Discrimination without disadvantage

The SEP’s article on discrimination talks of discrimination as involving the imposition of a relative disadvantage on a member of a group.

This seems incorrect. Suppose Bob refuses to hire Alice because Alice is a woman. But Bob’s workplace is such a toxic environment that one is better off being jobless than working for Bob, regardless of whether one is a man or a woman. Bob has paradigmatically discriminated against Alice, but he has not imposed a relative disadvantage on her.

One might object that losing an option is always a disadvantage. But that is false: some options are degrading and it is better not to have them.

Perhaps we should subjectivize the relative disadvantage and say that discrimination involves the imposition of what is believed or intended to be a relative disadvantage. Bob presumably doesn’t think that working for him is a disadvantage. But imagine that Bob has the sexist belief that women are better off as housewives, and further believes that being a housewife is as good for a woman as being an employee of Bob’s is for a man. Then Bob does not believe he is imposing a relative disadvantage and he is not intending to do so, but he is clearly discriminating.

I am not sure how to fix the account of discrimination.

Necessary and Sufficient Conditions for Domination Results for Proper Scoring Rules

The preprint is now up.

Abstract: Scoring rules measure the deviation between a probabilistic forecast and reality. Strictly proper scoring rules have the property that for any forecast, the mathematical expectation of the score of a forecast p by the lights of p is strictly better than the mathematical expectation of any other forecast q by the lights of p. Probabilistic forecasts need not satisfy the axioms of the probability calculus, but Predd, et al. (2009) have shown that given a finite sample space and any strictly proper additive and continuous scoring rule, the score for any forecast that does not satisfy the axioms of probability is strictly dominated by the score for some probabilistically consistent forecast. Recently, this result has been extended to non-additive continuous scoring rules. In this paper, a condition weaker than continuity is given that suffices for the result, and the condition is proved to be optimal.

Deserving the rewards of virtue

We have the intuition that when someone has worked uprightly and hard for something good and thereby gained it, they deserve their possession of it. What does that mean?

If Alice ran 100 meters faster than her opponents at the Olympics, she deserves a gold medal. In this case, it is clear what is meant by that: the organizers of the Olympics owe her a gold medal in just recognition of her achievement. Thus, Alice’s desert appears appears to be appropriately analyzable partly in terms of normative properties had by persons other than Alice. In Alice’s case, these properties are obligations of justice, but they could simply be reasons of justice. Thus, if someone has done something heroic and they receive a medal, the people giving the medal typically are not obligated to give it, but they do have reasons of justice to do so.

But there are cases that fit the opening intuition where it is harder to identify the other persons with the relevant normative properties. Suppose Bob spends his life pursuing virtue, and gains the rewards of a peaceful conscience and a gentle attitude to the failings of others. Like Alice’s gold medal, Bob’s rewards are deserved. But if we understand desert as in Alice’s case, as partly analyzable in terms of normative properties had by others, now we have a problem: Who is it that has reasons of justice to bestow these rewards on Bob?

We can try to analyze Bob’s desert by saying that we all have reasons of justice not to deprive him of these rewards. But that doesn’t seem quite right, especially in the case of the gentle attitude to the failings of others. For while some people gain that attitude through hard work, others have always had it. Those who have always had it do not deserve it, but it would still be unjust to deprive them of it.

The theist has a potential answer to the question: God had reasons of justice to bestow on Bob the rewards of virtue. Thus, while Alice deserved her gold medal from the Olympic committee and Carla (whom I have not described but you can fill in the story) deserved her Medal of Honor from the Government, Bob deserved his quiet conscience and “philosophical” outlook from God.

This solution, however, may sound wrong to many Christians, especially but not only Protestants. There seems to be a deep truth to Leszek Kolakowski’s book title God Owes Us Nothing. But recall that desert can also be partly grounded in non-obligating reasons of justice. One can hold that God owes us nothing but nonetheless think that when God bestowed on Bob the rewards of virtue (say, by designing and sustaining the world in such a way that often these rewards came to those who strove for virtue), God was doing so in response to non-obligating reasons of justice.

Objection: Let’s go back to Alice. Suppose that moments after she ran the race, a terrorist assassinated everyone on the Olympic Committee. It still seems right to say that Alice deserved a gold medal for her run, but no one had the correlate reason of justice to bestow it. Not even God, since it just doesn’t seem right to say that God has reasons of justice to ensure Olympic medals.

Response: Maybe. I am not sure. But think about the “Not even God” sentence in the objection. I think the intuition behind the “Not even God” undercuts the case. The reason why not even God had reasons of justice to ensure the medal was that Alice deserved a medal not from God but from the Olympic Committee. And this shows that her desert is grounded in the Olympic Committee, if only in a hypothetical way: Were they to continue existing, they would have reasons of justice to bestow on her the medal.

This suggests a different response that an atheist could give in the case of Bob: When we say that Bob deserves the rewards of virtue, maybe we mean hypothetically that if God existed, God would have reasons of justice to grant them. This does not strike me as a plausible analysis. If God doesn’t exist, the existence of God is a far-fetched and fantastical hypothesis. It is implausible that Bob’s ordinary case of desert be partly grounded in hypothetical obligations of a non-existent fantastical being. On the other hand, it is not crazy to think that Alice’s desert, in the exceptional case of the Olympic Committee being assassinated, be partly grounded in hypothetical obligations of a committee that had its existence suddenly cut short.

When bad things happen to bad people

Sometimes when bad things happen to bad people, we feel like they deserve them. Suppose Alice has broken into Bob’s house to poison his beloved dog, and on her way out she trips and breaks her leg. It sure seems that she got what she deserved.

But the bad that a bad person deserves is punishment: an intentional harsh treatment by a punisher. Our feeling that Alice got what she deserved can only be right if there is some supernatural judge—for clearly no natural judge is behind this—who as a punishment has either caused her to break her leg or at least intentionally failed to prevent the leg-breaking.

I think this provides us with a little bit of evidence that God exists. For a feeling that p is some evidence that p. So, the feeling that Alice got what she deserved is some evidence that she got what she deserved.

Moral risk

Say that an action is deontologically doubtful (DD) provided that the probability of the action being forbidden by the correct deontology is significant but less than 1/2.

There are cases where we clearly should not risk performing a DD action. A clear example is when you’re hunting and you see a shape that has a 40% chance of being human: you should not shoot. But notice that in this case, deontology need play no role: expected-utility reasoning tells you that you shouldn’t shoot.

There are, on the other hand, cases where you should take a significant risk of performing a DD action.

Beast Case: The shape in the distance has a 30% chance of being human and a 70% chance of being a beast that is going to devour a dozen people in your village if not shot by you right now. In that case, it seems it might well be permissible to shoot.

This suggests this principle:

1. If a DD action has significantly higher expected utility than refraining from the action, it is permissible to perform it.

But this is false. I will assume here the standard deontological claim that it is wrong to shoot one innocent to save two.

Villain Case: You are hunting and you see a dark shape in the woods. The shape has a 40% chance of being an innocent human and a 60% chance of being a log. A villain who is with you has just instructed a minion to go and check in a minute on the identity of the shape. If the shape turns out to be a human, the minion is to murder two innocents. You can’t kill the villain or the minion, as they have bulletproof jackets.

The expected utility of shooting is significantly higher than of refraining from the action. If you shoot, the expected lives lost are (0.4)(1)=0.4, and if you don’t shoot the expected lives lost are (0.4)(2)=0.8. So shooting has an expected utility that’s 0.4 lives better than not shooting. But it is also clear, assuming the deontological claim that it is wrong to kill one to save two, that it is wrong to shoot in this case.

What is different from the villain case and the dangerous beast case is that in the Villain Case, the difference in expected utilities comes precisely from the scenario where the shape is human. Intuition suggests we should tweak (1) to evaluate expected utilities in a way that ignores the good effects of deontologically forbidden things. This tweak does not affect the Beast Case, but it does affect the Villain Case, where the difference in utilities came precisely from counting the life-saving benefits of killing the human.

I don’t know how to precisely formulate the tweaked version of (1), and I don’t know if it is sufficiently strong to covere all cases.

Defining killing

My presentation on killing for the Bios Centre is now on YouTube.

Canadian doubles

Since the shutdowns of the spring, I’ve been playing more tennis, with my son and with graduate students. Sometimes you end up having three people wanting to play tennis, though, and what do you do?

The canned solution is Canadian doubles where you have two people on one side and one on the other, you get two points for winning on the singles side and one for winning on doubles, you rotate the players counterclockwise between games, and you end at some fixed number of points, say 11, by a lead of two. And the singles court is used on the singles side while the doubles court is used on the doubles side.

This is a good game: alternating between being a single playing facing off against two and playing as part of a team is fun. However, we noticed two difficulties. First, the standard rotation scheme has the result that each time one serves, one faces the same person. That reduces the variation. A fix of this is to depart from the counterclockwise rotation. A more serious problem is that towards the end of a match, when playing on the double side, one can have a perverse desire to lose the game. For imagine that your partner has 10 points, you have 8, and the third player has less than 8. Then if your side wins, your partner reaches 11 and wins the match. But if your side loses, you may still have a chance to win the match later. This can sap motivation.

Also missing from the standard Canadian doubles is that in the interest of trying all combinations, it would be nice to have the chance to be a singles server and a singles receiver in the same match.

So, after a number of iterations, here is improved Canadian doubles (I am Canadian, by the way). Instead of playing to a fixed score, you play three rounds of six games. Highest score wins. Ties are possible. You can end early if you can see that the number of games left is insufficient to change the ranking between the players.

The first and third rounds have serving from the doubles side. The second round has serving from the singles side. In each round, positions rotate in such a way that each of the six arrangements occurs once. Moreover, the positions are so arranged as to minimize the same player “being on the spot” too many times in a row. Thus, no one serves twice in a row or is in singles twice in a row, and we rule out the tiring sequence of singles, then serving, then singles again. I generated the sequences with a brute force python script.

Round 1: Service from doubles side

Server	Partner	Receiver
B	C	A
C	A	B
A	B	C
C	B	A
A	C	B
B	A	A

Round 2: Service from singles side

Server	Receiver-Deuce	Receiver-Ad
A	C	B
C	A	B
A	B	C
B	A	C
C	B	A
B	C	A

Round 3: Service from doubles side

Server	Partner	Receiver
C	B	A
B	A	C
A	C	B
B	C	A
A	B	C
C	A	B

Loving excessively and the existence of God

1. Francis of Assisi did not love nature excessively and Mother Teresa did not love the needy too much.

2. Francis loved nature as reflecting God and Mother Teresa loved the needy as images of God.

3. If God does not exist, then to love nature as reflecting God or to love someone as an image of God is to love something as better than it is.

4. To love something as better than it is is to love it excessively.

5. So, if God does not exist, Francis of Assisi loved nature excessively and Mother Teresa loved the needy too much. (2–4)

6. So, God exists. (1 and 5)

Nonadditive scoring rules and domination

I wrote a rough draft of a paper proving geometrically that any strictly proper scoring rule continuous on the probabilities has every score of a non-probability dominated by a score of a probability, without assuming additivity of score. My proof is very much geometric.

Notes: Richard Pettigrew first announced this result in a forthcoming paper, but his proof is flawed. Then Michael Nielsen found a proof in the special case of bounded scoring rules. Finally, Nielsen and I approximately simultaneously (within hours of each other) found quite different proofs without the assumption of boundedness (though there could still be problems in one or the other proof). Research continues regarding how far the condition of continuity can be weakened.

Nonadditive strictly proper scoring rules and arguments for probabilism

[This post uses the wrong concept of a strictly proper score. See the comments.]

A scoring rule for a credence assignment is a measure of the inaccuracy of the credences: the lower the value, the better.

A proper scoring rule is a scoring rule with the property that for each probabilistically consistent credence assignment P, the expected value according to P of the score for P is maximized at P. If it’s maximized uniquely at P, the scoring rule is said to be strictly proper.

A scoring rule is additive provided that it is the sum of scoring rules each of which depends only on the credence assigned to a single proposition and the truth value of that proposition.

The formal epistemology literature has a lot of discussion of a strict domination theorem that given an additive strictly proper scoring rule, you will do better to have a credence assignment that is probabilistically consistent: indeed, another credence assignment will give a better score in every possible world.

The assumption of strict propriety gets a fair amount of discussion. Not so the assumption of additivity.

It turns out that if you drop additivity, the theorem fails. Indeed: this is trivial. Consider any strictly proper scoring rule s, and modify it to a rule s^* that assigns the score −∞ to any inconsistent credence. Then any inconsistent credence receives the best possible score in every possible world. Moreover, s^* is still strictly proper if s is because the definition of strict propriety only involves the behavior of the scoring rule as applied to consistent credences, and hence s^* is strictly proper if and only if s is. And, of course, s^* is not additive.

But of course my rule s^* is very much ad hoc and it is gerrymandered to reward inconsistency. Can we make a non-additive scoring for which the domination theorem fails that lacks such gerrymandering and is somewhat natural?

I think so. Consider a finite probability space Ω, with n points ω₁, ..., ω_n in it. Now, consider a scoring rule generated as follows.

Say that a simple gamble g on Ω is an assignment of values to the n points. Let G be a set of simple gambles. Imagine an agent who decides which simple gamble g in G to take by the following natural method: she calculates ∑_iP({ω_i})g(ω_i), where P is her credence assignment, and chooses the gamble g that maximizes this sum. If there is a tie, she has some tie-resolution mechanism. Then, we can say that the G-score of her credences is the negative of the utility gained from the gamble she chose. In other words, her G-score at location ω_i is −g(ω_i) where g is a maximally auspicious gamble according to her credences.

It is easy to see that G-score is a proper score. Moreover, if there are never any ties in choosing the maximally auspicious gamble, the score is strictly proper.

This is a very natural way to generate a score: we generate a score by looking how well you would do when acting on the credences in the face of a practical decision. But any scores generated in this way will fail to satisfy the domination theorem. Here’s why: the scoring rule scores any inconsistent non-negative credence P that is non-zero on some singleton the same way as it scores the consistent credence P^* defined by P^*(A)=∑_ω ∈ AP({ω})/∑_{ω ∈ Ω}P({ω}). Thus, the domination theorem will fail to apply to any scoring rule generated in the above way, since the domination thing does not happen for consistent credences.

The only thing that remains is to check that there is some natural strictly proper rule that can be generated using the above method. Here’s one. Let G_n be the set of simple gambles that assign to the n points of Ω values that lie in the n-dimensional unit ball. In other words, each simple gamble g ∈ G_n is such that ∑_i(g(a_i))² ≤ 1.

A bit of easy constrained maximization using Lagrange multipliers shows that if P is a credence assignment on Ω such that P({ω_i}) ≠ 0 for at least one point ω_i ∈ Ω, then there is a unique maximally auspicious gamble g and it is given by g(ω_j)=P({ω_j})/(∑_i(P({ω_i}))²)^1/2. Because of the uniqueness, we have a strictly proper scoring rule.

The G_n-score of a credence assignment P is then s(P, ω_j)= − P({ω_j})/(∑_i(P({ω_i}))²)^1/2.

This looks fairly natural. The choice of G_n seems fairly natural as well. There is no gerrymandering going on. And yet the domination theorem fails for the G_n-score. (I think any strictly convex set of simple gambles works for G_n, actually.)

Thus, absent some good argument for why G_n-score is a bad way to score credences, it seems that the scoring rule domination argument isn’t persuasive.

More generally, consider any credence-based procedure for deciding between finite sets of gambles that has the following two properties:

1. The procedure yields a gamble that maximizes expected utility in the case of consistent credences, and

2. The procedure never recommends a gamble that is dominated by another gamble.

There are such procedures that apply to interesting classes of inconsistent credences and that are nonetheless pretty natural. Given any such procedure, we can extend it arbitrarily to apply to all inconsistent credences, we assign a score to a credence assignment as the negative of the value of the selected gamble, and we have a proper score to which the domination theorem doesn’t apply. And if make our set of gambles be the n-ball G_n, then the score is strictly proper.

Killing and letting die

1. It is murder to disconnect a patient who can only survive with a ventilator without consent and in order to inherit from them.

2. Every murder is a killing.

3. So, it is a killing to disconnect a patient who can only survive with a ventilator without consent and in order to inherit from them.

4. Whether an act is a killing does not depend on consent or intentions.

5. So, it is a killing to disconnect a patient who can only survive with a ventilator.

Of course, whether such a disconnection is permissible or not is a further question, since not every killing is wrong (e.g., an accidental killing need not be wrong).

Learning whether p by bringing it about that p

Alice is driving to an appointment she doesn’t care much about. She is, however, curious whether she will arrive on time. To satisfy her curiosity, she stops driving, since she knows that if she stops driving, she won’t arrive on time.

It seems a bit perverse to bring it about that p in order to know whether p. Yet there are cases where people do that.

A straightforward family of cases is very pragmatic. You can only make preparations for something if you know what will happen, so you force a particular thing to happen. For instance, you can only book vacation travel when you know where you will decide to go—so, you decide where to go.

One family of cases is linked to anxiety. Not knowing whether p can induce a lot of anxiety, and knowing for sure can relieve that anxiety. This is, presumably, one of the reasons why peopel turn themselves in for crimes: to relieve the anxiety of not knowing whether one will be arrested today, one ensures that one is arrested today.

Another family is scientific. One arranges a laboratory setup in part precisely to know what the experimental setup is like.

But the Alice case is different from all these. In all of the above cases, you seek knowledge whether p for the sake of something other than knowledge whether p: to buy plane tickets, to relieve anxiety, or to learn some other scientific facts.

What seems perverse, then, is to bring it about that p for the sake of knowing whether p for the sake of knowing (“[t]o satisfy her curiosity”, I said of Alice).

I wonder, now, whether Alice is really being perverse. Maybe it’s just this: there are very few things that we can bring about where there is significant non-instrumental value in knowing them. There is very little value in knowing whether one will arrive on time to the appointment apart from instrumental considerations. Most of the things knowledge of which has significant non-instrumental value are out of our hands: theological, philosophical and scientific facts. But if there is very little value, it’s not worth much trouble. If an appointment is of so little value that it’s worth missing it to know whether one will make it on time to it, it’s probably not worth going to in the first place!

Jan 26 Bios Centre Talk: Defining Murder

On January 26, 2021 at 18:30 GMT / 12:30 PM Central / 1:30 PM Eastern, I will be giving a work in progress Zoom talk on Defining Murder at the Bios Centre in London. I will have interesting cases, and various questions, but I don't know if I'll have any good answers.

Everyone is welcome, but you need to contact the organizer to sign up: amccarthy@bioscentre.org.

I can jump 100 feet up in the air

Consider a possible world w₁ which is just like the actual world, except in one respect. In w₁, in exactly a minute, I jump up with all my strength. And then consider a possible world w₂ which is just like w₁, but where moments after I leave the ground, a quantum fluctuation causes 99% of the earth’s mass to quantum tunnel far away. As a result, my jump takes me 100 feet in the air. (Then I start floating down, and eventually I die of lack of oxygen as the earth’s atmosphere seeps away.)

Here is something I do in w₂: I jump 100 feet in the air.

Now, from my actually doing something it follows that I was able to do it. Thus, in w₂, I have the ability to jump 100 feet in the air.

When do I have this ability? Presumably at the moment at which I am pushing myself off from the ground. For that is when I am acting. Once I leave the ground, the rest of the jump is up to air friction and gravity. So my ability to jump 100 feet in the air is something I have in w₂ prior to the catastrophic quantum fluctuation.

But w₁ is just like w₂ prior to that fluctuation. So, in w₁ I have the ability to jump 100 feet in the air. But whatever ability to jump I have in w₁ at the moment of jumping is one that I already had before I decided to jump. And before the decision to jump, world w₁ is just like the actual world. So in the actual world, I have the ability to jump 100 feet in the air.

Of course, my success in jumping 100 feet depends on quantum events turning out a certain way. But so does my success in jumping one foot in the air, and I would surely say that I have the ability to jump one foot. The only principled difference is that in the one foot case the quantum events are very likely to turn out to be cooperative.

The conclusion is paradoxical. What are we to make of it? I think it’s this. In ordinary language, if something is really unlikely, we say it’s impossible. Thus, we say that it’s impossible for me to beat Kasparov at chess. Strictly speaking, however, it’s quite possible, just very unlikely: there is enough randomness in my very poor chess play that I could easily make the kinds of moves Deep Blue made when it beat him. Similarly, when my ability to do something has extremely low reliability, we simply say that I do not have the ability.

One might think that the question of whether one is able to do something is really important for questions of moral responsibility. But if I am right in the above, then it’s not. Imagine that I could avert some tragedy only by jumping 100 feet in the air. I am no more responsible for failing to avert that tragedy than if the only way to avert it would be by squaring a circle. Yet I can jump 100 feet in the air, while no one can square a circle.

It seems, thus, that what matters for moral responsibility is not so much the answer to the question of whether one can do something, but rather answers to questions like:

1. How reliably can one do it?

2. How reliably does one think (or justifiably think or know) one can do it?

3. What would be the cost of doing it?

Sheep in sheep's clothing

Suppose you know the following facts. In County X, about 40% of sheep wear sheep costumes. There is also the occasional trickster who puts a sheep costume on a dog, but that’s really rare: so rare that 99.9% of animals that look like sheep are sheep, most of them being ordinary sheep but a large minority being sheep dressed up as sheep.

You know you’re in County X, and you come across a field with an animal that looks like a sheep. There are three possibilities:

1. It’s an ordinary sheep. Probability: 59.94%

2. It’s a sheep in sheep costume. Probability: 40.06%

3. It’s some other animal in sheep costume. Probability: 0.10%.

You’re justified in believing that (1) or (2) is the case, i.e., that the animal is a sheep. And if it turns out that you’re right, then I take it you know that it’s a sheep. You know this regardless of whether it’s an ordinary sheep or a sheep in sheep costume.

But now consider County Y which is much more like the real world. You know that in County Y, only about 0.1% of sheep wear sheep costumes. And there is the occasional trickster who puts a sheep costume on a dog. In County Y, once again, 99.9% of animals that look like sheep are sheep, and 99.9% of those are ordinary sheep without sheep’s costumes.

Now you know you’re in County Y and you come across an animal that looks like a sheep. You have three possibilities again, but with different probabilities:

4. It’s an ordinary sheep. Probability: 99.80%

5. It’s a sheep in sheep costume. Probability: 0.10%.

6. It’s some other animal in sheep costume. Probability: 0.10%.

In any case, the probability that it’s a sheep of some sort is 99.9%. It seems to me that just as in County X, in County Y you know that what you’re facing is a sheep regardless of whether it’s an ordinary sheep or a sheep in sheep costume.

But if what you’re facing is a sheep dressed up as a sheep, then you are in something very much like a standard Gettier case. So in some standard Gettier cases, if you reason probabilistically, it is possible to know.