Truthful job scheduling is a
mechanism design
Mechanism design (sometimes implementation theory or institution design) is a branch of economics and game theory. It studies how to construct rules—called Game form, mechanisms or institutions—that produce good outcomes according to Social ...
variant of the
job shop scheduling
Job-shop scheduling, the job-shop problem (JSP) or job-shop scheduling problem (JSSP) is an optimization problem in computer science and operations research. It is a variant of optimal job scheduling. In a general job scheduling problem, we are gi ...
problem from
operations research
Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods to improve management and ...
.
We have a project composed of several "jobs" (tasks). There are several workers. Each worker can do any job, but for each worker it takes a different amount of time to complete each job. Our goal is to allocate jobs to workers such that the total
makespan
In operations research
Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods t ...
of the project is minimized. In the standard job shop scheduling problem, the timings of all workers are known, so we have a standard optimization problem. In contrast, in the truthful job scheduling problem, the timings of the workers are not known. We ask each worker how much time he needs to do each job, but, the workers might lie to us. Therefore, we have to give the workers an incentive to tell us their true timings by paying them a certain amount of money. The challenge is to design a payment mechanism which is
incentive compatible
In game theory and economics, a mechanism is called incentive-compatible (IC) if every participant can achieve their own best outcome by reporting their true preferences. For example, there is incentive compatibility if high-risk clients are bette ...
.
The truthful job scheduling problem was introduced by Nisan and Ronen in their 1999 paper on
algorithmic mechanism design Algorithmic mechanism design (AMD) lies at the intersection of economic game theory, optimization, and computer science. The prototypical problem in mechanism design is to design a system for multiple self-interested participants, such that the part ...
.
Definitions
There are
jobs and
workers ("m" stands for "machine", since the problem comes from scheduling
jobs to computers). Worker
can do job
in time
. If worker
is assigned a set of jobs
, then he can execute them in time:
:
Given an allocation
of jobs to workers, The
makespan
In operations research
Operations research () (U.S. Air Force Specialty Code: Operations Analysis), often shortened to the initialism OR, is a branch of applied mathematics that deals with the development and application of analytical methods t ...
of a project is:
:
An ''optimal allocation'' is an allocation of jobs to workers in which the makespan is minimized. The minimum makespan is denoted by
.
A ''mechanism'' is a function that takes as input the matrix
(the time each worker needs to do each job) and returns as output:
* An allocation of jobs to workers,
;
* A payment to each worker,
.
The utility of worker
, under such mechanism, is:
:
I.e, the agent gains the payment, but loses the time that it spends in executing the tasks. Note that payment and time are measured in the same units (e.g., we can assume that the payments are in dollars and that each time-unit costs the worker one dollar).
A mechanism is called truthful (or
incentive compatible
In game theory and economics, a mechanism is called incentive-compatible (IC) if every participant can achieve their own best outcome by reporting their true preferences. For example, there is incentive compatibility if high-risk clients are bette ...
) if every worker can attain a maximum utility by reporting his true timing vector (i.e., no worker has an incentive to lie about his timings).
The ''approximation factor'' of a mechanism is the largest ratio between
and
(smaller is better; an approximation factor of 1 means that the mechanism is optimal).
The research on truthful job scheduling aims to find upper (positive) and lower (negative) bounds on approximation factors of truthful mechanisms.
Positive bound – m – VCG mechanism
The first solution that comes to mind is
VCG mechanism, which is a generic truthful mechanism. A VCG mechanism can be used to minimize the sum of costs. Here, we can use VCG to find an allocation which minimizes the "make-total", defined as:
:
Here, minimizing the sum can be done by simply allocating each job to the worker who needs the shortest time for that job. To keep the mechanism truthful, each worker that accepts a job is paid the second-shortest time for that job (like in a
Vickrey auction
A Vickrey auction or sealed-bid second-price auction (SBSPA) is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest ...
).
Let OPT be an allocation which minimizes the makespan. Then:
: