在這篇論文中，利用電腦模擬分析，我們研究懲罰輕重如何影響剪刀石頭布系統的穩定性。我們發現，依不同輕重的的懲處，系統喪失多樣性的平均時間隨著系統人口變化有很不一樣的趨勢。臨界的懲處值為該遊戲總得失為零的情況，重懲處的系統傾向不穩定，懲罰越重多樣性越快喪失，而輕懲處的系統多樣性保存時間與系統大小呈指數關係，所以在合理生物時間內，可以看作是穩定的生態系統。最後，我們呈現了一個可以同時描述這兩種不同趨勢的相的方程式。

In this thesis, we study the systems of simple rock-paper-scissors games with different punishments s for the loser of the games. In common RPS games, the loser’s punishment is the same as the winner’s gain. When we adjust the punishment to smaller or bigger quantities, the systems’ stabilities are dramatically changed with two different trends. We analyzed the mean extinction time and obtained two different kinds of extinction behavior separated by the critical point. For strong selection regime, the mean extinction time grows logarithmically and does not be prolonged much with larger population size. For weak selection regime, the mean extinction time grows exponentially as system size goes up and the ecosystem is exponentially protected from extinction. Furthermore, a universal scaling function for the two phases is presented at the end of the thesis.

1 Introduction 1
2 Evolutionary games and replicator equations 4
2.1 Rate equations and selection................... 4
2.2 Frequency-dependent fitness ................... 6
2.3 Fitness landscape......................... 7
2.4 Replicator equations ....................... 7
2.5 Game theory and payoff matrix ................. 9
3 Payoff matrix with cyclic domination 11
3.1 Equivalent properties....................... 12
3.2 Rescale the payoffs ........................ 13
3.3 Zero-sum game and nonzero-sum game . . . . . . . . . . . . . 14
3.4 Lotka-Volterra model and cyclic competition . . . . . . . . . . 16
4 From deterministic to stochastic 19
4.1 Extinction mechanisms and selection strength . . . . . . . . . 19
4.2 The critical population size in Traulsen’s paper . . . . . . . . 20
4.3 Moran process and birth-death processes . . . . . . . . . . . . 21
4.4 The RGB algorithm........................ 22
5 Simulation result 26
5.1 Three trends of extinction time with different “penalty” . . . . 26
5.2 Universal scaling function .................... 29
6 Conclusions 33
A The Operation of RGB algorithm in our simulation 35

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