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ECON 701 MODULE 7 EXERCISES

微观经济练习代写 Exercise 1 (The Monty Hall Game). The example we now discuss is now fairly well known. The name comes from an American television game show,

Exercise 1 (The Monty Hall Game). 微观经济练习代写

The example we now discuss is now fairly well known. The name comes from an American television game show, Let’s Make a Deal, and its host, Monty Hall, though the situation we describe, and all the variants described in the literature as the “Monty Hall Problem,” are not how the actual game show was played. We’ll give a very simple description of the game.

There are three boxes and one prize. Monty moves first and without being seen by the contestant puts the prize in one of the boxes. Rather than referring always to the contestant let’s give her a name and call her Alice. Alice then chooses one of the boxes. Monty then opens an empty box not chosen by Alice and shows her that it is empty and offers her the choice of switching from the box she had originally chosen to the other unopened box. Alice then either switches or doesn’t. If Alice ends up with the box containing the prize then Monty pays her the value of the prize, otherwise Monty doesn’t pay her anything.

It seems reasonable to assume that Alice preferes to win the prize than not. It probably is not good modeling to assume that Monty prefers not to award the prize, but it makes for a more useful example for our purposes to make this assumption. So we shall.

(i) 微观经济练习代写

A strategy is a complete plan of how to play the game. How many different strategies does each player have? Can you give a systemic way of labelling these strategies?

(ii) Give the normal or strategic form of this game. It’s probably easier to draw if you make Alice the row player, who we have been calling Player 1. You can either label the cells with payoffs, or label each cell with a W or L depending on whether Alice wins or not.

(iii) Are any rows dominated in the sense that whatever Monty’s strategy there is another row that’s always at least as good for Alice, and for some choices of Monty better? If so, delete these rows and give the simpler game.

(iv) I claim that it is an equilibrium that Monty plays each of his strategies with equal probability and Alice plays each of her strategies in which she always switches with equal probability. Explain why this is an equilibrium. Does it seem to you that there will be other equilibria?

Exercise 2. 微观经济练习代写

This question deals with the equilibria of two person games where each player has two or three strategies.

Graph the best reply correspondences for each player and mark on the graph all the mixed strategy equilibria, including the ones that are equivalent to pure strategy equilibria. Label your diagram completely, labelling the axes, any relevant values on each axis, and clearly indicating which is the graph of the best reply correspondence of Player 1 and which of Player 2.

(ii) Consider a two player game that has best replies as indicated in the following diagram.

Use the best reply diagrams to find all mixed strategy equilibria. Carefully explain the process of how you found the equilibria and explain why the process does find all equilibria.

Exercise 3. Consider the following two player game.

(i) Find all the pure strategy equilibria for this game.

(ii) Draw a simplex representing the mixed strategy space of Player 1 and carefully mark on it the regions where the three pure strategies of Player 2 are best, explaining how you found these regions and explicitly calculating the extreme points of the various regions.

(iii) Draw a simplex representing the mixed strategy space of Player 2 and carefully mark on it the regions where the three pure strategies of Player 1 are best, explicitly calculating the extreme points of the various regions.

(iv) Use the two diagrams that you have drawn to find all the mixed equilibria of this game.

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