THE APPENDIX FOR THE PAPER: INCENTIVE-AWARE JOB ALLOCATION FOR ONLINE SOCIAL CLOUDS. Appendix A 1

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1 TE APPENDIX FOR TE PAPER: INCENTIVE-AWARE JO ALLOCATION FOR ONLINE SOCIAL CLOUDS Yu Zhang, Mihaea van der Schaar Aendix A 1 1) Proof of Proosition 1 Given the SCP, each suier s decision robem can be formuated as a continuous-time Marov decision rocess hich is defined as foos: State s (, ) ; Action a ; Poicy ; State transition robabiity f ( s s, a) = (, ) f ( ); Vaue function U(,, ). Since this Marov decision rocess has a finite state sace and is ergodic, it is shon in [1] that there is aays a uniue otima oicy. We then rove statement (i) and (ii). First, suose that there is a air (, ) such that According to (A1), e have that If a suier taes a = 0 instead of - + d ' (, ) Î (0, ). U(,, )= y( ) c (, ) å (, V ) (, ). (A1) + since the suier i receive unishments henever it deviates from a (, ) hen its state is (, ), { (, )} does not change =. ence, its exected ongterm utiity becomes y () + (, ) V(, ) å hich is aays higher than + d '. U(,, ) According to the one-shot deviation rincie, is not the otima strategy, hich eads to a contradiction. Therefore, it is roved that (, ) Î {0, } aays hods. Therefore to rove statement (ii), e ony have to sho that if (, ) = 0 for some, then (, ) = 0 for any <. 1 The euations in the aendix are numbered in the format (A#) in order to differentiate ith the euations in the manuscrit hich are numbered in the format (#). 1

2 Suose that (, ) = 0for some. Then according to the one-shot deviation rincie, its exected ong-term utiity, i.e. y () + ((1- b) V(, ) + bv (0, ), is higher than the exected ong-term utiity hen it chooses to ay a =, i.e. y ()- c + å (,, ) V(, ). ere (,, a) denotes the reutation transition robabiity hen the action a is chosen. Therefore, e have that c > (1 - g )[ av ( + 1, ) + ( b-a) V(, ) -bv (0, )]. d + Without oss of generaity, e assume that ( - 1, ) =. Using the same argument, e have that c (1 - g )[ av (, ) + ( b-a) V( -1, ) -bv (0, )]. oever, it has been roved in [2] that d + given the reutation udate scheme (1) and the threshod-based ricing scheme y in the manuscrit, it is aays true that V( + 1, ) ³ V(, ), " and aso V( + 1, )- V(, ) ³ V(, )-V( - 1, ), ". Therefore, e have ' av( + 1, ) + ( b-a) V(, )- bv(0, ) ³ av(, ) + ( b-a) V( -1, )- bv(0, ).(A2) ence, e have a contradiction and ( - 1, ) = cannot hod. Since this concusion is vaid for a, e shoud have (, ) = 0 for any < if (, ) = 0, and statement (ii) foos. 2) Proof of Proosition 2 According to the one-shot deviation rincie, an SCP is sustainabe if and ony if U (, ) ³ U (, ) hods for any air (, ), i.e. coo dev I( ³ h) - c + å (, ) V(, ) ³ I( ³ h) + ((1- b) V(, ) + bv (max{ -1,0}, )) coo coo coo coo d + d + '. (A3) With sime maniuations on (A3), e obtain (8) and Proosition 2 foos. 3) Proof of Theorem 1 Let 1 - g P = [ a ({ + 1, }, ) + ( b - a ) (, )- b (0, )] V L V V, coo coo coo d + (A4) 2

3 denote a suier s incentive constraint at reutation, an SCP is sustainabe if and ony if P ³ c, for any and. First, it has been shon in [2] that given the reutation udate scheme (1) and the threshod-based ricing scheme y in the manuscrit, it is aays true that V V( + 1, ) ³ (, ), " and aso V( + 1, )- V(, ) ³ V(, )-V( - 1, ), ". Therefore, P monotonicay increases ith and, by transforg the above condition (A4), e have that that an SCP is sustainabe ony if 1 - g P = a [ V(1, )- V(0, )] ³ c 0, coo coo d + for any, hich is maximized hen h( y ) = 1. ence, it can be concuded that if an SCP ith h > 1 is sustainabe, then an SCP is aso sustainabe here its ricing threshod is h = 1 and the other arameters are the same ith. No e oo at the socia efare of a sustainabe SCP. Since the socia efare is roortiona to the fraction of suiers hose reutations are no ess than the threshod h, i.e. åh () = 1 - h () å. ence, it is aays true that () 1 (0) å h - h. Therefore to coo ³ h < h coo summarize, if there is an SCP ith h > 1 that is sustainabe, e can aays construct another sustainabe SCP here its ricing threshod is h = 1 and the other arameters are the same ith such that the socia efare under being 1 - h coo (0), hich is higher than that under. That is, the otima sustainabe SCP aays has h = 1. Aso, it shoud be noted that given a ricing threshod h, the exected one-stage game utiity of a suier hose reutation ³ h is aays max ( ) ( ) ³ h coo -c ò f d, hich is not infuenced by the seection of L. Therefore, P remains unchanged hen the vaue of L changes as ong as L ³ h. We can thus 0, concude that if an SCP is aso sustainabe ith its ricing threshod being h = 1 and L > 1, then an SCP ith L = 1 and a the other arameters same as is aso sustainabe. Given the fact that the socia efare under and are both 1 - h coo (0), e can concude that if sustainabe SCPs exist, then there is aays an otima SCP hich deivers the highest socia efare ith h = 1 and L > 1. Therefore, Theorem 1 foos. 4) Proof of Proosition 3 With to-eve reutation, an SCP is sustainabe if and ony if coo 3

4 and c (1 -g ) b[ V (, ) -V (, )], (A5) coo coo d + Substituting (9) into (A5) and (A6), e have c (1 -g ) a[ V (, ) -V (, )]. (A6) coo coo d + and c b, (A7) 1- g d + c a. (A8) 1- g d + ence, the imum of the RS of (A7) and (A8) is maximized hen a = 1 and b = 1 and conseuenty, hen c. (A9) 1- g d + hods for a, sustainabe SCPs exist. Since (A9) euas to (10), Proosition 3 foos. 5) Proof of Coroary 1 Consider a tas of oad K Î [ K, K ]. According to Proosition 4, the sufficient and necessary max condition for a suier oring on this tas to comy ith the SCP can be exressed as foos Kc. (A10) K( K)/ K (1 - e) ( K) d + K( K)/ K ence, it is obvious that hen ( K ), (1 -e) ( K) 0 and thus the LS of (A10) aroaches to infinity. Since the RS of (A10) is finite, (A10) can never be satisfied. As a resut, it can be concuded that suiers have sufficient incentive to comy ith the SCP in a tas of oad K hen ( K ) is smaer than some integer n K. Since this argument hods for a ossibe vaues of K, Coroary 1 foos ith nmax = max { n }. KÎ[ K, K ] max K 6) Proof of Theorem 2 From the manuscrit, e have the average ong-term utiity of a suier to be exressed as 4

5 and V (, ) = ( - c) + ( ( V ) (, ) + ( V ) (, )) coo coo coo coo coo (A11) ere V (, ) =- c+ ( ( V ) (, ) + ( V ) (, )). (A12) coo coo coo coo coo max ( ) (, ) ( ) 1 g g (1 b ) coo = ò f d = and co o max ( ) (, ) ( ) (1 )(1 ) coo = f d coo ò = - g - a + g. Meanhie, e have h ( ) (1 - g ) a coo ( ) = = ( ) + ( ) (1- g ) a+ g b coo coo the fooing otimization robem. ence the otima a and b can be soved in (1 - g ) a max ab, (1 - g ) a+ g b c st.. b," 1 - g d + c a, ". (A13) 1- g d + 0 a 1 0 b 1 Suose b > a, e then have b > a d + d [(1 -g )(1 - a ) + g (1 -b )] 1 - [(1 -g )(1 - a ) + g (1 -b )]. ence, it is obvious that there aays exist a sufficienty sma D such that the SCP is sti sustainabe hie the socia efare increases by reacing b ith b -D. This contradicts the fact that b is the otima SCP and hence e have b a aays hods. 5

6 Since b a, it is aays true that max max Î [, ] c = b 1- g d [(1-g )(1- a ) + g (1-b )] and c a. Using sime maniuation, it can be derived 1- g d [(1 -g )(1 - a ) + g (1 -b )] that a = 1 and b ì d ü + -g = max ï í ï ý. ence, Theorem 2 foos. [, Î max ] (1 -g ) -g c ïî ïþ 7) Proof of Proosition 4 According to (10), it is straightforard that there are to regions [, ] and [, ] such that a suier has sufficient incentive to cooerate if and ony if the ob oad Î [, ] hen its reutation is and Î [, ] hen its reutation is. If <, then the exected error robabiity g / 1 (1 e) K = - - is too arge such that a suier of reutation oses its incentive to cooerate. On the other hand, if >, the immediate cost for roviding resources is too arge such that a suier of reutation aso do not ant to cooerate. The same argument aies to [, ]. ence, for a tas of oad K, it is otima to divide the oad into as fe obs as ossibe such that the faiure robabiity of this tas is imized hie the resuting oad for each ob fas in the regions [, ] and [, ]. ence, e have and K / ( K ) {, } - >. K /( ( K ) 1) {, } Suose ( K) > ( K ) ith K < K, it is obvious that K / ( K ) ³ K /( ( K ) + 1) > K /( ( K ) + 1) > {, }. ence, suiers i not cooerate in this tas, hich contradicts the caim that ( K ) is the otima ob aocation for this tas. REFERENCES [1] A. Leizaroitz and A. J. Zasavsi, Uniueness and Stabiity of Otima Poicies of Finite State Marov Decision Processes, Mathematics of Oerations Research, 32, ,