The following material is flaggedBefore we start on this discussion, a vaguely relevant video. Spoilers for a certain VN/anime/manga series, even though it manages to not name many names.Green Level. It is intended to reflect material that the author believes to be a matter of consensus among experts in the field. This belief may be incorrect, however; and as the author is not an expert and does not have an expert fact-checking the article, errors may creep in.

But back to the question. Let's return to the game we started with:

A | B | |

A | (-1,1) | (1,-1) |

B | (1,-1) | (-1,1) |

So, since you lose if you pick A, and you lose if you pick B, what do you do?

You make your move based on something that cannot be predicted, something that even you cannot predict. You make your move random. Flip a coin, spin a roulette, or roll a die. You play what is called a

*mixed strategy*.

Now, let's put together a general form of this:

A | B | |

A | (-A,A) | (B,-B) |

B | (C,-C) | (-D,D) |

__>__0

So, if we play a mixed strategy, we need to figure out what odds we should give to each move (assuming that your opponent will assign the same odds):

Odds of Playing A | Outcome |

0% | -D |

25% | -(1/4*1/4)A+(1/4*3/4)B+(3/4*1/4)C-(3/4*3/4)D =-A/16+3B/16+3B/16-9D/16 |

50% | -(1/2*1/2)A+(1/2*1/2)B+(1/2*1/2)C-(1/2*1/2)D =-A/4+B/4+C/4-D/4 |

75% | -(3/4*3/4)A+(3/4*1/4)B+(1/4*3/4)C-(1/4*1/4)D =9A/16+3B/16+3C/16-D/16 |

100% | -A |

<<Metagames: The Punishing Prisoner's Dilemma | Game Theory | Metagames: The Instantaneous Punishing Prisoner's Dilemma>>

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