CS470/570 Intro to Artificial Intelligence Chapter 5: Game Playing Written Homework

Instructions: Answers to the questions below must be presented in hardcopy, on the due date noted on the course website. All submissions must:

• have solutions presented in the order listed here, and clearly labeled.
• A stapled submission with a neatly printing cover sheet is required.

Typed answers are preferable, but clear handwritten work is acceptable. Obviously, requested figures/diagrams could be cleanly hand-drawn.

The Problems:

Note to DrD for next time: This problem statement needs improved. Should GIVE them a (partial) TTT tree...that way (a) tree is smaller and (b) all have same tree, for easier grading.

5.9. (all parts) Exploring a simple game tree and applying pruning. Tic-Tac-Toe (TTT) is about the simplest game space one could consider but, even so, it is sizeable. I've added some clarification to what is required a few of the parts; for ones where I had nothing to add, just do them based on the book.

(a) Let's walk through this a bit more thoroughly: (a) first tell me how many possible boards positions there are, i.e., how big the state space of boards is for the 3x3 tic-tact-toe world. Explain your reasoning! (b) now answer the actual question: approximately how many possible *games of TTT* are possible? Explain your reasoning carefully.

(b) So you will draw a tree structure with a TTT board at every node.

(c). easy, nothing to add.

(d). Part (c) gave you board values at the leaves. This is asking you to run minimax on your game tree. Label the Min and Max levels, and show what values become attached to the non-leaf nodes as a result of MiniMax.

e) This part just asks you to essentially run Minimax again, but this time with alpha-beta pruning. Instead of "circle", let's mark pruned nodes by outlining them in red for easier reading. The last sentence about "optimal order" is important! It's saying that siblings are explored in such a way that would maximize pruning...so, depending how you generated/drew your tree, you may not be exploring branches in the order your drew them. Review 5.3.1 for a refresher on this "optimality".

5.16. (all parts) Expectimax: Minimax search in games of chance. This exercise gives you some hands-on experience with how minimax works when an element of chance is added.

a) It's just asking you to show Expectimax running on this tree. Label min,max, and chance levels clearly.

In addition to providing the figure it asks for, give a step-by-step explanation of how values are bubbled up the tree for the leftmost main branch only.Walk me through how the values for the first two triangle nodes are computed, how you got the value for the circle node above them, and then what the status of the root node becomes at that point in the process. You do not need to explicitly explain the other half of the search.

(b) and (c) It's asking you to consider the potential for pruning in this process. When it says "Given nodes 1-6", it means "Given that I have explored those nodes and know their values...but do not know values of 7 and 8". Indeed, it's asking you to ponder whether you really need to explore the remaining nodes (i.e. they could affect ultimate move choice) or not. Explain your answers carefully, i.e., could/could not exploring/not exploring affect move choice.

(d). Here it's just asking you to show the result of your analysis in parts b/c.

5.21 (all parts) Some thought problems. Make the assumption here that you have infinite time and can ply down to the end of the game (bottom of tree) in all cases. Obviously the T/F answer is the least interesting thing to me. What will matter is your analysis/explanation leading to that answer.