JOURNAL ()F THE EXPT,TRIMENTAL ANAL\5IS OF IIL,FL\VI()R 20t0, 94,373*386 N LIMBER 3 ( NOV!,NI IJI,]R)

CHESS AS A BEHAWORAL MODEL FOR COGNITNE SKILL RESEARCH: P,EVNW OF BLINDFOLD CI]ESS BY ELIOT HEAR\T AND.TOHN IA{OTT

FnqNr:is MEcnNnn

I'H!] MEC]HNER FOUNDATION

This rnultifaceted work on chess played without sight of the pieces is a sophisticated psychologist's exanrination ol this topic and of chess skill in gcneral, inchrding a detailed ancl comprchensive historical account. This review builds on Hearst and Knott's assertion that chess can provide a uniquely usefirl model fbr research on several issues in the area of cognitive skill and imagery. A key issue is the relationship betiveen viewing a stinmlus and mental imagery in the light of blindfbld chess masters' consistent rcports that they do not use or have images. This review also proposes a methodology fbr rneasuring and quantifyins an individual's skill shortfall frorn a theoretical maxirnum. 'l'his methodologF, based on a 1951 proposal by Claude Shannon, is applicable to any choice siluation in which all the available choices are known.'l'he proposed "Proficiency" rneasure reflects the ecluivalent number of "yes-no" questions that would have been required to arrive at a best choice, considering also the time consumed. As the measure prolides a valid and nonarbitrary way to compare difl'erent skills and the elfects of diff'erent independent variables on a siven skill, it may have a lvide range of applications in cognitive skill research, skill training, and education. Kq anrds: chess research, mental imagery, conceptualization, cognitive skill measurement, skill training, representations, r.isual perception

Perhaps the reason why chess so often this time as a behavior researcher (Hearst, fascinates psychologists is that along with 1979, 1988; Flcarst, Besley & Farthing, 1970; mathematics, music, and other arts, it has If earst & Jenkins, 1974) while a prof'essor at produced displays of virtuosity sornetimes the University of Missouri, Indiana University, viewed as pinnacles of human achievement. Ttre University of Calilbrnia at Berkclcy, One of the more dazz.Tilng feats of this sort is Columbia Universig', and The University of blindlbld chess-chess played without sight of Arizona. The other author, John Knott, is a the chessboard or pieces, with the players lifelong researcher- and prominent authority calling out their moves. Hearst and Knott show on blindfold chess.' us why this topic should interest not only chess The book clearly stands as the dcfinitive players but als<-r behavior researchers, neuro- compendium on the topic of blindfold chess. biologists, psycholosists, and educators. Chess players have already expressed admira- Eliot Hearst straddles the worlds of chess tion fbr the depth of its scholarship.z includ- and psychology at the highest levels. By age 21 ing the authors' painstaking and masteful he had already achieved national and interna- analysis of some 444 historically significant tional prominence as one of the rnost talented blindfold chess sames, and psychologically senior chess mastcrs of his generation. IIe is flavored biographical sketches of history's pictured in the 1952 photo (Figure 1), stand- grcatcst blindfold chess masters. ing, with thc three other members of the Behavior analysts will be particularly inter- Columbia College chess team of which he was ested in the chapters "Research on General the captain. In 1962, he captained the United Chess Skill" and "Psychological Stuclies and States Ol;'rnpic Chess Team. Commentaries on BlinclFold Chess" (pp. 151- Flearst did his graduate work at Columbia 190),3 which review the salient iesearch University under Professor W.N. Schoenfeld literature on those topics, includins the work and then achieved prominence once again but

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Jl3 374 FR4"VC1S MECIINER

F-ig. L T'he Clolumbia Clollege chess team of 1949-i952 after a radio match with Yale. Right to left: .fames Sher-win, I.lliot Hearst, Carl Burger, Franci.s Mechner (Courtesy of the Columbia Universiq, Archives). of Alfred Binet, Alfred Cleveland, several for cognitive skill research. Chess has, in fact, Russian psychologists, Adrian de Groot, IIer- been dubbed "the drosophila of cognition bert Simon, William Chase, Dennis IIolding, research and psychometrics" (Chase & Simon, Fernand Gobet, Neil Charness, Christopher 1973; Van der Maas & Wagenmakers, 2005; Chabris, and many others, including Flearst p.150). himself. The present review attempts to show how Chess as a Unique Research Mod.el the abovelisted features of chess provide a The book draws attention to a number of research methodology applicable to a widc issues in the realm of cognitive behavior that range of problems whose investigation re- have received little attention in the behavior quires quantitative measurement of skill and analytic literature, possibly for lack of a knowledge. methodology with which to address them Notable Findings (Foxx & Faw, 2000; Marr, 2003; Staddon, In their review of the literature, Hearst and 2001). The features of chess that should make Knott document a number of notable conclu- it interesting to behavior analysts as a model sions that rnay surprise chess players and non- for cognitive behavior research are these: The chess players alike: choices (chess moves) are discrete, involve purely cognitive behavior, are susceptible to . The gcneral memory of chess masters, registration and quantitative evaluation by including those able to play many blindfblcl computer, and the number of choices avail- games simultaneously, is no better than that able in a given position (approxirrrately 37 on of the average person. average) is convenient. These f'catures may be . Highly skilled players can form long-term some of the reasons why Hearst and Knott memories of full-board chess positions believe (pp. 150-151), as does fcrrmer worlcl within seconds of viewing them. chess champion Garry Kaspar<-rv (2010), that r lligh level chcss skill (not just blindfold chess can senre as a useful laboratorv model chess) requires a recognition-action reper- REVNW OFRLINDFOLD CHE,SS .)t!,

toire of some 50,000 to 100,000 fearures of squares, and the lengths and intersection chess positions and associated responses. squarcs of all the diagonals. Ilearst and Knott ' Some of the strongest masters lind the add that, "Geornetric knowledge of the actual sight of a chcss position to be more chessboard ... presumably underlies what an distracting than helpful when thinkine expert blindfold player rneans when he talks ahead during a game. about r.isualizing "lines of force" or "powers . Practicing blinclfold chess improves sighted of'a piece"-rather than seeing actual pieces chess skill. and colored squares in the mind's eye." . Sorne of the strongest blindfolcl chess mas- Viaoing and Visualizing ters claim that the strength of their blind- These reports point to the need for a fold play is similar to that of their siehted detailed examination of the relationshio be- play- twcen uieuing, in the sense of contemporane- ous perception of cxternal stimuli with light But the bombshcll, which prompts the pres- falling on the relina, ancl the beharior ent reexamination of "visualization," is this: gcnerally referred to as "visualization," or blindfokl chess masters consistentb reltctrt that what "mental imagery." \44rat do the two types of thq uisualize are not images of pieces rtr chessbonruIs, behavior have in cornmon and how do they but abstractions oJ th,ese with minimal or no physical cliffer?o Defining visualizing as "internal see- features. A q,pical report is, "I do not visualize ing" or "sceing a stimulus in its absence" real pieces but I know where thcy are." (Moore, 2008; Skinner, 1953, 1974) does not Chess Masters' Protocok Regarding Their "Visual- acombination or plan is in ous exteroceptivc stimulus while visualizinq is not. progress," "lincs of force," "pieces are only . fiiend or foe, carriers of particular actions," Viewins permits the stimulus to be scanned "sort of formless visions of the positions," and and interrogated regarding even its most so forth. Many of the masters report that they unimportant details (c.g., the colors, have no mental image at all (p.151)." shapcs, sizcs, or sur{ace characteristics of Such introspective reports and protocols the chess pieces, the source or level of the illumination), while visualizing regarding "private" cvents and processes does not. generally tend to be accorded low status as The Image Conjecture-

No€, 2004). It rnay also be based in large part uisualizing, mental imagtng and. representation,T on our subjective experience-the powerful all of which have this undesired connotation. introspective illusion thatwe "see" an internal The term should also distinguish between image "in the mind's eye." \Arhen applied to viewing and visualizing. The terrts conceptual chess, the Image conjecture would imply that ization or abstraction don't make that distirrc- blindfold masters "have" internal images of tion because they refer to behavior that occurs chess positions, but their protocols clearly state in both. that they do not. This quandary reminds us that when a The proposed conceptualization may seem scientific discipline's traditional terminologres radical. It rejects the common premise implied cannot accommodate a new recuirement. one in much of the psychology and neuroscience must choose between redefrning an old term literature, that "mental imagery," "visual and coining a new one, a common occurrence imagery," or "representation" is so elemental in the evolution of disciplines (Mechner, 2008, further' and self-evident as not to reouire pp. 236-237 ; Zental| Jackson-Sm ith, & Jagielo, definition or examination. Th-e alternative 1990). Imaging research may now be at this view offered here and developed in the point. sections that follow, is that "seeing" can occur The terms that fully circumvent the inherent only in the sense of viewing, where a visual dualistic mentalism of the Image conjecture stimulus can be scanned and interrogated, and are, paradoxically and ironically, mentalization cannot occur in "mental imaging," "visualiza- and mentalize. Sayrng that blindfold chess tion," or "seeing with the mind's eye," none masters mentalize chess positions avoids the of which permit scanning-interrogation. The implication that they "have a mental image," proposed conclusion is that such "visualiza- because it permits the (parsimonious) inter- tion" behavior actually consists exclusively of pretation, in conforrnity with the chess mas- the recall of prniously aquired concepts drawn ters' protocols, that only conceptual behavior from our preexisting behaaior repertoires. Our occurs in mentalization. The term mentaliza- subjective sensation of internally "seeing" this tion has the added virtue of being applicable "mental type of image" is purely an introspec- to all modalities-visual, auditory, tactile, tive illusion whose cornpelling power may be ollactt-rry. or kinesthetic. due to a kind of fusion or blending phenom- Mmtalizations Cmsist of Concepts enon-perceiving continuity where the stimu- The oresent formulation is based on the lus elements are, in fact, totally discrete. Other behavioi-analytic concept of "concept," de- examples of this q?e of illusion are our fined as "discrimination between classes and perception of srnooth continuous motion generalization within classes" (Hull, 1951; when we see the discrete still frames of a Keller & Schoenfeld, 1950)." Thus, a concep- movie-the frames being analogous to the tual triangle may be outlined or not, filled in recalled concepts, or our illusion of a likeness or not, of any color or size, right, acute or of a face when viewing a digitized array of a obtuse, verbally defrned, or a compound of all relatively small number of square black, white, of these attributes; the concept of a familiar and gray pixels. person may include features, clothing, facial A T erminolo$ml Quand.ary expressions, a compound all these; The present analysis thus requires a term or of of similarly, when chess masters mentalize a chess that does not connote an internal or mental conceptualize its image. The terms most often used in the position, they certain oI dpramic psychological and neuroscience literature are relational and attributes. The range and diversity of concepts (includ- "abstract" concepts 'The term "representation" has been used in a ing ones, and chained multiplicity of senses in cognitive ncuroscicncc-isomor- referred to as skills) are as great as our phic mappings (copies) of mental irnages, the reproduc- behavior repertoire itself, which includes all tion or retrieval of an image (Huk, 2008), and conceptual of our knowledge and memories, and these (Martin, The term confounds or abstract responses 2007). may be linked to different sensory modalities these usases, while our need here is to distinp;uish between them. And it is uncomfbrtable to apply the term "image" ^- to othe r sensory rnodalities, as in "auditory imagery" when "The term .oncEtis used here in the broad serrse that referrins to music or a voice (Smith, Reisberg, & Wilson, regards skills, whether motor or cognitive, as chains of I 992) . concepts (Mechner, 1967). REVNW OFBLINDFOLD CHESS

or combinations of them. Concepts linked to The scanning and interrogation behavior the visual modaliry include nor only thc basic that occurs in viewing involves eye movements hard-wired "vocabulary of vision" concepts that focus successively on iconically retained (edges, brightness, colors, contrasts, move- small-areas, largely those subtended by the ments, distance to and betlveen objects, fovea. The cues that guide eye movements to directional location, etc.; Zeki, 1978; Zeki et successive fbcus areas are provided mainly by al., 1991), but also the multiple slnthetic peripheral vision (Blackburn and Nguyen, complex human counterparts of the chick's 2002). The iconic memories of the cascading innate concept of the overhead silhor.rette of a succession of small focus areas permit them to hawk, and range from fast-learned concepts be blended into mosaic images sufficiently (e.g., the human face) to more slowly learned complete and coherent to be perceived as ones likc written words. Perhaps the richncss meaningful stimuli (Koch and Ullman, 1985). of our repertoires of visual concepts contrib- Without iconic memory, such blending would utes to the illusion of an internal image-the not be possible. illusion that may render somewhat counterin- Some features of such viewed images may be tuitive the notion that mcntalizations can be retained in longer-term memory as concepts accounted for {ully by concepts that are drawn when (a) attention is somehow directed to exclusively from the existing behavior reper- them, (b) an instruction or other qpe of toire.' An inciclental mcthodological bonus of continge ncy generates the behavior of scanning vrewing mentalization as consisting entirely of for a particular feature, or (c) an encountered such concepts is that doing so brings it into the stimulus feature is recognized as significant, purview of the behavioral disciplines related to based on a learning history (e.u., Bichot, Rossi, conceptualization and learning, including and Desimonel, 2005). Those are cases in which equivalence research, relational frame work, a stimulus feature mav be conceDtualized and skill training, etc. retainecl in longer-lerm memory. The Scaraing-Interrogation. Process in Vieuing Conceptualization in Vieuing and Mantalizing There is ample evidence (fiom tachistoscr-rpic Some conceptualizations can occur in cither and other studies) that subjects retain quasi- viewing or mentalizine, and somc (e.g., "ab- photographic images somewhat like irfter-imag- stract" or "verbal" oncs) can occur onlv in es, known as iconic memory, for approximately mentalizing. Concepttralizations that can oc- a quarter of a second after exposure of a visual cur in viewing when scanning and interrogat- stimulus (e.g., Sperling, 1960). During that ing the stimulus might take such forms as, "A brief time, such images can be scanned and fly just landed on the tip of the brown cow's "You interrogated almost as if they were viewed white left ear," or are frownins." If the se same stimuli. This biological phenomenon may have conceplualizations wert. lo occur in its evolutionary roots in locomotion behavior, mentalizing, they would not be the result of scanning-they which necessitates short-terrn recall of the would be drawn from the behavior repertoire. features of temain being traversed, and may In chess, if the conceptu- "Queens also play a role in readirrg, listening, scannins, alization are still on the board," occurred in viewing a etc. (Mechner, 2009). position, it would result from scanning and interrogating the chess- board; if it occurred in mentalization, it would " It is also worth noting in this corrnection that be imported f,rom the repertoire. rnentalizing is not limited to the recall of previously perceived stimuli. Most chess positions mentalized in Some concepts have physical ernbodiments blindfbld play or when thinking ahead in sighted play that can be viewed, and some do not. A (chess players usually call this "calculating") have not concept like "car," for instance, has a physical been seen befrrre. Similarly, we can easily mentalize novel embodiment, but the fact that it can be viewed sentences, melodies, or scenes that we have never actuallv seen or heard. That is because the conceDts that occur rrr liom the inside or the outside, front or side, mcntalization can consisl ol recombinaiiot)\. rcanangc- near or far, makes it a concept that could also ments, or s)'ntheses of I'eatures. components, or other be mentalized as an abstraction. Concepts that attributes of previous conceptualizations-visual, auditory, do not have physical embodiments and there- emotional, relational, ()r abstract ones. Much of what we call thinking, too, may consist of recombination, reassem- fore cannot be viewed (e.g., time, quantity, bly, and syntheses of such components into novel love), though usually termed abstract, are still configurations (Bar,2007; Mechner, 1994, pp. 10-11). concepts in the behavioral sense of the ternr. 378 FR.A^IUS MECHAER

MENTALIZING VIEWING . Retina is involved

Contemporaneous exteroceptive stimulus ls present

Can scan and interrogate the stimulus regarding details

I.-ig. ?. The relationship between viet'ing and mentalizinu

Important in the context of conceptualiza- only the conceptual behavior that viewing and tion that occurs in viewing is the fact that mentalization have in common. The Venn concepts already' in the behavior repertoire diagram in Figure 2 shows this relationship. can, and often do, override contemporaneous The Image Conjec,ture as an Empirical Proposi- perception (Brown, 1973; Carter & Werner, tiort 1978; Cumming, Berry'rnan, & Cohen, 1965; Although the Image conjecture is adequate- Mectrner 1994, pp. 33-34; Schoenfeld & ly refutcd hy thc scanning-interrogalion lesl. il Cumming, 1963; Wright & Curnming, can also be formulated as the empirical 1971)-what we perceivc is largely a function proposition (unsupported though it may be) of what we have learned to perceivc (Graham, that the iconic mcmory images on which 1951, pp. 911*915; Skinner, 1953; Woodworth viewing depends can be retained for hours or & Schlosberg, 1955, pp. 403-491). Thus days rather than just split seconds, and can be rrrentalizations that occur in viewine can scanncd and intcrrogated during that longer override the reality of what is actually therc, a time much like a viewed stimulus, a kind of phenomenon that explains many instances of "photographic mcmory" effect, as in "eidetic mispcrceptions and mistakes not only in chcss imagery." But the extreme rarity of abilities but also in all kinds of everyday situations. This like those exhibited by Kim Peek (Trcffert & phenomenon may also explain the fusion or Christensen, 2005) and some others (Treffert, blending effect that may be responsible for the 2009) (whose behavior when mentalizins "mental imaging" illusion. rcsembles the scanning and interrogation of What Vieuing and Mentalizing Haae in Common a complex visual stimuh.rs like a page of print), Many {MRI studies show that viewing and makes it unlikely that such a phenomenon is mentalizing activate similar brain areas, sug- involved in normal instances of mentalization. gesting that they involve at least sorne of the In addition, most mentalized chess positions same behavior (Borst & Kosslyn, 2008; Kolb & (as well as nonchess mentalizations) are novel, Wishaw, 2009; Kosslyn, 1980, 1994; Richard- whereas eidetic irnagery applies to previously son, 1999; \A/heeler, Peterson & Buckner, r.iewed stirnr"rli. Finally, the masters' protocol 2000).'0 For that behavior we need look no data are inconsistent with thc possibility that further than conceptualization. The similarity their mentalizations involve eidetic imagery. of the activated areas has sornetirnes been Alleged Euidence for the Image Conjecture misinterpreted as cvidence for the Image The Irnage conjecture and introspective conjecture (Edelman, Grill-Spcctor, Kushnir illusion may also drive somc common misin- & Malach, 1998; Kourtzi & Kanwisher, 2001), terpretations of imagery research data. Many but the parsimonious interpretation is that it is of thc studies often citcd as supporting thc Image conjecture, some of which are reviewecl (2008, 166-178), and tu by Hearst and Knott pp n.. -*rr" finclings halc also bcen rcportecl for by Kolb ancl Wisharv (2009, pp. 639-645), are audition (Intons-Petersen, 1992; Naatanen, i985; Reis- bere, 1992; Smith, Reisberg, & Wilson, 1992) ancl for open to altcrnative, more behavioral interpre- olfaction (Bensafi et al., 2003). tations. For example: A measure like tirne REVNW OF BLINDFOLD CHESS 379 needed for the mental rotation of a shape or another (or the same) player in a similar for finding a place on a recalled map does not position. The blindfold masters' protocols, differentiate between accessing an "internal (e.g., "knowing which combination or plan is image," and history-based conceptual process- in progress") suggest that the action can es. The reason is that time required for a consist of a sequence of moves as a unit mental rotation task can be interpreted as time (pp. 151-152,162-166; Chase and Sirnon). needed to complete a purely conceptual Mentalizing in Normal Chess 1'eu1i11s-sne that was learned via a history of Many strong chess masters report that they actual physical rotations of objects, with actually find it helpful to look away from the associated observation and conceptualization board when thinking ahead ("calculating") of the stimulus changes corresponding to each (pp. 9, 127, I5l,189-190). The reason for this fractional amount of rotation. The same type could be that when considering a move in of alternative interpretation would apply to the normal over-the-board play, the player typically Kosslp, Ball, and Reiser (1978) finding of tries to mentalize the position that will exist time-distance correlations in the reading of after the move is made while at the same tinre mentalized maps. Nor can eye movements still viewing the conflicting stimulus of the linked to spatial or geometric aspects of a about-to-be-moved piece in its original position. mentalized stimr.rlus (e.g., Donahoe & Palmer, But why would the helpfulness of looking 1994, pp. 253-256) be interpreted to mean away depend on skill level? Because when that an image is being scanned, because eye l<-roking away, the player relinquishes sight not movements, as well as other motor or verbal only of the about-to-be-moved piece but also of behavior, are often coordinatively linked to all the other pieces-the ones that would not conceptualization behavior that has spatial or move. Sight of these others rvould be helpful geometric aspects. Palmer (2010, pp. 38-39) only to the extent of the player's uncertainty makes the related ooint that more research regarding their placement when looking away attention should be directed to eye move- from the board. The stronger the player, the ments. smaller that uncertainty, the less help is Charness (1976, p.159) found that most derived from the sight of them, and therefore experienced chess players can form robust the greater the extent to which the interfer- long-term memories of chess positions after ence effects outweigh the value of viewing viewing them for only a few seconds, and ttrem and the board. reconstruct them much later after having The Matching of Mmtakzations spent the intervening tirrre in activities de- Relevant to llearst and Knott's discussion of signed to be interfering. But in interpreting Chase & Simon's (1973) "recognition-action" this finding it is important to note that the thesis is the suggestion that skilled perfor- masters retained only the ability to reconstruct mances (of which chess is an example) are the positions, not necessarily images of them, normally practiced and honed by matching a and that this ability can be based on concep- mentalized model (rather than a contempora- tual behavior exclusively. neous exteroceptive one) of the desired The'oRccognition-Action" Rcpertoire of Concepts performance (Mechncr, 1994, pp. 29-34). A large repertoire of piece configuration For instance, musical perlbrrners, especially concepts is evidently needed for both blincl- when practicing, try to match their mentaliza- fold and sighted chess (p. 9, 91, 127, 190). tion of the music as they want it to sound, and Chase and Simon (1973, pp.157-158) estimate a golfer may try to match his mentalization of that master-level play requires a "recognition- Tiger Woods' swing. Chess players, too, some- action" repertoire of 50,000 to 100,000 such times strive to match mentalized model behav- concepts (also called "chunks," "templates," ior as when they covertly ask themselves, or "patterns")." The "action" in such a "\A4rat would a grandmaster (or my coach) recognition-action concept can be a single do in this position?" or "\A/hat did I do in a move or a whole sequence of moves, a plan, an similar position in a previous game?" \Ahat is algorithm, or the recall of the behal'ior of matched in such cases is a mentalized behav- ioral event, either one that actually occurred B.l*ui,u. analysts would concept'alize it as a multiple (a recalled event), or one that never occurred, discrimination repertoire of that magnitude. as when a dancer mentalizes, say, a slinking

-tt 380 fI?/-VC7S MECHNER

tiger-a cornposite of conceptualized ele- Blakeslee, & Furst, 1986; Ryan & Simons, 1983; ments. Van Gog, Paas, Marcus, Alres, & Sweller, The Progression from Nouice to Expert 2009). Since sighted chess is at the cognitive Hearst and Knott report de Groot's 1965 extreme of the motor-cognitivc range and conclusion (p.54), later supported by others, depends heavily on mentalization skill, it that stronger players consider roughly the shoulcl be expected to beneflt significantly same number of moves as weaker players, but from playing blindfold chess, which amounts consider better ones, and faster. The explana- to pure mentalization practice. This wcluld tion may be the same as the one offerecl in the explain and support the claim by grandmasters motor performance literature for the changes and Hearst and Knott that blindfold play that occur as a performance progresses fiom improvt's siuhtt'd plal . novice to expert due to learnins and practic- A Shill Measurernent Meth,odolog ing: Performance components that functioned Many types of skill research requirc a independently and separately in the novice dependent variable that quantifies a choice's perfpawn-equivalents as the cognitivc (Gobet & Charness. 2006).'' arbitrary units of measurcnrent. They used ooMental Practice" and the Bmfix of Blindfold thcsc ratings to compare blindfolcl and sighted Play play for the frequency ancl magnitude of The benefits of practice accrue to the "mistakes" (there was no significant differ- "mistake" "pawn" particular skills that are practice6l-16 111e1e1 ence), defining as a 1.5 skill when motor skills are practiced, and to shortfall'" fiom best, and "blunders" as covert or cognitive skills when it is those that signifi cantly larser shortfalls. are practiced (Mechner, f994, p. 39). That is The Issue of a Measure's Arbitrariness why "mental" practice is most beneficial in In view of thc fact that every chess Dosition per{brmances that have significant cognitive must result in a win, loss, or clraw when components, and least benefrcial in perfor- played out to the end of the game, any rating mances that dcpend mainly on overt motor of the strength of a move or the magnitude behavior (p. 170; Fleuer, 1985; Ross, 1985; Ryan, of a mistake is necessarily arbitrary. Comput- er-generated ratings reflect the computer's . '' L,xperienced chess players can cite numerous c\am- evaluation of the position after it has ples of such larger units and linked actions. Examples: When the configuration is a queen's side pawn nrajoriq, -''The sense in which the term "best" is used here is consider strategics that will convert that majority into a simply that even in the occasional instances where there passed pawn; in certain posilions, assign high priority to exist still stronger moves, little would be gainecl by the placenient of a rook on an open file or orr the seventh including them, even if they could be fbund. When therc rank, especially when cioing so includes a threat. For are several such moves in a given position, any one ofthem expert players, such algorithms are unitary concepts would oualifv as "best." t'pur"rr-"quivalenl perceived in a split second. The authors cite studies that t'A shortfall fiom best" can bc discuss the thousands of hours of study needed to acnuirc thought of as a measure of how much better or worse a the necessary number of such rrnitary concepts ancl linked particular move makes the position, using the value of :r element-s for the achievcment of masterlevel skill. pawn as the unit of measurement. REVNW OF BLINDFOLD CHESS 381 calculated ahead a certain number of Requirements of ct Gmeral$ Usful Skill moves. But how many moves and moves of Meastre what strength? Because the answers to these To be useful, a measure of chess skill should questions are arbitrary, so are such ratings.l5 be applicable to the performance of an If ratings are arbitrary, they do not permit individual player facing particular positions. meaningful comparisons for different values The measure should quantify the magnitude of an independent variable, either within or of players' Skill deficits and pinpoint their between skill areas. To be usable for such nature, separately fr-rr different phases of the comparisons, the units of measurement must game and for different types of positions. The have the same meaning regardless of the measure proposed here meets these require- independent variable used and the skill area ments and can be used to study the effects of involved. such independent variables as training meth- Trad,itional Skill Rating Methods and Their od, sleep, or practice techniques. Limintions Measuring Skill Shor{all lrom a Theoretiml Skill or knowledge involving choice behavior Maximam (henceforth referred to simply as "Skill") is Claude Shannon (1951) in his paper "The often studied with "right-or-wrong" items Prediction and Entropy of Printed English" (Chase & Ericsson, 1982). Problems of the showed how a normal speaker of English can widely used "what's-the-best-move" t)?e are function as a human measuring instrument to normally scored either as "percent correct" or quanti8/ informational properties of letter with someone's subjective ratings of the sequences. He proposed itre ''entropy" l8 possible answers. But a wrong choice does measure to quantify a subject's information not provide information about the nature or deficit for each successive letter in such magnitude of the responsible Skill deficit, and sequences. The 1og2 of the number of tries an overall score does not identify the items needed to identify each letter, expressed as that presented problems. "bits of information," corresponds to the Hearst and Knott make fiequent reference number of yes-no questions the subject would to the numerical Skill ratings that reflect have needed to ask if he had used a yes-no players' past competitive performance results questioning strategy to identify the letter. against each other. Such ratings, widely used The method proposed here flips Shannon's r6 in compelitive games like chess (pla3, 163) (1951) procedure around, so that the entropy and go," being relative, are numerically measure is applied to thre subject's Skill deficitfor arbitrary and have no anchorage points-they each stimulus situation rather than to an float and drift. If all rated chess or go players attribute of the stimulus. Here a stimuius were to become stronger or weaker to the attribute is used as the known standard against same degree at the same time, their respective which the Skill deficit is measured. The sarlc ratings would not chanse. method is obviously applicable to any multiple- choice situation in which all the possible choices, including the best one, are known 15It might also be noted that the information such a and can be made available. rating conveys to a particular player depends on the The Uncertainty Measure Applied Choice plaver's skill level. For inslance, a top srandmaster might to interpret an advantage of 1.[i pawn-equivalents as an easy Situations win, and a relatively inexperienccd player might interpret The underlying rationale of the proposed it as a minor edge. 16The proccdure is that an individual's performance co-pclitive-per{brrnance-basecl rating formula in any choice situation can be expressed in internationally used in chess, der,ised by the Hungarian physicist Arpad Elo (1978), arbitrarilv assigns a rating terms of uncertainty regarding the best an- somewhat below I,000 to beginners, while players of world swer. As in Shannon's (1951) procedure, championship strength would usually achieve ratings in Uncertainty (" Lf ") is defined as the log2 of the-vicinity of 2,800. number of tries a particular individual would " In go, the compe titive-perfonnance-based rating scale used by the American Clo Association for amateur players need to get the beit answer, and is measured assigns aratineofaround 350 (35 knr) to rankbesinners and +800 (8 dan) to the strongest amatcurs. Professional o. i,rfbr-ation deficit, disorganization, disordcr, players are rated internationally on a separate scale lrom I unpredictability, or its inverse: nesentropy, degree of to 9. oreanization, or predictabilitv (Wierrer, 1948, 1950). -o 382 rTi,{A/C7S MECH},TER in bits. This t/ corresDonds to the number of greater the [I20 The formula for [/would thr,rs ves-no ouestions the individual would need if be tr - log2 n where n is the number of tries a question-asking procedure were applicable. needed toknd the best *,,rr".?t Here is the underlying reasoning: Suppose you A valid measure of ability to find best moves knew that a coin was tossed three times. To in given positions must reflect not just U but eliminate your uncertainty as to how it landed also time consumed, as a joint function of the each time, you rvould need to ask three yes-no two. Such a measure of achievement speed questions: (1) "Did it land heads on the first wcluld be a useful dependent variable fbr toss?" (2) "Did it land heads on the second studies in which the independent variable toss?" and (3) "Did it land heads on the third might be various tlpes of training procedures, toss?" That makes eight possible outcomes, test conditions, tl?es of positions, or playcr and the log2 of 8 is 3. Thus, your uncertainty U variables such as cxperience, Elo rating, was 3 bits. fatigue, ingested sut)stances, age, and so forth. The unique properfy of the l/measure is that The Proficienq Measure it is nonarbitrary-its units are always bits of We will apply the term "Proficiency" to such infcrrmation, regardless of the skill area. It is a measure. We would want the theoretical therefore applicable to a wide variety and range maximum Proficiency score of a player who of skills, knowledge, and choice situations, and finds the best move on the first try in zero thereby enables quantitatively meaningful com- seconds to be 1.00 and to decrease toward zero parisons among these. Arbitrary measures, in as the number of tries ri and the time contrast, do not permitvalid comparisons of the consumed I increases. Thus both n and f must tlpe one normally wishes to make. appcar in the denominator. There are several The U Measure Applied to Chess Shill formulas that meet these requirements but a a useful One of the reasons chess provides straightfor-ward one is 1/(1+ U'f") where [/ - of choice behavior is that the laboratory model log2n as discussed above, / is the tirne used fbr (1951) is readily applicable Shannon procedure thc n tries,?z and k is a scaling constant that chess programs can dcfine to it. Computer sets the weight assigned to the time factor in terms of maximum Skill level operationally in relation to the t/facio..Z3 Th,.tt number of tries best move (or the set of best moves when the can be traded off against thinking timc in a there is more than one) for any given position.le way that leaves the Proficiency score unaffect- is with a The procedure The subject presented ed-the player can think longer so as to chess pr-rsition on a computer screen and is require fewer tries, or Llse less thinking time instr-ucted to indicate the best move. After by tryng many choices quickly. The possibility try, the computer responds "correct" or each tleans that a player's "try again." The computer registers fhe of such tradeoffs Proficiency score for each position need not number of tries needed to reach a best move for that position (without counting repeats). For the hypothetical player who rvould 20A possible relinenrent of this procedrtre would give "partial trying ntoves that thc computcr always be correct on the first try, L/ would be credit" for considers second-bes1, third-best, etc. Such tries could, is Skill zero bits because the log of I zero-the fbr instance, be counted as fractional rather than whole level at which a player would need t<-r ask zero tries, with the amount of partial credit based on the yes-no questions. A less skilled player, who computer's evaluation of move rankings, although any might need, say, eight tries, would have a Uof evaluation function is inel'itably arbitrary. 21To eflect of clralcc whcn the nrtmber of (i.e., corresp<-rnding to three yes-no reflcct the 3 bits available choices is limited, as in chess where the average que stions). So, the greater the Skill deficit, the nrrrnber of possible choices may be 37, n can be assigrred an appropriate exponent (meaning that log2 n would have leA computer chess prograrn named Rybka, reputedly a coefficient), so that [/would approach its maximum of the strongest in existence, plays at a strength level close to one at whatever faster rate may be desired. 22T'he that of the stronsest hum:rn grandmasters, and programs computer software should subtract out the time reputed to be even stronger have more recently been consurned by the physical act of keying in each try. made available on the internet as free downloads. In the " Oth.. possible formulas for Proficiency that also meet case ofstanclard 19X II go, on the other hand, there are as the reqrrirements are l/ (l+U'ext) where e is Euler's yet no computer programs that can generate moves nrrmber, 1 / (n"'td) , rnd. 7 / (n"' s*t) . The derivation of the beyond the level ofweak players. However, there are now term za is s{/ : ,1.+s, and the 1.45 exponent can be computer progralns that can generate best rnoves for 9X9 increased to a to reflect the effect of chance when the go. number of choices is limited, as discussed earlier. REWEW Of. BLINDFOLD CHESS 383 be strongly affected by the amount of time tions with long time limits), or high (corre- spent-a "Proficiency constancy" that may sponding to rapid play conditions). In skill accommodate stylistic differences between areas other than games, such as timed tests, players in the way they use time. the ft setting could correspond to a fluency Nthough Proficiency may often be correlat- requirement. The ft setting also makes avail- ed with Elo rating, it measures something able a research methodology for ascertaining other than practical competitive playng the effects on Proficiency of various time- strength. Many players are weak in some types sensitive variables. If A is scaled along the of positions and strong in others, or may abscissa and Proficiency along the ordinate, a collapse in a competitive stress situation but be parameter would be represented by a family of brilliant at identifying best moves absent a cur-ves. Examples of parameters could be age, competitive contingency. Such players may rest, ingested substances, training variables, achieve high Proficiency scores but be weak practice, or type of Skill. in practical play. The performance Ievel of most chess players Proficimq Constanq and the h Setting depends on the time limits used (e.g., rapid The degree of Profrciency constancy de- play versus play under standard tournament pends on the A setting. If k is set at zero, time time limits), just as a runner's rating might plays no role at all, and the Proficiency score differ for sprinting versus marathon running. would then be maxirnized by thinking as long The A setting is analogous to a time limit in as possible on each try so as to minimize n. On that it sets the weight assigned to time in the other hand, if ft is set very high, Proficierrcy relation to [I would be maximized by trying every reason- Addressing Qtestions Specific to Chess able move as quickly as possible, as number of There are also many chess-specifrc research tries would then have little effect. In both of questions that can be addressed with the these cases, there would be no Proficiency Proficiency rneasure: Is the well documented constancy. Therefore, Proficiency constancy is peaking of strength in chess players' mid maximized at some intermecliate value of h.2a thirties (Elo, 1965) correlated with a peaking Here is one possible way to think about the ft of Proficiency? The Proficiency measure can setting: Many players may need four to eight also be used to compare players from different tries to frnd a best move, for a Uin the ranse of countries or clubs whose standards, unan- 2 to 3 bits. If k is set at 0.2 (the fifth root) and r chored as they are, tend to drift apart over is measured in seconds, / to the A power would time. \Arhen applied over long periods of time be 3 if the player spends 243 seconds (3 being the Proficiency measure would permit com- the fifth root of 243) and 2 if the player spends parisons of players who may not even have 32 seconds. With that A setting, the time factor been alive at lhe same time. would have a relatively small impact on The Proficiency measure can also be used to Proficiency-fwould vary by a factor of around investigate experimentally some of the claims I (243/32), while I to the ft power would vary and hypotheses cited by Hearst and Knott. For by a factor of only around 1.5 (3/2). With instance, is the chess masters' claim that their higher settings of k, for instance 0.3 or 0.5, the blindfold play is as strong as their sighted play time factor would hllo. ^ correspondingly matched by a corresponding similarity in their greater relative impact." blindfold and sighted Proficiency scores? In game applications like chess, A could be Appkcations in Other Skill Areas hwolaing set low (corresponding to tournament condi- Choice The methodology for measuring Proficiency 2a In developing the required sof'tware, a mathematically is applicable to the study of any game or Skill inclined programmer might consider using calculus of dornain defined by discrete time-constrained variations in programming the speed-accuracy trade-offs, arrd Lagrangian rnultiplier methods for the "optimality choices or decisions for which all available with constraints" problem-an optimization function that choices, including the best one, are known, as has been used in economics in connection with "utiliq' in chess. Examples are certain types of functions," in operations research, and in solving optimr identification, classification, or problem solv- zati_on in conlrol engincerine. problems ing tasks in business, legal, counseling, mili- '" Since A would obl'iously have differcnt efl'ects in other possible Proficiency fbrmulas, it would be assigned tary, and social situations, and in academic appropriately different values. subjects likc spelling, geography, mathematics, 384 FR4}/(,]1.i MECHNER

the sciences, certain verbal skills, literature, or mentalization skill might be improved by social studies. repetitive alternation between actually hearing Many performance skills involve mentaliza- a passage and mentalizing it. Similarly, the tion processes that are invisible to an obseryer q,pe of mentalization skill used in drawing (Mechner 1994, pp. 18-21), as when a chess from life or copying may be improved by trainer obseryes the trainee searching for a repetitive alternation between looking at the move. Data consisting of the trainee's incor- model and mentalizinu it. rect tries could provide the trainer with a Conchtsion valuable window into such mentalization Dro- This book can be expected to stand for a cesses-one that worrld he more inlormative long time as the definitive compendium on than the oral reports obtained in response to blindfold chess fbr chess players, chess histo- the "think out loud" tlpe of request that chess rians, and students of samcs. Many of thc trainers sometimcs use (de Groot, 1965; Sil- topics covered also have provocative implica- man, 1991). The same technique would be tions for conceptual and research issues in applicablc to thc coaching of any skill that behavior analysis, psychology, neurosciencc, involves choice. performance learning, training, and educa- Applications in Shill Areas that Do Not Inaolae tion. The present review explores these and Choice describes a possible behavior analytic method- Since a key component of chess Skill is the ology fcrr addressing them. \{hile there are , as ability to think ahead ("calculate"), which yet, few Skill areas in which quantitative requires mentalization of positions that may measurement methods have been dcveloped, occur several moves later, it follows that any the proposed methodology migtrt stimulatc irnprovement in mentalization skill would also such developrnent. In addition to its potential be reflected in the Proficiency measure. Thus uses in cducational testing and training, it rnay Proficiency could bc rcgarded as an indirect also expand the methodological armarnentar- mcasure of mentalization skill. This consider- ium of behavior analysis for the experimental ation becomes relevant in extending the st udy of' r'ogrritive behavior. methodology to Skill domains that do not involve choice behavior but that depend, as does chess Skill, on mentalization. REFE,RENCES Consider, for instance, musical composition Bar, M. (2007). The proactive brain: Using analogics and skill, depends mentalizatiorr which on auditory associations to gcnerate predictions. TrerLds irL Oogtti skills. Similarly, drawing from life depends in tite Sciences. 11. 280-289. part on the abiligv to mentalize the model Belen'kii, V. Y., Gurfinkel, V. S., & Pal'tsev, Y. I. (l9ti7). whcn the eyes are on the drawing rather than Elements of control of voluntary movements. lldrfilzls on the model. Unlike chess. neither of these ics, 12,154-761. Bennett, M. R., & H:rcker, P. M. S. (2003). The philosophical skills involves discrete or defrnable choices. founrlations oJ neurosciente- Oxford, London: Blackwell Although the procedure for measuring Profi- Publishing. ciency would therefore not bc directly appli- Bensafi, M., Porter, J., Pouliot, S., Mainland,.f.,.fohnson, cable to them, it may nonetheless be plausible B., Zclano, C., & ... Sobel, N. (2003). C)lfactomotor lo extraoolate to thern certain results obtained actir4tl' during imaeery rnirnics that during percep- tion. Nrdure Natrostieru:e, 6, 1142-1114. in chesJ Proficiency research. In thc 1960s I Bernshtcin, N. (1967). The t:oordination. and regukttion oJ' tested a mentalization exercise that consisted mouements - Oxfbrd: Permagon Press. of alternating several times between mentaliz- Bichot, N. P., Rossi, A. F., & l)esimonel, R. (2005). Parallel inu a particular sequence of chess moves and and serial neural mechanisms lbr visual search in actually seeing that same sequence played out Macaque Area V4. Scie'n,ce, 308,529 531. Blackburn, M. R., & Nguyen, H. G. (2002). Modelirrg the on the board. If the Proficiency measure now biological mechanisms of vision: Scan paths. Undnseo showed that this tlpe of exercise is, indeed, AI antl Robotics, Code 943, Naval Ocean Systems Center, effective for improving chess mentalization San Diego, Califbrnia, 92152-5000. skill, one might infer, by extrapolation, that Borst, (i., & Kosslyn, S. M. (200ti). Mental irnagery and similar t)?es of exercises would improve mental perception: Stmctural equivalence revealed by scanrrirrs processes. Memory untl Cognition, 36, mentalization skill in skill areas that don't 849-U62. involve choice, like musical composition skill Brown, R. (1973). A Jirst language: The early stages. and drawing from life. For example, musical Oambridge, MA: Haward Univcrsity Prcss. REVNW OFBLINDFOLD CHESS 385

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