## Question

Q1: Selling a partnership, independent values (Modified Gibbons 4.10)

Two partners must dissolve their partnership. Partner 1 currently owns share s of the partnership. Partner 2 owns share 1 - s. The game has the following sequence of actions:

1. Nature chooses each player’s valuation for owning the whole partnership, v1, v2 ∈ [0, 1]. Each value vi is independently and uniformly distributed on [0, 1]. Each player observes her own value, but not the value of the other player.

2. Partner 1 chooses a price, p ≥ 0.

3. Partner 2 chooses whether to buy or sell at the price named.

4. If 2 sells, payoffs are: (v1 - p(1 - s), p(1 - s)). If 2 buys, payoffs are: (ps, v2 - ps).

What is the (weak) perfect Bayesian equilibrium?

Q2: Selling a partnership, correlated values (Modified Gibbons 4.10)

Two partners must dissolve their partnership. Partner 1 currently owns share s of the partnership. Partner 2 owns share 1 - s. The game has the following sequence of actions:

1. Nature chooses the common valuation for owning the whole partnership, v ∈ [0, 1]. Let f denote the density function for v and F denote the distribution function. (If this is too hard, assume a uniform distribution on [0, 1].) Player 2 observes this value, but player 1 does not.

2. Partner 1 chooses a price, p ≥ 0.

3. Partner 2 chooses whether to buy or sell at the price named.

4. If 2 sells, payoffs are: (v - p(1 - s), p(1 - s)). If 2 buys, payoffs are: (ps, v - ps).

What is the (weak) perfect Bayesian equilibrium?

[Note: Both players get the same value from the firm! Player 2 observes this value at the beginning of the game, but player 1 does not. This is different from Q1, in which there are two different values.]

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