I Introduction
Recent years have witnessed the rapid proliferation of smartphones and wearable devices. They are equipped with a plethora of multimodal sensors and they possess powerful computation as well as communication capabilities. With these technological features, ordinary mobile users can actively monitor their surrounding environments such as temperature, noise, vibration, network connectivity and geographic position without demanding sophisticated instruments. This leads to a new paradigm of problemsolving known as participatory crowdsensing: a requester releases sensing tasks and collects contributed data from a number of mobile users that seek their individual benefits or the benefit of their community. Recent applications in [1, 2, 3, 19] adopted crowdsensing to perform the indoor localization, collection of location information and noise monitoring, etc.
The efficacy of a crowdsensing system heavily relies on the exerted efforts of mobile users. However, they are reluctant to share sensing capabilities due to the cost of energy, data traffic, time consumption, and risk of privacy leakage etc. A large body of studies have been devoted to developing efficient incentive mechanisms [4, 20, 21, 22, 23, 25, 26]
that can be roughly classified into two representative types:
allpay auction [4, 23] and Tullock contest [20, 25]. In the former, the requester usually plays the role of auctioneer, and each contributor bids his efforts to the requester. The winning contributor acquires the entire reward, while the efforts of a losing contributor cannot be reimbursed. The latter approach, unlike allpay auction, splits the total reward to all the contributors exerting positive efforts. Each contributor receives a fraction of the reward proportional to his efforts, and inversely proportional to the aggregate collected efforts.Incentive mechanisms based on auction theory are in general perfectly discriminatory, i.e. the best bidder wins the competition while the others lose for sure [25]. Owing to fear of sunk cost, allpay auction is inclined to discouraging the participation of relatively weak contributors [24]. In the light of its potential limitation, [20] proposed to utilize Tullock contest to maximize the total sensing time of contributors. Tullock contest is partially discriminatory so that each contributor gains a positive reward at the equilibrium if he exerts positive efforts. Though serving as a salient incentive mechanism for crowdsensing, the standard Tullock contest can be far away from optimality when the contributors are heterogeneous in their marginal costs of effort. Authors in [25] presented an optimal discrimination strategy for Tullock contest when the marginal costs of the users are heterogeneous in crowdsensing. The basic rationale is to enable a user of smaller marginal cost to exert more efforts under a Bayesian game framework.
In this paper, we consider partially discriminatory incentive mechanisms at a new regime, namely timeliness sensitive crowdsensing contest (TSCC). This is motivated by versatile realworld crowdsensing applications where the timely sensing results are more valuable to the requester. For instance, CityExplorer [15] is a gamebased crowdsensing system in which a winning player sets as many markers as possible in a citywide game area within a finite time period by taking photos and providing concrete information. OpenSense [7] enrols users to perform realtime air quality monitoring at different sites of a city. TruCentive [16], CrowdPark [17] and ParkNet [18] collect the timely parking information from drivers and distributes to those in need of it. NoiseTube [19] is a participatory sensing framework for monitoring ambient noise in an area of several square kilometers.
Though the above systems are designed for different purposes, they share a set of similar properties. Firstly, the contributors do not join instantly and simultaneously, while the requester desires large efforts from those who joins the sensing earlier. The joining time can be the timestamp of a mobile user passing by a given sensing venue, assuming that the task is announced at time 0. Secondly, the contributors do NOT exert efforts to change the joining times; in stead, they contribute efforts to execute the specific task. For instance, a driver will not drive his car out solely for the purpose of finding the availability of a parking lot. Executing a task such as taking and uploading photos consume a certain amount of resources. The cost of altering the joining time overwhelms that of task execution so that the joining time of the contributor is deemed as his property instead of his strategy to gain a larger share of the reward. Hence, the joining time and effort of a contributor are two orthogonal factors. To incentivize more efforts from early contributors, a naive approach is to raise the reward, which is obviously not desired by the requester. From this angle, a fundamental question arises: can the requester incentivize more efforts from early joining contributors by designing appropriate discrimination mechanisms other than increasing his budget for crowdsensing?
To answer this question, our first step is to examine in what form a feasible discrimination strategy should take so as to maximize the requester’s efficiency under the Tullock contest model. Here, the efficiency is defined as the aggregate weighted efforts brought by perunit of the requester’s budget. This metric shares the similar principle as those in [20, 25], but is more general than a specific utility function. We model the competition of contributors as a noncooperative game. Each contributor selfishly maximizes his payoff that is the difference between the received reward and cost of efforts. Our analysis on the unique Nash equilibrium (NE) reveals that the reward discrimination along with a “virtual nature contributor” can leverage the requester’s high efficiency and the simplicity of mechanism design: the reward discrimination endows an early contributor the larger maximum achievable reward, and vice versa; the nature player enables the requester to retain a fractional of reward when the number of contributors is small, thus improving his efficiency.
Once Tullock contest structure has been determined for crowdsensing, our second step is to design practical incentives with timeliness sensitivity. Two practical approaches are proposed, the earliest strategy where only a subset of earliest contributors are rewarded, and the termination time strategy
that any contributor later than a “deadline” will be ruled out in this sensing task. In both strategies, they compete under an incomplete information scenario because each of them only knows his own joining time, while is unaware of those of the others, or is impossible to predict the joining times of future players. Hence, we formulate each competition as a twostage Stackelberg Bayesian game, in which the requester announces the contest function to the contributors at StageI and they compete for the reward with the probability distribution of joining time at StageII. The existence and uniqueness of the Bayesian Nash equilibrium (BNE) are proved for each strategy. Based on these BNEs, the requester can compute the optimal number of rewarded contributors for the earliest
strategy and the optimal deadline for the termination time strategy.Our major contributions are summarized below:

To the best of our knowledge, this is the first attempt to explore the design space for incentivizing more efforts from the early joining contributors.

We show that the reward discrimination is suitable to elicit more efforts from early joining contributors. The NE of the Tullock contest is analyzed where a set of practical discrimination strategies are proposed to give preference to the early contributors.

We formulate Stackelberg Bayesian Nash games for the earliest and the termination time strategies. The optimal number of rewarded contributors in the former and the optimal deadline in the latter are presented.

The Stackelberg Bayesian Nash game framework is generalized to the open crowdsensing system where the arrival of contributors is governed by a Poisson process.

Extensive simulations manifest that the proposed strategies can greatly improve the efficiency of the requester. In particular, the distribution of joining times for the closed system is derived from the WiFi access data of students in a campus building.
The remainder of this paper is structured as follows. Second II presents the game model for participatory sensing with the timeliness consideration. A set of practical incentive strategies are proposed in section III. Section IV analyzes the Bayesian Nash equilibria of incentive strategies and presents the optimal parameter configurations. Section V extends the game framework to an open crowdsensing system with Poisson arrival of the contributors. The tracedriven experiments are performed in Section VI. Section VII reviews the stateoftheart work and Section VIII concludes this paper.
Ii System Model
In this section, we present a suit of game models for participatory sensing that take into account the joining time of contributors.
Iia Motivation and Basic Model
In most of crowdsourcing and mobile sensing applications, the requester pursue not only high quality but also timely efforts. However, recent research merely concentrates on the quality or effort while assuming that all the contributors participate in the crowdsensing simultaneously. In reality, the contributors more often join asynchronously. To harvest more efforts from the early joining contributors, the requester can increase the monetary reward, which is obviously undesirable. Our purpose here is to explore the design space for incentivizing timely and large efforts without increasing the requester’s reward. The timeliness sensitive crowdsensing contest (TSCC) possesses two key factors, one is the joining time and the other is the exerted effort. The joining time is the duration between the instant of task distribution and that of task completion. We make an important assumption as the following.
Assumption 1
The joining time and effort of a contributor are perfectly complimentary factors in the crowdsensing contest. The effort refers to various types of resources used to perform a sensing task, excluding those related to joining time. The joining time of a contributor is determined exogenously.
This assumption manifests that the joining time and effort CANNOT be confused together. As an example, in mobile sensing applications, the joining time can be the timestamp of a mobile user passing by the given sensing site, and the effort refers to energy expenditure or even monetary cost to execute this task. A rational contributor will decide how many efforts he should spend on the sensing task, knowing his joining time. In what follows, we deliver the mathematical model for the incentive mechanism design.
We consider a crowdsensing system with one requester and potential contributors. The requester releases a task at time 0 with a reward budget , and the contributors compete for this reward. We denote by the effort made by contributor , and by
his joining time. Define two vectors
and as and . The core of crowdsensing is the reward allocation mechanism that incentivizes the high effort from contributors, by taking the timeliness into consideration. We adopt a modified Tullock contest success function (CSF) to characterize the competition among contributors. Define and . Let be the reward obtained by the contributor, given the sets of joining time and effort level . There has(1) 
where , the maximum achievable reward of contributor , is a function of his joining time . In this Tullock crowdsensing contest, each contributor shares the total reward proportionally to his efforts, and inversely proportional to the aggregate efforts from all the contributors. The physical interpretation is that a contributor acquires a larger reward if he contributes more efforts, and a smaller reward if any of his opponent spends more.
Two new features are introduced beyond the standard Tullock model. One is the “reward discrimination” on the joining times, that is, if . An early joining contributor has a larger maximum achievable reward than a late one To be noted, there are two other discrimination rules named “weight discrimination” and “exponent discrimination”. However, only the reward discrimination is suitable for our problem where we leave the lengthy analysis in AppendixII. The other feature is to introduce a constant that if it is positive, the reward will not be completely assigned to the contributors. If is 0, when there are only a couple of contributors, the requester has to assign the whole reward even though a very small amount of efforts are collected. Actually, a nonzero is equivalent to adding a NATURE player, avoiding such adverse situations. A larger means that the requester retains a higher percentage of reward. Denote by the total reward paid to the contributors:
(2) 
In the incentive mechanism design, the requester needs to make the payment equal to the budget as the benchmark.
The payoff of contributor , , is denoted as the difference between his reward and the cost of efforts. Hence, there yields
(3) 
where the marginal cost of efforts is normalized as 1. When is not included in the CSF, we can rewrite (resp. ) as (resp. ) for simplicity. Concerning the requester’s utility, an immediate contribution is no longer identically important to a late one, even if their efforts are the same. Inspired by this observation, we transform the importance of timeliness to the requester into a positive weight (sometimes simplified as ) for contributor . If , there has . Define as the utility of the requester:
(4) 
Define as the requester’s efficiency that is the utility brought by perunit payment:
(5) 
The efficiency reflects the amount of utility brought by perunit reward, and serves as the metric of the requester to quantify the performance of incentive mechanisms. Using instead of provides an intuitive understanding on how good an incentive mechanism can achieve. An auxiliary metric is named as “discrimination gain” denoted by where and indicate the requester’s efficiencies without and with discrimination. In what follows, we formulate two different games to understand the competition of contributors, and explore the requester’s utility maximization strategy.
Remark 1: We consider a single requester because it is less likely that two requesters compete simultaneously over the same set of mobile users.
IiB G1: Complete Information of Joining Times
We formulate a noncooperative game to characterize the competition of contributors in which the joining time is the common knowledge. In practice, when a contributor exerts certain efforts, he only knows his own joining time, while not those of his opponents. However, the complete information game allows us to gain important insights of the incentive mechanism design in a more tractable way. The game G1 comprises three key elements:

Players: A set of potential contributors;

Strategies: The action of player is , ;

Payoffs: The payoff of player is , .
Each player selfishly maximizes his individual payoff. The outcome of their competition is depicted by the famous Nash Equilibrium as the following.
Definition 1
(G1 Nash Equilibrium) The strategy profile is a Nash equilibrium of G1 if there exists
(6) 
where is the set of efforts excluding .
Definition 2
(Individual Rationality) Each player will receive a nonnegative payoff if he exerts a positive amount of efforts.
IiC G2: Incomplete Information of Joining Times
The joining time is inherently a private information to the contributors. Especially, when the earliest contributor exerts his effort, he is by no means aware of the other’s joining time that have not taken place. We hereby formulate a twostage Stackelberg Bayesian game to characterize the crowdsensing contest where each player knows his exact joining time and the joining time distribution of the other contributors. In StageI, the requester announces the incentive mechanism and the joining time distribution so that his utility is optimized and the budget is balanced. In StageII, each contributor decides how many efforts to use to maximize his individual payoff.
StageII: Contributors’ Bayesian Game. Except that the players and the strategies are the same as those in G1, we append three different elements for this Bayesian game.

Types: The type of a contributor is the joining time ;

Probabilities: The types of all the contributors are drawn from an i.i.d. priori distribution ;

Payoffs: The payoff of a contributor is the expectation (occasionally written as ).
Each contributor chooses his effort to maximize his payoff with partial information. The equilibrium reached by the competing contributors is Bayesian Nash Equilibrium (BNE) stated as the following.
Definition 3
(Bayesian Nash Equilibrium) The strategy profile is a Bayesian Nash Equilibrium (BNE) of G2 if for all and , there has
(7) 
where denotes the set of strategies excluding , and denotes the expectation.
StageI: Requester’s CSF Choice. At StageI, the requester can configure the parameter and the functions for all so as to maximize his utility, given the budget . Formally, the equilibrium is the solution to the following optimization problem:
(8)  
The requester’s efficiency is optimized on the basis of the Bayesian Nash equilibrium at StageII.
IiD Comparison with Existing Models
Our game framework is closely related to but different from the pioneering studies in [20] and [25]. The major differences are summarized in three aspects:

Discrimination Rule. There does not exist a discrimination rule in the crowdsensing contest of [20]. In [25], the marginal cost of a contributor is his private knowledge and the discrimination is based on the marginal cost. We investigate the discrimination on the joining time of the contributors where the origin is the different valuation of the requester on perunit of effort at different joining times. The analytical framework in [25] does not apply to our problem and hence a new framework is necessary.

Closed and Open Systems. Only the closed system with a fixed number of contributors is studied in [20] and [25]. Our game framework is applicable to not only this closed system but also the open system with the arrival of contributors governed by a stochastic process. The closed system usually exhibits the busty arrivals of the contributors while the open system has a more stable arrival rate. We are lucky to see that the incentive mechanisms can be orchestrated under the same framework in these two divergent systems.

Objective Function. The objective of the requester in [20] and [25] is to maximize his total utility on the efforts minus the payment to the contributors. We define a new metric for the requester, namely the crowdsensing efficiency. It is not constrained to the absolute net utility in a particular setting, but is to quantify the utility of the requester brought by perunit of the budget.
Iii Timeliness Sensitive Crowdsourcing Contest with Complete Information
In this section, we analyze the competition of contributors with timeliness sensitivity, and present important insights in the design of practical incentive mechanisms.
Iiia Nash Equilibrium
According to the individual rationality property, a contributor will not exert efforts if his participation cannot bring positive payoff. Hence, we need to scrutinize the participation of players on top of a NE. For simplicity of notations, we perform a change of variables by setting (esp. excluding ), and . We observe that the utility function of each contributor is strictly concave in . Then, the first order condition describes the global maximum of with respect to ,
(9) 
where the equality holds upon . If this equality does not hold, the maximum is obtained at , in which contributor does not “participate” in the crowdsensing. Therefore, the best response of the contributor to is
(10) 
The pure strategy Nash equilibrium must have for each contributor. We have the following lemma with regard to the participation of contributors.
Lemma 1
(Principles of Participation) In the standard contest function, the participation of a contributor satisfies:

if , there must have ;

if , the contributor does not participate in the contest; when increases, the number of participating contributors decreases.
Proof: Please refer to AppendixI.
Based on the best response function in Eq.(10), there has
(11) 
when only the contributors from to exert positive effort at the NE. We hereby present a method to search the explicit NE within steps. The first step is to compute by assuming the participation of contributors at the NE. If is positive for all , the NE is obtained. Otherwise, if any is negative, this means that some of the contributors do not participate at this NE. By removing the concurrent contributors with the smallest , we proceed to search until all are positive for . The NE strategy is subsequently computed by
(12) 
The actual payment to the player is
(13) 
The detailed algorithm to find the unique NE is shown in Algorithm 1.
We next consider a special discrimination rule that the basic idea is to set the larger maximum rewards to the early contributors and 0 to late ones. Our purpose is to gain the important insights on the design of discrimination rules and on the parameter configuration.
MultiContributors: , and ; if .
When the maximum rewards of the early contributors are identical, and the remaining late ones are not rewarded, only the former ones participate at the NE. Their efforts at the NE are given by ()
(14) 
The total reward paid to the contributors is . Then, the NE efficiency of the requester is
(15) 
that yields The upper and lower bounds are obtained or approximated when is 0 or is approaching .
Special case 1): the requester is not sensitive to the joining time (i.e. for all ). The efficiency is greater than and any discrimination (i.e. ) yields a smaller efficiency.
Special case 2): the requester is interested in only the earliest two contributions (i.e. and for all ). The discrimination gain can be as high as with the discrimination rule .
IiiB Understanding the function of
Introducing adds the complexity to the analysis and design of the incentive mechanism. We hereby investigate the role of and provide a simple guideline of configuring through the above example.
(1) A positive may prevent the participation of contributors with low maximum rewards. When is below for a late contributor, he will not participate in the sensing at the NE. Hence, precludes the participation of late contributors so that the early contributors may exert more efforts.
(2) A positive has a potential to improve the efficiency of the given discrimination rule. According to Eq.(15), the efficiency is an increasing function of .
(3) When the number of contributors, , is large, the efficiency gain brought by becomes less remarkable.
Given equal to 2, the efficiency is obtained by . Then, is with , and approaches as is sufficiently close to . For equal to , is with , and approaches as is sufficiently close to .
(4) Choosing a large is beneficial to the requester, especially when only a few contributors participate in the competition. However, a large will reduce the number of participants at the NE. In realistic scenarios, the mobile users join the crowdsensing randomly so that the participation of contributors is very sensitive to the joining times in the presence of a large . Hence, is suggested to be a fixed fraction of simply for the purpose of avoiding the scenarios with only a couple of participants.
The KKT optimization is adopted for the optimal incentive mechanism design. Due to the page limit, we leave the detailed analysis in AppendixI.
Remark 2: The requester can orchestrate the reward discrimination scheme to maximize his crowdsensing efficiency, given the vector of the joining times. However, this is infeasible in reality because the joining time of a player is not known until he undertakes the sensing task. A practical incentive mechanism needs to be announced before the crowdsensing, and needs to be easily implemented.
IiiC Practical Reward Discrimination Strategies
We hereby turn the reward discrimination into reality, considering that each contributor is unaware of the joining time of other contributors. Three practical strategies are proposed.
Earliest Strategy: Let until be the joining times of all the contributors. Let denote the ordered values of . Then, are called the order statistics of . The earliest contributors will have a chance of being rewarded (i.e. ), and all other late contributors will not be rewarded (i.e. ).
Termination Time Strategy: The requester sets up a termination time so that only contributions before are rewarded. Similarly, we denote by the random joining time of all the contributors. If , there has , and otherwise.
Linearly Decreasing Strategy: The maximum reward function is set to for every player . The velocity can be tuned to further optimize the efficiency of the requester.
In the earliest strategy, a contributor is unaware of his ranking in terms of the joining time; in the termination time strategy, he is unaware of the number of opponents joining before the deadline; in the linearly decreasing strategy, he is unaware of the maximum achievable rewards of his opponents. These uncertainties lead to competitions with incomplete information.
Remark 3: The linearly decreasing strategy is yet a different implementation of the earliest strategy, which will be shown later on.
Iv Stackelberg Bayesian Game with Timeliness Sensitivity
In this section, we analyze the Stackelberg Bayesian Nash equilibrium (SBNE) of crowdsensing contests. Optimal strategies are proposed for the requester to maximize his efficiency.
Iva Earliest Strategy
In our context, all the contributors are not notified whether they are ranked as the earliest contributors. Hence, as the first step, we must compute the probability of being one of the earliest contributors if the contributor joins at time . Suppose that is the smallest joining time. Among all the contributors, there are contributors earlier than him and contributors later than him. Considering all the possible rankings of no larger than the place, the probability of being the earliest contributor is given by
(16) 
where is the probability of a contributor joining before . Note that in the earliest strategy, the number of participated contributors can be at most , though only the first of them are rewarded. Here, with certain abuse of notation, we can use a function to denote , and is continuous, differentiable and strictly decreasing w.r.t. . Then, the utility of contributor with joining time is simplified as
(17) 
For contribution , the joining time is his private information, while only the statistical joining time distribution of other contributions is known as a priori. In this situation, we model the crowdsourcing competition as a Bayesian game where the join time of a contributor is characterised as his type. We will apply the backward induction principle to solve the proposed Bayesian game, i.e., first analyzing StageII and then determining the optimal policy at StageI accordingly.
StageII: Finding Bayesian Nash Equilibrium. For any contributor other than , i.e. and , since each is drawn from the common distribution , the expected utility of contributor on all the combinations is given by
(18)  
As the first step, we analyze the existence and uniqueness of this Bayesian Nash equilibrium.
Theorem 1
(Existence and Uniqueness of Earliest BNE) The Earliest incentive strategy has a unique Bayesian Nash equilibrium (BNE) in the contest.
Proof: Please refer to AppendixI.
We now turn to establish basic properties of purestrategy Bayesian Nash equilibrium. As stated before, the Bayesian Nash equilibrium strategy satisfies
(19) 
Similar to the NE condition of the complete information game, for each contributor of the earliest scheme, the equilibrium condition is given by
(20) 
with equality in the situation . Otherwise, the contributor does not participate. In general, there does not admit a closeform solution to this system of (in)equations. However, we can still infer interesting properties of the Bayesian Nash equilibrium.
Theorem 2
(Participation and Bound under Earliest Strategy) In the Bayesian Nash equilibrium of Earliest scheme, contributor ’s strategy has the following property (with a meaningful condition ).

If , is positive with for all .

If , there exists a such that is 0 for while is positive and strictly decreasing for . Especially, is no larger than .

The upper bound of is for and is otherwise.
Proof: Please refer to AppendixI.
Remark 4: A contributor may not participate in the earliest contest if his joining time is later than a certain threshold.
StageI: Optimizing Requester’s Efficiency. In the earliest scheme, the requester announces his reward allocation function before the contest takes place. As mentioned before, once is chosen to be a fraction of the maximum reward , the efficiency is independent of . The requester only needs to configure an optimal to optimize his efficiency.
At the Bayesian Nash equilibrium, the amount of efforts exerted by contributor is . This contributor generates a utility of to the requester. Since is unknown to the requester, his expected utility on contributor ’s effort is obtained by
(21) 
Because the joining time is i.i.d. at all the contributors (the assumption of identical distribution is not obligatory), the expected total utility of the requester is
(22) 
For each set of joining time , the reward paid to all the contributors is
(23) 
at the BNE. Then, the expected reward paid by the requester takes the following integral form
(24) 
The requester needs to configure an appropriate such that the expected reward apportioned to the contributors is equal to the budget .
For each vector of joining time , the efficiency of the requester is given by
(25) 
Then, the expected efficiency is
(26)  
The optimal can be searched for no more than times in the above formula. The main complexity stems from the multidimensional integral in computing the BNE at stageII. This multidimensional integral can be substituted by the Monte Carlo sum via discretizing the integral zone into a large number of multidimensional cubes.
IvB Termination Time Strategy
In the termination time strategy, the requester only allocates rewards to the contributors who join the contest before time . Obviously, a contributor later than time will not participate in the contest. Given the distribution of joining times, a contributor arrives before time is computed as . In this game, the number of participating contributors is uncertain. Denote by the number of participating contributors in a pool of contributors in total. The probability that out of contributors join before the expiration time is given by
(27) 
due to their i.i.d. joining times. We analyze this Bayesian game using the backward induction as follows.
StageII: Finding Bayesian Nash Equilibrium. We analyze the expected utility of the contributor here. If he participates, there will be at most other contributors. To compute the expected utility, we exhaust all the possibilities in terms of the number of contributors in the contest. For a fixed maximum reward , his expected utility is obtained by
(28)  
When is zero, the contributor only competes with the Nature player. In this Bayesian game, each contributor aims to maximize his expected utility. To begin with, we show the existence of a unique Bayesian Nash equilibrium.
Theorem 3
(Existence and Uniqueness of BNE for Termination Time Strategy) The crowdsensing contest using termination time strategy admits a unique symmetric Bayesian Nash equilibrium.
Proof: Please refer to AppendixI.
The BNE is symmetric in which all the contributors joining before time exert the same amount of effort. According to the optimality conditions at the BNE, the following equations hold
(29) 
The above equation intuitively shows that an increase of induces more effort from the contributors. For the special case , admits a close form solution, that is, .
StageI: Optimizing Requester’s Efficiency. From the requester’s perspective, contributor with the joining time () generates a utility to him. For the special case that no contributor arrives before , the utility of the requester is 0 for sure, and his efficiency is set to 0 as a penalty of unsuccessful crowdsensing. Therefore, the expected total utility of the requester is obtained by
(30) 
The number of contributors whose joining times before time ranges from 0 to . The corresponding probability of having participants before is . A fraction of the total reward allocated by the requester can be solved as
(31) 
Here, is chosen to let be equal to the budget . One should be informed that the summation begins from the subscript , which is unlike (28). This is because Eq.(31) counts the allocated reward when there is at least one contributors.
We next compute the efficiency of the requester. Denote by the joining time of contributor before termination time . For each valid contributor, his joining time follows a conditional distribution of that of the original joining time,
(32) 
The pdf of is subsequently obtained by
(33) 
We then compute the efficiency of the requester for each vector of joining times
(34) 
when out of contributors are before and for . The expectation of incorporates two kinds of uncertainties; one is the number of valid contributors, and the other is the exact joining times of the valid contributors. With the detailed derivation in the AppendixI, we have
(35) 
The optimal termination time is chosen to maximize the efficiency of the requester, that is,
where and are functions of . The optimal can be approximated by enumerating a finite number of candidate termination times or by a bisection search.
IvC Linearly Decreasing Strategy
The requester sets as a linearly decreasing function where the velocity is tunable to maximize his efficiency. For the fixed , the Stackelberg Bayesian Nash equilibrium can be solved using the same approach as that of the earliest strategy. In other words, is in the earliest strategy and is in the linearly decreasing strategy. Another exception is that the earliest strategy needs to try different , while the linearly decreasing strategy needs to search the different velocity . One can choose a finite set of candidate beforehand and try them one by one.
IvD Comparisons
Both the earliest
and termination time strategies are practical simplifications to the original complicated reward allocation function. They constitute the contest with incomplete information model by using Bayesian game theory. Though their basic ideas are to incentivize the early joining contributors to exert more effort, they differ in several aspects.

The type of the earliest strategy is the joining time, while the type is the number of contributors joining before time in the termination time strategy (implicitly determined by their joining time).

A contributor arriving after a certain time may not exert efforts in the earliest strategy, while in the termination time strategy, all the contributors joining before exert the same amount of positive effort, and those arriving after will not spend any effort.

The earliest and the linearly decreasing strategies incur more complicated integral computations than the termination time strategy.
Remark 5: In the Bayesian game, each contributor is aware of the joining time distribution of all the others. When the joining time distributions of contributors are heterogeneous, announcing all these distributions may divulge the privacy of contributors, even though such information is anonymous. A feasible solution is to treat all of them homogeneous and to release the anonymous joining time distribution.
V Open Crowdsensing System
In this section, we generalize our game framework to incorporate the open crowdsensing system with external arriving contributors.
Va Modelling an Open Crowdsensing System
Our previous models consider a fixed number of potential contributors, where the joining time of each contributor is characterized by a certain distribution. This is actually a closed crowdsensing system. In some applications, the requester is open to the arriving contributors without keeping the information of their personal joining time. Each arriving contributor decides the amount of efforts for crowdsensing, knowing the incentive mechanism and his joining time. Here, we suppose that the potential contributors join the contest at a Poisson rate denoted by . The public information is thus this arrival rate, other than the joining time distributions of all the contributors.
Figure 1 illustrates the counting process where denotes the joining epoch of the contributor, and denotes the interarrival time between the contributor and the one. Note that the process starts at time 0 and that multiple arrivals cannot occur simultaneously (owing to the infinite divisibility of time). The arrival process can also be specified by two other stochastic processes. The first alternative is the sequence of interarrival times, , ,
. These are positive random variables defined in terms of the arrival epochs by
and for . Hence, given the , the arrival epochs are specified as(36) 
The joint distribution of
for all is sufficient to specify the arrival process where () is exponentially distributed. The second alternative for specifying an arrival process is the counting process
, where for each , the random variable is the number of arrivals up to and including time . The counting process is an uncountably infinite family of random variables . When examining a sequence of arrivals, we usually consider a truncated counting process with a fixed number of arrivals. We denote be the sequence of joining epochs, and denote be the vector of joining times of this sequence.The joint distribution of joining epochs is a known result in queueing theory [30]
. By convolving a number of exponential distributions, the joint probability density function (p.d.f.) of the sequence
is given by(37) 
Let be the joint density of conditioned on and time . Then, this density is constant over the region and has the expression
(38) 
Here, Eq. (37) is the joint density of where can be arbitrary in this sequence. Eq. (38) is usually the joint density of in a fixed duration from 0 to .
An interesting observation is that the joint p.d.f. is not explicitly related to until , but is implicitly coupled in the conditions
. The conditional joint density is uniformly distributed. For this Poisson process, the probability moment function for
(i.e. the number of arrivals in ) is given by(39) 
In what follows, we will investigate whether our analytical framework for the closed crowdsensing system can be generalized to an open system with external arrivals.
VB Earliestn Strategy of Open Crowdsensing System
VB1 StageII
If a contributor joins the contest at time , the probability of having no more than arrivals is given by
(40) 
Then, the maximum reward that the joining contributor can acquire is on average.
Consider a truncated counting system with a maximum number of contributors where is reasonably large. The basis of this simplification is that the joining of the potential contributors after contributor is usually too late from the requester’s angle. The payoff of the contributor arriving at is the same as that of Eq. (17) except for substituting in Eq. (16) by the one in Eq. (40). For the joining contributor, the p.d.f. of joining time distribution is given by Eq. (37).
When the contributor joins, he has no knowledge on the joining times of all the others. After isolating the arrival epoch from the arrival sequence, we assume the remaining sequence as the original Poisson counting system. This assumption slightly deviates from the original arrival sequence, while greatly reducing the complexity of modelling the expected utility of the contributor. Then, for the remaining contributors, the p.d.f. of contributor is computed by Eq. (37). The expected utility of contributor on all the combinations of is given by
(41)  
where there are integrals and is not regarded as an item in the range . The procedure of finding the Bayesian Nash equilibrium follows that of the closed crowdsensing system.
VB2 StageI
At the Bayesian Nash equilibrium, the joining contributor exerts efforts that yield the utility to the requester. Because each contributor makes decision independently, the requester sees a collection of contributions each of which depends on its own joining time. For contributors with the sequence of arrival epochs , the expected utility of the requester at the Bayesian Nash equilibrium is
(42)  
and the expected reward paid to contributors is
(43)  
For a vector of joining times , the efficiency of the requester is computed as
(44) 
Then, the expected efficiency is
(45)  
Compared with the closed crowdsensing system, the analytical framework of the open system remains the same. In fact, the open system is very similar to the closed system except that the joining time of an independent contributor is substituted by the joint distribution of all the incoming contributors.
VC Termination Time Strategy of Open Crowdsensing System
VC1 StageII
In an open crowdsensing system, the total number of contributors joining before the deadline can be arbitrarily large. To avoid the analytical complexity, we suppose that there are at most contributors. For each contributor arriving at
, he can estimate the probability of meeting other
contributors with the joining times before . In this situation, there are contributors in total conditioned on the existence of at least one contributor joining before . Then, there has
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