Consideration of Cycle in Efficiency of Minority Game T. Harada and T. Murata (Kansai University) Abstract In this study, we observe cycle in efficien

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Consideration of Cycle in Efficiency of Minority Game T. Harada and T. Murata (Kansai University) Abstract In this study, we observe cycle in efficiency of Minority Game. The Minority Game is a game when a group of participants (or agents) wins when it has a smaller number of participants. We realize simulations of the Minority Game with more than 100 million agents. From our simulation results, we observed a cycle that varies according to the size of memory of each agent. We show some simulation results showing those cycles. Key Words: Agent-based simulation, Large-scale simulation, Minority Game 1 Minority Game MG 1) (Agent-Based Simulation ABS) ABS ABS ABS ABS 2, 3) 4, 5) MG 10,001 50,001 1 6) MG 2 MG n 2 0 1 s m 0 1 MG m Table 1: Strategy tables of m = 3, s = 2 History Strategy 1 Strategy 2 Point 3 Point -1 0 0 0 0 1 0 0 1 1 1 0 1 0 0 1 0 1 1 0 0 1 0 0 1 0 1 0 1 0 0 1 1 0 1 1 1 1 1 0 1 m 1 1 1 Table 1 m = 3 s = 2 Table 1 1 2 0 0 1 1 010 Table 1 1 0 0 MG S k = 2 2m n s m S n = n s S k n = 101 m = 2 s = 2 S k = 2 22 = 16 S n = 101 2 = 202 第 5 回社会システム部会研究会 (2014 年 3 月 5 日 -7 日 沖縄 ) - 1 - PG0002/14/0000-0001

Fig. 1: Transition of Average Efficiency (n = 11 m = 10 s = 2) Fig. 3: Transition of Average Efficiency (n = 100, 001 m = 10 s = 2) Fig. 2: Transition of Average Efficiency (n = 1, 001 m = 10 s = 2) 3 3.1 n = 11 1, 001 100, 001 100, 000, 001 m = 10 s = 2 10 Fig. 1 n = 11 Fig. 2 n = 1001 Fig. 3 n = 100, 001 Fig. 4 n = 100, 000, 001 Fig. 1 Fig. 4 n = 11 n = 1, 001 n = 100, 001 n = 100, 000, 001 2,048 step 2,048 step 2048 step 3.2 n = 100, 001 m = 8 9 11 s = 2 10 Fig. 4: Transition of Average Efficiency (n = 100, 000, 001 m = 10 s = 2) Fig. 5 m = 8 Fig. 6 m = 9 Fig. 7 m = 11 m = 10 2,048 step m = 9 1,024 step m = 8 512 step m = 11 4,096 step 2 m+1 step n = 100, 001 m = 10 s = 1 16 64 10 Fig. 8 s = 1 Fig. 9 s = 16 Fig. 10 s = 64 s = 2 Fig. 3 2,048 step Fig. 8 s = 1 Fig. 9 Fig. 10 s = 2 2,048 step s = 1 s > 1 3.3-2 -

Fig. 5: Transition of Average Efficiency (n = 100, 001 m = 8 s = 2) Fig. 8: Transition of Average Efficiency (n = 100, 001 m = 10 s = 1) Fig. 6: Transition of Average Efficiency (n = 100, 001 m = 9 s = 2) Fig. 9: Transition of Average Efficiency (n = 100, 001 m = 10 s = 16) Fig. 7: Transition of Average Efficiency (n = 100, 001 m = 11 s = 2) n = 11 1, 001 100, 001 100, 000, 001 m = 10 s = 2 10 Fig. 11 Fig. 14 Fig. 11 Fig. 12 Fig. 10: Transition of Average Efficiency (n = 100, 001 m = 10 s = 64) Fig. 13 Fig. 14 2,048 step 2,048 step 0.2% - 3 -

Fig. 11: Transition of Ratio of Agents Who Switch Their Strategy Tables (n = 11 m = 10 s = 2) Fig. 13: Transition of Ratio of Agents Who Switch Their Strategy Tables (n = 100, 001 m = 10 s = 2) Fig. 12: Transition of Ratio of Agents Who Switch Their Strategy Tables (n = 1, 001 m = 10 s = 2) 4 0.5 n = 100, 001 2 m+1 step Savit 7) z 2 m /n z 0.5 z 0.5 z < 0.5 z < 0.5 z 0.5 Fig. 15 n = 11 101 1, 001 10, 001 100, 001 1, 000, 001 m = 1 2... 16 s = 2 10 1 z 2 m /n Savit 7) n = 11 25 101 1, 001 Fig. 15 n = 1, 000, 001 z 0.5 Fig. 14: Transition of Ratio of Agents Who Switch Their Strategy Tables (n = 100, 000, 001 m = 10 s = 2) z Fig. 16 n = 1, 001 s = 2 m = 2 3... 8 z < 0.5 1,000 1,000 4 step 4 2 m+1 Fig. 16 m < 6 z < 0.064 n = 1, 001 z 0.5 m = 8 9 10 s = 2 1,000 Fig. 17 Fig. 17 m = 8 Table 2 n = 1, 001 m = 2 3... 16 s = 2 z Table 2 z = 0.511 m = 9 z = 0.256 m = 8 Savit 7) Fig. 15 Fig. 18 n = 101-4 -

Fig. 15: Phase Transitions of Efficiency (s = 2) Fig. 17: Transition of Average Efficiency (n = 1, 001 m = 7 8 9 s = 2) Fig. 16: Transition of Average Efficiency (n = 1, 001 m = 2 3... 16 s = 2) Fig. 19 Fig. 20 n = 10, 001 Fig. 21 Fig. 22 n = 100, 001 n = 101 m = 2 n = 10, 001 m < 9 n = 100, 001 m < 12 n = 101 m = 2 z = 0.040 n = 10, 001 m = 8 z = 0.026 n = 100, 001 m = 11 z = 0.024 n = 10, 001 m = 9 z = 0.051 n = 100, 001 m = 12 z = 0.041 z < 0.041 4, 5) s = 2 Fig. 9 Fig. 10 5 MG 2 m+1 step Fig. 18: Transition of Average Efficiency (n = 101 m = 2 3 4 s = 2) s > 1 z < 0.041 4, 5) s > 2 s = 2 2 m+1 step 2 m+1 step 1) Damien Challet Yi-Cheng Zhang Emergence of cooperation and organization in an evolutionary game Physica A Vol.246 407/418 (1997) 2) D Vol. J90-D 2423/2431 (2007) 3) Dan Chen Georgios K. Theodoropoulos Stephen J. Turner Wentong Cai Robert Minson Yi Zhang Future Generation Computer Systems Vol. 24 658/671 (2008) - 5 -

Fig. 19: Transition of Average Efficiency (n = 10, 001 m = 2 3... 6 s = 2) Fig. 21: Transition of Average Efficiency (n = 100, 001 m = 2 3... 8 s = 2) Fig. 20: Transition of Average Efficiency (n = 10, 001 m = 7 8... 11 s = 2) 4) 27 1/4 (2013) 5) 4 41/46 (2013) 6) 29 441/446 (2013) 7) Robert Savit Radu Manuca and Rick Riolo Adaptive competition, market efficiency, and phase transitions Physical Review Letters Vol.82 No. 10 2203/2206 (1999) Fig. 22: Transition of Average Efficiency (n = 100, 001 m = 9 10... 12 s = 2) Table 2: z and σ 2 /n (n = 1, 001 m = 2 3... 16 s = 2) m z σ 2 /n 2 0.004 4.183 3 0.008 2.109 4 0.016 0.963 5 0.032 0.525 6 0.064 0.281 7 0.128 0.172 8 0.256 0.147 9 0.511 0.148 10 1.023 0.167 11 2.046 0.193 12 4.092 0.213 13 8.184 0.231 14 16.368 0.241 15 32.735 0.245 16 65.471 0.248-6 -

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