cite{zhang2017survey}. Moreover, we consider a practical case in which users have incomplete knowledge of other users' types and the contracts they select. Thus, the goal of each user is to select the most optimal contract based only on the distribution of the other users' types.

Next, we use the framework of contract theory to determine the pricing mechanism at the MNO that incite the users to accept being served over unlicensed channels. Then, we introduce Bayesian matching theory to propose a distributed algorithm that enables the users to achieve their requirement in terms of QoS. In other words, the matching algorithm will determine the amount of traffic that should be served over both licensed and unlicensed spectrum and thus determining the values of $\alpha_i$ and $\beta_i$ for all $i\in\mathcal{U}$.

\section{Contracts Design for Unlicensed Spectrum Pricing}
\label{contract}
To ensure that the users have an incentive to collaborate with the MNO by accepting to be served via unlicensed bands, the MNO must design feasible contracts that ensure the participation of the users by at least selecting one of the contracts that the MNO proposes. Moreover, the MNO must motivate each user to select the contract designed for its own type.

In contrast to classical contract models that are used to model resource allocation problems in wireless networks such as in \cite{gao2011spectrum,duan2014cooperative,zhang2015contract}, we consider a Bayesian model. In addition to information asymmetry that captures the unavailability of certain user types to the MNO, we assume that the type of every user as well as the contract selected by that user is not known to other users. Such information is important in our model due to the interdependence of the users' contracts that appears in the rates achievable by every user. Thus, a user has to determine it most optimal contract based only on the distribution of users types which is specific to our model. In such system, the feasibility conditions of the contracts are defined differently.

\subsection{Feasibility of a Contract}
In a Bayesian model, the MNO must design contracts for the users, that satisfy the two following constraints:
\begin{definition}
\emph{Truthfully implementable Bayesian strategies (TIBS) }: A contract $f(\boldsymbol{\theta})$ is truthfully implementable in Bayesian strategies if there exists a direct mechanism $\Gamma$ such that truth-telling is a Bayesian equilibrium. In other words, it is better for a user to select the contract defined for its type than selecting another contract, i.e.,  $\forall i\in\mathcal{N}$ and $\theta_i\in\Theta_i$,
\begin{equation}\label{IC}
\int_{\Theta_{-i}} u_i(\alpha_i,\theta_i,\boldsymbol{\theta}_{-i})\geq \int_{\Theta_{-i}}u_i(\alpha_i,\hat{\theta}_i,\boldsymbol{\theta}_{-i}),
\end{equation}
for all $\hat{\theta}_i\in\Theta_i$ and $\hat{\theta}_i\neq\theta_i$.
\end{definition}
 This condition ensures that none of the users can achieve a better performance in the network by changing its type. The second condition ensures that the contract selected by a user should guarantee that the expected utility of the user is nonnegative given the types distribution $\boldsymbol{\Theta}_{-i}$.
\begin{definition}
\emph{Interim Individually Rational Contract (IIR)}: A contract is interim individually rational if:
\begin{equation}\label{IR}
\int_{\Theta_{-i}} u_i(\alpha_i,\theta_i,\boldsymbol{\theta}_{-i})\geq 0, ~~ \forall i\in\mathcal{N}.
\end{equation}
\end{definition}
In a Bayesian model, we can state the following lemmas from the two previous conditions.
\begin{lemma}
For any feasible contract $f(\boldsymbol{\theta})=(\alpha_i(\boldsymbol{\theta}),\beta_i(\boldsymbol{\theta}),\pi_i(\boldsymbol{\theta})$), we have $\bar{\pi}_i(\theta_i)\geq \bar{\pi}_i(\hat{\theta}_i)$ if and only if $\bar{v}_i(\theta_i)\geq\bar{v}_i(\hat{\theta}_i)$ for all $\hat{\theta}_i\in\Theta_i$.
\end{lemma}
\begin{proof}
The proof is provided in Appendix A.
\end{proof}
Lemma 1 shows that a user that pays more for its traffic should receive higher QoS or performance level compared to users that pay less, and vice versa. Next lemma shows the performance that should be achieved by the users based on the price they pay.
\begin{lemma}
For any feasible contract $f(\boldsymbol{\theta})=(\alpha_i(\boldsymbol{\theta}),\beta_i(\boldsymbol{\theta}),\pi_i(\boldsymbol{\theta}))$, if $\theta_i\geq \theta_j$, then $\bar{v}_i(\alpha_i,\theta_i)\geq\bar{v}_i(\alpha_i,\theta_j)$.
\end{lemma}
\begin{proof}
We prove this lemma using the TIBS constraint. For a user $i$ of type $\theta_i$ and user $j$ of type $\theta_j$, we have:
\begin{equation}
\theta_i \bar{v}_i(\alpha_i,\theta_i)-\bar{\pi}_i(\theta_i)\geq \theta_i \bar{v}_i(\alpha_i,\theta_j)-\bar{\pi}_j(\theta_j),
\end{equation}
and,
\begin{equation}
\theta_j \bar{v}_i(\alpha_i,\theta_j)-\bar{\pi}_j(\theta_j)\geq \theta_j \bar{v}_i(\alpha_i,\theta_i)-\bar{\pi}_i(\theta_i).
\end{equation}
By adding the two inequalities, we get:
\begin{equation}
\theta_i \bar{v}_i(\alpha_i,\theta_i)+\theta_j \bar{v}_i(\alpha_i,\theta_j)\geq \theta_i \bar{v}_i(\alpha_i,\theta_j)+\theta_j \bar{v}_i(\alpha_i,\theta_i),
\end{equation}
which is equivalent to:
\begin{equation}
(\theta_i -\theta_j)\bar{v}_i(\alpha,\theta_i)\geq (\theta_i-\theta_j) \bar{v}_i(\alpha,\theta_j).
\end{equation}

\end{proof}
Lemma 2 shows that a user of a higher type should receive a higher valuation. Note that the highest expected performance can be achieved only if its traffic is served through the licensed bands. Thus, when the application of the user requires a high data rate, the fraction of served traffic over licensed spectrum should be larger.

From Lemma 1 and Lemma 2, we conclude that for a feasible contract, all contracts satisfy:
\begin{align}
0\leq \bar{\pi}_i(\theta_{1})\leq\bar{\pi}_i(\theta_{2})\leq...\leq\bar{\pi}_i(\theta_{K}),\\ 0\leq\bar{v}_i(\alpha_i,\theta_{1})\leq\bar{v}_i(\alpha_i,\theta_{2})\leq...\leq \bar{v}_i(\alpha_i,\theta_{K}).
\end{align}

\subsection{Incentive Mechanism Design}
At the MNO, the optimization problem can be formulated as follows:
\begin{equation}
\label{problem}
\begin{aligned}
& \underset{\{\alpha_i(\boldsymbol{\theta}),\boldsymbol{\pi}_i(\boldsymbol{\theta})\}_{\forall i}}{\text{max}}
& &\int_{\Theta}\sum_{i\in\mathcal{N}}\left[\pi_i(\theta_i,\boldsymbol{\theta}_{-i})-c_i(\theta_i,\boldsymbol{\theta}_{-i})\right]dP(\theta),  \\
& \text{s.t. }
& & (\ref{IC}), \forall i\in\mathcal{N}, \\
&&& (\ref{IR}), \forall i\in\mathcal{N},\\
&&& 0\leq\alpha_i(\boldsymbol{\theta})\leq 1,\forall i\in\mathcal{N},
\end{aligned}
\end{equation}
where the first two constraints represent the TIBS and IIR conditions that ensure the feasibility of a contract. In addition to the non-convexity of the formulated optimization problem, there is a large number of constraints that increase the complexity of the problem. To solve the problem (\ref{problem}), we first state the following equivalence result.
\begin{theorem}
\label{theorem}
A contract is both TIBS and IIR if and only if $\forall i\in\mathcal{N}$,
\begin{enumerate}
\item  $\bar{v}_i(\alpha_i,\theta_i)$ is nondecreasing,
\item $\bar{u}_i(\theta_i)=\bar{u}_i(\ubar{\theta}_i)+\int_{\ubar{\theta}_i}^{\theta_i}{\bar{v}_i(\alpha,\theta_i)(\eta)d\eta},
$ for all $\theta_i$, and
\item $\bar{u}_i(\ubar{\theta}_i)\geq 0$.
\end{enumerate}
\end{theorem}
\begin{proof}
The proof is provided in Appendix B.
\end{proof}
Now, we solve (\ref{problem}) by first writing the price as a function of the amount of traffic served through the licensed and unlicensed channels and give the following result.
\begin{proposition}
\label{prop}
Let $\{(\alpha^*_i,\pi^*_i)\}_{\forall i\in\mathcal{N}}$ be a the optimal contract with fixed time allocations $\{\alpha^*_i, \forall i\}$. The optimal price for every type is given by:
\begin{flalign}\label{price}
&\bar{\pi}^*_1=\theta_1\bar{v}_1(\alpha_i^*,\theta_i),\nonumber\\
&\bar{\pi}^*_i=\theta_i\bar{v}_i(\alpha_i^*,\theta_i)+\int_{\underline{\theta}_i}^{\theta_i}{\bar{v}_i(\alpha_i^*,\eta)d\eta}, \forall i\in\mathcal{N}.
\end{flalign}
\end{proposition}
\begin{proof}
The proof is provided in Appendix C.
\end{proof}
Having defined the closed-form expression of the price charged by the MNO for every user, we have to derive the optimal values of $\alpha^*$ to define the amount of traffic that should be served over both licensed and unlicensed channels. For this, based on Proposition \ref{prop}, we write the allocation problem of the licensed and unlicensed spectra by simplifying the initial problem (\ref{problem}) in which we replace the optimal price by its expression as derived in (\ref{price}). Thus, the allocation problem of the spectrum can be defined as follows:
\begin{equation}
\label{problem_simp}
\begin{aligned}
& \underset{\{\alpha_i(\boldsymbol{\theta})\}_{\forall i}}{\text{max}}
& &\int_{\Theta}\sum_{i\in\mathcal{N}}\Big[\theta_i\bar{v}_i(\alpha_i,\theta_i)-c_i(\theta_i,\boldsymbol{\theta}_{-i})  +\int_{\underline{\theta}_i}^{\theta_i}{\bar{v}_i(\alpha_i^*,\eta)d\eta}\Big]dP(\theta),  \\
& \text{subject to}& & 0\leq\alpha_i(\boldsymbol{\theta})\leq 1,\forall i\in\mathcal{N}.
\end{aligned}
\end{equation}
The assignment problem of the traffic to the licensed and unlicensed bands is equivalent to the Knapsack problem which is NP-hard. Thus, due to the interdependence between the amount of traffic served over the licensed and unlicensed bands for the users, it is difficult to solve (\ref{problem_simp}) using classical optimization approaches. To address this problem, we propose a novel approach to assign the files requested by the users to the licensed and unlicensed bands while accounting for the QoS requirement of each file. To this end, we use the framework of matching theory with incomplete information in which the two disjoint sets of BS-channel pairs and user-subfile pairs are matched to one another by the MNO \cite{liu2014stable,wang2013interdependent}. The files are assumed to be divided into subfiles of the same size. In this model, the types of the users are not known to the BSs neither to the MNO.
\section{Frequency Band Allocation as a Bayesian Matching Game}
\label{matching}
In this section, we formulate a matching game with incomplete information which is also known as \emph{Bayesian matching game} with externalities \cite{liu2014stable,wang2013interdependent,bodine2011peer}, to determine the fraction of each requested file that should be served over both licensed frequency bands and unlicensed channels. A matching game is defined by two-sets of players that evaluate one another using preference relations. The game is considered Bayesian because the BSs do not have information regarding users' types and only have the distribution of the types. Moreover, the assignment of the channels to the users is done by the principal which is not aware of the types of the users neither of their utility functions. The matching problem is considered with externalities due to the interdependence between users preferences. In fact, the performance that is achieved by a given user depends on the global amount of traffic that is served over the same frequency band. We define traffic of the users as set of file chunks $\mathcal{F}=\{f_{11},...,f_{1K_1},...,f_{F1},...,f_{1K_F}\}$ in which each file $f_l$ of size $L_{f_l}$ bits that is requested by a user, is divided into $K_f$ file chunks of size $L_{f}^{\text{min}}$ which we define as the minimum file size that we assign to a frequency band. Then, we formulate the problem of spectrum allocation in an LTE-U system as a Bayesian one-to-many matching game in which a set of frequency bands is assigned to a set of file chunks, where each file chunk is served over a single frequency band and by a single BS. In the model we consider, the users act on behalf of the files and the BSs act on behalf of the licensed and unlicensed channels.

Thus, to model the problem as a Bayesian one-to-many matching game \cite{gu2015matching}, we consider the two sets $\mathcal{A}\in\mathcal{N}\times\mathcal{F}$ of subfile-user pairs and $\mathcal{M}\in\mathcal{C}\times\mathcal{S}$ of licensed/unlicensed frequency band and BS pairs as two sets of players. We denote the set of licensed frequency bands by $\mathcal{C}_1$ and the set of unlicensed channels by $\mathcal{C}_2$ with $\mathcal{C}=\mathcal{C}_1\cup\mathcal{C}_2$. The matching is defined as an assignment of subfiles and users pairs in $\mathcal{A}$ to BSs and channels pairs in $\mathcal{M}$ in which the BSs act on behalf of the licensed and unlicensed channels as they are the ones to decide on which frequency band they serve their traffic. Moreover, the users act on behalf of the files and determine the BS and channel to accept or not. Depending on the achievable rate and the price charged by the MNO, every user $i$ builds a preference relation $\succ_{(i,f)}$ for its file chunk $f$ over subsets of BS and channel pairs in $\mathcal{M}$ and being unmatched $\emptyset$. Similarly, every BS builds a preference relation $\succ_{(s,c)}$ \cite{bodine2011peer} for its channel $c$ over the pairs of users and subfiles in $\mathcal{A}$. In classical matching games, the players define their preferences based on a utility function that measure their performance in the network when matched to any of the players in the opposite set. In our model, the utility of any BS or user depends on the traffic served by each BS and frequency band which is a function of users' type. Therefore, in the presence of types distribution, instead of computing the exact utilities, the players define their preferences based on the expected utilities.

The number of users that a BS can serve over a given frequency band depends on the total traffic load on each frequency band as well as the channel conditions. In a matching game, the capacity of a pair $(s,c)$ is known as a quota that we denote by $q_{(s,c)}$ \cite{Roth1990}. A one-to-many matching game can be defined as follows.
\begin{definition}
A \emph{one-to-many matching} $\mu$ is a mapping from the set $\mathcal{A}\cup \mathcal{M}$ into the set of all subsets of $\mathcal{A}\cup \mathcal{M}$ such that for every $m=(s,c) \in \mathcal{M}$ and $a=(i,f) \in \mathcal{A}$ \cite{Roth1991}:
\begin{enumerate}
\item $\mu (m)$ is contained in $\mathcal{A}$ and $\mu(a)$ is an element of $\mathcal{M}$;
\item $|\mu (m)| \leq q_{(s,c)}$ for all $(s,c)$ in $\mathcal{M}$;
\item $|\mu (a)| \leq 1$ for all $a$ in $\mathcal{A}$;
\item $m$ is in $\mu (a)$ if and only if $a$ is in $\mu(m)$,
\end{enumerate}
with $\mu (m)$ being the set of user-subfile pairs that is associated to channel $c$ and BS $s$ under matching $\mu$.
\end{definition}
This definition states that a matching is a one-to-many relation in the sense that each channel-BS pair is matched to a set of user-subfile pairs, and each subfile is served over a single frequency band and a single BS. Before setting an assignment of subfiles to frequency bands, each player needs to specify its \emph{preferences} over subsets of the opposite set based on its goal in the network. We use the notation $\mathcal{A}_1 \succ_{(s,c)} \mathcal{A}_2$ to represent that BS $s$ prefers to serve over frequency band $c$, the user-subfile pairs in the set $\mathcal{A}_1 \subseteq \mathcal{A}$ than to serve the ones in $\mathcal{A}_2 \subseteq \mathcal{A}$. A similar notation is used for the users to set a preference list over the opposite set of BS-channel pairs. Faced with a set $\mathcal{M}$ of possible partners, a user that requests a subfile $f$ can determine a preference relation over the frequency bands and BSs pairs. We denote by $(\mu,s_1,c_1)\succ_a (\mu,s_2,c_2)$ the preference of user-subfile pair $a=(i,f)$ when user $i$ prefers being served by BS $s_1$ over frequency band $c_1$ to being served by BS $s_2$ over frequency band $c_2$ given the current matching $\mu$, with $(\mu,s_1,c_1)\neq(\mu,s_2,c_2)$. Next, we define the preferences of the users and the BSs.

\emph{1) Preferences of users:} From the users' side, each user seeks to maximize its own individual expected utility function for a given requested subfile. Therefore, from the users' point of view, we use the difference between the achievable rate and the price charged by the MNO for the user as the utility function. Thus, using the channel coefficient estimation, and the distribution of other users types, a user $i$ determines its utility for every subfile it requests $f$ given the possible BS-channel pairs using the expected utility defined in (\ref{exp_utility_users}), and thus, builds a  preference vector $\xi_a$ with $a=(i,f), \forall f\in\mathcal{F}_i$ and $\mathcal{F}_i\subseteq\mathcal{F}$ being the set of file chunks that user $i$ requests.

\emph { 2) Preferences of the base stations:} We now define a novel preference scheme at the BSs to give priority to the users based on the type they reveal for the MNO.
A similar model is considered in \cite{semiari2014matching} in which it is shown that exploiting the information concealed in user' preferences offers considerable gains for the players. However, unlike \cite{semiari2014matching}, not only the preferences of users impact the utility of the BSs but also the types of users as discussed previously. Moreover, we account for the fact that users may manipulate the outcome of the matching process in their favor by misreporting their types \cite{wang2013interdependent}. Furthermore, unlike \cite{semiari2014matching}, the BSs can serve multiple files at a given frequency band which introduces externalities as the performance achievable by every user depends on the types of other users which indirectly determine the traffic load on each channel.

The utility of a BS when serving a pair $a=(i,f)$ of type $\theta_a$ over frequency band $c$ given that users reveal their real types, can be given by:
\begin{equation}
\gamma_{ma}=\pi_a(\theta_a,\boldsymbol{\theta}_{-a})-c_a(\mu(\boldsymbol{\theta}),\theta_a,\boldsymbol{\theta}_{-a}),
\end{equation}
where we replace $\alpha_i$ in the cost function by the current matching $\mu$ since the fraction of traffic that is served over each of the licensed and unlicensed bands is determined by the current matching outcome. The utility of the BS represents the difference between the price that the user pays and the cost for being served at its target QoS. Knowing that users have different QoS requirements, we introduce a priority vector $\boldsymbol{\phi}\in \{0,1\}^{1\times A}$ for the user-subfile pairs in $\mathcal{A}$. This vector allows the BSs to assign priorities for the user-subfile pairs. Depending on the priority given to a user-subfile pair, a BS will promote the utility of the particular user-subfile pair. If pair $a$ has another licensed frequency band option to which it can apply according to its preference vector, the BS sets $\boldsymbol{\phi}(a)=1$, otherwise $\boldsymbol{\phi}(a)=0$. Consequently, the priority vectors are different at each BS and frequency band pair. Thus, instead of ranking users by only accounting for the cost of serving the subfile of the user and the price charged by the MNO, each BS takes into account the QoS requirements of every subfile. The utility function of each BS-channel pair $m=(s,c)$ when serving a pair $a=(i,f)$ given the types distribution of the users and the current matching $\mu$ can be defined as follows:
\begin{align}
\label{pref_m}
\xi_{ma}(I,\mu,\boldsymbol{\theta})=\int_{\Theta}\Big[\epsilon \gamma_{ma}(\theta_a,\boldsymbol{\theta}_{-a})\Big]dP(\theta)+(1-\epsilon)\Psi_{ma}(\eta_a,I(\mu(\boldsymbol{\theta}),\boldsymbol{\theta}),\kappa_{si}),
\end{align}
where the BS's utility is a function of priority coefficient $\eta_a$, resemblance factor $\epsilon\in\{0,1\}$, and users' types. Clearly, user-subfile pair $a$ will be rejected if its utility is not one of the $q_m$ highest utilities. If two users are identified by the BS to which they apply, to have the same priority and the subfiles they request are of the same type, then $\epsilon=1$. Otherwise, the BS assigns $\epsilon=0$ to the utility of those two users. Function $\Psi_{ma}$ is given by:
\begin{align}
\Psi_{ma}(\eta_a,I(\mu(\boldsymbol{\theta}),\boldsymbol{\theta}))=&\int_{\Theta}\Big[\gamma_{ma}(\theta_a,\boldsymbol{\theta}_{-a})+G_{ma}\Big] dP(\theta),\nonumber\\
=\int_{\Theta}\gamma_{ma}(\theta_a,\boldsymbol{\theta}_{-a})&+\frac{c_i(\mu(\boldsymbol{\theta}),\theta_a,\boldsymbol{\theta}_{-a})}{\theta_a\eta_a}dP(\theta).
\end{align}
The promotion function $G_{ma}$ represents the amount of promotion given to each user. It enables the BSs to prioritize serving the files with high type and less remaining options, over the licensed bands as we show later. That is, a BS decreases the cost for serving this user based on user's priority $\eta_a$ and type $\theta_a$. The higher the priority or the type of a given subfile, the more promotion it receives from the BS. Basically, let $\eta_a\in\{\eta^{\prime},\eta^{\prime\prime},\eta^{\prime\prime\prime}\}$ indicate the first, second and third priority coefficient, respectively. Parameter $\epsilon$ is used to avoid prioritizing two subfiles that have the same priority, since the promotion is a function of the users types and channel estimations knowing that a single user may have multiple requests. Clearly, the proposed priorities allow to promote subfiles that are experiencing a lower data rate compared to their required QoS, thus, allowing them to have a better BS and spectrum association. The prioritizing procedure is described as follows.

For every frequency band $c$, a BS $s$ can group the user-subfile pairs into three priority classes as follows:\\
{\bf Priority 1:} This includes user-subfile pairs that have BS $s$ and channel $c$ as their first and their only remaining preference with $c$ being a licensed frequency band. Therefore, these remaining applicants have been accepted by BS $s$ and channel $c$ in the first iteration of proposals. That is, all $a$ have $\boldsymbol{\phi}_m(a)=0$ and $\xi_a(1)=(s,c)$.\\
{\bf Priority 2:} This includes user-subfile pairs for which BS $s$ and channel $c$ is not the first preference but it is the only remaining BS in the preference list that contains a licensed band. In other words, $\boldsymbol{\phi}_m(a)=0$ and $\xi_a(1)\neq(s,c)$ and $\exists~ m_1\neq m\in\mathcal{M} \text{ with } c_1\in\mathcal{C}_1 \text{ such that } (s_1,c_1)\succ_{a} (s,c)$.\\
{\bf Priority 3:} This includes user-subset pairs that, if and when rejected by BS $s$, they still have other choices in their preference list that include a licensed frequency band.

In such practical systems in which users only know the distribution of the other users' types, the goal of the users is to select their contracts so that they reach a Bayesian stable matching (BSM) that can be defined as follows \cite{xiao2015bayesiano,xiao2015bayesian}:
\begin{definition}
The BSM represents the state in which none of the users can improve its expected utility by selecting another contract given the distribution of other users' types.
\end{definition}
To reach the BSM outcome, next, we propose a distributed matching algorithm for the assignment of the unlicensed bands to the files requested by the users.

\emph{3) Proposed LTE-U Spectrum Allocation Algorithm:} In Table I, we introduce a spectrum allocation algorithm in an LTE-U system based on the deferred acceptance algorithm \cite{gu2015matching,roth1992two}. Unlike classical one-to-many matching games with externalities in which a swap-matching is defined as an outcome of the algorithm \cite{pantisano2013matching}, the difference is in the utility functions of the players. In fact, since the distribution of users types is known to the BSs, every BS and user can determine an estimation of the interference that any user may experience over a given frequency bands. Thus, the players do not need to update their utilities and preferences based on the current matching $\mu$ at every proposals period.

\begin{table}[!t]
\label{algorithm-table}
\footnotesize
  \centering
  \caption
    \vspace*{-0.3em}Proposed LTE-U spectrum allocation algorithm.}
    \begin{tabular}{p{14cm}}
      \hline
      \vspace{.1cm}
{\bf inputs:} $\mathcal{M}; \mathcal{A}$; $\forall i, \pi_i $; $P(\theta)$\\
\vspace{-0.1cm}\textbf{Phase 1 - Specification of the preferences}   \vspace*{.1em}\\
\hspace*{1em}- Each BS-channel pair and user-subfile pair set its preference list based on the utility functions  (\ref{pref_m}) and (\ref{exp_utility_users}), respectively.\vspace*{.1em}\\
\textbf{Phase 3 - Matching algorithm}   \vspace*{.1em}\\
\hspace*{1em}\textbf{Repeat}\vspace*{.1em}\\
\hspace*{2em}- For every subfile $f\in\mathcal{F}_i$, user $i$ sends its preference list to the next BS-band pair $m$ to which it has not applied yet. \vspace*{.3em}\\
\hspace*{2em}- Every BS $s$ updates its applicants list for every channel $c\in\mathcal{C}$ and assigns priorities to the users as defined in Section IV. 2. Then, the BS updates its preference list $\phi_m$ for every pair $m\in\{s\}\times\mathcal{C}$. \vspace*{.3em}\\
\hspace*{2em}- Every BS $s$ accepts the $q_{(s,c)}$ most preferred applicants for every channel $c\in\mathcal{C}$ and rejects the others.\vspace*{.3em}\\
\hspace*{2em}- The rejected user-subfile pairs remove the rejecting BS-channel pair from their list of preferences.\vspace*{.1em}\\
\hspace*{1em}\textbf{Until} no user-subfile pair is rejected.\vspace*{.1em}\\
{\bf Output:} Bayesian stable matching $\mu$.\\
   \hline
    \end{tabular}\label{tab:algo1}\vspace{-0.2cm}
\end{table}
\begin{proposition}
\label{theo_2}
The proposed spectrum allocation algorithm for LTE-U networks is guaranteed to converge to a Bayesian stable outcome.
\end{proposition}
\begin{proof}
The convergence of the algorithm is guaranteed due to the externalities that are canceled out in a matching game with incomplete information. In fact, given that the users estimate their utilities, they do not need to have the exact set of users and subfiles that are served over every channel. Moreover, the fact that the algorithm is based on the classical deferred acceptance algorithm guarantees the convergence to a stable outcome.
\end{proof}

\section{Simulation Results}
\label{sim}
 For our simulations, we consider 200 users unless stated otherwise, deployed in a $1\times1$ km area served by 20 BSs via 12 unlicensed channels and 120 backhaul resource blocks (BRBs). The users can request files of different QoS levels that we rank from 1 to 6 for the corresponding values $[200,250,350,450,550,650] \text{ Mbps}$. The network parameters are provided in Table II.
\begin{center}
\scriptsize
   \begin{table}[t!]
\caption{ Simulation parameters.}\label{tab:table}
 \begin{tabular}{|c|c||c|c|}
 \hline
Parameters & Values & Parameters & Values \\
 \hline
  Total number of BSs & $20$  &  Required rate per user& $\{0,0.2,0.4,0.5,0.6,0.7 \}$Mbps  \\
 \hline
  Total number of WAPs& 10 & WAP communication range & $90$ m \\
 \hline
Total number of unlicensed channels & $12$ &  BS communication range& $ 200$ m  \\
 \hline
Total number of licensed resource blocks & 120 &  Path loss exponent & 3  \\
 \hline
  Noise power& -174 dBm/Hz&  Total number of files & 100 \\
 \hline
Total bandwidth& 1 GHz & Size of a file   & 50 Mbits \\
 \hline

\end{tabular}
\end{table}
\end{center}
\vspace{-1cm}

Fig. \ref{res1} shows the mean achievable rate by the users in the scenario with complete information in which case the BSs know the rate requirement of the users and the Bayesian framework we considered. The performance of the users in the proposed mechanism approaches their performance in the case of complete information. The data rate of the users decreases in the network due to the interference that is larger when the number of users increases in the network. Moreover, the allocated resource blocks per user decreases as the number of users increases.

\begin{figure}
\centering
\includegraphics[scale=0.65]{comp-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Numerical results for the mean data rate per user as a function of the number of users.}\vspace{-0.3cm}
\label{res1}
\end{figure}
In Fig. \ref{res2}, we show the impact of the evolution of users' utility when they select different contracts. A given user achieves its highest utility when selecting the contracts designed for their own types. Indeed, when a user selects a contract of a type higher than its own type, the user cannot afford the resource blocks that are allocated to serve its content. However, when a user selects a contract of lower type, the resource blocks that are allocated to serve its content are not sufficient to meet its QoS requirements.
\begin{figure}
\centering
\includegraphics[scale=0.65]{type-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Users utility resulting from the proposed approach as a function of contracts' type.}\vspace{-0.3cm}
\label{res2}
\end{figure}

In Fig. \ref{res5}, we compare the amount of traffic that is served over unlicensed bands for the type 1 users and type 3 users. Depending on the case we consider, we add type 3 or type 1 users to the network. The results show that the amount of traffic offloaded to the unlicensed bands increases for both type 1 and type 3 users as the number of users increases in the system. However, more type 1 traffic is served over unlicensed bands as the requirement in terms of data rate is less critical compared to type 3 traffic and the amount of offloaded traffic remains the same when the number of users exceeds 600 due to the saturation of the unlicensed bands.

\begin{figure}
\centering
\includegraphics[scale=0.65]{traffic_LTEU-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Amount of traffic offloaded to the unlicensed bands as a function of the number of users.}\vspace{-0.3cm}
\label{res5}
\end{figure}

Fig. \ref{res3} shows the fraction of users that achieve their QoS requirement as a function of the number of users in the networks. In Fig. \ref{res3}, we compare the proposed mechanism with an allocation framework in which a random number of users accept to be served over the unlicensed channels. The fraction of users that achieve their QoS decreases by increasing the number of users in the networks because the licensed and unlicensed channels have to be shared among a large number of users which decreases their data rate. The proposed mechanism outperforms by up to 45\% the unaware QoS resource allocation algorithm. This is due to the random allocation of the licensed and unlicensed bands that does not account for the requirements of the users. Thus, the random allocation can result in users with high QoS requirements served via unlicensed channels.
\begin{figure}
\centering
\includegraphics[scale=0.6]{fraction-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Fraction of users that achieve their required QoS as a function of the number of users.}\vspace{-0.3cm}
\label{res3}
\end{figure}

In Fig. \ref{res4}, we compare the proposed contract-based pricing scheme with a uniform pricing mechanism in which the same price is fixed for serving the files. The results in Fig. \ref{res4} show the evolution of the users' utility as a function of the number of users in the system. The experienced utility per the users decreases when increasing the number of users. This is due to the interference from the BSs that impact the achievable data rate per user. When the network has more than 600 users, the difference between the two schemes becomes substantial as the average utility per user becomes twice higher than the utility they achieve under the uniform pricing scheme. This is due to the prices fixed by the MNO that are adapted to the requirement of the users which incite them to select the contract that offers the best possible data rate by accounting for other users requirements and traffic load.

\begin{figure}
\centering
\includegraphics[scale=0.7]{con_uniform-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Users' utility resulting from the proposed approach and a uniform pricing scheme.}\vspace{-0.3cm}
\label{res4}
\end{figure}

\begin{figure}
\centering
\includegraphics[scale=0.7]{LTEU-SBS-eps-converted-to.pdf}\vspace{-0.1cm}
\caption{Amount of traffic served over licensed bands as a function of the number of BSs.}\vspace{-0.3cm}
\label{res6}
\end{figure}
In Fig. \ref{res6}, we compare the amount of traffic that is served over licensed bands for type 1 users and type 3 users as a function of the number of BSs in the network. The results show that the amount of traffic served via the licensed bands decreases initially for both type 1 and type 3 users as the number of BSs becomes higher in the system. However, more type 1 traffic is served over licensed bands as the data rate requirement of type 1 users is less critical compared to type 3 traffic. Moreover, Fig. \ref{res6} shows that the amount of traffic served over the unlicensed bands remains relatively the same for both type 1 and type 3 users when the number of BSs exceeds 60 BSs. This is due to the limited licensed spectrum resources that are shared among all of the BSs.

\section{Conclusion}
\label{con}
In this paper, we have proposed a novel LTE-U incentive mechanism that motivates users to accept being served over unlicensed channel. To this end, we have used contract theory enabling every MNO to set a pricing mechanism for LTE-U users and then used matching theory to determine the fraction of LTE traffic that should be offloaded to the unlicensed bands while accounting for the QoS requirements of every content. The main advantage of using contract theory is the ability to model information asymmetry and incompleteness given that it is difficult for MNOs to gather information about users requirements. The outcome of the contract mechanism is a contract defined by the price charged by the MNO for serving the content as well as the amount of resource that is required to achieve the required QoS by the users. Moreover, we have proposed a distributed matching algorithm to allocate both licensed and unlicensed frequency bands to the users. Simulations results have shown that the traffic can be better allocated using the proposed mechanism and increase the fraction of users that are served at their requested QoS.

\begin{appendices}
\section{Proof of Lemma 1}
We divide the proof into two parts. First, we prove that if $\bar{\pi}_i(\theta_i)\geq \bar{\pi}_i(\hat{\theta}_i)$, then $\bar{v}_i(\theta_i)\geq \bar{v}_i(\hat{\theta}_i)$. Due to TIBS constraint in (4), we have:
\begin{equation}
\theta_i \bar{v}_i(\alpha,\theta_i)-\bar{\pi}_i(\theta_i)\geq \theta_i \bar{v}_i(\alpha,\hat{\theta}_i)-\bar{\pi}_i(\hat{\theta}_i),
\end{equation}
i.e.,
\begin{equation}
\theta_i\Big[ \bar{v}_i(\alpha,\theta_i)-\bar{v}_i(\alpha,\hat{\theta}_i)\Big]\geq -\bar{\pi}_i(\theta_i) -\bar{\pi}_i(\hat{\theta}_i).
\end{equation}
Since $\bar{\pi}_i(\theta_i)\geq\bar{\pi}_i$, we conclude:
\begin{equation}
\theta_i\Big[ \bar{v}_i(\alpha,\theta_i)-\bar{v}_i(\alpha,\hat{\theta}_i)\Big]\geq -\bar{\pi}_i(\theta_i) -\bar{\pi}_i(\hat{\theta}_i)\geq 0.
\end{equation}
and thus, $\bar{v}_i(\alpha,\theta_i)\geq \bar{v}_i(\alpha,\hat{\theta}_i)$. Next, we prove that if $\bar{v}_i(\alpha,\theta_i)\geq \bar{v}(\alpha,\hat{\theta}_i)$, then $\bar{\pi}_i(\theta_i)\geq \bar{\pi}_i(\hat{\theta}_i)$. Due to TIBS constraint in (4), we have
\begin{equation}
\theta_i\bar{v}_i(\alpha,\theta_i)-\bar{\pi}_i(\theta_i)\geq \theta_i\bar{v}_i(\alpha,\hat{\theta}_i)-\bar{\pi}_i(\hat{\theta}_i),
\end{equation}
which can be transformed to be
\begin{equation}
\bar{\pi}_i(\hat{\theta}_i)-\bar{\pi}_i(\theta_i)\geq \theta_i\Big[\bar{v}_i(\alpha,\hat{\theta}_i)-\bar{v}_i(\theta_i)\Big].
\end{equation}
Since $\bar{v}_i(\alpha,\hat{\theta}_i)\geq\bar{v}_i(\theta_i)$, we conclude:
\begin{equation}
\bar{\pi}_i(\hat{\theta}_i)-\bar{\pi}_i(\theta_i)\geq \theta_i\Big[\bar{v}_i(\hat{\theta}_i)-\bar{v}_i(\alpha,\theta_i)\Big]\geq0,
\end{equation}
and thus $\bar{\pi}_i(\hat{\theta}_i)\geq\bar{\pi}_i(\theta_i)$.

\section{Proof of Theorem1}
First part of the proof ``if-part": First, we show that 1) and 2) in Theorem \ref{theorem} imply TIBS. Take any two values $\theta_i,\hat{\theta}_i\in[\ubar{\theta}_i,\bar{\theta}_i]$ with $\theta_i>\hat{\theta_i}>\ubar{\theta}$. From 2) we have:
\begin{equation}
\bar{u}_i(\theta_i)-\int_{\bar{\theta}_i}^{\theta_i}\bar{v}_i(\alpha,\eta)d\eta=\bar{u}_i(\hat{\theta}_i)-\int_{\hat{\theta}_i}^{\hat{\theta}_i}\bar{v}_i(\alpha,\eta)d\eta.
\end{equation}
Rearranging terms and using 1) gives:
\begin{flalign}
\bar{u}_i(\theta_i)-\bar{u}_i(\hat{\theta}_i)=\int_{\hat{\theta}_i}^{\theta_i}\bar{v}_i(\alpha,\eta)d\eta\geq\int_{\hat{\theta}_i}^{\theta_i}\bar{v}_i(\alpha,\hat{\theta}_i)d\eta=(\theta_i-\hat{\theta}_i)\bar{v}_i(\alpha,\hat{\theta}_i).
\end{flalign}
Note that $\bar{v}_i(\alpha,\hat{\theta}_i)$ is a constant. Hence,
\begin{equation}\label{1}
\bar{u}_i(\theta_i)\geq\bar{u}_i(\hat{\theta}_i)+(\theta_i-\hat{\theta}_i)\bar{v}_i(\alpha,\hat{\theta}_i)=\theta_i\bar{v}_i(\alpha,\hat{\theta}_i)+\bar{\pi}_i(\hat{\theta}_i).
\end{equation}
Similarly, suppose that $\hat{\theta}_i\geq \theta_i\geq\ubar{\theta}_i$. By the same reasoning, we have
\begin{equation}\label{2}
\bar{u}_i(\hat{\theta}_i)\geq \bar{u}_i(\theta_i)+(\hat{\theta}_i-\theta_i)\bar{v}_i(\alpha,\theta_i)=(\hat{\theta}_i-\theta_i)\bar{v}_i(\alpha,\theta_i).
\end{equation}

Together, (\ref{1}) and (\ref{2}) imply TIBS. Next, we prove that 1), 2) and 3) in Theorem \ref{theorem} imply IIR. We do this by contradiction. Suppose the implication is wrong. Then, there exists some $\theta_i\geq\ubar{\theta}_i$ with $\bar{u}_i(\theta_i)\leq0$. We just established that 1) and 2) imply TIBS. However, TIBS in conjunction with 3) implies:
\begin{equation}
\bar{u}_i(\theta_i)\geq \theta_i\bar{v}_i(\alpha,\ubar{\theta}_i)-\bar{\pi}_i(\ubar{\theta}_i)>\ubar{\theta}_i\bar{v}_i(\alpha,\ubar{\theta}_i)-\bar{\pi}_i(\ubar{\theta}_i),
\end{equation}
a contradiction.

"Only if part": We now show that TIBS implies 1), and 2). For any $i\in\mathcal{N}$ and any two types $\theta_i, \hat{\theta}_i\in[\ubar{\theta}_i,\bar{\theta}_i]$, TIBS requires that,
\begin{equation}\label{3}
\bar{u}_i(\theta_i)\geq \theta_i\bar{v}_i(\alpha,\hat{\theta}_i)-\bar{\pi}_i(\hat{\theta}_i)=\bar{u}_i(\hat{\theta}_i)+(\theta_i-\hat{\theta}_i)\bar{v}_i(\alpha,\hat{\theta}_i),
\end{equation}
and,
\begin{equation}\label{4}
\bar{u}_i(\hat{\theta}_i)\geq \hat{\theta}_i\bar{v}_i(\alpha,\theta_i)-\bar{\pi}_i(\theta_i)=\bar{u}_i(\theta_i)+(\theta_i-\hat{\theta}_i)\bar{v}_i(\alpha,\theta_i).
\end{equation}
Suppose without loss of generality that $\theta_{i}\leq\hat{\theta}_i$. From (\ref{3}) and (\ref{4}), it follows that
\begin{equation}
\bar{v}_i(\alpha,\hat{\theta}_i)\geq \frac{\bar{u}_i(\theta_i)-\bar{u}(\hat{\theta}_i)}{\theta_i-\hat{\theta}_i}\geq \bar{v}_i(\alpha,\theta_i),
\end{equation}
which shows that $\bar{v}_i(.)$ is nondecreasing. Next, letting $\hat{\theta}_i\to\theta_i$, we obtain $\frac{d\bar{u}_i(\theta_i)}{d\theta_i}=\bar{v}_i(\theta_i)$ for all $\theta_i$. Integrating both sides over $[\ubar{\theta}_i,\theta_i]$ gives
\begin{equation}
\bar{u}_i(\theta_i)=\bar{u}(\theta_i)+\int_{\ubar{\theta}_i}^{\theta_i}\bar{v}_i(\alpha,\eta)d\eta,
\end{equation}
$\forall \theta_i$. Finally, note that IIR obviously implies 3) by definition of IIR.

\section{Proof of Proposition \ref{prop}}
\subsection{Feasibility conditions:} We first show how we derive the price in (\ref{price}). We replace conditions (\ref{IR}) and (\ref{IC}) in (\ref{problem}) by the conditions provided in Theorem 1:
\begin{equation}
\label{pb}
\begin{aligned}
& \underset{\{\alpha_i(\boldsymbol{\theta}),\boldsymbol{\pi}_i(\boldsymbol{\theta})\}_{\forall i}}{\text{max}}
& &\int_{\Theta}\sum_{i\in\mathcal{N}}\left[\pi_i(\theta_i,\boldsymbol{\theta}_{-i})-c_i(\theta_i,\boldsymbol{\theta}_{-i})\right]dP(\theta),  \\
& \text{subject to}
& &  \bar{v}_i(\theta_i) \text{ is nondecreasing}, \\
&&& \bar{u}_i(\theta_i)=\bar{u}_i(\ubar{\theta}_i)+\int_{\ubar{\theta}_i}^{\theta_i}{\bar{v}_i(\alpha,\eta)d\eta}, \forall \theta_i, i\in\mathcal{N},\\
&&& \bar{u}_i(\ubar{\theta}_i)\geq 0, \forall i\in\mathcal{N},\\
&&& 0\leq\alpha_i(\boldsymbol{\theta})\leq 1,\forall i\in\mathcal{N}.
\end{aligned}
\end{equation}
After doing some algebraic manipulations on the second constraint in (\ref{pb}), we have:
\begin{flalign}
\label{cons}
-\int_{\Theta}\pi_i(\boldsymbol{\theta})=\ubar{\theta}_iv_i(\alpha,\ubar{\theta}_i)-\pi_i(\ubar{\theta}_i)-\int_{\Theta}\pi_i(\boldsymbol{\theta})\left(\theta_i-\frac{1-P_i(\theta_i)}{p_i(\theta_i)}\right)dP(\theta).
\end{flalign}
Inserting (\ref{cons}) in the objective function of the optimization problem (\ref{pb}), the problem can be relaxed and written as:
\begin{equation}
\label{relax}
\begin{aligned}
& \underset{\{\alpha_i(\boldsymbol{\theta}),\boldsymbol{\pi}_i(\boldsymbol{\theta})\}_{\forall i}}{\text{max}}
& &\int_{\Theta}\sum_{i\in\mathcal{N}}\left[\pi_i(\theta_i,\boldsymbol{\theta}_{-i})-c_i(\theta_i,\boldsymbol{\theta}_{-i})\right]dP(\theta),  \\
& \text{subject to}
& &  \bar{v}_i(\alpha,\theta_i) \text{ is nondecreasing}, \\
&&& \bar{u}_i(\ubar{\theta}_i)\geq 0, \forall i\in\mathcal{N},\\
&&& 0\leq\alpha_i(\boldsymbol{\theta})\leq 1,\forall i\in\mathcal{N}.
\end{aligned}
\end{equation}

We can see that the second constrain in (\ref{relax}) must bind, thus we have:
\begin{equation}
\label{e}
\bar{u}_i(\ubar{\theta}_i)=0,
\end{equation}
for all $i\in\mathcal{N}$. By Theorem \ref{theorem}, we can write (\ref{e}) as:
\begin{equation}
\bar{\pi}_i^*(\theta_i)= \theta_i\bar{v}_i(,\alpha_i^*,\theta_i)+\int_{\ubar{\theta}_i}^{\theta_i}\bar{v}_i(\alpha_i^*,\eta)d\eta,
\end{equation}
where we used $u_i(\theta_i)=\theta_i\bar{v}_i(\alpha_i,\theta_i)-\bar{\pi}_i(\theta_i)$. Next, we proof the optimality and the uniqueness of this solution.

\subsection{Proof of optimality:} We first show that the optimal price in (\ref{prop}) maximizes the utility of the MNO assuming that the amount of traffic served over the licensed and unlicensed bands is fixed to $\{\alpha_i^*, \forall i\}$. Suppose that that there exists another feasible price $\{\tilde{\pi}_i, \forall i\}$ which achieves a better solution than $\{\pi^*_i,\forall i\}$ in (\ref{prop}). Since the expected utility of the MNO given by the objective function in (\ref{problem}) is an increasing function of the total sum of the prices $\sum_{i\in\mathcal{N}}\pi_i$ payed by the users, we must have that $\sum_{i\in\mathcal{N}}\tilde{\pi}_i>\sum_{i\in\mathcal{N}}\pi^*_i$. Thus, for at least one user of type $\theta_i$, we have $\tilde{\pi}_i>\pi^*_i$ which is equivalent to $\overline{\tilde{\pi}}_i>\overline{\pi^*}_i$ because we have $\bar{\pi}_i(\theta_i,\boldsymbol{\theta}_{-i})=\int_{\Theta_{-i}}\pi_i(\theta_i,\boldsymbol{\theta}_{-i})dP_{-i}(\boldsymbol{\theta}_{-i})$.

If $i=1$, then $\overline{\tilde{\pi}}_1>\overline{\pi^*}_1$. Since $\overline{\pi^*}_i=\theta_1\bar{v}_1(\alpha_i,\theta_i)$, then $\overline{\tilde{\pi}}_1>\theta_1\bar{v}_1(\alpha_i,\theta_i)$. However, it violates the IIR constraint for type $\theta_1$. Thus, we have to check if for any $i>1$ the supposition can be satisfied.

Since $\{\tilde{\pi}_i,\forall i\}$ is feasible, then $\{\tilde{\pi}_i,\forall i\}$ must satisfy the right inequality of  condition 3) in Theorem \ref{theorem}. Thus, we have:
\begin{equation}
\overline{\tilde{\pi}}_i\leq\overline{\tilde{\pi}}_{i-1}+\theta_i(\bar{v}_i(\alpha_i,\theta)-\bar{v}_{i-1}(\alpha_{i-1},\theta_{i-1})).
\end{equation}
By substituting $\theta_i(\bar{v}_i(\alpha_i,\theta)-\bar{v}_{i-1}(\alpha_{i-1},\theta_{i-1}))=\overline{\pi_i^*}-\overline{\pi^*}_{i-1}$  as in (\ref{prop}) into this inequality, we have
\begin{equation}
\overline{\tilde{\pi}}_{i-1}>\overline{\pi^*}_{i-1}.
\end{equation}
Using the above argument repeatedly, we finally obtain that
\begin{equation}
\overline{\tilde{\pi}}_1>\overline{\pi^*}_1=\theta_1\bar{v}_1(\alpha_1,\theta_1),
\end{equation}
which violates the IIR constraint for type-$\theta_1$ again.
\subsection{Proof of uniqueness:} Here, we prove that the price defined in (\ref{prop}) is the unique solution that maximizes the objective function in (\ref{problem}). We also prove by contradiction. Suppose that there exists another $\{\hat{\pi}_i,\forall i\}\neq\{\pi_i^*,\forall i\}$ such that $\sum_{i\in\mathcal{N}}{\overline{\hat{\pi}}_i}=\sum_{i\in\mathcal{N}}{\overline{\pi^*}_i}$ in the objective function of (\ref{problem}).

Thus, there is at least one user for which the price satisfies, $\overline{\hat{\pi}}_i<\overline{\pi^*}_i$ and one user for which $\overline{\hat{\pi}}_l>\overline{\pi^*}_l$. We can focus on type $\theta_l$ and $\overline{\hat{\pi}}_l>\overline{\pi^*}_l$.
By using the same argument as before, we have $\overline{\hat{\pi}}_1>\overline{\pi^*}_1=\theta_1\bar{v}_1(\alpha_1,\theta_1)$ which violates the IIR constraint for type $\theta_1$.

\end{appendices}

\bibliographystyle{IEEEtran}

redpajama_set_name: RedPajamaArXiv


Over the past few decades, the welfare sector in Sweden has been increasingly exposed to competition. Even though this change has caused an intense debate, the consequences for the workers have gained less interest. The aim of this study is to investigate how the system of customer choice (LOV) in home based elderly care affects the workers' employment and working conditions. The study draws mainly upon experiences from three larger municipalities in Central Sweden. The source material is primarily in-depth interviews with employees, trade union representatives, employers and local government officials. The study has an intersectional approach, and the analysis also relates to the scholarly debate on quasi markets, rationality, flexibility and precarious employment.
The results show, among other things, that the customer choice model, including the reimbursement system in use to pay the providers, implies structural risks for temporary employment, reduced manning and stress. Since the system does not include any price competition, the providers are supposed to compete on quality. In fact, however, they also tend to compete by reducing labour costs. The reimbursement system also presupposes rigorous planning and increased and individualised (often digitalised) control of the workers. Moreover, the providers seek to maximise the time the workers are in direct contact with the customers/users, which is the time they can invoice. This means less time for transports, personal meetings, occupational health and safety training and so on. The customer-focused rhetoric in this system helps to legitimize this practice.
Another finding is that many small service providers operating within this quasimarket lack collective agreements. The workers can sometimes be subjected to severe abuse, for example through cash-in-hand payment and work in the grey area between regular employment and self-employment. The employers often use workers who are participating in public labour market programs as cheap labour. Since many companies use foreign languages, ethnicity and culture for competitive advantage, workers with an immigrant background are most at risk of being abused. Even though the local authorities have the main responsibility for the elderly care, they strictly maintain that issues regarding employment and working conditions should be handled by the social partners alone. A conclusion is that trade unions, employers and the local governments need to establish a tripartite cooperation in order to combat unfair competition, labour abuse and the undercutting of wages and labour standards.


Phalaenopsis sumatrana Korth & Rchb. f.
Etymology: From Sumatra island were the species is native.
Synonyms: Phalaenopsis gersenii Rolfe -- Orchid Rev. xxv. 226 (1917).
Phalaenopsis zebrina Teijsm. & Binn. -- Natuurk. Tijdschr. Ned.-Indië xxiv. (1862) 319.
Polychilos sumatrana (Rchb.) Shim, Malayan Nat. Journ. 36(1):26. 1982.
Primary hybrids: You can view Phalaenopsis sumatrana primary hybrids by species from the graphic page or via the following alphabetical listing.


I ran into Tony winner, Cady Huffman, in December at Gypsy of the Year. She choreographed the opening number and I was dancing in piece by Kristine Bergasse with a cast of my contemporaries. It was great catching up with some old pals. The best part was partnering the spectacular Karine Plantadit - we've known each other for years but never had the chance to dance together. Cady is producing and starring, with Vienne Cox, in a new webseries call Cady Did - which will premiere later this year. At rehearsal for Gypsy she asked me to play her agent in the first episode. Who would refuse? She did a great job producing this show - very professional. Felt just like a regular SAG gig and playing opposite the lady herself was a total hoot. Can't wait to watch the first season.


Vibe Marketplace has a local history of providing quality home decor combined with excellent customer service. We pride ourselves on our craftsmanship with a friendly, fast follow up to questions. We know you're making an investment and we're honored to have a small impact on making your house a home.
Our custom handmade sliding barn doors and barn door packages fit into just about any decor... from rustic farmhouse to modern decor. They are sure to draw attention and are an eye popping focal point in your home. Barn doors are excellent for bedrooms, bathrooms, living rooms, laundry rooms and pantries. Use them as rustic queen and king headboards or wall decor... so many options!
We start by hand selecting each board from a local lumber mill. We commonly use pine, however cedar, oak and alder are also available for an additional cost. We take pride in the details of creating a unique, custom piece for you...we sand and stain each board. When building your sliding barn door we use both wood glue and finishing nails for durability and longevity. We stain both sides of your wood door however pattern/design are only on one side, leaving the paneling in the back. We'll gladly fill nail holes for an addl' $60. PLEASE NOTE: Wood fill for holes will still be slightly noticeable as the fill doesn't stain exactly like real wood.
We want your sliding barn door to be a fully custom piece and we give you the option of how your sliding door is sealed. We normally use polyurethane semi-gloss however we also offer satin and glossy finishes as well. Installation holes are not pre drilled, this is something I normally advise doing when barn door hardware is installed. Basic tools needed to install hardware; Drill, bits, level and ratchet are helpful. Any other questions, please ask!
Local orders will receive FREE install!
• Or Just to say hi!
We strive to construct your order as quickly as possible, however orders are fulfilled on a first come first serve basis and can take up to 20 business days to ship. Need a rush and need it sooner than that? Ask about expediting your order. NOTE: Our lead time is currently 4-6 weeks. If you have a "need by" date your working against, please let us know and we will do our best to accommodate you.
WE ACTUALLY DON'T CHARGE SHIPPING!
Shipping is free for all doors regardless of location. We know this can be a hinderance to non-local buyers, and we strive to make it as easy for a customer to receive one of our custom doors whether you're on the east coast, west coast or anywhere in between! Want it for local use? No problem, we'll have your item packaged and ready for pick up on an agreed upon date!


Super cute and trendy subtle gladiator style flats with zip up heel on back. These will look great when paired with your favorite cuffed jeans and your perfect graphic t-shirt or dress these up for work with a super stylish dress and add a matching, classy statement necklace! Shop away, you sassy thing! Fit true to size. (Colors: Tan and Pale Yellow with Gold Accents).
*Please Note: This sassy style is considered FINAL SALE.


Defragmented is a lightning-fast cyberpunk action RPG. Unravel a power struggle for the future of evolution through strategic combat, unique character classes and a pulsing synth soundtrack. Welcome to Entropolis – New World, Bad Blood.
Defragmented is a clever mix of varied RPG mechanics and hectic action-oriented shooting. While the camera needs a bit of work to use properly, you can still enjoy the great visuals and the stellar soundtrack.


Data Warehouse Services – TribolaTech Inc.
Organizations rely on data warehouse platforms to deliver actionable, timely, and trustworthy intelligence. Enterprise Data Warehouse (EDW) technology organizes and aggregates analytical data from various data sources and serves as a central repository for operations and analysis. It offers . Tribolatech helps businesses improve performance by designing, developing and implementing data warehousing solutions. Through our technical expertise, in-database analytics, predictive models, and embedded business algorithms we can help organizations drive better business decisions.
With data explosion at its peak, storing valuable data is a primary concern for many organizations. We work with business to determine the optimal method that best suits your needs. Our partners offer "hybrid" deployments, in which cloud software is hosted on an organization's private servers. Whether you are looking to lower your infrastructure costs or to optimize cloud strategy, TribolaTech is there every step of the way.
Let TribolaTech take the guesswork out of the equation. While a data lake has nearly unlimited potential but requires complex transformations before achieving insights. On the other hand, data warehouse requires significant investment in advance, yet in return delivers the ability to easily organize and analyze everything. Our solutions will ensure that you get the best results from all your data sources.
Security is often the top concern considering how critical the information stored including company financials, corporate trade secrets, employee information, client lists and more. Most reputable cloud vendors have strict standards in place to keep data safe that are compliant to the regulatory standards enforced by world organizations. TribolaTech data experts are well versed with data regulations and norms and will work to ensure your data is safe, secure and compliant.
You have questions, we have answers. With lot of data comes lots of questions! What is the nature of my data? Do I need Real Time or is batch better? Structured or Unstructured? What About Data Volatility? How clean is my data? TribolaTech has solutions for your questions and solutions to tackle your problems.


text:
Online Laboratory Magazine
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Lodz University of Technology - Institute of Polymer and D...
Triple Quad LC-MS for detection of luciferin-6'-p-nitrobenzyl ether - a footprint of peroxynitrite
Radoslaw Podsiadly, Aleksandra Grzelakowska, Lodz University of Technology - Institute of Polymer and Dye Technology
Jacek Zielonka, Medical College of Wisconsin
Peroxynitrite (ONOO-), a highly reactive oxidizing and nitrating agent is generated via a spontaneous, diffusion-controlled reaction between nitric oxide (·NO) and superoxide radical anion (O2·-). Peroxynitrite has been implicated as a key pathophysiological intermediate in various diseases, including acute and chronic inflammatory processes, diabetes, sepsis, ischemia-reperfusion, atherosclerosis and neurodegenerative disorders as well as chemotherapy-induced nephrotoxicity.
Low steady state concentrations and unfavorable spectroscopic properties make direct detection of peroxynitrite in cells practically impossible. One of the methods to detect ONOO- is to use a chemical probe which is selective and/or forms a footprint product after reaction with this oxidant.
Oxidation of arylboronic acid
Over the last decade, it has been demonstrated that several oxidants such as hydrogen peroxide, hypochlorous acid, protein hydroperoxides and peroxynitrite convert arylboronic acids (ArB(OH)2) to phenolic products ArOH. However, the reactivity of boronates with peroxynitrite (ONOO-) is of special interest. Not only the rate constant of the reaction of boronate probes with ONOO- is the highest of all tested biological oxidants, but the reaction also typically leads to the formation of minor but ONOO--specific products (ArNO2, ArH) in addition to the major phenolic product (ArOH). Since the minor pathway is specific only for the reaction of boronates with peroxynitrite, formation of these products (ArNO2, ArH) may serve as a unique footprint providing a diagnostic marker for peroxynitrite formation in cells-free and cellular systems.
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When I first heard about a trail 50 miler in the Philadelphia city limits, I was shocked and curious. Being in the 5th largest city in the country has its perks, but that doesn't always include "plethora of trails to run." There are beautiful trails here, for sure, but none that compare to those out west. I knew that I had to talk with the Race Director (RD) of the race and learn more about the series of races he puts on each year.
Stephan Weiss started running like many people, hoping to get into better shape and to quit smoking. His father-in-law passed away from lung cancer and with kids of his own, Weiss wanted to get into shape and live a long, healthy life. He started out building his mileage on the road, like a lot of runners (myself included). Within 18 months, he went from not running to toeing the line to the Boston Marathon. Speedy for sure (marathon PR is 2:46:28 at the 2010 Steamtown Marathon), he found out about local trail races and spent more time racing and running on the trails.
After another local race, the Blues Cruise 50k, was out of a RD, he and his wife decided to tackle the job and found themselves loving it. He decided to put on a yearly race series with a German spin (Weiss was born in Germany and spent most of his life there) and created the Überendurance Sports, a group that put on multiple races throughout the year, ranging from a 10 miler to a 50 miler, with other distance and timed races in between.
I asked Weiss about his ultra on a trail in the city of Philadelphia and he said he wasn't trying to make a statement about trail running in Philadelphia, but that the park was local to him, he knew the trails, and felt it was a good place to hold races. The Dirty German Endurance Fest, which has 25k, 50k, and 50 mile options, is held in Pennypack Park, a park established in 1905 and popular to many trail runners and outgoers alike. The course allows beginners to dip their toes into trail running, but allows others looking for a distance challenge to have a fun day. The course is a figure 8 that is 25k long with plenty of aid stations and a fairly non-technical course for beginners and experienced runners alike.
When I asked Weiss what makes this race special, he brought up that his wife is a pastry chef and that her goodies will be found at the finish line. A German-themed race with homemade baked goods from a professional chef? Be still my heart. There is also a German-themed aid station and in previous years, runners have raced in lederhosen and beer girl dresses. Age group winners can also go home with an authentic German weather house and I know many local runners who signed up for the race JUST to win this.
When it comes to advice, Weiss says, especially for beginners, to not expect it to be like your typical road race. With bigger road races, runners zoom in and out of aid stations. He said to stop, enjoy the aid station, allow yourself a few seconds to fill up your water bottle, eat snacks, and talk with the volunteers. He brings up something that I loved so much when I first started running ultras: it's so much more about slowing down, appreciating your experience, absorbing all the positive energy coming from other racers and volunteers, and mainly, having a great day!
Up to 500 racers will toe the line of the Dirty German Endurance Fest, where I will be running my first stand-alone 50 miler. I may or may not be in a beer maid outfit (leaning toward definitely), but I am eager to add to the growing trail running and ultra running community here in the City of Brotherly Love.
For more details on this race, which is being held May 18th, and their other races, stop by their website to check it out!


text:
Here's how to help Uvalde shooting victims, survivors and their families
written by MyParisTexas | May 27, 2022
Nineteen children and two adults were killed in the second-deadliest school shooting in U.S. history. Here are some ways people across the world can help the grieving community of Uvalde in the aftermath.
While families and community members in Uvalde process their grief after the worst school shooting in Texas history, they are also dealing with medical, funeral and legal expenses. Here's how you can help.
Belong, a division of SJRC Texas, a nonprofit that cares for children and families that have faced trauma, put together a list of resources for those affected by the Uvalde shooting, including financial, medical, mental health and grief support services.
Monetary donations
Crowdfunding platform GoFundMe set up a page with verified fundraisers put together by family members of shooting victims and nonprofit organizations.
The Uvalde Consolidated Independent School District created a bank account at First State Bank of Uvalde where people can send funds directly to shooting victims and their families. Funds can be sent electronically through Zelle using the email robbschoolmemorialfund@gmail.com or through the mail at the address 200 East Nopal St., Uvalde, Texas 78801. Make checks payable to the "Robb School Memorial Fund."
The City of Uvalde set up a separate fund to which people can mail checks payable to the "Robb School Memorial Fund" to P.O. Box 799, Uvalde, Texas 78802.
University Health San Antonio has also set up the Uvalde Victim Relief Fund for shooting victims and their families on their website. To donate to this fund, people should select "SP-Uvalde Victims Relief Fund" under designation when making a one-time or recurring donation.
The League of United Latin American Citizens is also fundraising for survivors and families of Uvalde shooting victims, saying that 100% of one-time or recurring donations will go directly to those affected by the shooting.
The South Texas Blood and Tissue Center and University Health San Antonio are asking for blood donations to help supplement survivors of the shooting. People can schedule an appointment with the center here or with University Health here.
Roger Ruiz, a senior communications specialist for the center, said although blood donation appointments are booked through next week, people need to continue to donate blood beyond this tragedy, as centers across the country have experienced a blood shortage for about two years now.
"Just because we saw 600 donors doesn't mean that we have a seven-day supply of blood on the shelves," Ruiz said. "You can remember on top of what we're sending out with Uvalde, we still have to keep our orders with all our hospitals because there's still patients in our area that still need transfusions, so that's why it's so important to have an adequate blood supply."
The center is asking people who have made appointments to call the center at 210-757-9509 if they can't make it so they can schedule another donor.
We Are Blood, a nonprofit blood donation center that serves Central Texas, is partnering with AT&T to host a blood drive at the Texas Capitol on Thursday from 9 a.m. to 3 p.m. in honor of the Uvalde shooting victims. Two buses will be parked at 1220 Colorado Street to take blood donations from walk-ins only, no appointments. While We Are Blood has been in contact with the South Texas Blood and Tissue Center, the center has not asked We Are Blood to send blood to Uvalde as of Wednesday afternoon, said Nick Canedo, the vice president of community engagement for We Are Blood. But We Are Blood remains on standby to assist if needed, Canedo said.
Local funeral parlors are offering free funeral services to the families of those who were killed in the shooting.
The Rushing-Estes-Knowles Mortuary said in a Facebook post their professionals are helping law enforcement at Robb Elementary School and that families would not be charged for their services.
The Hillcrest Memorial Funeral Home also said they are assisting families affected by the Uvalde shooting at no cost.
The San Antonio Legal Services Association called on Facebook for volunteer attorneys to offer free legal services to victims' families and survivors.
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Photo: Bobby Kramer, 84, and Charlie Garvan, 52, sit next to each other as they donate blood at the Herby Ham Activity Center on Wednesday, May 25, 2022, in Uvalde. Community members gathered to donate blood after a school shooting took place Tuesday.
BY EMILY HERNANDEZ with Texas Tribune.