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# Identification and analysis of non-additive measures

Research project The amount of work per hour produced by two people is not additive. Two individuals never work exactly the same amount. To build successful applications, we need to better understand the non-additive measures and provide both theoretical results and algorithms.

In mathematics, the concept of measure is a generalisation of length, area and volume, and a measure is additive in most mathematical problems. If we have two boxes, then the volume they occupy is the sum of the volume of each box. The same applies to probability; if we have a 1/6 chance of having a 3 when we roll a die, and a 1/6 chance of having a 6, then we have a 2/6 chance of having either a 3 or a 6. But non-additive measures? The amount of labour per hour produced by two people is not additive – together they may be more effectively than when alone, or may be less effective!

## Project overview

### Project period:

2024-01-01 2025-12-17

### Participating departments and units at Umeå University

Department of Computing Science

### External funding

The Kempe Foundation

## How do we identify and analyse a measure?

The non-additive measures are studied in mathematics, economics and computing science, especially in the field of artificial intelligence. Yet, the theory behind these measures is not as well developed as the theory of probabilities. To build successful applications, we need to better understand non-additive measures and also provide theoretical results and algorithms.

### Inspired by probability and measure theory

Probability and measurement theories give inspiration for the type of mathematical tools we want to develop for non-additive measures. One type of these measures - distorted probabilities - is used to model how people make decisions. They're useful for modelling people who are risk averse and people who are risk seeking. Distorted probabilities are one of the simplest non-additive measures and other measures exist. However, two problems are important:

• How can we identify a measure?
For example, which measure models a set of workers? which measure accurately represents how a person makes a decision? and which measure should be used in a machine learning system?
• How can we analyse a measure?
How to compare two different measures? If we identify measures automatically, analysis and comparison are required.

### With the support of the Kempe Foundations

Our research in the area of non-additive measures is mainly theoretical and developed through international co-operation. This research project is funded by Kempestiftelserna.

### Facts

Professor Vicenç Torra is a leading researcher in the field of data protection, approximate reasoning and decision making. At the Department of Computing Science, he is head of the research group NAUSICA: PrivAcy-AWare traNSparent deCIsions. Both the department and Professor Vicenç Torra are affiliated with WASP, the Wallenberg AI, Autonomous Systems and Software Program, Sweden's single largest research programme.

## External funding

Latest update: 2024-05-28