Each theory-agent Aj is endowed with a share of the decision maker’s resources in χi proportional to the decision maker’s credence at the time of χn in the corresponding moral theory Tj .
So, that, effectively, resource shares always track credences in theories. Wouldn’t this be unfair to the theories/theory-agents who invest disproportionately in growing or preserving their resources?
The theories/theory-agents could coordinate to save and invest to avoid tragedy of the commons, but some may still be more interested in saving/investing than others, and will effectively be forced to transfer wealth to those that spend more. Some may even be willing to put themselves in net debt, but this debt could be partially transferred to others.
Like in your example, I also don’t think we’d want to leave all of the gains a theory-agent made with that theory-agent if their theory lost almost all of its support. Taking only from their initial endowment doesn’t seem to go far enough.
There are probably different ways to redistribute resources after credences change, but I imagine they’ll all be problematic or at least have weird consequences, even if not objectionable per se.
Some ideas and problems in this section of the comment.
(1) The approach that seemed at first most obvious to me is to rescale resources proportionally with the relative changes to credences at each step:
Suppose your current shares of resources are s1,s2,...,sn across theories T1,T2,...,Tn. These should sum to 1. They need not necessarily be nonnegative, in case we want to allow net debts. Suppose your credences have been multiplied by f1,f2,...,fn since you last checked and redistributed. Then, you would want to distribute resources proportionally to the products f1s1,f2s2,...,fnsn. This may not sum to 1, so you have to normalize by the dot product, f1s1+f2s2+...+fnsn to get the actual shares.
I expect the renormalization to lead to problems. If your credences in T1 increases, and your credence in T2 decreases by the same in absolute terms, but your credence in T3 is unaffected, it seems weird and potentially problematic for this to affect T3’s resources, especially if it means losing resources.
Also, if a theory with net negative resources (net debt) has credence in it increased, it shouldn’t be forced further into debt, and it should normally gain resources on net. In those cases, it’s not clear what we should do and why. We could just replace fj with 1/fj or 1, before renormalizing.
(2) Instead then, we might adjust resource shares identically with our absolute credence adjustments, a1,an,...,an, i.e. if the credence in theory j has changed in absolute terms from cj to cj+aj, theory j’s resources should change from sj to sj+aj. Because a1+a2+...+an=0 (including if we add or drop theories), we wouldn’t need to renormalize, and theories whose credences don’t change aren’t affected. This seems more natural and less ad hoc than the first approach.
But, we may sometimes have sj+aj<0, even with cj+aj>0,sj>0, because the theory-agent has been spending down its resources or growing them less than others, and a theory with net positive resources and still positive credence in it is forced into debt or to give up the rest of its resources, which doesn’t seem fair. It’s not clear how much it should lose.
(3) Another approach would be to let the theory-agents decide ahead of time how resources should be redistributed with changes in credences, by some kind of vote (moral parliament). We don’t need to decide this for them. But this could disproportionately favour more popular views (or coalitions of views). You could have them vote as if they don’t know the credences in and resources with each theory (or as if they assume they’d only get a small fraction of the resources) and impartially across theories to prevent some biased rules and tyranny of the majority.
(4) Or we could use ad hoc fixes to the first two and hack them together. In (2), don’t let a theory that had net positive resources go into debt because of credence changes, but allow it to hit 0, and just redistribute less to the theories that should have gained, in proportion to how much they should have gained (from the theory that hit 0, ideally). Then we could average this with some version of (1).
At a minimum, I’d probably want:
If the credence in a theory hasn’t decreased, it shouldn’t lose resources when redistributing.
In particular, if a theory-agent puts itself into net debt, it shouldn’t get to transfer that debt to others just because the credence in that theory decreased. Or, maybe this should be an exception to the rule, but it could incentivize debt-taking in a bad way.
A theory shouldn’t be put into debt or left with no resources after redistribution with credence changes, if it had positive net resources just before and still has positive overall credence in it after.
Changes in resources should scale with the changes in credences and go in the same direction (or not change, in some cases). Substantial changes in credences should normally lead to substantial changes in resources, and small changes in credences should lead to small (or 0) changes in resources. If a theory had positive resources before redistribution and lost (almost) all credence in it, it should transfer (almost) all of its resources to others (maybe paying debts to other theory-agents first?).
In the paper, you write:
So, that, effectively, resource shares always track credences in theories. Wouldn’t this be unfair to the theories/theory-agents who invest disproportionately in growing or preserving their resources?
The theories/theory-agents could coordinate to save and invest to avoid tragedy of the commons, but some may still be more interested in saving/investing than others, and will effectively be forced to transfer wealth to those that spend more. Some may even be willing to put themselves in net debt, but this debt could be partially transferred to others.
Like in your example, I also don’t think we’d want to leave all of the gains a theory-agent made with that theory-agent if their theory lost almost all of its support. Taking only from their initial endowment doesn’t seem to go far enough.
There are probably different ways to redistribute resources after credences change, but I imagine they’ll all be problematic or at least have weird consequences, even if not objectionable per se.
Some ideas and problems in this section of the comment.
(1) The approach that seemed at first most obvious to me is to rescale resources proportionally with the relative changes to credences at each step:
Suppose your current shares of resources are s1,s2,...,sn across theories T1,T2,...,Tn. These should sum to 1. They need not necessarily be nonnegative, in case we want to allow net debts. Suppose your credences have been multiplied by f1,f2,...,fn since you last checked and redistributed. Then, you would want to distribute resources proportionally to the products f1s1,f2s2,...,fnsn. This may not sum to 1, so you have to normalize by the dot product, f1s1+f2s2+...+fnsn to get the actual shares.
I expect the renormalization to lead to problems. If your credences in T1 increases, and your credence in T2 decreases by the same in absolute terms, but your credence in T3 is unaffected, it seems weird and potentially problematic for this to affect T3’s resources, especially if it means losing resources.
Also, if a theory with net negative resources (net debt) has credence in it increased, it shouldn’t be forced further into debt, and it should normally gain resources on net. In those cases, it’s not clear what we should do and why. We could just replace fj with 1/fj or 1, before renormalizing.
(2) Instead then, we might adjust resource shares identically with our absolute credence adjustments, a1,an,...,an, i.e. if the credence in theory j has changed in absolute terms from cj to cj+aj, theory j’s resources should change from sj to sj+aj. Because a1+a2+...+an=0 (including if we add or drop theories), we wouldn’t need to renormalize, and theories whose credences don’t change aren’t affected. This seems more natural and less ad hoc than the first approach.
But, we may sometimes have sj+aj<0, even with cj+aj>0,sj>0, because the theory-agent has been spending down its resources or growing them less than others, and a theory with net positive resources and still positive credence in it is forced into debt or to give up the rest of its resources, which doesn’t seem fair. It’s not clear how much it should lose.
(3) Another approach would be to let the theory-agents decide ahead of time how resources should be redistributed with changes in credences, by some kind of vote (moral parliament). We don’t need to decide this for them. But this could disproportionately favour more popular views (or coalitions of views). You could have them vote as if they don’t know the credences in and resources with each theory (or as if they assume they’d only get a small fraction of the resources) and impartially across theories to prevent some biased rules and tyranny of the majority.
(4) Or we could use ad hoc fixes to the first two and hack them together. In (2), don’t let a theory that had net positive resources go into debt because of credence changes, but allow it to hit 0, and just redistribute less to the theories that should have gained, in proportion to how much they should have gained (from the theory that hit 0, ideally). Then we could average this with some version of (1).
At a minimum, I’d probably want:
If the credence in a theory hasn’t decreased, it shouldn’t lose resources when redistributing.
In particular, if a theory-agent puts itself into net debt, it shouldn’t get to transfer that debt to others just because the credence in that theory decreased. Or, maybe this should be an exception to the rule, but it could incentivize debt-taking in a bad way.
A theory shouldn’t be put into debt or left with no resources after redistribution with credence changes, if it had positive net resources just before and still has positive overall credence in it after.
Changes in resources should scale with the changes in credences and go in the same direction (or not change, in some cases). Substantial changes in credences should normally lead to substantial changes in resources, and small changes in credences should lead to small (or 0) changes in resources. If a theory had positive resources before redistribution and lost (almost) all credence in it, it should transfer (almost) all of its resources to others (maybe paying debts to other theory-agents first?).