Liability games

A firm has liabilities towards a group of creditors. We analyze the question of how to distribute the asset value of the firm among the creditors and the firm itself. Compared to standard bankruptcy games as studied in the game theory literature, we introduce the firm as an explicit player and define a new class of transferable utility games called liability games. Liability games are superadditive, constant sum, partially convex, and partially concave. The core of a liability game is empty if and only if the firm is insolvent and has multiple positive liabilities. We analyze the nucleolus of the game and show that allocating the asset value of the firm using the nucleolus satisfies efficiency, non-negativity, and liabilities boundedness. We prove that at the nucleolus, the firm gets a strictly higher amount than its stand-alone value if and only if the firm is insolvent and has multiple positive liabilities. The firm is using the threat to pay others to get debt forgiveness and is able to keep a positive amount of its assets. We provide conditions under which the nucleolus coincides with a generalized truncated proportional rule, assigning a non-negative payment to the firm and distributing the remainder in proportion to the liabilities, truncated by the asset value of the firm.