3 MAX X O X O X . . . MAX (X) O A 1 A 2 A 3 3 2 2 X O X - PDF document

deterministic chance perfect information chess, checkers, backgammon go, othello monopoly imperfect information battleships, bridge, poker, scrabble blind tictactoe nuclear war MAX (X) X X X MIN (O) X X X X X X 3 MAX X O X O X . . . MAX (X) O A 1 A 2 A 3 3 2 2 X O X X O X O . . . MIN MIN (O) X X A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 . . . . . . . . . . . . 3 12 8 2 4 6 14 5 2 . . . X O X X O X X O X TERMINAL O X O O X X O X X O X O O Utility − 1 0 +1

3 3 MAX MAX 3 3 2 MIN MIN X X 3 12 8 3 12 8 2 3 3 MAX MAX 3 2 14 3 2 14 5 MIN MIN X X X X 3 12 8 2 14 3 12 8 2 14 5 3 3 MAX MAX MIN 3 2 14 5 2 MIN .. .. .. X X MAX 3 12 8 2 14 5 2 MIN V

Black to move White to move White slightly better Black winning MAX 1 2 1 20 MIN 1 2 2 4 1 20 20 400

5 0 1 2 3 4 6 7 8 9 10 11 12 MAX 3 − 1 CHANCE 0.5 0.5 0.5 0.5 2 4 0 − 2 MIN 2 4 7 4 6 0 5 − 2 25 24 23 22 21 20 19 18 17 16 15 14 13 MAX 2.1 1.3 21 40.9 DICE .9 .1 .9 .1 .9 .1 .9 .1 2 3 1 4 20 30 1 400 MIN 2 2 3 3 1 1 4 4 20 20 30 30 1 1 400 400