TL;DR: If you define “badness” as 1 - V, then a horizontal shear means that instead of optimizing for Goodness, you optimize for Goodness/Badness (with an affine transform of Badness). The JB rotation is where you take the ratio and affine transform both Goodness and Badness.
After a JB rotation of 45 degrees like dealt with in the post, the expected value will be
where EV is the pre rotation value. This is monotonic, except for the part where it blows up at EV = 1 and swaps sign as you cross it (corresponding to the problem of a vertical vector). [1]
You can check that it gives the right decision rule because if you compare two fractions, you’ll get
only when , or equivalently . For the same reason, if you decide between probability distributions (“gambles” or “lotteries”) then you get the same thing.
This suggests the following transformation:
that is, just a horizontal shear (the vertical shear corresponds to adding a constant to the utility function). This is chosen so that the expected values are
If you add in a factor to allow for all shears, then you can use the fact that you can get any rotation in three shears (via HVH for some V,H vertical and horizontal shears) to get any JB rotations this way.
What’s going on here? Well with a normalization of the values, the quantity 1 - EV could be interpreted as “badness” or “imperfection”. You can still call it “badness” even with non-normalized values, but it sounds nicer if they do actually add to 1. [2]
A horizontal shear converts from optimizing for Goodness to optimizing for Goodness/Badness. Basically, the P here is keeping track of the normalizing factor needed to scale Q into an expected utility—so if you scale P, then you can take ratios of expected utility. Thus to get the ratio Goodness/Badness, you need to scale P in the opposite way—that is, you multiply by Badness. So our agent regards badder things as more likely, though note that the values/shouldnesses change differently (in that shouldnes doesn’t change).
If you take a different scale factor for the shear, then you are just affine transforming badness (because ).
What’s a Jeffrey Bolker rotation? It’s when you affine transform both goodness and badness, and look at the ratio.
As a final note: Our good poet Robert Frost doesn’t have to be an evidential decision theorist. He could instead look at P(X|do(A)) and Q(X|do(A)) = P(X|do(A)) V(X), where do(A) can mean the causal counterfactual, the logical counterfactual, or whatever else you want in some new decision theory. If you have an intrinsic cost/reward for doing an action, you can put it in a two-argument function V(X|do(A)) = V(X,A). Note that in the evidential case, you can replace the conditional probabilities with joint probabilities, because the ratio is all that matters anyways; so either way you really have joint/two argument probabilities/values/shouldness.
By the by, you may think “hyperbola!” and I think you’d be right: (P + Q)/(P—Q) definitely looks like some of the hyperbolic rotation stuff… except the ratio is not actually preserved, it’s just the ordering that is. But, well, it is suspicious, and hyperbolas do involve products, and other similar quantities, so I get the sense there’s something here (besides the fact that the new value is literally tracing out a hyperbola).
An alternative that is not nicer in my opinion: 1/V_new = 1/V_old − 1, that is, you are taking the reciprocals and shifting to get a new reciprocal utility. If the values were in the range [0,1] then the reciprocals would be in the range [1,infinity], so subtracting one brings us back to [0,infinity], which reciprocates to [0,infinity] again. That’s sorta a justification for subtracting 1.
Aha! (1+V)/(1-V) looks like it’s decaying to −1 at negative V, and blows up at V=1 - meaning it’s risk loving. So we went from risk neutral to risk loving, and to compensate we made bad outcomes more likely.
That is, the analogy to the risk neutral measure seems to check out: a JB rotation increases your risk tolerance, and compensates by increasing the probability of the bad outcomes.
Specifically, if you take the derivatives you get 2/(1-V)^2 V’ for the first and 2/(1-V)^2 (V″ + 2V’^2) for the second. The (negative) ratio is -V″/V’ − 2 V’, and -V″/V’ is often used as a measure of risk aversion. If instead we divide by V’^2 we’d get -V″/V’^2 − 2. A log utility a + bln(x) has a constant value of -V″/V’^2 equal to 1/b, so this is saying that we reduce this measurement of risk aversion by a constant.
I think you should be able to write the risk neutral measure from finance this way, but it’s not obvious to me how to do so, and I don’t know enough mathematical finance to easily deconfuse myself here.
We can also go backwards. So, basically, if M is “money”, then a risk averse utility of M/(1+M) with probability (1+M) P will take the same actions as someone with utility linear in money.
TL;DR: If you define “badness” as 1 - V, then a horizontal shear means that instead of optimizing for Goodness, you optimize for Goodness/Badness (with an affine transform of Badness). The JB rotation is where you take the ratio and affine transform both Goodness and Badness.
After a JB rotation of 45 degrees like dealt with in the post, the expected value will be
where EV is the pre rotation value. This is monotonic, except for the part where it blows up at EV = 1 and swaps sign as you cross it (corresponding to the problem of a vertical vector). [1]
You can check that it gives the right decision rule because if you compare two fractions, you’ll get
only when
This suggests the following transformation:
that is, just a horizontal shear (the vertical shear corresponds to adding a constant to the utility function). This is chosen so that the expected values are
If you add in a factor to allow for all shears, then you can use the fact that you can get any rotation in three shears (via HVH for some V,H vertical and horizontal shears) to get any JB rotations this way.
What’s going on here? Well with a normalization of the values, the quantity 1 - EV could be interpreted as “badness” or “imperfection”. You can still call it “badness” even with non-normalized values, but it sounds nicer if they do actually add to 1. [2]
A horizontal shear converts from optimizing for Goodness to optimizing for Goodness/Badness. Basically, the P here is keeping track of the normalizing factor needed to scale Q into an expected utility—so if you scale P, then you can take ratios of expected utility. Thus to get the ratio Goodness/Badness, you need to scale P in the opposite way—that is, you multiply by Badness. So our agent regards badder things as more likely, though note that the values/shouldnesses change differently (in that shouldnes doesn’t change).
If you take a different scale factor for the shear, then you are just affine transforming badness (because
What’s a Jeffrey Bolker rotation? It’s when you affine transform both goodness and badness, and look at the ratio.
As a final note: Our good poet Robert Frost doesn’t have to be an evidential decision theorist. He could instead look at P(X|do(A)) and Q(X|do(A)) = P(X|do(A)) V(X), where do(A) can mean the causal counterfactual, the logical counterfactual, or whatever else you want in some new decision theory. If you have an intrinsic cost/reward for doing an action, you can put it in a two-argument function V(X|do(A)) = V(X,A). Note that in the evidential case, you can replace the conditional probabilities with joint probabilities, because the ratio is all that matters anyways; so either way you really have joint/two argument probabilities/values/shouldness.
By the by, you may think “hyperbola!” and I think you’d be right: (P + Q)/(P—Q) definitely looks like some of the hyperbolic rotation stuff… except the ratio is not actually preserved, it’s just the ordering that is. But, well, it is suspicious, and hyperbolas do involve products, and other similar quantities, so I get the sense there’s something here (besides the fact that the new value is literally tracing out a hyperbola).
An alternative that is not nicer in my opinion: 1/V_new = 1/V_old − 1, that is, you are taking the reciprocals and shifting to get a new reciprocal utility. If the values were in the range [0,1] then the reciprocals would be in the range [1,infinity], so subtracting one brings us back to [0,infinity], which reciprocates to [0,infinity] again. That’s sorta a justification for subtracting 1.
Aha! (1+V)/(1-V) looks like it’s decaying to −1 at negative V, and blows up at V=1 - meaning it’s risk loving. So we went from risk neutral to risk loving, and to compensate we made bad outcomes more likely.
That is, the analogy to the risk neutral measure seems to check out: a JB rotation increases your risk tolerance, and compensates by increasing the probability of the bad outcomes.
Specifically, if you take the derivatives you get 2/(1-V)^2 V’ for the first and 2/(1-V)^2 (V″ + 2V’^2) for the second. The (negative) ratio is -V″/V’ − 2 V’, and -V″/V’ is often used as a measure of risk aversion. If instead we divide by V’^2 we’d get -V″/V’^2 − 2. A log utility a + bln(x) has a constant value of -V″/V’^2 equal to 1/b, so this is saying that we reduce this measurement of risk aversion by a constant.
I think you should be able to write the risk neutral measure from finance this way, but it’s not obvious to me how to do so, and I don’t know enough mathematical finance to easily deconfuse myself here.
We can also go backwards. So, basically, if M is “money”, then a risk averse utility of M/(1+M) with probability (1+M) P will take the same actions as someone with utility linear in money.