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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?

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  1. So how should I draw the extensive form and how can you make the beliefs clear? What you wan't to use in your extensive form drawing is a dotted line which stands for "information set". Look at the corresponding example on Wikipedia, which is equivalent to your information problem. Knowing this, drawing the extensive form gets easy...
  2. About the strategic form, do you need to put everything in one matrix? Yes, exactly. Software Inc has exactly two pure strategies (Put effort or don't put effort), Hardware Inc has four:

    • Put effort if part is OK and Put effort if part is not OK
    • Put effort if part is OK and don't Put effort if part is not OK
    • Don't put effort if part is OK and Put effort if part is not OK
    • Don't put effort if part is OK and don't Put effort if part is not OK

    Draw a 4x2 matrix with these pure strategies and fill in the utilities for each player for each of the 8 possible outcomes, which you can easily make with the informations given in the text. Sidenote: As the second player (software guy) does not know the action of the player before him (hardware guy), you can look at his decision just as you would in a simultaneous move game :)

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  • $\begingroup$ I know that I should use a dotted line, but should this dotted line connect all the 4 nodes? Because that makes the drawing rather unclear. At the entries of the matrix, should you use the expected value? $\endgroup$ – DeanTheMachine Jan 2 '15 at 11:29
  • $\begingroup$ @RubenMeijs In an extensive form, when an agent has to choose for an action, all the states (notes) which would look identical to the agent, i.e. where he has no way to know in which of them he is, are part of the same information set - you have to connect all nodes in an information set together (even if there are more then two and this might look strange to you). As for the matrix entry, you put the expected utilities (not values!) in this particular outcome in the matrix, just as in every strategic form game. See: link $\endgroup$ – miwe Jan 2 '15 at 12:21

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