# Arbitrage Equilibrium, Invariance, and the Emergence of Spontaneous Order in the Dynamics of Bird-like Agents

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## Abstract

**:**

## 1. Self-Organization in Active Matter: Background

## 2. Statistical Teleodynamics of Flocking: A Game-Theoretic Formulation

#### 2.1. Agent’s Utility: Position Dependence

#### 2.2. Agent’s Utility: Velocity Dependence

#### 2.3. Agent’s Effective Utility

## 3. Results

#### 3.1. Stability of the Arbitrage Equilibrium

**Disturbance 1**: Velocity disturbance, where each agent’s velocity is changed to a random orientation and magnitude.**Disturbance 2**: Position disturbance, where each agent’s position is randomly changed.**Disturbance 3**: Position and velocity disturbance, where both position and velocity vectors are changed.

## 4. Conclusions

## 5. Methods

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Effective utility and its derivative as a function of the number of neighbors, ${n}_{i}$, for different values of alignment ${l}_{i}$ ($\alpha ,\beta ,\gamma ,\delta =0.5,0.005,0.25,1$). There are two points where the derivative of the effective utility is zero for different alignments.

**Figure 2.**Trajectory of the agents and the corresponding phase portrait in ${n}_{i}-{l}_{i}$ space for the average number of neighbors of each individual agent during the course of the simulation, corresponding to (

**a**) Reynolds’ boids for $a=0.5,b=0.01,c=0.5$ and (

**b**) Utility-driven agents for $\alpha =0.5,$$\beta =0.01,\gamma =0.25,\delta =1$.

**Figure 3.**Trajectory of the average of number of neighbors of each agent and average alignment of the agents in the ${n}_{i}$, ${l}_{i}$ phase space, and corresponding estimated averages for the (

**a**) Reynolds’ model (

**b**) Utility-driven model ($\delta =1$).

**Figure 4.**Histogram of the utility (${h}_{i}$) of the agents for $\alpha ,\beta ,\gamma ,\delta =0.5,0.005,0.25,1$ corresponding to (

**a**) $\Delta t=0.01$, (

**b**) $\Delta t=0.1$ and (

**c**) $\Delta t=0.5$. Dashed line shows the average utility at a particular time.

**Figure 5.**The trajectory of the agents as a function of time and the corresponding phase portraits for $\alpha ,\beta ,\gamma ,\delta =0.5,0.005,0.25,1$ for stability analysis case studies (

**a**) Disturbance 1, (

**b**) Disturbance 2, and (

**c**) Disturbance 3. The disturbances occur after equilibrium at a time-step of 101 (in red). The configuration at the end of the simulation is also shown is shown (in green).

Utility of Cohesion | $\alpha {n}_{i}$ | Benefit of having ${n}_{i}$ neighbors |

Disutility of Congestion | $-\beta {n}_{i}^{2}$ | Cost of crowding by ${n}_{i}$ neighbors |

Utility of Alignment | $\gamma {\sum}_{j}{n}_{ij}{\mathbf{s}}_{i}\xb7{\mathbf{s}}_{j}$ | Benefit of being aligned with the neighbors |

Disutility of Competition | $-ln{n}_{i}$ | Cost of competition from the ${n}_{i}$ neighbors |

Entropic restlessness |

Configuration | Predicted Flock Size | Observed Flock Size |
---|---|---|

$\alpha =0.1,\beta =0.005,\gamma =0.25$ | 31.9 | 25.3 ± 8.0 |

$\alpha =0.3,\beta =0.005,\gamma =0.25$ | 53.1 | 50.1 ± 16.0 |

$\alpha =0.5,\beta =0.005,\gamma =0.02$ | 50.0 | 44.5 ± 13.4 |

**Table 3.**Utility of different percentiles of the agents at the 1000th time-step corresponding to Figure 4.

Population | Time-Step Size, $\Delta \mathit{t}$ | Average Utility |
---|---|---|

Top 1% | 0.01 | 23.79 |

0.1 | 23.80 | |

0.5 | 23.76 | |

Top 1–10% | 0.01 | 23.77 |

0.1 | 23.78 | |

0.5 | 23.68 | |

Top 10–50% | 0.01 | 23.60 |

0.1 | 23.51 | |

0.5 | 23.35 | |

Top 50–75% | 0.01 | 22.83 |

0.1 | 22.64 | |

0.5 | 22.58 | |

Bottom $50\%$ | 0.01 | 18.59 |

0.1 | 15.59 | |

0.5 | 17.91 |

Domain | System | Utility Function (${\mathit{h}}_{\mathit{i}}$) |
---|---|---|

Physics | Thermodynamic game | $-\beta ln{E}_{i}-ln{n}_{i}$ |

Biology | Bacterial chemotaxis | $\alpha {c}_{i}-ln{n}_{i}$ |

Ecology | Ant crater formation | $b-{\displaystyle \frac{\omega {r}_{i}^{a}}{a}}-ln{n}_{i}$ |

Sociology | Segregation dynamics | $\eta {n}_{i}-\xi {n}_{i}^{2}+ln(H-{n}_{i})-ln{n}_{i}$ |

Economics | Income game | $\alpha ln{S}_{i}-\beta {\left(ln{S}_{i}\right)}^{2}-ln{n}_{i}$ |

Ecology | Garuds game | $\alpha {n}_{i}-\beta {n}_{i}^{2}+\gamma {n}_{i}{l}_{i}-ln{n}_{i}$ |

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**MDPI and ACS Style**

Sivaram, A.; Venkatasubramanian, V.
Arbitrage Equilibrium, Invariance, and the Emergence of Spontaneous Order in the Dynamics of Bird-like Agents. *Entropy* **2023**, *25*, 1043.
https://doi.org/10.3390/e25071043

**AMA Style**

Sivaram A, Venkatasubramanian V.
Arbitrage Equilibrium, Invariance, and the Emergence of Spontaneous Order in the Dynamics of Bird-like Agents. *Entropy*. 2023; 25(7):1043.
https://doi.org/10.3390/e25071043

**Chicago/Turabian Style**

Sivaram, Abhishek, and Venkat Venkatasubramanian.
2023. "Arbitrage Equilibrium, Invariance, and the Emergence of Spontaneous Order in the Dynamics of Bird-like Agents" *Entropy* 25, no. 7: 1043.
https://doi.org/10.3390/e25071043