Preemptive Investment Consider the Market Entry game we saw in class: an Entrant must choose whether to go “in or out”of the market. If the Entrant goes In, the Incumbent must choose whether to Fight or Not.

Choosing In involves a fixed, sunk cost of 5 for the Entrant.

The following payoffs do not include the sunk cost. If the Entrant goes In and the Incumbent fights, they both receive a payoff of 4. If the Incumbent does not fight, they both receive a payoff of 10. If the Entrant stays out, it saves the fixed cost and gets 0, while the Incumbent gets 20.

1. (a) Draw the game tree and the payoffs at each terminal node.

2. (b) Find the backwards-induction equilibrium.

3. (c) Now imagine that before the game is played, the Incumbent can make an in- vestment. Each investment opportunity entails an up-front cost and modifies the payoffs in the ensuing game. The Incumbent must choose one of the three following options:

o Lobby for tighter regulation. This costs 2 to the incumbent, and increases the Entrants fixed cost of choosing In from 5 to 8.

o Improve its technology so to better tolerate a price war. This costs 8 to the Incumbent, and increases its payoff of Fighting from 4 to 12: (No change to the other payoffs).

o Advertising campaign that shifts 2 units of profits from the Entrant to the Incumbent whenever the Entrant chooses In. The campaign costs 1.

Evaluate the three investment opportunities and find the one with the highest re- turn to the Incumbent. (Hint: do not draw out a huge three with the Incumbents initial move, just modify the tree in (a) with the new payoffs.)

Audition Game (vaguely based on TV shows) Two venture capitalists want to fund a project. There are two possible projects, and each VC only has resources to fund one. Two entrepreneurs arrive sequentially and pitch their ideas to the VCs (who are sitting in the same room). Each entrepreneur s idea has a net present value V that is uniformly distributed between 0 and 10. VCs dont know the exact value of an idea until they hear the pitch. Upon hearing the 1st entrepreneurs pitch, both VCs must simultaneously announce yes or no. If only one says yes, she earns the value V of that idea, and leaves the game. If both say yes, a 50:50 coin flip determines which VC gets to fund the entrepreneur, and which one stays in the game. Whoever is left in the game (i.e., one or both VCs) gets to hear the second entrepreneurs pitch, and decide whether to fund it. There is no value for a VC in ending the game without funding a project.

1. (a) Suppose for a moment there was only one VC. Use backwards induction. Which ideas should he fund in the second period if he says no to the first one?

2. (b) Consider the single VCs plan of action for the whole game. Fill the blanks. “In the first period, I will fund the following ideas: _____________.”

3. (c) Now consider the game with two VCs. In order to apply backwards induction, you must compute the following two key values: what is the expected payoff of a VC who enters the second period alone (which occurs if the other VC funds the first idea)? And what is the expected payoff of both VCs entering the second period (which occurs if both say no to the first idea)?

4. (d) Suppose the first idea comes in, and both VCs realize its worth V = 8. Find the backwards-induction equilibria of the game.

5. (e) Now suppose the first idea is worth 2. Find the backwards-induction equilibria of the game.

6. (f) Finally, suppose the first idea is worth 4. Find the backwards-induction equilibria of the game.

7.(g) Consider each VCs equilibrium plan of action for the whole game. Fill the blanks. If I ever get to the second idea, I will say ____________; and I will say yes to the first idea if _________________.

**Subject Mathematics Game Theory**