In the contest design problem, initiated by Lazear and Rosen (JPE’81), there are n strategic contestants, each of whom decides an effort level. A contest designer with a fixed budget must then design a mechanism that allocates a prize pi to the i-th rank based on the outcome, to incentivize contestants to exert higher costly efforts and induce high-quality outcomes.

We study the bidding problem in repeated uniform price multi-unit auctions from the perspective of a single *value-maximizing* buyer who aims to maximize their cumulative value over $T$ rounds while adhering to return-on-investment (RoI) constraints in each round. Buyers adopt $m$-*uniform bidding* format, where they submit $m$ bid-quantity pairs $(b_i, q_i)$ to demand $q_i$ units at bid $b_i$. We introduce *safe* bidding strategies as those that satisfy RoI constraints in every auction, regardless of competing bids. We show that these strategies depend only on the bidder’s valuation curve, and the bidder can focus on a finite subset of this class without loss of generality. While the number of strategies in this subset is exponential in $m$, we develop a polynomial-time algorithm to learn the optimal safe strategy that achieves sublinear regret in the online setting, where regret is measured against a clairvoyant benchmark that knows the competing bids *a priori* and selects a fixed hindsight optimal safe strategy. We then evaluate the performance of safe strategies against a clairvoyant that selects the optimal strategy from a richer class of strategies in the online setting. In this scenario, we compute the *richness ratio*, $\alpha\in(0, 1]$ for the class of strategies chosen by the clairvoyant and show that our algorithm, designed to learn safe strategies, achieves $\alpha$-approximate sublinear regret against these stronger benchmarks. Experiments on semi-synthetic data from real-world auctions show that safe strategies substantially outperform the derived theoretical bounds, making them quite appealing in practice.

Motivated by applications such as cloud platforms allocating GPUs to users or governments deploying mobile health units across competing regions, we study the constrained dynamic allocation of a reusable resource to a group of strategic agents. Our objective is to simultaneously (i) maximize social welfare, (ii) satisfy multi-dimensional long-term cost constraints, and (iii) incentivize truthful reporting. We begin by numerically evaluating primal-dual methods widely used in constrained online optimization and find them to be highly fragile in strategic settings -- agents can easily manipulate their reports to distort future dual updates for future gain. To address this vulnerability, we develop an incentive-aware framework that makes primal-dual methods robust to strategic behavior. Our primal-side design combines epoch-based lazy updates -- discouraging agents from distorting dual updates -- with dual-adjust pricing and randomized exploration techniques that extract approximately truthful signals for learning. On the dual side, we design a novel online learning subroutine to resolve a circular dependency between actions and predictions; this makes our mechanism achieve $\tilde{\mathcal{O}}(\sqrt{T})$ social welfare regret (where $T$ is the number of allocation rounds), satisfies all cost constraints, and ensures incentive alignment. This $\tilde{\mathcal{O}}(\sqrt{T})$ performance matches that of non-strategic allocation approaches while additionally exhibiting robustness to strategic agents.

Today's online platforms heavily lean on algorithmic recommendations for bolstering user engagement and driving revenue. However, these recommendations can impact multiple stakeholders simultaneously---the platform, items (sellers), and users (customers)---each with their unique objectives, making it difficult to find the right middle ground that accommodates all stakeholders. To address this, we introduce a novel fair recommendation framework, Problem (FAIR), that flexibly balances multi-stakeholder interests via a constrained optimization formulation. We next explore Problem (FAIR) in a dynamic online setting where data uncertainty further adds complexity, and propose a low-regret algorithm FORM that concurrently performs real-time learning and fair recommendations, two tasks that are often at odds. Via both theoretical analysis and a numerical case study on real-world data, we demonstrate the efficacy of our framework and method in maintaining platform revenue while ensuring desired levels of fairness for both items and users.

Online advertising channels commonly focus on maximizing total advertiser welfare to enhance channel health, and previous literature has studied augmenting ad auctions with machine learning predictions on advertiser values (also known asmachine-learned advice ) to improve total welfare. Yet, such improvements could come at the cost of individual bidders' welfare and do not shed light on how particular advertiser bidding strategies impact welfare. Motivated by this, we present an analysis on an individual bidder's welfare loss in the autobidding world for auctions with and without machine-learned advice, and also uncover how advertiser strategies relate to such losses. In particular, we demonstrate how ad platforms can utilize ML advice to improve welfare guarantee on the aggregate and individual bidder level by setting ML advice as personalized reserve prices when the platform consists ofautobidders who maximize value while respecting a return on ad spend (ROAS) constraint. Under parallel VCG auctions with such ML advice-based reserves, we present a worst-case welfare lower-bound guarantee for an individual autobidder, and show that the lower-bound guarantee is positively correlated with ML advice quality as well as the scale of bids induced by the autobidder's bidding strategies. Further, we show that no truthful, and possibly randomized mechanism with anonymous allocations can achieve universally better individual welfare guarantees than VCG, in the presence of personalized reserves based on ML-advice of equal quality. Moreover, we extend our individual welfare guarantee results to generalized first price (GFP) and generalized second price (GSP) auctions. Finally, we present numerical studies using semi-synthetic data derived from ad auction logs of a search ad platform to showcase improvements in individual welfare when setting personalized reserve prices with ML-advice.

We consider a dynamic pricing problem for repeated contextual second-price auctions with multiple strategic buyers who aim to maximize their long-term time discounted utility. The seller has limited information on buyers’ overall demand curves which depends on a non-parametric market-noise distribution, and buyers may potentially submit corrupted bids (relative to true valuations) to manipulate the seller’s pricing policy for more favorable reserve prices in the future. We focus on designing the seller’s learning policy to set contextual reserve prices where the seller’s goal is to minimize regret compared to the revenue of a benchmark clairvoyant policy that has full information of buyers’ demand. We propose a policy with a phased-structure that incorporates randomized “isolation” periods, during which a buyer is randomly chosen to solely participate in the auction. We show that this design allows the seller to control the number of periods in which buyers significantly corrupt their bids. We then prove that our policy enjoys a T-period regret of $O(\sqrt{T})$ facing strategic buyers. Finally, we conduct numerical simulations to compare our proposed algorithm to standard pricing policies. Our numerical results show that our algorithm outperforms these policies under various buyer bidding behavior.

Internet advertisers (buyers) repeatedly procure ad impressions from ad platforms (sellers) with the aim to maximize total conversion (i.e. ad value) while respecting both budget and return-on-investment (ROI) constraints for efficient utilization of limited monetary resources. Facing such a constrained buyer who aims to learn her optimal strategy to acquire impressions, we study from a seller’s perspective how to learn and price ad impressions through repeated posted price mechanisms to maximize revenue. For this two-sided learning setup, we propose a learning algorithm for the seller that utilizes an episodic binary-search procedure to identify a revenue-optimal selling price. We show that such a simple learning algorithm enjoys low seller regret when within each episode, the budget and ROI constrained buyer approximately best responds to the posted price. We present simple yet natural buyer’s bidding algorithms under which the buyer approximately best responds while satisfying budget and ROI constraints, leading to a low regret for our proposed seller pricing algorithm. The design of our seller algorithm is motivated by the fact that the seller’s revenue function admits a bell-shaped structure when the buyer best responds to prices under budget and ROI constraints, enabling our seller algorithm to identify revenue-optimal selling prices efficiently.

In digital online advertising, advertisers procure ad impressions simultaneously on multiple platforms, or so-called channels, such as Google Ads, Meta Ads Manager, etc., each of which consists of numerous ad auctions. We study how an advertiser maximizes total conversion (e.g. ad clicks) while satisfying aggregate return-on-investment (ROI) and budget constraints across all channels. In practice, an advertiser does not have control over, and thus cannot globally optimize, which individual ad auctions she participates in for each channel, and instead authorizes a channel to procure impressions on her behalf: the advertiser can only utilize two levers on each channel, namely setting a per-channel budget and per-channel target ROI. In this work, we first analyze the effectiveness of each of these levers for solving the advertiser's global multi-channel problem. We show that when an advertiser only optimizes over per-channel ROIs, her total conversion can be arbitrarily worse than what she could have obtained in the global problem. Further, we show that the advertiser can achieve the global optimal conversion when she only optimizes over per-channel budgets. In light of this finding, under a bandit feedback setting that mimics real-world scenarios where advertisers have limited information on ad auctions in each channels and how channels procure ads, we present an efficient learning algorithm that produces per-channel budgets whose resulting conversion approximates that of the global optimal problem.

In a carbon auction, licenses for CO2 emissions are allocated among multiple interested players. Inspired by this setting, we consider repeated multi-unit auctions with uniform pricing, which are widely used in practice. Our contribution is to analyze these auctions in both the offline and online settings, by designing efficient bidding algorithms with low regret and giving regret lower bounds. We also analyze the quality of the equilibria in two main variants of the auction, finding that one variant is susceptible to collusion among the bidders while the other is not.

Traditional multi-armed bandit (MAB) frameworks, predominantly examined under stochastic or adversarial settings, often overlook the temporal dynamics inherent in many real-world applications such as recommendation systems and online advertising. This paper introduces a novel non-stationary MAB framework that captures the temporal structure of these real-world dynamics through an auto-regressive (AR) reward structure. We propose an algorithm that integrates two key mechanisms: (i) an alternation mechanism adept at leveraging temporal dependencies to dynamically balance exploration and exploitation, and (ii) a restarting mechanism designed to discard out-of-date information. Our algorithm achieves a regret upper bound that nearly matches the lower bound, with regret measured against a robust dynamic benchmark. Finally, via a real-world case study on tourism demand prediction, we demonstrate both the efficacy of our algorithm and the broader applicability of our techniques to more complex, rapidly evolving time series.

The second price auction has been the prevalent auction format used by advertising exchanges because of its simplicity and desirable incentive properties. However, even with an optimized choice of reserve prices, this auction is not revenue optimal when the bidders are heterogeneous and their valuation distributions differ significantly. In order to optimize the revenue of advertising exchanges, we propose an auction format called the boosted second price auction, which assigns a boost value to each bidder. The auction favors bidders with higher boost values and allocates the item to the bidder with the highest boosted bid. We propose a data-driven approach to optimize boost values using the previous bids of the bidders. Our analysis of auction data from Google's online advertising exchange shows that the boosted second price auction with data-optimized boost values outperforms the second price auction and empirical Myerson auction by up to 6% and 3%, respectively.

In many online platforms, customers' decisions are substantially influenced by product rankings as most customers only examine a few top-ranked products. Concurrently, such platforms also use the same data corresponding to customers' actions to learn how these products must be ranked or ordered. These interactions in the underlying learning process, however, may incentivize sellers to artificially inflate their position by employing fake users, as exemplified by the emergence of click farms. Motivated by such fraudulent behavior, we study the ranking problem of a platform that faces a mixture of real and fake users who are indistinguishable from one another. We first show that existing learning algorithms---that are optimal in the absence of fake users---may converge to highly sub-optimal rankings under manipulation by fake users. To overcome this deficiency, we develop efficient learning algorithms under two informational environments: in the first setting, the platform is aware of the number of fake users, and in the second setting, it is agnostic to the number of fake users. For both these environments, we prove that our algorithms converge to the optimal ranking, while being robust to the aforementioned fraudulent behavior; we also present worst-case performance guarantees for our methods, and show that they significantly outperform existing algorithms. At a high level, our work employs several novel approaches to guarantee robustness such as: (i) constructing product-ordering graphs that encode the pairwise relationships between products inferred from the customers' actions; and (ii) implementing multiple levels of learning with a judicious amount of bi-directional cross-learning between levels. Overall, our results indicate that online platforms can effectively combat fraudulent users without incurring large costs by designing new learning algorithms that guarantee efficient convergence even when the platform is completely oblivious to the number and identity of the fake users.

Motivated by online decision-making in time-varying combinatorial environments, we study the problem of transforming offline algorithms to their online counterparts. We focus on offline combinatorial problems that are amenable to a constant factor approximation using a greedy algorithm that is robust to local errors. For such problems, we provide a general framework that efficiently transforms offline robust greedy algorithms to online ones using Blackwell approachability.

We combine theory and empirics to (i) show that some buyers in online advertising markets are financially constrained and (ii) demonstrate how to design auctions that take into account such financial constraints. We use data from a field experiment where reserve prices were randomized on Google’s advertising exchange (AdX). We find that, contrary to the predictions of classical auction theory, a significant set of buyers lowers their bids when reserve prices go up. We show that this behavior can be explained if we assume buyers have constraints on their minimum return on investment (ROI). We proceed to design auctions for ROI-constrained buyers. We show that optimal auctions for symmetric ROI-constrained buyers are either second-price auctions with reduced reserve prices or subsidized second-price auctions. For asymmetric buyers, the optimal auction involves a modification of virtual values. Going back to the data, we show that using ROI-aware optimal auctions can lead to large revenue gains and large welfare gains for buyers.

We study long-run market stability for repeated price competitions between two firms, where consumer demand depends on firms' posted prices and consumers’ price expectations called reference prices. Consumers' reference prices vary over time according to a memory-based dynamic, which is a weighted average of all historical prices. We focus on the setting where firms are not aware of demand functions and how reference prices are formed but have access to an oracle that provides a measure of consumers' responsiveness to the current posted prices. We show that if the firms run no-regret algorithms, in particular, online mirror descent (OMD), with decreasing step sizes, the market stabilizes in the sense that firms' prices and reference prices converge to a stable Nash Equilibrium (SNE). Interestingly, we also show that there exist constant step sizes under which the market stabilizes. We further characterize the rate of convergence to the SNE for both decreasing and constant OMD step sizes.

On online platforms, consumers face an abundance of options that are displayed in the form of a position ranking. Only products placed in the first few positions are readily accessible to the consumer, and she needs to exert effort to access more options. For such platforms, we develop a two-stage sequential search model where in the first stage, the consumer sequentially screens positions to observe the preference weight of the products placed in them and forms a consideration set. In the second stage, she observes the additional idiosyncratic utility that she can derive from each product and chooses the highest-utility product within her consideration set. For this model, we first characterize the optimal sequential search policy of a welfare-maximizing consumer. We then study how platforms with different objectives should rank products. We focus on two objectives: (i) maximizing the platform's market share and (ii) maximizing the consumer's welfare. Somewhat surprisingly, we show that ranking products in decreasing order of their preference weights does not necessarily maximize market share or consumer welfare. Such a ranking may shorten the consumer's consideration set due to the externality effect of high-positioned products on low-positioned ones, leading to insufficient screening. We then show that both problems---maximizing market share and maximizing consumer welfare---are NP-complete. We develop novel near-optimal polynomial-time ranking algorithms for each objective. Further, we show that even though ranking products in decreasing order of their preference weights is suboptimal, such a ranking enjoys strong performance guarantees for both objectives. We complement our theoretical developments with numerical studies using synthetic data in which we show (1) that heuristic versions of our algorithms that do not rely on model primitives perform well and (2) that our model can be effectively estimated using a maximum likelihood estimator.

In the classical contextual bandits problem, in each round $t$, a learner observes some context $c$, chooses some action $a$ to perform, and receives some reward $r_{a,t}(c)$. We consider the variant of this problem where in addition to receiving the reward $r_{a,t}(c)$, the learner also learns the values of $r_{a,t}(c')$ for all other contexts $c'$; i.e., the rewards that would have been achieved by performing that action under different contexts. This variant arises in several strategic settings, such as learning how to bid in non-truthful repeated auctions (in this setting the context is the decision maker's private valuation for each auction). We call this problem the contextual bandits problem with cross-learning. The best algorithms for the classical contextual bandits problem achieve $\tilde{O}(\sqrt{CKT})$ regret against all stationary policies, where $C$ is the number of contexts, $K$ the number of actions, and $T$ the number of rounds. We demonstrate algorithms for the contextual bandits problem with cross-learning that remove the dependence on $C$ and achieve regret $\tilde{O}(\sqrt{KT})$ (when contexts are stochastic with known distribution), $\tilde{O}(K^{1/3}T^{2/3})$ (when contexts are stochastic with unknown distribution), and $\tilde{O}(\sqrt{KT})$ (when contexts are adversarial but rewards are stochastic). We simulate our algorithms on real auction data from an ad exchange running first-price auctions (showing that they outperform traditional contextual bandit algorithms).

Motivated by pricing in ad exchange markets, we consider the problem of robust learning of reserve prices against strategic buyers in repeated contextual second-price auctions. Buyers' valuations \new{for} an item depend on the context that describes the item. However, the seller is not aware of the relationship between the context and buyers' valuations, i.e., buyers' preferences. The seller's goal is to design a learning policy to set reserve prices via observing the past sales data, and her objective is to minimize her regret for revenue, where the regret is computed against a clairvoyant policy that knows buyers' heterogeneous preferences. Given the seller's goal, utility-maximizing buyers have the incentive to bid untruthfully in order to manipulate the seller's learning policy. We propose two learning policies that are robust to such strategic behavior. These policies use the outcomes of the auctions, rather than the submitted bids, to estimate the preferences while controlling the long-term effect of the outcome of each auction on the future reserve prices. The first policy called Contextual Robust Pricing (CORP) is designed for the setting where the market noise distribution is known to the seller and achieves a T-period regret of $O(d\log(Td) \log (T))$, where $d$ is the dimension of {the} contextual information. The second policy, which is a variant of the first policy, is called Stable CORP (SCORP). This policy is tailored to the setting where the market noise distribution is unknown to the seller and belongs to an ambiguity set. We show that the SCORP policy has a T-period regret of $O(\sqrt{d\log(Td)}\;T^{2/3})$.

Data marketplaces provide platforms for management of large data sets. The data markets are rapidly growing, yet the pricing strategies for data and data analytics are not yet well-understood. In this paper, we explore some of the pricing schemes applicable to data marketplaces in the context of transportation traffic data. This includes historical and real-time freeway and arterial congestion data. We investigate pricing raw sensor data vs. processed information (e.g, prediction of traffic patterns or route planning services) and show that, under natural assumptions, the raw data should be priced higher than processed information.

We analyze a novel architecture for caching popular video content to enable wireless device-to-device (D2D) collaboration. We focus on the asymptotic scaling characteristics and show how they depend on video content popularity statistics. We identify a fundamental conflict between collaboration distance and interference and show how to optimize the transmission power to maximize frequency reuse. Our main result is a closed form expression of the optimal collaboration distance as a function of the model parameters. Under the common assumption of a Zipf distribution for content reuse, we show that if the Zipf exponent is greater than 1, it is possible to have a number of D2D interference-free collaboration pairs that scales linearly in the number of nodes. If the Zipf exponent is smaller than 1, we identify the best possible scaling in the number of D2D collaborating links. Surprisingly, a very simple distributed caching policy achieves the optimal scaling behavior.

We propose a new scheme for increasing the throughput of video files in cellular communications systems. This scheme exploits (1) the redundancy of user requests as well as (2) the considerable storage capacity of smartphones and tablets. Users cache popular video files and-after receiving requests from other users-serve these requests via device-to-device localized transmissions. The file placement is optimal when a central control knows a priori the locations of wireless devices when file requests occur. However, even a purely random caching scheme shows only a minor performance loss compared to such a “genie-aided” scheme. We then analyze the optimal collaboration distance, trading off frequency reuse with the probability of finding a requested file within the collaboration distance. We show that an improvement of spectral efficiency of one to two orders of magnitude is possible, even if there is not very high redundancy in video requests.

Motivated by the availability of real-time data on customer characteristics, we consider the problem of personalizing the assortment of products to each arriving customer. For an arriving customer of type z, the company must decide, in real-time, on the assortment of products to offer. Given the offered assortment, the customers make choices on which products to buy, if any, according to a general choice model that is specific to each customer type. Our goal is to develop a revenue-maximizing policy that determines the assortment to offer to each arriving customer, taking into account the customer type and the current inventories.

Video on-demand streaming from Internet-based servers is becoming one of the most important services offered by wireless networks today. In order to improve the area spectral efficiency of video transmission in cellular systems, small cells heterogeneous architectures (e.g., femtocells, WiFi off-loading) are being proposed, such that video traffic to nomadic users can be handled by short-range links to the nearest small cell access points (referred to as “helpers”). As the helper deployment density increases, the backhaul capacity becomes the system bottleneck. In order to alleviate such bottleneck we propose a system where helpers with low-rate backhaul but high storage capacity cache popular video files. Files not available from helpers are transmitted by the cellular base station. We analyze the optimum way of assigning files to the helpers, in order to minimize the expected downloading time for files. We distinguish between the uncoded case (where only complete files are stored) and the coded case, where segments of Fountain-encoded versions of the video files are stored at helpers. We show that the uncoded optimum file assignment is NP-hard, and develop a greedy strategy that is provably within a factor 2 of the optimum. Further, for a special case we provide an efficient algorithm achieving a provably better approximation ratio of 1-(1-1/d )d, where d is the maximum number of helpers a user can be connected to. We also show that the coded optimum cache assignment problem is convex that can be further reduced to a linear program. We present numerical results comparing the proposed schemes.

Video is the main driver for the inexorable increase in wireless data traffic. In this paper we analyze a new architecture in which device-to-device (D2D) communications is used to drastically increase the capacity of cellular networks for video transmission. Users cache popular video files and - after receiving requests from other users - serve these requests via D2D localized transmissions; the short range of the D2D transmission enables frequency reuse within the cell. We analyze the scaling behavior of the throughput with the number of devices per cell. The user content request statistics, as well as the caching distribution, are modeled by a Zipf distribution with parameters γrand γc, respectively. For the practically important case γr0> 1, we derive a closed form expression for the scaling behavior of the number of D2D links that coexist without interference. Our analysis relies on a novel Poisson approximation result for wireless networks obtained through the Chen-Stein Method.

We propose a new scheme for increasing the throughput of video files in cellular communications systems. This scheme exploits (i) the redundancy of user requests as well as (ii) the considerable storage capacity of smartphones and tablets. Users cache popular video files and - after receiving requests from other users - serve these requests via device-to-device localized transmissions. We investigate what is the optimal collaboration distance, trading off frequency reuse with the probability of finding a requested file within the collaboration distance. We show that an improvement of spectral efficiency of one to two orders of magnitude is possible, even if there is not very high redundancy in video requests.

We suggest a novel approach to handle the ongoing explosive increase in the demand for video content in mobile devices. We envision femtocell-like base stations, which we call helpers, with weak backhaul links but large storage capabilities. These helpers form a wireless distributed caching network that assists the macro base station by handling requests of popular files that have been cached. We formalize the wireless distributed caching optimization problem for the case that files are encoded using fountain/MDS codes. We express the problem as a convex optimization. By adding additional variables we reduce it to a linear program. On the practical side, we present a detailed simulation of a university campus scenario covered by a single 3GPP LTE R8 cell and several helper nodes using a simplified 802.11n protocol. We use a real campus trace of video requests and show how distributed caching can increase the number of served users by as much as 600-700%.

We suggest a novel approach to handle the ongoing explosive increase in the demand for video content in wireless/mobile devices. We envision femtocell-like base stations, which we call helpers, with weak backhaul links but large storage capacity. These helpers form a wireless distributed caching network that assists the macro base station by handling requests of popular files that have been cached. Due to the short distances between helpers and requesting devices, the transmission of cached files can be done very efficiently. A key question for such a system is the wireless distributed caching problem, i.e., which files should be cached by which helpers. If every mobile device has only access to a exactly one helper, then clearly each helper should cache the same files, namely the most popular ones. However, for the case that each mobile device can access multiple caches, the assignment of files to helpers becomes nontrivial. The theoretical contribution of our paper lies in (i) formalizing the distributed caching problem, (ii) showing that this problem is NP-hard, and (iii) presenting approximation algorithms that lie within a constant factor of the theoretical optimum. On the practical side, we present a detailed simulation of a university campus scenario covered by a single 3GPP LTE R8 cell and several helpers using a simplified 802.11n protocol. We use a real campus trace of video requests and show how distributed caching can increase the number served users by as much as 400 - 500%.

We introduce a novel wireless device-to-device (D2D) collaboration architecture that exploits distributed storage of popular content to enable frequency reuse. We identify a fundamental conflict between collaboration distance and interference and show how to optimize the transmission power to maximize frequency reuse. Our analysis depends on the user content request statistics which are modeled by a Zipf distribution. Our main result is a closed form expression of the optimal collaboration distance as a function of the content reuse distribution parameters. We show that if the Zipf exponent of the content reuse distribution is greater than 1, it is possible to have a number of D2D interference-free collaboration pairs that scales linearly in the number of nodes. If the Zipf exponent is smaller than 1, we identify the best possible scaling in the number of D2D collaborating links. Surprisingly, a very simple distributed caching policy achieves the optimal scaling behavior and therefore there is no need to centrally coordinate what each node is caching.

In this paper, we propose a novel cognitive cooperative relaying scheme using multi-carrier transmission in which cognitive users assist primary users by relaying their information using Decode and Forward (DF) strategy. The best cognitive user is selected as a relay for a Primary User (PU). Outage probability of the PU is investigated. Our analyses show substantial improvements in the outage probability of the PU when using the proposed cooperative scheme compared to non-cooperative one. We also perform several simulations to corroborate our theoretical results.