# Does the “Free Lunch” Come with Fries?

So, I’ve been thinking about getting into the consulting business (primarily in the U.S.), and a few high-profile wealth-management firms have advised me to move to the States in order to take advantage of its favorable tax plan. Saving money on taxes always sounds like a pretty good strategy, but is moving to a different country the best option?

We’ll get to that…but, first, we require a brief digression (or two).

Some of you might have heard of something called “Purchasing Power Parity” (PPP). It’s a concept from economic theory that “compares different…currencies through a…’basket of goods’ approach. [T]wo currencies are in equilibrium…when a…basket of goods (taking into account the exchange rate) is priced the same in both countries” (Investopedia). PPP basically compares the exchange rate ($r_x$) with the quotient of prices (E) for an identical item (or a basket of items) sold in both countries. It’s a speculative measure designed to “predict” the movement of the value of one currency against another. So, if a particular Einstein bobblehead costs $p_u = 9.99$ USD and $p_c = 15.99$ CAD, $E=p_c p_u^{-1}\approx 1.6$ USD, which is significantly higher than the then-current exchange rate of (about) 1.29 USD. This means Canadians will prefer to purchase the bobblehead in the U.S. (@ $12.89) and use the other three-plus dollars for something else. In short, the Canadian price is too high, suggesting the USD will be appreciate (over some time frame) until E equals the exchange rate. When $E\neq r_x$, we speculate the possible existence of an arbitrage opportunity. The concept of arbitrage is hardly new, yet it continues to drive the boldest (and, at times, most reckless) investment strategies on Wall Street and beyond. It’s often said, colloquially, that “there ain’t no such thing as a free lunch” (TANSTAAFL); the proven concept of arbitrage, however, completely belies that claim. I offer a quick and tailored primer for skeptical readers. #### “TISATAAFL” or “How to Guarantee a Gambling Profit Using Mathematics” In very basic terms, a betting-arbitrage opportunity arises when the sum of the bettors’ odds of a successful outcome derived from the gaming odds is less than one.* In a two-system bet, this is very simply calculated as $\Big[\Big(1-a_1(a_1+b_1)^{-1}\Big)+\Big(1-a_2(a_2+b_2)^{-1}\Big)\Big]<1$, where the $a_1/b_1$ odds of Bettor 1 to win are $a_1(a_1+b_1)^{-1}$. The second bet placed with Bettor 2 follows similarly. An example: Suppose James desires to place a wager on the 2018 college football national title game between Alabama and Clemson. His brother, Joshua, is giving 2/1 odds on Alabama, and Michael is giving 1.25/1 odds on Clemson. James wants to determine if this represents a genuine arbitrage opportunity. The 2/1 line means Joshua believes Alabama has a 2/3 probability of winning (i.e., 2/(2+1)), and the 1.25 (= 5/4)/1 line means Michael believes Clemson has a 5/9 probability of winning (i.e., (5/4)/[(5/4)+1]). Calculating (their belief of) the probability that James will win each bet is $(1 - 2/3) + (1 - 5/9) = (1/3) + (4/9) = 7/9 <1$. It is, in fact, an arbitrage opportunity. How does arbitrage work? Let’s say James’s gambling allowance happens to be 100 dollars in total, and he’ll bet x dollars on Alabama and 100 – x dollars on Clemson. Let’s imagine Alabama wins. An Alabama victory means James loses –x dollars to Joshua and wins 5(100 – x)/4 dollars from Michael. This yields the first linear profit curve (red line, below): $-\frac{9}{4} x+ 125=0$. Of course, we want the profit to be greater than zero, so we set the LHS of the equation accordingly and solve, yielding $x<\frac{500}{9}$. Now, imagine Clemson wins. This would mean James loses -(100 – x) dollars to Michael and wins 2x dollars from Joshua, the sum of which yields the second linear profit curve (blue line, below): $3x-100>0$ such that $x>\frac{100}{3}$. Thus, the amount of money James needs to wager with Josh (x) and Michael (100 – x) to guarantee a profit no matter which team wins the game lies within the global inequality derived from both equations: $\frac{100}{3}. This is a range from (roughly) 33.34 dollars to 55.56 dollars. Curious readers will want to know two things: (1) the maximum possible profit based on the gambling allowance and (2) the optimum wager James should place on Alabama to generate that amount. We’ve jumped the gun a bit by providing the above graphic, but the answer involves solving the system of linear equations in the previous paragraph; the linear profit curves cross each other at an equilibrium point—recall the supply-and-demand curves of elementary economics—and it is this intersection that represents the Cartesian coordinate that (a) reveals the maximum possible profit ($f(x_m)$) and (b) the optimum bet ($x := x_m$) that guarantees that maximum profit amount. Fortunately, as a general principle, we don’t need to graph these functions. Setting both equations equal to each other and solving for $x_m$ is sufficient: Solving $-\frac{9}{4}x_m+ 125 = 3x_m - 100$ gives us an optimum bet value of $x_m\approx 42.86$ dollars on Alabama and 100 – 42.87 = 57.14 dollars on Clemson. This betting profile generates a maximum guaranteed profit (p) of $p := f(x_m)\approx 28.57$ dollars based on a 100-dollar total wager—no matter which team wins the game. The above graphic provides the relevant visual representation. This should give you a general sense of the power and seductiveness arbitrage offers and why it’s essentially the Holy Grail of any investment strategy. (For some readers, it might be cool enough to know arbitrage exists, and you may want to make a few bets with your friends. But do the math first!) To put it simply, there’s no better option available to you than the one that generates a financial profit no matter the outcome. (Sorry, Milton Friedman!) One might ask what this has to do with PPP or moving to a foreign country. It involves the notion of currency arbitrage as a “free lunch.” Recall the Canadian who was interested in the Einstein bobblehead. She essentially earns three dollars by making the (online) purchase from the States. It’s as if she bought the bobblehead in Canada and the government deposited three dollars into her account. Unfortunately, PPP doesn’t tell us when to make the purchase (we need the exchange rate for that), but we use that information to make inferences, like whether we’ll save money if we buy a book from Spokane rather than Vancouver. PPP, however, only involves “tradeable” commodities. “Immobile goods” like real estate and services are inaccessible to PPP calculations. One such “inaccessible” item is tax liability. The professional advice I received was simple: Move to the United States in order to avail yourself of the more attractive federal tax rates. But can I get a “free lunch” by staying in my country of residence? PPP tells me whether there exists a currency imbalance, not whether I should move. An approach that does help me make this determination involves what I will call the “Net-Purchasing-Power Index” (NPPI). NPPI simply calculates the exchange rate ($r_q$) that represents the equilibrium point between two baskets of post-tax income portfolios and compares it to the current exchange rate ($r_x$). We begin with first principles—the technical definition of net income—and derive the NPPI from there:Here, $a$ is the total value of the income portfolio in the U.S., $r_q$ is (again) the equilibrium rate, $t_c$ is the relevant federal tax rate in the target country, and $t_u$ is the relevant federal tax rate in the U.S. Our goal is to calculate the NPPI by calculating the quotient of $r_q$ and $r_x$. When NPPI < 1, the exchange rate is greater than equilibrium rate, and we have the potential for an arbitrage opportunity—but not yet a guarantee. For that, we need to do a bit more work. As the NPPI tends to zero (i.e., as the exchange rate gets larger), the portions of our potential free lunch grow significantly. Let’s walk through an example. Suppose an analysis suggests my U.S. corporation will generate$500,000 USD in consulting fees in 2018. Conventional wisdom, as we’ve seen, suggests relocating to the U.S. That is, a $500,000 portfolio at a U.S. federal tax rate of 39% leaves me with$500,000(1 – 0.39) = $305,000 dollars if I move to the States. If I bring that money into a target country with a federal tax rate of, say, 47%, I’ll only have an after-tax amount of$265,000. It seems as if I’m losing money by choosing not to move. But what about $r_x$? Let’s imagine the exchange rate between the U.S. and the target country is $r_x\approx 1.10$ USD. Is an arbitrage opportunity possible? Using the equation above, $r_q\approx 1.151$, which is the rate that “equalizes” the post-exchange purchasing power between both countries. Because $r_q r_x^{-1}>1$, I would be (really) be losing money. Clearly, the after-tax, after-conversion portfolio of $291,500 is less than the$305,000 I’d be able to spend on goods and services in the U.S. If I think the extra $13,500 I’d save by moving to the U.S. is worth the time and effort, I should relocate. But what if $r_x\approx 1.2532$. Then, $r_q r_x^{-1} < 1$ and I’ve have a real chance to make some free money by staying put: In this case, I create$332,098 by bringing my U.S. income into the target country, and I enjoy a net-purchasing power of +$27,098. (Assume I’ve taken advantage of the legal means to minimize double-taxation issues.) But isn’t this a guaranteed arbitrage opportunity? No, because we haven’t accounted for price differentials. What if prices are much more expensive in the target country? That is, what if PPP is severely unbalanced, as in the bobblehead example? In that case, the increased prices eat away at any NPPI surplus, though if we’re dealing with tradeable commodities, as we saw earlier, one would simply purchase those items from the States. Unfortunately, importing goods isn’t always a guarantor of profits. Let’s say, for argument’s sake, the amount of income we’re dealing with is$100. If $r_x\approx 1.2532$, then NPPI < 1 and I’m left with $61 living in the U.S. and$66.41 in the target country. After buying the bobblehead at $9.99 USD, I have$51.01 in the U.S. and $50.42 if I buy it in the target country ($66.41 – $15.99). NPPI < 1, but I’ve still lost money. (For simplicity sake, assume the sales tax is equal and the federal tax rate applies to our$100 conversion.) This means I have to import the bobblehead from the States to have a chance at maintaining my NPPI advantage, and, in this case, I do still come out ahead: $66.41 – 9.99(1.2532) =$53.89, which means staying in the target country is still $2.88 better than if I’d moved to the U.S. and paid for the bobblehead in USD. It’s a very slight advantage, but that’s only because the amounts we’re dealing with are small. As the portfolio (a) grows, so does the advantage. (This ineluctably leads to the notion of leverage as an investment strategy, but we won’t address that here.) Unfortunately, as the price grows, the advantage decreases, and if the price is high enough, choosing not to move becomes a disadvantage. The question, then, becomes this: Is there any way to evaluate an arbitrage opportunity given a specific constellation of values for the variables we’ve been discussing? Yes, there is. Such an evaluation involves solving a linear optimization problem that accounts for price levels. I will call this the Currency Arbitrage Price (CAP), and it utilizes both NPPI and PPP values. In what follows, however, we assume NPPI < 1. (Recall that if NPPI > 1, then no arbitrage opportunity is possible.) So, what do we need to know? We need to determine the maximum price level of a specific item in the target country that guarantees a post-purchase profit. We can calculate this by adding our price variables to the calculation of $r_q$. Solving the necessary inequality for $p_c$, we have:Notice the cancellation that occurs when $\text{PPI}\cdot p_u \to p_c$. This last inequality tells us how much an (identical) item needs to cost in the target country in order to guarantee a profit given the other variables. We can visualize this inequality by graphing the linear CAP function $f(p_c)=a((1-t_c)r_x - (1-t_u)) + p_u - p_c$, and we guarantee arbitrage when $p_c \geq 0$ and $f(p_c) > 0$. Armed with this information, let’s revisit the$100/bobblehead example. Solving the above inequality gives us $p_c = 15.41$. This means we are guaranteed an arbitrage opportunity when the bobblehead price is $9.99 in the U.S. and less than$15.41 in the target country. Let’s imagine it’s priced in the clearance bin (in the target country) at $11.99. In the U.S., paying in USD, we’d be left with the usual after-tax, after-purchase amount of$51.01, but in the target country, we’d now have an after-tax, after-purchase balance of $54.42. Despite the disparity in currency valuations and the higher tax rate, we enjoy an overall profit, which is an increase from the earlier amount of$53.89 we gained from importing.

Free lunch.

We can do a bit more. Imagine the target country decides all bobbleheads should be $11.99, and the U.S. decides it must reduce bobblehead prices to stay competitive. We love this Einstein bobblehead so much that we want to send it to all our friends. But the U.S. price keeps falling. How long can we purchase the bobblehead at$11.99 in the target country until we lose our arbitrage advantage? In other words, at what U.S. price does our profit reach zero? To solve this problem, we simply solve the above inequality for $p_u$. This gives us:

$p_u > a((1-t_c)r_x - (1-t_u))-p_c$.

The function for $p_u$ follows similarly. As long as the U.S. price is greater than $p_u$, we retain our arbitrage advantage. So, if bobblehead prices remain fixed at $11.99 in the target country, the U.S. price can fall to$6.57 per unit and we’ll still earn a profit (as small as it might be at that price). You don’t need any extra information to calculate $p_u$. PPI gives us the prices we need, and the values for all the other variables—exchange and tax rates—are easily accessible to the public. We’re simply doing some basic algebraic shuffling.

If we factor sales tax into the price differential, we add a layer of complexity to the problem of quantifying arbitrage. If $s_u$ and $s_c$ are the sales-tax rates in the U.S. and Canada, respectively, then our profit curves become

$f(p_c) = \left(1+s_u\right)p_u + a\left(\left(1-t_c\right)r_x -\left(1-t_u\right)\right) -\left(1+s_c\right)p_c$

for the price in Canada and

$f(p_u) = -a\left(\left(1-t_c\right)r_x -\left(1-t_u\right)\right)+\left(1+s_c\right)p_c -\left(1+s_u\right)p_u$

for the price in the U.S., respectively. In this more complex case, imagine we import $5000 USD at an exchange rate of $r_x=1.2532$ with federal income-tax rates of $t_u=0.10$ and $t_c=0.15$ and sales-tax rates of $s_u=0.0875$ and $s_c=0.12$ in the U.S. and Canada, respectively. We note that NPPI < 1, and we want to purchase a new computer where $p_u=2500$ USD and $p_c=3499$ CAD. Do we have an arbitrage opportunity? Unfortunately, we don’t—not until the Canadian price is reduced to less than$3,165.

The function for the Canadian price (above) reveals this upper bound when $p_c =0$ (solid green). As you can see from the graph, we lose money at the current price ratio ($1407.22 –$1781.25 = -$374.03). This is shown by the gap between the red- and blue-dashed lines transversed by the constant function that represents the U.S. price $x=p_u$; this is the difference between $f(p_u)$ values of the individual profit curves (and not the above functions that arise from setting those equations equal to each other and solving for $p_c$ and $p_u$, which are represented by the solid lines on the graphs). Though we lose money if we purchase the computer in Canada at the current price of$3,499, we do gain a profit of $137.71 by importing it from the U.S. But let’s imagine we choose to wait for a local sale, and the Best Buy in Vancouver reduces the price to$2,999. Now, we do have an arbitrage opportunity:

We’ve now earned a better-than-importing profit of $1967.22 –$1781.25 = $185.97, despite the higher federal- and sales-tax rates, by bringing in the USD-based income and paying for the computer in CAD. The graph also reveals that we’ll continue to generate a profit until the U.S. price—in response to Canada’s competitiveness—drops to about$2,329 (purple), at which point the individual (dashed) profit curves intersect with each other at $x=p_u$ and the total profit drops to zero: Both the Canadian and U.S. consumers, at that point, would be left with a remaining balance of \$1967.22 after purchasing the computer.

### Conclusion

So, that’s it. Currency arbitrage in a nutshell. Perhaps something like this exists somewhere in the literature—I imagine it might, even though I’ve never seen it explicitly during my study of economics—but we offer it here in the event it will pique general interest. It might be beneficial to review the basic process involved in calculating the CAP for a given portfolio combined with a certain collection of data points:

(1) Calculate the equilibrium rate $r_q$
(2) Confirm NPPI < 1
(3) Determine $p_c$ and $p_u$
(4) Purchase “identical items” in the target country if the price less than $p_c$
(5) Purchase (4)’s items freely until its price in the host country reaches $p_u$ (assume $p_c$ remains constant)

Anticipated objections to the CAP model:

(I) Availability of identical goods

If a tradeable good in the target country is truly unique, you couldn’t have purchased it anywhere else; the notion of PPP is simply unimportant in those cases. Of course, you will have to decide whether you wish to (or, for some reason, must) pay for that uniqueness or if you’d prefer to choose an item that closely (but not precisely) matches the one you’re considering, assuming such an item is available. As far as price modeling is concerned, very closely matched items can be (and probably should be) considered “identical.” Variations among packages of Bic pens, for example, probably don’t mean very much with respect to the sticker price.

(II) PPP applicability

PPP only accounts for so-called tradeable goods, but it is possible to compare “immobile” goods using a number of objective metrics. For homes and real estate, for example, we could use price/sq.ft., location, year of construction, amenities, projected repairs, and many other measures of objective value. Much like the issue of identical goods, then, we can gain a pretty good comparison between immobile goods between countries that will allow us to use a generalized approach to PPP. Value is in the eye of the beholder, which means an eye toward equality of value between such goods is achievable.

So, what about the big question: Should I move to the U.S. based on my fanciful financial projections or remain in the target country and bring the money here? Well, if NPPI < 1, which means the exchange rate outstrips the taxation gap, then I’m guaranteed a free lunch (or two) as long as I purchase (near-)identical goods that fall below the $p_c$ upper bound. If I can do that through importing goods with a favorable exchange rate or by taking advantage of cheaper relative prices in the target country given a certain sales-tax profile, then it’s in my interest to eschew the idea of relocating, even though the tax rates are more favorable in the States.

* In wagers like these, everyone must hold their money until after the event is completed. In this way, an arbitrageur can cover her losses with her winnings and keep the remaining profit. Online betting sites require you to front the money as you make the wager, which is why this arbitrage strategy won’t work in those cases.

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# Can We Quantify Certain Kinds of Ethical Choices?

In his book Ethics in the Real World, the renowned philosopher Peter Singer proposes a metric for ethical risk informed by his (generally held) worldview of consequentialism (i.e., the idea that the consequence of an act determines the ethical value of that act). Singer states that, generally speaking, “we can measure how bad a particular risk is by multiplying the probability of the bad outcome by how bad the outcome would be” (183). Thus, an act is considered more ethical if it offers less general risk (for death, for torture, for financial waste, for suffering, for climate change, etc.) than an alternative act. We can model Singer’s non-mathematical comments by the very simple product $\text{S}_n = p_n\sigma_n$ where $\text{S}_n$ is the “Singer risk” for the nth event and $\sigma_n$ and $p_n$ are the outcome and probability for the nth event, respectively. Note that $\text{S}_n$ is really just an area calculation in $\mathbb{R}^2$ with the “sides” of the rectangle defined as the two variables in question; the larger the area, the greater the risk. Simple, right? We will return to the concept of area later.

But is this a viable model for risk? Forget for a moment about other kinds of ethical choices we make that have less definitive outcomes—a decision to break a friend’s confidentiality, defending a colleague from a false accusation that risks alienation among one’s coworkers, telling the truth despite hurting someone’s feelings, etc. Limited to the probability of “bad outcomes” we can quantify, however, does Singer’s product capture a quantification of ethical risk in a real and intuitive way? Is the ethical value of an act, in general, determined by the consequence(s) of that act? At a first glance, it seems we shouldn’t take Singer’s metric too seriously—and, perhaps, he doesn’t either—because it immediately strikes the reader as an inadequate method to quantify ethical risk in any meaningful way. How can we, to imagine one easy example, compare the loss of life between, and among, different demographics? Is it more ethical to prevent the death of a child if that preventive measure causes the death of, say, an elderly person? Five elderly people? What if it caused the death of a young, female professional at the height of her earning and reproductive powers? Is it even possible to balance those scales when making a risk assessment?

Even if we could achieve some sort of balance involving what I will call congruent cases (i.e., outcomes that involve a single parameter: the number of people harmed, the tonnage of CO2 released into the atmosphere, etc.), we’re still left with the much more difficult problem of quantifying incongruent outcomes: Is the ethical risk for blindness and malnutrition in third-world countries equal to that of domestic homelessness and drug addiction? Is rolling back the pursuit of nuclear energy (and the problems associated with managing its toxic, immutable waste) on par with diminishing our carbon footprint by reducing CO2 levels? If it is, how can we model that risk relationship? If not, why not, and how do we build into Singer’s model an objective and unbiased evaluation of those disparities? Assuming we accept Singer’s basic design, it would be an extraordinarily difficult task to “nondimensionalize,” as it were, the innumerable combinations of outcomes that would necessarily inform our decision-making process. If Singer’s model—and the philosophical platform of consequentialism, in general—has any hope of offering even a partial solution to the important kinds of ethical dilemmas he raises in his book, it must be able to handle the complexities involved in comparing these kinds of incongruities. But for the sake of argument, let’s set aside those additional complexities—as well as general critiques of consequentialism—and address the model in its most simplified form: a risk metric as a simple product limited to congruent outcomes.

Singer’s basic approach isn’t entirely without some precedent. Financial risk models, for example, involve (the sum of the) products of probabilities and returns, but they are couched within much larger mathematical and statistical machinery and require several additional calculations (e.g., expected rate of return, variance, etc.). The expected value E(X) of a continuous random variable involves the integration of the product of the random variable and the PDF, which has attached to it certain conditions (only positive values, total integration equals 1). There are other examples. Singer, however, argues that a quantification of risk could be limited to the product of an outcome and the probability that outcome occurs, and it is the validity of this basic approach we will challenge.

In light of this very narrow definition of ethical risk, then, consider the following thought experiment, couched in the form of a poll question, that was posted on three different FB groups:

Which of the following options would you consider to be the ethically superior choice?
(1) Ten people are killed if you roll a three with a ten-sided die.

(2) One person is killed if you fail to roll a three with (a different) ten-sided die.

Here, we set two independent, stochastic events (very nearly) equal to each other, though it seems clear they’re not equal ethically; in doing so, we hoped to investigate whether people would respond to the quantification of risk, as defined by Singer, or, perhaps, something else. (I’ve reasonably defined “how bad the outcome would be” simply by the number of people who would be killed.) Contrary to predictions based on Singer’s metric, a sizable majority of people (33/44 = 0.75) selected option 1 as the more ethical choice, even though (a) option 2 actually offers slightly LESS risk, which makes it the preferred choice according to Singer’s model, and (b) the number of people at risk for harm in option 1 is ten times greater.

So, what happened?

Most people seem to have responded not to Singer’s risk metric but to the probability of the outcomes. The risk of ten people dying ($\text{S}_1=1$) is very much mitigated by the fact that there’s a 90-percent chance nothing happens and the ten people at risk will remain unharmed. This stands in sharp relief with option 2 ($\text{S}_2=0.9$), where there exists a 90-percent chance the person at risk will be killed, despite the fact that the total number of people at risk is one-tenth that of option 1. It seems the pollsters simplified the ethical dilemma by focusing on the probabilities of the outcomes, as if the poll options were as follows:

(1) There’s a 90% chance no one dies.
(2) There’s a 10% chance no one dies.

Notice the sigma values have vanished. Risk has now been reduced to reflect the p-values for harm, as if participants (subconsciously) treated Singer’s metric like a function $f:\mathbb{R}^+_0\to\mathbb{R}$ defined by $f(p_n,\sigma_n) = p_n\sigma_n$ and evaluated the poll options as $\partial_{\sigma_n}f$. (Because $\sigma_1\neq\sigma_2$, we can’t simply cancel the outcomes.) This result is not particularly surprising. Most participants seemed to follow a probabilistic risk-aversion strategy rather than an outcome-averse one, but it’s an evaluation process that’s clearly not linked to Singer’s consequentialism, which demands a deference to the fact that $\text{S}_2 < \text{S}_1$. That is, the poll results reify the notion that ethical preferences might very well engender greater risk according to Singer’s model.

One might imagine what the polling would have looked like if it followed Singer’s metric. Perhaps everyone would have picked option 2, the result of privileging $\sigma$ regardless of the associated probabilities and/or recognizing it offers less overall risk (0.9 < 1.0). In another scenario, the polling might have been split almost equally between both options, reflecting the (near) equality of risk between the two options. The next question seems inevitable: How could we equate these outcomes in the minds of pollsters? How much would we have to increase the value of $\sigma_1$ such that people felt the objective evaluation of both risks were, in fact, very much the same, where it made essentially no difference (in terms of ethical risk) whether we chose option 1 or 2? Perhaps the relatively low p-value for option 1 would overwhelm any value we could assign to $\sigma_1$. Perhaps there’s more structure to perceptions involving probability-outcome relationships; for example, they might be inversely proportional to each other ($p_1\propto\sigma_1^{-1}$). It’s difficult to speculate. If Singer’s model were more robust, we could simply solve the equation for the appropriate variable and calculate the perfect balance of risk, much like we attempt to do in finance or economics. Unfortunately, like so many mathematical models in other fields, things aren’t never quite so simple.

So, what do we do when mathematical equality doesn’t transpose to psychological or ethical “equality”? How can we make sense of two poll options with essentially equal risk that engender such a divergent response? Fortunately, we can use some tools from linear algebra to help us explore the degree to which two risk values—as Singer products with congruent outcomes—are (dis)similar. Assume a nonsingular risk matrix A is a 2 x 2 matrix whose entries are defined as follows: $a_{11}=p_i$$a_{12}=\sigma_i$$a_{21}=p_j$, and $a_{22}=\sigma_j$ where u $= [p_1\,\,\sigma_1]$ and v $= [p_2\,\,\sigma_2]$. The length of the cross product of (these risk) vectors u, v is equal to the absolute value of det A:

$\omega = \Vert \textbf{u}\times\textbf{v}\Vert = \left |\,\text{det}\!\begin{bmatrix}p_i & \sigma_i\\ p_j & \sigma_j\end{bmatrix}\right|=\left|\,p_i\sigma_j - p_j\sigma_i\right |$ .

The value of omega reveals a relationship between risk vectors. The determinant of a 2 x 2 matrix, if it exists, can be thought of as the area of a projected parallelogram in $\mathbb{R}^2$ delimited by its vectors—in this case, u and v. The greater the $\omega$ value, the greater the area of the projected parallelogram and the larger the dissimilarity between Singer risks. Though $\text{S}_1\approx\text{S}_2$ according to the Singer metric, $\omega = 8.9$, revealing the relationship is not nearly as close as the risk products suggest. This result might also proffer a partial explanation for the poll results, which, despite near equality in risk values, are heavily skewed toward option 1. Perhaps $\omega$ responds in some way to the pollsters’ decision to privilege likelihood over outcome. For a quick comparison, consider $\omega=1.1$ when $p_1=0.5$$\sigma_1=6$$p_2=0.6$, and $\sigma_2=5$. Here, the Singer risks are equal (3), yet even while comparing two events with identical risk products, the sensitivity of $\omega$ is able to differentiate between them. That may be a helpful and quick initial guide when comparing the risk of two congruent ethical choices.

### Preference Rules

At this point, we might be inclined to consider the feasibility of certain kinds of “preference rules” (PR) with respect to ethical risk; that is, are there any ways to make an objectively unequivocal decision between Singer risks given certain values? The short answer: Yes, there are, and we list three such rules (PR1-3) that will always hold in any Singer-risk comparison. We also include two “derived preference rules” (DPR) that similarly hold in any situation:

PR1: $p_i = p_j\to \min\,(\sigma_i,\sigma_j)$
PR2: $\sigma_i = \sigma_j\to \min\,(p_i,p_j)$
PR3: $\left((p_i < p_j) \land (\sigma_i < \sigma_j)\right) \to \text{S}_i$

DPR1:  $\left(\left(p_i\sigma_j < p_j\sigma_i\right) \land \left(p_i p_j^{-1} > 1\right)\right) \to \text{S}_j$
DPR2:  $\left(\left(p_i\sigma_j < p_j\sigma_i\right) \land \left(p_i p_j^{-1} < k^{-1}\right)\right) \to \text{S}_i$

PR1-3 are almost insultingly obvious, and for those unfamiliar with the symbols of formal logic, I offer an informal exposition. PR1 states that if the probabilities between two Singer risks are equal, we will prefer the smaller outcome, which is equivalent to preferring the smaller Singer-risk value. (Remember, we’re limiting our investigation to risk products with congruent outcomes.) PR2 simply reverses the issue addressed in PR1: If the outcomes of two Singer-risk values are the same, we will prefer the smaller probability, where, again, we’re preferring the smaller Singer-risk value. PR3 formalizes the concept inherent in PR1-2: If both the probability and the outcome of a Singer-risk value are smaller than those of a second Singer-risk value, we will prefer, as we should expect, the smaller Singer-risk value. These rules are inviolable and will obviously hold in all cases.

The DPRs are only slightly less obvious, and we only construct them because they relate to our earlier exploration of omega. DPR1 says that if $p_i\sigma_j$ is the smaller risk-matrix value and $p_i p_j^{-1}$ is greater than 1, prefer $\text{S}_j$. This is a convenient rule if you’re given det A products and the associated probabilities. The proof for this is trivial.

Proof (direct): Suppose A is a 2 x 2 nonsingular (i.e., $\text{det} A\neq 0$) risk matrix such that $p_1\sigma_2 < p_2\sigma_1$. Then, $p_1 p_2^{-1}<\sigma_1\sigma_2^{-1}$. If $p_1 p_2^{-1} > 1$, then $p_1 >p_2$. But $\sigma_1\sigma_2^{-1}>p_1 p_2^{-1}$, which means $\sigma_1\sigma_2^{-1}>1$ and $\sigma_1 > \sigma_2$$\Box$

Thus, we will prefer the smaller Singer-risk value as prescribed by PR3. Unfortunately, a proof for DPR2 must use a different approach, but we can at least state it as follows: Suppose A is a 2 x 2 nonsingular risk matrix such that $p_i\sigma_j < p_j\sigma_i$ and the ratio of probabilities, $p_ip_j^{-1}$, is less than 1/k, then prefer Singer risk $\text{S}_i$. The same definitions from DPR1 apply here as well. One might have already asked the obvious question: Whence k? It arises in the process of transforming the principal inequality to an equality:We know $k > 1$, so $k^{-1} < 1$ and we’re now in a position to formalize a proof for DPR2.

Proof (direct): We need to show that if $p_1 p_2^{-1} < k^{-1}$, then $\sigma_2 > \sigma_1$ given the det A inequality. By transforming the former inequality into one we can use, we see that $p_1 p_2^{-1} < k^{-1}$ becomes $-\ln p_1p_2^{-1} > \ln k$ by the properties of logarithms, and it’s no coincidence this latter inequality involves the last two (RHS) terms of the final equality displayed above. It is the case that $\sigma_2 > \sigma_1$ if $-\ln p_1p_2^{-1}>\ln k$ because of the signs of the terms. Simplifying, we have which only holds when $\sigma_2 > \sigma_1$, as desired.  $\Box$

Because $p_2 > p_1$, we will prefer $\text{S}_1$ as prescribed by PR3. How do these DPRs relate to the poll options? We have 0.1(1) < 0.9(10) and 0.1 < 0.9, so we need to determine if 0.1/0.9 < 1/k. In this case, 1/k is smaller, which confirms our intuition that PR3 can’t be invoked in the poll-question case: $j > i$ with respect to p-values, and $i > j$ in terms of outcomes. Unfortunately, we cannot establish a similarly inviolable preference rule for these kinds of mismatched inequalities, for it is this mismatched relationship that makes difficult the process of determining the ethical preference between single products involving congruent outcomes.

### Area as a Quantification for Risk

What about Singer’s implied use of “area” as a metric? We’ve already mentioned its very limited scope prevents it from being a comprehensive model, but the fact that his model quantifies risk as an area calculation is not, ipso facto, a problem. There exists a long and storied tradition in mathematics, for example, in which area calculations are the very calculations we want: probability densities, work, distance, center-of-mass problems, kinetic energy, average value of a function, and arc length are only a few examples. We’ll see a few more shortly. Within the context of that rich tradition, then, we can imagine a number of other area-based models in an effort to uncover a metric that (at least) betrays the poll results. Part of what complicates matters is that the probability values of “Singer risks” aren’t built upon the same mathematical infrastructure. In other words, we cannot directly compare probabilities within the same distribution. We’d like to be able to do so, but if we view $\sigma_n$ as a continuous random variable, the associated PDF functions cannot be equal. This can be seen with even a quick glance at the poll options: What probability distribution, for example, decreases probability values as we increase the area under the distribution curve? A cohesive PDF in the case of the Singer metric would have to yield both 0.9 at x = 1 and 0.1 at x = 10. If there is such a distribution, I’m not aware of it. Of course, the lack of a distinct PDF is mitigated by the reality that our ethical dilemma isn’t entirely random. Yes, there is a stochastic process attached to an impending if-then action, but that’s not the same thing as having a truly random variable.

We consider three area calculations as integrations of functions $f : \mathbb{R}^+\to\mathbb{R}$ whose definitions are implicitly defined below:

1.  $\int_0^{\sigma_n} c_n x\,\,\mathrm{d}x = p_n\int_0^{\sigma_n} x\,\,\mathrm{d}x$

The quantification of risk is now the area under the above linear function where $p_n=c_n \in (0,1)$ is simply the slope of the function. Option 2 is far less risky (0.45 << 5) because risk now involves the quadratic growth of the outcome. A mental visualization of the (respective areas under the) graphs of these Singer metrics will be enough to convince anyone of the inadmissibility of this approach as a viable model. As we’ve seen, most people seemed to respond to the likelihood of the event and not the value of $\sigma$, as if the outcomes were largely irrelevant to the decision-making process, yet the linearity of an outcome-based design dramatically increases sensitivity to $\sigma$; we’re simply privileging the wrong variable.

2.  $\iint\limits_Rf(x,y)\,\,\mathrm{d}A=\int_0^{p_n}\!\!\int_0^{\sigma_n}\!xy\,\,\mathrm{d}x\,\mathrm{d}y$

A volume calculation in $\mathbb{R}^3$ does a better job of approximating the Singer-risk values, but it also fails to model the poll results. The integration is straightforward and simplifies to $\frac{1}{4}(p_n\sigma_n)^2$. This approach only slightly reduces the value of the quadratic growth in the previous example by squaring the p-value, but it’s not enough of a reduction in most cases. (Recall from analysis that if $a >1$, then $(a^{-n})\to 0$ as $n\to\infty$.) Even though this new model tightens the risk difference between both options (+0.0475 vs. +0.1), it still suggests option 2 offers slightly less risk: $\text{S}_1 = 0.25$ and $\text{S}_2 = 0.2025$.

3.  $\iint\limits_Gg(x,y,f(x,y))\left(\partial_x^2f+\partial_y^2f+1\right)^{1/2}\!\mathrm{d}A=\!\left(\frac{1}{4}\sigma_n^{-2}+\frac{5}{4}\right)^{1/2}\!\!\int_0^{p_n}\!\int_0^{\sigma_n}\left(xy+\frac{x}{2\sigma_n}-\frac{y}{2}+1\right)\mathrm{d}y\,\mathrm{d}x$

Here, the surface-area calculation in $\mathbb{R}^3$ also fails to model the poll results. We (somewhat arbitrarily) choose the plane $z=f(x,y)=x(2\sigma_n)^{-1} - y/2 +1$ in the hope of striking a better balance between probabilities and outcomes. Simplifying and solving leaves us with the following product:
Here, like the other two approaches, option 1 remains the riskier option: $\text{S}_1\approx 1.7$ and $\text{S}_2\approx 1.4$. The risk gap between options, however, has now widened compared to the volume calculation, and we still have the quadratic growth of $\sigma$ built into the model. An alternative iterated integral—namely, $\int_0^{p_n}\!\int_0^{\sigma_n}xyz\,\sec\gamma\,\mathrm{d}y\,\mathrm{d}x$—shrinks this gap ($\text{S}_1\approx 0.36$ and $\text{S}_2\approx 0.26$), but it still fails to track the majority decision to treat option 1 as the more ethical choice. Thus, in every case, the models we’ve explored produce a risk value for option 2 that is less than option 1. This is disappointing, but we only intend to offer a brief investigation into the possibility of an alternative model. A fully realized and robust design is well beyond the scope of a blog post, so I will leave it to interested readers to pursue a viable solution, including a better motivation for z, concerning the kinds of ethical problems we’re investigating here.

### Conclusion

After all this, though, we’ve neglected to ask, perhaps, the most crucial question: Is the concept of risk a vitally important and pervasive consideration? To this, we must offer a full-throated “yes!” We need only remind the reader that notions of risk aren’t, as this post might suggest to some, mere fodder for a tiresome intellectual and mathematical exercise; we as a society make many decisions based on quantifications of risk—from actuaries calculating life expectancy for insurance policies and the beta risk of financial investments to disaster management and the cost-benefit analysis involved with safety recalls. And though pure notions of ethical risk are absent in most of these examples, we still very much engage in just the kinds of stochastic events reified in the poll options—where lives can, and often do, (literally) hang in the balance; there are probabilities associated with dying an unnatural death by being hit by a drunk driver, with the collapse of hedge funds holding severely over-leveraged arbitrage portfolios, with the flooding and destruction of Florida’s coastal cities during hurricane season, and with how many people might be killed by driving Company X’s new car. But that’s not all: We use these quantifications to draft legislation, evaluate legal settlements, decide how maintenance funds are allocated, and design the constellation of foods and products your children will put in their mouths. Sometimes, such decisions can lead to subversive, and even illegal, acts.

Yet despite the vertiginous ubiquity of risk assessments that swirl around us, many people simply refused to choose a poll option on the grounds of some misguided moral indignation (“The only ethical choice is not to choose!”). Perhaps their reluctance involves the potential dread that comes with an increased awareness of self, that adjudicating a tough ethical decision requires a prism through which some are afraid to see themselves. It takes the willingness of an honest and restless soul to subject oneself to such psychic refractions.

That we should all have such courage.

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# A Proposed Proof for the Existence of God

Assume it is impossible to prove God does not exist. Then the probability that God exists, $p(\text{G})$, however minuscule, is greater than zero: $p(\text{G}) = ab^{-1} \in (0,1)$. Also assume, as many important physicists and cosmologists do, that (1) the multiverse exists and is composed of an infinite number of independent universes and (2) our current universe is but one of those infinite universes existing in the multiverse.*

If the probability of the non-existence of God, denoted $p(-\text{G})$, in some universe is defined as

$p(-\text{G}) = (1 - ab^{-1})\in\left(0,1\right)$

then as the number of universes (n) approaches infinity,

$\lim_{n \rightarrow \infty} (1 - ab^{-1})^n = 0$.

That is, the sequence $\left(1-ab^{-1}\right)^n\to 0$ as $n\to\infty$. Any event that can happen will ineluctably happen given enough trials. This means God must exist in at least one universe within the multiverse, and if He does, then He must exist in all universes, including our universe, because omnipresence is a necessary condition for God to exist.

Q.E.D.

* This is certainly a reasonable, if not ubiquitously held, concept that follows from the mathematics of inflationary theory. In Our Mathematical Universe, for example, Max Tegmark suggests if “inflation…made our space infinite[, t]hen there are infinitely many Level I parallel universes” (121).

Standard

# The Myth of Altruism

The American Heritage Dictionary (2011) defines “altruism” as “selflessness.” If one accepts that standard definition, then it seems reasonable to view an “altruistic act” as one that fails to produce a net gain in personal benefit for the actor subsequent to its completion. (Here, we privilege psychological altruism as opposed to biological altruism, which is often dismissed by the “selfish gene” theory of Darwinian selection and notions of reproductive fitness.) Most people, however, assume psychologically-based altruistic acts exist because they believe an act that does not demand or expect overt reciprocity or recognition by the recipient (or others) is so defined. But is this view sufficiently comprehensive, and is it really possible to behave toward others in a way that is completely devoid of self? Is self-interest an ineluctable process with respect to volitional acts of kindness? Here, we explore the likelihood of engaging in an authentically selfless act and capturing true altruism, in general. (Note: For those averse to mathematical jargon, feel free to skip to the paragraph that begins with “[A]t this stage” to get a basic understanding of orthogonality and then move to the next section, “Semantic States as ‘Intrinsic Desires’,” without losing much traction.)

##### The Model

Imagine for a moment every potential (positive) outcome that could emerge as a result of performing some act—say, holding the door for an elderly person. You might receive a “thank you,” a smile from an approving onlooker, someone reciprocating in kind, a feeling you’ve done what your parents (or your religious upbringing) might have expected you to do, perhaps even a monetary reward—whatever. (Note: We assume there will never be an eager desire or expectation for negative consequences, so we require all outcomes to be positive, beneficial events. Of course, a comprehensive model would also include the desire to avoid negative consequences—the ignominy of failing to return a wallet or aiding a helpless animal (an example we will revisit later)—but these can be transformed into positive statements that avoid the unnecessary complications associated with the contrapositive form.)

We suppose there are n outcomes, and we can imagine each outcome enjoys a certain probability of occurring. We will call this the potential vector $\mathbf{p}$, the components of which are simply the probabilities that each outcome (ordered 1 through n) will occur:

$\mathbf{p} = [p(1), p(2), p(3),\dots,p(n-1),p(n)]$

and $0\leq p(i)\leq 1$ where $\sum_{i=1}^n p(i)$ does not have to equal 1 because events are independent and more than a single outcome is possible. (You might, for example, receive both a “thank you” and a dollar bill for holding the door for an elderly woman.) So, the vector $\mathbf{p}$ represents the agglomeration of the discrete probabilities of every positive thing that could occur to one’s benefit by engaging in the act.

Consider, now, another vector, $\mathbf{q}$, that represents the constellation of desires and expectations for the possible outcomes enumerated in $\mathbf{p}$. That is, if $\mathbf{q} = [q(1),q(2),q(3),\dots,q(n-1),q(n)]$, then $q(i)$ catalogs the interest and desire in outcome $p(i)$. (It might be convenient to imagine $\mathbf{q}$ as a binary vector of length n and an element of $\text{R}_2^n$, but we will be better to treat $\mathbf{q}$ vectors as a subset of the parent vector space $\text{R}^n$ to which $\mathbf{p}$ belongs.) In other words, $q(i) = 0,1$: either you desire the outcome (whose probability is denoted by) $p(i)$ or you don’t. (There are no “probabilities of expectation or desire” in our model.) We will soon see how these vectors address our larger problem of quantifying acts of altruism.

The point $\text{Q}$ in $\text{R}^n$ is determined by $\mathbf{q}$, and we want to establish a plane parallel to (and including) $\mathbf{q}$ with normal vector $\mathbf{p}$. Define a point X generated by a vector $\mathbf{x} = t\mathbf{q}$ where the scalar $t>1$ and $\mathbf{x} = [c_1,c_2,c_3,\dots,c_{n-1},c_n]$. If $\mathbf{p}$ is a normal vector of $\mathbf{x} - \mathbf{q}$, then the normal-form equation of the plane is given by $\mathbf{p}\cdot(\mathbf{x} - \mathbf{q})=0$, and its general equation is

$\sum_{i=1}^n p(i)c_i = p(1)c_1 + p(2)c_2 + \dots + p(n-1)c_{n-1} + p(n)c_n=0$.

We now have a foundation upon which to establish a basic, quantifiable metric for altruism. If we assume, as we did above, that an altruistic act benefits the recipient and fails to generate any positive benefits for the actor, then such an act must involve potential and expectation vectors whose scalar product equals zero, which means they stand in an orthogonal (i.e., right-angle) relationship to each other. It is interesting to note there are only two possible avenues for $\mathbf{p}$$\mathbf{q}$ orthogonality within our model: (a) the actor desires and/or expects absolutely no rewards (i.e., $\mathbf{q}=0$), which is the singular and generally understood notion of altruism, and (b) the actor only desires and/or expects rewards that are simply impossible (i.e., $p(i)=0$ where $q(i)=1$). (We will assume $\mathbf{p}\neq0$.) In all other cases, the scalar product will be greater than zero, violating the altruism requirement that there be no benefit to the actor. Framed another way, (the vector of) an altruistic act forms part of a basis for a subspace in $\text{R}^n$.

At this stage, it might be beneficial to pause and walk through a very easy example. Imagine there are only three possible outcomes for buying someone their morning coffee at Starbucks: (1) the recipient says “thank you,” (2) someone buys your coffee for you (“paying it forward”), and (3) the person offers to pay your mortgage. A reasonable potential vector might be [0.9, 0.5, 0]—i.e., there’s a 90% chance you’ll get a “thank you,” a 50% chance someone else will buy your coffee for you, and a zero-percent chance this person will pay your mortgage. Now, assume your expectation vector for those outcomes is [1, 0, 0]—you expect people to say “thank you” when someone does something nice for them, but you don’t expect someone to buy your coffee or pay your mortgage as a result. The scalar product is greater than zero ($0.9(1) + 0.5(0) + 0^2 = 0.9$), which means the act of buying the coffee fails to meet the requirement for altruism (i.e., the potential vector is not orthogonal to the plane that includes Q and X = tq). In this example, as we’ve seen in the general case, the only way buying the coffee could have been an altruistic act is if (a) the actor expects or desires no outcome at all or (b) the actor expected or desired her mortgage to be paid (and nothing else). We will discuss later the reasonableness of the former scenario. (It might also be interesting to note the model can quantify the degree to which an act is altruistic.)

The above formalism will work in every case where there is a single, fixed potential vector and a specified constellation of expectations; curious readers, however, might be interested in cases where there exists a non-scalar-multiple range of expectations (i.e., when X $=\mathbf{x}\neq t\mathbf{q}$ for some scalar t), and we can dispatch the formalism fairly quickly. In these cases, orthogonality would involve a specific potential vector and a plane involving the displacement of expectation vectors. The vector form of this plane is $\mathbf{x}=\mathbf{q} + t_1\mathbf{u} + t_2\mathbf{v}$, and direction vectors $\mathbf{u}$,$\mathbf{v}$ are defined as follows:

$\mathbf{u}=\overrightarrow{QS}=[s(1)-q(1),s(2)-q(2),\ldots,s(n-1)-q(n-1),s(n)-q(n)]$

with $\mathbf{v}$ defined similarly for points Q and R; $t_i$ are scalars (possibly understood as time per some unit of measurement for a transition vector), and points S and R of the direction vectors are necessarily located on the plane in question. Unpacking the vector form of the equation yields the following matrix equation:

$\begin{bmatrix}c_1\\c_2\\c_3\\ \vdots\\c_{n-1}\\c_n\end{bmatrix}=\begin{bmatrix}q(1)\\q(2)\\q(3)\\ \vdots\\q(n-1)\\q(n)\end{bmatrix}+t_1\begin{bmatrix}s(1)-q(1)\\s(2)-q(2)\\s(3)-q(3)\\ \vdots\\s(n-1)-q(n-1)\\s(n)-q(n)\end{bmatrix}+t_2\begin{bmatrix}r(1)-q(1)\\r(2)-q(2)\\r(3)-q(3)\\ \vdots\\r(n-1)-q(n-1)\\r(n)-q(n)\end{bmatrix}$

whose parametric equations are

$\begin{matrix}c_1=q(1)+t_1[s(1)-q(1)]+t_2[r(1)-q(1)]\\ \vdots\\ c_n=q(n)+t_1[s(n)-q(n)]+t_2[r(n)-q(n)].\end{matrix}$

It’s not at all clear how one might interpret “altruistic orthogonality” between a potential vector and a transition or range (i.e., subtraction) vector of expectations within this alternate plane, but it will be enough for now to consider its normal vectors—one at Q and, if we wish, one at X (through the appropriate mathematical adjustments)—as secondary altruistic events orthogonal to the relevant plane intersections:

$p_1(1)c_1 - p_2(1)c_1 + p_1(2)c_2 - p_2(2)c_2 + \dots + p_1(n)c_n - p_2(n)c_n = 0.$

##### Semantic States as ‘Intrinsic Desires’

To this point, we’ve established a very simple mathematical model that allows us to quantify a notion of altruism, but even this model hinges on the likelihood that one’s expectation vector equals zero: an actor neither expects nor desires any outcome or benefit from engaging in the act. This seems plausible for events we can recognize and catalog (e.g., reciprocal acts of kindness, expressions of affirmation, etc.), but what about the internal motivations—philosophers refer to these as—that very often drive our decision-making process? What can we say about acts that resonate with these subjective, internal motivations like religious upbringing, a generic sense of rectitude, cultural conditioning, or the Golden Rule? These intrinsic desires must also be included in the collection of benefits we might expect to gain from engaging in an act and, thus, must be included in the set of components of potential outcomes. If you’ve been following the above mathematical discussion, such internal states guarantee non-orthogonality; that is, they secure a scalar for $\mathbf{p}\cdot\mathbf{q}$ because $p_k,q_k >0$ for some internal state k. This means internal states belie a genuine act of altruism. It is important to note, too, these acts are closely associated with notions of social exchange theory, where (1) “assets” and “liabilities” are not necessarily objective, quantifiable things (e.g., wealth, beauty, education, etc.) and (2) one’s decisions often work toward shrinking the gap between the perceived self and ideal self. (See, particularly, Murstein, 1971.) In considering the context of altruism, internal states combine these exchange features: An act that aligns with some intrinsic desire will bring the actor closer to the vision of his or her ideal self, which, in turn, will be subjectively perceived and experienced as an asset. Altruism is perforce banished in the process.

So, the question then becomes: Is it possible to act in a way that is completely devoid of both a desire for external rewards and any motivation involving intrinsic desires, internal states that provide (what we will conveniently call) semantic assets? As I hope I’ve shown, yes, it is (mathematically) possible—and in light of that, then, I might have been better served placing quotes around the word myth in the title—but we must also ask ourselves the following question: How likely it is that an act would be genuinely altruistic given our model? If we imagine secondary (non-scalar) planes $P_1, P_2,\dots, P_n$ composed of expectation vectors from arbitrary points $p_1,p_2,\dots,p_n$ (with $p_j \in P_j$) parallel to the x-axis, as described above, then it is easy to see there are a countably infinite number of planes orthogonal to the relevant potential vector. (Assume $q\neq 0$ because if q is the zero vector, it is orthogonal to every plane.) But there are an (uncountably) infinite number of angles $0<\theta<\pi$ and $\theta\neq\pi/2$, which means there exists a far greater number of planes that are non-orthogonal to a given potential vector, but this only considers $\theta$ rotations in $\text{R}^2$ as a two-dimensional slice of our outcome space $\text{R}^n$. As you might be able to visualize, the number of non-orthogonal planes grows considerably if we include $\theta$ rotations in $\text{R}^3$. Within the context of three dimensions, and to get a general sense of the unlikelihood of acquiring random orthogonality, suppose there exists a secondary plane, as described above, for every integer-based value of $0<\theta<\pi$ (and $\theta\neq\pi/2$) with rotations in $\text{R}^2$; then the probability of a potential vector being orthogonal to a randomly chosen plane $P_j$ of independent expectation vectors is highly improbable: p = 1/178 = 0.00561797753, a value significant to eleven digits. If we include $\text{R}^3$ rotations to those already permitted, the p-value for random orthogonality decreases to 0.00001564896, which is a value so small as to be essentially nonexistent. So, although altruism is theoretically possible because our model admits the potential for orthogonality, our model also suggests such acts are quite unlikely, especially for large n. For philosophically sophisticated readers, the model supports the theory of psychological altruism (henceforth ‘PA’) that informs the vast majority of decisions we make in response to others, but based on p-values associated with the prescribed model, I would argue we’re probably closer to Thomas Hobbes’s understanding of psychological egoism (henceforth ‘PE’), even though the admission of orthogonality subverts the totalitarianism and inflexibility inherent within PE.

One final thought explicates the obvious problem with our discussion to this point: There isn’t any way to quantify probabilities of potential outcomes based on events that haven’t yet happened, even though we know intuitively such probabilities, outcomes, and expectations exist. To be sure, the concept of altruism is palpably more philosophical or psychological or Darwinian than mathematical, but our model is successful in its attempt to provide a skeletal structure to a set of disembodied, intrinsic desires—to posit our choices are, far more often than they are not, means to ends (whether external or internal) rather than selfless, other-directed ends in themselves.

##### Some Philosophical Criticisms

Philosophical inquiry concerning altruism is rich and varied. Aristotle believed the concept of altruism—the specific word was not coined until 1851 by Auguste Comte—was an outward-directed moral good that benefited oneself, the benefits accruing in proportion to the number of acts committed. Epicurus argued that selfless acts should be directed toward friends, yet he viewed friendship as the “greatest means of attaining pleasure.” Kant held for acts that belied self-interest but argued, curiously, they could also emerge from a sense of duty and obligation. Thomas Hobbes rejected the notion of altruism altogether; for him, every act is pregnant with self-interest, and the notion of selflessness is an unnatural one. Nietzsche felt altruistic acts were degrading to the self and sabotaged each person’s obligation to pursue self-improvement and enlightenment. Emmanuel Levinas argued individuals are not ends in themselves and that our priority should be (and can only be!) acting benevolently and selflessly towards others—an argument that fails to address the conflict inherent in engaging with a social contract where each individual is also a receiving “other.” (This is the problem with utilitarian-based approaches to altruism, in general.) Despite the varied historical analyses, nearly every modern philosopher (according to most accounts) rejects the notion of psychological egoism—the notion that every act is driven by benefits to self—and accepts, as our model admits, that altruism does motivate a certain number of volitional acts. But because our model suggests very low p-values for PA, it seems prudent to address some of the specific arguments against a prevalent, if not unshirted, egoism.

1. Taking the blue pill: Testing for ‘I-desires’

Consider the following story:

Mr. Lincoln once remarked to a fellow passenger…that all men were prompted by selfishness in doing good. His [companion] was antagonizing this position when they were passing over a corduroy bridge that spanned a slough. As they crossed this bridge they espied an old razor-backed sow on the bank making a terrible noise because her pigs had got into the slough and were in danger of drowning. [M]r. Lincoln called out, ‘Driver can’t you stop just a moment?’ Then Mr. Lincoln jumped out, ran back and lifted the little pigs out of the mud….When he returned, his companion remarked: ‘Now Abe, where does selfishness come in on this little episode?’ ‘Why, bless your soul, Ed, that was the very essence of selfishness. I should have had no peace of mind all day had I gone on and left that suffering old sow worrying over those pigs.’ [Feinberg, Psychological Altruism]

The author continues:

What is the content of his desire? Feinberg thinks he must really desire the well-being of the pigs; it is incoherent to think otherwise. But that doesn’t seem right. Feinberg says that he is not indifferent to them, and of course, that is right, since he is moved by their plight. But it could be that he desires to help them simply because their suffering causes him to feel uncomfortable (there is a brute causal connection) and the only way he has to relieve this discomfort is to help them. Then he would, at bottom be moved by an I-desire (‘I desire that I no longer feel uncomfortable’), and the desire would be egoistic. Here is a test to see whether the desire is basically an I-desire. Suppose that he could simply have taken a pill that quietened the worry, and so stopped him being uncomfortable, and taking the pill would have been easier than helping the pigs. Would he have taken the pill and left the pigs to their fate? If so, the desire is indeed an I-desire. There is nothing incoherent about this….We can apply similar tests generally. Whenever it is suggested that an apparently altruistic motivation is really egoistic, since it [is] underpinned by an I-desire, imagine a way in which the I-desire could be satisfied without the apparently altruistic desire being satisfied. Would the agent be happy with this? If they would, then it is indeed an egoistic desire. if not, it isn’t.

This is a powerful argument. If one could take a pill—say, a tranquilizer—that would relieve the actor from the discomfort of engaging the pigs’ distress, which is the assumed motivation for saving the pigs according to the (apocryphal?) anecdote, then the volitional act of getting out of the coach and saving the pigs must then be considered a genuinely altruistic act because it is directed toward the welfare of the pigs and is, by definition, not an “I-desire.” But this analysis makes two very large assumptions: (1) there is a singular motivation behind an act and (2) we can whisk away a proposed motivation by some physical or mystical means. To be sure, there could be more than one operative motivation for an action—say, avoiding discomfort and receiving a psychosocial reward—and the thought-experiment of a pill removing the impetus to act does not apply in all cases. Suppose, for example, one only desires to avoid the pigs’ death and not the precursor of their suffering. Is it meaningful to imagine the possibility of a magical pill that could avoid the pigs’ death? If by the “pill test” we intend to eviscerate any and all possible motivations by some fantastic means, then we really haven’t said much at all. We’ve only argued the obvious tautology: that things would be different if things were different. (Note: the conditional A –> A is always true, which means A <–> A is, too.) Could we, for example, apply this test to our earlier coffee experiment? Imagine our protagonist could take a pill that would, by acting on neurochemical transmitters, magically satisfy her expectation and desire for being thanked for purchasing the coffee. Can we really say her motivation is now altruistic, presumably because the pill has rendered an objective “thank you” from the recipient unnecessary? In terms of our mathematical model, does the pill create a zero expectation vector? It’s quite difficult to imagine this is the case; the motivation—that is, the expectation of, and desire for, a “thank you”—is not eliminated because it is fulfilled by a different mechanism.

2. Primary object vs. Secondary possessor

As a doctor who desires to cure my patient, I do not desire pleasure; I desire that my patient be made better. In other words, as a doctor, not all my particular desires have as their object some facet of myself; my desire for the well-being of my patient does not aim at alteration in myself but in another. My desire is other-regarding; its object is external to myself. Of course, pleasure may arise from my satisfied desire in such cases, though equally it may not; but my desire is not aimed at my own pleasure. The same is true of happiness or interest: my satisfied desire may make me happy or further my interest, but these are not the objects of my desire. Here, [Joseph] Butler simply notices that desires have possessors – those whose desires they are – and if satisfied desires produce happiness, their possessors experience it. The object of a desire can thus be distinguished from the possessor of the desire: if, as a doctor, my desire is satisfied, I may be made happy as a result; but neither happiness nor any other state of myself is the object of my desire. That object is other-regarding, my patient’s well-being. Without some more sophisticated account, psychological egoism is false. [See Butler, J. (1726) Fifteen Sermons Preached at the Rolls Chapel, London]

Here, the author errs not in assuming pleasure can be a residual feature of helping his patients—it can be—but in presuming his desire for the well-being of others is a first cause. It is likely that such a desire originates from a desire to fulfill the Hippocratic oath, to avoid imposing harm, which demands professional and moral commitments from a good physician. The desire to be (seen as) a good physician, which requires a (“contrapositive”) desire to avoid harming patients, is clearly a motivation directed toward self. Receiving a “thank you” for buying someone’s coffee might create a feeling of pleasure within the actor (in response to the pleasure felt and/or exhibited by the recipient), but the pleasure of the recipient is not necessarily (and is unlikely to be) a first cause. If it were a first (and only) cause, then all the components of the expectation vector would be zero and the act would be considered altruistic. Notice we must qualify that if-then statement with the word “only” because our model treats such secondary “I-desires” as unique components of the expectation vector. (“Do I desire the feeling of pleasure that will result in pleasing someone else when I buy him or her coffee?”) We will set aside the notion that an expectation of a residual pleasurable feeling in response to another’s pleasure is not necessarily an intrinsic desire. I can expect to feel good in response to doing X without desiring, or being motivated by, that feeling—this is the heart of the author’s argument—but if any part of the motivation for buying the coffee involves a desire to receive pleasure—even if the first cause involves a desire for the pleasure of others—then the act cannot truly be cataloged as altruistic because, as mentioned above, it must occupy a component within q. The issue of desire, then, requires an investigation into first causes (i.e., “ultimate”) motivations, and the logical fallacy of Joseph Butler’s argument (against what is actually psychological hedonism) demands it.

3. Sacrifice or pain

Also taken from the above link:

A simple argument against psychological egoism is that it seems obviously false….Hume rhetorically asks, ‘What interest can a fond mother have in view, who loses her health by assiduous attendance on her sick child, and afterwards [sic] languishes and dies of grief, when freed, by its death, from the slavery of that attendance?’ Building on this observation, Hume takes the ‘most obvious objection’ to psychological egoism.[A]s it is contrary to common feeling and our most unprejudiced notions, there is required the highest stretch of philosophy to establish so extraordinary a paradox. To the most careless observer there appear to be such dispositions as benevolence and generosity; such affections as love, friendship, compassion, gratitude. […] And as this is the obvious appearance of things, it must be admitted, till some hypothesis be discovered, which by penetrating deeper into human nature, may prove the former affections to be nothing but modifications of the latter. Here Hume is offering a burden-shifting argument.  The idea is that psychological egoism is implausible on its face, offering strained accounts of apparently altruistic actions. So the burden of proof is on the egoist to show us why we should believe the view.

Sociologist Emile Durkheim argued that altruism involves voluntary acts of “self-destruction for no personal benefit,” and like Levinas, Durkheim believed selflessness was informed by a utilitarian morality despite his belief that duty, obligation, and obedience to authority were also counted among selfless acts. The notion of sacrifice is perhaps the most convincing counterpoint to overriding claims to egoism. It is difficult to imagine a scenario, all things being equal, where sacrifice (and especially pain) would be a desired outcome. It would seem that a decision to act in the face of personal sacrifice, loss, or physical pain would almost certainly guarantee a genuine expression of altruism, yet we must again confront the issue of first causes. In the case of the assiduous mother, sacrifice might service an intrinsic (and “ultimate”) desire to be considered a good mother. In the context of social-exchange theory, the asset of being (perceived as) a good mother outweighs the liability inherent within self-sacrifice. Sacrifice, after all, is what good mothers do, and being a good mother resonates more closely with the ideal self, as well as society’s coeval definition of what it means to be a “good mother.” In a desire to “do the right thing” and “be a good mother,” then, she chooses sacrifice. It is the desire for rectitude (perceived or real) and the positive perception of one’s approach to motherhood, not solely the sacrifice itself, that becomes the galvanizing force behind the act. First causes very often answer the following question: “What would a good [insert category or group to which membership is desired] do?”

What of pain? We can imagine a scenario in which a captured soldier is being tortured in the hope he or she will reveal critical military secrets. Is the soldier acting altruistically by enduring intense pain rather than revealing the desired secrets? We can’t say it is impossible, but, here, the aegis of a first cause likely revolves around pride or honor; to use our interrogative test for first causes: “Remaining true to a superordinate code is what [respected and honorable soldiers] do.” They certainly don’t dishonor themselves by betraying others, even when it’s in one’s best interest to do so. Recalling Durkheim’s definition, obedience (as distinct from the obligatory notion of duty) also plays an active role here: Honorable soldiers are required to obey the established military code of conduct, so the choice to endure pain might be motivated by a desire to be (seen as) an obedient and compliant soldier who respects the code rather than (merely) an honorable person, though these two things are nearly inextricably enmeshed. To highlight a relevant religious example, Jesus’ sacrifice on the cross might not be considered a truly altruistic act if the then-operative value metric privileged a desire to be viewed by the Father as a good, obedient Son, who was willing to sacrifice Himself for humanity, above the sacrifice (and pain) associated with the crucifixion. (This is an example where the general criticism of Durkheim’s “utilitarian” altruism fails; Jesus did not receive from His utilitarian sacrifice in the way mankind did.) These are complex motivations that require careful parsing, but there’s one thing we do know: If neither sacrifice nor pain can be related to any sort of intrinsic desire that satisfies the above interrogative test, then it probably should be classified as altruistic, even though, as our model suggests, this is not likely to be the case.

4. Self-awareness

Given the arguments, it is still unclear why we should consider psychological egoism to be obviously untrue.  One might appeal to introspection or common sense, but neither is particularly powerful. First, the consensus among psychologists is that a great number of our mental states, even our motives, are not accessible to consciousness or cannot reliably be reported…through the use of introspection. While introspection, to some extent, may be a decent source of knowledge of our own minds, it is fairly suspect to reject an empirical claim about potentially unconscious motivations….Second, shifting the burden of proof based on common sense is rather limited. Sober and Wilson…go so far as to say that we have ‘no business taking common sense at face value’ in the context of an empirical hypothesis. Even if we disagree with their claim and allow a larger role for shifting burdens of proof via common sense, it still may have limited use, especially when the common sense view might be reasonably cast as supporting either position in the egoism-altruism debate.  Here, instead of appeals to common sense, it would be of greater use to employ more secure philosophical arguments and rigorous empirical evidence.

In other words, we cannot trust thought processes in evaluating our motivations to act. We might think we’re acting altruistically—without any expectations or desires—but we are often mistaken because, as our earlier examples have shown, we fail to appreciate the locus of first causes. (It is also probably true, for better or worse, that most people prefer to think of themselves more highly than they ought—a process that better approaches exchange ideas of the ideal self in choosing how and when to act.) Jeff Schloss, the T.B. Walker Chair of Natural and Behavioral Sciences at Westmont College, suggests precisely this when he states that “people can really intend to act without conscious expectation of return, but that [things like intrinsic desires] could still be motivating certain actions.” The interrogative test seems like one easy way to clarify our subjective intuitions surrounding what motivates our actions, but we need more tools. Our model seems to argue that the burden of proof for altruism rests with the actor—“proving,” without resorting to introspection, one’s expectation vector really is zero—rather than “proving” the opposite, that egoism is the standard construct. Our proposed p-values based on the mathematics of our model strongly suggest the unlikelihood of a genuine altruism for a random act (especially for large n), but despite the highly suggestive nature of the probability values, it is unlikely they rise to the level of “empirical evidence.”

##### Conclusion

Though I’ve done a little work in a fun attempt to convince you genuine altruism is a rather rare occurrence, generally speaking, it should be said that even if my basic conceit is accurate, this is not a bad thing! The “intrinsic desires” and (internal) social exchanges that often motivate our decision-making process (1) lead to an increase in the number of desirable behaviors and (2) afford us an opportunity to better align our actions (and ourselves) with a subjective vision of an “ideal self.” We should note, too, the “subjective ideal self” is frequently a reflection of an “objective ideal ([of] self)” constructed and maintained by coeval social constructs. This is a positive outcome, for if we only acted in accordance with genuine altruism, there would be a tragic contraction of good (acts) in the world. Choosing to act kindly toward others based on a private desire that references and reinforces self in a highly abstract way stands as a testament to the evolutionary psychosocial sophistication of humans, and it evinces the kind of higher-order thinking required to assimilate into, and function within, the complex interpersonal dynamic demanded by modern society. We should consider such sophistication to be a moral and ethical victory rather than the evidence of some degenerate social contract surreptitiously pursued by selfish persons.

References:

Bernard Murstein (Ed.). (1971). Theories of Attraction and Love. New York, NY: Springer Publishing Company, Inc.

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# To Infinity…and Beyond!

In researching Gödel’s Incompleteness Theorem, I stumbled upon an article that stated no one has proven a line can extend infinitely in both directions. This is shocking, if it’s true, and after a quick Google search, I couldn’t seem to find anything that contradicts the claim. So, in the spirit of intellectual adventure, I’ll offer a fun proof-esque idea here.

Consider a line segment of length $\ell$ that is measured in some standard unit of distance/length (e.g., inches, miles, nanometers, etc.). We convert the length of $\ell$—whatever units and length we’ve chosen (say, 0.5298 meters)—into a fictitious unit of measurement we’ll call Hoppes (hpe) [pronounced HOP-ease]. So, now, one should consider the length of $\ell$ to be 2 hpe such that $\ell$/2 = 1 hpe. We then add some fraction (of the length of) $\ell$ to (both ends of) itself, and let’s say the fraction of $\ell$ we’ll use, call it $a$, is 3$\ell$/4, which equals 3/2 hpe. The process by which we will add $a$ to $\ell$ will be governed by the following geometric series:

$s_n(a) = 1+a+a^2+a^3+\dots+a^{n-1} = (1-a^n)(1-a)^{-1}=a^{n-1}(a-1)^{-1}$.

Let us add the terms of $s_n(a)$ to both sides of $\ell$; first, we add 1 hpe to both sides ($\ell=4$ hpe), then 3/2 hpe ($\ell=7$ hpe, then 9/4 hpe ($\ell=23/2$ hpe)and so forth. If we keep adding to $\ell$ units of hpe based on the series $s_n(a)$, then we’re guaranteed a line that extends infinitely in both directions because $\lim_{n\rightarrow\infty} (a^{n}-1)(a-1)^{-1} = \infty$ when $\vert a\vert \geq 1$.

Now, suppose we assume it is impossible to extend our line segment infinitely in both directions. Then $s_n(a)$ must converge to $(1-a)^{-1}$, giving us a total length of $2+(1-a)^{-1}$ hpe for $\ell$, because $\lim_{n\rightarrow\infty} 1-a^{n}=1$, which is only possible when $\vert a\vert < 1$. (We cannot have a negative length, so $a\in \text{R}^+_0$.) But this contradicts our $\vert a\vert$ value of 3/2 hpe above, which means the series $s_n(a)$ is divergent.  $\Box$

N.B. Some might raise the “problem” of an infinite number of discrete points that composes a line (segment), recalling the philosophical thorniness of Zeno’s (dichotomy) paradox; this is resolved, however, by similarly invoking the concept of limits (and is confirmed by our experience of traversing complete distances!):

$\sum_{i=1}^{\infty} (1/2)^i=\frac{1}{2}\sum_{i=0}^{\infty} (1/2)^i=\frac{1}{2} s_n (\frac{1}{2})=\frac{1}{2}\Big( 1+\frac{1}{2}+(\frac{1}{2})^2+\cdots\Big)=\frac{1}{2}\Big(\frac{1}{1-\frac{1}{2}}\Big) = 1$,

a single unit we can set equal to our initial line segment $\ell$ with length 2 hpe.

Special thanks to my great friend, Tim Hoppe, for giving me permission to use his name as an abstract unit of measurement.

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MATHEMATICS

# Dr. Who, Fibonacci’s Rabbits, and the Wasp Apocalypse (Updated)

Chapter XII from Fibonacci’s Liber abaci describes the following scenario:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?

Solving this riddle, of course, yields the famous Fibonacci sequence:

$\text{F}_n = 1,1,2,3,5,8,13,21,34,55,89,144,223,377,610,987,1597,2584,4181,6765...$

where the nth term in the sequence is the sum of the previous two terms (n-1 and n-2). That is, $\text{F}_n = \text{F}_{n-1} + \text{F}_{n-2}$ where n > 2.

What is much less well known is that a renegade group of rabbits escaped this enclosure and were later captured after trying to overrun a garden owned by a resident in a nearby town. These rabbits, they discovered, have a unique physiology: they procreate at a much faster rate. We’ll call this the “renegade sequence”:

$\text{R}_n = 1,2,4,6,15,18,46,50,115,120,....$

$\text{Figure 1:}$ $g(x) = 0.7131e^{0.5477x}$

In the original storyline for “The Ark in Space” (1975), Dr. Who battles the Wirrn, “a wasp creature [that lays] its eggs inside cryo-preserved humans”; it just so happens the Wirrn’s reproductive pattern follows $\text{R}_n$ (but at a much faster rate—in thousands of births per hour), and after traveling back to the twelfth century to recover the rabbits in order to study their anomalous physiology, Dr. Who tries unsuccessfully to unravel the sequence, a necessary step toward predicting the exact date the Wirrns will take over the earth while also determining whether a proposed vaccine is guaranteed to work fast enough to avert human extinction. Unfortunately, everyone dies (including Dr. Who, whose regenerative powers are neutralized by a wasp-like venom), the invading creatures repopulate the earth, and the series ends. (Cooler heads prevailed, though, and the story was rewritten.)

Figure 1 shows $\text{R}_n$ as a function of time (in n hours) plotted with the regression line (red) defined in the caption (R^2 = 0.9829). Before he received the fatal sting, however, Dr. Who realized the renegade sequence can be derived from an underlying sequence, call it $\text{S}_n$, defined by $\text{R}_n = \sum_{i=1}^n \text{S}_{i}$, but he was unable to define an equation that calculates $\text{R}_n$ precisely, one that predicts the total growth of Wirrns for any hour n after the initial infection.

Challenge: Find an equation that calculates any n in $\text{R}_n$ by defining a function $f : \text{Z}^+_0\to \text{Z}$ that generates $\text{S}_n$.

(I developed this sequence from my work on stock-market trends. I’ll post the answer in an update TBD.)

The answer, based on $\text{S}_n$, is

$\text{R}_n = \sum_{i=1}^n \lceil{\frac{x_i}{2}}\rceil^{3-2\bar{x_i}} + 0^{\bar{x_i}}$

where $\bar{x} \equiv x\pmod{2}$.

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# Toward a quantification of intellectual disciplines

As a mathematician, I often find myself taking the STEM side of the STEM-versus-liberal-arts-and-humanities debate—this should come as no surprise to readers of this blog—and my principal conceit, that of a general claim to marginal productivity, quite often (and surprisingly, to me) underwhelms my opponents. So, I’ve been thinking about how we might (objectively) quantify the value of a discipline. May we argue, if we can, that quantum mechanics is “more important” than, say, the study of Victorian-period literature? Is the philosophy of mind as essential as the macroeconomics of international trade? Are composers of dodecaphonic concert music as indispensable to the socioeconomic fabric as historians of WWII? Is it really possible to make such comparisons, and should we be making them at all? Are all intellectual pursuits equally justified? If so, why should that be the case, and if not, how can society differentiate among so many disparate modes of inquiry?

To that end, then, I’ve quickly drafted eleven basic categories I believe can aid us in the quantification of an intellectual pursuit:

I. Demand

This will perforce involve a few slippery statistical calculations: average annual salary (scaled to cost-of-living expenses) for similar degree holders (e.g., BSc, PhD, etc.), the size of associated university departments, job-placement rates among graduates with the same terminal degree, the number of relevant publications (both popular and academic), and anything that betrays a clear supply-and-demand approach to the activities of participants within a discipline and the output they generate.

II. Influence

How fertile is the (inter-field) progeny of research? How often are articles cited by other disciplines? Do the articles, conferences, and symposia affect a diverse collection of academic research in different fields with perhaps sweeping consequences, or does the intellectual offspring of an academic discipline rarely push beyond the confines of its participants?

III. Difficulty

What is the effort required for mastery and original contribution? In general, we place a greater value on things that take increased effort to attain. It’s easier, for example, to eat a pizza than to acquire rock-hard abs. (As an aside, and apart from coeval psychosexual aspects of attraction—obesity was considered a desirable trait during the twelfth to fifteenth centuries because it signified wealth and power—being fit holds greater societal value because it, among other things, represents the more difficult, ascetic path, which suggests something of an evolutionary advantage.) Average time to graduation, the number of prerequisite courses for degree candidacy, and the rigor of standardized tests might also play a useful role here.

IV. Applicability

How practical is the discipline’s intellectual import? How much utility does it possess? Does it (at least, eventually) lead to a general increase in the quality of life for the general population (e.g., the creation of plastics), or is it limited in its scope and interest only to those persons with a direct relationship to its machinery (e.g., non-commutative transformational symmetries in the development of Mozart’s Piano Sonata no. 12 in F major K. 332)? A less diplomatic characterization might involve asking the simple question: Who cares?

V. Recognition

Disciplines and academic fields that enjoy major prizes (e.g., Nobel, Pulitzer, Fields, Abel, etc.) must often succumb to more rigorous scrutiny and peer-reviewed analysis than those disciplines whose metrics more heavily rely upon the opinion of a small cadre of informed peers and the publish-or-perish repositories of second-tier journals willing to print marginal material. This isn’t a rigid metric, of course: Many economists now reject the Nobel-winning efficient-market hypothesis, and the LTCM debacle of the late 90s revealed the hidden perniciousness crouching behind the Black-Scholes equation, which also earned its creators a Nobel prize. (Perhaps these examples suggest something deficient about economics.) In general, though, winning a major international prize is a highly valued accomplishment that validates one’s work as enduring and important.

VI. Objectivity

Can we prove the propositions of an academic discipline, or are its claims wholly unfalsifiable? Is the machinery of an intellectual discipline largely based upon subjective and intuitive interpretation or rigorously defined axioms? Can the value and importance of a conceit change if coeval opinion modulates its position? It seems desirable to prefer an objective and provable claim to one based on subjectivity and a mushy, ever-changing worldview.

VII. Future value

What is the potential influence surrounding the field’s unsolved problems? Do experts generally believe resolving those issues might eventually lead to significant breakthroughs (or possibly chaos!), or will the discipline’s elusive solutions effectuate only incremental and localized progress when viewed through the widest possible lens?

VIII. Connectivity

What might be the long-range repercussions of eliminating a discipline? Would anyone beyond its active members notice its absence? How essential is its intellectual currency to our current socioeconomic infrastructure? One or two generations removed from our own? There exists inherent value in the indispensable.

IX. Ubiquity

How many colleges and universities offer formal, on-campus degrees in the field? Is its study limited to regional or localized interests, or is it embraced by a truly international collective? Wider academic availability, regardless of where you live, suggests a greater general value.

X. Labor mobility

Is employment contingent upon a specific geographic area or narrowly defined economies? Does an intellectual discipline provide global opportunity? Do gender gaps or racial-bias issues exist that might impede entry for qualified candidates? How flexible is the discipline’s intellectual infrastructure? Do the skills you acquire permit productivity within a range of disparate occupations and applications, or do they translate poorly to other sectors of the the labor market because graduates are pigeonholed into a singular intellectual activity?

Can you find meaningful employment without going to graduate school, or must you finish a PhD in order to be gainfully employed? There are certain exceptions, of course: brain surgeons, for example, enjoy a very limited employment landscape—and earning anything less than an M.D. degree means you can’t practice medicine—but this is an example of an outlier that offer counterbalancing compensation within the larger model.

XI. Automation

What is the probability a discipline will be automated in the future? Can your field easily be replaced by a robot or a sufficiently robust AI (or even new advances in classical computer algorithms) in the next 15 years? (Luddites beware.)

__________

Not perfect, but it’s a pretty good start, I think. The list strikes a decent balance across disciplines and, taken as a whole, doesn’t necessarily privilege any particular field. A communications major, for example, might score near the top in labor mobility, automation, and ubiquity but very low in difficulty and prize recognition (and likely most other categories, too). I also eliminated certain obvious categories (like historical import) because the history of our intellectual landscape has often been marked by hysteria, inaccuracy, and misinformation. To privilege, say, music(ology) because of its membership to the quadrivium when most people believed part of its importance revolved around its ability to affect the four humors seems unhelpful. (It also seems unfair to penalize, say, risk analysts because the stock market didn’t exist in the sixth century.)

Where we go from here is anyone’s guess. Specific quantifying methods might only require the most obvious metric: a function $f : \text{R}^n\to \text{R}$ with a series of weightings where n is the total number of individual categories, $c_i$, and the total value of a discipline, $v_j$, is calculated by a geometric mean, provided no category can have a value of zero: $v_j = \left(\prod_{i=1}^n c_i\right)^{1/n}.$