# Harnessing the Computational Power of Fluids for Optimization of Collective Decision Making

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## Abstract

**:**

## 1. Introduction

#### 1.1. Competitive Multi-Armed Bandit Problem (CBP)

#### 1.2. TOW Dynamics

#### 1.3. The TOW Bombe

## 2. Results for CBP

**SM**) is a state in which the maximum amount of total reward is obtained by all the players. In this problem, the social maximum corresponds to a segregation state in which the players choose the top three distinct machines ($C,D,E$), respectively; there are six segregation states indicated by

**SM**in the Tables. In contrast, the Nash equilibrium (

**NE**) is a state in which all the players choose machine E independent of others’ decisions; machine E gives the reward with the highest probability, when each player behaves selfishly.

## 3. Results for the Extended Prisoner’s Dilemma Game

- A: keep silent;
- B: confess (implicate him- or herself);
- C: implicate the next person (circulative as 1,2,3,1,2,3,⋯);
- D: implicate the third person (circulative as 1,2,3,1,2,3,⋯);
- E: implicate both of the others.

- the set of reward probabilities for the charges $(0,0,0)$ is ($R2$, $R2$, $R2$);
- the $(1,1,1)$ → ($R1$, $R1$, $R1$);
- the $(2,1,1)$ or $(1,2,1)$ or $(1,1,2)$ → (R, R, R): the social maximum;
- the $(2,2,2)$ → (P,P,P): the Nash equilibrium.

## 4. Conclusions

## 5. Discussion

## Methods

#### The Weighting Parameter ω

#### TOW Dynamics for General BP

#### Generating Methods of Fluctuations

#### Internal Fixed Fluctuations

#### Internal Random Fluctuations

- r is a random value from $[0,1]$. We call this “seed”.
- There are $NM$ ($=15$) possibilities for a seed position. Choose the seed position (${i}_{0}$, ${k}_{0}$) randomly from ${i}_{0}$ = $1,\cdots ,3$ and ${k}_{0}$ = $1,\cdots ,5$ and place the seed r at the point,$$Shee{t}_{({i}_{0},{k}_{0})}=r.$$
- All elements of the ${k}_{0}$th column other than (${i}_{o}$, ${k}_{0}$) are substituted with $-0.5\ast r$.
- All elements of the ${i}_{0}$-th row other than (${i}_{o}$, ${k}_{0}$) are substituted with $-0.25\ast r$.
- All remaining elements are substituted with $r/8.0$.
- The matrix sheet is summed up in a summation matrix $Su{m}_{(i,k)}$.
- Repeat from two to six for D times. Here, D is a parameter.

#### Internal M-Random Fluctuations (Exponential)

- For each player i, independent random value ${r}_{i}$ is generated from $[0,1]$. We call these “seeds”.
- There are ${N}^{M}$ ($=125$) possibilities for a seed position pattern. For each player i, choose the seed position (i, ${k}_{0}\left(i\right)$) randomly from ${k}_{0}\left(i\right)$ = $1,\cdots ,5$ and place the seed ${r}_{i}$ at the point$$Shee{t}_{(i,{k}_{0}\left(i\right))}={r}_{i}.$$
- For each i, all elements of the ${k}_{0}\left(i\right)$-th column other than (i, ${k}_{0}\left(i\right)$) are substituted with $-0.5\ast {r}_{i}$.
- All remaining elements of the 1th row are substituted with $-0.50\ast ({r}_{1}-0.50\ast {r}_{2}-0.50\ast {r}_{3})$.
- All remaining elements of the 2th row are substituted with $-0.50\ast ({r}_{2}-0.50\ast {r}_{1}-0.50\ast {r}_{3})$.
- All remaining elements of the 3th row are substituted with $-0.50\ast ({r}_{3}-0.50\ast {r}_{1}-0.50\ast {r}_{2})$.
- The matrix sheet is summed up in a summation matrix $Su{m}_{(i,k)}$.
- Repeat from two to seven for D times. Here, D is a parameter.

#### External Oscillations

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Competitive bandit problem (CBP). (

**a**) segregation state; (

**b**) collision state; and (

**c**) payoff matrix for player 1 (player 2).

**Figure 3.**(

**a**) scores of the TOW bombe in the typical example where (${P}_{A}$, ${P}_{B}$, ${P}_{C}$, ${P}_{D}$, ${P}_{E}$) = ($0.03$, $0.05$, $0.1$, $0.2$, $0.9$); (

**b**) sample averages of total scores of the TOW bombe in the case where (${P}_{A}$, ${P}_{B}$, ${P}_{C}$, ${P}_{D}$, ${P}_{E}$) = ($0.03$, $0.05$, $0.1$, $0.2$, $0.9$); and (

**c**) sample averages of mean distance between player’s scores in the case where (${P}_{A}$, ${P}_{B}$, ${P}_{C}$, ${P}_{D}$, ${P}_{E}$) = ($0.03$, $0.05$, $0.1$, $0.2$, $0.9$).

**Table 1.**Payoff matrix of the case where (${P}_{C}$, ${P}_{D}$, ${P}_{E}$)$\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$($0.1$, $0.2$, $0.9$), player 3 chooses C.

Player 2: C | Player 2: D | Player 2: E | |
---|---|---|---|

player 1: C | $1/30$, $1/30$, $1/30$ | $0.05$, $0.2$, $0.05$ | $0.05$, $0.9$, $0.05$ |

player 1: D | $0.2$, $0.05$, $0.05$ | $0.1$, $0.1$, $0.1$ | $0.2$, $0.9$, $0.1$ SM |

player 1: E | $0.9$, $0.05$, $0.05$ | $0.9$, $0.2$, $0.1$ SM | $0.45$, $0.45$, $0.1$ |

**Table 2.**Payoff matrix of the case where (${P}_{C}$, ${P}_{D}$, ${P}_{E}$)$\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$($0.1$, $0.2$, $0.9$), player 3 chooses D.

Player 2: C | Player 2: D | Player 2: E | |
---|---|---|---|

player 1: C | $0.05$, $0.05$, $0.2$ | $0.1$, $0.1$, $0.1$ | $0.1$, $0.9$, $0.2$ SM |

player 1: D | $0.1$, $0.1$, $0.1$ | $2/30$, $2/30$, $2/30$ | $0.1$, $0.9$, $0.1$ |

player 1: E | $0.9$, $0.1$, $0.2$ SM | $0.9$, $0.1$, $0.1$ | $0.45$, $0.45$, $0.2$ |

**Table 3.**Payoff matrix of the case where (${P}_{C}$, ${P}_{D}$, ${P}_{E}$)$\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}$($0.1$, $0.2$, $0.9$), player 3 chooses E.

Player 2: C | Player 2: D | Player 2: E | |
---|---|---|---|

player 1: C | $0.05$, $0.05$, $0.9$ | $0.1$, $0.2$, $0.9$ SM | $0.1$, $0.45$, $0.45$ |

player 1: D | $0.2$, $0.1$, $0.9$ SM | $0.1$, $0.1$, $0.9$ | $0.2$, $0.45$, $0.45$ |

player 1: E | $0.45$, $0.1$, $0.45$ | $0.45$, $0.2$, $0.45$ | $0.3$, $0.3$, $0.3$ NE |

Selection Pattern | Degree of Charges | Probability |
---|---|---|

( A, A, A ) | ( 0, 0, 0 ) | 0.55 0.55 0.55 |

( A, A, B ) | ( 0, 0, 1 ) | 0.73 0.73 0.40 |

( A, A, C ) | ( 1, 0, 0 ) | 0.40 0.73 0.73 |

( A, A, D ) | ( 0, 1, 0 ) | 0.73 0.40 0.73 |

( A, A, E ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( A, B, A ) | ( 0, 1, 0 ) | 0.73 0.40 0.73 |

( A, B, B ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( A, B, C ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( A, B, D ) | ( 0, 2, 0 ) | 0.76 0.30 0.76 |

( A, B, E ) | ( 1, 2, 0 ) | 0.73 0.30 0.76 |

( A, C, A ) | ( 0, 0, 1 ) | 0.73 0.73 0.40 |

( A, C, B ) | ( 0, 0, 2 ) | 0.76 0.76 0.30 |

( A, C, C ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( A, C, D ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( A, C, E ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( A, D, A ) | ( 1, 0, 0 ) | 0.40 0.73 0.73 |

( A, D, B ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( A, D, C ) | ( 2, 0, 0 ) | 0.30 0.76 0.76 |

( A, D, D ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( A, D, E ) | ( 2, 1, 0 ) | 0.30 0.73 0.76 |

( A, E, A ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( A, E, B ) | ( 1, 0, 2 ) | 0.73 0.76 0.30 |

( A, E, C ) | ( 2, 0, 1 ) | 0.30 0.76 0.73 |

( A, E, D ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( A, E, E ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( B, A, A ) | ( 1, 0, 0 ) | 0.40 0.73 0.73 |

( B, A, B ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( B, A, C ) | ( 2, 0, 0 ) | 0.30 0.76 0.76 |

( B, A, D ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( B, A, E ) | ( 2, 1, 0 ) | 0.30 0.73 0.76 |

( B, B, A ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( B, B, B ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( B, B, C ) | ( 2, 1, 0 ) | 0.30 0.73 0.76 |

( B, B, D ) | ( 1, 2, 0 ) | 0.73 0.30 0.76 |

( B, B, E ) | ( 2, 2, 0 ) | 0.30 0.30 0.76 |

( B, C, A ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( B, C, B ) | ( 1, 0, 2 ) | 0.73 0.76 0.30 |

( B, C, C ) | ( 2, 0, 1 ) | 0.30 0.76 0.73 |

( B, C, D ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( B, C, E ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( B, D, A ) | ( 2, 0, 0 ) | 0.30 0.76 0.76 |

( B, D, B ) | ( 2, 0, 1 ) | 0.30 0.76 0.73 |

( B, D, C ) | ( 3, 0, 0 ) | 0.20 0.79 0.79 |

( B, D, D ) | ( 2, 1, 0 ) | 0.30 0.73 0.76 |

( B, D, E ) | ( 3, 1, 0 ) | 0.20 0.76 0.79 |

( B, E, A ) | ( 2, 0, 1 ) | 0.30 0.76 0.73 |

( B, E, B ) | ( 2, 0, 2 ) | 0.30 0.76 0.30 |

( B, E, C ) | ( 3, 0, 1 ) | 0.20 0.79 0.76 |

( B, E, D ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( B, E, E ) | ( 3, 1, 1 ) | 0.30 0.76 0.76 |

( C, A, A ) | ( 0, 1, 0 ) | 0.73 0.40 0.73 |

( C, A, B ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( C, A, C ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( C, A, D ) | ( 0, 2, 0 ) | 0.76 0.30 0.76 |

( C, A, E ) | ( 1, 2, 0 ) | 0.73 0.30 0.76 |

( C, B, A ) | ( 0, 2, 0 ) | 0.76 0.30 0.76 |

( C, B, B ) | ( 0, 2, 1 ) | 0.76 0.30 0.73 |

( C, B, C ) | ( 1, 2, 0 ) | 0.73 0.30 0.76 |

( C, B, D ) | ( 0, 3, 0 ) | 0.79 0.20 0.79 |

( C, B, E ) | ( 1, 3, 0 ) | 0.76 0.20 0.79 |

( C, C, A ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( C, C, B ) | ( 0, 1, 2 ) | 0.76 0.73 0.30 |

( C, C, C ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( C, C, D ) | ( 0, 2, 1 ) | 0.76 0.30 0.73 |

( C, C, E ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( C, D, A ) | ( 1, 1, 0 ) | 0.40 0.40 0.73 |

( C, D, B ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( C, D, C ) | ( 2, 1, 0 ) | 0.30 0.73 0.76 |

( C, D, D ) | ( 1, 2, 0 ) | 0.73 0.30 0.76 |

( C, D, E ) | ( 2, 2, 0 ) | 0.30 0.30 0.76 |

( C, E, A ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( C, E, B ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( C, E, C ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( C, E, D ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( C, E, E ) | ( 2, 2, 1 ) | 0.40 0.40 0.73 |

( D, A, A ) | ( 0, 0, 1 ) | 0.73 0.73 0.40 |

( D, A, B ) | ( 0, 0, 2 ) | 0.76 0.76 0.30 |

( D, A, C ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( D, A, D ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( D, A, E ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( D, B, A ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( D, B, B ) | ( 0, 1, 2 ) | 0.76 0.73 0.30 |

( D, B, C ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( D, B, D ) | ( 0, 2, 1 ) | 0.76 0.30 0.73 |

( D, B, E ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( D, C, A ) | ( 0, 0, 2 ) | 0.76 0.76 0.30 |

( D, C, B ) | ( 0, 0, 3 ) | 0.79 0.79 0.20 |

( D, C, C ) | ( 1, 0, 2 ) | 0.73 0.76 0.30 |

( D, C, D ) | ( 0, 1, 2 ) | 0.76 0.73 0.30 |

( D, C, E ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( D, D, A ) | ( 1, 0, 1 ) | 0.40 0.73 0.40 |

( D, D, B ) | ( 1, 0, 2 ) | 0.73 0.76 0.30 |

( D, D, C ) | ( 2, 0, 1 ) | 0.30 0.76 0.73 |

( D, D, D ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( D, D, E ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( D, E, A ) | ( 1, 0, 2 ) | 0.73 0.76 0.30 |

( D, E, B ) | ( 1, 0, 3 ) | 0.76 0.79 0.20 |

( D, E, C ) | ( 2, 0, 2 ) | 0.30 0.76 0.30 |

( D, E, D ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( D, E, E ) | ( 2, 1, 2 ) | 0.40 0.73 0.40 |

( E, A, A ) | ( 0, 1, 1 ) | 0.73 0.40 0.40 |

( E, A, B ) | ( 0, 1, 2 ) | 0.76 0.73 0.30 |

( E, A, C ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( E, A, D ) | ( 0, 2, 1 ) | 0.76 0.30 0.73 |

( E, A, E ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( E, B, A ) | ( 0, 2, 1 ) | 0.76 0.30 0.73 |

( E, B, B ) | ( 0, 2, 2 ) | 0.76 0.30 0.30 |

( E, B, C ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( E, B, D ) | ( 0, 3, 1 ) | 0.79 0.20 0.76 |

( E, B, E ) | ( 1, 3, 1 ) | 0.76 0.30 0.76 |

( E, C, A ) | ( 0, 1, 2 ) | 0.76 0.73 0.30 |

( E, C, B ) | ( 0, 1, 3 ) | 0.79 0.76 0.20 |

( E, C, C ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( E, C, D ) | ( 0, 2, 2 ) | 0.76 0.30 0.30 |

( E, C, E ) | ( 1, 2, 2 ) | 0.73 0.40 0.40 |

( E, D, A ) | ( 1, 1, 1 ) | 0.60 0.60 0.60 |

( E, D, B ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( E, D, C ) | ( 2, 1, 1 ) | 0.70 0.70 0.70 |

( E, D, D ) | ( 1, 2, 1 ) | 0.70 0.70 0.70 |

( E, D, E ) | ( 2, 2, 1 ) | 0.40 0.40 0.73 |

( E, E, A ) | ( 1, 1, 2 ) | 0.70 0.70 0.70 |

( E, E, B ) | ( 1, 1, 3 ) | 0.76 0.76 0.30 |

( E, E, C ) | ( 2, 1, 2 ) | 0.40 0.73 0.40 |

( E, E, D ) | ( 1, 2, 2 ) | 0.73 0.40 0.40 |

( E, E, E ) | ( 2, 2, 2 ) | 0.50 0.50 0.50 |

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Kim, S.-J.; Naruse, M.; Aono, M.
Harnessing the Computational Power of Fluids for Optimization of Collective Decision Making. *Philosophies* **2016**, *1*, 245-260.
https://doi.org/10.3390/philosophies1030245

**AMA Style**

Kim S-J, Naruse M, Aono M.
Harnessing the Computational Power of Fluids for Optimization of Collective Decision Making. *Philosophies*. 2016; 1(3):245-260.
https://doi.org/10.3390/philosophies1030245

**Chicago/Turabian Style**

Kim, Song-Ju, Makoto Naruse, and Masashi Aono.
2016. "Harnessing the Computational Power of Fluids for Optimization of Collective Decision Making" *Philosophies* 1, no. 3: 245-260.
https://doi.org/10.3390/philosophies1030245