# Quantifying and Interpreting Connection Strength in Macro- and Microscopic Systems: Lessons from Bell’s Approach

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. A Macroscopic Game

## 3. Winning by Cheating

**Cheat**

**1:**

**Cheat**

**2:**

#### 3.1. Measure ${\mu}_{1}$

**Example**

**1.**

#### 3.2. Measure ${\mu}_{2}$

**Example**

**2.**

#### 3.3. Measuring Connection Strength

**Theorem**

**1.**

**Proof.**

## 4. Winning with Additional Resources from Nature: Entangled Particles

## 5. Social and Financial Systems

## 6. Discussion: Connection Strength with Bell’s Approach

#### 6.1. Connection Strength in Macroscopic Systems; Elements of Reality

#### 6.2. The Hidden Variable Space

#### 6.3. Causality, Connection Strength and Explainability in Machine Learning

#### 6.4. Connection Strength in Quantum Systems, Free Choice and Locality

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) The flow of information in the macroscopic game is shown by dotted lines in Panel (

**a**). The flow of information does not necessarily represent causal influences as Alice and Bob may generate their answers $a,b$ by using strategies that ignore information that they receive. (

**b**) The right-hand panel shows the intended separation between private (insider) information (which should only be available for a) by a Chinese Wall in the financial example described in Section 5. Public information is not shown as it is available to all participants. The red arrow shows a breach of the Chinese Wall.

**Table 1.**The following table illustrates a specific strategy, using Cheat 1 and communication in each round.

x | Message | Regime | a | b | $\mathit{a}\mathit{b}$ |
---|---|---|---|---|---|

0 | None | $xy\in \{00,01\}$ | $+1$ | $+1$ | $+1$ |

1 | Yes | $xy=10$ | $+1$ | $+1$ | $+1$ |

1 | Yes | $xy=11$ | $+1$ | $-1$ | $-1$ |

**Table 2.**The following table illustrates a specific strategy using Cheat 2, where both Alice and Bob know the regime in each round.

x | y | Regime Known | a | b | $\mathit{ab}$ |
---|---|---|---|---|---|

0 | 0 | yes | $+1$ | $+1$ | $+1$ |

0 | 1 | yes | $+1$ | $+1$ | $+1$ |

1 | 0 | yes | $+1$ | $+1$ | $+1$ |

1 | 1 | yes | $+1$ | $-1$ | $-1$ |

**Table 3.**The following table shows the different situations for a specific strategy using Cheat 1, where Alice sends text messages in some rounds in which she has received the question $x=1$.

x | Message | Regime | a | b | $\mathit{ab}$ | Round In |
---|---|---|---|---|---|---|

0 | None | $xy\in \{00,01,10,11\}$ | $+1$ | $+1$ | $+1$ | $\overline{T}$ |

1 | Yes | $xy=10$ | $+1$ | $+1$ | $+1$ | T |

1 | Yes | $xy=11$ | $+1$ | $-1$ | $-1$ | T |

**Table 4.**The following table illustrates a specific strategy using Cheat 2, where Alice and Bob know the regime in some rounds.

x | y | Regime Known | a | b | $\mathit{ab}$ | Round In |
---|---|---|---|---|---|---|

0 | 0 | irrelevant | $+1$ | $+1$ | $+1$ | $K\cup \overline{K}$ |

0 | 1 | irrelevant | $+1$ | $+1$ | $+1$ | $K\cup \overline{K}$ |

1 | 0 | irrelevant | $+1$ | $+1$ | $+1$ | $K\cup \overline{K}$ |

1 | 1 | no | $+1$ | $+1$ | $+1$ | $\overline{K}$ |

1 | 1 | yes | $+1$ | $-1$ | $-1$ | K |

**Table 5.**The following table illustrates the behavior of Alice and Bob. They both conduct only typical activities, except Bob behaves atypically on days when he receives confirmation from someone with private information confirming that market rumors are true.

x | y | Situation | a | b | $\mathit{ab}$ |
---|---|---|---|---|---|

0 | 0 | no large order, no rumors | $+1$ | $+1$ | $+1$ |

0 | 1 | no large order, but (unfounded) rumors | $+1$ | $+1$ | $+1$ |

1 | 0 | large order, but no rumors | $+1$ | $+1$ | $+1$ |

1 | 1 | large order and rumors, but no confirmation to Bob | $+1$ | $+1$ | $+1$ |

1 | 1 | large order and rumors and confirmation to Bob (“K”) | $+1$ | $-1$ | $-1$ |

**Table 6.**The following table illustrates the behavior in the case that Bob sometimes reacts to rumors by executing unusual transactions in the market and never receives information from the private side.

x | y | Situation | a | b | $\mathit{ab}$ | Prob. |
---|---|---|---|---|---|---|

0 | 0 | no large order, no rumors | $+1$ | $+1$ | $+1$ | $100\%$ |

0 | 1 | no large order, but (unfounded) rumors | $+1$ | $-1$ | $-1$ | p |

0 | 1 | no large order, but (unfounded) rumors | $+1$ | $+1$ | $+1$ | $1-p$ |

1 | 0 | large order, but no rumors | $+1$ | $+1$ | $+1$ | $100\%$ |

1 | 1 | large order, (true) rumors | $+1$ | $-1$ | $-1$ | ${p}^{\prime}$ |

1 | 1 | large order, (true) rumors | $+1$ | $+1$ | $+1$ | $1-{p}^{\prime}$ |

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Gallus, C.; Blasiak, P.; Pothos, E.M.
Quantifying and Interpreting Connection Strength in Macro- and Microscopic Systems: Lessons from Bell’s Approach. *Entropy* **2022**, *24*, 364.
https://doi.org/10.3390/e24030364

**AMA Style**

Gallus C, Blasiak P, Pothos EM.
Quantifying and Interpreting Connection Strength in Macro- and Microscopic Systems: Lessons from Bell’s Approach. *Entropy*. 2022; 24(3):364.
https://doi.org/10.3390/e24030364

**Chicago/Turabian Style**

Gallus, Christoph, Pawel Blasiak, and Emmanuel M. Pothos.
2022. "Quantifying and Interpreting Connection Strength in Macro- and Microscopic Systems: Lessons from Bell’s Approach" *Entropy* 24, no. 3: 364.
https://doi.org/10.3390/e24030364