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Influence of Experts' Domain-specific Knowledge on Risk Taking in Adversarial Situations


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Experts play a considerable role in society, as they have to evaluate the risk of policies in many fields of social life and in adversarial situations (e.g., the military). Yet, the influence of expertise on risk taking in adversarial situations has received little attention. An examination of the strategies used by chess players ranging from amateurs to masters in competitive games (n = 73,341) revealed an unexpected pattern of results. First, the majority of players favored the riskier strategy. This result is in line with the literature on economic decision making that indicates a tendency to take risks in situations where the outcome can be either positive or negative. More surprising is our second finding: As skill increased, the majority of players still adopted a risk seeking attitude but the proportion of players taking a more conservative approach increased. This result would tend to indicate that experts making decisions with impact on their own life become increasingly risk averse. Overall, our findings indicate that knowledge moderates but does not eliminate risk taking behavior. They also highlight that risk taking in adversarial situations might result from a complex set of factors. Further research should establish which psychological processes drive players to adopt a risk taking or conservative strategy in their games.
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Journal of Expertise / June 2020 / vol. 3, no. 2
Influence of Experts’ Domain-specific
Knowledge on Risk Taking in Adversarial
Philippe Chassy1 and Fernand Gobet2
1 Mathematical Psychology lab, University of Liverpool, UK
2 London School of Economics and Political Science, UK
Correspondence: Philippe Chassy,
Experts play a considerable role in society, as they have to evaluate the risk of policies in many fields of
social life and in adversarial situations (e.g., the military). Yet, the influence of expertise on risk taking
in adversarial situations has received little attention. An examination of the strategies used by chess
players ranging from amateurs to masters in competitive games (n = 73,341) revealed an unexpected
pattern of results. First, the majority of players favored the riskier strategy. This result is in line with the
literature on economic decision making that indicates a tendency to take risks in situations where the
outcome can be either positive or negative. More surprising is our second finding: As skill increased, the
majority of players still adopted a risk seeking attitude but the proportion of players taking a more
conservative approach increased. This result would tend to indicate that experts making decisions with
impact on their own life become increasingly risk averse. Overall, our findings indicate that knowledge
moderates but does not eliminate risk taking behavior. They also highlight that risk taking in adversarial
situations might result from a complex set of factors. Further research should establish which
psychological processes drive players to adopt a risk taking or conservative strategy in their games.
Risk, expertise, uncertainty, strategy, adversarial situations
Evolutionary pressures drive individuals to
compete for resources. Occasionally, people or
groups decide to engage in high-risk operations to
win over the competition. The confrontation,
whether direct or indirect, will have significant
consequences for all parties involved. At the
group level, high-risk operations are common in
institutionalized activities such as politics,
business, and the military. Competition can be
observed at the individual level in both direct
confrontations such as in sports and indirect
oppositions such as promotion at work. In these
real-life situations, characterized by potential loss,
people face uncertain environments where risks
cannot be evaluated precisely. Due to this
uncertainty, society relies on experts to inform
and often establish strategies (Knighton, 2004;
Vertzberger, 1995). Despite its importance, the
actual influence of experts on risk taking in high-
risk situations has not received much attention in
the scientific literature (Gobet, 2016). Yet as
experts inform key decisions in society, there is a
need to understand how experts evaluate risk and
set their attitude with respect to potential losses.
In the present paper, we evaluate the influence of
expert knowledge on strategic risk by examining
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Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 89
Journal of Expertise / June 2020 / vol. 3, no.2
decisions at the individual level. To ensure that
decisions are concrete, we selected a domain
where deciders incur the consequences of their
Attitude to risk has been extensively explored
in laboratory conditions with experiments
involving economic decisions (Kahneman &
Tversky, 1979; Tversky & Kahneman, 1992). In
situations that can have either a positive or a
negative outcome, the bulk of the evidence points
to individuals being risk seeking (Birnbaum,
2008; Ert & Erev, 2013). Perceived risk has been
shown to depend upon how the decider frames the
situation (Birnbaum, 2008; Kahneman & Tversky,
1984). Crucially, it is during the phase of building
a representation of the situation that experts’
domain-specific knowledge provides them with a
substantial advantage over non-experts. This
representation, built nearly instantaneously
(Chassy & Gobet, 2011a), enables experts to solve
simple problems correctly within a few seconds
(Van der Maas & Wagenmakers, 2005).
Considering that recognition of strategic features
is rapid and immediately offers potential solutions
(Bilalić et al., 2009; Chassy, 2013), the influence
of such domain-specific knowledge should be
sizable on risks estimates. It is known that people
tend to avoid situations that might involve
potential loss even if there is the possibility of a
gain (Ert & Erev 2008). But, once they face the
prospect of a potential loss individuals become
risk seeking (Scholer et al., 2010). The immediate
prospect of the loss promotes risk seeking
behaviora mechanism that is served by specific
brain regions (Tom et al., 2007). The evidence
from various domains of expertise, on the other
hand, indicates that experts tend to perceive less
risk than laypeople when evaluating the level of
risks in potentially damaging technologies (Slovic
et al., 1995). A study from Savadori et al. (2004)
has provided evidence that experts do not base
their judgement on the same cues as laypeople.
Taken together, these studies indicate that
domain-specific knowledge modifies the
construction of the problem situation, and by
doing so impacts on the final estimate of risk.
These studies, however, do not put the expert
in a situation wherein they would incur the
consequences of their decisions. In addition, and
crucially, they refer to hypothetical situations,
where experts advise a policy on a fictitious
experimental scenario. Testing risk taking in high-
risk confrontations requires meeting several
methodological criteria: a real-life situation to
ensure ecological validity of the findings, a
domain of expertise that puts the experts in a
position wherein risk is quantifiable, a situation
that is competitive, and, crucially, decisions that
will impact on the decider.
Chess is one of the very few domains that
meets all these methodological criteria. It is a
zero-sum game where the gain of one player
corresponds to the loss of the other player.
Players’ skill level is measured precisely and
quantitatively by the Elo rating system (Elo,
1978). This feature, which is nearly unique in
research into expertise, is particularly useful as, in
adversarial situations, the opponent’s actions
might affect the course of events and with it the
outcome. Knowing the opponent’s level of
expertise provides important information when
framing the situation. Three additional features
contribute to the use of chess as a fitting domain
for studying risk. First, it is a visuospatial game
played with the same rules worldwide, which
avoids biases linked to language (Casasanto,
2008). Second, the presence of databases
including games played all over the world ensures
that the results will not be biased by cultural
approaches to risk (e.g., Li et al., 2009). Last and
most importantly, the Elo rating, initially
developed as a measure of expertise, is regarded
by chess players as a reward system. For most if
not all players, Elo points are at the core of their
chess life, as they not only indicate the level of
expertise (a source of prestige) but also are the
key factor upon which players are invited to play
in tournaments, to participate in team
competitions, and to deliver lectures.
In this context, the literature indicating
dominance of loss aversion in mixed gambles (Ert
& Erev, 2008; Scholer et al., 2010) implies that
experts take the risky option to avoid high losses
when they face weaker opponents. The attitude of
the weaker players is subtler. The larger the rating
difference with the better player, the higher the
gain in case of a win but also the higher the
probability of losing small. If the weaker player
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 90
Journal of Expertise / June 2020 / vol. 3, no.2
focuses on the potential gain (winning a large
number of Elo points), then a risk-taking strategy
might be undertaken; however, if the player
focuses on the low probability of winning, then a
more conservative strategy might be undertaken.
To understand which factor is the most important
to non-experts, Slezak and Sigman (2012)
examined the speed-accuracy tradeoff of non-
expert players (rating < 2000 Elo) in games where
the thinking time was limited to three minutes for
each player. The results indicated that weaker
players tended to focus on accuracy by playing
slower. Players are thus sensitive to risk,
becoming more conservative when meeting better
rated opponents.
Our study aimed at expanding on this initial
finding by examining the decisions of experts. We
examined the strategies chosen by chess players
during competitive events that took place during
one entire year all over the world. Our analysis is
based on the first move played in the game, a
move that is often chosen by players before the
game. At the beginning of the game, there is a
definite number of strategies to choose from
(Matanović et al., 1971). Different strategies
involve different levels of risk. The fact that
players decide the strategy they are going to play
beforehand is demonstrated by the fact that they
know the first moves by rote memory (Chassy &
Gobet, 2011b) and that they can identify the
strategy upon mere recognition of the position
(Chassy, 2013).
Based on the reviewed literature, we put
forward three hypotheses. First, considering that
chess players willingly engage in a game which in
essence is a battle, we would expect the players to
be predominantly risk takers in this context.
Second, the theoretical considerations discussed
above suggest that experts’ domain-specific
knowledge will enable them to generate a more
accurate representation of the situation and thus to
evaluate risk better; thus, based on the fact that all
players are essentially sensation seekers (Joireman
et al., 2002), we predicted that experts would be
more risk taking than non-experts. Third, we
would predict that players adapt to their opponent
levels of skill so as to minimize loss.
We used the games from a commercial chess
database (Fritz Database, version 12; Morsch,
2009). We selected all the games played over a
year, namely from April 1, 2008, to March 31,
2009, by players whose rating spanned the
1600-2399 Elo range. The final sample
consisted of 73,341 games. Players were
assigned to one of four groups: amateurs (1600-
1799 Elo, M = 1719.18; SD = 56.57), club
players (1800-1999 Elo, M = 1915.62;
SD = 56.05), candidate masters (2000-2199 Elo,
M = 2103.34; SD = 57.21), or masters (2200-
2399 Elo, M = 2292.64; SD = 57.16). (To
minimize ambiguity between the candidate-
master and the master classes, we will refer to
the candidate-master class as the candidate
class.) The cutoff point at 2,000 Elo corresponds
to the definition of chess expertise in the
scientific literature (Elo, 1978). Hence, our
sample was made of two non-expert and two
expert groups. The four selected skill levels
ensured that a sufficiently large number of
games were used and had the advantage that the
levels occupied adjacent positions in the rating
scale, which made comparisons easier.
Measure of Risk
Although other measures exist, it is common in
research on judgment and decision making to
use standard deviation (σ) around the expected
value (μ) to define risk (Rothschild & Stiglitz,
1970; Damodaran, 2007). The application of
this definition to risk taking in chess has already
proven useful to analyze attitude to risk in
adversarial situations (Chassy & Gobet, 2015).
After partitioning the data set as a function of
the first move, we calculated the variance
around the mean outcome for each skill to
determine their level of risk (see Appendix 1).
Based on an extensive chess literature (e.g.,
Matanović, et al., 1971), one can categorize the
first moves of the game in two main groups:
open games (1.e4) and closed games (1.d4, 1.c4,
and 1.Nf3). Prior to our sample year, these four
moves account for 96% of all openings on a
period covering 5 years (2003-2007), for the
same player range (1600-2399). To estimate risk
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 91
Journal of Expertise / June 2020 / vol. 3, no.2
directly from the empirical data, we used the
games in the Fritz database (Big Database
2010). For all games where moves were
traceable during the period of 2002-2007, we
analyzed the pattern of wins, draws, and losses.
The time window 2002-2007 was selected so as
to be close to the period being analyzed and to
be sufficiently large to enable accurate
estimates. Games with only one move or
incomplete games were removed from the
database (n < 1%). Risk, defined as variance
around the mean outcome, was estimated by
analyzing 1,474,378 games. For all skill levels
pooled, σ was 42.69% for open games and
41.59% for closed games; the difference is
statistically highly reliable given the large
number of observations (see below). In line with
the definition of risk, as compared to closed
games open games increase the chances of both
winning (39.58% vs 38.91%) and losing
(33.66% vs 30.93%) while decreasing the
chances of drawing (26.75% vs 30.16%). Our
empirical definition of risk is in line with the
subjective experience of chess authorities for
over a century (e.g., Aagard, 2002; Gunsberg,
1901). Since open games involve more risks,
they were labeled as “risky”, and closed games
were labeled as “conservative”. The reader
should bear in mind the fact that these labels
refer to relative risk, since conservative
openings still entail some degree of risk.
To shed light on the factors influencing the
choice of risky or conservative strategies by
chess players, we calculated the payoff for each
game. We followed the mathematical
procedures developed by Elo (1978). The first
step consisted in computing the rating difference
between the player initiating the game
(henceforth, first player) and the player playing
the second move (henceforth, second player),
see equation 1 below. Then, we calculated the
probability of winning of the first player in each
game (see equation 2). The results were checked
against the table of probabilities provided by the
International Chess Federation (FIDE).
The second step consisted in calculating the
potential change in rating (see equations 3 and 4
below). As indicated by FIDE, the k parameter
is adjusted to various situation (e.g., age of the
player, stage of career). To facilitate
comparisons, we used k = 20 for the four skill
levels. As a control for the correctness of our
calculations of the payoffs and probabilities, we
calculated the utility of each game (Von
Neumann & Morgenstern, 2007). All games
should have a zero value as chess is, by
definition, a zero-sum game; this theoretical
assumption was verified (see Appendix 2).
Rating difference: = Elosecond Elofirst
Probability of winning of the first player: p = (1+10/400)-1
Payoff of the game (win): Payoff = k(1 p)
Payoff of the game (loss): Payoff = k(0 p)
The data show that the risky strategy was
selected more often (54.40%) than the
conservative strategy (45.60%); a difference that
that is statistically significant as indicated by a
chi-square analysis on frequencies,
χ²(1, n = 73,341) = 569.184, p < .05, φ = .09.
This result supports the hypothesis that chess
players are predominantly risk seeking. To
explore whether the level of expertise influences
risk taking, we compared the frequencies of
occurrence of risky and conservative systems as
a function of skill level. Figure 1 shows that, at
all risk levels, risky strategies dominate
conservative strategies. A key finding is that, as
their skill increases, players tend to be more
conservative. The difference between levels of
expertise is statistically significant,
χ²(3, n = 73,341) = 117.742, p < .05 , φ = .04.
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 92
Journal of Expertise / June 2020 / vol. 3, no.2
Figure 1. Percentage use of the risky strategy for each group of skill.
We then analyzed whether players are
influenced by the level of the opponent in their
attitude to risk. This is a crucial analysis since it
shows whether players adapt to the opponent’s
level of skill. Figure 2 shows the percentage of
first players who selected a risky strategy as a
function of the level of skill of their opponents.
While variations in the use of risky strategies
can be noted within a class, Figure 2 confirms a
global trend of decreasing risk as the skill level
of a player increases. But Figure 2 also
highlights two noticeable situations: Amateurs
meeting Masters and vice-versa. Amateurs, like
Masters, turn more cautious than against any
other opposition when they meet the opposite
end of the spectrum of skill.
Figure 2. Risk taking of the first player as a function of the skill level of the first player and the opponent.
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 93
Journal of Expertise / June 2020 / vol. 3, no.2
Table 1 reports, for each of the 16 conditions,
the average payoff and probability, the
percentage of risky strategy, and the χ2 test of
independence. Payoff and probability
characterize the situation that players were
facing. Their decision, which is whether to
adopt a conservative or risky strategy, is
reflected in the average percentage of risky
strategies used. To illustrate with an example,
we consider the case of first players at club level
facing a second player at master level. In the
case of a win, the club players could win 17.45
Elo points on average and in the case of a loss,
they incurred an average loss of 2.55 Elo points;
the low probability of winning (p = .13) and the
high probability of losing (p = .87) defined the
challenge these players were facing (see
Appendix 2). In these conditions, club players
decided to use the risky strategy in 55.90% of
the games; a choice that departs significantly
from an equal distribution of conservative and
risky strategies, as indicated by the associated
significant χ² value of 36.82.
Table 1. Mean values for the payoffs and probabilities in case of victory or defeat of the first player.
Note: * Significant at p < .05, n: number of games
The independence tests reported in Table 1
indicate for each of the 16 conditions whether the
distribution of games using risky or conservative
strategy departs from chance or not. In fourteen
conditions out of 16, chess players significantly
favored the risky strategy over the conservative
one. Only when amateurs and masters met did the
players not show this preference. As indicated in
Table 1, the significant differences are found in
the two conditions that have the lower number
of cases, casting a shadow on the validity of the
This paper has investigated risk taking in chess,
a zero-sum confrontational situation. We have
analyzed the strategic choices made by players
of four different levels of skill (amateur, club,
candidate, and master level). The analysis was
conducted on 73,341 chess games that were
played across the globe in an entire year. The
results support the hypothesis that chess players,
whether expert or not, tend in general to choose
the riskier option at the beginning of the game.
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 94
Journal of Expertise / June 2020 / vol. 3, no.2
The second result of interest is that risk levels
are inversely proportional to skill. Finally, due
to differences in statistical power across
conditions, we cannot come to a definite
conclusion as to whether our data support our
third hypothesis; it remains unclear whether
players adapt to the opponent. Our findings
inform the current literature on risk taking and
open new avenues for investigating decision
making processes in adversarial situations.
As predicted by Tversky and Kahneman’s
(1992) theory of decision making, loss aversion
drove a majority of players to use more risk-
seeking strategies. Yet, a substantial 45.60% of
the players favored conservative strategies, a
figure highlighting the specificity of
confrontational situations. The dynamics
underlying the choice of a strategy, whether
conservative or risky, emerge from interaction
between numerous factors. One factor that
might play a key role is players’ personality. It
is known that children who play chess score
higher on the Big Five dimensions of
Intellect/openness and Energy/extraversion than
children who do not play chess (Bilalić et al.,
2007), although this pattern of results was not
found with adult chess players (Vollstädt-Klein
et al., 2010). Another personality trait
potentially playing a role in determining the
level of risk undertaken by players is sensation
seeking (Horvath & Zuckerman, 1993)the
keenness to engage in activities yielding intense
experiences. Chess players have been shown to
score higher than the general population on
sensation seeking (Joireman et al., 2002), a trait
that is unsurprisingly correlated with risk taking
behaviors (Kern et al., 2014). We speculate that
chess players’ attitude to risk results from the
balance between the natural tendency to avoid
loss, on the one hand, and chess players’
personalities, on the other hand. The net result
of these opposing forces, which vary from
player to player, determines whether a player
will be driven primarily by loss aversion and
choose the conservative option or by sensation
seeking and thus be more risk taking.
Our results highlight that experts use more
conservative strategies. This key result is
unexpected since it stands in stark contrast with
previous literature. In domains of expertise that
do not include competition or confrontations,
estimates of risk by experts have been lower
than estimates of risk by laypeople (Slovic, et
al., 1995). Since experts perceive less risk in a
given situation, they usually are more risk
taking than non-experts who are more
conservative because of the higher level of
perceived risk. Assuming that experts have on
average the same level of risk tolerance would
lead naturally to the conclusion that they might
take more risks. Another factor that might
increase risk taking in confrontational situations
is self-confidence (Krueger & Dickson, 1994).
Since experts in various domains demonstrate
high levels of confidence (Shanteau, 1988), we
could reasonably think that the level of risk
would be higher than for laypeople. Our results
suggest otherwise. It appears that, in
confrontational situations, loss aversion
outweighs self-confidence and risk seeking, and
this is more the case with experts than with club
players. We would attribute this counter-
intuitive result to the fact that our study is
strictly ecological. Most studies conducted in
the field of risk and risk taking are laboratory
manipulations of fictitious costs and gains.
Although these studies are informative about the
cognitive processes underpinning the evaluation
of costs, the deciders do not incur any real
penalty in case of losses. The chess players in
our sample were putting their Elo rating at stake
in each game. The reality of a potential loss has
made chess players relatively conservative, thus
showing sensitivity to potential loss.
Paradoxically, chess players’ conservatism
might directly stem from their huge amount of
domain-specific knowledge. The representation
of the situation, which is richer for experts
(Campitelli & Gobet, 2004), drives them to
consider more potential outcomes and thus get a
more accurate representation of uncertainty; this
in turn might generate more loss aversion. A
second factor potentially accounting for the
conservative attitude of many players is the fact
that the situation is confrontational. While most
experiments conducted on risk-taking provide
probabilities of losing, they are set as gambles
where the loss of the decider is not necessarily
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 95
Journal of Expertise / June 2020 / vol. 3, no.2
beneficial to a competitor. In a chess game,
there is a will of the opponent to win, and thus if
nothing is done, the game will be lost. As it
entails a threat, this confrontational situation
might make players more prudent.
Our third hypothesis stating that players
adapt risk to the opponent’s skill level has found
some supportive evidence in the statistical
analysis of the distribution of risky and
conservative strategies; however, as highlighted
above, in the two cases where statistically
significant differences were not observed, the
number of players was drastically smaller than
in the other fourteen conditions. Thus, there is
an issue of statistical power. On the one hand,
statistics do not disprove our views. There is
some, even if fractional, evidence that amateurs
and master players adapt to their opponent when
the Elo rating difference is at its highest.
However, the analysis relies on a relatively
limited number of cases as compared to other
conditions. In several of the other conditions,
the same proportion of conservative strategies
revealed a statistical difference between
conservative and risky strategies favoring the
latter ones. So, on the other hand, we have not
reached the point where evidence is
undisputable. As a consequence, it is not clear
whether there is a definite trend or whether our
data are an artefact reflecting random
fluctuations in the use of conservative strategies.
We have to conclude at this point that the debate
remains open and must be investigated in future
The present study has limitations that should
be kept in mind for a correct interpretation of
the results. An important aspect of the study was
that damage was limited to chess rating. Even
though Elo ratings impact on status and
potential income, they are definitely different
from physical damage and our results are not
predictive, for example, of the levels of risk in
combat situations. Second, the effects sizes are
small; thus, while the results are theoretically
important, their practical implications might be
more limited. A third factor to bear in mind is
that the culture of the domain plays a role in
biasing non-experts’ risk attitudes. In chess,
many chess books value openings that lead to
aggressive, high-risk situations for the first
player (e.g., Levy & Keene, 1976). The first
move 1.e4, which was found risk-seeking in our
study, was even promoted by a world champion
as the best move (Fischer, 1995). This type of
advice and the social pressure that might
accompany it could bias non-experts to take
more risks. As noted by Pleskac and Hertwig
(2014), social norms can influence risk taking,
and chess is certainly a domain where the norm
is to take risks. Fourth, though competitive, non-
expert players might approach the game paying
less attention to the potential outcome, as it
would not impact them as much as it would
impact professional players. Lastly, we would
like to mention that our measure of risk based
on the first move is somewhat arbitrary. Our use
of the first move has the merit of measuring risk
ecologically, but players’ attitude to risk might
be more subtle than a binary choice between
risky and conservative options.
A new picture of risk taking in adversarial
situations emerges. Whereas, in line with
theoretical predictions, chess players are
predominantly risk taking when playing chess,
the results also brought to light the fact that a
significant proportion of players chose the more
conservative option. Such a finding calls for a
revision of risk-taking theories that predict
unconditional loss-averse behaviors. With
respect to the influence of domain-specific
knowledge, we have showed that experts in
adversarial situations are more conservative,
thus suggesting that domain-specific knowledge
increases loss aversion. The extent to which this
finding applies to other adversarial situations is
left undetermined but opens an avenue for
future investigations on the relationship between
domain-specific knowledge and attitude to risk.
In conclusion, experts’ strategic thinking
integrates two factors: domain-specific
information which makes experts more
conservative and the level of expertise of the
opponent which can make experts more risk
Author’s Declarations
The authors declare that there are no personal or
financial conflicts of interest regarding the
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 96
Journal of Expertise / June 2020 / vol. 3, no.2
research in this article.
The authors declare that they conducted the
research reported in this article in accordance with
the Ethical Principlesof the Journal of Expertise.
The authors declare that they are not able to make
the dataset publicly available but are able to
provide it upon request.
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Acceptance: 4 May 2020
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 98
Journal of Expertise / June 2020 / vol. 3, no.2
Appendix 1. Calculation of risk
Risk is calculated as variation around the mean. For a sample of n observations of outcome, risk is calculated
in three steps:
̅ = 𝑥
σ2 = ∑(x-x
̅)2 / n
Risk = σ2
In full agreement with the regulations of the International Chess Federation (Section 11.1 of the FIDE
Laws of Chess [see]), a win is coded as 1, a loss as
0 and a draw as 0.5. Following the same regulations, the expected result is the win probability plus half the
probability of drawing.
Hence an opening move that would lead to 40% wins, 30% draws and 30% losses on a sample of
n = 1,000 games would have a mean expectancy of
M = (.40 1) + (.30 1/2) = .55 thus entailing a risk variance of
σ2R = [(1 .55)2 400 + (.5 .55)2 300 + (0 .55)2 300)] / 1000 = 0.1725 and so,
risk = 0.1725 = 0.415331.
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 99
Journal of Expertise / June 2020 / vol. 3, no.2
Appendix 2: Zero-Sum Game and Distribution of Payoffs in the Sample
Chess is a zero-sum game, so the gain of one player is the loss of the other. For example, let us consider
player A, rated 1800 Elo, playing against player B, rated 2200 Elo. By following the formulas provided
by Elo (1978) and rounding numbers to the nearest integer, we obtain the following information: Player
A has a 9% chance of winning 18 Elo and a 91% chance of losing 2 Elo. In case of a win, Player A wins
18 Elo and Player B loses 18 Elo. In case of loss, Player A loses 2 Elo and Player B wins 2 Elo. In both
cases, the loss of a player equals the gain of the other, but the stakes differ according to the outcome of
the game.
In our sample of 73,431 games, the combination of four levels of skill for the first player and four
levels of skill of the second player generates 16 conditions. The probabilities of winning and losing for
the first player are depicted in Figure A1. The probability distribution of winning as calculated from the
data is in dark green and the probability distribution of losing in light green. Figure A1 shows that the
probabilities for the win and loss prospects are mirror images of one another as they naturally sum to 1.
Figure A2 depicts the distribution of payoff in Elo points in case of gains (dark blue) and loss (light
blue). Density for payoffs exactly shows the distribution of potential gains and losses for all 73,341
games. Taken together, Figures A1 and A2 depict the prospects that the players were facing.
In all games, following the formulas provided by Elo (1978), we found that the added expected
utilities for each outcome cancelled each other, confirming that chess is a zero-sum game.
Figure A1. Probability distribution of winning (dark green) and losing (light green) for the first player.
Chassy and Gobet (2020) Expertise and Risk in Adversarial Situations 100
Journal of Expertise / June 2020 / vol. 3, no.2
Figure A2. Distribution of payoffs as a function of the skill of the first and second players
for both winning (dark blue) and losing (light blue), from the first player standpoint.
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"This is the classic work upon which modern-day game theory is based. What began more than sixty years ago as a modest proposal that a mathematician and an economist write a short paper together blossomed, in 1944, when Princeton University Press published Theory of Games and Economic Behavior. In it, John von Neumann and Oskar Morgenstern conceived a groundbreaking mathematical theory of economic and social organization, based on a theory of games of strategy. Not only would this revolutionize economics, but the entirely new field of scientific inquiry it yielded--game theory--has since been widely used to analyze a host of real-world phenomena from arms races to optimal policy choices of presidential candidates, from vaccination policy to major league baseball salary negotiations. And it is today established throughout both the social sciences and a wide range of other sciences. This sixtieth anniversary edition includes not only the original text but also an introduction by Harold Kuhn, an afterword by Ariel Rubinstein, and reviews and articles on the book that appeared at the time of its original publication in the New York Times, tthe American Economic Review, and a variety of other publications. Together, these writings provide readers a matchless opportunity to more fully appreciate a work whose influence will yet resound for generations to come.