Game Theoretic Analysis of Tree Based Referrals for Crowd Sensing Social Systems with Passive Rewards

0.18 MB·

by  Kundan Kandhway

ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
Game Theoretic Analysis of Tree Based Referrals
for Crowd Sensing Social Systems with Passive
Rewards
Kundan Kandhway and Bhushan Kotnis
Department of Electronic Systems Engineering, Indian Institute of Science
Email: {kundan,bkotnis}@dese.iisc.ernet.in
Abstract—Participatory crowd sensing social systems rely on thechildnodes,aretermedasgeometricincentivemechanisms
6
the participation of large number of individuals. Since humans [5] and such indirect rewards are called passive rewards.
1
arestrategicbynature,effectiveincentivemechanismsareneeded
0 Geometric incentive mechanisms have also been used with
to encourage participation. A popular mechanism to recruit
2 success in participatory crowd sourcing tasks. The team that
individuals is through referrals and passive incentives such as
n geometricincentivemechanismsusedbythewinningteaminthe won the 2009 DARPA Network Challenge used a geometric
a 2009 DARPA Network Challenge and in multi level marketing incentive mechanism to recruit a large number of volunteers
J schemes. The effect of such recruitment schemes on the effort to complete the task [6]. The challenge involved locating the
7 put in by recruited strategic individualsis not clear. This paper positions of 10 large red weather balloons scattered in the
2 attempts to fill this gap. Given a referral tree and the direct
continental United States in the shortest possible time [6].
and passive reward mechanism, we formulate a network game
] where agents compete for finishing crowd sensing tasks. We The prize winning team developed a mechanism where an
T characterize the Nash equilibrium efforts put in by the agents individualwhofindsthe locationofthe balloonpasseshalf of
G andderiveclosedformexpressionsforthesame.Wediscoverfree her reward to her parent (recruiter), who in turn passes half
riding behavior among nodes who obtain large passive rewards.
. of it to her parent, and so on. Thus, an ancestor who is k hop
s This work has implications on designing effective recruitment
away from the node who finished the task gets 1 part of
c mechanisms for crowd sourced tasks. For example, usage of 2k+1
[ geometric incentive mechanisms to recruit large number of thereward(themechanismmayrestrictrewardsharingtoonly

individuals may not result in proportionate effort because of a few levels). A mathematical analysis of the of this strategy
1
v free riding. revealedthatfor a rationalindividualrecruitingthe maximum
5 numberof people she knowsis the best response strategy[7].
0 I. INTRODUCTION Clearly for such a large search task, a large number of
5 The widespread presence of smart phones, wearables, GPS recruitsis key. However,that is not the only importantfactor;
7
devices and other hand held sensors have enabled individuals the efforts put in by those individuals to finish the task is
0
. to collect valuable data from their surrounding environments. equally critical. Particularly, the incentive mechanism which
1
Analyzingsuchcrowdsenseddatacanhelpmonitorurbanand was used to recruit individuals should also ensure that it
0
industrial pollution levels [1], improve our understanding of does not disincentivize individuals to work hard. The effect
6
1 urban traffic patterns [2], and can even prove to be useful for of geometric incentive mechanisms, that were used to recruit
: surveillanceandemergencyresponse[3].Somecrowdsensing individuals, on the efforts put in by them to finish the task is
v
tasks, such as traffic monitoring, can be performed using not clear. In this paper we aim to fill that gap.
i
X opportunistic sensing that does not actively involve the user, By formulating a network game we show the effect geo-
r while otherssuchas pollutionmonitoringor surveillancemay metric incentive mechanisms have on the efforts put in by
a
require active participation of the user [4]. Such participatory rational agents to finish the task. We analyze the behavior of
sensing systems, due to the human in the loop, raises unique agents connected over a referral network (tree) and compute
challengessuchasrecruitingandincentivizingalargenumber an equilibrium effort profile. Our analysis technique is game
of individuals to participate in the sensing activity. theoretic, i.e., we model each individual as a strategic agent
Offeringreferralrewardsforrecruitingfriendstosignupfor who wants to maximize her utility in a given situation. The
completinga task is widely used in multi levelmarketing [5]. equilibrium we consider is a pure strategy Nash equilibrium
Such mechanisms generate referral trees where a parent node (PSNE), and we show that it always exists and is unique for
represents an individual that has recruited or referred another interesting model parameters. More importantly, our results
individualwhoisrepresentedasachildtothereferringparent show that while geometric incentive schemes are an excellent
node. In such schemes, apart from rewarding individuals for tool for recruiting individuals they may indeed disincentivize
finishing the tasks, additional rewards, in the form of incen- a few individuals from putting any effort.
tives, are provided to encourage individuals to recruit people
fromtheirsocialconnections.A popularstrategyistoprovide II. MODEL
A. The Task Arrival and Processing Model
a proportion of the reward earned by recruited individuals to
the recruitingindividualrecursively.Such mechanisms,where Letthe setofindividualsbe denotedbyN ={1,2,...,n},
the parent node obtains a proportion of the reward earned by and connected over a hierarchical directed tree T. The di-
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].
ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
rection of the edges are from the root towards the leaves. The indirect incentives are shared with other players as
We assume that if an individual performs a crowd sourced specified by the reward sharing matrix ∆ = [δ ] where δ
ij ij
task, she gets a direct incentive, and the nodes along the path is the fraction of the total reward received by i when j grabs
from the individual to the root receive an indirect incentive. the task. Note that δ = γ, ∀i ∈ N. Since the network is a
ii
We assume that the tasks arrive at a Poisson rate λ, which directedtree,ifj doesnotappearinsubtreeofithenδ =0.
ij
is a common knowledge, and are queued up until served. We call the matrix ∆ monotone non-increasing if δ > δ ,
ij ij′
We assume that each agent competes for the task (as is the whenever the hop-distance dist (i,j) < dist (i,j′), and
T T
case in multi level marketing and the 2009 DARPA Network anonymousif the δ depends only on the distance of i and j
ij
Challenge). Each node i attempts to capture a task from the onT,dist (i,j),andnotontheidentitiesofthenodes.Hence
T
work queue according to a Poisson process with rate λ . If these are reasonable assumptions to make.
i
the total reward of a task is R, we assume that a node gets a While some of our results extend to general networks and
direct incentiveof γ·R if she grabsthe job (γ ∈(0,1)), with general∆matrices,inthispaper,we focusonmonotonenon-
the rest shared as indirect incentives. The cost of maintaining increasingandanonymousrewardsharingmatrices.Toensure
anattemptrateλ isC·λ , whereC >0isaknownconstant. that the total reward is bounded above by R, we assume
i i
The task is assigned exclusively to the agent who attempts it δ ≤ 1, ∀j ∈ N. Let us denote T as the subtree
k∈N kj i
firstafterthetaskarrives.We assumetheagentshaveuniform rooted at i. Combining the direct and indirect rewards and
P
skillofperformingatask.Thetimetocompleteataskissmall costs, the expected utility of agent i∈N given by,
compared to the inter-attempt times of any agent and hence u (λ ,λ ), λ >λ,
u (λ ,λ )= i,1 i −i k∈N k (1)
the task completion time is ignored. i i −i u (λ ,λ ), λ ≤λ,
(cid:26) i,2 i −i Pk∈N k
We can model the task arrival and departure in a server- where λ =(λ ,...,λ ,λ ,...,λ ), and,
−i 1 i−1 i+1 nP
qcouneusoelidmaoteddelsewrvhiecreerattheeoafrrthivealenrtairtee nisetwλo(r>k is0) ajn∈dNλthje, ui,1(λi,λ−i):=λγRPk∈λNi λk +λjR∈TXi\{δi}ijPk∈λNj λk −Cλi, (2)
becausethesuperpositionofPoissonprocessesisPoissonwith
the rate being the sum of the rates [8]. When agentPi tries to ui,2(λi,λ−i):=γRλi+λRXδijλj−Cλi. (3)
capturea jobfromthetask queue,theprobabilitythatshe can j∈Ti\{i}
In both (2) and (3), the first term on the RHS denotes the
grab the job is given by λi/ j∈Nλj. This is because the expectedutilityofagentiduetoherowneffortλi,thesecond
inter-attempttimes are exponentiallydistributed for a Poisson
termdenotesthe indirectutility comingfromthe effortsλ of
P j
process, and using the memoryless property of exponential
all the nodes j ∈T \{i}, and the third term denotes the cost
i
randomvariables[9].Thereare tworegimesthe system could
to maintain the effort. The utility model is parametrized by
operate in.
the reward R, reward sharing matrix ∆, cost C, and tree T.
Case 1: When λ > λ, the queueing process is a
j∈N j C. Effort Sharing and Effort Zones
positive recurrent Markov chain and all tasks will be served,
and the output pProcess would also be Poisson with rate λ The effortsharing functionf is a recursivefunction,which
is computed for a given tree T and a reward sharing matrix
(Burke’s Theorem [8]), the rate at which agent i would grab
∆, in a bottom up fashion, i.e., from the leaves towards the
the task is therefore given by λ · λ / λ . Hence, the
i j∈N j
direct reward to i is λγR·λ / λ . root.LetusdenotethesetofdirectedtreesbyT.Foranytree,
Case2:When λ ≤λi,theqj∈uNeuePijngprocessisanullor f is initialized to 1 for all the leaves, and then computed as
transient Markovjc∈hNainjand witPh high probability (probability follows.
approaching unPity) the queue will be non-empty at a steady Effort Sharing Function: A mapping f : T ×[0,1)n×n →
state [9].Insucha setting,anyagentiwhoattemptstograba [0,1] given by the recursive formula,
task actually gets a task, and hence the direct reward of agent f(T ,∆)=max 0,1− δ ·f(T ,∆) . (4)
i j∈Ti\{i} ij j
i would be γR·λ .
i This function is manximum when i is a leaf, and doecreases
If a node j hits a task, she receives a ‘direct’ reward, and P
for nodes with large subtrees below them (i.e., the ones
each node i on the directed path from the root to j receives
with large passive rewards). In Sec. III we will see that, for
an ‘indirect’ reward. Since the efforts are costly (Cλ ), each
i interesting modelparameters, the effortequilibriumlevels are
agenthas to decide on the effortsto maximizeher net payoff,
proportional to this function, leading to smaller effort levels
i.e., (direct + indirect) reward minus the cost. If there are
for nodes with large subtrees below them. Based on this we
indirect rewards, a node will reduce efforts to reduce costs. partition the space of parameters V := {v = (R,∆,C,T)}
Thisinducesagamebetweenthenodes.Thestrategyofplayer
into four regions:
i is to choose the attempt rate λi ∈ Si = [0,∞). In the Region I: R = v: γ·R ∈[0,1) (5)
following section, we derive an expression for the expected 1 C
utility for an agent i. Region II: R =nv: γ·R =1 o (6)
2 C
B. Indirect Reward and the Utility Model
Region III: R =nv: γ·R ∈ 1o, j∈Nf(Tj,∆) (7)
The task arrival process and its incentive sharing scheme 3 C Pj∈Nf(Tj,∆)−1
induce competition among the nodes. Direct incentives are Region IV: R =nv: γ·R >(cid:16) jP∈Nf(Tj,∆) io (8)
earned when an agent grabs a task. 4 C Pj∈Nf(Tj,∆)−1
n P o
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].
ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
III. ANALYTICALRESULTS
We compute the PSNE for the four regions R to R in
1 4
Eqn (5)–(8) (hence a PSNE always exists). The results serve
to predict the efforts put in by strategic agents connected via
a network for the given reward sharing matrix ∆.
Thefirstresultshowsthatu isconcaveinλ .Thishelpsin
i i
calculationofthePSNE asitisautilitymaximizationprocess
for all agents.
Lemma III.1. u (λ ,λ ) is a concave function in λ .
i i −i i
Proof: We show this in two steps.
Fig. 1: Graphical illustration of the parameter space and the Nash Step 1: u (λ ,λ ) is a linear function in λ , which is
i,2 i −i i
equilibria characterization.
a special case of a concave function. Now we show that
u (λ ,λ ) is concave in λ . Differentiating Eq. (2),
We will analyze PSNE effort profiles for the game in the i,1 i −i i
wexhhoibleitspadriafmfeeretenrtsbpeahcaev.ioWreinweialclhseoefitnheSseecr.eIgIiIotnhsa(tatnhdehgeanmcee ∂∂λiui,1(λi,λ−i)= λγR(cid:16)Pj6=i(λjj−∈PNjλ∈jT)i2\{i}δijλj(cid:17) −C (13)
the regions are divided in this way). and by taking the second derivaPtive we get,
ELf:f=ort{λZo=ne(sλ:1W,.e..p,aλrnti)ti}onintthoefospuarcreegoifonags:ents’ effort profile ∂∂λ22iui,1(λi,λ−i)= −2λγR(cid:16)Pj6=(iλjj∈−NPλjj∈)T3i\{i}δijλj(cid:17) (14)
Z = λ: λ <λ Zone 1, (9) Since λ ≥ 0 ∀i, λ − P δ λ ≥ 0, this is
1 j∈N j i j6=i j j∈Ti\{i} ij j
Z =nλ:P λ =λo Zone 2, (10) due to the fact th(cid:16)atPδij ∈ [0,1P) and |{j ∈ N(cid:17): j 6= i}| ≥
2 j∈N j |T \{i}|. Hence, u (λ ,λ ) is concave in λ .
i i,1 i −i i
Z =nλ∈PZ :λ + o δ λ Step 2: We observe that, ui(λi,λ−i) =
3 2 i j∈Ti\{i} ij j min{u (λ ,λ ),u (λ ,λ )}. This is because that when
i,1 i −i i,2 i −i
≥nλ 1− C ,∀ iP∈N Zone 3, (11) λ < 1, u (.) < u (.) and when λ ≥ 1,
γR j∈Nλj i,1 i,2 j∈Nλj
Z = λ(cid:16): (cid:17)λ >λ o Zone 4. (12) uPi,1(.)≥ui,2(.) fromEquations(1),(2) and(3)P.Thefunction
4 j∈N j ui(·) is the minimum of two concave functions and hence is
Notice that Zn3 ⊆PZ2. We defineoit separately to show a result concave. (cid:3)
that characterizes the set of Nash equilibria for region R3. If The next result characterizes the PSNE in region R1.
we consider a single server abstraction of the entire network,
then the arrival rate λ sees a consolidated service rate of Theorem III.2 (PSNE in R1). If v ∈ R1, then the PSNE
effort profile is unique, and is given by λ∗ =(0,...,0).
λ .Iftheservicerateissmaller,equal,orlargerthanthe
i∈N i
arrival rate, then according to the conditions (9–12), the task Proof: When v ∈ R , we argue that all players will put
P 1
queue would be either over loaded (Zone 1), critically loaded zero effort in Nash Equilibrium. Suppose λ∗ ∈Z , it implies
4
(Zone 2, Zone 3), or under loaded (Zone 4) respectively. u (λ∗) = u (λ∗). Taking derivative of the utility at λ∗, we
i i,1
ThecrowdsourcingmanagerwouldliketooperateinZ4,so get,
athnadttohveerreaislonnogapcecruimoduloaftitoimneo,fatallskinsc.oming tasks are served ∂∂λ∗iui,1(λ∗i,λ∗−i)= λγR(cid:16)Pj6=iλ∗j −Pλj∗∈T2i\{i}δijλ∗j(cid:17) −C
Weseefrom(1)thatthezonescorrespondtodifferentutility j∈N j
structures, λ(cid:16)∗P− (cid:17) δ λ∗
(i) u (λ)=uT (λ), if λ∈Z ; (ii) u (λ)=uT (λ), if = λ ·γR· j6=i j j∈Ti\{i} ij j −C
λ∈iZ ∪Z .i,1 4 i i,2 j∈Nλ∗j! (cid:16)P Pj∈Nλ∗j (cid:17)
1 2
Inthefollowingsection,weshowthatineachoftheregions P <1 P ≤1
R1 to R4, the PSNE are mapped to the zones Z1 to Z4, <γ|R−{Cz<0.} | {z }
sometimesuniquely.ThePSNEalwaysexists,whichisanon- Thisisa contradictionforλ∗ to beanequilibrium,sinceeach
trivialresult.Fig. 1 graphicallyillustrates theparameterspace playeriwouldbebetteroffbydecreasingtheireffortfromλ∗.
i
and the main results on the characterization of the equilibria Hence λ∗ ∈/ Z . So, λ∗ ∈ Z ∪Z , which implies u (λ∗)=
4 1 2 i
(detailed analysis is carried out in Sec. III). It gives the u (λ∗). But ∂ u = γR−C < 0, since v ∈ R . Thus
intuition that since the reward to cost ratio R/C is small in λi∗,2=0 for all ∂iλ∈i Ni,2. 1 (cid:3)
i
regionsR toR ,thesumoftheequilibriumeffortscouldfall The intuition is that in R , the reward to cost ration R/C
1 3 1
below the incoming rate λ. However, in region R , the R/C issmallenough.Hence,noindividualgetsanypositivepayoff
4
ratio is large enough to ensure a unique PSNE, in which the by putting any positive effort.In R , the R/C ratio reachesa
2
sum equilibriumeffortcan efficiently serve the incomingtask critical value, where there exists multiple PSNE. The agents
stream. collectivelyare indifferentbetween justmeetingthe incoming
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].
ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
rate λand keepingthe sumeffortsmaller thanthis. Hencewe λ∗ ≥0. This leads us to (from Eq. (13)),
i
get the following theorem. 2
λγR
Theorem III.3 (PSNE in R2). If v ∈ R2, then any λ∗ ∈  λ∗j = C  λ∗j − λ∗jδij (17)
Z1∪Z2 is a PSNE. jX∈N Xj6=i j∈XTi\{i}
Foragiventreeanditsequilibriumprofileλ∗,let  λ∗ =
Proof: Case 1: Suppose λ∗ ∈Z . Then u (λ∗)= u (λ∗) j∈N j
4 i i,1 x (variable substitution), then manipulation of (17) leads to,
and we repeat the analysis of Theorem III.2 to show that P
x2C
∂ u (λ∗,λ∗ )<0,∀ i∈N, even for v ∈R . Hence it is λ∗+ λ∗δ =x− , ∀ i ∈N (18)
n∂oλt∗iani,1equiilibr−iuimasplayerswillkeepdecreasing2theirefforts. i j∈XTi\{i} j ij λγR
Case 2:Whenλ∗ ∈Z ∪Z , we haveu (λ∗)=u (λ∗)and We doanothervariablesubstitutiontodenotetheRHSofEqn
1 2 i i,2
∂ u (λ∗,λ∗ )=0,∀ i∈N. Thus, u (λ ,λ∗ ) is constant (18) by y (≥0 since LHS is ≥0). That is,
∂λ∗ i,2 i −i i i −i
whien λ∗ ≤ λ, for all i ∈ N. Hence each element of x2C
j∈N j y =x− . (19)
the set Z ∪Z is a PSNE. (cid:3) λγR
1 2
P
For the sake of convenience, we will discuss region R4, Lemma III.7. λ∗ =yf(T ,∆), ∀i∈N.
before R . In the final region R , the ratio R/C crosses i i
3 4
a minimum threshold, which guarantees that the equilibrium Proof: We provethis claim via inductiononthe levelsofT.
effort profile is unique and sufficient to serve the incoming Let the depth of T be D.
task rate, i.e., j∈Nλ∗j > λ. We show this using a few From Eqn (18), λ∗i + j∈Ti\{i}δijλ∗j =y,∀ i∈N. From
intermediate lemmas. Cases 1 and 2 above,
P P
Lemma III.4. If v ∈ R4, and if there exists a PSNE effort λ∗ =max 0,y− δ λ∗ ∀ i∈N (20)
profile λ∗, it cannot lie in Z ∪Z . i  ij j
1 2
j∈XTi\{i}
Proof: We prove this via contradiction. Suppose ∃ a PSNE Step 1: For an arbitrarynodej at levelD, from (20), λ∗ =
j
effort profile λ∗ ∈ Z1 ∪ Z2, i.e., j∈Nλ∗j ≤ λ. Hence y. Hence, the proposition is true as f(Tj,∆)=1 for a leaf.
u (λ∗,λ∗ ) = u (λ∗,λ∗ ),∀ i ∈ N. But, ∂ u (λ∗) = Now,selectanarbitrarynodei(whichisnotaleaf)atlevel
γRi −i C−i> 0,∀i,2i ∈i N−i(differentiPating Eq∂.λ(i3)i),.2This is D−1.From(20)weget,λ∗i =max(0,y− j∈Ti\{i}δijy)=
because, v ∈ R4 and from Eqn (8) we know that γCR > ymax(0,1− j∈Ti\{i}δij1)=yf(Ti,∆).P
aPtPjλ∈jN∈∗N.f(SfT(ojT,,j∆,i∆)−)is1 b>ett1e.rHofefncbey, uinticlriteyasoinfgihiissinefcfroeratsifnrgominλλ∗i, levSetleDp 2−: lL.eCtPoλn∗jsi=deryafn(Tajr,b∆itr)arbyentorudee ifoartalellvenloDdes−jlu−pt1o.
i From (20) and (4),
which is a contradiction to the fact that λ∗ is a PSNE. (cid:3)
Corollary III.5. For v ∈ R4, if there exists a PSNE effort λ∗i =max0,y− y·f(Tj,∆)δij=yf(Ti,∆) (21)
profile λ∗ then ui(λ∗)=ui,1(λ∗), ∀ i∈N. which concludes thej∈inXTid\u{ci}tion.  (cid:3)
LemmaIII.6. Iftheutilitystructureui isgivenbyui,1, ∀i∈ To find an expression for PSNE we now evaluate y. The
N, thenthere existsa uniquePSNE effortprofile λ∗ givenby, sum of efforts of all the players is defined as x. Hence,
wheλre∗it=heλfCγuRnc(cid:18)tio(PnPjf∈j∈NNisff(d(TTejjfi,∆,n∆)e)−d)12i(cid:19)n f(4()T.i,∆),∀ i∈N, (15) xexp=resysPioxnj=∈aNndfs(oTλljv∗,i∆n=g).λfoγSrRuxbstyiiteultdjin∈sg,Nffo(rTyj,∆fro)m−(119) in(t2h2is)
Proof: If a PSNE effort profile (λ∗,λ∗ ) exists in j C f(T ,∆)
i −i j∈N P j∈N j !
the given game, then it must satisfy, u (λ∗,λ∗ ) ≥ X
u (λ ,λ∗ ),∀λ ∈ S = [0,∞),∀i ∈ Ni.,1Thiis i−miplies, Using (19) we get, P
i,1 i −i i i
λ∗i = arg max ui,1(λi,λ∗−i),∀ i∈N. y= λγR j∈Nf(Tj,∆)−1 (23)
λi∈Si=[0,∞) C  2
ThusinordertofindtheNashequilibriumwehavetosolve P j∈Nf(Tj,∆)
 
the following optimization problem for each i∈N. Combining (23) and thecl(cid:16)aim above, the P(cid:17)SNE is given by,
P
max u (λ ,λ∗ ) min −u (λ ,λ∗ )
s.t. λi λii,1≥0i, −i (cid:27) ⇒ s.t. λi −λii,1≤0i −i λ∗i = λCγR j∈Nf(Tj,∆)−12f(Ti,∆),∀ i∈N
(16) P j∈Nf(Tj,∆)
 
Duetoconcavityofui,1,thisisaconvexoptimizationproblem KKT equationsled(cid:16)Pto a unique so(cid:17)lution of the optimization
with linear constraints, which can be solved using KKT problem, hence PSNE is unique. (cid:3)
theorem. At the minimizer λ∗ of problem (16), ∃ µ ∈ R
such that, (i) µ ≥ 0, (ii) − ∂i u (λ∗,λ∗ )− µ = 0, (iii) Theorem III.8 (PSNE in R4). If v∈R4, then there exists a
−µλ∗ =0, (iv) −λ∗ ≤0. ∂λi i,1 i −i unique PSNE effort profile λ∗ given by Eqn (15), which lies
Casei1: µ>0⇒λ∗i =0 and in this case ∂ u =−µ≤0. in Z4.
i ∂λi i,1
Case 2: µ = 0 ⇒ ∂∂λiui,1(λ∗i,λ∗−i) = 0 and in this case Proof: From Corollary III.5 when v ∈ R4, ui(λ∗) =
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].
ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
u (λ∗). From Lemma III.6 with this utility function, the Z ⊆ Z , the equality in the third line comes by replacing
i,1 3 2
unique PSNE effort profile is given by Eqn (15). We use this λ∗ =λ in Eq. (13), and the inequality in the third line
j∈N j
expression to compute the sum λ∗, and substitute the isobtainedbyreorganizingtheexpressionin Eqn(11). Hence
value of γR/C from (8) to get j∈N λj∗ > λ ⇒ λ∗ ∈ Z . Pfor each agent i, the utility is maximized at λ∗, when other
Pj∈N j (cid:3)4 playersareplayingtheequilibriumstrategy.So,λ∗ isaPSNE.
In Z , the sum of the equilibrPium efforts is more than λ. (⇒):We aregiventhatv∈R ,andλ∗ isaPSNE.We first
4 3
Hence, it shows that in order to meet the goal of efficiently show that λ∗ should necessarily be in Z . We show this via
2
serving the incoming tasks, the reward to cost ratio has to contradiction. Let us consider the two following cases.
be above a threshold given by (8). We also see that when Case 1: Let λ∗ be the PSNE if it exists such that λ∗ ∈Z ,
1
the parameters are in R , the equilibrium effort of agent i is that is, λ∗ <λ then, u (λ∗)=u (λ∗),∀ i∈N. But,
4 j∈N j i i,2
proNpoowrtiownealdtioscfu(sTsi,r∆eg)i.on R3. In region R3, the reward to o∂f∂λiiuiis,2i(Pnλcr)e=asiγng·Rin−λCi >at0λ,∗∀. Sio∈,Ni is(fbroetmter(7o)f)f.Hbyenincec,reuatsiliintyg
cost ratio is a little larger than R2. However, it is still not his effortfrom λi, which is a contradictionto the fact that λ∗
enoughto guaranteethesum equilibriumeffortlevelsto cross is a PSNE.
λ (which was the case in R4). In particular, we show that it Case 2: Let λ∗ be the PSNE if it exists such that λ∗ ∈Z3,
is necessary and sufficient for the equilibriumeffortprofile to thatis, λ∗ >λthen,u (λ∗)=u (λ∗),∀i∈N.Thus
j∈N j i i,1
LliveemimnaZ3II.IT.9o.sIhfovw∈thRat3,wZe3neisednoZn3emtoptbye. nonempty. Ewqhnich(2i2Ps)aiscovnatlriadd.icUtisoinn.gH(e7n)cien,i(f2t2h)ewPeSNgEetePffojr∈tNprλo∗jfile≤λλ∗
exists, it must lie in Z .
2
Proof: Theproofisconstructive.Letusconsiderthefollow- Now,sinceλ∗ ∈Z ,theutilitywillhaveatransitionatthis
2
ing summation: S = λ 1− C f(T ,∆). Substitut- point. We know that λ∗ is a PSNE, hence the utility must be
γR j∈N j
ing C ≥ j∈Nf(Tj,∆)(cid:16)−1 (usin(cid:17)gP(7)), we get S ≤ λ. Now, maximized at this point for all agents i ∈ N, given that the
γR P j∈Nf(Tj,∆) otheragentsstick tothe equilibriumstrategies.Hence,itmust
let us construPct a λ, such that λ = λ 1− C f(T ,∆)+ hold that,
i γR i
Hβie(nλc−e,Sb)y,c∀oins∈truNc,tiwonh,ere,βi ≥λ0=, ∀λi⇒∈(cid:16)Nλ∈anZd (cid:17).Wni=e1sβeei =tha1t,. ∂∂λiui(λ∗i−,λ∗−i)= ∂∂λiui,2(λ∗i,λ∗−i)≥0, and (24)
Pi∈N i 2P C ∂∂λiui(λ∗i+,λ∗−i)= ∂∂λiui,1(λ∗i,λ∗−i)≤0. (25)
λi+ δijλj =f(Ti,∆)+ δijf(Tj,∆)·λ(cid:18)1− γR(cid:19) Condition (24) yields, γ·R−C ≥0, which is satisfied since
j∈TXi\{i}  ≥1,usingdje∈finTXitiio\n{ofi}ffromEq.(4)  v ∈ R3. Condition (25) yields, γR(cid:16)λ−λ∗i−Pλj∈Ti\{i}δijλ∗j(cid:17) −
| {z } C ≥ 0, since we have shown that λ∗ ∈ Z . Reorganizing
2
+|βi+j∈CTXi\{i≥}{z0δijβj(λ−S}) tHheiTsnocines,euqλmu∗aml∈iatyZr,i3zw.eethgeist,sλe∗ict+ionP, ijn∈Tthi\e{im}δoidjeλl∗jc≥onsλid(cid:16)e1re−d,γCpRu(cid:3)(cid:17)re.
≥λ 1− strategy Nash equilibria always exist. The regions correspond
(cid:18) γR(cid:19)
to different reward to cost ratio, and the results predict the
So, λ∈Z3, hence Z3 is nonempty. (cid:3) effortsexpectedfromrationalagentswhentheratioliesinone
oftheseregions.Itgivesa measureofhowlargeγR/C needs
Theorem III.10 (Necessary and Sufficient Condition for
PSNE in R ). If v ∈ R , an effort profile λ∗ is a PSNE, tobeinordertoefficientlyservetheincomingtaskprocess(to
3 3
if and only if λ∗ ∈Z . operatein Z4). The results suggestthatnodeshavingnetwork
3
advantages due to recruiting other nodes free ride on others.
Proof: (⇐): We show that if λ∗ ∈ Z3, it is a PSNE. From Thus, a geometric incentive mechanism may disincentivize
LemmaIII.9,Z3 6=φ, sowecanpicka λ∗ ∈Z3.Since,λ∗ ∈ nodes that have large subtrees from putting efforts.
Z implies λ∗ ∈Z , the utility undergoes a transition at this
3 2
point, u (λ∗−,λ∗ ) = u (λ∗−,λ∗ ), and u (λ∗+,λ∗ ) = IV. NUMERICAL RESULTS
i i −i i,2 i −i i i −i
u (λ∗+,λ∗ ). Since u is continuous but not differentiable WenumericallycomputethePSNE(usingTheoremIII.8)to
i,1 i −i i
at λ∗, we look at the left and right derivatives at this point. illustrate the effect of the referral tree on individualefforts in
region R — the most interesting region where all incoming
4
∂ ∂
∂λ∗ui(λ∗i−,λ∗−i)= ∂λ∗ui,2(λ∗i−,λ∗−i)=γR−C >0, tasks are served. We consider a referral tree shown in Fig.
i i 2a. To illustrate the analytical results we enforce the structure
∂ ∂
∂λ∗ui(λ∗i+,λ∗−i)= ∂λ∗ui,1(λ∗i+,λ∗−i) in Eqn (26) on the reward sharing matrix ∆ = [δij]. This is
i i motivated by the geometric incentive mechanism used by the
= γR λ−λ∗i − j∈Ti\{i}δijλ∗j −C ≤0 winning team in the DARPA Network challenge [6].
(cid:16) Pλ (cid:17) 1 distT(i,j)+1 j ∈T ,
δij = a i (26)
The first inequality comes because v ∈ R3. Since λ∗ ∈ ( (cid:0)0 (cid:1) otherwise.
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].
ThispaperwaspresentedinSocialNetworkingWorkshopatInternationalConferenceonCommunicationSystemsandNetworks(COMSNETS),Bangalore,India,January2016.
a=2 a=2 a=3 a=3
Node 1 0.25 0.5 0.25 0.5
0.2 0.4 0.2 0.4
Node 2
Node 3 Node 4 Node 5 effort00.1.15 pected utility00..23 effort00.1.15 pected utility00..23
ex ex
Node 6 Node 7 0.05 0.1 0.05 0.1
0 0 0 0
Node 8 12345678 12345678 12345678 12345678
players players players players
(a)Referral Tree. (b)Effortandrewards,a=2. (c)Effortandrewards, a=3.
Fig. 2: Referral tree and PSNE effort/reward profiles for the referral tree shown in Fig.2a. λ=0.2, R=15, C =1.
rewardsifanyagentintheirsubtreefinishesatask.Weprovide
1.5
a=2 acompleteanalysisandaclosedformsolutionforallpossible
a=3
a=4 system parameters. We compute the PSNE effort profile for
thecompleteparameterspaceandshowthatthePSNEalways
ort
eff exists. In some regions of the parameter space it is unique,
m
u 1 while in others it is not. Our results uncover free riding
n s
ea behavioramongnodeswhoobtain largepassive rewardsfrom
m
their subtrees. This has implications on crowd sourced tasks
such as the DARPA Network challenge. In particular, usage
of geometric incentivemechanismsto recruitlarge numberof
0.5
0 20 40 60 80 100 individuals may not result in proportionate effort due to free
number of nodes
riding.
Fig.3:Meansumeffortvs.numberofnodesforvaryinga.Treesare
ACKNOWLEDGEMENT
generatedrandomly.Resultsobtainedbyaveragingover500samples.
Thisworkwasapartofthecourseprojectforthe“GameTheory”
λ=0.2, R=15, C =1.
course which the authors took during January–April 2012 session
at IISc. Authors thank the instructor Prof. Y Narahari and the TA
Swaprava Nath for introducing them to DARPA Network Challenge
Here, a>1 andγ =δ =1/a. The PSNE effortprofilesand
ii and for their comments on improving the presentation of this paper.
expected utility are shown in Fig. 2b and Fig. 2c. As seen in
both the figures, node 2 who has three children does not put REFERENCES
any effort due to passive rewards. In the a = 2 case, node 4 [1] R.K.Ganti,F.Ye,andH.Lei,“Mobilecrowdsensing: currentstateand
doesnotputanyeffort,butwhenthepassiverewardisreduced futurechallenges,”IEEECommunicationsMagazine,vol.49,no.11,pp.
32–39,2011.
byincreasinga to3,node4 putsinsomeeffort.Althoughthe [2] B.Pan,Y.Zheng,D.Wilkie, andC.Shahabi, “Crowdsensing oftraffic
sum effort is higher when a = 2, the variability in the effort anomalies based onhuman mobility and social media,” inInternational
among the nodes is also high compared to a=3. Conference on Advances in Geographic Information Systems, 2013, pp.
344–353.
Fig. 3 illustrates the effect of a on sum effort. Trees were [3] E. T.-H. Chu, Y.-L. Chen, J.-Y. Lin, and J. W. Liu, “Crowdsourcing
generated randomly. To generate a tree of size N we carry support system for disaster surveillance and response,” in International
SymposiumonWirelessPersonalMultimediaCommunications,2012,pp.
out the following process recursively. Given a tree of size
21–25.
i, the (i + 1)th node chooses a parent at random from i [4] P.Dutta, P. M. Aoki, N. Kumar, A. Mainwaring, C. Myers, W.Willett,
nodestoattachitself.Wediscoverthatthesumeffortsaturates and A. Woodruff, “Common sense: participatory urban sensing using
a network of handheld air quality monitors,” in ACM Conference on
as number of nodes in the tree increases. This suggests
EmbeddedNetworked SensorSystems,2009,pp.349–350.
that recruiting large number of individuals using geometric [5] Y. Emek, R. Karidi, M. Tennenholtz, and A. Zohar, “Mechanisms for
incentivesmaynotyield proportionalincreasein productivity. multi-level marketing,” in ACM Conference on Electronic Commerce,
2011,pp.209–218.
[6] G. Pickard, W. Pan, I. Rahwan, M. Cebrian, R. Crane, A. Madan, and
V. CONCLUSIONS A. Pentland, “Time-critical social mobilization,” Science, vol. 334, no.
6055,pp.509–512,2011.
In this paper we performed a game theoretic analysis on
[7] G. Pickard, I. Rahwan, W. Pan, M. Cebria´n, R. Crane, A. Madan,
the effortsputin by individualsrecruitedusinga referraltree, and A. Pentland, “Time critical social mobilization: The darpa network
which is a popular crowd sensing mechanism for recruiting challenge winningstrategy,” arXivpreprintarXiv:1008.3172, 2010.
[8] D.P.Bertsekas,R.G.Gallager,andP.Humblet,Datanetworks. Prentice-
individuals. We propose a queuing model with Poisson task
HallInternational NewJersey,1992,vol.2.
arrivalswhereagentscompetetofinishthetask.Agentsreceive [9] R.W.Wolff,Stochasticmodellingandthetheoryofqueues. Englewood
not only direct rewards for finishing a task, but also indirect Cliffs,NJ,1989,vol.96.
Copyright(c)2016IEEE.Personaluseispermitted.Foranyotherpurposes,[e-mail].