Chapter 4. Concluding Remarks
A.3 Fitness of strategies
A.3.1 Simple Social Norm
A.3.1 Simple Social Norm
Figure A.3.1: The calculation of the expected payoffs of strategies for the Simple Social Norm (GGBG).
According to the rule of game in the simple social norm, there are three strategies
as choice. The agent has chance of being a donor and chance of being a recipient. On one hand, when the agent be a donor, the ratio he meets good (G) recipient is and meets bad (B) recipient is , and may take cooperation (C) action cost , , or 0 cost; On the other hand, when the agent be a recipient, he meets CC, CD, DD players with probabilities , respectively, and is expected to obtain , , or 0 revenue, respectively. So we calculate the expected payoff in CC, CD, DD, respectively. The interpretation form is .
The different expected payoff effect in three strategies is in donor action. The CC donor will always cooperate no matter the recipient whose reputation is good or bad;
The CD donor will cooperate only when the recipient is good, otherwise they will no cooperate and no cost; The DD donor will never donate and no cost.
Due to the asynchronous updating method, the expected payoff of each strategy in each agent is separately calculated. Then we sum up and take average for the total expected payoff each agent in each strategy, respectively. The expected payoff
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calculation way in simple social norm is the same in later weakly augmented social norm and strongly augmented social norm.
The expected payoff of strategy CC is:
The expected payoff of strategy CD is:
The expected payoff of strategy DD is:
To sum up, the expected payoff of all strategies in the simple social norm (GGBG) is:
, , ,
The calculation process is in Appendix 2.1
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A.3.2 Weakly Augmented Norm
In simple social norm, the cooperation (C) strategy has a cooperation cost in donor and cooperation benefit in recipient. The same calculation rule in weakly augmented social norm, the punishment (P) strategy has a punishment cost α in donor and punishment fine β in recipient.
Figure A.3.2: Calculation of the expected revenue of strategies for the Weakly Augmented Social Norm (GGBGBG).
The calculation of the expected revenue of strategies for the weakly augmented
social norm (GGBGBG) in CC, CD, CP and DD strategies are similar with simple social norm but add punishment action. In the payoff tree, the upper branch represents the donor branch and the lower branch represents the recipient branch. The donor branch keeps the same structure but the recipient branch turns to be more options. The lower branches that agents meet CC, CD, CP and DD respectively in CC,
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CD, CP and DD strategies are the same in recipient’s revenue. While the upper branches that agents donate or not are different in donor’s cost. In the lower branch, for the recipient encounters with the CC or DD agents, this reputation is irrelevant, while it matters for the encounter with the CD or CP agents. Expected payoff depends on both the donor’s action and recipient’s reputation in all branches.
The expected payoff CC, CD, CP and DD in weakly augmented social norm (GGBGBG) are summarized as follows:
Substitution previously calculated values of g:
, ,
,
The calculation process is in Appendix 2.2
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A.3.3 Strongly Augmented Norm
In simple social norm, the cooperation (C) strategy has a cooperation cost in donor and cooperation benefit in recipient. The same calculation rule in strongly augmented social norm, the punishment (P) strategy has a punishment cost α in donor and punishment fine β in recipient.
Figure A.3.3: Calculation of the expected revenue of strategies for the Strongly Augmented Social Norm (GGBBBG).
The calculation of the expected revenue of strategies for the weakly augmented
social norm (GGBBBG) in CC, CD, CP and DD strategies are similar with weakly augmented social norm with punishment action addition. In the payoff tree, the upper branch represents the donor branch and the lower branch represents the recipient branch. The donor branch and recipient branch keep the same structure. The lower branches that agents meet CC, CD, CP and DD respectively in CC, CD,
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CP and DD strategies are the same in recipient’s revenue. While the upper branches that agents donate or not are different in donor’s cost. In the lower branch, for the recipient encounters with the CC or DD agents, this reputation is irrelevant, while it matters for the encounter with the CD or CP agents. Expected payoff depends on both the donor’s action and recipient’s reputation in all branches.
The expected payoff CC, CD, CP and DD in strongly augmented social norm (GGBBBG) are summarized as follows:
Substitution previously calculated values of g:
, , , ,
, The calculation process is in Appendix 2.3
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