# Working with Convex Responses: Antifragility from Finance to Oncology

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## Abstract

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## 1. Introduction: Where the Idea of Antifragility Came From

- The expectation must be conditioned on absence of “blowup”, that is, the left tail of the distribution must be constrained (see Geman et al., 2015) [4], which involves all higher moments of $f\left(x\right)$.
- The payoff functions are almost never monotone.
- Taking into account higher moments of the distributions is analogous to going beyond second-order effects: third, fourth, etc.

Simply, a coffee cup on a table suffers more from large deviations than from the cumulative effect of some shocks—conditional on being unbroken, it has to suffer more from “tail” events than regular ones around the center of the distribution, the ‘at-the-money’ category. This is the case of elements of nature that have survived: conditional on being in existence, then the class of events around the mean should matter considerably less than tail events, particularly when the probabilities decline faster than the inverse of the harm, which is the case of all used monomodal probability distributions. Further, what has exposure to tail events suffers from uncertainty; typically, when systems—a building, a bridge, a nuclear plant, an airplane, or a bank balance sheet—are made robust to a certain level of variability and stress but may fail or collapse if this level is exceeded, then they are particularly fragile to uncertainty about the distribution of the stressor, hence to model error, as this uncertainty increases the probability of dipping below the robustness level, bringing a higher probability of collapse. In the opposite case, the natural selection of an evolutionary process is particularly antifragile, indeed a more volatile environment increases the relative survival rate of robust species and eliminates those whose superiority over other species is highly dependent on environmental parameters.

## 2. Medicine and Convexity

- A convex response to energy balance over a fixed time window necessarily implies gains from intermittent fasting in some situations and under some strict conditions (that is, higher variance in the distribution of nutrients) over some range within the limits of that time window;
- The presence of metabolic problems in populations that have a steady supply of food intake, as well as evidence of human fitness to an environment that provides moderate variations in the availability of food, both necessarily imply a concave response to food within a range and time frame.

#### Antifragility in Treatment Scheduling

## 3. Antifragility in Oncology

#### 3.1. Defining (Local) Fragility in Oncology

#### 3.2. Fragility and Taylor Series Approximations

#### 3.3. Fragility and Finite Differences

#### 3.4. Applications of Hill Function

#### 3.5. The First-Order Sigmoid Curve

- Pure sigmoids with smoothness characteristics expressed in trigonometric or exponential form, $f:\mathbb{R}\to [0,1]$:$$f\left(x\right)=\frac{1}{2}tanh\left(\right)open="("\; close=")">\frac{\kappa x}{\pi}$$$$f\left(x\right)=\frac{1}{1-{e}^{-ax}}$$
- Gompertz functions (a vague classification that includes above curves but can also mean special functions).
- Special functions with support in $\mathbb{R}$ such as the error function $f:\mathbb{R}\to [0,1]$$$f\left(x\right)=-\frac{1}{2}\mathrm{erfc}\left(\right)open="("\; close=")">-\frac{x}{\sqrt{2}}$$
- Special functions with support in $[0,1]$, such as $f:[0,1]\to [0,1]$$$f\left(x\right)={I}_{x}(a,b),$$
- Special functions with support in $[0,\infty )$$$f\left(x\right)=Q\left(\right)open="("\; close=")">a,0,\frac{x}{b}$$
- Piecewise sigmoids, such as the CDF of the Student distribution$$f\left(x\right)=\left(\right)open="\{"\; close>\begin{array}{c}\frac{1}{2}{I}_{\frac{\alpha}{{x}^{2}+\alpha}}\left(\right)open="("\; close=")">\frac{\alpha}{2},\frac{1}{2}\hfill \\ x\le 0\hfill \end{array}\frac{1}{2}\left(\right)open="("\; close=")">{I}_{\frac{{x}^{2}}{{x}^{2}+\alpha}}\left(\right)open="("\; close=")">\frac{1}{2},\frac{\alpha}{2}+1\hfill $$

#### 3.6. Some Necessary Relations Leading to a Sigmoid Curve

## 4. The Generalized Dose–Response Curve

- ${S}^{N}$(−∞) = ${k}_{L}$, and
- ${S}^{N}$($+\infty $) = ${k}_{R}$, and (equivalently for the first and last of the following conditions)
- $\frac{{\partial}^{2}{S}^{N}}{\partial {x}^{2}}$≥ 0 for x∈ (−∞, ${k}_{1}$), $\frac{{\partial}^{2}{S}^{N}}{\partial {x}^{2}}$< 0 for x∈ (${k}_{2}$, ${k}_{>2}$), and $\frac{{\partial}^{2}{S}^{N}}{\partial {x}^{2}}$≥ 0 for x∈ (${k}_{>2}$, ∞), with ${k}_{1}>{k}_{2}\ge \dots \ge {k}_{N}$.

#### Antifragility and Heterogeneity

## 5. Nonlinearities and Medical Iatrogenics

- Convexity for a dose-response function increases fragility (from the expansion of the left tail in response to the increase in the scale of the distribution).
- Detection of a nonlinearity allows the prediction of fragility and helps formulate probabilistic decisions without much knowledge of the probability distribution beyond minimum standard attributes.
- The presence of concavity in the tails of the distribution implies a silent risk.

#### 5.1. Effect Reversal

The standard model currently in use applies a linear scale, extrapolating cancer risk from high doses to low doses of ionizing radiation. However, our discovery of DSB clustering over such large distances casts considerable doubts on the general assumption that risk to ionizing radiation is proportional to dose, and instead provides a mechanism that could more accurately address risk dose dependency of ionizing radiation.

#### 5.2. Nonlinearity of NNT and the Consequences

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Antifragility Indirectly Detected in the Various Literature

Field | Papers |
---|---|

Mithridatization and hormesis | Kaiser (2003) [16], Rattan (2008) [46], Calabrese and Baldwin (2002, 2003a, 2003b) [17,47,48], Aruguman et al. (2006) [49]. |

Caloric restriction and hormesis | Martin, Mattson et al. (2006) [12] |

Cancer treatment and fasting | Longo et al. (2010) [50], Safdie et al. (2009) [51], Raffaghelo et al. (2010), [52], Lee et al. (2012) [53] |

Aging and intermittence | Fontana et al. [54] |

For brain effects | Anson, Guo, et al. (2003) [55], Halagappa, Guo, et al. (2007) [56], Stranahan and Mattson (2012) [57]. The long-held belief that the brain needed glucose, not ketones, and that the brain does not go through autophagy, has been progressively replaced. |

Yeast and longevity under restriction | Fabrizio et al. (2001) [58]; SIRT1, Longo et al. (2006) [59], Michan et al. (2010) [60] |

Diabetes, remission or reversal | Taylor (2008) [61], Lim et al. (2011) [62], Boucher et al. (2004) [63]; diabetes management by diet alone, early insights in Wilson et al. (1980) [64]. Couzin (2008) [65] gives insight that blood sugar stabilization does not have the effect anticipated (there need to be stressors). The ACCORD study (Action to Control Cardiovascular Risk in Diabetes) found no benefits from lowering blood glucose levels. Synthesis, Skyler et al. (2009) [66], old methods, Westman and Vernon (2008) [67]. Bariatric (or other) surgery as alternative to intermittent fasting: Pories (1995) [68], Guidone et al. (2006) [69], Rubino et al. 2006 [70] |

Ramadan and effect of fasting | Trabelsi et al. (2012) [71], Akanji et al. (2012). Note that the Ramadan time window is short (12 to 17 h) and possibly fraught with overeating so conclusions need to take into account energy balance and that the considered effect is at the low-frequency part of the timescale. |

Caloric restriction | Harrison (1984), Wiendruch (1996), Pischon (2008) |

Autophagy for cancer | Kondo et al. (2005) [72] |

Autophagy (general) | Danchin et al. (2011) [73], He et al. (2012) [74] |

Fractional dosage | Wu et al. (2016) [75] |

Jensen’s inequality in exercise | Many such as Schnohr and Marott (2011) [76], intermittent extremes vs. moderate physical activity. |

Cluster of ailments | Yaffe and Blackwell (2004) [77], Alzheimer and hyperinsulenemia, Razay and Wilcock (1994) [78]; Luchsinger, Tang, et al. (2002) [79], Luchsinger Tang et al. (2004) [80] Janson, Laedtke, et al. (2004) [81]. |

Benefits ofsome type ofstress (and convexity of the effect) | For the different results from the two types of stressors, short and chronic, Dhabar (2009) “A hassle a day may keep the pathogens away: the fight-or-flight stress response and the augmentation of immune function” [82]. For the benefits of stress on boosting immunity and cancer resistance (squamous cell carcinoma), Dhabhar et al. (2010) [83], Dhabhar et al. (2012) [84], Ansbacher et al. (2013) [85] |

Iatrogenics of hygiene and systematic elimination of germs | Rook (2011) [86], Rook (2012) [87] (auto-immune diseases from absence of stressors), Mégraud and Lamouliatte (1992) [88] for Helyobacter Pilori and incidence of cancer. |

## Appendix B. Simple Convexity and Its Effects

## Appendix C. Relaxing the Assumption of Fixed Treatment Schedules

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**Figure 1.**Fragility below level K as indicative of survival. It is not quite symmetric because global antifragility is conditioned on tail robustness (“to do well, one must first survive”). The Taleb and Douady (2013) [2] paper shows that the gap between ${\int}^{K}f(x,\sigma )dx$ and ${\int}^{K}f(x,\sigma +\Delta )dx$, where $\sigma $ is the scale of the distribution, is proportional to the concavity of $f\left(x\right)$. Hence, without knowing the distribution (PDF above), one can gauge such an effect by looking at the nonlinearity of $f(.)$ below the threshold K.

**Figure 2.**Simple (first-order) nonincreasing or nondecreasing sigmoids, defined as floored and capped increasing functions. They map to the payoff in finance of a binary option with time left to expiration. As the sigmoid loses smoothness (with the decreased time to expiration), it becomes, at the limit, a Heaviside function, see Figure 3.

**Figure 4.**Example treatment-scheduling protocols. (

**A**) Input distribution of dosing is typically unimodal (“even”) or bimodal (“uneven”). (

**B**) Protocols are typically fixed, with doses administered at regular intervals. It may be feasible to temporarily increase the dose (green), with periodic treatment holidays. (

**C**) Even treatment is optimal to maximize response for concavity; uneven for convexity.

**Figure 5.**These three graphs (related to the convex (concave) transformations of random variables) summarize and simplify our main idea; they show how we can go from the reaction or dose response $S\left(x\right)$, combined with the probability distribution of x, to the probability distribution of $S\left(x\right)$ and its properties: mean, expected benefits or harm, variance of $S\left(x\right)$. Thus, we can play with the various parameters that can affect $S\left(x\right)$ and those that can affect the distribution of x, and extract results from the output. $S\left(x\right)$, as we show, can take different forms (we chose a monotone convex or concave $S\left(x\right)$, but a second-order mixed sigmoid can also be used).

**Figure 6.**The second derivative is an approximation for fragility for low values of h. (

**A**) Hill function, $H\left(x\right)$ (Equation (11)) shown for $n=10$, ${E}_{0}=0$, ${E}_{1}=100$, and $C=10$. Analytically derived second derivative (Equation (12)) is shown in the bottom panel. (

**B**) Difference between fragility and second derivative at various dose values (red to blue) corresponding to panel A. As $h\to 0$, the error approaches zero: $F(x,h)-{h}^{2}\frac{{d}^{2}H}{d{x}^{2}}\to 0$.

**Figure 7.**(

**A**) How a fractional intervention is more effective to surpass a threshold than a constant dosage of the same average. This is akin to stochastic resonance (in physics), by which the presence of noise causes the signal to rise above the detection threshold. For instance, genetically modified BT crops produce a constant level of pesticide, which appears to be much less effective than occasional manual interventions to add doses to conventional plants. The same may apply to antibiotics, chemotherapy, and radiation therapy [32]. (

**B**) How more variance impacts the exceedance over the threshold. If threshold ≥ mean, we have convexity, and the variance increases the payoff more than variations in the mean. Such an effect is proportional to the remoteness of such threshold. Note that the harm function is defined as positive.

**Figure 8.**(

**Left**) A time series illustration of how a higher variance (hence scale), given the same mean, allow more spikes, hence an antifragile effect. We have random paths of two gamma distributions of the same mean, different variances, ${X}_{1}\sim G(1,1)$ and ${X}_{2}\sim G(\frac{1}{10},10)$, showing higher spikes and maxima for ${X}_{2}$. The effect depends on the norm ${\left|\right|.\left|\right|}_{\infty}$, more sensitive to tail events, even more than just the scale which is related to the norm ${\left|\right|.\left|\right|}_{2}$. (

**Right**) Representation of antifragility of (

**Left**) in distribution space: we show the probability of exceeding a certain threshold for a variable, as a function of $\sigma $, the scale of the distribution, while keeping the mean constant.

**Figure 9.**(

**A**) Every (relatively) smooth dose response with a floor has to be initially convex, hence prefers variations. (

**B**) Every (relatively) smooth dose response with a ceiling has to be concave while approaching the ceiling, hence prefers stability.

**Figure 11.**Generalizing the Dose–Response Curve, ${S}^{2}\left(\right)open="("\; close=")">x;{a}_{1},{a}_{2},{b}_{1},{b}_{2},{c}_{1},{c}_{2}$ The convex part in the increasing section is what we call “antifragile”.

**Figure 12.**Relationship between convexity and mixed, heterogeneous populations. (

**A**) Dose response shown for sensitive (green) and resistant (red) cell lines. When mixed, dose response is a weighted average of each (Equation 15; black). (

**B**) Fragility shown for sensitive (green) and resistant (red) cell lines. When mixed, fragility (black) switches from locally convex to locally concave multiple times.

**Figure 13.**Drug benefits when convex to numbers needed to treat (NNT) in the left part, with gross iatrogenics invariant to condition (the constant line). We are assuming a standard sigmoidal benefit function.

**Figure 14.**Unseen risks and mild gains: translation of Figure 13 into a probabilistic representation, showing to the skewness of a decision involving iatrogenics when the condition is mild. This also gives the intuition of the Taleb and Douady [2] translation theorems from concavity for $S\left(x\right)$ into probabilistic attributes.

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## Share and Cite

**MDPI and ACS Style**

Taleb, N.N.; West, J.
Working with Convex Responses: Antifragility from Finance to Oncology. *Entropy* **2023**, *25*, 343.
https://doi.org/10.3390/e25020343

**AMA Style**

Taleb NN, West J.
Working with Convex Responses: Antifragility from Finance to Oncology. *Entropy*. 2023; 25(2):343.
https://doi.org/10.3390/e25020343

**Chicago/Turabian Style**

Taleb, Nassim Nicholas, and Jeffrey West.
2023. "Working with Convex Responses: Antifragility from Finance to Oncology" *Entropy* 25, no. 2: 343.
https://doi.org/10.3390/e25020343