Game Theory with Applications to Finance and Marketing

game tree again, and so on and so forth. Because we are given a finiteextensive game with perfect information, the above procedure willeventually reach the beginning of the game tree, thereby determiningan equilibrium path, which by definition is a pure strategy SPNE. Thisprocedure is usually referred to as Kuhn-Zermelo algorithm. Observethat if no two terminal nodes give any player the same payoff, the obtainedSPNE is unique.Example 4: Every NE in the extensive game depicted in section 30 isalso an SPNE.Example 5: Chen is the owner-manager of a firm. The debt due oneyear from now has a face value equal to $10. The total assets in placeworth only $8. But, just now, Chen found an investment opportunitywith NPV=x > 1, which does not need any extra investment otherthan the current assets in the firm. Chen comes to his creditor andasks the latter to reduce the face value of debt by $1. He claims (heis really bad) that he will not take the investment project unless thecreditor is willing to reduce the face value as he wants.(i) Suppose x > 2. Show that there is an NE in which the creditoragrees to reduce the face value of debt and Chen makes the investment.(ii) Show that the NE in (i) is not an SPNE, because it involves incrediblethreat from Chen.(iii) How may your conclusion about (ii) change if x ∈ (1, 2]?(iv) Define bankruptcy as a state where the firm’s net worth dropsto zero. In case of (iii), conclude that Chen’s company has not gonebankrupt.39. Example 5 shows that in a dynamic game, only SPNEs are reasonableoutcomes among NEs. Now we apply the procedure of backward inductionto solve the SPNE for the Stackelberg game. First, consider thesubgame starting with firm 2’s decision. A subgame is distinguishedby firm 1’ choice of q 1 . Given q 1 , firm 2 has infinite possible strategiesq 2 (·), but which one is the best? Of course, the profit maximizingstrategy is the best, and henceq 2 = 1 − q 1.221