EXAMPLE 3.19 (EXERCISE 8 ON PAGE 288). An interaction between two players who are considering a possible business partnership

(1) The players simultaneously choose whether to make an investment; investment entails a personal cost of 3; not investing costs nothing

(2-1) After observing the investment choices, the players jointly decide whether to form a partnership firm and, if so, divide the profit (16 if both invested; 12 if one invested) equally

8 if neither invested

(2-2) If they decide not to work together, each player obtains x – 3 if invested, and zero otherwise

player1 | player2 | P | S | |

C1 | I | i | 5, 5 | x-3, x-3 |

C2 | I | n | 3, 6 | x-3, 0 |

C3 | N | i | 6, 3 | 0, x-3 |

C4 | N | n | 4, 4 | 0, 0 |

in x > 6, equilibrium is (I, i)

1 \ 2 | i | n |

I | 5, 5 | x-3, 0 |

N | 0, x-3 | 4, 4 |

in x < 6

1 \ 2 | i | n |

I | 5, 5 | 3, 6 |

N | 6, 3 | 4, 4 |

(a) What outcome maximizes the joint value?

(b) Describe conditions on x such that there is an eqbm in which both players invest.

(c) Briefly provide some intuition for your answers to (a) and (b) in relation to be “hold-up” problem.

- Reference: Chang-Koo Chi, (27/50) Game Theory and Applications 7 – Hold-up problem exercise, Jul 8, 2020, https://youtu.be/aBhlI8-lgfk