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in rock, paper, scissors
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then Pi is (1/3, 1/3, 1/3)
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and Pi of rock, so Pi(R)--is a 1/3.
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So without belaboring it,
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that's all I'm doing here, is developing some notation.
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Let's immediately encounter two things you might have questions about.
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So the first is, that in principle Pi(Si) could be zero.
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Just because I'm playing a mixed strategy,
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it doesn't mean I have to involve all of my strategies.
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I could be playing a mixed strategy on two of my strategies
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and leave the other one with zero probability.
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So, for example, again in rock, paper, scissors,