[MIRIx Cambridge MA] Limiting resource allocation with bounded utility functions and conceptual uncertainty

Vika

This is a result from the first MIRIx Cambridge workshop (coauthored with Janos and Jim).

One potential problem with bounded utility functions is: what happens when the bound is nearly reached? A bounded utility maximizer will get progressively more and more risk averse as it gets closer to its bound. We decided to investigate what risks it might fear. We used a toy model with a bounded-utility chocolate maximizer, and considered what happens to its resource allocation in the limit as resources go to infinity.

We use "chocolate maximizer'' as conceptual shorthand meaning an agent that we model as though it has a single simple value with a positive long-run marginal resource cost, but only as a simplifying assumption. This is as opposed to a paperclip maximizer, where the inappropriate simplicity is implied to be part of the world, not just part of the model.

Conceptual uncertainty

We found that if a bounded utility function approaches its bound too fast, this has surprising pathological results when mixed with logical uncertainty. Consider a bounded-utility chocolate maximizer, with philosophical uncertainty about what chocolate is. It has a central concept of chocolate $C_1$ , and there are classes of mutated versions of the concept of chocolate $C_2, C_3, \dots, C_n$ at varying distances from the central concept, such that the probability that the true chocolate is in class $C_i$ is proportional to $i^{-\alpha}$ (i.e. following a power law).

Suppose also that utility is bounded using a sigmoid function $U(x)=S(x)$ , where x is the amount of chocolate produced. In the limit as resources go to infinity, what fraction of those resources will be spent on the central class $C_1$ ? That depends which sigmoid function is used, and in particular, how quickly it approaches the utility bound.

Example 1: exponential sigmoid

Suppose we allocate $r_{i}$ resources to class $C_i$ , with $\sum_{i}r_{i}=r$ for total resource r. Let $U(r_i) = 1-e^{-r_{i}}$ .

Then the optimal resource allocation is

$\begin{align*} \vec{r} & =\arg\max\sum_{i=1}^{n}P(C_{i})U(r_{i})\\ & =\arg\max\sum_{i=1}^{n}ci^{-\alpha}(1-e^{-r_{i}}) \end{align*}$

Using Lagrange multipliers, we obtain for all i,

$\begin{align*} 0 & =\frac{\partial}{\partial r_{i}}\left[\sum_{i=1}^{n}ci^{-\alpha}(1-e^{-r_{i}})+\lambda(\sum_{i=1}^{n}r_{i}-r)\right]\\ & =-ci^{-\alpha}e^{-r_{i}}+\lambda \end{align*}$

Then,

$\begin{align*} r_{i} & =-\alpha\log i+(\log c-\log\lambda)\\ r & =-\alpha\sum_{j=1}^{n}\log j+n(\log c-\log\lambda)\\ \frac{1}{n}(r+\alpha\sum_{j=1}^{n}\log j) & =\log c-\log\lambda\\ r_{i} & =-\alpha\log i+\frac{1}{n}\left(r+\alpha\sum_{j=1}^{n}\log j\right)\\ \frac{r_{i}}{r} & =\frac{1}{n}+\frac{\alpha}{r}\left(\frac{1}{n} \sum_{j=1}^{n}\log j-\log i\right)\\ \lim_{r\rightarrow\infty}\frac{r_{i}}{r} & =\frac{1}{n} \end{align*}$

Thus, the resources will be evenly distributed among all the classes as r increases. This is bad, because the resource fraction for the central class $C_1$ goes to 0 as we increase the number of classes.

EDITED: Addendum on asymptotics

Since we have both r and n going to infinity, we can specify their relationship more precisely. We assume that n is the highest number of classes that are assigned nonnegative resources for a given value of r:

$\frac{1}{n}+\frac{\alpha}{r}\left(\frac{1}{n} \sum_{j=1}^{n}\log j-\log n\right) \geq 0$

$\begin{align*} \log n &\leq \frac{r}{n\alpha}+ \frac{1}{n}\sum_{j=1}^{n}\log j \\ &= \frac{r}{n\alpha}+\frac{1}{n}\log(n!)\\ &\approx \frac{r}{n\alpha}+ \log n - 1 + \frac{1}{n}O(\log n)\\ \end{align*}$

Thus,

$\frac{1}{n}\left(\frac{r}{\alpha} + O(\log n)\right)\geq 1$

so the highest class index that gets nonnegative resources satisfies

$n-O(\log n)\approx \frac{r}{\alpha}$

Example 2: arctan sigmoid

Let $U(r_i) = \arctan(r_{i})$ .

The optimal resource allocation is

$\begin{align*} \vec{r} & =\arg\max\sum_{i=1}^{n}P(C_{i})U(r_{i})\\ & =\arg\max\sum_{i=1}^{n}ci^{-\alpha}\arctan(r_{i}) \end{align*}$

Using Lagrange multipliers, we obtain for all i,

$\begin{align*} 0 & =\frac{\partial}{\partial r_{i}}\left[\sum_{i=1}^{n}ci^{-\alpha}\arctan(r_{i})-\lambda(\sum_{i=1}^{n}r_{i}-r)\right]\\ & =ci^{-\alpha}\frac{1}{1+r_{i}^{2}}-\lambda \end{align*}$

Then,

$\begin{align*} r_{i} & =\sqrt{\frac{c}{\lambda}i^{-\alpha}-1}\approx\sqrt{\frac{c}{\lambda}}i^{-\alpha/2}\\ r & =\sqrt{\frac{c}{\lambda}}\sum_{i=1}^{n}i^{-\alpha/2}<\sqrt{\frac{c}{\lambda}}\zeta(\alpha/2)\\ \frac{r_{i}}{r} & >\frac{i^{-\alpha/2}}{\zeta(\alpha/2)}\\ \frac{r_{0}}{r} & >\frac{1}{\zeta(\alpha/2)} \end{align*}$

Thus, for $\alpha>2$ the limit of the resource fraction for the central class $C_1$ is finite and positive.

Conclusion

The arctan sigmoid results in a better limiting resource allocation than the exponential sigmoid, because it has heavier tails (for sufficiently large $\alpha$ ). Thus, it matters which bounding sigmoid function you choose.

This is a result from the first MIRIx Cambridge workshop (coauthored with Janos and Jim).

Conceptual uncertainty

Example 1: exponential sigmoid

Suppose we allocate $r_{i}$ resources to class $C_i$ , with $\sum_{i}r_{i}=r$ for total resource r. Let $U(r_i) = 1-e^{-r_{i}}$ .

Then the optimal resource allocation is

$\begin{align*} \vec{r} & =\arg\max\sum_{i=1}^{n}P(C_{i})U(r_{i})\\ & =\arg\max\sum_{i=1}^{n}ci^{-\alpha}(1-e^{-r_{i}}) \end{align*}$

Using Lagrange multipliers, we obtain for all i,

$\begin{align*} 0 & =\frac{\partial}{\partial r_{i}}\left[\sum_{i=1}^{n}ci^{-\alpha}(1-e^{-r_{i}})+\lambda(\sum_{i=1}^{n}r_{i}-r)\right]\\ & =-ci^{-\alpha}e^{-r_{i}}+\lambda \end{align*}$

Then,

EDITED: Addendum on asymptotics

$\frac{1}{n}+\frac{\alpha}{r}\left(\frac{1}{n} \sum_{j=1}^{n}\log j-\log n\right) \geq 0$

Thus,

$\frac{1}{n}\left(\frac{r}{\alpha} + O(\log n)\right)\geq 1$

so the highest class index that gets nonnegative resources satisfies

$n-O(\log n)\approx \frac{r}{\alpha}$

Example 2: arctan sigmoid

Let $U(r_i) = \arctan(r_{i})$ .

The optimal resource allocation is

$\begin{align*} \vec{r} & =\arg\max\sum_{i=1}^{n}P(C_{i})U(r_{i})\\ & =\arg\max\sum_{i=1}^{n}ci^{-\alpha}\arctan(r_{i}) \end{align*}$

Using Lagrange multipliers, we obtain for all i,

$\begin{align*} 0 & =\frac{\partial}{\partial r_{i}}\left[\sum_{i=1}^{n}ci^{-\alpha}\arctan(r_{i})-\lambda(\sum_{i=1}^{n}r_{i}-r)\right]\\ & =ci^{-\alpha}\frac{1}{1+r_{i}^{2}}-\lambda \end{align*}$

Then,

Thus, for $\alpha>2$ the limit of the resource fraction for the central class $C_1$ is finite and positive.

Conclusion

This is bad, because the resource fraction for the true class goes to 0 as we increase the number of classes.

I think this is an important point you make, but... you seem to be looking at r/n as both n and r go to infinity, which makes my brain hurt. Whether this is a problem seems to depend on how fast r and n go to infinity, as if, for instance, r=n^2 then as n goes to infinity, ri/r goes to 0 and ri goes to infinity. If an increase in physical resources leads to a greater than linear increase in concept space (due to having more processing power) then this would become problematic.

Also I am confused because at first C1 is described as the "central concept of chocolate" the concept most probable to represent actual chocolate, and later C1 is described as the "true class" as if the mode of a probability distribution has probability 1.

See Addendum on asymptotics in the post.

1Vika12y

Fixed the sloppy terminology about the central / true concept, thanks for pointing it out!

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[MIRIx Cambridge MA] Limiting resource allocation with bounded utility functions and conceptual uncertainty

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Conceptual uncertainty

Example 1: exponential sigmoid

EDITED: Addendum on asymptotics

Example 2: arctan sigmoid

Conclusion

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[MIRIx Cambridge MA] Limiting resource allocation with bounded utility functions and conceptual uncertainty

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Conceptual uncertainty

Example 1: exponential sigmoid

EDITED: Addendum on asymptotics

Example 2: arctan sigmoid

Conclusion

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