We consider mechanisms without payments for the problem of scheduling unrelated machines. Specifically, we consider truthful in expectation randomized mechanisms under the assumption that a machine (player) is bound by its reports: when a machine lies and reports value $\tilde{t}_{ij}$ for a task instead of the actual one t ij , it will execute for time $\tilde{t}_{ij}$ if it gets the task (unless the declared value $\tilde{t}_{ij}$ is less than the actual value t ij , in which case, it will execute for time t ij ). Our main technical result is an optimal mechanism for one task and n players which has approximation ratio (n+1)/2. We also provide a matching lower bound, showing that no other truthful mechanism can achieve a better approximation ratio. This immediately gives an approximation ratio of (n+1)/2 and n(n+1)/2 for social cost and makespan minimization, respectively, for any number of tasks. We also study the price of anarchy of natural algorithms.

/pdf/scheduling-without-payments-1ipex18uqt.pdf

Scheduling Without Payments

Winner-imposing strategyproof mechanisms for multiple Facility Location games

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou's network. We improve upon the value 4/3 by means of Coordination Mechanisms. 
We increase the latency functions of the edges in the network, i.e., if $\ell_e(x)$ is the latency function of an edge $e$, we replace it by $\hat{\ell}_e(x)$ with $\ell_e(x) \le \hat{\ell}_e(x)$ for all $x$. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if $\CM(r)$ denotes the cost of the worst Nash flow in the modified network for rate $r$ and $\Copt(r)$ denotes the cost of the optimal flow in the original network for the same rate then [\ePoA = \max_{r \ge 0} \frac{\CM(r)}{\Copt(r)}.] 
We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4 = 1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.

Improving the Price of Anarchy for Selfish Routing via Coordination Mechanisms

We reconsider the well-studied Selfish Routing game with affine latency functions. The Price of Anarchy for this class of games takes maximum value 4/3; this maximum is attained already for a simple network of two parallel links, known as Pigou’s network. We improve upon the value 4/3 by means of Coordination Mechanisms. We increase the latency functions of the edges in the network, i.e., if l
 e
 (x) is the latency function of an edge e, we replace it by $\hat{\ell}_{e}(x)$
 with $\ell_{e}(x) \le \hat{\ell}_{e}(x)$
 for all x. Then an adversary fixes a demand rate as input. The engineered Price of Anarchy of the mechanism is defined as the worst-case ratio of the Nash social cost in the modified network over the optimal social cost in the original network. Formally, if $\hat{C}_{N} (r)$
 denotes the cost of the worst Nash flow in the modified network for rate r and C
 opt
 (r) denotes the cost of the optimal flow in the original network for the same rate then $$\mathit{ePoA} = \max_{r \ge 0} \frac{\hat{C}_N(r)}{C_{\mathit{opt}}(r)}. $$
 We first exhibit a simple coordination mechanism that achieves for any network of parallel links an engineered Price of Anarchy strictly less than 4/3. For the case of two parallel links our basic mechanism gives 5/4=1.25. Then, for the case of two parallel links, we describe an optimal mechanism; its engineered Price of Anarchy lies between 1.191 and 1.192.

We consider mechanisms without payments for the problem of scheduling unrelated machines. Specifically, we consider truthful in expectation randomized mechanisms under the assumption that a machine (player) is bound by its reports: when a machine lies and reports value tij for a task instead of the actual one tij, it will execute for time tij if it gets the task--unless the declared value tij is less than the actual value tij, in which case, it will execute for time tij. Our main technical result is an optimal mechanism for one task and n players which has approximation ratio (n + 1)/2. We also provide a matching lower bound, showing that no other truthful mechanism can achieve a better approximation ratio. This immediately gives an approximation ratio of (n + 1)/2 and n(n + 1)/2 for social cost and makespan minimization, respectively, for any number of tasks.

/pdf/scheduling-without-payments-4jl0rbezhr.pdf

Scheduling without payments

We consider the problem of designing truthful mechanisms for scheduling n tasks on a set of m parallel related machines in order to minimize the makespan. In what follows, we consider that each task is owned by a selfish agent. This is a variant of the KP-model introduced by Koutsoupias and Papadimitriou (Proc. of STACS 1999, pp. 404---413, 1999) (and of the CKN-model of Christodoulou et al. in Proc. of ICALP 2004, pp. 345---357, 2004) in which the agents cannot choose the machine on which their tasks will be executed. This is done by a centralized authority, the scheduler. However, the agents may manipulate the scheduler by providing false information regarding the length of their tasks. We introduce the notion of increasing algorithm and a simple reduction that transforms any increasing algorithm into a truthful one. Furthermore, we show that some of the classical scheduling algorithms are indeed increasing: the LPT algorithm, the PTAS of Graham (SIAM J. Appl. Math. 17(2):416---429, 1969) in the case of two machines, as well as a simple PTAS for the case of m machines, with m a fixed constant. Our results yield a randomized r(1+?)-approximation algorithm where r is the ratio between the largest and the smallest speed of the related machines. Furthermore, by combining our approach with the classical result of Shmoys et al. (SIAM J. Comput. 24(6):1313---1331, 1995), we obtain a randomized 2r(1+?)-competitive algorithm. It has to be noticed that these results are obtained without payments, unlike most of the existing works in the field of Mechanism Design. Finally, we show that if payments are allowed then our approach gives a (1+?)-algorithm for the off-line case with related machines.

On truthfulness and approximation for scheduling selfish tasks

We consider the problem of designing truthful mechanisms for scheduling n tasks on a set of m parallel (identical or related) machines in order to minimize the makespan. Each task is owned by a selfish agent whose aim is to minimize the completion time of her task. We focus on the following process: at first the agents declare the length of their tasks, then given these bids the scheduler allocates the tasks to the machines. The objective of the scheduler is to minimize the makespan, i.e. the date at which the last task finishes its execution, while the aim of each agent is to minimize its completion time and thus an agent may lie if by doing so, her task can finish earlier. More formally, we are given m machines M1, M2, . . . , Mm with speeds s1, s2, . . . , sm, respectively. We assume that s1 ≤ s2 ≤ . . . ≤ sm and we denote by r = sm/s1 the ratio between the largest and smallest speeds. We are also given a set of tasks H = {T1, . . . , Tn}. For every i ≤ n, we denote by l(Ti), or sometimes li, the length of task Ti and C(Ti) its completion time. If task Ti is scheduled on machine Mj , its processing time is l(Ti)/sj . We will say that a task Ti is larger than a task Tj if and only if l(Ti) > l(Tj) or (l(Ti) = l(Tj) and i > j). Our goal is to design scheduling algorithms which minimize the makespan (maxj{C(Tj)}) and that simultaneously prevent the agents from reporting false values about their length. The problem is denoted by Q||Cmax for the case with different speeds, and by P ||Cmax when the machines are identical.

Alex-Ariel Tchetgnia

Papers

On truthfulness and approximation for scheduling selfish tasks

On the truthfulness and the approximation for scheduling selfish tasks