Mere addition paradox

The mere addition paradox, also known as the repugnant conclusion, is a problem in ethics, identified by Derek Parfit and discussed in his book Reasons and Persons. The paradox identifies the mutual incompatibility of four intuitively compelling assertions about the relative value of populations.

1. The paradox
Consider the four populations depicted in the following diagram: A, A+, B− and B. Each bar represents a distinct group of people, with the groups size represented by the bars width and the happiness of each of the groups members represented by the bars height. Unlike A and B, A+ and B− are complex populations, each comprising two distinct groups of people.
How do these populations compare in value?
Parfit observes that i) A+ seems no worse than A. This is because the people in A are no worse-off in A+, while the additional people who exist in A+ are better off in A+ compared to A if it is stipulated that their lives are good enough that living them is better than not existing.
Next, Parfit suggests that ii) B− seems better than A+. This is because B− has greater total and average happiness than A+.
Then, he notes that iii) B seems equally as good as B−, as the only difference between B− and B is that the two groups in B− are merged to form one group in B.
Together, these three comparisons entail that B is better than A. However, Parfit observes that when we directly compare a population with high average happiness and B a population with lower average happiness, but more total happiness because of its larger population, it may seem that B can be worse than A.
Thus, there is a paradox. The following intuitively plausible claims are jointly incompatible: a that A+ is no worse than A, b that B− is better than A+, c that B− is equally as good as B, and d that B can be worse than A.

2. Criticisms and responses
Some scholars, such as Larry Temkin and Stuart Rachels, argue that the apparent inconsistency between the four claims just outlined relies on the assumption that the "better than" relation is transitive. We may resolve the inconsistency, thus, by rejecting the assumption. On this view, from the fact that A+ is no worse than A and that B− is better than A+ it simply does not follow that B− is better than A.
Torbjorn Tannsjo argues that the intuition that B is worse than A is wrong. While the lives of those in B are worse than those in A, there are more of them and thus the collective value of B is greater than A. Michael Huemer also argues that the repugnant conclusion is not repugnant and that normal intuition is wrong.
However, Parfit argues that the above discussion fails to appreciate the true source of repugnance. He claims that on the face of it, it may not be absurd to think that B is better than A. Suppose, then, that B is in fact better than A, as Huemer argues. It follows that this revised intuition must hold in subsequent iterations of the original steps. For example, the next iteration would add even more people to B+, and then take the average of the total happiness, resulting in C-. If these steps are repeated over and over, the eventual result will be Z, a massive population with the minimum level of average happiness; this would be a population in which every member is leading a life barely worth living. Parfit claims that it is Z that is the repugnant conclusion.

3. Alternative usage
An alternative use of the term mere addition paradox was presented in a paper by Hassoun in 2010. It identifies paradoxical reasoning that occurs when certain statistical measures are used to calculate results over a population. For example, if a group of 100 people together control $100 worth of resources, the average wealth per capita is $1. If a single rich person then arrives with 1 million dollars, then the total group of 101 people controls $1.000.100, making average wealth per capita $9.901, implying a drastic shift away from poverty even though nothing has changed for the original 100 people. Hassoun defines a no mere addition axiom to be used for judging such statistical measures: "merely adding a rich person to a population should not decrease poverty" although acknowledging that in actual practice adding rich people to a population may provide some benefit to the whole population.
This same argument can be generalized to many cases where proportional statistics are used: for example, a game sold on a download service may be considered a failure if less than 20% of those who download the demo then purchase the game. Thus, if 10.000 people download the demo of a game and 2.000 buy it, the game is a borderline success; however, it would be rendered a failure by an extra 500 people downloading the demo and not buying, even though this "mere addition" changes nothing with regard to income or consumer satisfaction from the previous situation.