Assignment 2: Multi-Agent Pac-Man

Pac-Man, now with ghosts.
Minimax, Expectimax,


In this assignment, you will design agents for the classic version of Pac-Man, including ghosts. Along the way, you will implement both minimax and expectimax search and try your hand at evaluation function design.

The code base has not changed much from the previous assignment, but please start with a fresh installation, rather than intermingling files from project 1. You can, however, use your and in any way you want.

The code for this project contains the following files, available as a zip archive.

Key files to read Where all of your multi-agent search agents will reside. The main file that runs Pac-Man games. This file also describes a Pac-Man GameState type, which you will use extensively in this project The logic behind how the Pac-Man world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid. Useful data structures for implementing search algorithms.
Files you can ignore Graphics for Pac-Man Support for Pac-Man graphics ASCII graphics for Pac-Man Agents to control ghosts Keyboard interfaces to control Pac-Man Code for reading layout files and storing their contents


What to submit: You will fill in portions of during the assignment. You should submit this file with your code and comments. You may also submit supporting files (like, etc.) that you use in your code. Please do not change the other files in this distribution or submit any of our original files other than

Evaluation: Your code will be autograded for technical correctness. Please do not change the names of any provided functions or classes within the code, or you will wreak havoc on the autograder. However, the correctness of your implementation -- not the autograder's judgements -- will be the final judge of your score. If necessary, we will review and grade assignments individually to ensure that you receive due credit for your work.

Getting Help: You are not alone! If you find yourself stuck on something, contact the course staff for help. Office hours, section, and the newsgroup are there for your support; please use them. If you can't make our office hours, let us know and we will schedule more. We want these projects to be rewarding and instructional, not frustrating and demoralizing. But, we don't know when or how to help unless you ask.

Piazza: Post your questions (but not project solutions) on Piazza. Please be careful not to post spoilers.


Multi-Agent Pac-Man

First, play a game of classic Pac-Man:

Now, run the provided ReflexAgent in
python -p ReflexAgent
Note that it plays quite poorly even on simple layouts:
python -p ReflexAgent -l testClassic
Inspect its code (in and make sure you understand what it's doing.

Question 1 (10 points)  Improve the ReflexAgent in to play respectably. The provided reflex agent code provides some helpful examples of methods that query the GameState for information. A capable reflex agent will have to consider both food locations and ghost locations to perform well. Your agent should easily and reliably clear the testClassic layout:

python -p ReflexAgent -l testClassic
Try out your reflex agent on the default mediumClassic layout with one ghost or two (and animation off to speed up the display):
python --frameTime 0 -p ReflexAgent -k 1
python --frameTime 0 -p ReflexAgent -k 2
How does your agent fare? It will likely often die with 2 ghosts on the default board, unless your evaluation function is quite good.

Note: you can never have more ghosts than the layout permits.

Note: As features, try the reciprocal of important values (such as distance to food) rather than just the values themselves.

Note: The evaluation function you're writing is evaluating state-action pairs; in later parts of the project, you'll be evaluating states.

Options: Default ghosts are random; you can also play for fun with slightly smarter directional ghosts using -g DirectionalGhost. If the randomness is preventing you from telling whether your agent is improving, you can use -f to run with a fixed random seed (same random choices every game). You can also play multiple games in a row with -n. Turn off graphics with -q to run lots of games quickly.

The autograder will check that your agent can rapidly clear the openClassic layout ten times without dying more than twice or thrashing around infinitely (i.e. repeatedly moving back and forth between two positions, making no progress).

python -p ReflexAgent -l openClassic -n 10 -q

Don't spend too much time on this question, though, as the meat of the project lies ahead.

Question 2 (10 points) Now you will write an adversarial search agent in the provided MinimaxAgent class stub in Your minimax agent should work with any number of ghosts, so you'll have to write an algorithm that is slightly more general than what appears in the textbook. In particular, your minimax tree will have multiple min layers (one for each ghost) for every max layer.

Your code should also expand the game tree to an arbitrary depth. Score the leaves of your minimax tree with the supplied self.evaluationFunction, which defaults to scoreEvaluationFunction. MinimaxAgent extends MultiAgentAgent, which gives access to self.depth and self.evaluationFunction. Make sure your minimax code makes reference to these two variables where appropriate as these variables are populated in response to command line options.

Important: A single search ply is considered to be one Pac-Man move and all the ghosts' responses, so depth 2 search will involve Pac-Man and each ghost moving two times.

Hints and Observations

Question 3 (10 points) Make a new agent that uses alpha-beta pruning to explore the minimax tree more efficiently, in AlphaBetaAgent. Again, your algorithm will be slightly more general than the pseudo-code in the textbook, so part of the challenge is to extend the alpha-beta pruning logic appropriately to multiple minimizer agents.

You should see a speed-up (perhaps depth 3 alpha-beta will run as fast as depth 2 minimax). Ideally, depth 3 on smallClassic should run in just a few seconds per move or faster.

python -p AlphaBetaAgent -a depth=3 -l smallClassic

The AlphaBetaAgent minimax values should be identical to the MinimaxAgent minimax values, although the actions it selects can vary because of different tie-breaking behavior. Again, the minimax values of the initial state in the minimaxClassic layout are 9, 8, 7 and -492 for depths 1, 2, 3 and 4 respectively.

Question 4 (10 points) Random ghosts are of course not optimal minimax agents, and so modeling them with minimax search may not be appropriate. Fill in ExpectimaxAgent, where your agent agent will no longer take the min over all ghost actions, but the expectation according to your agent's model of how the ghosts act. To simplify your code, assume you will only be running against RandomGhost ghosts, which choose amongst their getLegalActions uniformly at random.

You should now observe a more cavalier approach in close quarters with ghosts. In particular, if Pac-Man perceives that he could be trapped but might escape to grab a few more pieces of food, he'll at least try. Investigate the results of these two scenarios:

python -p AlphaBetaAgent -l trappedClassic -a depth=3 -q -n 10
python -p ExpectimaxAgent -l trappedClassic -a depth=3 -q -n 10
You should find that your ExpectimaxAgent wins about half the time, while your AlphaBetaAgent always loses. Make sure you understand why the behavior here differs from the minimax case.

Question 5 (10 points) An interesting property of minimax search is that any order preserving transformation of the leaf values can't change the optimal action at the root. Prove this by induction on the depth of the tree. If you're rusty on how to do an induction proof, please check with us. (Here's what we mean by an order preserving transformation: Suppose the leaf states are labeled 1...n, and have values x1...xn. An order preserving transformation will assign values y1...yn, such that xi < xj iff yi < yj.

Question 6 (10 points) In class, we mentioned that reducing the branching factor by a square root effectively doubled the search horizon (depth). Rigorously prove that this is true assuming that there is a fixed, maximum frontier size that is allowed in both cases. (Alternatively, you might assume a fixed budget of node expansions, but then the result will be true only in a big O sense, and not exactly.) Note that even though this question is motivated by a comment about alpha-beta, it is just a question about how trees change when you change the branching factor. You don't need to worry about the tree being uneven due to pruning.)

Question 7 (10 points) For this problem we will consider the search tree in Figure 5.2 from the text, except that we will swap the min and max nodes. (a) Provide the minimax values for the nodes labeled A, B, C and D. (b) Indicate which branches would be pruned by alpha-beta (assuming depth first, left to right expansion). (c) What is the maximum number of nodes that could ever be pruned (considering all possible leave values) in a tree with this structure?