# 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

## References

- Mesquita BBDe. The Predictioneer’s Game; Random House Inc.: New York, NY, USA, 2009. [Google Scholar]
- Narendra, K.S.; Member, S.; Thathachar, M.A.L. Learning automata—A survey. IEEE Trans. Syst. Man Cybern.
**1974**, SMC-4, 323–334. [Google Scholar] [CrossRef] - Fudenberg, D.; Levine, D.K. The Theory of Learning in Games; The MIT Press: Cambridge, MA, USA, 1998. [Google Scholar]
- Marden, J.R.; Young, H.P.; Arslan, G.; Shamma, J. Payoff based dynamics for multiplayer weakly acyclic games. SIAM J. Control Optim.
**2009**, 48, 373–396. [Google Scholar] [CrossRef] - Turing, A.M. On computable numbers, with an application to the Entscheidungsproblem. Proc. Lond. Math. Soc.
**1936**, 42, 230–265. [Google Scholar] - Turing, A.M. Computability and λ-definability. J. Symb. Log.
**1937**, 2, 153–163. [Google Scholar] [CrossRef] - Moore, C. A complex legacy. Nat. Phys.
**2011**, 7, 828–830. [Google Scholar] [CrossRef] - Feynman, R.P. Feynman Lectures on Computation; Perseus Books: New York, NY, USA, 1996. [Google Scholar]
- Roughgarden, T. Selfish Routing and the Price of Anarchy; The MIT Press: Cambridge, MA, USA, 2005. [Google Scholar]
- Nisan, N.; Roughgarden, T.; Tardos, E.; Vazirani, V.V. Algorithmic Game Theory; Cambridge University Press: New York, NY, USA, 2007. [Google Scholar]
- Kim, S.-J.; Aono, M.; Hara, M. Tug-of-war model for multi-armed bandit problem. In Unconventional Computation; LNCS 6079; Calude, C.S., Hagiya, M., Morita, K., Rozenberg, G., Timmis, J., Eds.; Springer: Berlin/Heidelberg, Germany, 2010; pp. 69–80. [Google Scholar]
- Kim, S.-J.; Aono, M.; Hara, M. Tug-of-war model for two-bandit problem: Nonlocally correlated parallel exploration via resource conservation. BioSystems
**2010**, 101, 29–36. [Google Scholar] [CrossRef] [PubMed] - Kim, S.-J.; Aono, M. Amoeba-inspired algorithm for cognitive medium access. NOLTA
**2014**, 5, 198–209. [Google Scholar] [CrossRef] - Kim, S.-J.; Aono, M.; Nameda, E. Efficient decision-making by volume-conserving physical object. New J. Phys.
**2015**, 17, 083023. [Google Scholar] [CrossRef] - Auer, P.; Cesa-Bianchi, N.; Fischer, P. Finite-time analysis of the multiarmed bandit problem. Mach. Learn.
**2002**, 47, 235–256. [Google Scholar] [CrossRef] - Kocsis, L.; Szepesvári, C. Bandit based monte-carlo planning. In Proceedings of the 17th European Conference on Machine Learning, Berlin, Germany, 18–22 September 2006; LNAI 4212. Springer: Berlin/Heidelberg, Germany, 2006; pp. 282–293. [Google Scholar]
- Gelly, S.; Wang, Y.; Munos, R.; Teytaud, O. Modification of UCT with Patterns in Monte-Carlo Go. (Research Report) RR-6062. 2006, pp. 1–19. Available online: https://hal.inria.fr/inria-00117266/document (accessed on 6 December 2016).
- Lai, L.; Jiang, H.; Poor, H.V. Medium access in cognitive radio networks: A competitive multi-armed bandit framework. In Proceedings of the IEEE 42nd Asilomar Conference on Signals, System and Computers, Pacific Grove, CA, USA, 26–29 October 2008; pp. 98–102.
- Lai, L.; Gamal, H.E.; Jiang, H.; Poor, H.V. Cognitive medium access: Exploration, exploitation, and competition. IEEE Trans. Mob. Comput.
**2011**, 10, 239–253. [Google Scholar] - Agarwal, D.; Chen, B.-C.; Elango, P. Explore/exploit schemes for web content optimization. In Proceedings of the Ninth IEEE International Conference on Data Mining, Miami, FL, USA, 6–9 December 2009.
- Davies, D. The Bombe—A remarkable logic machine. Cryptologia
**1999**, 23, 108–138. [Google Scholar] [CrossRef] - Kim, S.-J.; Aono, M. Decision maker using coupled incompressible-fluid cylinders. Adv. Sci. Technol. Environmentol.
**2015**, B11, 41–45. [Google Scholar] - Helbing, D.; Yu, W. The outbreak of cooperation among success-driven individuals under noisy conditions. Proc. Natl. Acad. Sci. USA
**2009**, 106, 3680–3685. [Google Scholar] [CrossRef] [PubMed] - Arrow, K.J. A difficulty in the concept of social welfare. J. Political Econ.
**1950**, 58, 328–346. [Google Scholar] [CrossRef] - Kim, S.-J.; Naruse, M.; Aono, M.; Ohtsu, M.; Hara, M. Decision maker based on nanoscale photo-excitation transfer. Sci. Rep.
**2013**, 3, 2370. [Google Scholar] [CrossRef] [PubMed] - Naruse, M.; Nomura, W.; Aono, M.; Ohtsu, M.; Sonnefraud, Y.; Drezet, A.; Huant, S.; Kim, S.-J. Decision making based on optical excitation transfer via near-field interactions between quantum dots. J. Appl. Phys.
**2014**, 116, 154303. [Google Scholar] [CrossRef][Green Version] - Naruse, M.; Berthel, M.; Drezet, A.; Huant, S.; Aono, M.; Hori, H.; Kim, S.-J. Single photon decision maker. Sci. Rep.
**2015**, 5, 13253. [Google Scholar] [CrossRef] [PubMed] - Kim, S.-J.; Tsuruoka, T.; Hasegawa, T.; Aono, M.; Terabe, K.; Aono, M. Decision maker based on atomic switches. AIMS Mater. Sci.
**2016**, 3, 245–259. [Google Scholar] [CrossRef]

**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