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 (1984). The paradox identifies the mutual incompatibility of four intuitively compelling assertions about the relative value of populations. Parfit’s original formulation of the repugnant conclusion is that “For any perfectly equal population with very high positive welfare, there is a population with very low positive welfare which is better, other things being equal.”

The paradox
Consider the four populations depicted in the following diagram: A, A+, B− and B. Each bar represents a distinct group of people. The bar's width represents group size while the bar's height represents group happiness. Unlike A and B, A+ and B− are complex populations, each comprising two distinct groups of people. It is also stipulated that the lives of the members of each group are good enough that they would rather be alive than not exist.



Parfit makes the following three suggestions regarding the value of the populations:


 * 1. 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, since it is stipulated that their lives are good enough that it is better for them to be alive than to not exist.


 * 2. B− seems better than A+. This is because B− has greater total and average happiness than A+.


 * 3. 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 also observes the following:


 * 4. When we directly compare A (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: (1) that A+ is no worse than A, (2) that B− is better than A+, (3) that B− is as good as B, and (4) that B can be worse than A.

Responses
Some scholars, such as Larry Temkin and Stuart Rachels, argue that inconsistencies between the four claims (above) relies on the assumption that the "better than" relation is transitive. The inconsistency could then be resolved by rejecting the assumption. According to this view, although A+ is no worse than A, and B− is better than A+, it does not follow that B− is better than A.

Torbjörn Tännsjö 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.

However, Parfit argues that the above discussion fails to appreciate the true source of repugnance. He claims that initially, it may not be wrong to think that B is better than A. Assuming this is true, 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.

A number of philosophers (including Torbjörn Tännsjö, Yew-Kwang Ng, Hilary Greaves and Toby Ord) have agreed that avoiding the repugnant conclusion is not a necessary property of a satisfactory theory of population ethics.

Variants
The Very Repugnant Conclusion is a property of theories of population ethics that is a stronger version of the repugnant conclusion. Informally, the very repugnant conclusion is that in some theories of population ethics, equal populations with very high welfare are better than extremely large populations that are composed of many lives barely worth living, and a large number of lives with very high negative welfare. All leading theories of population ethics imply the very repugnant conclusion.

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 video game sold on a download service may be considered a failure if less than 20% of those who download the game 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.