Problems and Solutions in Infinite Ethics

9 Xodarap 04 January 2015 02:06PM

(Crossposted from the EA forum.)

Summary: The universe may very well be infinite, and hence contain an infinite amount of happiness and sadness. This causes several problems for altruists; for example: we can plausibly only affect a finite subset of the universe, and an infinite quantity of happiness is unchanged by the addition or subtraction of a finite amount of happiness. This would imply that all forms of altruism are equally ineffective.

Like everything in life, the canonical reference in philosophy about this problem was written by Nick Bostrom. However, I found that an area of economics known as "sustainable development" has actually made much further progress on this subject than the philosophy world. In this post I go over some of what I consider to be the most interesting results.

NB: This assumes a lot of mathematical literacy and familiarity with the subject matter, and hence isn't targeted to a general audience. Most people will probably prefer to read my other posts:


1. Summary of the most interesting results

  1. There’s no ethical system which incorporates all the things we might want.
  2. Even if we have pretty minimal requirements, satisfactory ethical systems might exist but we can’t prove their existence, much less actually construct them
  3. Discounted utilitarianism, whereby we value people less just because they are further away in time, is actually a pretty reasonable thing despite philosophers considering it ridiculous.
    1. (I consider this to be the first reasonable argument for locavorism I've ever heard)

2. Definitions

In general, we consider a population to consist of an infinite utility vector (u0,u1,…) where ui is the aggregate utility of the generation alive at time i. Utility is a bounded real number (the fact that economists assume utility to be bounded confused me for a long time!). Our goal is to find a preference ordering over the set of all utility vectors which is in some sense “reasonable”. While philosophers have understood for a long time that finding such an ordering is difficult, I will present several theorems which show that it is in fact impossible.

Due to a lack of latex support I’m going to give English-language definitions and results instead of math-ey ones; interested people should look at the papers themselves anyway.

3. Impossibility Results

3.1 Definitions

  • Strong Pareto: if you can make a generation better off, and none worse off, you should.
  • Weak Pareto: if you can make every generation better off, you should.
  • Intergenerational equity: utility vectors are unchanged in value by any permutation of their components.
    • There is an important distinction here between allowing a finite number of elements to be permuted and an infinite number; I will refer to the former as “finite intergenerational equity” and the latter as just “intergenerational equity”
  • Ethical relation: one which obeys both weak Pareto and intergenerational equity
  • Social welfare function: an order-preserving function from the set of populations (utility vectors) to the real numbers

3.2 Diamond-Basu-Mitra Impossibility Result1

  1. There is no social welfare function which obeys Strong Pareto and finite intergenerational equity. This means that any sort of utilitarianism won’t work, unless we look outside the real numbers.

3.3 Zame's impossibility result2

  1. If an ordering obeys intergenerational equity over [0,1]N, then almost always we can’t tell which of two populations is better 
    1. (i.e. the set of populations {X,Y: neither X<Y nor X>Y} has outer measure one)
  2. The existence of an ethical preference relation on [0,1]N is independent of ZF plus the axiom of choice

4. Possibility Results

We’ve just shown that it’s impossible to construct or even prove the existence of any useful ethical system. But not all hope is lost!

The important idea here is that of a “subrelation”: < is a subrelation to <’ if x<y implies x<’y.

Our arguments will work like this:

Suppose we could extend utilitarianism to the infinite case. (We don't, of course, know that we can extend utilitarianism to the infinite case. But suppose we could.) Then A, B and C must follow.

Technically: suppose utilitarianism is a subrelation of <. Then < must have properties A, B and C.

Everything in this section comes from (3), which is a great review of the literature.

4.1 Definition

  • Utilitarianism: we extend the standard total utilitarianism ordering to infinite populations in the following way: suppose there is some time T after which every generation in X is at least as well off as every generation in Y, and that the total utility in X before T is at least as good as the total utility in Y before T. Then X is at least as good as Y.
    • Note that this is not a complete ordering! In fact, as per Zame’s result above, the set of populations it can meaningfully speak about has measure zero.
  • Partial translation scale invariance: suppose after some time T, X and Y become the same. Then we can add any arbitrary utility vector A to both X and Y without changing the ordering. (I.e. X > Y ó X+A > Y+A)

4.2 Theorem

  1. Utilitarianism is a subrelation of > if and only if > satisfies strong Pareto, finite intergenerational equity and partial translation scale invariance.
    1. This means that if we want to extend utilitarianism to the infinite case, we can’t use a social welfare function, as per the above Basu-Mitra result

4.3 Definition

  • Overtaking utilitarianism: suppose there is some point T after which the total utility of the first N generations in X is always greater than the total utility of the first N generations in Y (given N > T). Then X is better than Y.
    • Note that utilitarianism is a subrelation of overtaking utilitarianism
  • Weak limiting preference: suppose that for any time T, X truncated at time T is better than Y truncated at time T. Then X is better than Y.

4.4 Theorem

  1. Overtaking utilitarianism is a subrelation of < if and only if < satisfies strong Pareto, finite intergenerational equity, partial translation scale invariance, and weak limiting preference

4.5 Definition

  • Discounted utilitarianism: the utility of a population is the sum of its components, discounted by how far away in time they are
  • Separability:
    • Separable present: if you can improve the first T generations without affecting the rest, you should
    • Separable future: if you can improve everything after the first T generations without affecting the rest, you should
  • Stationarity: preferences are time invariant
  • Weak sensitivity: for any utility vector, we can modify its first generation somehow to make it better

4.6 Theorem

  1. The only continuous, monotonic relation which obeys weak sensitivity, stationary, and separability is discounted utilitarianism

4.7 Definition

  • Dictatorship of the present: there’s some time T after which changing the utility of generations doesn’t matter

4.8 Theorem

  1. Discounted utilitarianism results in a dictatorship of the present. (Remember that each generation’s utility is assumed to be bounded!)

4.9 Definition

  • Sustainable preference: a continuous ordering which doesn’t have a dictatorship of the present but follows strong Pareto and separability.

4.10 Theorem

  1. The only ordering which is sustainable is to take discounted utilitarianism and add an “asymptotic” part which ensures that infinitely long changes in utility matter. (Of course, finite changes in utility still won't matter.)

5. Conclusion

I hope I've convinced you that there's a "there" there: infinite ethics is something that people can make progress on, and it seems that most of the progress is being made in the field of sustainable development.

Fun fact: the author of the last theorem (the one which defined "sustainable") was one of the lead economists on the Kyoto protocol. Who says infinite ethics is impractical?

6. References

  1. Basu, Kaushik, and Tapan Mitra. "Aggregating infinite utility streams with intergenerational equity: the impossibility of being Paretian." Econometrica 71.5 (2003): 1557-1563. http://folk.uio.no/gasheim/zB%26M2003.pdf
  2. Zame, William R. "Can intergenerational equity be operationalized?." (2007).  https://tspace.library.utoronto.ca/bitstream/1807/9745/1/1204.pdf
  3. Asheim, Geir B. "Intergenerational equity." Annu. Rev. Econ. 2.1 (2010): 197-222.http://folk.uio.no/gasheim/A-ARE10.pdf

Political Skills which Increase Income

57 Xodarap 02 March 2014 05:56PM

Summary: This article is intended for those who are "earning to give" (i.e. maximize income so that it can be donated to charity). It is basically an annotated bibliography of a few recent meta-analyses of predictors of income.

Key Results

  • The degree to which management “sponsors” your career development is an important predictor of your salary, as is how skilled you are politically.

  • Despite the stereotype of a silver-tongued salesman preying on people’s biases, rational appeals are generally the best tactic.

  • After rationality, the best tactics are types of ingratiation, including flattery and acting modest.

Ng et al. performed a metastudy of over 200 individual studies of objective and subjective career success. Here are the variables they found best correlated with salary:

Predictor

Correlation

Political Knowledge & Skills

0.29

Education Level

0.29

Cognitive Ability (as measured by standardized tests)

0.27

Age

0.26

Training and Skill Development Opportunities

0.24

Hours Worked

0.24

Career Sponsorship

0.22

(all significant at p = .05)


(For reference, the “Big 5” personality traits all have a correlation under 0.12.)

Before we go on, a few caveats: while these correlations are significant and important, none are overwhelming (the authors cite Cohen as saying the range 0.24-0.36 is “medium” and correlations over 0.37 are “large”). Also, in addition to the usual correlation/causation concerns, there is lots of cross-correlation: e.g. older people might have greater political knowledge but less education, thereby confusing things. For a discussion of moderating variables, see the paper itself.

Career Sponsorship

There are two broad models of career advancement: contest-mobility and sponsorship-mobility. They are best illustrated with an example.

Suppose Peter and Penelope are both equally talented entry-level employees. Under the contest-mobility model, they would both be equally likely to get a raise or promotion, because they are equally skilled.

Sponsorship-mobility theorists argue that even if Peter and Penelope are equally talented, it’s likely that one of them will catch the eye of senior management. Perhaps it’s due to one of them having an early success by chance, making a joke in a meeting, or simply just having a more memorable name, like Penelope. This person will be singled out for additional training and job opportunities. Because of this, they’ll have greater success in the company, which will lead to more opportunities etc. As a result, their initial small discrepancy in attention gets multiplied into a large differential.

The authors of the metastudy found that self-reported sponsorship levels (i.e. how much you feel the management of your company “sponsors” you) have a significant, although moderate, relationship to salary. Therefore, the level at which you currently feel sponsored in your job should be a factor when you consider alternate opportunities.

The Dilbert Effect

The strongest predictor of salary (tied with education level) is what the authors politely term “Political Knowledge & Skills” - less politely, how good you are at manipulating others.

Several popular books (such as Cialdini’s Influence) on the subject of influencing others exist, and the study of these “influence tactics” in business stretches back 30 years to Kipnis, Schmidt and Wilkinson. Recently, Higgins et al. reviewed 23 individual studies of these tactics and how they relate to career success. Their results:


Tactic

Correlation

Definition (From Higgins et al.)

Rationality

0.26

Using data and information to make a logical argument supporting one's request

Ingratiation

0.23

Using behaviors designed to increase the target's liking of oneself or to make oneself appear friendly in order to get what one wants

Upward Appeal

0.05

Relying on the chain of command, calling in superiors to help get one's way

Self-Promotion

0.01

Attempting to create an appearance of competence or that you are capable

of completing a task

Assertiveness

-0.02

Using a forceful manner to get what one wants

Exchange

-0.03

Making an explicit offer to do something for another in exchange for their doing what

one wants

(Only ingratiation and rationality are significant.)

This site has a lot of information on how to make rational appeals, so I will focus on the less-talked-about ingratiation techniques.

How to be Ingratiating

Gordon analyzed 69 studies of ingratiation and found the following. (Unlike the previous two sections, success here is measured in lab tests as well as in career advancement. However, similar but less comprehensive results have been found in terms of career success):

Tactic

Weighted Effectiveness (Cohen’s d difference between control and intervention)

Description

Other Enhancement

0.31

Flattery

Opinion Conformity

0.23

“Go along to get along”

Self-presentation

0.15

Any of the following tactics: Self-promotion, self-deprecation, apologies, positive nonverbal displays and name usage

Combination

0.10

Includes studies where the participants weren’t told which strategy to use, in addition to when they were instructed to use multiple strategies

Rendering Favors

0.05


Self-presentation is split further:

Tactic

Weighted Effect Size

Comment

Modesty

0.77


Apology

0.59

Apologizing for poor performance

Generic

0.28

When the participant is told in generic terms to improve their self-presentation

Nonverbal behavior and name usage

-0.14

Nonverbal behavior includes things like wearing perfume. Name usage means referring to people by name instead of a pronoun.

Self-promotion

-0.17

 

 

Moderators

One important moderator is the direction of the appeal. If you are talking to your boss, your tactics should be different than if you’re talking to a subordinate. Other-enhancement (flattery) is always the best tactic no matter who you’re talking to, but when talking to superiors it’s by far the best. When talking to those at similar levels to you, opinion conformity comes close to flattery, and the other techniques aren't far behind.

Unsurprisingly, when the target realizes you’re being ingratiating, the tactic is less effective. (Although effectiveness doesn’t go to zero - even when people realize you’re flattering them just to suck up, they generally still appreciate it.) Also, women are better at being ingratiating than men, and men are more influenced by these ingratiating tactics than women. The most important caveat is that lab studies find much larger effect sizes than in the field, to the extent that the average field effect for the ingratiating tactics is negative. This is probably due to the fact that lab experiments can be better controlled.

Conclusion

It’s unlikely that a silver-tongued receptionist will out-earn an introverted engineer. But simple techniques like flattery and attempting to get "sponsored" can appreciably improve returns, to the extent that political skills are one of the strongest predictors of salaries.

 

I would like to thank Brian Tomasik and Gina Stuessy for reading early drafts of this article.

References

Cohen, Jacob. Statistical power analysis for the behavioral sciences. Psychology Press, 1988.

 

Gordon, Randall A. "Impact of ingratiation on judgments and evaluations: A meta-analytic investigation." Journal of Personality and Social Psychology 71.1 (1996): 54.

 

Higgins, Chad A., Timothy A. Judge, and Gerald R. Ferris. "Influence tactics and work outcomes: a meta‐analysis." Journal of Organizational Behavior 24.1 (2003): 89-106.

 

Judge, Timothy A., and Robert D. Bretz Jr. "Political influence behavior and career success." Journal of Management 20.1 (1994): 43-65.

 

Kipnis, David, Stuart M. Schmidt, and Ian Wilkinson. "Intraorganizational influence tactics: Explorations in getting one's way." Journal of Applied psychology 65.4 (1980): 440.


Ng, Thomas WH, et al. "Predictors of objective and subjective career success: A meta‐analysis." Personnel psychology 58.2 (2005): 367-408.

A Pure Math Argument for Total Utilitarianism

-5 Xodarap 27 October 2013 05:05PM

Summary: I sketch an argument that population ethics should, in a certain technical sense, be similar to addition. I show that a surprising theorem of Hölder's implies that this means that we should be total utilitarians.

Addition is a very special operation. Despite the wide variety of esoteric mathematical objects known to us today, none of them have the basic desirable properties of grade-school arithmetic.

This fact was intuited by 19th century philosophers in the development of what we now call "total" utilitarianism. In this ethical system, we can assign each person a real number to indicate their welfare, and the value of an entire population is the sum of each individuals' welfare.

Using modern mathematics, we can now prove the intuition of Mills and Bentham: because addition is so special, any ethical system which is in a certain technical sense "reasonable" is equivalent to total utilitarianism.

What do we mean by ethics?


The most basic premise is that we have some way of ordering individual lives. 

We don't need to say how much better some life is than another, we just need to be able to put them in order. We might have some uncertainty as to which of two lives is better:


In this case, we aren't certain if "Medium" or "Medium 2" is better. However, we know they're both better than "Bad" and worse than "Good".

In the case when we always know which of two lives is better, we say that lives are totally ordered. If there is uncertainty, we say they are lattice ordered.

In either case, we require that the ranking remain consistent when we add people to the population. Here we add a person of "Medium" utility to each population:


The ranking on the right side of the figure above is legitimate because it keeps the order - if some life X is worse than Y, then (X + Medium) is still worse than (Y + Medium). This ranking below for example would fail that:


This ranking is inconsistent because it sometimes says that "Bad" is worse than "Medium" and other times says "Bad" is better than "Medium". A basic principle of ethics is that rankings should be consistent, and so rankings like the latter are excluded.

Increasing population size


The most obvious way of defining an ethics of populations is to just take an ordering of individual lives and "glue them together" in an order-preserving way, like I did above. This generates what mathematicians would call the free group. (The only tricky part is that we need good and bad lives to "cancel out", something which I've talked about before.)

It turns out that merely gluing populations together in this way gives us a highly structured object known as a "lattice-ordered group". Here is a snippet of the resulting lattice:


This ranking is similar to what philosophers often call "Dominance" - if everyone in population P is better off than everyone in population Q, then P is better than Q. However, this is somewhat stronger - it allows us to compare populations of different sizes, something that the traditional dominance criterion doesn't let us do.

Let's take a minute to think about what we've done. Using only the fact that individuals' lives can be ordered and the requirement that population ethics respects this ordering in a certain technical sense, we've derived a robust population ethics, about which we can prove many interesting things.

Getting to total utilitarianism


One obvious facet of the above ranking is that it's not total. For example, we don't know if "Very Good" is better than "Good, Good", i.e. if it's better to have welfare "spread out" across multiple people, or concentrated in one. This obviously prohibits us from claiming that we've derived total utilitarianism, because under that system we always know which is better.

However, we can still derive a form of total utilitarianism which is equivalent in a large set of scenarios. To do so, we need to use the idea of an embedding. This is merely a way of assigning each welfare level a number. Here is an example embedding:

  • Medium = 1
  • Good = 2
  • Very Good = 3

Here's that same ordering, except I've tagged each population with the total "utility" resulting from that embedding:


This is clearly not identical to total utilitarianism - "Very Good" has a higher total utility than "Medium, Medium" but we don't know which is better, for example.

However, this ranking never disagrees with total utilitarianism - there is never a case where P is better than Q yet P has less total utility than Q.

Due to a surprising theorem of Holder which I have discussed before, as long as we disallow "infinitely good" populations, there is always some embedding like this. Thus, we can say that:
Total utilitarianism is the moral "baseline". There might be circumstances where we are uncertain whether or not P is better than Q, but if we are certain, then it must be that P has greater total utility than Q.

An application

Here is one consequence of these results. Many people, including myself, have the intuition that inequality is bad. In fact, it is so bad that there are circumstances where increasing equality is good even if people are, on average, worse off.

If we accept the premises of this blog post, this intuition simply cannot be correct. If the inequitable society has greater total utility, it must be at least as good as the equitable one.

Concluding remarks

There are certain restrictions we want the "addition" of a person to a population to obey. It turns out that there is only one way to obey them: by using grade school addition, i.e. total utilitarianism.
[For those interested in the technical result: Holder showed that any archimedean l-group is l-isomorphic to a subgroup of (R,+). The proof can be found in Glass' Partially Ordered Groups as Corollary 4.1.4. This article was originally posted here.]