the assignment problem econ

The Assignment Problem

Cite this chapter.

• Horst Siebert 3  

211 Accesses

1 Citations

The assignment problem can be understood as the issue how decision rights are allocated to the agents and organizational subunits of a society. In this wide interpretation, the assignment problem refers to three different, albeit related phenomena: i) the allocation of property rights to individuals, i.e. households, and to firms, ii) the allocation of decisions rights to the government (including its subunits) relative to the private sector, and, within the sphere of government, iii) the attribution of policy instruments to economic policy agents. Economic policy agents are the different layers of government, the central bank, and trade unions to name the most important ones. 1 Assigning decisions rights is not only a national problem but refers to international decision rules as well.

We can think of the assignment problem as an optimizing problem in which decision rights are allocated in such a way that a goal function is maximized. In this fundamental issue of institutional economics, the goal function is complex, including values such as individual freedom and equity as well as the economic criterion of efficiency. Phrasing the assignment problem in these terms, I am putting the problem as if we were to reinvent mankind’s institutional arrangements from scratch. This is, of course, only an abstract and a rather theoretical approach. In the real world, the allocation of property rights to agents is given, and the given allocation changes only marginally except for major upheavals as in the shift from communist central planning to the market economy. In the following we look for some principles for the allocation of decision rights.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

• Available as PDF
• Read on any device
• Instant download
• Own it forever
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Unable to display preview.  Download preview PDF.

Bergsten, C. F. (1999) America and Europe: Clash of the Titans? Foreign Affairs 76 (2), 20–49.

Article   Google Scholar  

Coase, R. (1960) The Problem of Social Cost. Journal of Law and Economics 3 , 1–44.

Coeuré, B., and J. Pisani-Ferry (2000) The Euro, Yen, and Dollar: Making the Case Against Benign Neglect. In: Kenen, P. B., and A. K. Swoboda (eds.), Reforming the International Monetary and Financial System. International Monetary Fund, Washington, D.C.

Google Scholar  

Furubotn, E. G., and S. Pejovich (1972) Property Rights and Economic Theory. A Survey of Recent Literature. Journal of Economic Literature 10 , 1137–1162.

Kenen, P. (2000) The International Economy. 4th edition. Cambridge: University Press.

Book   Google Scholar  

Sachverständigenrat zur Begutachtung der gesamtwirtschaftlichen Entwicklung (1967) Stabilität im Wachstum. Stuttgart und Mainz.

Siebert, H. (1993) Principles of the Economic System in the Federal Republic — An Economist’s View. In: Kirchhof, P., and D. P. Kommers (eds.), Germany and its Basic Law. Baden-Baden.

Siebert, H. (1996) Institutionelle Arrangements für die Zuweisung von Opportunitätskosten. In: Immenga, U., W. Möschel, and D. Reuter (Hrsg.), Festschrift für Ernst-Joachim Mestmäcker zum siebzigsten Geburtstag. Nomos, Baden-Baden.

Siebert, H. (1997) Der Verlust des ordnungspolitischen Denkens. Universitas 52 (11), 1021–1029.

Siebert, H. (1999) Improving the World’s Financial Architecture. The Role of the IMF. Kiel Discussion Papers 351. Kiel Institute of World Economics.

Siebert, H. (2000a) Außenwirtschaft. 7th edition. Lucius & Lucius, Stuttgart.

Siebert, H. (2000b) The Case for Benign Neglect. In: Kenen, P. B., and A. K. Swoboda (eds.), Reforming the International Monetary and Financial System. International Monetary Fund, Washington, D.C.

Siebert, H. (2000c) The Japanese Bubble: Some Lessons for International Macroeconomic Policy Coordination. Aussenwirtschaft 55 (2000), 233–250.

Tichy, G. (1999) Konjunkturpolitik: Quantitative Stablisierungspolitik bei Unsicherheit. 4th edition. Springer, Berlin.

Download references

Author information

Authors and affiliations.

President of the Kiel Institute of World Economics, Chair of Theoretical Economics at the University of Kiel, Member of the German Council of Economic Advisers, Germany

Horst Siebert

You can also search for this author in PubMed   Google Scholar

Editor information

Editors and affiliations.

Institut f. Statistik und Mathematische Wirtschaftstheorie, Universität Karlsruhe, Zirkel 2, Rechenzentrum, 76128, Karlsruhe, Germany

Siegfried K. Berninghaus ( Professor Dr. ) ( Professor Dr. )

Fachbereich Wirtschaftswissenschaften VWL/Außenwirtschaft, Universität Osnabrück, Rolandstr. 8, 49069, Osnabrück, Germany

Michael Braulke ( Professor Dr. ) ( Professor Dr. )

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this chapter

Siebert, H. (2001). The Assignment Problem. In: Berninghaus, S.K., Braulke, M. (eds) Beiträge zur Mikro- und zur Makroökonomik. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-56606-6_35

Download citation

DOI : https://doi.org/10.1007/978-3-642-56606-6_35

Publisher Name : Springer, Berlin, Heidelberg

Print ISBN : 978-3-642-62679-1

Online ISBN : 978-3-642-56606-6

eBook Packages : Springer Book Archive

Assignment problem

The tinbergen rule, adjustment in the swan diagram, effective market classification, allowing for shocks.

• Expenditure changing and expenditure switching: Internal and External Balan ...
• Expenditure changing and expenditure switching
• Expenditure changing and expenditure switching: Effects of Expenditure Swit ...
• Equilibrium exchange rate: New Open Economy Macroeconomic Class of Models
• Expenditure changing and expenditure switching: Effects of Expenditure Chan ...
• Marshall-Lerner condition: Importance of the Marshall-Lerner Condition
• International Monetary Fund conditionality: Analytical Tools
• J-curve effect: Estimating the J-Curve
• Mundell-Fleming model: Policy Shocks
• Comments: (0)
• Concepts and Principles
• Models and Theory
• Institutions and Agreements
• Policies and Instruments
• Analysis and Tools
• Sectors and Special Issues

Help | Advanced Search

Economics > Theoretical Economics

Title: fair congested assignment problem.

Abstract: We propose a fair and efficient solution for assigning agents to m posts subject to congestion, when agents care about both their post and its congestion. Examples include assigning jobs to busy servers, students to crowded schools or crowded classes, commuters to congested routes, workers to crowded office spaces or to team projects etc... Congestion is anonymous (it only depends on the number n of agents in a given post). A canonical interpretation of ex ante fairness allows each agent to choose m post-specific caps on the congestion they tolerate: these requests are mutually feasible if and only if the sum of the caps is n. For ex post fairness we impose a competitive requirement close to envy freeness: taking the congestion profile as given each agent is assigned to one of her best posts. If a competitive assignment exists, it delivers unique congestion and welfare profiles and is also efficient and ex ante fair. In a fractional (randomised or time sharing) version of our model, a unique competitive congestion profile always exists. It is approximately implemented by a mixture of ex post deterministic assignments: with an approxination factor equal to the largest utility loss from one more unit of congestion, the latter deliver identical welfare profiles and are weakly efficient. Our approach to ex ante fairness generalises to the model where each agent's congestion is weighted. Now the caps on posts depend only upon own weight and total congestion, not on the number of other agents contributing to it. Remarkably in both models these caps are feasible if and only if they give to each agent the right to veto all but (1/m) of their feasible allocations.

Submission history

Access paper:.

• HTML (experimental)
• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .

Assignment problems and their application in economics

Publication or External Link

Four assignment problems are introduced in this thesis, and they are approached

based on the context they are presented in. The underlying graphs of

the assignment problems in this thesis are in most cases bipartite graphs with two

sets of vertices corresponding to the agents and the resources. An edge might show

the interest of an agent in a resource or willingness of a manufacturer to produce

the corresponding product of a market, to name a few examples.

The rst problem studied in this thesis is a two-stage stochastic matching

problem in both online and oine versions. In this work, which is presented in

Chapter 2 of this thesis, a coordinator tries to benet by having access to the

statistics of the future price discounts which can be completely unpredictable for

individual customers. In our model, individual risk-averse customers want to book

hotel rooms for their future vacation; however, they are unwilling to leave booking to

the last minute which might result in huge savings for them since they have to take

the risk of all the hotel rooms being sold out. Instead of taking this risk, individual

customers make contracts with a coordinator who can spread the risk over many

such cases and also has more information on the probability distribution of the future

prices. In the rst stage, the coordinator agrees to serve some buyers, and then in

the second stage, once the nal prices have been revealed, he books rooms for them

just as he promised. An agreement between the coordinator and each buyer consists

of a set of acceptable hotels for the customer and a single price. Two models for

this problem are investigated. In the rst model, the details of the agreements are

proposed by the buyer, and we propose a bicriteria-style approximation algorithm

that gives a constant-factor approximation to the objective function by allowing a

bounded fraction of our hotel bookings to overlap. In the second model, the details

of the agreements are proposed by the coordinator, and we show the prices yielding

the optimal prot up to a small additive loss can be found by a polynomial time

In the third chapter of this thesis, two versions of the online matching problem

are analyzed with a similar technique. Online matching problems have been studied

by many researchers recently due to their direct application in online advertisement

systems such as Google Adwords. In the online bipartite matching problem, the

vertices of one side are known in advance; however, the vertices of the other side

arrive one by one, and reveal their adjacent vertices on the oine side only upon

arrival. Each vertex can only be matched to an unmatched vertex once it arrives and

we cannot match or rematch the online vertex in the future. In the online matching

problem with free disposal, we have the option to rematch an already matched oine

vertex only if we eliminate its previous online match from the graph. The goal is to

maximize the expected size of the matching. We propose a randomized algorithm

that achieves a ratio greater than 0:5 if the online nodes have bounded degree. The

other problem studied in the third chapter is the edge-weighted oblivious matching in

which the weights of all the edges in the underlying graph are known but existence

of each edge is only revealed upon probing that edge. The weighted version of

the problem has applications in pay-per-click online advertisements, in which the

revenue for a click on a particular ad is known, but it is unknown whether the user

will actually click on that ad. Using a similar technique, we develop an algorithm

with approximation factor greater than 0:5 for this problem too.

In Chapter 4, a generalized version of the Cournot Competition (a foundational

model in economics) is studied. In the traditional version, rms are competing in

a single market with one heterogeneous good, and their strategy is the quantity

of good they produce. The price of the good is an inverse function of the total

quantity produced in the market, and the cost of production for each rm in each

market increases with the quantity it produces. We study Cournot Competition on

a bipartite network of rms and markets. The edges in this network demonstrate

access of a rm to a market. The price of the good in each market is again an

inverse function of the quantity of the good produced by the rms, and the cost of

production for each rm is a function of its production in dierent markets. Our

goal is to give polynomial time algorithms to nd the quantity produced by rms

in each market at the equilibrium for generalized cost and price functions.

The nal chapter of this thesis is on analyzing a problem faced by online

marketplaces such as Amazon and ebay which deal with huge datasets registering

transaction of merchandises between many buyers and sellers. As the size of datasets

grow, it is important that the algorithms become more selective in the amount of

data they store. Our goal is to develop pricing algorithms for social welfare (or

revenue) maximization that are appropriate for use with the massive datasets in

these networks. We specially focus on the streaming setting, the common model

for big data analysis. Furthermore, we include hardness results (lower bounds)

on the minimum amount of memory needed to calculate the exact prices and also

present algorithms which are more space ecient than the given lower bounds but

approximate the optimum prices for the goods besides the revenue or the social

welfare of the mechanism.

URI (handle)

Collections.

How to Solve the Assignment Problem: A Complete Guide

Table of Contents

Assignment problem is a special type of linear programming problem that deals with assigning a number of resources to an equal number of tasks in the most efficient way. The goal is to minimize the total cost of assignments while ensuring that each task is assigned to only one resource and each resource is assigned to only one task. In this blog, we will discuss the solution of the assignment problem using the Hungarian method, which is a popular algorithm for solving the problem.

Understanding the Assignment Problem

Before we dive into the solution, it is important to understand the problem itself. In the assignment problem, we have a matrix of costs, where each row represents a resource and each column represents a task. The objective is to assign each resource to a task in such a way that the total cost of assignments is minimized. However, there are certain constraints that need to be satisfied – each resource can be assigned to only one task and each task can be assigned to only one resource.

Solving the Assignment Problem

There are various methods for solving the assignment problem, including the Hungarian method, the brute force method, and the auction algorithm. Here, we will focus on the steps involved in solving the assignment problem using the Hungarian method, which is the most commonly used and efficient method.

Step 1: Set up the cost matrix

The first step in solving the assignment problem is to set up the cost matrix, which represents the cost of assigning a task to an agent. The matrix should be square and have the same number of rows and columns as the number of tasks and agents, respectively.

Step 2: Subtract the smallest element from each row and column

To simplify the calculations, we need to reduce the size of the cost matrix by subtracting the smallest element from each row and column. This step is called matrix reduction.

Step 3: Cover all zeros with the minimum number of lines

The next step is to cover all zeros in the matrix with the minimum number of horizontal and vertical lines. This step is called matrix covering.

Step 4: Test for optimality and adjust the matrix

To test for optimality, we need to calculate the minimum number of lines required to cover all zeros in the matrix. If the number of lines equals the number of rows or columns, the solution is optimal. If not, we need to adjust the matrix and repeat steps 3 and 4 until we get an optimal solution.

Step 5: Assign the tasks to the agents

The final step is to assign the tasks to the agents based on the optimal solution obtained in step 4. This will give us the most cost-effective or profit-maximizing assignment.

Solution of the Assignment Problem using the Hungarian Method

The Hungarian method is an algorithm that uses a step-by-step approach to find the optimal assignment. The algorithm consists of the following steps:

• Subtract the smallest entry in each row from all the entries of the row.
• Subtract the smallest entry in each column from all the entries of the column.
• Draw the minimum number of lines to cover all zeros in the matrix. If the number of lines drawn is equal to the number of rows, we have an optimal solution. If not, go to step 4.
• Determine the smallest entry not covered by any line. Subtract it from all uncovered entries and add it to all entries covered by two lines. Go to step 3.

The above steps are repeated until an optimal solution is obtained. The optimal solution will have all zeros covered by the minimum number of lines. The assignments can be made by selecting the rows and columns with a single zero in the final matrix.

Applications of the Assignment Problem

The assignment problem has various applications in different fields, including computer science, economics, logistics, and management. In this section, we will provide some examples of how the assignment problem is used in real-life situations.

Applications in Computer Science

The assignment problem can be used in computer science to allocate resources to different tasks, such as allocating memory to processes or assigning threads to processors.

Applications in Economics

The assignment problem can be used in economics to allocate resources to different agents, such as allocating workers to jobs or assigning projects to contractors.

Applications in Logistics

The assignment problem can be used in logistics to allocate resources to different activities, such as allocating vehicles to routes or assigning warehouses to customers.

Applications in Management

The assignment problem can be used in management to allocate resources to different projects, such as allocating employees to tasks or assigning budgets to departments.

Let’s consider the following scenario: a manager needs to assign three employees to three different tasks. Each employee has different skills, and each task requires specific skills. The manager wants to minimize the total time it takes to complete all the tasks. The skills and the time required for each task are given in the table below:

The assignment problem is to determine which employee should be assigned to which task to minimize the total time required. To solve this problem, we can use the Hungarian method, which we discussed in the previous blog.

Using the Hungarian method, we first subtract the smallest entry in each row from all the entries of the row:

Next, we subtract the smallest entry in each column from all the entries of the column:

We draw the minimum number of lines to cover all the zeros in the matrix, which in this case is three:

Since the number of lines is equal to the number of rows, we have an optimal solution. The assignments can be made by selecting the rows and columns with a single zero in the final matrix. In this case, the optimal assignments are:

• Emp 1 to Task 3
• Emp 2 to Task 2
• Emp 3 to Task 1

This assignment results in a total time of 9 units.

I hope this example helps you better understand the assignment problem and how to solve it using the Hungarian method.

Solving the assignment problem may seem daunting, but with the right approach, it can be a straightforward process. By following the steps outlined in this guide, you can confidently tackle any assignment problem that comes your way.

How useful was this post?

Click on a star to rate it!

Average rating 0 / 5. Vote count: 0

No votes so far! Be the first to rate this post.

We are sorry that this post was not useful for you! 😔

Let us improve this post!

Tell us how we can improve this post?

Operations Research

1 Operations Research-An Overview

• History of O.R.
• Approach, Techniques and Tools
• Phases and Processes of O.R. Study
• Typical Applications of O.R
• Limitations of Operations Research
• Models in Operations Research
• O.R. in real world

2 Linear Programming: Formulation and Graphical Method

• General formulation of Linear Programming Problem
• Optimisation Models
• Basics of Graphic Method
• Important steps to draw graph
• Multiple, Unbounded Solution and Infeasible Problems
• Solving Linear Programming Graphically Using Computer
• Application of Linear Programming in Business and Industry

3 Linear Programming-Simplex Method

• Principle of Simplex Method
• Computational aspect of Simplex Method
• Simplex Method with several Decision Variables
• Two Phase and M-method
• Multiple Solution, Unbounded Solution and Infeasible Problem
• Sensitivity Analysis
• Dual Linear Programming Problem

4 Transportation Problem

• Basic Feasible Solution of a Transportation Problem
• Modified Distribution Method
• Stepping Stone Method
• Unbalanced Transportation Problem
• Degenerate Transportation Problem
• Transhipment Problem
• Maximisation in a Transportation Problem

5 Assignment Problem

• Solution of the Assignment Problem
• Unbalanced Assignment Problem
• Problem with some Infeasible Assignments
• Maximisation in an Assignment Problem
• Crew Assignment Problem

6 Application of Excel Solver to Solve LPP

• Building Excel model for solving LP: An Illustrative Example

7 Goal Programming

• Concepts of goal programming
• Goal programming model formulation
• Graphical method of goal programming
• The simplex method of goal programming
• Using Excel Solver to Solve Goal Programming Models
• Application areas of goal programming

8 Integer Programming

• Some Integer Programming Formulation Techniques
• Binary Representation of General Integer Variables
• Unimodularity
• Cutting Plane Method
• Branch and Bound Method
• Solver Solution

9 Dynamic Programming

• Dynamic Programming Methodology: An Example
• Definitions and Notations
• Dynamic Programming Applications

10 Non-Linear Programming

• Solution of a Non-linear Programming Problem
• Convex and Concave Functions
• Kuhn-Tucker Conditions for Constrained Optimisation
• Quadratic Programming
• Separable Programming
• NLP Models with Solver

11 Introduction to game theory and its Applications

• Important terms in Game Theory
• Saddle points
• Mixed strategies: Games without saddle points
• 2 x n games
• Exploiting an opponent’s mistakes

12 Monte Carlo Simulation

• Reasons for using simulation
• Monte Carlo simulation
• Limitations of simulation
• Steps in the simulation process
• Some practical applications of simulation
• Two typical examples of hand-computed simulation
• Computer simulation

13 Queueing Models

• Characteristics of a queueing model
• Notations and Symbols
• Statistical methods in queueing
• The M/M/I System
• The M/M/C System
• The M/Ek/I System
• Decision problems in queueing

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

3.8: Assignment- Problem Set — Economic Thinking

• Last updated
• Save as PDF
• Page ID 48298

Click the following link to download the problem set for this module:

• Economic Thinking Problem Set
• Economic Thinking Problem Set. Provided by : Lumen Learning. License : CC BY: Attribution

• Share full article

Advertisement

Supported by

Panera Will Discontinue Charged Lemonade Drinks

The highly caffeinated line of Charged drinks at Panera Bread was the subject of two wrongful-death lawsuits.

By Amanda Holpuch

Panera Bread will stop selling its highly caffeinated fruit-flavored drinks, which were the subject of lawsuits by people who said the drinks had caused health problems, including two deaths.

The drinks, known as Charged Lemonade and Charged Sips, will be removed from the menu, a source familiar with the decision said on Tuesday.

A regular size of the Charged drinks, which come in three flavors, has at least 155 milligrams of caffeine, while the large sizes have at least 233 milligrams, according to Panera’s website .

According to the Food and Drug Administration , most “healthy adults” can safely consume up to 400 milligrams of caffeine per day, or about four or five cups of regular coffee, depending on the brand and roast.

In response to a question from The New York Times asking if the Charged drinks were being discontinued, Panera Bread provided a statement about its “menu transformation.”

After surveying more than 30,000 customers, the statement said, the company was “focusing next on the broad array of beverages we know our guests desire,” including low-caffeine options.

It was not clear when the Charged drinks would be removed from the menu.

The Charged Lemonade attracted widespread attention and media coverage after a video was posted in December 2022 on TikTok by a user who said that she routinely drank the beverage but was shocked to learn how much caffeine it contained.

In the past year, several lawsuits have been filed against Panera by people who said that the caffeinated drinks caused their loved ones’ deaths or personal health problems.

The parents of a college student with a heart condition said in a lawsuit filed in October that their daughter, Sarah Katz , 21, died the same day that she drank Panera’s Charged Lemonade in Philadelphia in September 2022.

Dennis Brown, 46, died in October 2023 after suffering a “cardiac event” while walking home from a Panera Bread in Fleming Island, Fla., after drinking three servings of Charged Lemonade, according to a lawsuit filed in December by his mother, sister and brother.

After the lawsuit was filed by Mr. Brown’s family, Panera said it stood by the safety of its products.

Amanda Holpuch covers breaking news and other topics. More about Amanda Holpuch

Browse Course Material

Course info, instructors.

• Prof. Esther Duflo
• Prof. Benjamin Olken

Departments

As taught in.

• Developmental Economics
• Microeconomics

Learning Resource Types

Development economics, assignments.

Students were asked to complete the following assignments for the course:

• Problem Sets
• Replication Assignment : Write an essay that reviews, replicates, and extends an empirical paper in one of the topics covered in the course.
• Research Proposal : The goal of this research proposal is to give you a “jump start” on working on a topic that you could (ideally) work on for your second-year paper (or a future paper for your dissertation).

You are leaving MIT OpenCourseWare