# 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

- Toner, J.; Tu, Y.; Ramaswamy, S. Hydrodynamics and phases of flocks. Ann. Phys.
**2005**, 318, 170–244. [Google Scholar] [CrossRef][Green Version] - Narayan, V.; Ramaswamy, S.; Menon, N. Long-lived giant number fluctuations in a swarming granular nematic. Science
**2007**, 317, 105–108. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ramaswamy, S. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys.
**2010**, 1, 323–345. [Google Scholar] [CrossRef][Green Version] - Marchetti, M.C.; Joanny, J.F.; Ramaswamy, S.; Liverpool, T.B.; Prost, J.; Rao, M.; Simha, R.A. Hydrodynamics of soft active matter. Rev. Mod. Phys.
**2013**, 85, 1143. [Google Scholar] [CrossRef][Green Version] - Cates, M.E.; Tailleur, J. When are active Brownian particles and run-and-tumble particles equivalent? Consequences for motility-induced phase separation. EPL Europhys. Lett.
**2013**, 101, 20010. [Google Scholar] [CrossRef][Green Version] - Cates, M.E.; Tailleur, J. Motility-induced phase separation. Annu. Rev. Condens. Matter Phys.
**2015**, 6, 219–244. [Google Scholar] [CrossRef][Green Version] - Gonnella, G.; Marenduzzo, D.; Suma, A.; Tiribocchi, A. Motility-induced phase separation and coarsening in active matter. Comptes Rendus Phys.
**2015**, 16, 316–331. [Google Scholar] [CrossRef][Green Version] - Berkowitz, R. Active Particles Map to Passive Random Walks. Physics
**2020**, 13, s146. [Google Scholar] [CrossRef] - O’Byrne, J.; Tailleur, J. Lamellar to Micellar Phases and Beyond: When Tactic Active Systems Admit Free Energy Functionals. Phys. Rev. Lett.
**2020**, 125, 208003. [Google Scholar] [CrossRef] - Reynolds, C.W. Flocks, herds and schools: A distributed behavioral model. In Proceedings of the 14th annual Conference on Computer Graphics and Interactive Techniques, Anaheim, CA, USA, 27–31 July 1987; pp. 25–34. [Google Scholar]
- Vicsek, T.; Czirók, A.; Ben-Jacob, E.; Cohen, I.; Shochet, O. Novel type of phase transition in a system of self-driven particles. Phys. Rev. Lett.
**1995**, 75, 1226. [Google Scholar] [CrossRef][Green Version] - Ballerini, M.; Cabibbo, N.; Candelier, R.; Cavagna, A.; Cisbani, E.; Giardina, I.; Lecomte, V.; Orlandi, A.; Parisi, G.; Procaccini, A.; et al. Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study. Proc. Natl. Acad. Sci. USA
**2008**, 105, 1232–1237. [Google Scholar] [CrossRef] [PubMed] - Bialek, W.; Cavagna, A.; Giardina, I.; Mora, T.; Silvestri, E.; Viale, M.; Walczak, A.M. Statistical mechanics for natural flocks of birds. Proc. Natl. Acad. Sci. USA
**2012**, 109, 4786–4791. [Google Scholar] [CrossRef] [PubMed] - Bialek, W.; Cavagna, A.; Giardina, I.; Mora, T.; Pohl, O.; Silvestri, E.; Viale, M.; Walczak, A.M. Social interactions dominate speed control in poising natural flocks near criticality. Proc. Natl. Acad. Sci. USA
**2014**, 111, 7212–7217. [Google Scholar] [CrossRef] [PubMed] - Venkatasubramanian, V.; Luo, Y.; Sethuraman, J. How much inequality in income is fair?: A microeconomic game theoretic perspective. Phys. A Stat. Mech. Its Appl.
**2015**, 435, 120–138. [Google Scholar] [CrossRef][Green Version] - Venkatasubramanian, V. Statistical teleodynamics: Toward a theory of emergence. Langmuir
**2017**, 33, 11703–11718. [Google Scholar] [CrossRef] - Venkatasubramanian, V. How Much Inequality Is Fair?: Mathematical Principles of a Moral, Optimal, and Stable Capitalist Society; Columbia University Press: New York, NY, USA, 2017. [Google Scholar]
- Venkatasubramanian, V.; Luo, Y. How much income inequality is fair? Nash bargaining solution and its connection to entropy. In Network Theory and Agent-Based Modeling in Economics and Finance; Springer: Berlin/Heidelberg, Germany, 2019; pp. 159–174. [Google Scholar]
- Kanbur, R.; Venkatasubramanian, V. Occupational arbitrage equilibrium as an entropy maximizing solution. Eur. Phys. J. Spec. Top.
**2020**, 229, 1661–1673. [Google Scholar] [CrossRef] - Venkatasubramanian, V.; Sivaram, A.; Das, L. A unified theory of emergent equilibrium phenomena in active and passive matter. Comput. Chem. Eng.
**2022**, 164, 107887. [Google Scholar] [CrossRef] - Rosenthal, R.W. A class of games possessing pure-strategy Nash equilibria. Int. J. Game Theory
**1973**, 2, 65–67. [Google Scholar] [CrossRef] - Sandholm, W.H. Population Games and Evolutionary Dynamics; MIT Press: Cambridge, MA, USA, 2010. [Google Scholar]
- Easley, D.; Kleinberg, J. Networks, Crowds, and Markets; Cambridge University Press: Cambridge, UK, 2010; Volume 8. [Google Scholar]
- Monderer, D.; Shapley, L.S. Potential games. Games Econ. Behav.
**1996**, 14, 124–143. [Google Scholar] [CrossRef] - Grégoire, G.; Chaté, H. Onset of collective and cohesive motion. Phys. Rev. Lett.
**2004**, 92, 025702. [Google Scholar] [CrossRef][Green Version] - Durve, M.; Peruani, F.; Celani, A. Learning to flock through reinforcement. Phys. Rev. E
**2020**, 102, 012601. [Google Scholar] [CrossRef] [PubMed] - Katare, S.; Venkatasubramanian, V. An agent-based learning framework for modeling microbial growth. Eng. Appl. Artif. Intell.
**2001**, 14, 715–726. [Google Scholar] [CrossRef] - Seborg, D.E.; Edgar, T.F.; Mellichamp, D.A.; Doyle, F.J., III. Process Dynamics and Control; John Wiley & Sons: Hoboken, NJ, USA, 2016. [Google Scholar]
- Åström, K.J.; Murray, R.M. Feedback Systems: An Introduction for Scientists and Engineers; Princeton University Press: Princeton, NJ, USA, 2021. [Google Scholar]

**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