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 Information Author(s): Herbert A. Simon Source: Proceedings of the National Academy of Sciences of the United States of America, Vol. 71, No. 6 (Jun., 1974), pp. 2276-2279 Published by: National Academy of Sciences Stable URL: http://www.jstor.org/stable/63405 . Accessed: 03/05/2014 09:14 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access to Proceedings of the National Academy of Sciences of the United States of America. http://www.jstor.org This content downloaded from 130.132.123.28 on Sat, 3 May 2014 09:14:26 AM All use subject to JSTOR Terms and Conditions

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Page 1: The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact [email protected].

.

National Academy of Sciences is collaborating with JSTOR to digitize, preserve and extend access toProceedings of the National Academy of Sciences of the United States of America.

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Page 2: The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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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Page 3: The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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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Page 4: The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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, program for choosing moves. (The program must not be too idiosyncratic, else he would not be a grandmaster!) This program will some- times lead him to choose the correct move in a problematic

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

y of win by player 1 *

ids minus number of tails

0 -2 -4 -6 -8 -10

0.172 - - - - 63 0.144 0.035 - 44 0.110 0.016 0 - - -

13 0.063 0 0 0 - -

50 0 0 0 0 0 0 0 0 0 0 0

plays) remaining and net excess of heads. N = 5, k = 2.

situation, sometimes the incorrect move. Different players of the same strength will make the same fraction of errors, but may make their errors in different positions.

Of course the assumption that only fifty-fifty situations are problematic is not essential, but for the moment, it simplifies matters to think of a game as consisting of a relatively small number of choice situations in which there is one chance in two, on average, of going wrong, whereas in all other positions the correct move will be made with certainty.

Within this framework, we can define "losing" or "winning" move in chess, and we can also estimate about how many such moves there are in a game. I will continue, for the present, to speak of two players who have about the same strength.

If a player makes a sufficient number of correct moves while his opponent is making mistakes, sooner or later he will have a W' position, and then a W* position, and finally an actual win. The final outcome of the game, W, D, or L, is determined by the actual number of mistakes each player makes. For the game lattice of Fig. 1, the function is shown in Fig. 2. In this game, the player who makes two fewer mistakes than his op- ponent wins; if neither achieves this margin, the game is a draw. We might say here that a player makes a losiong move when he makes his second mistake; but there is really ino more reason to call this second mistake the losing move than to apply that label to the first mistake.

The theory of gamblers' ruin throws further light on the situation. Represent the game as a series of tosses of a (un- biased) coin. If the final number of heads exceeds the number of tails by more than k, the game is W; if it falls short by more than k, the game is L; otherwise the game is D. Now, we can compute by the binomial theorem the probability of W or L, given the total number of coins, the number of coins remaining to be tossed, and the excess (or deficiency) of heads over tails. Table I gives the probabilities of a win by the first player for various values of the latter two parameters in a game with five moves (10 coins), and with an excess of two heads re- quired for a win. The numbers in this table are not entirely unrealistic as a description of grandmaster chess.

Chess is probably not quite as symmetrical a game, how- ever, as the abstract game used for illustration. It is usually thought that, in play between nearly equal players, White has the better chance to win; and that Black should not be able to get more than a draw. To represent the asymmetry, set the initial value of the game equal to some small positive number, and adjust k accordingly. The values, V0 = 1, k = 1 might be realistic for chess between strong players-that is White will win if he makes one less mistake than Black, but he will lose only if he makes two more mistakes than Black.

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Page 5: The Psychological Concept of ``Losing Move'' in a Game of Perfect Information

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

With these numbers, and N = 10, games between even players would then result in wins for White in seven or eight games out of 20; in draws in about nine games out of 20; and losses for White in three or four games out of 20.

Relation of psychological to game-theoretical notion of "losing move"

In game theory, the value of a position is determined by working backward from the end of the game. We can readily define a value, for the formal model of a game that has been proposed here, that resembles the game-theoretic value more closely than does V.

Call the new value, U. If a game is L*, then U -= -1; if the game is W*, then U = 1; if the game is D*, U = 0. Now, for any position that is not definitely won, lost, or drawn, set U equal to the arithmetic mean of U for the two positions that result from its alternative moves. U then is equal to the ex- pected value of the position. The underlying probabilities are precisely the probabilities of the corresponding gamblers' ruin problem (taking account, in this case, of draws as well as wins and losses).

Now an error is any move that lowers the expected value of the player's position. Leaving aside draws for a moment, we can say that if a player's U is negative, his judgements on subsequent problematic positions will have to be better than his opponent's if he is to equalize the position. If U is already a sizeable negative fraction, a single error may lead to a L* position. This evaluation represents very well the subjective feeling that players in a "minus" position commonly have, that they are walking a tightrope, where only a faultless per- formance can save them from disaster.

To reflect fully all the nuances of the meanings of advantage and disadvantage in an actual chess game, we would certaiinly have to introduce additional complications into the model. I will merely mention two such possibilities.

Time Limits. Chess is played under strict time limits (e.g., 1 hr for 20 moves). A move may be obvious if the player has lots of time to consider it, but highly problematic if he has to make his choice very quickly. A sizeable fraction of the serious er- rors made by grandmasters are made when they are under time pressure.

Frequency of Problematic Positions. The model assumes that the two players face problematic positions alternately-the first player, then the second player, and so on. In actual play, problematic positions may face the one player more often than the other. One possible meaning of the term "initiative," as it is used by chess players, is that the player with the initiative can present his opponent with problematic positions, but not vice versa. Then, in terms of expected values, the player with initiative would have the larger expected value.

Play Between Unequal Players. It is not hard to generalize the model to take account of differences in skill between the players. First define the abstract game for the weaker player, by selecting the choice points at which he will not do better than chance. (For players in certain ranges of skill, the poorer move may be chosen more often than the better one.)

At many of the choice points that are toss-ups for the weaker player the stronger player will know very well (or at

The Concept of Losing Move 2279

least with a high probability) which is the correct move. The simplest way to represent his greater strength is to suppose that he always makes the right choice at this subset of choice points, hence has fewer opportunities for error than his op- ponent. This is equivalent, in the gamblers' ruin representa- tion of the problem, to assuming that the stronger player starts out with a positive score equal to the number of choice points that are problematic for the weaker player, but not for him. Assuming as before that a surplus of two or three cor- rect decisions is enough to decide the game of chess, it can be seen that there need to be only a small number of such asym- metrical choice points virtually to guarantee victory to the stronger player.

The plausibility of this representation is supported by the well-known phenomena of simultaneous play by stronger against weaker players. The stronger player can win against an almost arbitrary number of weaker opponents, taking only a few seconds for each move. Masters who are able to do this report that they simply make "standard" moves until they notice that their opponents have made a mistake, then move to exploit the mistake. What is a "mistake" to the master is a wrong choice made by the weaker player at a point where the latter, but not the former, was unable to discriminate correctly among alternatives.

Conclusion

From a game-theoretic standpoint, chess is a trivial 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 for the player. This concept of "error" is very different from the con- cept actually used by chess players.

To introduce for purposes of psychological theory notions of "error" and "losing move" into a theory of chess, we must take into account the fallibility of the players (men or com- puters) as infortnation processing systems. One way to do this is to distinguish between "pobvioiis" and "problematic" moves, and to take account of the probability that a correct choice will be made in a problematic situation.

There are several ways to formalize this idea. One is to represent chess by an idealized game of moves in a lattice. Adding the further assumption that in a problematic situation the player on move will make the correct choice with a speci- fied probability, we can define various functions (for example, our functions V and U), to measure the value of a position. HIigher values correspond to higher probabilities that the player will wino in those positions. An error or losing move is then simply one that reduces the value substantially.

This formnalization appears to capture, at least to a first ap- proximation, the notion of "losing move," as that phrase is actually used by chess-players, and annotators of games.

I am grateful to Hans Berliner and John Gaschnig for helpful comments on an earlier draft of this paper. This research was supported in part by Research Grant MH-07722 from the National Institute of Mental Health and in part by the Advanced Research Projects Agency of the Office of the Secretary of De- fense (F44620-70-C-0107) which is monitored by the Air Force Office of Scientific Research.

1. Alexander, C. H. O'D. (1972) Fischer v. Spassky (Vintage Books, New York).

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