Lower Bound for Envy-Free and Truthful Makespan Approximation on Related Machines

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1 Lower Bound for Envy-Free nd Truthful Mespn Approximtion on Relted Mchines Lis Fleischer Zhenghui Wng July 14, 211 Abstrct We study problems of scheduling jobs on relted mchines so s to minimize the mespn in the setting where mchines re strtegic gents. In this problem, ech job j hs length l j nd ech mchine i hs privte speed t i. The running time of job j on mchine i is t il j. We see mechnism tht obtins speed bids of mchines nd then ssign jobs nd pyments to mchines so tht the mchines hve incentive to report true speeds nd the lloction nd pyments re lso envy-free. We show tht 1. A deterministic envy-free, truthful, individully rtionl, nd nonymous mechnism cnnot pproximte the mespn strictly better thn 2 1/m, where m is the number of mchines. This result contrsts with prior wor giving deterministic PTAS for envy-free nonymous ssignment nd distinct deterministic PTAS for truthful nonymous mechnism. 2. For two mchines of different speeds, the unique deterministic sclble lloction of ny envy-free, truthful, individully rtionl, nd nonymous mechnism is to llocte ll jobs to the quicest mchine. This lloction is the sme s tht of the VCG mechnism, yielding 2-pproximtion to the minimum mespn. 3. No pyments cn me ny of the prior published monotone nd loclly efficient lloctions tht yield better thn n m-pproximtion for Q C mx [1, 3, 5, 9, 13] truthful, envy-free, individully rtionl, nd nonymous mechnism. This wor ws supported in prt by NSF grnts CCF nd CCF Contct informtion: {lf,zhenghui}@cs.drtmouth.edu. Dept of Computer Science, Drtmouth College, Hnover, NH 3755, USA. An extended bstrct will pper in SAGT

2 1 Introduction We study problems of scheduling jobs on relted mchines so s to minimize the mespn (i.e. Q C mx ) in strtegic environment. Ech job j hs length l j nd ech mchine i hs privte speed t i, which is only nown by tht mchine. The speed t i is the time it tes mchine i to process one unit length of job t i is the inverse of the usul sense of speed. The running time of job j on mchine i is t i l j. A single job cnnot be performed by more thn one mchine (indivisible), but multiple jobs cn be ssigned to single mchine. The worlod of mchine is the totl length of jobs ssigned to tht mchine nd the cost is the running time of its worlod. The scheduler would lie to schedule jobs to complete in minimum time, but hs to py mchines to run jobs. The utility of mchine is the difference between the pyment to the mchine nd its cost. The mechnism used by the scheduler ss the mchines for their speeds nd then determines n lloction of jobs to mchines nd pyments to mchines. Idelly, the mechnism should be fir nd efficient. To ccomplish this, the following fetures of mechnism re desirble. Individully rtionl A mechnism is individully rtionl (IR), if no gent gets negtive utility when reporting his true privte informtion, since rtionl gent will refuse the lloction nd pyment if his utility is negtive. In order tht ech mchine ccepts its lloction nd pyment, the pyment to mchine should exceed its cost of executing the jobs. Truthful A mechnism is truthful or incentive comptible (IC), if ech gent mximizes his utility by reporting his true privte informtion. Under truth-telling, it is esier for the designer to design nd nlyze mechnisms, since gents dominnt strtegies re nown by the designer. In truthful mechnism, n gent does not need to compute the strtegy mximizing his utility, since it is simpler to report his true informtion. Envy-free A mechnism is envy-free (EF), if no gent cn improve his utility by switching his lloction nd pyment with tht of nother. Envy-freeness is strong concept of firness [1, 11]: ech gent is hppiest with his lloction nd pyment. Prior wor on envy-free mechnisms for mespn pproximtion problems ssumes tht ll mchine speeds re public nowledge [6, 15]. We ssume tht the speed of mchine is privte informtion of tht mchine. This ssumption mes it hrder to chieve envy-freeness. Only if the mechnism is lso truthful, cn the mechnism designer ensure tht the lloction is truly envy-free. In this pper, we prove results bout nonymous mechnisms. A mechnism is nonymous, roughly speing, if when two gents switch their bids, their llocted jobs nd pyments lso switch. This mens the lloction nd pyments depend only on the gents bids, not on their nmes. Anonymous mechnisms re of interest in this problem for two resons. On the one hnd, to the best of our nowledge, ll polynomiltime mechnisms for Q C mx re nonymous [1, 3, 5, 9, 15]. On the other hnd, in ddition to envy-freeness, nonymity cn be viewed s n dditionl chrcteristic of firness [4]. We lso study sclble lloctions. Sclbility mens tht multiplying the speeds by the sme positive constnt does not chnge the lloction. Intuitively, the lloction function should not depend on the units in which the speed re mesured, nd hence sclbility is nturl notion. But lloctions bsed on rounded speeds of mchines re typiclly not sclble [1, 5, 13]. The truthful mechnisms nd envy-free mechnisms for Q C mx re both well-understood. There is pyment scheme to me n lloction truthful if nd only if the lloction is monotone decresing [3]. For Q C mx, n lloction is monotone decresing if no mchine gets more worlod by bidding slower speed thn its true speed. On the other hnd, mechnism for Q C mx cn be envy-free if nd only if its lloction is loclly efficient [15]. An lloction is loclly efficient if mchine never gets less worlod thn slower one. The complexity of truthful mechnisms nd, seprtely, envy-free mechnisms hve been completely settled. Q C mx is strongly NP-hrd, so there is no FPTAS for this problem, ssuming P NP. On the other hnd, there is deterministic monotone PTAS [5] nd distinct deterministic loclly efficient PTAS [15]. This implies the existence of truthful mechnisms nd distinct envy-free mechnisms tht pproximte 2

3 the mespn rbitrrily closely. However, neither of these pyment functions me the mechnisms both truthful nd envy-free. The VCG mechnism for Q C mx is truthful, envy-free, individully rtionl, nd nonymous [8]. However, since the VCG mechnism mximizes the socil welfre (i.e. minimizing the totl running time), it lwys lloctes ll jobs to the quicest mchines, yielding m-pproximtion of mespn for m mchines in the worst cse. So question is whether there is truthful, envy-free, individully rtionl nd nonymous mechnism tht pproximtes the mespn better thn the VCG mechnism. Since there lredy exists mny lloction functions tht re both monotone nd loclly efficient, one nturl step to nswer this question could be checing whether some of these lloction functions dmit truthful nd envy-free pyments. Our Results. We show tht 1. A deterministic envy-free, truthful, individully rtionl, nd nonymous mechnism cnnot pproximte the mespn strictly better thn 2 1/m, where m is the number of mchines. (Section 3). This result contrsts with prior results [5, 15] discussed bove. 2. For two mchines of different speeds, the unique deterministic sclble lloction of ny envy-free, truthful, individully rtionl, nd nonymous mechnism is to llocte ll jobs to the quicest mchine. (Section 5). This lloction is the sme s tht of the VCG mechnism, yielding 2-pproximtion of mespn for this cse. 3. No pyments cn me ny of the prior published monotone nd loclly efficient lloctions tht yield better thn n m-pproximtion for Q C mx [1, 3, 5, 9, 13] truthful, envy-free, individully rtionl, nd nonymous mechnism. Relted Wor. Hochbum nd Shmoys [12] give PTAS for Q C mx. Andelmn, Azr, nd Sorni [1] give 5-pproximtion deterministic truthful mechnism. Kovács improves the pproximtion rtio to 3 [13] nd then to 2.8 [14]. Rndomiztion hs been successfully pplied to this problem. Archer nd Trdos [3] give 3-pproximte rndomized mechnism, which is improved to 2 in [2]. Dhngwtnoti [9] et. l. give monotone rndomized PTAS. All these rndomized mechnisms re truthful-in-expecttion. However, we cn show tht no pyment function cn form truthful, envy-free, individully rtionl nd nonymous mechnism with ny lloction function of these mechnisms. We give proof for deterministic lloction [13] in Section 5 nd nother one for rndomized lloction [3] in Appendix B. When plyers hve different finite vlution spces, it is nown tht monotone nd loclly efficient lloction function my not dmit prices to form simultneously truthful nd envy-free mechnism for llocting goods mong plyers [7]. In this pper, we consider mechnisms where ll plyers hve identicl infinite vlution spces. Cohen et. l. [8] study the truthful nd envy-free mechnisms on combintoril uctions with dditive vlutions where gents hve upper cpcity on the number of items they cn receive. They see truthful nd envy-free mechnisms tht mximize socil welfre nd show tht VCG with Clre Pivot pyments is envy-free if gents cpcities re ll equl. Their result cn be interpreted in our setting by viewing tht ech gent hs the sme cpcity n nd the vlution of ech gent is the reverse of its cost. So their result implies tht the VCG mechnism for Q C mx is truthful nd envy-free; but the VCG mechnism does not give good pproximtion gurntee for mespn. 2 Preliminries There re m mchines nd n jobs. Ech gent will report bid b i R to the mechnism. Let t denote the vector of true speeds nd b the vector of bids. A mechnism consists of pir of functions (w, p). An lloction w mps vector of bids to vector of llocted worlod, where w i (b) is the worlod of gent i. For ll bid vectors b, w(b) must correspond 3

4 to vlid job ssignment. An lloction w is clled sclble if w i (b) = w i (c b) for ll bid vectors b, ll i {1... m} nd ll sclrs c >. A pyment p mps vector of bids to vector of pyments, i.e. p i (b) is the pyment to gent i. The cost mchine i incurs by the ssigned jobs is t i w i (b). Mchine i s privte vlue t i mesures its cost per unit wor. Ech mchine i ttempts to mximize its utility, u i (t i, b) := p i (b) t i w i (b). The mespn of lloction w(b) is defined s mx i w i (b) t i. A mechnism (w, p) is c-pproximte if for ll bids b nd vlues t, the mespn of the lloction given by w is within c times the mespn of the optiml lloction, i.e., mx i w i (b) t i c OPT (t), where OPT (t) is the minimum mespn for mchines with speeds t. Vector b is sometimes written s (b i, b i ), where b i is the vector of bids, not including gent i. A mechnism (w, p) is truthful or incentive comptible, if ech gent i mximizes his utility by bidding his true vlue t i, i.e., for ll gent i, ll possible t i, b i nd b i, p i (t i, b i ) t i w i (t i, b i ) p i (b i, b i ) t i w i (b i, b i ) A mechnism (w, p) is individully rtionl, if gents who bid truthfully never incur negtive utility, i.e. u i (t i, (t i, b i )) for ll gents i, true vlue t i nd other gents bids b i. A mechnism (w, p) is envy-free if no gent wishes to switch his lloction nd pyment with nother. For ll i, j {1,..., m} nd ll bids b, p i (b) b i w i (b) p j (b) b i w j (b). Notice tht we use bids b insted of the true speeds t in this definition, becuse mechnism cn determine the envy-free lloction only bsed on the bids. However, mechnism cn ensure the outcome is envy-free, only if it is lso truthful. A mechnism (w, p) is nonymous if for every bid vector b = (b 1,..., b m ), every such tht b is unique nd every l, w l (..., b 1, b l, b +1,..., b l 1, b, b l+1,...) = w (b) nd p l (..., b 1, b l, b +1,..., b l 1, b, b l+1,...) = p (b). The condition tht b is unique is importnt, becuse in some cse the mechnism my hve to llocte jobs of different lengths to gents with the sme bids. If mechnism (w, p) is nonymous nd the bid of n gent is unique, the worlod of tht gent stys the sme no mtter how tht gent is indexed. So we cn write w i (b i, b i ) simply s w(b i, b i ) for unique b i to represent the worlod of gent i. Similrly, we cn write p i (b i, b i ) simply s p(b i, b i ) for unique b i. Chrcteriztion of truthful mechnisms Lemm 1 ([3]). The lloction w(b) dmits truthful pyment scheme if nd only if w is monotone decresing, i.e., w i (b i, b i) w i (b i, b i ) for ll i, b i, b i b i. In this cse, the mechnism is truthful if nd only if the pyments stisfy where the h i s cn be rbitrry functions. bi p i (b i, b i ) = h i (b i ) + b i w i (b i, b i ) w i (u, b i ) du, i (1) By Lemm 1, the only flexibility in designing the truthful pyments for lloction w is to choose the terms h i (b i ). The utility of truth-telling gent i is h i (b i ) b i w i(u, b i )du, becuse his cost is t i w i (t i, b i ), which cncels out the second term in the pyment formul. Thus, in order to me the mechnism individully 4

5 rtionl, the term h i (b i ) should be t lest b i w i(u, b i ) du for ny b i. Since b i cn be rbitrrily lrge, h i should stisfy h i (b i ) Chrcteriztion of envy-free mechnisms w i (u, b i ) du, i, b i. (2) An lloction function w is envy-free implementble if there exists pyment function p such tht the mechnism M = (w, p) is envy-free. An lloction function w is loclly efficient if for ll bids b, nd ll permuttions π of {1,, m}, m m b i w i (b) b i w π(i) (b). i=1 Lemm 2 ([15]). Alloction w is envy-free implementble if nd only if w is loclly efficient. The proof of sufficiency constructs pyment scheme tht ensures the envy-freeness for ny loclly efficient lloction w. Specificlly, ssuming b 1 b 2... b m, the pyments for relted mchines re the following: { b 1 w 1 (b) for i = 1 p i (b) = p i 1 (b) + b i (w i (b) w i 1 (b)) for i {2,..., m} These pyments re not truthful pyments, since p 1 (b) is clerly not in the form of (1). But the set of envy-free pyments is convex polytope for fixed w, since pyments stisfying liner constrints i, j p i (b) b i w i (b) p j (b) b i w j (b) re envy-free. So there could be other pyments tht re both envy-free nd truthful. i=1 3 Lower Bound on Anonymous Mechnisms In this section, we will prove n pproximtion lower bound for truthful, envy-free, individully rtionl, nd nonymous mechnisms. Theorem 3. Let M = (w, p) be deterministic, truthful, envy-free, individully rtionl, nd nonymous mechnism. Then M is not c-pproximte for c < 2 1 m. Since the only flexibility when designing pyments in truthful mechnism is to choose the h i s, we need to now wht ind of h i s re required for envy-free nonymous mechnisms. The following two lemms give necessry conditions on h i s. Lemm 4. If mechnism (w, p) is both truthful nd nonymous, then there is function h such tht h i (v) = h(v) in (1) for ll bid vector v R m 1 + nd mchines i. Proof. Let β be rel number such tht β < min j v j. For ll i {1,..., m 1}, define vector b (i) = (v 1,..., v i 1, β, v i,..., v m 1 ), b (i) = (v 1,..., v i, β, v i+1,..., v m 1 ). Since v = b (i) i = b (i) (i+1) nd M is nonymous, it must be tht p i (b (i) ) = p i+1 (b (i) ) nd w i (b (i) ) = w i+1 (b (i) ). Since α < v j for ny < α < β, j {1... m 1}, we lso hve w i (α, v) = w i+1 (α, v) by nonymity. Thus, for truthful pyments, we hve β β h i (v) = p i (b (i) ) βw i (b (i) ) + w i (α, v) dα = p i+1 (b (i) ) βw i+1 (b (i) ) + w i+1 (α, v) dα = h i+1 (v). Lemm 5. Let L = l. If mechnism M = (w, p) is truthful, envy-free, nd nonymous, then for ll t R m + nd i, j {1,..., m}. h(t i ) h(t j ) L t i + (t j t i )w i (t), (3) 5

6 Proof. If mchine j does not envy mchine i, then p j (t) t j w j (t) p i (t) t j w i (t). Using (1) to substitute in for p i nd p j, nd Lemm 4, this yields ( tj ) ( ti ) h(t j ) + t j w j (t) w(x, t j ) dx t j w j (t) h(t i ) + t i w i (t) w(x, t i ) dx t j w i (t). Rerrnging terms gives h(t i ) h(t j ) ti ti tj w(x, t i ) dx tj L dx dx + (t j t i )w i (t) = L t i + (t j t i )w i (t). w(x, t j )dx + (t j t i )w i (t) Proof of Theorem 3. Consider n = m jobs of length l = (1,..., 1, m). Let L := 2m 1 denote the totl length of the jobs. Define speed vector t = (mα,..., mα, α), where α is rel number tht only depends on m nd c nd will be determined t the end of this section. We will show tht if M is deterministic, truthful, envy-free, individully rtionl nd nonymous, it should llocte ll jobs to the quicest mchine in this instnce. Clim 6. For speed vector t = (mα,..., mα, α) nd jobs l = (1,..., 1, m), if M is c-pproximte nd w i (t) 1 for some i {1,..., m 1}, then h(t 1 ) (L + m 1 Lc ) α. Proof. Since M is truthful nd individully rtionl, inequlity (2) pplies, nd h(t 1 ) w(x, t 1 ) dx α Lc α w(x, t 1 ) dx + w(x, t 1 ) dx + α Lc mα α w(x, t 1 ) dx. Apply M to vector (x, t 1 ). By the locl efficiency of w, job m should be ssigned to the quicest mchine. So for x < α, w(x, t 1 ) l m. When x < α Lc, ll the jobs should be ssigned to the mchine with speed x for mespn less thn α/c. Otherwise the mespn is t lest α, contrdicting M is c-pproximte. Since w i (mα, t i ) 1 for some i {1,..., m 1}, monotonicity implies w i (x, t 1 ) 1 for ll x (α, mα). Since x (α, mα) is unique in vector (x, t 1 ), we get w(x, t 1 ) 1 by nonymity. Thus h(t 1 ) α Lc α L dx + m dx + α Lc mα α 1 dx = 1 c α + mα m Lc m 1 α + mα α = (L + Lc )α. Let t = (1, mα,..., mα, α). Applying M to t, Lemm 5 implies h(t 1) h(t m) L + (α 1)w 1 (t ) L α. Since t 1 = t 1, this implies h(t 1 ) L α + h(t m). (4) Clim 7. If M is c-pproximte, then h(t 1 ) < L α+f(m, c), where f(m, c) = γ m 1 L+h(γ m 2, γ m 1,..., γ, 1) nd γ = cl + ɛ for some < ɛ < 1. 6

7 Proof. Define speed vector t (i) = (γ i 1, γ i 2,..., γ, 1, mα,..., mα) of length m for i 2, where γ = cl + ɛ for some < ɛ < 1. Let us consider the lloction M mes to mchine 1 for bid vector t (i). The speed of mchine 1 is γ i 1 γ for i 2. The speed of mchine i is 1. The mespn of llocting ll jobs to mchine i is L while the mespn of llocting t lest one job to mchine 1 is t lest γ = cl + ɛ. Since M is c-pproximte, this mens w 1 (t (i) ) =. Using Lemm 5, we hve h(t (i) 1 ) h(t(i) m) t (i) 1 L + (t(i) m t (i) 1 )w 1(t (i) ) = γ i 1 L. Since t (i) m = t (i+1) 1, this implies h(t (i) 1 ) h(t(i+1) 1 ) γ i 1 L for i {2,..., m 1}. Summing up these inequlities on ll i, we hve m 1 h(t (2) 1 ) h(t(m) 1 ) L γ i 1 < γ m 1 L Since t m = t (2) 1, we get h(t m) < γ m 1 L + h(t (m) ) = f(m, c). Plugging this into (4) yields 1 i=2 h(t 1 ) < L α + f(m, c). (5) To complete the proof of Theorem 3, consider speed vector t with α = m 1f(m, c). If mechnism M does not llocte ll jobs to mchine m, then w i (t) 1 for some i {1,..., m 1}. Then Clim 6 implies tht h(t 1 ) α L + f(m, c), contrdicting (5). So M must llocte ll jobs to mchine m in this cse, yielding mespn of (2m 1)α while the mespn of the schedule tht ssigns job j to mchine j for ll j is mα. Thus, M is c-pproximte for some c 2 1/m. Lc 4 Chrcterizing Sclble Mechnisms on Two Mchines We show tht nown monotone nd loclly efficient lloctions do not hve pyments to form truthful, envy-free, individully rtionl, nd nonymous mechnisms. (See Section 5 nd Appendix B.) In this section, we will show tht for two mchines, there is just one deterministic sclble lloction tht cn be mde truthful, envy-free, individully rtionl, nd nonymous. This lloction turns out to be the sme lloction s the VCG mechnism. Lemm 8. Let w be deterministic nd sclble lloction function for 2 mchines. For some > 1, if w(x, ) > for ll > nd x <, then there is some g() > such tht Proof. For < x <, let x = 2 t. w(x, ) dx w(, x) dx + g(). w(x, ) dx = = = w( 2 t, )( 2 t 2 ) dt 2 w(, t) dt t w(, t) dt + t w(, t) dt t2 (integrte by substitution) (w is sclble) 7

8 For < t < nd > 1, we hve t 2 /( )2 = Therefore, 42 (+1) > 1. For < t <, we hve 2 t 2 1. w(x, ) dx = ( + 1) 2 ( 4 2 ( + 1) 2 1 w(, t) dt + ) w(, t) dt + w(, t) dt w(, t) dt We lso hve +1 2 w(, t) dt = +1 2 w(1, t +1 ) dt = 2 1 w(1, y) dy. The first equlity follows the sclbility of w nd we get the second equlity by substituting t with y. Since w(x, ) > for ll > nd x <, we hve w(1, y) > for y > 1/. In sum, w(x, ) dx ( ) 4 w(, x) dx + g(), where g() = 2 +1 (+1) w(1, y) dy >. 1 Theorem 9. Let M = (w, p) be deterministic, truthful, envy-free, individully rtionl, nd nonymous. If w is sclble, then for two mchines of different speeds, w lloctes ll jobs to the quicest mchine. Proof. Let L denote the totl length of jobs. First, consider two mchines of speed t 1 = 1 nd t 2 = ( > 1). Since M is truthful, envy-free, nd nonymous, by Lemm 5, we hve h() h(1) L + ( 1)L = L (6) Since w is individully rtionl, h(1) w(x, 1). We will show tht w(, ) = for ny > 1. For contrdiction, ssume w(r, ) > for some r > 1. Let be such tht w(x, ) > for x < nd w(x, ) = for x >. By monotonicity, such exists. By the ssumption tht w(r, ) > for some r > 1, we now tht > 1. Since w is sclble, we hve for ny x >, w(y, ) = w(, x) = L if y/ = /x, i.e. y = 2 /x < /. Therefore, h() = w(x, ) dx L dx + w(x, ) dx + L + w(x, ) dx + w(x, ) dx L + (w(x, ) + w(, x)) dx + g() = L + L dx + g() = (L + g()) w(, x) dx + g() (Lemm 8) Te > h(1)/g(). We hve h() > L + h(1) from (7). This contrdicts (6). (7) 5 Pyments for Known Alloction Rules Although the VCG mechnism is truthful, envy-free, individully rtionl, nd nonymous, it does not hve good pproximtion gurntee for mespn. In [13], the LPT* lgorithm is described nd shown to 8

9 be monotone decresing. In this section, we will show tht LPT* is loclly efficient nd no pyment function cn form n envy-free, truthful, individully rtionl, nd nonymous mechnism with the LPT* lgorithm. We lso prove similr result for rndomized mechnisms in Appendix B: the rndomized 2-pproximtion lgorithm in [2, 3], whose expected lloction is monotone decresing nd loclly efficient, dmits no pyment function tht cn me it simultneously truthful-in-expecttion, envy-free-in-expecttion, individully rtionl, nd nonymous. We cn show similr results with similr proofs for the lloctions in [1, 5, 9]. The LPT* lgorithm is the following: Let w j i be the worlod of mchine i before job j is ssigned. Assume the jobs re indexed so tht l 1 l 2... l m. Note tht this lgorithm rounds the speeds nd hence is not sclble. Algorithm 1 LPT* Algorithm 1: Define rounded speed of mchine i to be s i := 2 log bi. 2: for j = 1 to m do 3: Assign job j to mchine i tht minimizes (w j i + l j) s i. 4: end for 5: Among mchines of sme rounded speed, reorder bundles on these mchines so tht mchine with smller bid gets more jobs. Lemm 1. LPT* is loclly efficient. Proof. We need to show tht w i (b) w (b) for ny b i > b. If s i = s, then step 5 ensures w i (b) w (b). So suppose s i > s. Consider the lst job, j, ssigned to mchine i. Since job j is ssigned to mchine i rther thn mchine, it should be tht (w j i + l j)s i (w j + l j)s, where w j i is the worlod of mchine i before job j is ssigned. Thus, w j + l j s i s (w j i + l j) 2(w j i + l j). Tht is w j 2wj i + l j. Since w (b) w j nd w i(b) = w j i + l j, we get w (b) w i (b). Theorem 11. There is no pyment function tht will me LPT* simultneously truthful, envy-free, individully rtionl, nd nonymous. Proof. Let w denote the lloction of the LPT* lgorithm. For contrdiction, ssume there exists pyment function p such tht mechnism M = (w, p) is truthful, envy-free, individully rtionl, nd nonymous. Apply M to the problem of two jobs with lengths l 1 = 2 nd l 2 = 1, nd two mchines with speeds t 1 = 1, t 2 = where > 1 nd is power of 2. By Lemm 5, we hve Since M is individully rtionl, we lso hve h() 4 h() h(1) 3 + ( 1) 3 = 3. (8) w(x, ) dx w(x, ) dx + 4 w(x, ) dx w( 4, ) + w(x, ) dx + w(2, ), 4 w(x, ) dx where the lst inequlity follows the monotonicity of w. Since is power of 2, the LPT* lgorithm ensures w( 4, ) = 3 nd w(2, ) = 1. Since w is loclly efficient, for ny 4 < x <, mchine with speed x gets t lest job one, i.e., w(x, ) 2. Therefore, h() = Now te = 8h(1), we hve h() 13 4 = 26h(1) nd h() h(1) + 3 = 25h(1) from (8), contrdiction. 9

10 Theorem 11 implies tht locl efficiency nd monotonicity of n lloction re not sufficient for the existence of pyment function to form n envy-free, truthful, individully rtionl, nd nonymous mechnism. This insufficiency of monotonicity nd locl efficiency still holds, even if the lloction function is lso sclble. See Proposition 13 in Appendix C for more detils. 6 Open Questions In this pper, we estblish n pproximtion lower bound 2 1/m for ny deterministic, envy-free, truthful, individully rtionl, nd nonymous mechnism while the upper bound is m given by the VCG mechnism. So one open question is whether the VCG mechnism is the best mong ll truthful nd envy-free mechnisms. The proof of Lemm 5 implicitly gives chrcteriztion of mechnisms tht re truthful, envy-free, nd nonymous. Idelly, there would be chrcteriztion of lloctions for which there exist prices tht me the resulting mechnism truthful nd envy-free. Another interesting question is whether there is chrcteriztion of truthful nd envy-free mechnisms for Q C mx. References [1] Nir Andelmn, Yossi Azr, nd Motti Sorni. Truthful pproximtion mechnisms for scheduling selfish relted mchines. 22nd Annul Symposium on Theoreticl Aspects of Computer Science, pges 69 82, 25. [2] Aron Archer. Mechnisms for discrete optimiztion with rtionl gents. PhD thesis, Ithc, NY, USA, 24. [3] Aron Archer nd Ev Trdos. Truthful mechnisms for one-prmeter gents. In Proceedings of the 42nd Annul Symposium on Foundtions of Computer Science, pges , 21. [4] Iti Ashlgi, Shhr Dobzinsi, nd Ron Lvi. An optiml lower bound for nonymous scheduling mechnisms. In Proceedings of the 1th ACM Conference on Electronic Commerce, pges , 29. [5] George Christodoulou nd Annmári Kovács. A deterministic truthful PTAS for scheduling relted mchines. In Proceedings of the Twenty-First Annul ACM-SIAM Symposium on Discrete Algorithms, pges , 21. [6] Edith Cohen, Michl Feldmn, Amos Fit, Him Kpln, nd Svetln Olonetsy. Envy-free mespn pproximtion: extended bstrct. In Proceedings of the 11th ACM Conference on Electronic Commerce, pges , 21. [7] Edith Cohen, Michl Feldmn, Amos Fit, Him Kpln, nd Svetln Olonetsy. On the interply between incentive comptibility nd envy freeness [8] Edith Cohen, Michl Feldmn, Amos Fit, Him Kpln, nd Svetln Olonetsy. Truth nd envy in cpcitted lloction gmes [9] Peerpong Dhngwtnoti, Shhr Dobzinsi, Shddin Dughmi, nd Tim Roughgrden. Truthful pproximtion schemes for single-prmeter gents. In Proceedings of 49th Annul Symposium on Foundtions of Computer Science, pges 15 24, 28. [1] L. Dubins nd E. Spnier. How to cut ce firly. Americn Mthemticl Monthly, 68:1 17, [11] Duncn Foley. Resource lloction nd the public sector. Yle Economic Essys, 7:45 98, [12] Dorit S. Hochbum nd Dvid B. Shmoys. A polynomil pproximtion scheme for scheduling on uniform processors: Using the dul pproximtion pproch. SIAM J. Comput., 17: ,

11 [13] Annmári Kovács. Fst monotone 3-pproximtion lgorithm for scheduling relted mchines. Algorithms - ESA 25: 13th Annul Europen Symposium, pges , 25. [14] Annmári Kovács. Tighter pproximtion bounds for LPT scheduling in two specil cses. J. of Discrete Algorithms, 7:327 34, September 29. [15] Ahuv Mu Alem. On multi-dimensionl envy-free mechnisms. In Proceedings of the 1st Interntionl Conference on Algorithmic Decision Theory, pges , 29. Appendix A Rndomized mechnisms In this section, we define the rndomized mechnisms for Q C mx in our setting. A rndomized mechnism (w, p) is truthful-in-expecttion, if ech gent i mximizes his expected utility by bidding his true vlue t i, i.e., for ll gent i, ll possible t i, b i nd b i, E[p i (t i, b i ) t i w i (t i, b i )] E[p i (b i, b i ) t i w i (b i, b i )]. A rndomized mechnism (w, p) is envy-free-in-expecttion if no gent wishes to switch his expected lloction nd expected pyment with nother, i.e., for ll i, j {1,..., m} nd ll bids b, E[p i (b)] b i E[w i (b)] E[p j (b)] b i E[w j (b)]. A rndomized mechnism (w, p) is nonymous if for every bid vector b = (b 1,..., b m ), every such tht b is unique nd every l, E[w l (..., b 1, b l, b +1,..., b l 1, b, b l+1,...)] = E[w (b)] nd E[p l (..., b 1, b l, b +1,..., b l 1, b, b l+1,...)] = E[p (b)]. So we see rndomized mechnisms tht re truthful-in-expecttion, envy-free-in-expecttion, individully rtionl, nd nonymous. For rndomized mechnism tht is truthful-in-expecttion, Lemm 1 still holds fter replcing ll w i (b) with E[w i (b)] [3]. By Lemm 1, the pyments re deterministic of rndomized mechnism tht is truthful-in-expecttion. It is esy to chec tht (2), Lemm 4 nd Lemm 5 still hold fter replcing the lloction of mchine with its expected lloction for ll mchines. B In [3], rndomized 3-pproximtion lgorithm (see Algorithm 2) is described nd its expected lloction is shown to be monotone decresing. In this section, we will show tht its expected lloction is loclly efficient nd there is no pyment tht mes it truthful-in-expecttion, envy-free-in-expecttion, individully rtionl nd nonymous mechnism. Theorem 12. There is no pyment function tht will me Algorithm 2 simultneously truthful-in-expecttion, envy-free-in-expecttion, individully rtionl, nd nonymous. Proof. Let w denote the lloction of Algorithm 2. For contrdiction, ssume there exists pyment function p such tht mechnism M = (w, p) is truthful-in-expecttion, envy-free-in-expecttion, individully rtionl nd nonymous. Apply M to the problem of two jobs with lengths l 1 = 2 nd l 2 = 1, nd two mchines with speeds t 1 = 1, t 2 = where > 1. By Lemm 5, we hve h() h(1) 3 + ( 1) 3 = 3, (9) 11

12 Algorithm 2 Rndomized 3-pproximtion Algorithm by Archer nd Trdos [3] Require: Jobs nd mchines re indexed so tht l 1 l 2... l n nd b 1 b 2,..., b m. 1: Compute lower bound of mespn T LB (b) = mx j min mx i { b i l j, j =1 l i r=1 1 b r 2: For ech mchine i, crete bin i of size s i (b) = T LB (b)/b i. 3: Assign jobs 1, 2,..., ( 1) to bin 1, where is the first job tht would cuse the bin to overflow. Then ssign to bin 1 piece of job exctly s lrge s the remining cpcity of bin 1. Continue by ssigning jobs to bin 2, strting with the rest of job nd so on, until ll jobs re ssigned. 4: For ech job j, ssign job j to mchine i with probbility equl to the proportion of job j tht is frctionlly ssigned to bin i. } where h(1) w(x, 1) dx. The expected worlod of mchine i is equl to the size of bin i by step 4, i.e., E[w i (b)] = s i (b). Algorithm 2 lso ensures tht for ll c >, s i (c b) = T LB(c b) c b i = c T LB(b) c b i = s i (b). So we get E[w i (c b)] = E[w i (b)] for ll c >. This implies, for two mchines, E[w(x, )] dx = E[w( x, 1)] dx = E[w(t, 1)] dt, where the lst equlity is obtined by substituting x with t. Since M is individully rtionl, we hve h() E[w(x, )] dx = E[w(x, 1)] dx = (3.5 + ln 3 ln 2), where we get the lst eqution by computing E[w(x, 1)] dx using the lloction of Algorithm 2. Now te = 2h(1), we hve h() (3.5 + ln 3 ln 2) > 7h(1), contrdicting (9). Since the expected worlod of mchine i equls to s i (b) = T LB (b)/b i, we get E[w i (b)] = T LB (b)/b i T LB (b)/b j = E[w j (b)] for b i b j. So the expected lloction of Algorithm 2 is loclly efficient. C Proposition 13. Locl efficiency, monotonicity nd sclbility of n lloction re not sufficient for the existence of pyment function to form n envy-free, truthful, individully rtionl, nd nonymous mechnism. Proof. Define lloction w for 2 mchines to be the lloction tht minimizes the mespn. If there re more thn one such lloctions, let w be the one tht lso minimizes the totl completion time. It is esy to verify tht w is unique. Hence w is well-defined nd nonymous. w is lso loclly efficient nd sclble, since it minimizes the mespn. Now, we show tht w is monotone. Let the lloction of w for 2 mchines with bids (b 1, b 2 ) be O = (L 1, L 2 ). Assume w.l.o.g tht b 1 L 1 b 2 L 2. If mchine 1 increses its bid, its lloction cn only go down. Consider the lloction of w: O = (L 1, L 2) for 2 mchines with bids (b 1, b 2), where b 2 > b 2. For contrdiction, ssume L 2 > L 2. So L 1 < L 1. If the mespn of O is b 1 L 1, i.e. b 1 L 1 b 2L 2, then we hve b 2 L 2 < b 2L 2 b 1 L 1 < b 1 L 1. So mx{b 2 L 2, b 1 L 1} < b 1 L 1. It contrdicts the optimlity of O. If the mespn of O is b 2L 2, i.e. b 2L 2 b 1 L 1, then b 2L 2 mx{b 1 L 1, b 2L 2 } by the optimlity of O. Since b 2L 2 > b 2L 2, we hve b 2L 2 b 1 L 1. So b 2 L 2 < b 1 L 1. Since b 1 L 1 < b 1 L 1, we hve mx{b 2 L 2, b 1 L 1} < b 1 L 1. It contrdicts the optimlity of O. Therefore, L 2 L 2 nd w is monotone. Since w is different from the VCG lloction for 2 mchines, this theorem follows from Theorem 9. 12

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