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# What is the subgame perfect equilibrium for Sony and Columbia pictures?

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Consider the following game between Sony, a manufacturer of video cassette players, and Columbia Pictures, a movie studio. Each firm must decide whether to use the VHS or Beta format - Sony to make video players, Columbia to release its movies for rental or purchase.

Columbia Pictures
Beta VHS
Sony Beta 20,10 0,0
VHS 0,0 10,20

a) Suppose this is a simultaneous-move, one-shot game. Restrict attention to pure strategies. Does either firm have a dominant strategy? Identify the Nash equilibrium or equilibria for this game.
b) Now suppose this is a sequential-move game in which Sony moves first. Write down a complete list of Sony's possible strategies, and of Columbia's possible strategies.
c) Draw an extensive form for the sequential-move game described in part b). Make sure you identify which player is "moving" at each decision node, and indicate the payoffs to each possible outcome of the game.
d) For the sequential-move game described in part b), identify a Nash equilibrium that is subgame perfect, and a Nash equilibrium that is not subgame perfect.

Consider the following modification of the game described in Problem 3: Suppose now that Dana (a stronger negotiator than Blair) decides to present Blair with a take-it-or-leave-it offer. Dana will offer Blair a particular share, X_B, of the proceeds, and will keep the rest (X_D = 200 - X_B). Blair can choose to accept the offer, in which case Blair and Dana will split the \$200 according to Dana's proposal, or to reject it - ion which case Mordecai the Mediator gets the \$200.
a) Describe the Nash equilibrium (or equilibria) for theis sequential-move game. Explain your reasoning.
b) Identify a subgame perfect Nash equilibrium for this game. Explain your reasoning.
c) Do you think that the subgame perfect Nash equilibrium you identifies in part b) is the most likely outcome of this game? Briefly explain.

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#### Solution Preview

Question 3

Section a
An inspection of the payoff matrix shows that there are two pure-strategy Nash equilibria in this game: [Beta, Beta] and [VHS, VHS] (the first action inside the brackets refers to the one Sony takes). The easiest way to see this is that, at these points, neither player has an incentive to unilaterally deviate from the strategy they're choosing. For example, let's take the [Beta, Beta] equilibrium. If any one player switches to VHS, his payoff will be 0, so it will not be a rational move. A similar thing happens at the [VHS, VHS] equilibrium. For analogous reasons, it's clear that neither [Beta, VHS] nor [VHS, Beta] are equilibria.

Section b
If this is a sequential game in which Sony plays first (always restricting the analysis to pure strategies), Sony has two possible strategies: Beta and VHS.

Columbia, on the other hand, has four possible strategies:

- Play Beta if Sony plays Beta and play Beta if Sony plays VHS
- Play Beta if Sony plays Beta and play VHS if Sony plays VHS
- Play VHS if Sony plays Beta and play VHS if Sony plays VHS
- Play VHS if Sony plays Beta and play Beta if Sony plays VHS

Section c
The drawing is in the attached jpg file.

Section d
The equilibrium [VHS, VHS] is not subgame perfect. Let's put ...

#### Solution Summary

This solution assess the subgame perfect equilibrium in 905 words.

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