Pac-Man, now with ghosts.
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
searchAgents.py in any way you want.
The code for this project contains the following files, available as a zip archive.
|multiAgents.py||Where all of your multi-agent search agents will reside.|
|pacman.py||The main file that runs Pac-Man games. This file also describes a Pac-Man |
|game.py||The logic behind how the Pac-Man world works. This file describes several supporting types like AgentState, Agent, Direction, and Grid.|
|util.py||Useful data structures for implementing search algorithms.|
|graphicsDisplay.py||Graphics for Pac-Man|
|graphicsUtils.py||Support for Pac-Man graphics|
|textDisplay.py||ASCII graphics for Pac-Man|
|ghostAgents.py||Agents to control ghosts|
|keyboardAgents.py||Keyboard interfaces to control Pac-Man|
|layout.py||Code for reading layout files and storing their contents|
What to submit: You will fill in portions of multiAgents.py
during the assignment. You should submit this file with your code and comments. You may also submit supporting files (like search.py, 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.
First, play a game of classic Pac-Man:
python pacman.pyNow, run the provided ReflexAgent in multiAgents.py:
python pacman.py -p ReflexAgentNote that it plays quite poorly even on simple layouts:
python pacman.py -p ReflexAgent -l testClassicInspect its code (in multiAgents.py) and make sure you understand what it's doing.
Question 1 (10 points) Improve the ReflexAgent in multiAgents.py 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 pacman.py -p ReflexAgent -l testClassicTry out your reflex agent on the default mediumClassic layout with one ghost or two (and animation off to speed up the display):
python pacman.py --frameTime 0 -p ReflexAgent -k 1
python pacman.py --frameTime 0 -p ReflexAgent -k 2How 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 pacman.py -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
multiAgents.py. 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
MultiAgentAgent, which gives access to
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
self.evaluationFunction). You shouldn't change this function, but recognize that now we're evaluating *states* rather than actions, as we were for the reflex agent. Look-ahead agents evaluate future states whereas reflex agents evaluate actions from the current state.
minimaxClassiclayout are 9, 8, 7, -492 for depths 1, 2, 3 and 4 respectively. Note that your minimax agent will often win (665/1000 games for us) despite the dire prediction of depth 4 minimax.
python pacman.py -p MinimaxAgent -l minimaxClassic -a depth=4
Directions.STOPaction from Pac-Man's list of possible actions. Depth 2 should be pretty quick, but depth 3 or 4 will be slow. Don't worry, the next question will speed up the search somewhat.
GameStates, either passed in to
getActionor generated via
GameState.generateSuccessor. In this project, you will not be abstracting to simplified states.
mediumClassic(the default), you'll find Pac-Man to be good at not dying, but quite bad at winning. He'll often thrash around without making progress. He might even thrash around right next to a dot without eating it because he doesn't know where he'd go after eating that dot.
python pacman.py -p MinimaxAgent -l trappedClassic -a depth=3Make sure you understand why Pac-Man rushes the closest ghost in this case.
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 pacman.py -p AlphaBetaAgent -a depth=3 -l smallClassic
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 pacman.py -p AlphaBetaAgent -l trappedClassic -a depth=3 -q -n 10
python pacman.py -p ExpectimaxAgent -l trappedClassic -a depth=3 -q -n 10You should find that your
ExpectimaxAgentwins about half the time, while your
AlphaBetaAgentalways 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?