Artificial Intelligence CS482, CS682, MW 1 2:15, SEM 201, MS 227 Prerequisites: 302, 365 Instructor: Sushil Louis, [email protected], http://www.cse.unr.edu/~sushil Games and game trees Multi-agent systems + competitive environment games and adversarial search

In game theory any multiagent environment is a game as long as each agent has significant impact on others In AI many games were Game theoretically: Deterministic, Turn taking, Two-player, Zerosum, Perfect information AI: deterministic, fully observable environments in which two agents act alternately and utility values at the end are equal but opposite. One wins the other loses

Chess, Checkers Not Poker, backgammon, Game types Starcraft? Counterstrike? Halo? WoW? Search in Games

Tic-Tac-Toe Two player, deterministic, small tree Two players: Max versus Min Approximately: 9! tree nodes Tic-Tac-Toe

Minimax search Minimax algorithm 3 player Minimax Two player minimax reduces to one number because utilities are opposite knowing one is enough But there should actually be a vector of two utilities with

player choosing to maximize their utility at their turn So with three players you have a 3 vector Alliances? Minimax properties Complete? Only if tree is finite Note: A finite strategy can exist for an infinite tree!

Optimal? Yes, against an optimal opponent! Otherwise, hmmmm Time Complexity? O( Space Complexity?

O(bm) Chess: b ~= 35, m ~= 100 for reasonable games Exact solution still completely infeasible Alpha-beta pruning

Alpha-beta Alpha-beta Alpha-beta Alpha-beta

Alpha-beta Alpha is the best value (for Max) found so far at any choice point along the path for Max Best means highest If utility v is worse than alpha, max will avoid it Beta is the best value (for Min) found so far at any choice point along the path for Min

Best means lowest If utility v is larger than beta, min will avoid it Alpha-beta algorithm Alpha beta example Minimax(root)

= max (min (3, 12, 8), min(2, x, y), min (14, 5, 2)) = max(3, min(2, x, y), 2) = max(3, aValue <= 2, 2) =3

Alpha-beta pruning analysis Alpha-beta pruning can reduce the effective branching factor Alpha-beta prunings effectiveness is heavily dependent on MOVE ORDERING 14, 5, 2 versus 2, 5, 14 If we can order moves well O(

Which is O(() Effective branching factor then become square root of b For chess this is huge from 35 to 6 Alpha-beta can solve a tree twice as deep as minimax in the same amount of time! Chess: Try captures first, then threats, then forward moves, then backward moves comes close to b = 12

Imperfect information You still cannot reach all leaves of the chess search tree! What can we do? Go as deep as you can, then Utility Value = Evaluate(Current Board) Proposed in 1950 by Claude Shannon Apply an evaluation function to non-terminal nodes

Use a cutoff test to decide when to stop expanding nodes and apply the evaluation function Evaluation function Must order nodes in the same way as the utility function Wins > Draws > Losses Fast

Otherwise it is better to search deeper and get more information For non-terminal states, high evaluations should mean higher probability of winning Chess is not a chancy game But computational limitations make eval function chancy! Which is better?

Evaluation functions A function of board features Use proportions of board-states with winning, losing, and drawing states to compute probabilities.

72% winning (1.0) 20% draws (0.0) 8% losses (0.5) Then: evalFunction(board state) = (0.72 * 1) + (0.2 * 0) + (0.08 * 0.5) Use a weighted linear sum of board features (Can also use non-linear f)

Chess book: pawn = 1, bishop/knight = 3, rook = 5, queen = 9 Good pawn structure = A, king safety = B evalFunction(board state) = * pawns + * bishops + * knight + * rook + + * good pawn structure + . All this information for chess comes from centuries of human expertise For new games?

When do we cutoff search Quiescence Horizon effect and singular extension Forward pruning Beam search ProbCut learn from experience to reduce the chance that

good moves will be pruned Like alpha-beta but prunes nodes that are probably outside the current alpha-beta window Othello Combine all these techniques plus Table lookups

Chess Openings (perhaps upto 10 moves) Endings (5, 6 pieces left) King-Rook versus King (KRK) King-Bishop-Knight versus King (KBNK) Checkers Is solved!

Stochastic Games Chance is involved (Backgammon, Dominoes, ) Increases depth if modeled like: Simple example (coin flipping)

Expected value minimax Backgammon With chance, exact values matter Fog of War Use belief states to represent the set of states you could be in

given all the percepts so far Kriegspiel You can only see your pieces Judge says: Ok, illegal, check, What is a belief state? Card Games

Consider all possible deals of a deck of cards, solve each deal as a fully observable game, then choose best move averaged over all deals Computationally infeasible but: Let us do Monte Carlo approximation Deal a 100 deals, a 1000 deals, whatever is computational feasible Choose best outcome move

Read section 5.7 state of the art game programs Errors in evaluation functions! Summary Games are fun to work on They give insight on several important issues in AI

Perfection is unattainable approximate Think about what to think about Uncertainty constrains assignment of values to states Optimal decisions depend on information state, not real state

Games are to AI as grand prix racing is to automobile design Searching with Nondeterministic actions In the past, we knew what state we were in and a solution was a path from root to goal. Now, how do you find paths when the environment is partially observable or non-deterministic or both and you dont know

what state you are in? You make contingency plans If in state x then y You use percepts I did an action with a non-deterministic result, percepts can tell me which result actually occurred

Erratic Vacuum cleaners Suck Sometimes cleans adjacent square Sometimes deposits dirt in current square

Transition Model Result Results Suck({1}) {5, 7} Erratic Vacuum cleaners Sometimes cleans adjacent square Sometimes deposits dirt in current square

Solution [Suck, if State == 5 then [Right, Suck] else []] Solutions are trees! Not sequences Solutions are nested if-then-else Many problems in the real world are of this type because exact prediction is impossible

Keep your eyes open when you drive/walk/fly And-Or search trees And-Or search trees Or nodes (Suck or Right) And node {5, 7} Results And-Or tree solution is a subtree:

Goal node at every leaf One action at each Or-node Includes every outcome branch for And Same as: [Suck, if State == 5 then [Right, Suck] else []]

Remember the simple problem solving agent? And-Or problem solver If there is a non-cyclic solution it must be findable from the earlier occurrence of state in

path (Completeness) Recursive, breadth-first. Can use breadth-first, Slippery vacuum worlds Movement actions sometimes fail and leave you in the same location No acyclic solutions!

Labels enable cycles [Suck, L1: Right, if State == 5 then L1 else Suck] Search Problem solving by searching for a solution in a space of possible solutions Uninformed versus Informed search Local search

Atomic representation of state Solutions are fixed sequences of actions With non-deterministic environment solutions are trees with labels Quiz (20 minutes) Types of task environments

Task Env Soccer Explore Titan Shopping for used AI books on the Net Playing tennis Playing tennis

against a wall Performing a high jump Knitting a sweater Bidding on an item in an auction

Observable Agents Deterministic Episodic

Static Discrete