# pragmatic encroachment, thick credence, and portfolio diversification

For some time, I’ve had the idea that a theory of “thick credences” that are rationally sensitive to both evidential and practical factors can be helpful in clarifying some fundamental issues in epistemology as well as useful in economic modeling. Over the past week, I’ve been working on a revision of a paper on pragmatic encroachment (the thesis that non-evidential factors, such as the stakes in a decision, should properly influence what a subject knows) that applies a theory of thick credences, which model degrees of belief using imprecise probabilities. According to proponents of thick, as opposed to sharp, credences, a degree of belief should have an upper and lower bound, rather than one specific value. For example, if I told you an urn has some proportion of red balls between $\frac{1}{4}$ and $\frac{3}{4}$. If you have thick credences your degree of belief a ball drawn randomly from the urn will be red could be an interval, $[\frac{1}{4}, \frac{3}{4}]$ naturally, but if you have sharp credences you’ll need to commit to a precise value, probably $\frac{1}{2}$. In the case where you have the thick credence we write $\underline{C}(r) = \frac{1}{4}$ and $\overline{C}(r) = \frac{3}{4}$ for the lower and upper credence. The thesis I defend in my paper is that the thickness of a credence is something that can rationally dilate or contract depending on practical circumstances. In this post I want to expand on an example from the paper, using an objection to pragmatic encroachment that was put forth by Baron Reed as a launching off point, that I think gives a nice rational psychology of portfolio diversification. I am not claiming that this is better than other accounts, just that it’s nifty. Some of this post is word for word from the paper I’m working on, but I’m going to fill in more detail than the paper demands.

Here is Reed’s objection to pragmatic encroachment:

I have a broker who is extremely reliable at picking stocks. She tells me that a biotech stock, BXD, is a good long term investment and that she can move a fourth of my assets into BXD stock. Given her testimony, I know that it will go up in value, so I agree. An hour later, she tells me that she can now move another fourth of my assets into BXD. I know it will go up in value, so I agree again. An hour later, the same thing happens. When she calls me for the fourth time, she offers to move my remaining assets into BXD. But she also points out that I would then have all of my assets tied up in a single stock, which is a very risky thing to do. The stakes have become too high, and so it’s not rational for me to buy more shares of BXD. Given RKP, this means that I no longer know that the stock will go up in value. I no longer have the knowledge that would permit me to keep the stock, so I tell my broker to sell all of it in favor of other investments I know to be safe. After an hour, she calls back to remind me that BXD is an excellent long term investment. Having sold all my shares, this is no longer a high stakes proposition for me. I reflect that she is reliable in her stock tips, and I again come to know that BXD will go up in value. So, I take her up on her offer to move a fourth of my assets into BXD. And so on.

B. Reed. “Practical matters do not affect whether you know.” In J. T. M. Steup and E. Sosa, editors, Contemporary Debates in Epistemology, 2nd. ed. Wiley-Blackwell, 2013.

The case Reed presents is complicated in ways that the following toy example will simplify, but I think that the example illustrates a kind of doxastic equilibrium for pragmatically dilating thick credences that dodges the worry that pragmatic encroachment will make you a money pump for your broker. Reed’s idea is that if one can lose knowledge that BXD is a good bet just by having the magnitude of one’s investment in it increase, then one will become a money pump for broker’s fees. I wish to argue, on the contrary, that appeal to pragmatic dilation of thick credences can offer a fine-grained account of the rationality portfolio diversification.

Consider the following scenario. I have on offer two gambles. Gamble A costs $0.95 and pays$1.00 if $p$ is true. Gamble T costs $0.95 and pays$1.00 if $q$ is true. Accordingly, if I know the odds corresponding to the probability $\frac{95}{100}$ to be advantageous in each case, I should buy into each prospect. $\frac{95}{100}$ is a  kind of threshold. If I have a sharp credence below that threshold, for either proposition, then I shouldn’t take the corresponding bet; above, I should. If I have a thick credence, I claim that if my lower bound falls below that threshold, then I don’t believe that it’s a good bet, and my epistemological claim is that whether such beliefs are knowledge is a function of stakes.

Now suppose I have the following testimony. Aldo asserts $p$‘ and Tomis asserts $q$‘. How shall I allocate funds to each prospect? Assuming that I take Aldo and Tomis to be equally reliable, it may seem clear that I should divide my purse evenly, but what if, perhaps knowing Tomis a bit more personally, I take him to be a bit more reliable. In this circumstance, maybe I should put my entire purse on $q$, according to decision-making with sharp credences $C(p)$ and $C(q)$. Assuming $\frac{95}{100} < C(p) < C(q)$, I regard each gamble as desirable, but it seems that I should regard Gamble T to be more desirable in virtue of my greater confidence in Tomis, in virtue of which Gamble T will always have higher expected utility than Gamble A.

I think the conclusion that greater confidence in Tomis implies not diversifying is counter-intuitive. I think we can describe a pragmatically dilating thick credence that nicely captures the intuition that I should somehow split my purse. I assume that in the situation described above, absent any independent evidence for $\neg p$ or $\neg q$, I should have $\overline{C}(p) = \overline{C}(q) = 1$. Taking Tomis to be more reliable than Aldo (but both to be sufficiently reliable that Gamble A and Gamble T are initially desirable) and before putting anything into either prospect, I should have $\frac{95}{100} < \underline{C}(p) < \underline{C}(q) < 1$. I am suggesting that $\underline{C}(p)$ and $\underline{C}(q)$ should be decreasing functions of $S_A$ and $S_T$, the stakes invested in Gamble A and Gamble T, respectively. Accordingly, my greater confidence in Tomis would be reflected by the inequality $\underline{C}_{S_A}(p) < \underline{C}_{S_T}(q)$ when $S_A = S_T$, but when $S_A < S_T$ the hypothesis of stakes-sensitive pragmatic dilation allows that inequality to be reversed so that $\underline{C}_{S_A}(p) > \underline{C}_{S_T}(q)$. If my purse is large enough, and I can buy as much as I want, my uniquely preferred portfolio $\{S_A, S_T\}$ is the one that satisfies the condition $\underline{C}_{S_A}(p) = \underline{C}_{S_T}(q) = \frac{95}{100}$.

The forgoing presentation is perhaps a little symbol-laden but the idea is intuitive. I have no reason to think either is wrong, so my upper credence on their testimony is 1, but I know reliable people sometimes are wrong, so my lower credences are below 1. My higher confidence in Tomis is reflected in my having a higher lower credence in $q$, initially. Since I’m more confident in Tomis, I’ll start by buying Gamble T, but as I get more and more of my purse into T my credence in $q$ gradually dilates and the lower bound on my credence in $q$ gradually falls below my lower bound for $p$. At this point, investing in Gamble A becomes more advantageous than investing in T, by my lights. There’s going to be a set of pairs of stakes $\{S_A, S_T\}$ where there’s a sort of equilibrium, adding more on either end of the scale tips the balance if it’s not matched (not necessarily “pound for pound”, however) on the other side. If I put enough into each prospect, I will reach a moment where each lower credence falls to the threshold $\frac{95}{100}$ and I should invest no further in either gamble.

We can be more exact by actually defining the lower credences as decreasing functions of stakes. Each function should be defined over the domain $[0, \infty]$ and restricted to the range $[0, 1]$. Given these constraints, the following functions are somewhat natural, though of course not uniquely entailed.

• $\underline{C}(p) = 0.97 - 0.97(1 - \frac{1}{S_A + 1})$
• $\underline{C}(q) = 0.99 - 0.99(1 - \frac{1}{S_T + 1})$

These are nice, but a slight adjustment makes the x-axis more reasonable as a representation of dollar stakes. So I came up with:

• $\underline{C}(p) = 0.97 - 0.97(1 - \frac{1}{\frac{S_A}{10^6} + 1})$
• $\underline{C}(q) = 0.99 - 0.99(1 - \frac{1}{\frac{S_T}{10^6} + 1})$

Graphing these next to each other gives the following, where the y axis represents the lower bounds on the thick credences in $p$ and $q$ and the x axis represents the stake invested:

If you draw a horizontal line that intersects both the red and blue lines, you’re in a kind of equilibrium, corresponding to the levels of investment indicated on the x axis below those points of intersection; increasing your stake in either gamble immediately makes the other one more desirable. Until your lower credences fall below $0.95$ each gamble is believed to be desirable to you and you should be buying, not selling. At the unique equilibrium defined by $\underline{C}_{S_A}(p) = \underline{C}_{S_T}(q) = \frac{95}{100}$, you should be neither buying nor selling. So the money-pump dynamic implied by Reed’s objection does not arise. Moreover, we have a nifty account of why and how funds should be allocated even to prospects one is less confident of than others in a portfolio. Obviously, a lot will depend on how lower credences are defined as dilating functions of stakes. The function given in the toy example given above is not plausible for someone that doesn’t have $\approx$ \$65,000 to invest. Furthermore, any actual portfolio allocation problem is going to be way more complicated because there will be a distribution of possible outcomes, not an all or nothing gamble, and dependencies on far more propositions than just one. And, there are obviously other frameworks for incorporating risk aversion into a portfolio through diversification. I’m not trying to claim too much here, but I think it’s nifty, and it answers Reed’s objection, and maybe it’s a framework that could be pushed further.