CSE 311A: Introduction to Intelligent Agents Using Science Fiction Spring 2020 Homework 4 Due: 11:59pm, April 7, 2020 (on Canvas) Provide your answers to the following questions in a PDF file and upload it to Canvas by the deadline. Question 1 [35 points] Read the short story \Little Lost Robot" by Isaac Asimov. You can access it (as well as other short stories in the \I, Robot" collection) through this link: https://www.cse.wustl.edu/~wyeoh/courses/cse311a/2020spring/ docs/Asimov-Isaac-I-Robot.pdf 1a) In the story, Nestor hides itself among the other NS-2 robots, and they all did not reveal Nestor to Dr. Calvin. Model the decision- making process of Nestor and all the other NS-2 robots as a (regular) simultaneous game, where the actions of Nestor are to either (a) reveal itself to Dr. Calvin or (b) stay hidden; and the actions of the other NS-2 robots are to either (a) reveal Nestor to Dr. Calvin or (b) keep Nestor hidden. 1 Provide reasons for your choice of the payoff function for this game. Make sure that they are consistent with the short story. Also provide the (pure or mixed) Nash equilibrium to your game. Question 2 [35 points] In the movie The Dark Knight, there was a scene where the Joker informed passengers on two ferries that there are explosives on both ferries and the detonator for the explosive on their ferry is with passengers on the other ferry. They have a choice to detonate or not detonate the other ferry. Additionally, he informed them that if neither ferry is detonated by midnight, then he would detonate both ferries. 2a) Represent this scene as accurately as possible using game theory (e.g., as some variant of Prisoner's Dilemma). In other words, your game should represent at least the following features of the scene: • All three players in the game (ferry1, ferry2, Joker). • The payoff of each player, as a function of the action of all players. Be sure to justify your payoffs. 2b) In your game above, what are the best response actions for each possible combination of the opponents' actions (it is possible to have more than one best response action if they have the same highest payoff)? Are there any dominating actions? Recall that an action is a dominating action if it is a best response action for all possible combinations of the opponents' actions. 2c) In your game above, is there a pure Nash equilibria? If so, what is it? If not, why isn't there one? 2d) Answer one of the two questions below: • If there is a pure Nash equilibria in your game, did the outcome in the movie correspond to one of your equilibria? If not, why is that so? 2 • If there is no pure Nash equilibria in your game, explain the reason for the outcome in the movie. Question 3 [30 points] Player 2 Rock Paper Scissors Rock 0, 0 -1, 1 1, -1 Player 1 Paper 1, -1 0, 0 -1, 1 Scissors -1, 1 1, -1 0, 0 Consider the Rock-Paper-Scissors game below, where each player can play Rock, Paper, or Scissors. The payoffs are in the form of (x, y), where Player 1 gets x and Player 2 gets y. Let P (xR) be the probability that Player 1 plays Rock, P (xP ) be the prob- ability of playing Paper, and P (xS) be the probability of playing Scissors. Also, let P (yR) be the probability that Player 2 plays Rock, P (yP ) be the probability of playing Paper, and P (yS) be the probability of playing Scissors. Identify a mixed Nash equilibria in the game above and represent it using the probability notations above. i.e., what are the probabilities P (xR), P (xP ), P (xS), P (yR), P (yP ), and P (yS)? Show how you arrived at those probabili- ties. 3.