**Note to the Teacher:** ..........

Prerequisites: ....

The extensive form of the case q=k=4 is shown below:

Ann has two information sets, holding a Q or a K, with two moves each, therefore she has four pure strategies "EE", "ED", DE", and "DD", where "E" stands for exchange and "D" ("don't") for no exchange. Beth has six information sets: Holding a "Q" with Ann having a "Q", or holding a "Q" with Ann holding a "K", or holding a "K", with Ann holding a "Q", or holding a "K" with Ann holding a "K", or holding a "Q" and not knowing what Ann has (since she didn't exchange), or holding a "K" with not knowing what Ann holds (since Ann didn't exchange). In each of these six information sets she has two options, "E" and "D", therefore Beth has 2^{6}=64 pure strategies. Therefore the normal form of the game is
a 4 × 64 matrix.

However, the game has four perfect-information-subgames which can be analyzed easily using Backwards Induction. They start at the four positions where Beth has to move and knows both of their cards since Ann has exchanged. In each of these cases it is clear what Beth has to do. We cut these subgames, and put the expected payoffs in to get the following truncated extensive form:

Here Beth has only two information sets left, and four pure strategies: "EE", "ED", "DE", and "DD". We get the following 4 × 4 matrix as normal form:

Ann has two information sets, holding a Q or a K, with two moves each, therefore she has four pure strategies "EE", "ED", DE", and "DD", where "E" stands for exchange and "D" ("don't") for no exchange. Beth has six information sets: Holding a "Q" with Ann having a "Q", or holding a "Q" with Ann holding a "K", or holding a "K", with Ann holding a "Q", or holding a "K" with Ann holding a "K", or holding a "Q" and not knowing what Ann has (since she didn't exchange), or holding a "K" with not knowing what Ann holds (since Ann didn't exchange). In each of these six information sets she has two options, "E" and "D", therefore Beth has 2

However, the game has four perfect-information-subgames which can be analyzed easily using Backwards Induction. They start at the four positions where Beth has to move and knows both of their cards since Ann has exchanged. In each of these cases it is clear what Beth has to do. We cut these subgames, and put the expected payoffs in to get the following truncated extensive form:

Here Beth has only two information sets left, and four pure strategies: "EE", "ED", "DE", and "DD". We get the following 4 × 4 matrix as normal form:

EE | ED | DE | DD | |

EE | -2/7 | -2/7 | -2/7 | -2/7 |

ED | 2/7 | 1/7 | 3/7 | 2/7 |

DE | -4/7 | -5/7 | -3/7 | -4/7 |

DD | 0 | -2/7 | 2/7 | 0 |

...................
Here is the truncated extensive form of MINI LE HER 3(4,4,4):

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