Answer to Question #253590 in Macroeconomics for robot

Question

Roger and Rafael play a game with the following rules. Roger is given $250 to divide between himself and Rafael. Rafael does not get to choose but he can reject Roger’s offer if he does not like it. If Rafael rejects, both get nothing. If Rafael accepts, both get the split that Roger decided.

a. What is this game called?

b. Find all Nash equilibria for this game.

c. When this game is played in the real world, do the predictions in part 1b materialize? Why/why not?

d. Are all Nash equilibria in part 1b Pareto Optimal? Explain

Answer 1

Given, 

Two Players- Roger and Rafael

Roger is given $250 to divide between himself and Rafael

Rafael is not allowed to choose 

However, he can reject Rojer's offer

If Rafael rejects Roger's offer the both get nothing and if Rafael accepts, they get the split that Roger decided.

a.

We know that there are two players-Roger and Rafael. Roger is given $250 to be split among the two players, however, we know that Rafael is not allowed to choose. He can only reject Roger's offer.

This kind of game is known as Dictator Game. In this kind of game one player receives an endowment which he is to split with the other player, however he decides how much the other player receives. 

b.

We know Roger receives $250 to be split between himself and Rafael

Let us assume that Roger decides to keep $150 for himself and give $100 to Rafael. 

The pay-off matrix can be as follows :

Now, 

If Rafael Accepts the offer, then Roger also Accepts the offer and receives a pay off of 150 or Roger can also Reject the offer and get the same pay-off

If Rafael Rejects the offer, Roger also Rejects the offer and gets a Pay-off of 0

Now, If Roger Accepts the offer, Rafael also Accepts the offer and receives a pay-off of 100

If Roger Rejects the offer, Rafael still accepts the offer and receives a pay-off of 100.

Therefore, the Nash Equilibrium is obtained when both Roger and Rafael Accept the offer and receive pay-offs (150,100) and if Roger rejects the pay-off but Rafael accepts the pay-off and receives the pay-off (150,100)

c.

When this game is played in the real world, the predictions in part (b) materializes since it is a Dictator Game, the Dictator would like to have the larger share of the endowment and if the other player is also okay with his share, he accepts the offer. This means that the Dictator or Roger would want the maximum pay-off. 

d.

The Nash Equilibrium obtained is Pareto Optimal since it gives the highest pay-offs to both the Players Roger and Rafael. Roger receives the highest pay-off of $150 whereas, Rafael receives the highest pay-off of $100.