There's a game theory concept for this called Shapley value which allocates credit in a way that does sum to 1, but takes into account a bunch of people each independently being required for all of the impact to happen.

Consider some project worked on by multiple organisations A, B, C and D. The benefit of the project is x. Each of the organisations is a necessary condition of the benefit x. The counterfactual impact of A is x; the counterfactual impact of B is x; etc. Despite this, the counterfactual impact of A, B, C, and D acting together is not 4*x, rather it is x.

In this example, Shapley value would give everyone x/4 of the credit, adding together to x as we'd naturally expect.

The alternative approach (which I argue is wrong) is to say that each of the n A voters is counterfactually responsible for 1/n of the $10bn benefit. Suppose there are 10m A voters. Then each A voterâ€™s counterfactual social impact is 1/10m$10bn = $1000. But on this approach the common EA view that it is rational for individuals to vote as long as the probability of being decisive is not too small, is wrong. Suppose the ex ante chance of being decisive is 1/1m. Then the expected value of Emma voting is a mere 1/1m$1000 = $0.001. On the correct approach, the expected value of Emma voting is 1/10m*$10bn = $1000. If voting takes 5 minutes, this is obviously a worthwhile investment for the benevolent voter, as per common EA wisdom.

This doesn't really seem like an argument to me... it seems like you start from the premise that voting must be rational, and that something like Shapley value would make it irrational, and thus that can't be the case. But this seems to me to be assuming the conclusion?

I guess this is a case where the expected value of the action and the Shapley value are not the same, because in one case you're analyzing an entire system and in the other case you're analyzing the individual action. But just as it may seem weird that Shapley value assigns such a small value to each vote, the expected value approach is basically saying that the first few votes have a value of literally 0, which also seems nonsensical to me.

Comment author:Halstead
27 May 2018 04:04:14PM
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1 point
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It needs to be explained why there is a paradox. I have not yet seen an explanation of why there might be thought to be one. EAs are concerned with having counterfactual impact. If you were a necessary condition of some benefit B occurring, then you have had counterfactual impact.

Re voting I'm appealing to how almost everyone in the academic literature assesses the expected value of voting, which is not by dividing the total value by each voter. I'm also appealing to a common EA idea which is discussed by Parfit and mentioned in Will's book, which is that voting is sometimes rational for altruistic voters. On your approach, it would pretty much always be irrational to vote even if the social benefits were extremely large: every social benefit would always be divided by the number of decisive voters, and so would be divided by many millions in any large election

I don't understand why the expected value approach says that the first few votes have a value of 0. Also, the ordering in which votes are cast is completely irrelevant to judging a voter's counterfactual imapct because all votes are indistinguishable wrt causing the outcome: it doesn't matter if I voted first and Emma voted last, we would still be decisive voters.

Comment author:rohinmshah
27 May 2018 07:08:23PM
4 points
[-]

It's not a paradox. The problem is just that, if everyone thought this way, we would get suboptimal outcomes -- so maybe we should figure out how to avoid that.

Suppose there are three possible outcomes:
P has cost $2000 and gives 15 utility to the world
Q has cost $1000 and gives 10 utility to the world
R has cost $1000 and gives 10 utility to the world

Suppose Alice and Bob each have $1000 to donate. Consider two scenarios:

Scenario 1: Both Alice and Bob give $1000 to P. The world gets 15 more utility. Both Alice and Bob are counterfactually responsible for giving 15 utility to the world.

Scenario 2: Alice gives $1000 to Q and Bob gives $1000 to R. The world gets 20 more utility. Both Alice and Bob are counterfactually responsible for giving 10 utility to the world.

From the world's perspective, scenario 2 is better. However, from Alice and Bob's individual perspective (if they are maximizing their own counterfactual impact), scenario 1 is better. This seems wrong, we'd want to somehow coordinate so that we achieve scenario 2 instead of scenario 1.

Are you neglecting to count the negative impact from causing other people to do the suboptimal thing? If I use my funds to set up an exploding matching grant that will divert the funds of other donors from better things too a less effective charity, that is a negative part of my impact.

Comment author:rohinmshah
29 May 2018 05:13:07PM
2 points
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Yes, that's right. I agree that a perfect calculation of your counterfactual impact would do the right thing in this scenario, and probably all scenarios. This is an empirical claim that the actual impact calculations that meta-orgs do are of the form that I wrote in my previous comment.

For example, consider the impact calculations that GWWC and other meta orgs have. If those impact calculations (with their current methodologies) showed a ratio of 1.1:1, that seems nominally worthwhile (you still have the multiplicative impact), but I would expect that it would be better to give directly to charities to avoid effects like the ones Joey talked about in his post.

A true full counterfactual impact calculation would consider the world in which GWWC just sends the money straight to charities and convinces other meta orgs to do the same, at which point they see that more money gets donated to charities in total, and so they all close operations and send money straight to charities. I'm arguing that this doesn't happen in practice. (I think Joey and Peter are arguing the same thing.)

Comment author:Halstead
28 May 2018 11:04:46AM
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1 point
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Firstly, people who believe in the correct account of counterfactual impact would have incentives to coordinate in the case you outline. Alice would maximise her counterfactual impact (defined as I define it) by coordinating with Bob on project R. The counterfactual impact of her coordinating with Bob would be +5 utility compared to scenario 1. There is no puzzle here.

Secondly, dividing counterfactual impact by contribution does not solve all these coordination problems. If everyone thought as per the Shapely value, then no rational altruists would ever vote, even when the true theory dictates that the expected value of doing so was very high.

Also consider the $1bn benefits case outlined above. Suppose that the situation is as described above but my action costs $2 and I take one billionth of the credit for the success of the project. In that case, the Shapely-adjusted benefits of my action would be $1 and the costs $2, so my action would not be worthwhile. I would therefore leave $1bn of value on the table.

Comment author:rohinmshah
29 May 2018 06:28:50PM
1 point
[-]

For the first point, see my response to Carl above. I think you're right in theory, but in practice it's still a problem.

For the second point, I agree with Flodorner that you would either use the Shapley value, or you would use the probability of changing the outcome, not both. I don't know much about Shapley values, but I suspect I would agree with you that they are suboptimal in many cases. I don't think there is a good theoretical solution besides "consider every possible outcome and choose the best one" which we obviously can't do as humans. Shapley values are one tractable way of attacking the problem without having to think about all possible worlds, but I'm not surprised that there are cases where they fail. I'm advocating for "think about this scenario", not "use Shapley values".

I think the $1bn benefits case is a good example of a pathological case where Shapley values fail horribly (assuming they do what you say they do, again, I don't know much about them).

My overall position is something like "In the real world when we can't consider all possibilities, one common failure mode in impact calculations is the failure to consider the scenario in which all the participants who contributed to this outcome instead do other altruistic things with their money".

Comment author:Flodorner
29 May 2018 08:51:16PM
0 points
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At this point, i think that to analyze the $1bn case correctly, you'd have to substract everyone's opportunity cost in the calculation of the shapley value (if you want to use it here). This way, the example should yield what we expect.

I might do a more general writeup about shapley values, their advantages, disadvantages and when it makes sense to use them, if i find the time to read a bit more about the topic first.

## Comments (20)

Best*2 points [-]There's a game theory concept for this called Shapley value which allocates credit in a way that does sum to 1, but takes into account a bunch of people each independently being required for all of the impact to happen.

In this example, Shapley value would give everyone x/4 of the credit, adding together to x as we'd naturally expect.

This doesn't really seem like an argument to me... it seems like you start from the premise that voting must be rational, and that something like Shapley value would make it irrational, and thus that can't be the case. But this seems to me to be assuming the conclusion?

I guess this is a case where the expected value of the action and the Shapley value are not the same, because in one case you're analyzing an entire system and in the other case you're analyzing the individual action. But just as it may seem weird that Shapley value assigns such a small value to each vote, the expected value approach is basically saying that the first few votes have a value of literally 0, which also seems nonsensical to me.

*1 point [-]It needs to be explained why there is a paradox. I have not yet seen an explanation of why there might be thought to be one. EAs are concerned with having counterfactual impact. If you were a necessary condition of some benefit B occurring, then you have had counterfactual impact.

Re voting I'm appealing to how almost everyone in the academic literature assesses the expected value of voting, which is not by dividing the total value by each voter. I'm also appealing to a common EA idea which is discussed by Parfit and mentioned in Will's book, which is that voting is sometimes rational for altruistic voters. On your approach, it would pretty much always be irrational to vote even if the social benefits were extremely large: every social benefit would always be divided by the number of decisive voters, and so would be divided by many millions in any large election

I don't understand why the expected value approach says that the first few votes have a value of 0. Also, the ordering in which votes are cast is completely irrelevant to judging a voter's counterfactual imapct because all votes are indistinguishable wrt causing the outcome: it doesn't matter if I voted first and Emma voted last, we would still be decisive voters.

It's not a paradox. The problem is just that, if everyone thought this way, we would get suboptimal outcomes -- so maybe we should figure out how to avoid that.

Suppose there are three possible outcomes: P has cost $2000 and gives 15 utility to the world Q has cost $1000 and gives 10 utility to the world R has cost $1000 and gives 10 utility to the world

Suppose Alice and Bob each have $1000 to donate. Consider two scenarios:

Scenario 1: Both Alice and Bob give $1000 to P. The world gets 15 more utility. Both Alice and Bob are counterfactually responsible for giving 15 utility to the world.

Scenario 2: Alice gives $1000 to Q and Bob gives $1000 to R. The world gets 20 more utility. Both Alice and Bob are counterfactually responsible for giving 10 utility to the world.

From the world's perspective, scenario 2 is better. However, from Alice and Bob's individual perspective (if they are maximizing their own counterfactual impact), scenario 1 is better. This seems wrong, we'd want to somehow coordinate so that we achieve scenario 2 instead of scenario 1.

Are you neglecting to count the negative impact from causing other people to do the suboptimal thing? If I use my funds to set up an exploding matching grant that will divert the funds of other donors from better things too a less effective charity, that is a negative part of my impact.

Yes, that's right. I agree that a perfect calculation of your counterfactual impact would do the right thing in this scenario, and probably all scenarios. This is an empirical claim that the actual impact calculations that meta-orgs do are of the form that I wrote in my previous comment.

For example, consider the impact calculations that GWWC and other meta orgs have. If those impact calculations (with their current methodologies) showed a ratio of 1.1:1, that seems nominally worthwhile (you still have the multiplicative impact), but I would expect that it would be better to give directly to charities to avoid effects like the ones Joey talked about in his post.

A true full counterfactual impact calculation would consider the world in which GWWC just sends the money straight to charities and convinces other meta orgs to do the same, at which point they see that more money gets donated to charities in total, and so they all close operations and send money straight to charities. I'm arguing that this doesn't happen in practice. (I think Joey and Peter are arguing the same thing.)

*1 point [-]Firstly, people who believe in the correct account of counterfactual impact would have incentives to coordinate in the case you outline. Alice would maximise her counterfactual impact (defined as I define it) by coordinating with Bob on project R. The counterfactual impact of her coordinating with Bob would be +5 utility compared to scenario 1. There is no puzzle here.

Secondly, dividing counterfactual impact by contribution does not solve all these coordination problems. If everyone thought as per the Shapely value, then no rational altruists would ever vote, even when the true theory dictates that the expected value of doing so was very high.

Also consider the $1bn benefits case outlined above. Suppose that the situation is as described above but my action costs $2 and I take one billionth of the credit for the success of the project. In that case, the Shapely-adjusted benefits of my action would be $1 and the costs $2, so my action would not be worthwhile. I would therefore leave $1bn of value on the table.

For the first point, see my response to Carl above. I think you're right in theory, but in practice it's still a problem.

For the second point, I agree with Flodorner that you would either use the Shapley value, or you would use the probability of changing the outcome, not both. I don't know much about Shapley values, but I suspect I would agree with you that they are suboptimal in many cases. I don't think there is a good theoretical solution besides "consider every possible outcome and choose the best one" which we obviously can't do as humans. Shapley values are one tractable way of attacking the problem without having to think about all possible worlds, but I'm not surprised that there are cases where they fail. I'm advocating for "think about this scenario", not "use Shapley values".

I think the $1bn benefits case is a good example of a pathological case where Shapley values fail horribly (assuming they do what you say they do, again, I don't know much about them).

My overall position is something like "In the real world when we can't consider all possibilities, one common failure mode in impact calculations is the failure to consider the scenario in which

allthe participants who contributed to this outcome instead do other altruistic things with their money".At this point, i think that to analyze the $1bn case correctly, you'd have to substract everyone's opportunity cost in the calculation of the shapley value (if you want to use it here). This way, the example should yield what we expect.

I might do a more general writeup about shapley values, their advantages, disadvantages and when it makes sense to use them, if i find the time to read a bit more about the topic first.

^ This is what I wanted to say, but even better than how I was going to say it.