Mechanism Design for Facility Location Problems Assignment

Assignment Task

1. Introduction

Algorithm mechanism design has an essential position in the field of computer science and finance research. A classic problem in the algorithm mechanism design is the facility location problem. In the classic problem of facility location, it is necessary to determine a group of potential sites to build facilities to meet agents' needs. In short, the facility is on a real line, and the designer has to choose where the facility is; each agent's cost is the distance from her location to the facility`s location.

When solving the facility location problem, we need to design a deterministic mechanism or randomized mechanism, respectively. The deterministic mechanism means that the output of the mechanism about facility location is an exact location. In comparison, the randomized mechanism means that there are two or more outputs about facility location. They respectively have different probability.

We need strategy-proof of the mechanism we have designed, i.e, prove that the mechanism we have designed ensures that everyone involved is telling the truth. After ensuring that, our design mechanism is strategy-proof. The ultimate goal of our mechanism is to minimize the social cost and maximize cost. The Social cost is the sum of all agents' costs, and the maximum cost is the maximum cost that is compared for all agents.

In the paper of [Procaccia and Tennenholtz], they introduced the concept of ‘Approximation mechanism design without money’. The approximation ratio consists of two parts: the total distance or maximum distance calculated by the output of the optimal situation as the denominator. The total distance or maximum distance calculated by the output of our mechanism as the denominator. It expresses the relationship between the optimal situation and our mechanism in terms of fractions.

In addition to locating the facility, the problem spills over into other areas as well. Even the issues are not geographical, from simply choosing the temperature in a classroom to choosing a committee to represent different political points of view. Because of its practical significance, various facility location problems have long attracted attention from different fields such as operations research, theoretical computer science, economics and algorithm game theory.

2. Central research problem/questions to be answered

Since the facility location problem has many different research directions, different researchers have different research directions in a primary setting. For example, the facility located at a real line, located in a tree structure. Currently, our main research direction is ‘Heterogeneous facility location without money’ [Serafino and Ventre], which means two different facilities are located on the real line. The input of our designed mechanism is the preference of agents to these two facilities, and the output is the location of these two facilities. In order to get the approximation ratio, we need to find its maximum and minimum, respectively, which in the term of mechanism design is the lower bound and upper bound. Our current central research problem is based on the research of [Serafino and Ventre]. Devise strategy-proof mechanisms to improve their demonstration of the lower bound and upper bound of total distance and maximize distance in the deterministic mechanism or randomized.

3. Methodological considerations

Our main methodological tools rely on Algorithmic Game Theory and Mechanism Design. In summary, our research followed the following process:

3.1 After reading the relevant papers, find the points that can be improved in these articles.

3.2 After finding the feasible points, put forward the deterministic mechanisms and randomized mechanisms and show strategy-proof.

3.3 After the completion of strategy proof, calculate the lower bound and upper bound on the approximation ratio. We mainly based on 'Approximate Mechanism Design Without Money'[Procaccia and Tennenholze] and 'Heterogeneous Facility Location Without Money' [Serafino and Ventre] at this stage of the research. To research ‘Approximate Mechanism Design without Money of the Facility Location Problem’.

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