# Achieving Perfect Coordination amongst Agents in the Co-Action Minority Game

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## Abstract

**:**

## 1. Introduction

## 2. Definition of the Model

## 3. Optimality of the Cyclic State

## 4. A Coordination Strategy to Reach the Periodic State

- If $r=1$, clearly, the only agent is assigned the available ID, and he knows his ID, and the algorithm ends.
- If $r>1$, the agents that are not to be assigned IDs in this round continue with the same choice as previous day, until the end of the algorithm. The agents use their personal random number generators to break this set of r agents into two smaller roughly equal sets, of sizes ${j}_{1}$ and $r-{j}_{1}$, where ${j}_{1}$ is a random variable. Now, the algorithm recursively assigns to the first set the IDs from $R+1$ to $R+{j}_{1}$, and then the remaining IDs from $R+{j}_{1}+1$ to $R+r$ to the second set, and the algorithm ends.

## 5. Expected Time to Reach the Cyclic State

## 6. Summary and Concluding Remarks

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References and Note

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**Figure 1.**Ranking of N = 11 agents using the proposed algorithm. The numerals above the agents indicates their assigned IDs.

**Figure 2.**Numerically determined exact values of ${T}_{n}$ for $n\le 30$. The equation of the approximate linear fit here is $y=1.4449x-1.0451$.

**Figure 3.**Log-periodic oscillations in the function ${H}^{*}\left(y\right)$ as a function of ${log}_{2}y$, determined by numerically summing the series in Equation (14), about the mean value $1.44269504089$. Note the small amplitude of the oscillations.

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**MDPI and ACS Style**

Rajpal, H.; Dhar, D.
Achieving Perfect Coordination amongst Agents in the Co-Action Minority Game. *Games* **2018**, *9*, 27.
https://doi.org/10.3390/g9020027

**AMA Style**

Rajpal H, Dhar D.
Achieving Perfect Coordination amongst Agents in the Co-Action Minority Game. *Games*. 2018; 9(2):27.
https://doi.org/10.3390/g9020027

**Chicago/Turabian Style**

Rajpal, Hardik, and Deepak Dhar.
2018. "Achieving Perfect Coordination amongst Agents in the Co-Action Minority Game" *Games* 9, no. 2: 27.
https://doi.org/10.3390/g9020027