I have to solve the following problem for my gametheory course:
Software Inc. and Hardware Inc. are in a joint venture together. The parts used can be defective or not; the probability of defective parts is 0.7, and this is commonly known before the start of the game. Each can exert either high or low effort, which is equivalent to costs of 20 and 0. Hardware moves first, but Software cannot observe his effort. Revenues are split equally at the end. If both firms exert low effort, total profits are 100. If the parts are defective, the total profit is 100; otherwise (i.e. if the parts are not defective), if both exert high effort, profit is 200. But if only one player does, profit is 100 with probability 0.9 and 200 with probability 0.1. Hardware discovers the truth about the parts by observation before he chooses effort, but software does not.
a) Determine the extensive form of this game. Is this a signaling game?
b) Determine the strategic form of this game.
c) Compute the (pure) Nash equilibria. Which one(s) is (are) subgame perfect? Perfect Bayesian?
I have tried to draw the extensive form, but I have no clue on how to show that Software Inc. doesn't know at which decision node he is and how I should call his believes. Because according to the story, Software doesn't know at any which of the 4 decision nodes he is (he doesn't know Hardware's decision and he doesn't know the outcome of the chance move). So how should I draw the extensive form and how can you make the beliefs clear?
Then about the strategic form, do you need to put everything in one matrix? If so how would you do this?