ArticlePDF Available

Application of the generalized matching law to chess openings: A gambit analysis

Authors:

Abstract

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 openings 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.
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 sacricing a chess piece for an opponent for capture to achieve long-term
positional advantages. One of the most popular gambits is called the Queens Gambit and
involves White offering a pawn to Black, which will open a lane for Whites Queen if accepted
by Black. In the present study, the generalized matching law (GML) was applied to chess open-
ings involving the Queens Gambit using over 71,000 archived chess games. Overall, chess
playersopening moves involving the Queens Gambit exhibited orderly matching as predicted
by the GML, and the GML accounted for more variance in playerschess 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 modied 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) reects sensitivity to reinforce-
ment, and the intercept (b) reects 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, 111 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,
Critcheld, Hitt, & Higgins, 2009; Falligant,
Boomhower, & Pence, 2016; Reed,
Critcheld, & 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 & Critcheld, 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
Queens Gambit (e.g., Kasparov & Keene,
1994; Ramiz, 2006). To offer the Queens
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 Queens
Gambit was not offered versus games in which
it was offered relative to the ratio of victories
accrued without the Queens 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 Whites use of the Queens
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 sufcient information
to identify the players and were retained for
further analysis, during which a range of game-
specic variables were extracted. These variables
included the name of the White player,
whether that player offered the Queens 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 Queens Gambit, in
how many games their gambits were accepted,
and the number of games won and lost with
and without the Queens Gambit. The pro-
gram R 3.5.3 (R Core Team, 2017) generated
all statistical analyses and gures.
Minimum game requirement. A preliminary
analysis of the database revealed that players
offered the Queens Gambit in approximately
11% of all games. This implies players with a
small number of games in the database may
not have had sufcient opportunities to offer
and benet from the Queens 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 Queens Gambit. If black
takes the pawn offered by white, the gambit is accepted.
If black makes any other move, the gambit is denied.
QG = Queens 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 Queens 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 Queens Gambits, assuming players offered
the Queens 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 nal 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 Queens 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
Queens 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-
fectmatching; values signicantly 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 playersbehavior con-
forms more strongly to the matching law as
they are exposed to additional gamesand 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 playerslast100gameswere
included for this analysis.
Of concern, aggregate data may produce illu-
sory matching effects and suggest functional
responsereinforcer 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-Queens gambit openings and non-Queens
gambit victories were treated as the numerator in the out-
come and predictor variables because they were more
common than Queens gambit openings and victories
(respectively). The logarithms of their ratios will thus be
positive, greatly simplifying the visual analysis of subse-
quent gures. 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
signicant 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
Queens Gambit reects 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 Queens
Gambit. To address this concern, consider
three quantities: (1) the probability a player
offers the Queens Gambit in the next game
after winning with it in the previous game,
(2) the probability a player offers the Queens
Gambit in the next game after winning with a
non-Queens Gambit opening in the previous
game, and (3) the difference, D, between the
two. If the Queens Gambit has no effect on
victories, D = 0. However, if there is differen-
tial reinforcement for using the Queens 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 Queens
Gambit. We also took the event records
(games) of the n = 100 players above and shuf-
ed 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 Queens 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 Queens Gambit offered it in proportionally
more games (df = 346, R
2
= .71). There was a
small, but statistically signicant, bias (i.e.,
regression intercept) toward offering non-
QueensGambitopenings(b= 0.10, SE = 0.03
p< .001) and a modest, but statistically signi-
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 condence
in the accuracy of those slopes and intercepts.
To interpret these values in practical terms, rst
note that the intercept of a regression represents
the model predicted value of an outcome (the
Queens Gambit offer ratio), when the predictor
Figure 2. Whites 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 players
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 t to the data
(estimated using Ordinary Least Squares). The shaded rib-
bon around the line of best t represents the 95% con-
dence interval for the slope of that line. R2 is the
proportion variance explained by the line of best t.
5MATCHING AND CHESS
variable (the log ratio of wins with and without the
Queens Gambit) is held at 0. Because both the
outcome and predictor variable in this model are
in log
10
units, a players data will occupy point
0 of the horizontal axis of Figure 2 if they have
won exactly as many games with the Queens
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 playersindividual matching performance. Each point represents a 50-game block of a players
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 perfectmatching. 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 Queens Gambit are expected to offer
openings other than the Queens 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-Queens
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-Queens Gambit offerings.
Figure 3 depicts example performance from
ve individual tournament-level players. As
shown in the gure, players generally exhibit
correspondence between their relative rate of
victory with Queens 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 Queens 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. Specically, 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 playersbehavior
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
truematching 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 Queens Gam-
bit offerings for a given rate of reinforcement
(i.e., relative success with the Queens 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
Queens 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 Queens
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 shufed 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 Queens Gam-
bit and victory is incredibly low.
DISCUSSION
We evaluated the applicability of the GML
to chess playersuse of the Queens Gambit
during chess openings across a large sample of
chess games played by experienced chess
players. Across players, the GML accounted for
a signicant amount of the variance in players
use of the Queens Gambit. As the relative
number of games won in which White offered
the Queens Gambit increased, the relative
number of games in which White offered the
Queens 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
playersvictories. Consistent with results from
Alferink et al. (2009), the GML accounted for
a greater proportion of variance in playersuse
of the Queens Gambit as the relative experi-
ence levels of the players increased. Sensitivity
estimates suggest playersreinforcement sensi-
tivity (i.e., sensitivity to games won) when
offering the Queens 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-Queens Gambit openings
relative to Queens Gambit openings given the
experienced rates of wins.
Overall, the GML successfully described chess
playersuse of the Queens 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 t 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 ndings
do not explain why matching occurs in the con-
text of chess openings involving the Queens
Gambit, and the descriptive nature of the
methodology used in the present study cannot
identify causal relations between sensitivity,
bias, and choice for Queens Gambit and non-
Queens Gambit openings yielding wins and
losses. Future research should evaluate how
Blacks responses to Whites Queen Gambit
(i.e., accepting vs. denying the gambit) conform
to the GML, in addition to assessing how non-
gambitchessopenings(e.g.,theRuy Lopez,
the English) also conform to the predictions of
theGML.Broadly,ourndings 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 benet
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.
REFERENCES
Alferink, L. A., Critcheld, T. S., Hitt, J. L., &
Higgins, W. J. (2009). Generality of the matching
law as a descriptor of shot selection in basketball.
Journal of Applied Behavior Analysis,42, 595608.
https://doi.org/10.1901/jaba.2009.42-595.
Baum, W. M. (1974). On two types of deviation from
the matching law: Bias and undermatching. Journal
of the Experimental Analysis of Behavior,22, 231242.
https://doi.org/10.1901/jeab.1974.22-231.
Blasius, B., & Tönjes, R. (2009). Zipfs law in the popu-
larity distribution of chess openings. Physical Review
Letters,103,15. https://doi.org/10.1103/
PhysRevLett.103.218701.
Borrero,J.C.,Crisolo,S.S.,Tu,Q.,Rieland,W.A.,
Ross,N.A.,Francisco,M.T.,&Yamamoto,K.Y.
(2007). An application of the matching law to
social dynamics. Journal of Applied Behavior Analy-
sis,40, 589601. https://doi.org/10.1901/jaba.
2007.589-601.
Borrero, J. C., & Vollmer, T. R. (2002). An application
of the matching law to severe problem behavior. Jour-
nal of Applied Behavior Analysis,35,1327. https://
doi.org/10.1901/jaba.2002.35-13.
Bulow, P. J., & Meller, P. J. (1998). Predicting teenage
girlssexual activity and contraception use: An appli-
cation of matching law. Journal of Community Psy-
chology,26, 581596. https://doi.org/10.1002/(SICI)
1520-6629(199811)26:6<581::AID-JCOP5>3.0.CO;
2-Y.
Casella, G., & Berger, R. L. (2001). Statistical inference
(2nd ed.). Pacic Grove, CA: Duxbury Press.
Cohen, J. (1992). A power primer. Psychological Bulletin,
112, 155159. https://doi.org/10.1037/0033-2909.
112.1.155.
Falligant, J. M., Boomhower, S. R., & Pence, S. T.
(2016). Application of the generalized matching law
to point-after-touchdown conversions and kicker
selection in college football. Psychology of Sport and
Exercise,26, 149153. https://doi.org/10.1016/j.
psych-sport.2016.07.006.
Herrnstein, R. J. (1961). Relative and absolute strength
of response as a function of frequency of reinforce-
ment. Journal of the Experimental Analysis of Behav-
ior,4, 267. https://doi.org/10.1901/je-ab.1961.4-
267.
Herrnstein, R. J. (1970). On the law of effect. Journal of
the Experimental Analysis of Behavior,13, 243266.
https://doi.org/10.1901/jeab.1970.13-243.
Kasparov, G., & Keene, R. (1994). Batsford chess openings
2. New York, NY: Henry Holt.
Mace, F. C., & Critcheld, T. S. (2010). Translational
research in behavior analysis: Historical traditions and
imperative for the future. Journal of the Experimental
Analysis of Behavior,93, 293312. https://doi.org/10.
1901/jeab.2010.93-293.
Mace, F. C., Neef, N. A., Shade, D., & Mauro, B. C.
(1994). Limited matching on concurrent-schedule
reinforcement of academic behavior. Journal of
Applied Behavior Analysis,27, 585596. https://doi.
org/10.1901/jaba.1994.27-585.
Maslov, S. (2009). Power laws in chess. Physics,2, 97.
https://doi.org/10.1103/Physics.2.97.
McDowell, J. J. (2013). On the theoretical and empirical
status of the matching law and matching theory. Psy-
chological Bulletin,139, 10001028. https://doi.org/
10.1037/a0029924.
Mechner, F. (2010). Chess as a behavioral model for cog-
nitive skill research: Review of blindfold chess by
Eliot Hearst and John Knott. Journal of the Experi-
mental Analysis of Behavior,94, 373386. https://doi.
org/10.1901/jeab.2010.94-373.
Ramiz, A. (2006). An investigation into the openings used
by top 100 chess players. International Journal of Per-
formance Analysis in Sport,6, 149160. https://doi.
org/10.1080/24748668.2006.11868363.
R Core Team. (2017). R: a language and environment for
statistical computing. Vienna, Austria: R Foundation
for Statistical Computing. Retrieved from. http://
www.Rproject.org/.
Reed, D. D., Critcheld, T. S., & Martens, B. K. (2006).
The generalized matching law in elite sport competi-
tion: Football play calling as operant choice. Journal
of Applied Behavior Analysis,39, 281297. https://
doi.org/10.1901/jaba.2006.146-05.
Romanowich, P., Bourret, J., & Vollmer, T. R. (2007).
Further analysis of the matching law to describe two-
and three-point shot allocation by professional bas-
ketball players. Journal of Applied Behavior Analysis,
40, 311315. https://doi.org/10.1901/jaba.2007.
119-05.
Schenk, M. J., & Reed, D. D. (2019). Experimental eval-
uation of matching via a commercially available bas-
ketball video game. Journal of Applied Behavior
Analysis. Advance online publication. https://doi.org/
10.1002/jaba.551
IAN CERO and JOHN MICHAEL FALLIGANT10
SCID (2017). SCID - Chess Database Software (Version
4.6.4) [Data le]. Retrieved November 4, 2017, from
http://scid.sourceforge.net/
Shannon, C. E. (1950). Programming a computer for
playing chess. The London, Edinburgh, and Dublin Phil-
osophical Magazine and Journal of Science,41,256275.
https://doi.org/10.1080/14786445008521796.
Vollmer, T. R. (2011). Three variations of translational
research: Comments on Critcheld (2011). The Behavior
Analyst,34,3135. https://doi.org/10.1007/bf03392231.
Vollmer, T. R., & Bourret, J. (2000). An application of
the matching law to evaluate the allocation of two-
and three-point shots by college basketball players.
Journal of Applied Behavior Analysis,33, 137150.
https://doi.org/10.1901/jaba.2000.33-137.
Received November 26, 2017
Final acceptance May 15, 2019
Action Editor, Jason Bourret
11MATCHING AND CHESS
... First examined in basic laboratory contexts, researchers have increasingly applied the GML to naturalistic, socially significant human behavior (Borrero & Vollmer, 2002). The generality of the GML has also been assessed in describing complex operant behavior in competitive contexts such as basketball (e.g., Alferink et al., 2009), baseball (e.g., Poling et al., 2011), football (e.g., Reed et al., 2006), hockey (Seniuk et al., 2015), martial arts (Seniuk et al., 2019), chess (Cero & Falligant, 2019), volleyball (Rotta et al., 2020), and videogames (Schenk & Reed, 2019). However, in many naturally occurring contexts, there are more than two concurrently available response alternatives/schedules of reinforcement. ...
... Even within the same skill level (e.g., all experts) or level of competition (e.g., Division I American collegiate sports), there may be differences in sensitivity/bias for players or teams who compete within the same level of competition (cf. Alferink et al., 2009;Cero & Falligant, 2019). Thus, it is unknown whether skill-level specific differences would be evident using the multialternative GML or if the GML would yield a robust descriptor of all pitchers' pitch selections (regardless of relative skill level). ...
... minimum possible rates of target pitches thrown (B i ) for a given rate of reinforcement (R i ) using as described in Herrnstein (1970). To the extent that these minimum and maximum bounds are not very close together, the chances of forced matching are low (Cero & Falligant, 2019;Cox et al., 2017). ...
Article
Full-text available
Cox et al. (2017) successfully applied the multialternative version of the generalized matching law (GML) to pitch selection among a sample of Major League Baseball (MLB) pitchers. The purpose of the present study was to replicate and extend these findings by fitting the multialternative GML to pitch data among a sample of MLB pitchers with varying levels of success in the major leagues. We also examined how matching parameters changed as a function of novel antecedent game contexts such as the infield shift, game location, and number of times the pitcher faced the batters in the batting order. These results replicate the findings from Cox et al. and suggest the multialternative GML is a robust descriptor of pitch selection among MLB pitchers. Together, these findings further extend the generality of the multialternative GML to naturalistic, non-laboratory environments.
... He proposed a simple rule called the matching law stating that the proportion of time or responses that an animal allocates to an option or action matches the proportion of reinforcement they receive from those options or actions 1 . The matching law has been shown to explain global choice behavior across many species 2 including pigeons 3-6 , mice [7][8][9] , rats [10][11][12] , monkeys [13][14][15][16][17][18] , and humans [19][20][21][22][23][24] , in a wide range of tasks, including concurrent variable interval, concurrent variable ratio, probabilistic reversal learning, and so forth. A common finding in most studies, however, has been that animals undermatch, corresponding to selection of the better stimulus or action less than it is prescribed by the matching law. ...
... In RL2 + LM+ and RL2 + CM+, the decision values are computed as in Eqs. (24), (25), respectively, however, ω LM and ω CM only range from 0 to 1 instead of −1 to 1. ...
Article
Full-text available
For decades, behavioral scientists have used the matching law to quantify how animals distribute their choices between multiple options in response to reinforcement they receive. More recently, many reinforcement learning (RL) models have been developed to explain choice by integrating reward feedback over time. Despite reasonable success of RL models in capturing choice on a trial-by-trial basis, these models cannot capture variability in matching behavior. To address this, we developed metrics based on information theory and applied them to choice data from dynamic learning tasks in mice and monkeys. We found that a single entropy-based metric can explain 50% and 41% of variance in matching in mice and monkeys, respectively. We then used limitations of existing RL models in capturing entropy-based metrics to construct more accurate models of choice. Together, our entropy-based metrics provide a model-free tool to predict adaptive choice behavior and reveal underlying neural mechanisms. Animals distribute their choices between alternative options according to relative reinforcement they receive from those options (matching law). Here, the authors propose metrics based on information theory that can predict this global behavioral rule based on local response to reward feedback.
... The matching law is one quantitative model of behavior that has been demonstrated to provide a good description of many applied phenomena such as problem behavior (Borrero et al., 2010;Borrero & Vollmer, 2002;Kronfli et al., 2021;McDowell, 1982McDowell, , 1988 and response allocation in sports (Alferink et al., 2009;Cero & Falligant, 2019;Cox et al., 2021;Cox et al., 2017;Reed et al., 2006;Romanowich et al., 2007;Rotta et al., 2020;Vollmer & Bourret, 2000). The matching law has also been applied to social behavior such as conversation (Borrero et al., 2007;Conger & Killeen, 1974;McDowell & Caron, 2010;Pierce et al., 1981). ...
Article
Full-text available
Recent research has developed and evaluated assessments of sociability in which time allocation near or away from an adult who initiates social interactions is used to characterize the participant as social, indifferent, or avoidant of social interaction. Though these qualitative outcomes have been useful, no studies have evaluated methods of obtaining more quantitative measures of sociability. The matching law has been demonstrated to describe a wide range of human behavior and may also be useful in describing social time allocation. We adapted the matching law and assessment of sociability procedures with the aim of providing a more precise, quantitative measure of sociability. We fitted the matching equation to the social time allocation data of 8 children with autism spectrum disorder. The equation was effective in quantifying sociability, accounted for a large proportion of variance in participants' behavior, did so equally well for participants who were social and avoidant, and provided a more sensitive measure relative to those used in previous research. The implications of this methodology, its potential utility, and directions for future research are discussed.
... Extensive empirical support exists for the GML from laboratory and nonlaboratory research (see McDowell, 2013). For example, the GML accounts well for choice across an array of operants such as lever pressing in rodents (e.g., Boomhower & Newland, 2016), key pressing or mouse clicking in undergraduates (e.g., Klapes et al., 2020;Madden & Perone, 1999), play selection and performance in a number of amateur (e.g., Alferink et al., 2009;Romanowich et al., 2007;Rotta et al., 2020) and professional (e.g., Cox et al., 2017;Reed et al., 2006) sports contexts, gambit selection among expert chess players (Cero & Falligant, 2019), conversation allocation among young adults (Borrero et al., 2007), and much more. ...
Article
The generalized matching law (GML) has been used to describe the behavior of individual organisms in operant chambers, artificial environments, and nonlaboratory human settings. Most of these analyses have used a handful of participants to determine how well the GML describes choice in the experimental arrangement or how some experimental manipulation influences estimated matching parameters. Though the GML accounts very well for choice in a variety of contexts, the generality of the GML to all individuals in a population is unknown. That is, no known studies have used the GML to describe the individual behavior of all individuals in a population. This is likely because the data from every individual in the population has not historically been available or because time and computational constraints made population-level analyses prohibitive. In this study, we use open data on baseball pitches to provide an example of how big data methods can be combined with the GML to: (1) scale within-subjects designs to the population level; (2) track individual members of a population over time; (3) easily segment the population into subgroups for further analyses within and between groups; and (4) compare GML fits and estimated parameters to performance. These were accomplished for each of 2,374 individuals in a population using 8,467,473 observations of behavior-environment relationships spanning 11 years. In total, this study is a proof of concept for how behavior analysts can use data-science techniques to extend individual-level quantitative analyses of behavior to the population-level focused on domains of social relevance.
... He proposed a simple rule called the matching law stating that the proportion of time or responses that an animal allocates to an option or action matches the proportion of reinforcement they receive from those options or actions (Herrnstein, 1961). The matching law has been shown to explain global choice behavior across many species (Williams, 1988) including pigeons (de Villiers and Herrnstein, 1976;William, 1979;Mazur, 1981;Villarreal et al., 2019), mice (Gallistel et al., 2007;Fonseca et al., 2015;Bari et al., 2019), rats (Gallistel, 1994;Belke and Belliveau, 2001;Lee et al., 2017), monkeys (Anderson et al., 2002;Sugrue et al., 2004;Lau and Glimcher, 2005;Kubanek and Snyder, 2015;Tsutsui et al., 2016;Soltani et al., 2021), and humans (Schroeder and Holland, 1969;Pierce and Epling, 1983;Beardsley and McDowell, 1992;Savastano and Fantino, 1994;Vullings and Madelain, 2018;Cero and Falligant, 2020), in a wide range of tasks, including concurrent variable interval, concurrent variable ratio, probabilistic reversal learning, and so forth. A common finding in most studies, however, has been that animals undermatch, corresponding to selection of the better option/action less than it is prescribed by the matching law. ...
Preprint
Full-text available
For decades, behavioral scientists have used the matching law to quantify how animals distribute their choices between multiple options in response to reinforcement they receive. More recently, many reinforcement learning (RL) models have been developed to explain choice by integrating reward feedback over time. Despite reasonable success of RL models in capturing choice on a trial-by-trial basis, these models cannot capture variability in matching. To address this, we developed novel metrics based on information theory and applied them to choice data from dynamic learning tasks in mice and monkeys. We found that a single entropy-based metric can explain 50% and 41% of variance in matching in mice and monkeys, respectively. We then used limitations of existing RL models in capturing entropy-based metrics to construct a more accurate model of choice. Together, our novel entropy-based metrics provide a powerful, model-free tool to predict adaptive choice behavior and reveal underlying neural mechanisms.
... The settings included, but are not restricted to, foraging in natural habitats [32], football stadiums (the decision to go for a one-point or two-point conversion, [33]), baseball diamonds (pitch selection, [34]), basketball courts (two-vs. three-point shots, [35]), and chess tournaments (the decision to use the Queen's gambit opening, [36]). The rewards included, but are not restricted to, food, money, brain stimulation, drugs, alcohol, and social approval (e.g., [28,37,38]). ...
Article
In keeping with the goals of this Special Issue, this paper poses the following questions: What are addiction's non-eliminable features and can they be explained by one or more general principles? I have added the qualifier "distinctive" to these goals, as in "distinctive non-eliminable features." The result is a highly heterogeneous list, which includes features of addiction's natural history, such as its high remission rates, its unique idioms (e.g., "kicking the habit"), and its patented interventions, such as Alcoholics Anonymous. I show that each of these distinctive features reflects how individuals make choices. In particular, they reflect the competing claims of two basic choice processes: global maximizing of the sort assumed in introductory economics text books and Herrnstein's matching law, which has empirical rather than theoretical roots. These are basic choice processes, which apply to all decision making, not just drugs and not just addicts. Nevertheless, they can result in addiction when one of the options has the capacity to undermine the value of competing interests and undermine global maximizing. Conversely, the analyses also show that the two basic choice processes combine so as to predict that addiction is a semi-stable state that is biased to resolve in favor of remission. These predictions are supported by the high rates of addiction, by the high rates of remission from addiction, and by the fact that remission is often unassisted or "spontaneous." The analyses fail to support the idea that pathological psychological processes lead to addiction. Rather they show that addiction emerges from the interactions of normal choice processes and the behaviorally toxic effects of drugs.
... In subsequent experiments, matching was found in rats, monkeys, cows, humans, and even insects (e.g., Davison & McCarthy, 1988;De Carlo & Abramson, 2012;Grace & Hucks, 2013). The settings have included but are not restricted to lever pressing in experimental chambers, signal detection tasks, foraging in natural settings, the opening moves in chess games, two-versus three-point shots in college basketball, and point-after-touchdown plays in the NFL (e.g., Cero & Falligant, 2020;Falligant, Boomhower, & Pence, 2016;Houston, 1986;Vollmer & Bourret, 2000). The rewards have included but are not restricted to food, money, brain stimulation, drugs, alcohol, and social approval (e.g., Conger & Killeen, 1974). ...
Article
Full-text available
The matching law describes the allocation of behavior over a wide range of settings, including laboratory experimental chambers, forest foraging patches, sports arenas, and board games. Interestingly, matching persists in settings in which economic analyses predict quite different distributions of behavior, and it also differs systematically from probability matching. We tested whether the matching law also describes the allocation of covert cognitive processes. Sixty-four participants viewed 2, small, vertically arranged adjacent stimuli that projected an image that fit within the fovea. A trial version of the reward contingencies used in matching law experiments determined which stimulus was the target. For example, in 1 condition, the top stimulus was the target 3 times more frequently than the bottom stimulus. However, the amount of time the stimuli were available was tailored to each participant so that they were not able to make use of the information in both stimuli even though an eye-tracking experiment confirmed that they saw both. The implication of this restriction is that participants had to decide which stimulus to attend to prior to each trial. The only available objective basis for this decision was the relative frequencies that a stimulus was the target. The matching law predicted the correlation between the relative frequency that a stimulus was the target and the proportion of trials that it was attended to. The results support the claim that the matching law is a general choice principle-one that describes the allocation of covert mental processes as well as overt behavioral responses. (PsycInfo Database Record (c) 2020 APA, all rights reserved).
Article
Full-text available
Three adolescent students with special educational needs were given a choice between completing one of two available sets of math problems. Reinforcers (nickels) across these alternatives were arranged systematically in separate experimental phases according to three different concurrent variable-interval schedules (reinforcement ratios of 2:1, 6:1, and 12:1). Time allocated to the two stacks of math problems stood in linear relationship to the reinforcement rate obtained from each stack, although substantial undermatching and bias were observed for all subjects. However, changes in the schedules were not followed by changes in allocation patterns until adjunct procedures (e.g., changeover delays, limited holds, timers, and demonstrations) were introduced. The necessity of adjunct procedures in establishing matching in applied situations is discussed as a limitation to quantitative applications of the matching law in applied behavior analysis.
Article
Full-text available
This multifaceted work on chess played without sight of the pieces is a sophisticated psychologist's examination of this topic and of chess skill in general, including a detailed and comprehensive historical account. This review builds on Hearst and Knott's assertion that chess can provide a uniquely useful model for research on several issues in the area of cognitive skill and imagery. A key issue is the relationship between viewing a stimulus and mental imagery in the light of blindfold chess masters' consistent reports that they do not use or have images. This review also proposes a methodology for measuring and quantifying an individual's skill shortfall from a theoretical maximum. This methodology, based on a 1951 proposal by Claude Shannon, is applicable to any choice situation in which all the available choices are known. The proposed “Proficiency” measure reflects the equivalent number of “yes—no” questions that would have been required to arrive at a best choice, considering also the time consumed. As the measure provides a valid and nonarbitrary way to compare different skills and the effects of different independent variables on a given skill, it may have a wide range of applications in cognitive skill research, skill training, and education.
Article
Full-text available
Article
Many recent nonlaboratory‐based quantitative analyses of behavior have relied on archival competitive sporting data. However, the ratio‐based reinforcement schedules in most athletic competitions and the correlational nature of archival data analyses raise concern over the contributions of such findings to the behavior analytic literature. The current experiment evaluated whether matching in a human operant paradigm would approximate matching observed in nonlaboratory‐based sports data. To this end, we used in‐game attributes to parametrically manipulate 2‐ and 3‐point shooting in a commercially available basketball video game. The behavior of 6 of 9 participants conformed to the generalized matching equation. These results suggest matching in sporting contexts may be a product of restricted nonindependent concurrent random‐ratio schedules. Implications of this experiment, including those in applied behavior analysis and potential influence on gamification, are discussed.
Article
The aim of the our study is an ınvestıgatıon ınto the openıngs used by top 100 chess players. The chess games (n=2046) that 70 players, who took part in Top 100 list, played during January-April 2006 period were examined. The frequencies and percentages of the openings used were calculated. Separating the chess players into five different age groups, the frequency values and percentages of the openings they used, and the winning, defeating, and drawing were calculated using the Chi-Square Test and evaluated statistically. The most frequently played openings were as follows respectively: Sicilian defence (n=476; 23,3%), Queen’s Gambit (n=326; 1,9%), Ruy Lopez/Spanish Game (n=252; 12,3%). The kinds of openings, Semi-Open Games (33 %) are the most frequent and Other Black Response To d4 (4,1 %) is the least frequently played. According to the kinds of openings, there are statistically significant differences between the drawings, winnings and defeatings obtained (p<0.05). The kinds of openings played according to age groups, Semi-Open Games are preferred most in the 20 years old or younger, 21 to 30 years old, and 31 to 40 years old age groups, Indian Systems are preferred most in the 41 to 50 years old age group (p<0.05). It can be claimed that Top 100 players prefer Semi-Open Games most; the winning, defeating, and drawing percentages change depending on the increasing age and the kinds of openings played; and the best opening is Caro-Kann for Whites, French Defence for Blacks, and Petrof’s Defence for a draw.
Article
This paper is concerned with the problem of constructing a computing routine or “program” for a modern general purpose computer which will enable it to play chess. Although perhaps of no practical importance, the question is of theoretical interest, and it is hoped that a satisfactory solution of this problem will act as a wedge in attacking other problems of a similar nature and of greater significance. Some possibilities in this direction are:- (1) Machines for designing filters, equalizers, etc. (2) Machines for designing relay and switching circuits. (3) Machines which will handle routing of telephone calls based on the individual circumstances rather than by fixed patterns. (4) Machines for performing symbolic (non-numerical) mathematical operations. (5) Machines capable of translating from one language to another. (6) Machines for making strategic decisions in simplified military operations. (7) Machines capable of orchestrating a melody. (8) Machines capable of logical deduction.