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Application of the generalized matching law to chess openings:

A gambit analysis

IAN CERO AND JOHN MICHAEL FALLIGANT

AUBURN UNIVERSITY

During the opening moves of a chess game, a player (typically White) may offer a number of

gambits, which involve sacriﬁcing a chess piece for an opponent for capture to achieve long-term

positional advantages. One of the most popular gambits is called the Queen’s Gambit and

involves White offering a pawn to Black, which will open a lane for White’s Queen if accepted

by Black. In the present study, the generalized matching law (GML) was applied to chess open-

ings involving the Queen’s Gambit using over 71,000 archived chess games. Overall, chess

players’opening moves involving the Queen’s Gambit exhibited orderly matching as predicted

by the GML, and the GML accounted for more variance in players’chess decision making as

their relative playing experience increased. This study provides support for the generality of the

GML and its application to complex operant behavior outside of laboratory contexts.

Key words: chess, choice, decision-making, generalized matching law

Organisms allocate their behavior among

concurrently available response alternatives

as a function of the relative reinforcement

the alternatives produce (e.g., Hernstein,

1970). The matching law is a mathematical

description of this relation between relative

responding and rates of reinforcement

(Hernstein, 1961). A modiﬁed version of the

matching law, referred to as the generalized

matching law (GML; Baum, 1974), can

describe systematic deviation from strict

matching in terms of the sensitivity of behavior

to relative reinforcement rate or as a function

of variables other than rate of reinforcement

(e.g., response effort, reinforcement quality).

The GML predicts that relative response alloca-

tion varies linearly with relative reinforcement

rate when log transformed, and is expressed by

the following relation:

log B1

B2

=alog R1

R2

+logb

where B

1

represents the rate of responding on

one response alternative and B

2

represents the

rate of responding on the second response alter-

native; R

1

and R

2

represent the relative rates of

reinforcement for those alternatives. The slope

of the line (a) reﬂects sensitivity to reinforce-

ment, and the intercept (b) reﬂects bias for one

of the response alternatives when equality of

reinforcement would predict indifference

between the response choices (e.g., Baum,

1974). If behavior matches reinforcement per-

fectly, the slope of the function, a, equals one

and the intercept, log b, is zero. Sensitivity to

reinforcement more (a> 1) or less (a<1)

extreme than strict matching would predict is

known as overmatching or undermatching,

respectively (see McDowell, 2013).

The GML has described complex operant

behavior in a variety of contexts, including

severe problem behavior (Borrero & Vollmer,

2002), conversation allocation (Borrero et al.,

Ian Cero is now at the University of Rochester Medical

School. John Michael Falligant is now at the Kennedy

Krieger Institute and Johns Hopkins University School of

Medicine.The authors would like to thank Jason Bourret

for his helpful feedback on this manuscript.

Correspondence concerning this article should be

addressed to Ian Cero, Department of Psychiatry, Univer-

sity of Rochester Medical School, Rochester, NY 14642.

Email: ian_cero@urmc.rochester.edu

doi: 10.1002/jaba.612

JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2019, 9999, 1–11 NUMBER 9999 ()

© 2019 Society for the Experimental Analysis of Behavior

1

2007), academic behavior (Mace, Neef,

Shade, & Mauro, 1994), risky sexual behavior

(Bulow & Meller, 1998), and both simulated

(Schenk & Reed, 2019) and nonsimulated

sport-related behavior (e.g., Alferink,

Critchﬁeld, Hitt, & Higgins, 2009; Falligant,

Boomhower, & Pence, 2016; Reed,

Critchﬁeld, & Martens, 2006; Vollmer &

Bourret, 2000). For example, in the context of

sports, research has shown that both collegiate

and professional basketball players attempt

more three-point shots (relative to two-point

shots) as the relative number of three-point

shots scored increases (Vollmer & Bourret,

2000). Research has also shown that shot selec-

tion is sensitive to alterations in the distance of

the three-point line (that arose from rule

changes in 1994 and 1997) in that the relative

number of three-point shot attempts increased

as the relative number of three-point shots

scored increased (Romanowich, Bourret, &

Vollmer, 2007). Other variables, such as a high

team success rate, more competitive NCAA

divisions (i.e., Division I and Division II), and

whether players are starters (as opposed to sub-

stitutes) are associated with increased sensitivity

of shot selection to relative rates of shots made

(Alferink et al., 2009). Thus, a wealth of

research suggests parameters of the GML are

sensitive to subtle variations in complex oper-

ant behavior across a variety of populations and

contextual variables.

Importantly, the GML has advanced the

analysis of choice occurring in both laboratory

and naturalistic environments, serving as a

powerful vehicle of translational behavioral

research (see Mace & Critchﬁeld, 2010;

Vollmer, 2011). Continuing to evaluate the

utility of the GML in describing complex

choice behavior has the potential to advance

the study of behavior analysis. For example, the

extent to which this choice model accurately

predicts behavior in other complex activities,

such as chess, where the number of legal moves

a player may make during a game is extremely

large (10

120

; Shannon, 1950), is unknown.

Chess can serve as a unique model for behav-

ioral research (see Mechner, 2010), and there is

a considerable body of quantitative research

evaluating various aspects of the game of chess,

such as the power-law distribution of chess

openings (Blasius & Tönjes, 2009). Thus, a

behavior-analytic assessment of chess openings

using the GML may speak to the generality of

the GML in describing complex choice behav-

ior, highlighting the applicability of basic

behavioral concepts to novel, naturally occur-

ring operant phenomena.

During the opening moves of a chess game,

a player (typically White) may offer a number

of gambits, which involves presenting material

to an opponent for capture in order to gain a

positional advantage. In other words, gambits

(if accepted by the opponent) typically result in

a short-term loss of chess pieces, but long-term

positional advantages. One of the most popular

gambits offered during chess openings is the

Queen’s Gambit (e.g., Kasparov & Keene,

1994; Ramiz, 2006). To offer the Queen’s

Gambit, White opens by advancing the pawn

in front of the King two spaces (1.d4) and

Black counters by advancing the pawn in front

of the King two spaces as well (1.d5). White

then sets up the gambit by moving the pawn in

front of the Queen two spaces (2.c4), as Black

can then accept the gambit and capture the c4

pawn and gain a material advantage over White

(see Figure 1). If accepted, White can more eas-

ily develop his/her powerful pieces and occupy

centrally located squares in the board for a

long-term strategic advantage. If declined,

Black does not cede the positional advantage to

White, but foregoes a material advantage and

allows White to advance two pawns to the cen-

ter of the board.

The goal of the present study was to analyze

the ratio of chess games in which the Queen’s

Gambit was not offered versus games in which

it was offered relative to the ratio of victories

accrued without the Queen’s Gambit to

IAN CERO and JOHN MICHAEL FALLIGANT2

victories accrued with it using a sample of

71,716 archived chess games played by

348 tournament chess players. In other words,

the purpose of the present study was to assess

the degree to which White’s use of the Queen’s

Gambit is accounted for by the GML. Addi-

tionally, given that previous research has

suggested differences in skill or experience

levels are associated with changes in parameters

of the GML (i.e., sensitivity and bias) within

sports contexts (e.g., Alferink et al., 2009), dif-

ferences in sensitivity, bias, and variance

accounted for by the GML across chess players’

experience levels were assessed.

METHOD

Data Source

All data were acquired through ScidBase

(Scid version 4.6.4; SCID, 2017), a large

online database of chess games that has been

utilized for multiple studies of statistical phe-

nomena in the game (Blasius & Tönjes, 2009;

Maslov, 2009). Each record in this database

represents a single game of chess archived in

Portable Game Notation (PGN), a digital-

friendly format for documenting chess play. A

single PGN record typically includes a variety

of information about a particular match,

including the names of the players, the date

and location of the match, the moves executed

by each player in a standardized algebraic for-

mat, and other contextual information about

the match (e.g., commentary, player rankings).

Data Acquisition and Processing

On November 4, 2017, the experimenters

downloaded every available record in the stan-

dard database (n= 127,810). Of these initial

games, 108,008 included sufﬁcient information

to identify the players and were retained for

further analysis, during which a range of game-

speciﬁc variables were extracted. These variables

included the name of the White player,

whether that player offered the Queen’s Gam-

bit to Black, whether Black accepted the gam-

bit, and whether the White player ultimately

won the match. The experimenters then calcu-

lated aggregate statistics for each player

(n= 1,336), including the total number of

games played as White, the number of games

in which they offered the Queen’s Gambit, in

how many games their gambits were accepted,

and the number of games won and lost with

and without the Queen’s Gambit. The pro-

gram R 3.5.3 (R Core Team, 2017) generated

all statistical analyses and ﬁgures.

Minimum game requirement. A preliminary

analysis of the database revealed that players

offered the Queen’s Gambit in approximately

11% of all games. This implies players with a

small number of games in the database may

not have had sufﬁcient opportunities to offer

and beneﬁt from the Queen’s Gambit,

preventing reliable assessment of their confor-

mity to the GML. To address this limitation, a

prospective power analysis was conducted to

Figure 1. Illustration of the Queen’s Gambit. If black

takes the pawn offered by white, the gambit is accepted.

If black makes any other move, the gambit is denied.

QG = Queen’s Gambit.

3MATCHING AND CHESS

estimate the number of observed games a player

would need to have played in order for the

probability of offering 5 Queen’s Gambits to

reach at least .80 (i.e., the typical power thresh-

old; Cohen, 1992). This was achieved using

the Negative Binomial distribution, which is

commonly implemented for sampling questions

of this kind (Casella & Berger, 2001). The

results indicated that any given player would

need to play 61 games before that player would

have at least an 80% chance of producing

5 Queen’s Gambits, assuming players offered

the Queen’s Gambit in 10% of games on aver-

age (rounded down from 11% to be conserva-

tive). Thus, the games of players that had at

least 61 games in the database were included in

the analysis, resulting in a ﬁnal dataset of

71,716 games played by 348 players (mean

games per player = 206.08, SD = 147.48, min.

= 61, max. = 858).

Analytic Procedure

Research on the GML has shown that

behavior allocation is often characterized both

by a baseline bias favoring one alternative over

its counterparts and by deviation in sensitivity

(over or under) to relative reinforcement from

the prediction of strict matching (McDowell,

2013). To estimate each of these parameters

among chess players, we conducted a linear

regression analysis. The outcome variable was

the ratio of games in which a White player did

not offer the Queen’s Gambit to Games in

which White did offer the gambit (i.e., a

behavior ratio).

1

The predictor variable was the

ratio of victories White accrued without the

Queen’s gambit to victories accrued with it

(i.e., a reinforcement ratio). Prior to analysis,

each of these quantities were converted to log

10

units to linearize their relationship. In the

resulting regression model, the intercept repre-

sents the bias parameter of the GML and the

slope represents the sensitivity (McDowell,

2013). Thus, a regression model with an inter-

cept of 0 and a slope of 1 would represent “per-

fect”matching; values signiﬁcantly different

from these would represent deviations from the

strict interpretation of the matching law that

are greater than would be expected by chance.

To analyze changing levels of conformity to the

GML as the number of games played increased,

players were later broken into brackets based

on the number of games they had played. This

analysis was reconducted on each bracket, sepa-

rately, and regression results across brackets

were compared. In the second regression analy-

sis, the possibility that players’behavior con-

forms more strongly to the matching law as

they are exposed to additional games—and thus

opportunities for reinforcement (i.e., winning)—

was also considered. This was achieved by recon-

ducting the previous GML regression analysis,

but with players grouped by the number of

games played. Note that, to ensure a balanced

analysis, only a player’slast100gameswere

included for this analysis.

Of concern, aggregate data may produce illu-

sory matching effects and suggest functional

response–reinforcer relations at a molar level

even when there is no differential reinforce-

ment to facilitate learning. For example, con-

sider a hypothetical group of players who

intermittently choose to rotate all their pieces

by 90 degrees before some of their games.

Although such behavior should have no impact

on a victory (and thus putative reinforcement),

it is still the case that players who perform the

rotation in twice as many games will accrue

twice as many wins with it (holding the base

rate for chess wins constant across groups).

This will produce a matching pattern similar to

our observed results in Figure 2, as well as a

1

The non-Queen’s gambit openings and non-Queen’s

gambit victories were treated as the numerator in the out-

come and predictor variables because they were more

common than Queen’s gambit openings and victories

(respectively). The logarithms of their ratios will thus be

positive, greatly simplifying the visual analysis of subse-

quent ﬁgures. Note, the conclusions produced from the

regression analysis described here will be the same, regard-

less of the numerator/denominator choice.

IAN CERO and JOHN MICHAEL FALLIGANT4

signiﬁcant and positive regression slope, even

though there is no possibility of learning

through differential reinforcement (i.e., no

functional relation between piece rotation and

victory). Note that this is true of individual-

level time series as well. A player who exhibits

variable preference for the piece turning strat-

egy over time will still produce an illusory

matching-like pattern similar to those of the

players in Figure 3. Thus, it is possible the

Queen’s Gambit reﬂects the same illusory phe-

nomenon. In other words, our observed results

may be a function of base rates for the proba-

bility of a victory, and do not necessarily imply

a functional relation that players have learned

through differential success with the Queen’s

Gambit. To address this concern, consider

three quantities: (1) the probability a player

offers the Queen’s Gambit in the next game

after winning with it in the previous game,

(2) the probability a player offers the Queen’s

Gambit in the next game after winning with a

non-Queen’s Gambit opening in the previous

game, and (3) the difference, D, between the

two. If the Queen’s Gambit has no effect on

victories, D = 0. However, if there is differen-

tial reinforcement for using the Queen’s Gam-

bit, D > 0. We randomly sampled n = 100

players who had played at least 60 games (per

our power analysis above) and had accrued at

least one win with and without the Queen’s

Gambit. We also took the event records

(games) of the n = 100 players above and shuf-

ﬂed them randomly (within players) n = 1,000

times, recording the resulting D

sim

for each

iteration. This strategy mirrors a scenario where

the probability a player uses the Queen’s Gam-

bit in the next round has nothing to do with

the previous round (i.e., a context where no

learning can occur).

RESULTS

Results from regression analysis revealed that

the behavior of chess players conforms to the pri-

mary prediction of the GML. As shown in

Figure 2, players with more previous wins from

the Queen’s Gambit offered it in proportionally

more games (df = 346, R

2

= .71). There was a

small, but statistically signiﬁcant, bias (i.e.,

regression intercept) toward offering non-

Queen’sGambitopenings(b= 0.10, SE = 0.03

p< .001) and a modest, but statistically signiﬁ-

cant, under-sensitivity to relative victories

(a= 0.85, SE = 0.03, p< .001). Recall that the

standard error (SE) is an estimate of the variation

in slopes (b) and intercepts (a) that would be

expected from repeated sampling. When the SE

is small, as is the case here, it increases conﬁdence

in the accuracy of those slopes and intercepts.

To interpret these values in practical terms, ﬁrst

note that the intercept of a regression represents

the model predicted value of an outcome (the

Queen’s Gambit offer ratio), when the predictor

Figure 2. White’s opening chess moves conform to

the generalized matching law (GML). Points represent

White players. The size of each point grows in proportion

to the number of games completed by the player in the

database. Vertical axis represents the ratio of each player’s

non-Queens Gambit (QG) openings to QG openings

(log10 scale); horizontal axis represents the ratio of victo-

ries each player accrued through non-QG openings rela-

tive to QG openings (log10 scale). The dashed line

represents theoretically perfect conformity to the GML;

the solid line represents the line of best ﬁt to the data

(estimated using Ordinary Least Squares). The shaded rib-

bon around the line of best ﬁt represents the 95% conﬁ-

dence interval for the slope of that line. R2 is the

proportion variance explained by the line of best ﬁt.

5MATCHING AND CHESS

variable (the log ratio of wins with and without the

Queen’s Gambit) is held at 0. Because both the

outcome and predictor variable in this model are

in log

10

units, a player’s data will occupy point

0 of the horizontal axis of Figure 2 if they have

won exactly as many games with the Queen’s

Gambit as without (i.e., log 10 X

X

=log

10 1ðÞ= 0).

Thus, a predicted log

10

ratio of 0.10 implies

Figure 3. Selected players’individual matching performance. Each point represents a 50-game block of a player’s

recorded history in the dataset. The horizontal axis represents the log-10 ratio of Non-QG victories to QG victories for

White in each block. The vertical axis represents the log-10 ratio of Non-QG openings to openings in which White

offered the QG. The dashed lines represent theoretically “perfect”matching. Solid lines represent regression slopes for

each individual player. Note that because these points are arrayed relative to QG offers and victories, they do not depict

a temporal sequence (i.e., moving farther right does not indicate a later game block). VAR = variance explained, b= sensi-

tivity (regression slope), a= bias (regression intercept).

IAN CERO and JOHN MICHAEL FALLIGANT6

that players who win as often as they lose with

the Queen’s Gambit are expected to offer

openings other than the Queen’s Gambit

10

0.10

= 1.26 times more often than perfect

matching would predict. Similarly, the sensitiv-

ity parameter of the model is represented by its

slope of .85, which is again in log

10

units. This

implies that a 10-fold increase in non-Queen’s

Gambit victories (i.e., a log

10

-unit increase of

1.0) would yield only a 10

0.85

= 7.08-fold

increase in non-Queen’s Gambit offerings.

Figure 3 depicts example performance from

ﬁve individual tournament-level players. As

shown in the ﬁgure, players generally exhibit

correspondence between their relative rate of

victory with Queen’s Gambit (dashed line) and

the rate at which they offered it (solid line) in

each block. This is true despite variable usage

and success with the Queen’s Gambit across

players (different average height on the vertical

axis) and across time for each player.

As shown in Figure 4, the variance explained

by the GML increases with the number of

games played, starting as low as R

2

= .30

(df = 79) for players with 100 to 200 games

and increasing to R

2

= .59 (df = 24) for players

with more than 400 games. Changes in bias

and sensitivity were also observed as players

experienced more games. Speciﬁcally, bias was

reduced from .34 in the least experienced

players to .18 in the most experienced. Like-

wise, sensitivity increased from .55 in the least

experienced to .88 in the most. Together, these

general patterns indicate players’behavior

increasingly conforms to the GML as they play

more games.

Notably, it is possible that the reinforcement

schedule alone could have produced the

matching results above. That is, a “trivially

true”matching pattern can occur in observed

data when the number of possible responses is

not meaningfully greater than the number of

reinforcement occasions. To guard against this

risk, Equations 2 and 3 from Herrnstein

(1970) were used to calculate the minimum

and maximum possible rates of Queen’s Gam-

bit offerings for a given rate of reinforcement

(i.e., relative success with the Queen’s Gambit).

If these two bounds were very close together,

there would be no possible deviation from

matching. However, as shown in Figure 5, this

was not the case. Instead, the observed rates of

Queen’s Gambit offerings clearly diverge from

both their maximum and minimum possible

values, suggesting (a) the reinforcement sched-

ule of wins and losses with the gambit indeed

allowed for potentially poor matching, but

(b) players seldom exhibited such poor

matching.

In this group, the observed D

obs

= .07 sug-

gests there is a measurably higher observed

probability a player will offer the Queen’s

Gambit after winning with it than after win-

ning with an alternative opening. However, it

is plausible that the true population D = 0 and

these players just coincidentally evidenced a dif-

ferent value. As described above, the event

records of 100 players were shufﬂed randomly

1,000 times, and the resulting D

sim

was

recorded for each iteration. Across all simula-

tions, the mean D

sim

= .01 (SD = .02) and not

a single simulation surpassed the D

obs

= .07

(p< .001), suggesting that the probability of

observed results occurring in the absence of a

functional relation between the Queen’s Gam-

bit and victory is incredibly low.

DISCUSSION

We evaluated the applicability of the GML

to chess players’use of the Queen’s Gambit

during chess openings across a large sample of

chess games played by experienced chess

players. Across players, the GML accounted for

a signiﬁcant amount of the variance in players’

use of the Queen’s Gambit. As the relative

number of games won in which White offered

the Queen’s Gambit increased, the relative

number of games in which White offered the

Queen’s Gambit increased proportionally.

7MATCHING AND CHESS

Follow-up analyses implied this proportional

increase was highly unlikely to be the result of

forced matching due to a constrained reinforce-

ment schedule or from the mere base-rate of

players’victories. Consistent with results from

Alferink et al. (2009), the GML accounted for

a greater proportion of variance in players’use

of the Queen’s Gambit as the relative experi-

ence levels of the players increased. Sensitivity

estimates suggest players’reinforcement sensi-

tivity (i.e., sensitivity to games won) when

offering the Queen’s Gambit approached opti-

mal levels for nonlaboratory investigations of

the GML (McDowell, 2013), though players

demonstrated slight undermatching. Bias esti-

mates were low, suggesting only a modest pref-

erence for non-Queen’s Gambit openings

relative to Queen’s Gambit openings given the

experienced rates of wins.

Overall, the GML successfully described chess

players’use of the Queen’s Gambit as a chess

opening across a large sample of games, and the

Figure 4. Bias, sensitivity and proportion of variance explained (R2) by the line of best ﬁt as a function of total

games White has played. Note the change in vertical scale across panels. Game ranges are given in traditional mathemat-

ical notation, where a square bracket is inclusive, and a curved bracket is exclusive (e.g., “[100, 200)”indicates players

with at least 100 games, but strictly less than 200). Note, as the number of games played increases, the number of

players in that bracket decreases (i.e., n [100, 200) = 81, n [200, 300) = 41, n [300, 400) = 23, n [400, 900) = 24).

IAN CERO and JOHN MICHAEL FALLIGANT8

GML accounted for more variance in players’

decision-making behavior as their relative experi-

ence increased. However, the current ﬁndings

do not explain why matching occurs in the con-

text of chess openings involving the Queen’s

Gambit, and the descriptive nature of the

methodology used in the present study cannot

identify causal relations between sensitivity,

bias, and choice for Queen’s Gambit and non-

Queen’s Gambit openings yielding wins and

losses. Future research should evaluate how

Black’s responses to White’s Queen Gambit

(i.e., accepting vs. denying the gambit) conform

to the GML, in addition to assessing how non-

gambitchessopenings(e.g.,the“Ruy Lopez,”

the “English”) also conform to the predictions of

theGML.Broadly,ourﬁndings suggest that the

matching law may be useful in the context of

teaching chess, as well as evaluating chess strate-

gies and tactical decisions. There may be a beneﬁt

to using bias and sensitivity parameters as depen-

dent variables for improving chess play. That is,

one could study their own chess games (or the

previous games of their opponents) to assess

undermatching/overmatching or systematic

responding within molecular (i.e., gambits,

move-by-move decisions) and molar (e.g., middle

game, end game) move sequences.

Results from this project add to the consid-

erable basic and applied literature highlight-

ing the empirical utility and descriptive

power of the matching law. Matching pro-

cesses are ubiquitous in complex human

behavior, allowing researchers to use the

matching law to study operant behavior

across varied laboratory and real-world con-

texts. An analytical factotum, the matching

law may be used to study a wide range of

topographically or functionally distinct

behaviors within a single individual or across

Figure 5. Observed matching (dark triangles) by maximum and minimum possible matching (light circles) for each

player. Optimal matching is given by the dashed line. Maximum and minimum possible matching were calculated using

Equations 2 and 3 from Herrnstein (1970).

9MATCHING AND CHESS

a large number of individuals (e.g., 71,716

chess games played by 348 chess players).

The current study serves as a useful proof of

concept for future researchers interested in

using the matching law within “big data”

contexts, highlighting the requisite steps and

procedural logic necessary to evaluate the

presence of functional relations (and not nec-

essarily matching as a forced property of a

schedule) when using aggregated data and

unique schedules of reinforcement.

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Received November 26, 2017

Final acceptance May 15, 2019

Action Editor, Jason Bourret

11MATCHING AND CHESS