The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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  • The Psychological Concept of ``Losing Move'' in a Game of Perfect InformationAuthor(s): Herbert A. SimonSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 71, No. 6 (Jun., 1974), pp. 2276-2279Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/63405 .Accessed: 03/05/2014 09:14

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  • Proc. Nat. Acad. Sci. USA Vol. 71, No. 6, pp. 2276-2279, June 1974

    The Psychological Concept of "Losing Mo (game theory/problem solving/choice under uncertainty,

    HERBERT A. SIMON

    Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213

    Contributed by Herbert A. Simon, March 29, 1974

    ABSTRACT From a game-theoretic standpoint, in any two-person game of perfect information, each position is won, lost, or drawn, and a move is to be considered an error only when it transfers the game from a more favorable to a less favorable state. A psychological concept of error is quite different, in that it must take into account the fallibility of the players as information processing systems. This paper introduces a psychological concept of error in such games based on the distinction between "obvious" and "problematic" moves, and proposes a formalism for this concept that appears to capture, at least to a first approximation, the notion of "losing move" as that phrase is actually used by players in games like chess.

    When a game between two equally matched chess grandmas- ters is won by one of them, the question to be answered is: "Which was the losing move?" It is widely believed, though it has never been proved, that with correct play on both sides a game of chess should be drawn. Hence, when a game is not

    drawn, it must be because the losing player at some point made a move that was not the best.

    The losing move is difficult to identify from both a practical and a conceptual standpoint. It is hard to identify practically, because to find the error means to find a move better than the one made by the grandmaster and to verify that it is better. Of

    course, since match and tournament chess games are played under time constraints, and annotators can take as much time as they like to carry out their analyses-as well as having the

    hindsight knowledge of how the game came out-the prob- lem is only difficult, not necessarily hopeless.

    But practical identification of the losing moves in chess

    games does not settle the questions. What is meant by a losing move? And is the losing move unique; may there be more than one in a game?

    From a strict game-theoretical standpoint, since chess is a

    game of perfect information, and since a chess game has one of three outcomes-Win (W), Lose (L), Draw (D)-each actual or possible move in a game can (in principle) be as- signed one of those values without ambiguity, by minimax-

    ing backward on the game' tree from the terminal branches. If the true value of the game is in fact D, then we could define a losing move as one that transforms the value from D to L. (Since a player, under our assumptions, can only transform the game from D to D, or from D to L, but never from D to W, there is no corresponding definition for "winning move.")

    This simple definition for "losing move" does not, however, conform very well with chess experience. Comments on chess games that one or the other player has "a better game" or "a plus" do not usually mean that the advantaged player can force a win-i.e., that he is in a W position. On the contrary,

    22

    ye" in a Game of Perfect Information (gamblers' ruin)

    the disadvantaged player often can obtain a draw if he finds the best moves in the subsequent sequence of moves.

    In annotations of chess games, moves that represent serious errors are usually marked with a question mark. However, a move may be adjudged "inferior" by the annotator even when he does not affix a question mark to it. In C. H. O'D. Alexan- der's book, Fischer v. Spassky, (1) for example, one or more question marks are affixed to moves in 16 of the 20 games. The four games where no moves were marked were all draws. In the nine games that were not drawn, the loser received an excess of question marks in all cases: one extra mark in three

    games, two in five, and four in one game. In the remaining seven draws, the question marks were evenly divided (one mark for each player) in four games, one player had an excess of one in two, and one player had an excess of two in one game. These statistics, which are not atypical of match and tourna- ment books, suggest that the player with one extra question mark is as likely as not to lose, anod with two or more extra

    question marks will very probably lose. The total number of moves adjudged by Alexander to be

    inferior, marked aind not, was approximately twice the number to which question marks were affixed, so that taking all of these into account would increase to three or four the number of "errors" required to move a position from D to L. We will not be interested here in the exact number, but rather in un- derstanding what is meant by "error" in this context.

    Since, in the strict game-theoretic formulation, a game of

    perfect information can only be W, L, or D, clearly some addi- tional concepts must be introduced if we are to speak of a

    player having an advantage, but not a winning advantage. Moreover, these concepts must be psychological in nature, relating to the chess knowledge of the player, and his actual

    ability as a finite information processor to foresee the conse-

    quences of his moves. Both of these notions relate to the

    psychological and not the game-theoretic aspects of the gane. At a superficial level, we could simply speak of the probability that, if two players of more or less equal strength were placed in the position in question, one or the other of them would win. There are several difficulties with this interpretation. For one thing, if we compared samples of games betweetn players of different strengths, we would almost certainly find that the probability was a function both of their relative strengths and of their absolute strengths. But however the statistics worked out, they would still not tell us, at any deeper level, what is meant for a player to have an "advantage" in a chess game if the position is D.

    In games between nearly equal grandmasters, the problem appears event a little more complicated. We have seen that it is

    76

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  • Proc. Nat. Acad. Sci. USA 71 (1974)

    L D

    -k -10 -8 -6 -4 -2 0

    FIG. 1. Lattice for an idealized game. / \/', path for

    seldom that the annotator discovers only one of the loser's moves that he believes can be improved. More often, he finds two or more. What is more, he often finds that the winner, too, did not always make the best move. Sometimes the game seems to proceed from a situation where it is W (or at least a plus) for one player to a situation where it is ID (after the leading player has made a weak move), to a later situation where it is again W, or where it.ha-sturned to L for the player who was originally ahead.

    A formal model of the game

    To explore these psychological constructs further, we will de- fine an idealized game which, like chess, has the three possible outcomes, W, L, or D. In this game, each time he is on move, the player has a choice between exactly two alternatives. Each position has an objective numerical value, V, but this value is not known to the players, who can only. estimate it oni the basis of their knowledge of the game. A player will always strive to choose the alternative that has the higher value, and his move can be evaluated objectively in terms of its actual effect on that value.

    The value 0 is assigned to the initial positionI. At each branch in the binary game tree described above, the value is increased by 1 if one branch (call it the right-hand branch) is chosen, and is decreased by I if the other branch (call it the left-hand branch) is chosen. All the terminals of the tree are 2N branches deep. All terminals with values in the range of -k to +k are D; terminals with values below --k are L for the first player; while terminals with values above +k are W for the first player.

    This idealized game can be represented by a lattice, as in Fig. 1, simply by identifying as identical all positions at a giveno depth that have the same value. A move to the upper left is an incorrect move if made by the first player, a correct move if made by the second player; and vice versa for a move to the upper right. The starting position, with value 0, is at the bottom of the lattice; the terminal W, D, and L positions, together with their values, at the top of the lattice. If both

    pllayers make only correct moves, they will proceed along the shortest zig-zag connecting the starting position with the terminal vertically above it.

    The Concept of Losing Move 2277

    w

    +k I I Move by 2 4 6 8 10 Player

    , ,- error 2 -7 ^~~~~~~~~~~~1

    perfect play; ,, " , path for game 1 won by player 1.

    Consider the situation after each player has made m moves. The value of the position arrived at must lie in the range -2m < V < +2m. The lower limit will be reached only if the first player has always chosen the poorer of the two moves open to him, while the second player has always chosen the better of the moves open to him. If, now, V + 2(N - m) < -k, the first player must lose, for his final score will be less than -k even if all his remaining choices are correct, and all his opponent's are wrong. If, on the other hand, V - 2(N - m) > +-k, then the first player must win, even if he makes no more correct choices, while his opponent makes all correct choices. In all other cases, it is still possible for the value of the game to change.

    If V + 2(N - m) < -k, the game is definitely lost (L*); if V - 2(N - m) > -k, the game is definitely won (W*). If +k - 2(N - m) > V > -k + 2(N - m), we will say that the game is definitely drawn (D*). In all other cases, we will say it is undecided. The values W* and L* correspond closely with the game-theoretical notions of W and L, respectively.

    Suppose that V < -k, but the game is not L*. Then if both players make the same proportion of correct moves in the remainder of the game, it will be lost by the first player. We will say that such a game is probably lost (L'). Symmetrically, if V > k, but the game is not W*, we will say that it is probably won (W').

    No. of Errors No. of Errors Player 2 Player 1

    0 1 2 3 4 5 6 7 8 9 10 0 D 1) L L L L L L L L L 1 D D D L L L L L L L L 2 W I) DDL L L L L L L 3 WWDDDL L L L L L 4 W W WDD D L L L LL 5 WWWW ) D 1) L LL L 6 WWWWWI) D D DL L 7 WWWWWWDDDLL 8 W W WWWWWD I) DL 9 W W W W W W W W D 1) D

    10 W W W W W W W W W D D

    FIG. 2. Outcome of game (as function of number of errors of each player). W, won by player 1; 1), drawn; L, lost by player 1.

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  • 2278 Psychology: Simon

    TABLE 1. Probabilit

    Number Number of het coins

    remaining 10 8 6 4 2

    10 - -- 8 .-- 0.3 6 - - 0.657 0.3 4 - - 0.938 0.688 O. 3 2 - 1.000 1.000 0.750 0.2 0 1.000 1.000 1.000 1.000 0

    * Calculated as a problem of gamblers' ruin, given number of coins <

    Finally, we can define a plus and a minus as follows. Select a proper fraction, a, 0 < a < 1. The first player has a plus whenever V > ak/2m, and a minus whenever V < -ak/2m. To see the meaning of this, consider the limit, where a = 1. V = k/2m means that the first player will win if he maintains his past relative advantage in correct choices over the second player. Now, to take a smaller value of a to define the bound- ary, is to say that a player has a plus (or minus) if he has a certain relative advantage, but not large enough to win the game.

    Another possible definition of plus and minus is even simpler. We can say that the first player has a plus whenever k > V > k', where k' is a constant equal to some fraction of k. Then, the first player, to win, must increase his advantage by the fixed amount, k - k'. The notions of "plus" and "minus" are not used in exactly the same way by all annotators, but these alternative definitions might reasonably be expected to span the common uses.

    Interpretation of the model

    How does the game of chess relate to this abstract model? When chess is played between grandmasters, many moves are unproblematical to the players-the correct move is more or less obvious to them, and all players at the level would choose the same move. Perhaps half or more of the moves in a well- fought game are "obvious" in this sense. In all other cases, the grandmaster must carry out a more or less elaborate problem- solving activity in order to select what he regards as the best move. His choice does not usually lie among a large number of moves-seldom, except in the opening, are there more than two or three real contenders. However, the difference in pro- mise between a pair of moves may be very hard to assess, and strong players may disagree as to which is the better. These are the moves over which players must ponder, and on which the game depends; and these are the only moves that require attention from the annotator (unless he is explaining the game to weaker players. What is an "obvious" move among grand- masters may be quite inexplicable to weaker players.)

    That a choice between two moves is problematic to a grand- master means that if a grandmaster is chosen at random and placed before the position, there is approximately a fifty- fifty probability that he will make one or the other choice. There is no assertion here that a grandmaster's choice con- tains any element of randomness. Each grandmaster is as- sumed to have a determinate, but idiosyncratic, prog...

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