# The More You Know, the More You Can Grow: An Information Theoretic Approach to Growth in the Information Age

## Abstract

**:**

## 1. Introduction

#### 1.1. Relation to Previous Work

#### 1.1.1. Evolutionary Economics: Decomposing Growth Descriptively

#### 1.1.2. Portfolio Theory: Optimizing Growth

#### 1.1.3. Economic Decision Theory: Interpreting Information

#### 1.2. Main Contributions

#### 1.2.1. Combining the Descriptive and the Optimal

#### 1.2.2. Growth as a Communication Process

## 2. Method: Fitness as Informational Fit

^{+}indicates the average updated generation after reproduction (while no superscript refers to the average distribution before updating).

#### 2.1. Decomposing Growth into Bits

#### 2.1.1. Benchmark of the Noiseless Channel

#### 2.1.2. Constraint of the Mixed Fitness Landscape

#### 2.1.3. Remaining Environmental Uncertainty

#### 2.1.4. Directed Selection

#### 2.1.5. Fitness Optimization

#### 2.2. Special Cases

#### 2.2.1. Kelly’s Setup

#### 2.2.2. Non-Diagonal Fitness Matrices

#### 2.2.3. End Result of Selection in Stationary Environments

#### 2.2.4. Perfect Foresight

#### 2.3. The More Populations Know, the More They Can Grow

## 3. Results: Empirical Applications

#### 3.1. Global Resources: Informing Division of Labor

#### 3.2. Big Data: Informing Business Growth Strategies

## 4. Discussion

## Supplementary Materials

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Google Trends data for the search engine terms ‘chocolate’ and ‘diet’ from May 2004 to February 2015. Vertical shadings indicate different environmental states (see Results section).

**Figure 2.**The communication channel between the environment and the average updated population. (

**a**) Representation as a traditional fitness matrix for the binary case. The fitness values in brackets show the case of the diagonal fitness matrix with type fitness ${}^{d}W$. (

**b**) Representation as a noiseless communication channel with transition probabilities ${p}^{+}({g}^{+}|e)$ for the binary case. The diagonal fitness matrix results in the noiseless channel, where only the identity transitions are non-zero: ${p}^{+}({g}^{+}=i|e=i)>0$, for all $i$.

**Figure 3.**The typical sets of the environmental states $E$ and the average updated future generation ${G}^{+}$, both over a large number of periods $t$. The transmission over the channel between the environment and the average updated population induces uncertainty to the identification of each environmental state during reception by the population. The uncertainty that the environmental state $(e=1)$ is sent over the channel is the conditional entropy of ${G}^{+}$: $H({G}^{+}|(e=1))$. According to the asymptotic equipartition property, there are approximately ${2}^{H({G}^{+}|(e=1))}$ of those. The total number of typical ${G}^{+}$ sequences is $\approx {2}^{tH({G}^{+})}$. Restricting ourselves to the subset of channel input such that the corresponding typical output sets do not overlap (see also Supplementary Section 1), we can bound the number of non-confusable inputs by dividing the size of the typical output set by the size of each typical-output-given-typical-input set: ${2}^{tH({G}^{+}|E)}$. The total number of disjoint and non-confusable sets is less than or equal to: ${2}^{t(H({G}^{+})-H({G}^{+}|E))}={2}^{t\text{}I({G}^{+};E)}$.

**Figure 4.**Venn diagram/I-diagram representation of mutual information. (

**a**) Optimal bet-hedging in Kelly’s case of the diagonal fitness matrix. Mutual information can be calculated as the difference between uncertainties: $H(E)-H(E|C)=I(E;C)$. It is always nonnegative in the two-variable case, as conditioning reduces uncertainty. (

**b**) Optimal bet-hedging with mixed non-diagonal fitness matrix, inside the region of bet-hedging. Also in the three variable case the circles are entropies and the intersections mutual information. One way to calculate the joint intersection of all three variables is: $I({G}^{+};E;C)=H(E)-H(E|{G}^{+})-\text{}I({G}^{+};E\text{}|C)$. In the case of bet-hedging inside the bet-hedging region the three involved variables form a Markov chain $E\leftrightarrow {G}^{+}\leftrightarrow C$. This implies that $E$ and $C$ do not have any mutual information outside of ${G}^{+}$ (${G}^{+}$ absorbs all common structure through optimal growth); or $I(E;C\text{}|{G}^{+})=0$. This can be shown by the reformulation $I(E;C\text{}|{G}^{+})=H(E|{G}^{+})-H(E\text{}|C,{G}^{+})$ (which holds in general). It shows that $H(E|{G}^{+})=H(E\text{}|C,{G}^{+})$. This means that from the perspective of the updated population, additional cues do not affect the perceived distribution of the environment (in the case of bet-hedging inside the bet-hedging region, compare with values in Table 2). A perfect cue in terms of a Venn diagram representation would imply a picture in Figure 4b similar to the complete overlap shown in Figure 4a, with the difference that $C$ and ${G}^{+}$ are switched. From Markovity it follows that in this case the uncertainty of the updated population cannot be smaller than the entropy of the cue, as it is completely absorbed through updating: $H({G}^{+})\ge H(E)=H(C)$. This follows from the data processing inequality [14]: $H({G}^{+})\ge I({G}^{+};E)\ge I({G}^{+};C)=H(E)=H(C)$.

**Figure 6.**Empirical growth of Google Trends data for the search engine terms “chocolate” and “diet” from May 2004 to February 2015, and optimized growth when following different bet-hedging strategies.

**Table 1.**Comparison of the descriptive decomposition (Equation (2)) and its special cases (Equations (6)–(10).

Log of Population Growth | Noiseless Channel | Fitness Landscape Constraint | Remaining Environmental Uncertainty | Directed Selection | |
---|---|---|---|---|---|

$\mathrm{log}\stackrel{\overline{\xaf}}{W}=$ | ${E}_{e}\left[\mathrm{log}{}_{hyp}{}^{d}W\right]$ | $-\text{}{D}_{KL}({P}^{+}(e|g)\parallel M(e|g))$ | $-H(E|{G}^{+})$ | $-\text{}{D}_{KL}({P}^{+}(g,e)\parallel P(g,e))$ | Equation (2) |

Kelly’s case no bet-hedging | $={E}_{e}\left[\mathrm{log}{}^{d}W\right]$ | + 0 | $-H(E)$ | $-{D}_{KL}(P(e)||P(g))$ | Equation (8) |

Kelly’s case with bet-hedging | $={E}_{e}\left[\mathrm{log}{}^{d}W\right]$ | + 0 | $-H(E)$ | +0 | Equation (7) |

optimal inside bet-hedging region | $={E}_{e}\left[\mathrm{log}{}_{hyp}{}^{d}W\right]$ | + 0 | $-H(E|{G}_{s}^{+})$ | $-\text{}I({G}_{s}^{+};E)$ | Equation (6) |

$={E}_{e}\left[\mathrm{log}{}^{d}W\right]$ | + 0 | $+\text{}\mathbf{0}$ | $-H(E)$ | Equation (7) | |

stable shares outside bet-hedging region | $={E}_{e}\left[\mathrm{log}{}^{d}W\right]$ | $-\text{}{D}_{KL}({P}_{s}^{+}(e|g)\parallel M(e|g))$ | $-H(E)$ | + 0 | Equation (9) |

optimal with perfect cue | $={E}_{e}\left[\mathrm{log}{}^{d}W\right]$ | $-\text{}{D}_{KL}({P}_{s}^{+}(e|g)\parallel M(e|g))$ | $+\text{}0$ | $+\text{}0$ | Equation (10) |

**Table 2.**Decomposition of global resource extraction between 1900 and 1998 into different cases. $H,\text{}{D}_{KL}$ and $I$ measured in bits.

$\mathbf{log}\stackrel{\overline{\xaf}}{\mathit{W}}$ | = | ${\mathit{E}}_{\mathit{e}}\left[\mathbf{log}{}_{\mathit{h}\mathit{y}\mathit{p}}{}^{\mathit{d}}\mathit{W}\right]$ | $-{\mathit{D}}_{\mathit{K}\mathit{L}}({\mathit{P}}^{+}(\mathit{e}|\mathit{g})\parallel \mathit{M}(\mathit{e}|\mathit{g}))$ | $-\mathit{H}(\mathit{E}|{\mathit{G}}^{+})$ | $-{\mathit{D}}_{\mathit{K}\mathit{L}}({\mathit{P}}^{+}(\mathit{g},\mathit{e})\parallel \mathit{P}(\mathit{g},\mathit{e}))$ | |
---|---|---|---|---|---|---|

Equation (2) descriptive | 0.02803 | = | 1.02702 | − 0.00005 | − 0.99873 | − 0.00021 |

Equation (6) optimal inside bet-hedging region | 0.02822 | = | 1.02702 | + 0 | − 0.99867 | $-\text{}I({G}_{s}^{+};E)$ − 0.000123 |

Equation (11) bet-hedging with cue WW2 in b-h region | 0.02823 | = | 1.02702 | + 0 | − 0.99867 | $-\text{}I({G}_{s}^{+};E\text{}|C)$ − 0.000111 |

**Table 3.**Exploration of different cases of the growth of Google Trends data for the search engine terms “chocolate” and “diet” from May 2004 to February 2015.

$\mathbf{log}\stackrel{\overline{\xaf}}{\mathit{W}}$ | = | ${\mathit{E}}_{\mathit{e}}\left[\mathbf{log}{}_{\mathit{h}\mathit{y}\mathit{p}}{}^{\mathit{d}}\mathit{W}\right]$ | $-\text{}{\mathit{D}}_{\mathit{K}\mathit{L}}({\mathit{P}}^{+}(\mathit{e}|\mathit{g})\parallel \mathit{M}(\mathit{e}|\mathit{g}))$ | $-\mathit{H}(\mathit{E}|{\mathit{G}}^{+})$ | $-\text{}{\mathit{D}}_{\mathit{K}\mathit{L}}({\mathit{P}}^{+}(\mathit{g},\mathit{e})\parallel \mathit{P}(\mathit{g},\mathit{e}))$ | |
---|---|---|---|---|---|---|

Equation (2) descriptive | 0 | = | 0.98474 | − 0.00074 | −0.98207 | − 0.00193 |

Equation (6) optimal in bet-hedging region | 0.00229 | = | + 0 | − 0.98073 | $-\text{}I({G}_{s}^{+};E)$ − 0.00173 | |

With cue Sept-Dec. | 0.00554 | = | − 0.00154 | − 0.97718 | − 0.00048 | |

With cues Sept-Dec. & Jan. | 0.00987 | = | − 0.00783 | − 0.96703 | + 0 | |

Equation (10) optimal with perfect cue | 0.07421 | = | − 0.91053 | + 0 | + 0 |

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Hilbert, M.
The More You Know, the More You Can Grow: An Information Theoretic Approach to Growth in the Information Age. *Entropy* **2017**, *19*, 82.
https://doi.org/10.3390/e19020082

**AMA Style**

Hilbert M.
The More You Know, the More You Can Grow: An Information Theoretic Approach to Growth in the Information Age. *Entropy*. 2017; 19(2):82.
https://doi.org/10.3390/e19020082

**Chicago/Turabian Style**

Hilbert, Martin.
2017. "The More You Know, the More You Can Grow: An Information Theoretic Approach to Growth in the Information Age" *Entropy* 19, no. 2: 82.
https://doi.org/10.3390/e19020082