Abstract-Type-reduction of type-2 fuzzy sets is considered to be a defuzzification bottleneck because of the computational complexity involved in the process of type-reduction. In this research, we prove that the closed-form Nie-Tan operator, which outputs the average of the upper and lower bounds of the footprint of uncertainty, is actually an accurate method for defuzzifing interval type-2 fuzzy sets.

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Abstract-The iterated prisoner's dilemma is an ideal model for the evolution of cooperation among the payoff-maximizing individuals. It has attracted wide interest to develop novel strategies since the success of tit-for-tat in Axelrod's iterated prisoner's dilemma competitions. Every strategy for iterated prisoner's dilemma utilizes a certain length of historical interactions with the opponent, which is regarded as the size of the memory, in making its choices. Intuitively, longer memory strategies must have an advantage over shorter memory strategies. In practice, however, most of the well-known strategies are short memory strategies that utilize only the recent history of previous interactions. In this paper, the effect of the memory size of strategies on their evolutionary stability in both infinite length and indefinite length n-person iterated prisoner's dilemma is studied. Based on the concept of a counter strategy, we develop a theoretical methodology to evaluate the evolutionary stability of strategies and prove that longer memory strategies outperform shorter memory strategies statistically in the sense of evolutionary stability. We also give an example of a memory-two strategy to show how the theoretical study of evolutionary stability assists in developing novel strategies.

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