Question

Princess Leia and her army of rebels are closing in on the Emperor’s forces, who are led by Darth Vader, for one final battle. With probability α Darth Vader will win the battle. If Darth Vader wins, he receives payoff WD and if the rebels lose they receive payoff LR. The rebels will fight no matter what if Darth Vader fights, because they intentionally blew up their ship after landing on the battle field. If Darth Vader fights and loses, the rebels can choose to execute him, or can choose to imprison him with payoffs ED, ER, ID, and IR, respectively. If Darth Vader surrenders then the rebels can murder him or forgive him. This will lead to payoffs of MD, MR, FDand FR, respectively. (a) Draw the game tree. State the conditions that will lead to an SPE in which: (b) Darth Vader will surrender and is murdered; (c) Darth Vader will fight and is imprisoned if he loses; (d) Darth Vader will fight and is executed if he loses. (e) For part (d) explain the conditions under which this is an example of what game theorists refer to as a “commitment problem” using the phrase “Pareto Efficient” in your explanation.

Answer 1

Answer a :-

DARTH VADER SURRENDER MURDER FIGHT FORGIVE REBELS (MR)/ REBELS (FR) WINS DARTH VADER (MD) LOSES DARTH VADER (FO ) REBELS VADER (LR) DARTU (LR) MD) EXECUTED PRISONED 1 ) DASMAS DARTH VADER (ED) REBELS (ID) (ER )

Answer b :-

If Darth Vader had surrendered and would have been murdered then it's pay off would be equal to MD while the Rebels payoff would be equal to to MR.

Answer c :-

If Darth Vader would have lost in the fight and would have been imprisoned then and his payoff would be equal to ID and the Rebels payoff would be equal to IR .

Answer d :-

If Darth Vader would have lost and would have been executed his payoff would have been equal to ED and the Rebels payoff would have been equal to ER.

Answer e :-

In Game Theory a commitment problem is a problem when two players would be better off in the present scenario by committing themselves to a Cooperative relationship in the future , but the benefits of cooperation and the present are not realised by the players and no mutually beneficial agreement is signed between them.

In the part d , the Army of Rebels had an opportunity to imprison or to execute Darth Vader when he lost the fight . however they choose to execute him instead of imprisoning him by only concentrating on the present scenario of benefits and completely ignoring the friendly relations that could have been formed if they would have been imprisoned for sometime and released by the army of Rebels on certain conditions that are beneficial for the Rebel Army.