Context effects in games: Local versus global sequential effects on choice in the prisoner's dilemma game

Judgment and Decision Making, Vol. 2, No. 5, December 2007, pp. 380–389
Context effects in games: Local versus global sequential effects on
choice in the prisoner’s dilemma game
Ivo Vlaev? and Nick Chater
Department of Psychology
University College London
Abstract
We report an experiment exploring sequential context effects on strategy choices in one-shot Prisoner’s Dilemma (PD)
game. Rapoport and Chammah (1965) have shown that some PDs are cooperative and lead to high cooperation rate,
whereas others are uncooperative. Participants played very cooperative and very uncooperative games, against anony-
mous partners. The order in which these games were played affected their cooperation rate by producing perceptual
contrast, which appeared only between the trials, but not between two separate sequences of games. These ?ndings sug-
gest that people may not have stable perceptions of absolute cooperativeness. Instead, they judge the cooperativeness of
each fresh game only in relation to the previous game. The observed effects suggest that the principles underlying judg-
ments about highly abstract magnitudes such as cooperativeness may be similar to principles governing the perception
of sensory magnitudes.
Keywords: decision making; cooperation; prisoner’s dilemma; context effects.
1 Introduction
been developed, in which prospects are judged in re-
lation to one another, such as the stochastic difference
Most applications of standard normative models, whether
model (González-Vallejo, 2002), multialternative deci-
of individual or strategic decision making, make the ba-
sion ?eld theory (Roe, Busemeyer, & Townsend, 2001),
sic assumption that each risky prospect or game is con-
the componential-context model (Tversky & Simonson,
sidered separately and the resulting choice should be
1993) and decision-by-sampling (Stewart, Chater, &
based only on the attributes of the particular prospect or
Brown, 2006). These theories all have in common the
game (Fudenberg & Tirole, 1991; Kreps, 1990). The
idea the mere presence of an option in a choice set
validity of this, and related, independence assumptions
may change the way another option is judged; or, more
has been challenged extensively in the past especially
broadly, that preferences are constructed afresh in the
in the context of individual decision making under risk.
light of the salient options in each situation or the recent
Thus, Allais (1953) ?rst demonstrated behavior violat-
past. Thus, preference is constructed rather than revealed
ing the independence axiom of expected utility theory
(see Slovic, 1995).
(here the independence is between mutually exclusive
possible outcomes). Later on, regret theory (Loomes &
In an attempt to ground this constructivist idea onto
Sugden, 1982) showed how regret can modify the util-
some fundamental properties of the perceptual system,
ity of an outcome that results from a particular choice
Stewart, Chater, Stott, and Reimers (2003) describe a
depending on the outcomes that would have resulted
phenomenon called prospect relativity: that the perceived
from other choices in the choice set. Recent psycho-
value of a risky prospect (e.g., “p chance of x”) is rela-
logical theories of individual decision making have also
tive to other prospects with which it is presented. Sim-
ilar effects were also found in ?nancial (saving and in-
?This work was conducted while Ivo Vlaev was a doctoral student at
vestment) decision making under risk (Vlaev, Chater, &
the Department of Experimental Psychology, University of Oxford. We
Stewart, 2007a; 2007b). These prospect relativity effects
thank Neil Stewart for his comments and Daniel Zizzo for providing the
experimental economics laboratory at the Department of Economics,
are counter to expected utility theory, the basic norma-
University of Oxford. We would also like to thank two anonymous ref-
tive principle for individual choice, which assumes that
erees and especially the editor of this journal, Jonathan Baron, for work-
the perceived value of each prospect should be depen-
ing out the details of the regression analyses and other invaluable sug-
dent only on its own attributes. Stewart et al. (2003)
gestions. Addressed: Ivo Vlaev, Department of Psychology, University
College London, 26 Bedford Way, London, WC1H 0AP, United King-
suggested that this phenomenon arises because of way
dom. Email: i.vlaev@ucl.ac.uk, n.chater@ucl.ac.uk
in which the magnitudes that de?ne the prospects are de-
380

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
381
?ned, and that the phenomenon has a common origin with
sion making. Therefore, we focus here on testing only
similar psychophysical effects on perception of sensory
(perceptual types of) sequential effects in game playing
magnitudes like brightness and loudness (Garner, 1954;
(which are typically played one at a time and hence se-
Laming, 1997; Lockhead, 1995).
quential in nature). We were particularly interested in
investigating whether the sequential context effects in
games are similar to the context effects observed in per-
1.1 Sequential vs. simultaneous context
ceptual judgment tasks like magnitude estimation (Stew-
Stewart et al. (2003) also found that only the simulta-
art, Brown, & Chater, 2002; Ward & Lockhead, 1970,
neously considered choice options affect the decisions
1971).
about risky prospects, without ?nding evidence for se-
In previous research, we have already found that the
quential effects. Birnbaum (1992), however, demon-
attributes of the previously seen games in?uenced the de-
strated such sequential effects by showing that skewing
cisions in the current game (Vlaev & Chater, 2006). In
the distribution of certainty equivalents offered for sim-
particular, we investigated choice and predictions about
ple gambles, whilst holding the range constant, in?u-
the choices of other players in PD game during one-shot
enced the selection of a certainty equivalent. (In making
plays with different, anonymous opponents on each trial.
a certainty equivalent judgment, participants select from
Participants played a sequence of games with varying de-
a set of options the amount of money for certain that is
grees of cooperativeness, as measured by Rapoport and
worth the same to them as a single chance to play the
Chammah’s (1965) Cooperation Index (described below).
prospect.) When the certainty equivalent options were
The cooperativeness of the games in each condition was
positively skewed (i.e., more small values), gambles were
varied and the results demonstrate that the average co-
overvalued compared to the negatively skewed context,
operation rate and the predicted cooperation of the other
consistent with range-frequency theory (Parducci, 1965,
player in each game strongly depended on the range and
1974).
the skew of the distribution of the cooperativeness of the
There has been a wealth of publications on various
preceding games. That is, we found that the representa-
other types of sequential effects on decision making. In
tion of the cooperativeness of the current game depends
repeated games, Rapoport, Stein, Parco, and Nicholas
on the distribution of cooperation indices of games that
(2003) showed the effect of the outcomes of whole series
the player has previously played. In particular, people
of previous games. Knez and Camerer (2000) showed
tend to contrast the current game with the other games
that cooperation in social dilemmas can be increased
in the sequence depending on the position of the current
by preceding play of coordination games. Thus they
game in the range, and also the rank order of the previous
demonstrated that cooperation in one game spills over
games on the cooperation index scale.
into cooperation in Prisoners’ Dilemma game. Mellers,
Schwartz, Ho, and Ritov (1997) report the effect of se-
1.2 Local vs. global context
ries of gains vs. series of losses vs. a mixture. Oth-
ers have exhaustively analyzed the dynamics of learning
In the study reported here, we again investigated how the
over series of games (Erev & Roth, 1998; Camerer &
cooperativeness of previous games in?uenced decisions
Ho, 1999). Thaler, Tversky, Kahneman, and Schwartz
in PD game, but this time we compared the impact of
(1997) have clearly shown the impact of trends (series of
global vs. local sequential context. That is, suppose that
gains vs. series of losses) in investors’ decision-making.
a person plays a long sequence of highly uncooperative
For example, Loewenstein and Thaler (1989), Loewen-
games. During this sequence, players are likely to de-
stein and Prelec (1992), and Frederick, Loewenstein, and
fect. How will they react on suddenly encountering more
O’Donoghue (2002) have carried out extensive research
cooperative games? If past history is the main factor de-
on choice over time, showing large and complex effects
termining behavior, they might be expected to continue
of short-term and long-term time-contexts. Other stud-
defect and to expect defection of their opponents. A past
ies have also shown that people frequently cooperate in
history of uncooperative games may have created “cyni-
scenarios which appear to have a one-shot Prisoners’
cism” in the face of a more cooperative game. However,
Dilemma structure (see, e.g., Dawes & Thaler, 1988).
if cooperativeness is represented in the same way as sen-
However, none of these studies on history dependence
sory magnitudes, the opposite possibility arises — that
(apart from Birnbaum, 1992, and Stewart et al., 2003)
the new more cooperative games are viewed as “espe-
used theoretical accounts based on some theory of the
cially” cooperative, in contrast to the prior uncooperative
fundamental perceptual mechanisms. Also, few existing
games. Hence, players might be expected to cooperate
studies have investigated whether perceptual context ef-
more. Thus, instead of cynicism, the player is “grate-
fects like prospect relativity also hold during choice under
ful for small mercies,” after a history of uncooperative
uncertainty in the context of interactive (strategic) deci-
games.

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
382
Player 2
observed in magnitude estimation tasks when the range
Cooperate Defect
of the stimuli is increased, or when stimuli are presented
in a larger set versus in isolation, is not perceptual but
Cooperate
C,C
S,T
Player 1
judgmental. In particular, models of absolute identi?ca-
Defect
T,S
D,D
tion, which assume that the locus of the limit in informa-
tion transmitted is perceptual, fail to predict (or require
Figure 1: Model of the Prisoner’s Dilemma Game.
modi?cation to predict) that channel capacity will remain
severely limited even for very large stimulus spacings.
In the RJM, the limit in channel capacity is not percep-
Our previous work (Vlaev & Chater, 2006) has shown
tual. Stewart, Brown, and Chater (2005) show that using
that such contrast effects can occur, when games of differ-
difference information optimally within a limited capac-
ent levels of cooperativeness are mixed together. But can
ity provides an account of the limit in information trans-
the effect of contrast (i.e., viewing a moderately cooper-
mitted. In summary, in the assumed absence of stable,
ative game as more cooperative, given a history of less
long-term absolute magnitudes, the representation of the
cooperative games) overcome the cynicism that might be
difference between the stimulus on the current trial and
expected to follow from a history of negative feedback?
the stimulus on the preceding trial is used in conjunction
A further question concerns how the effects of percep-
with the feedback from the previous trial to produce a re-
tual contrast affect an entire sequence of future games;
sponse.
our previous work focussed on the impact of context on
This new theoretical framework has yet to be explored
a single game. If effects of past games are highly lo-
in the domain of strategic games. If the principles under-
cal, then although we might observe context effects when
lying judgments about abstract magnitudes such as “co-
there is a switch between, say, less and more coopera-
operativeness” are similar to principles governing the per-
tive games, this effect will rapidly disappear. If, though,
ception of sensory magnitudes (as demonstrated by Vlaev
context effects are long-lasting, then the cooperativeness
& Chater, 2006), then it is very likely that the predic-
of previous games may have a substantial in?uence on
tions of the relative judgment model will also hold for
a long sequence of future play. Note that a substantial
strategic decision making in games. In particular, we do
in?uence here constitutes a global contrast effect that is
not expect to ?nd signi?cant long lasting (global) effects
larger than the local contrast effect, or, at least a statisti-
between whole sequences of games (i.e., a lack of last-
cally signi?cant effect on a whole sequence of future play
ing effects of stimuli across longer time periods). Instead
(i.e., in terms of cooperation rate).
we expect to ?nd only local context effects between cur-
With psychophysical stimuli, local effects appear
rent and previous games in a sequence (conforming to the
strongest — indeed a good deal of performance in ab-
RJM proposed by Stewart, Brown & Chater, 2005).
solute judgment tasks can be explained with reference to
In this experiment, similarly to the Vlaev and Chater
only the previous two stimuli as shown recently by Stew-
(2006) study, the stimulus context was assumed to be
art, Brown and Chater (2005). In particular, previous psy-
the cooperativeness of the previously played games in
chophysical models assume that identi?cation is achieved
the sequence, while the dependent variable was the par-
using long-term representations of absolute magnitudes.
ticipants’ cooperation rate. Games’ cooperativeness was
Stewart, Brown and Chater (2005) propose an alternative
expected to be judged similarly to other perceptual at-
relative judgment model (RJM) in which the elemental
tributes such as pitch and loudness.
perceptual units are representations of the differences be-
tween current and previous stimuli. These differences are
1.3 The cooperation index scale
used to respond without using long-term representations
of absolute magnitudes. The logic of RJM is based on
Figure 1 illustrates the structure of PD, which is the sub-
two main assumptions. The ?rst assumption is that judg-
ject of a vast literature in economics, behavioral decision
ment is relative and not absolute, which is based on abun-
making, and cognitive and social psychology. The game
dant evidence that long-term representation of attributes
is de?ned by the chain of inequalities T > C > D > S,
such as pitch and loudness may be very poor (see Stewart,
where C is the payoff if both cooperate, D is the payoff if
Brown, & Chater, 2005, for an extensive review of this
both defect, T is the payoff if one player defects and the
evidence). Models that use long-term representations of
other cooperates (often called the temptation payoff), S is
absolute magnitudes of stimulus values (as either exem-
the payoff if one player cooperates and the other defects
plars, anchors, or criteria) do not capture the sequential
(and this payoff is often called the sucker payoff).
effects adequately. In the RJM, judgment is instead rela-
In order to manipulate the cooperativeness of the
tive to the immediately preceding stimulus. The second
games in each session we used a measure developed
assumption is that the locus of the limit in performance
by Rapoport and Chammah (1965), who investigated

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
383
Other
Table 1: Prisoner’s Dilemma games used in the study.
Cooperate Defect
Cooperate
10,10
0,11
Outcomes
You
Defect
11,0
1,1
Game’s CI
CC
ST
TS
DD
Figure 2: Game matrix of a Prisoner’s Dilemma game
.1
?5,?5
?11, 0
0,?11
?6,?6
with a cooperation index .8.
.1 (x4)
?20,?20
?44, 0
0,?44
?24,?24
.1 (x7)
?35,?35
?77, 0
0,?77
?42,?42
Other
.1 (x10)
?50,?50
?110, 0
0,?110
?60,?60
Cooperate Defect
.8
10, 10
0, 11
11, 0
1, 1
Cooperate
?5,?5
?11,0
You
.8 (x4)
40, 40
0, 44
44, 0
4, 4
Defect
0,?11
?6,?6
.8 (x7)
70, 70
0, 77
77, 0
7, 7
Figure 3: Game matrix of a Prisoner’s Dilemma game
.8 (x10)
100, 100
0, 110
110, 0
10, 10
with a cooperation index .1.
Note. The payoffs in each cell of the table are as indicated
in the cells of the game matrix shown in Figure 1.
whether certain structural properties of the game will af-
fect people’s propensity to cooperate. They derived a co-
operation index (CI) for predicting the probability that
people will cooperate, de?ned by the ratio: (C - D)/(T -
because people compare and contrast the current game
S).
only with the previous one, or whether these effects can
In principle, the CI ranges from 0 to 1, where CIs close
also appear on a larger scale when there is (implicit) com-
to 1 characterise games in which cooperation is probable;
parison between two separate sequences of games. For
and CIs close to 0 characterise games in which defection
this study, there were only two types of games, very co-
is probable. For example, Figure 2 shows a game with
operative ones with index .8 and very uncooperative ones
a high cooperation index for which T = 11 > C = 10 >
with index .1, and we tested whether the order in which
D = 1 > S = 0 (see Figure 2). Thus, CI is (10–1)/11–0
these games were played affected people’s choices and
= 9/11, which is approximately .8. By contrast, Figure 3
predictions.
shows an uncooperative game, in which T = 0 > C = –5
In order to further accentuate the difference between
> D = –6 > S = –11. Thus, CI is (–5–(–6))/(–5–(–6)) =
the two game types, and hence to maximize the impact
1/11, which is approximately .1. In a seminal experimen-
of context effects, the uncooperative games were given
tal study, Rapoport and Chammah (1965) demonstrated
negative payoffs. Thus we expected to provoke stronger
a roughly linear relationship between the cooperation in-
perceptual dissociation between the cooperative and un-
dex and the cooperation rate: people tended to cooperate
cooperative (positive and negative) games, which could
more when playing games with a higher index.
further enforce the contrast between them (although their
In the experiment presented here, we used these two
strategic structure is identical). There were three order
levels of the CI in order to test whether participants’ co-
conditions in this study. For one group, the cooperative
operation rate strongly depend on the cooperativeness of
(positive) games were played ?rst and then the uncoop-
the preceding games. Thus, the present research departs
erative (negative) ones; for the other, the uncooperative
fundamentally from previous work in game theoretic de-
(negative) games were played ?rst and then the cooper-
cision making by trying to model the highly ?exible and
ative (positive) ones. These two conditions were testing
contextually variable way in which people represent mag-
whether the perceptual context effects appear globally be-
nitudes such as cooperativeness (and payoffs and proba-
tween two whole sequences of rounds. A third group of
bilities in this respect), rather than assuming that these
participants played a random mixture of the two game
attributes can be represented on stable internal psycho-
types where the perceptual context effects were expected
logical scales.
to appear locally between two neighboring rounds, i.e.,
on a round-by-round basis as predicted by the relative
judgment model (Stewart, Brown, & Chater, 2005). In
2 Experiment
the following sections, these ordering conditions are de-
noted in terms of the cooperativeness of the games in the
In this experiment, we decided to test whether the sequen-
sequence as High-Low, Low-High, and Mixed condition,
tial context effects caused by the previous games emerge
respectively.

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
384
2.1 Method
the nature of the argument we make here about the per-
ceptual context sensitivity in strategic decision making
2.1.1 Participants
that should, according to normative principles, be purely
Twenty participants took part in each of the three con-
dependent on the current trial.
ditions (groups) of this study (so there were 60 partic-
ipants in total) recruited from the University of Oxford
2.1.3 Procedure
student population via the experimental economics re-
search group mailing list of people who have asked to
The experiment was conducted interactively in separate
be contacted participated in this experiment. All partici-
groups for each condition. In the conditions with separate
pants were paid £3 plus performance related winnings of
sequence for the positive and the negative games, the par-
up to £3.
ticipants played each game type separately for 48 rounds
(i.e., 96 rounds in total). In the Mixed condition, the game
types were presented in a random order, which was the
2.1.2 Design
same for everyone. The participants were informed that
There were two types of PD game in this experiment –
they would play 96 rounds of the game and on each round
very cooperative game with cooperation index .8 and very
of the game they would play against a randomly selected
uncooperative game with cooperation index .1. Figure 2
player from their group. This random matching aimed to
presents a game matrix of a very cooperative game with
make it impossible to infer the strategy of the other player
index .8.
from the history of the game, and thus to control for pos-
In order to make the payoffs to be of comparable mag-
sible learning affects during the play. Thus we also aimed
nitudes in the two game types and therefore of equal im-
to prevent people from learning a model of their oppo-
portance for the players, we decided that the uncoopera-
nent, which is another signi?cant contextual factor that
tive game (offering negative payoffs) should offer simi-
has been shown to affect strategic behavior (see Pruitt &
lar absolute amounts as the cooperative game. Figure 3
Kimmel, 1977, for a review).
presents the matrix of the uncooperative PD game with
Each condition consisted of a sequence of rounds of
index .1, which offers negative payoffs (i.e., the strategic
PD game in which players make their choices simulta-
structure of the game remains unchanged).
neously. On each round of the game the participants
In order to control for the effects related to the abso-
were presented with a matrix of the game on the com-
lute magnitude of the received payoff from each round,
puter screen and they had to choose their decision strat-
the initial payoffs of each game, which were between 0
egy (1 or 2). We used the abstract label “1” to denote the
and 20, were multiplied by factors of 1, 4, 7 and 10, so
cooperative response and “2” for the uncooperative one
?nally there were four versions of each game index in
in order not to prime certain social values in the group,
terms of the magnitudes of the payoffs. Table 1 presents
which might induce certain strategies that could addi-
the eight games used in this study. Each game type (CI)
tionally bias the results. After both players in each pair
is presented as a separate row in the table, and each cell
have made their decisions the round ended and they were
presents the outcome payoffs for the two players. The
informed on the screen about the decision made by the
cells and the payoffs were organized according to the
other player, and their received payoffs from the game.
cells of the abstract game matrix shown in Figure 1.
Thus, the participants were paid for their participation in
The cooperative games involved only positive payoffs
cash according to their performance. At the end of the ex-
and the uncooperative games included negative payoffs
periment, the accumulated score in points was transferred
in order to make the distinction between the games very
into cash according to an exchange rate.
explicit, and thus to maximize any contrast effects that
might affect participants’ choices. In this respect, our
2.2 Results
goal was to demonstrate with this experiment the very
existence and the power of the effect, which from a nor-
The cooperation rates were averaged for every partic-
mative point of view should not exist. In other words,
ipant separately over each game type in every condi-
we aimed to demonstrate a game type effects without dis-
tion and these averaged results are presented in Figure
tinguishing those two factors (cooperativeness vs. pay-
4. In the Low-High condition the average cooperation
off sign). Of course, other versions of the design could
rate changed in the second half of the session after the
have cooperative games with negative payoffs and non-
participants started to play the positive games (?rst play-
cooperative ones with positive payoffs, or all games to be
ing the negative ones), jumping from 18% to 50%. In the
either positive or all negative, and it would be interesting
High-Low condition, the cooperation rate for the positive
to explore these versions in future research. Considera-
games (played ?rst) is 33% and drops down to 18% in the
tion of these further cases, however, should not change
second half when people start to play the negative games.

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
385
Figure 4: Cooperation rate for each game type in the three experimental conditions. (Error bars are standard error of
the mean.)
In the Mixed condition, the cooperation rate was 71% for
condition, would also result in a signi?cant interaction
the positive games and 18% for the negative ones. This
effect. The lack of statistical effect, however, suggests
result indicates that in all conditions the participants re-
that, although people were sensitive to the cooperative-
acted to the change from negative to positive games and
ness of the two game types, their cooperativeness was not
vice versa. The average cooperation in the positive games
affected by the change from positive to negative and neg-
in the Mixed condition was signi?cantly higher than the
ative to positive games respectively. In other words, this
cooperation in the positive games in both the High-Low
is an indicator of a lack of global context effect.
condition, t(38) = 4.65, p < .0001, and the Low-High con-
When we added the Mixed condition as a third
dition, t(38) = 2.31, p = .0264. This result is evidence
between-subject factor, then there was signi?cant effect
for the power of the contrast effect when the games are
of both the game type, F(1,57) = 55.5, p < .0001, and
mixed between trials and hence the comparison is on trial
the interaction between the game type and the experi-
by trial basis.
mental condition, F(2,57) = 5.91, p = .0046. This can
The cooperation rates were averaged for every partici-
only be explained by a very strong local contrast effect
pant separately over each game index and were analyzed
on the positive games in the Mixed condition (the neg-
in a repeated-measures analysis of variance with game
ative games have the same cooperation level as in the
type (positive vs. negative) as a within-subjects factor and
other conditions), which is due to the local trial-to-trial
the experimental condition (High-Low vs. Low-High) as
context effect between the games in this condition. We
a between-subjects factor. There was a signi?cant effect
also tested Mixed condition versus a combination of the
of the within-subject factor game type (indicating that
other two conditions (i.e., “blocked”). Thus, instead of
there was signi?cant change in cooperation from posi-
having three levels of the factor condition (H-L, L-H, and
tive to negative games and vice versa), F(1,38) = 16.5,
Mixed) we run an analysis with only two levels (H-L&L-
p < .0001. However, the interaction between the game
H vs. Mixed). Again, there was signi?cant effect of the
type and the experimental condition (the between-subject
interaction between the game type (H vs. L) and the ex-
factor) was not statistically signi?cant, F(1,38) = 2.12, p
perimental condition, F(1,58) = 9.22, p = .0036, which
= .154. If there is a signi?cant global context effect be-
corroborated our initial conclusion.
tween the two sequences of rounds (i.e., Low-High vs.
If context effects exist and are only transitory, one
High-Low condition), then there should be signi?cant in-
should also observe a lack of global effect by analyzing
teraction effect due to effect of the condition on cooper-
the time series of choices. To test this hypothesis, we
ation in the two game types. Note that a signi?cant con-
run two extra regression analyses of trends — one in the
trast effect on the positive games, which is implied by the
Mixed condition and second in the two other conditions
higher cooperation in the positive games in the Low-High
(Low-High vs. High-Low).

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
386
2.2.1 Mixed condition trends
going on in this complex strategic interaction.)
The trend analysis in the Mixed condition also demon-
We looked directly at sequential effects in the Mixed con-
strated that our experiment has the power to detect dif-
dition by regressing, for each subject, the response to
ferences that we did not ?nd within the conditions (con-
each round on the type of the previous round, the one be-
versely, the failure to ?nd signi?cant contrast effect here
fore that, and so on. Finding greater coef?cients for more
is not due to lack of statistical power, since the experi-
recent rounds would support our conclusion. For exam-
ment did detect the context effect at the aggregate level
ple, round n-3 was expected to have little effect, because
of the results). Therefore, this trend analysis cannot re-
its effect would be diluted by rounds n-2 and n-1. We run
ject our hypothesis that subjects are evaluating the cur-
a separate regression for each subject, and then averaged
rent payoffs (games) by comparing them to the context
the coef?cients across all participants in this condition.
set up by very recent payoffs. The next analysis aimed to
We used a one-sample t-test to test whether the average
support this hypothesis.
coef?cients (for each factor) differ from zero (indicating
the average size of the effect). However, the coef?cients
2.2.2 Low-High vs. High-Low trends
at t, t-1, and t-2 were not signi?cantly higher than zero
(inclusion of past choices as predictors did not change the
If context effects are only transitory, we should see this
results). This result suggests that there is no context ef-
effect by looking at the time series of choices in these
fect within the mixed condition, while our model predicts
two conditions too. In particular, a difference between
negative (contrast) effect. However, another interpreta-
Low-High and High-Low would exist, but only at early
tion is that, because of the contrast between the two game
rounds after the conditions have switched within subjects.
types, the subjects quickly started to coordinate their re-
To test this hypothesis, we ?t a model to each subject
sponses in the ?rst few rounds (in this condition). Note
in the Low-High and High-Low conditions respectively
that PD can be seen as a coordination game. Any change
with the following components:
in the environment that changes the ease with which co-
a: general linear trend over the 96 trials to capture the
ordination can occur can affect cooperation frequencies.
general effect of repetition;
It seems likely that coordination was easier in the Mixed
b: dummy variable for condition, High vs. Low (allow-
condition. For example, the simple rule “cooperate in
ing that the condition does matter, although appar-
the positive payoff game, defect otherwise” could easily
ently it doesn’t);
emerge. This rule cannot easily emerge in cases where
c: dummy variable for the ?rst 2 rounds in each block
only one type of game is repeatedly played. Thus, dif-
of 48 (after the transition) in order to capture the ini-
ferences in ease of learning to coordinate between treat-
tial learning period;
ments could explain lack of effect of previous rounds on
d: dummy variable for the ?rst 6 rounds in each block
the current choice in the Mixed condition (while such co-
of 48 rounds after the transition in order to capture
ordination can still be caused by the local contrast in the
immediate context effects (in each condition).
?rst rounds of the session).
This “coordination” hypothesis would imply a positive
The issue was whether b (global context effect) and d (lo-
interaction between the current game and trials. That is,
cal context effect) depend on the order of the conditions
the current game (at t) would have more effect as the sub-
(High, Low). The idea here was to remove the effect
jects learned. To answer this question, we split the data
of general changes over time by modeling those changes
into two halves (rounds 1–48, and rounds 49–96) and re-
within each block of 48 trials (but removing the ?rst few
gressed the current choice only on the current game in
trials in that block). That is, if there is only a local context
each half. Higher coef?cient for the current game in the
effect, then d in Low-High should be signi?cantly higher
second half would indicate that the subjects were learn-
than in High-Low, but no signi?cant difference should be
ing to conditionalise more on the game (which imply less
observed between b across the two conditions.
contrast effect of the previous game). Indeed, the average
Indeed, we observed such pattern: b(High-Low) = 1.34
coef?cients for current game in the ?rst half was 9.98,
and B (Low-High) = 2.91, but this difference was not
which is lower than in the second half (18.4) by the fac-
statistically signi?cant, t(38) = 0.47, p = .6383; how-
tor of two and this difference was marginally signi?cant,
ever, d(High-Low) = -5.93 was signi?cantly higher than
t(19) = 2.00, p = .0604. This result suggests that players
d(Low-High) = 2.78, t(38) = 2.47, p = .0182. (The other
learn to coordinate as the game progresses, which washes
two parameters, a and c, did not differ signi?cantly across
out the contrast effect with the previous game. This result
the two conditions.)
could also explain the sustained strong contrast between
These results indicate that the transition effect is sig-
the two game-types observed in this condition. (In this
ni?cant only in the very ?rst few rounds (i.e., showing
respect, we never claimed that there are no other effects
a temporary contrast effect in shifting from high to low

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
387
or low to high), while there is no global effect of con-
the stable ˜18% cooperation rate across all conditions is
text (the context effect seems to disappear after the ?rst
not suf?cient to rule out the signi?cant context effect ob-
six rounds). This is in line with the relative judgment
served in the Mixed condition.
model (Stewart, Brown, & Chater, 2005), in which ef-
fects from previous rounds t-2 and t-3 are allowed to have
3.1 Theoretical accounts
some residual effect, but to a much lesser degree. (Note
that currently this is an intensely debated issue in the ab-
Our results demonstrate strong evidence that people (ex-
solute identi?cation literature: see Stewart, 2007.) This
plicitly or implicitly) compare the games between the tri-
regression evidence also brings additional support for our
als, but the data do not indicate a long term representa-
hypothesis about the contextual locality of the effects,
tion of previous game sequences. This result is in line
which also supported by the aggregate level analysis and
with the predictions of the (psychophysical) relative judg-
the weak learning trends in the two control conditions.
ment model (Stewart, Brown, & Chater, 2005), which
postulates that judgments are based the differences be-
tween current and previous stimuli without using long-
3 General discussion
term representations of absolute magnitudes. Thus, we
also demonstrated that variation in local context would
The results clearly show sequential local context effects
have a bigger effect on judgment than global context,
on choice in strategic game playing, which coordinated
which is a main prediction of the model, and has not been
responses and created a long-lasting increase in cooper-
tested before in the context of strategic interaction (Vlaev
ation in the Mixed condition where the contrast between
& Chater, 2006, do not differentiate between such local
the two game types interleaved on a trial by trial basis.
and global context effects).
When the games were divided in two separate sequences,
In order to provide an account of the results presented
the negative games in the ?rst half of the session did not
in this article, we could assume that people are unable
produce a long-lasting contrast effect with the positive
to make reliable judgements of absolute cooperativeness,
games in the second half. As a consequence, the latter
because they do not have direct access to internal repre-
were no more cooperative than normal. Also the coop-
sentation and information about absolute magnitudes in
erative games in the ?rst half of the session did not pro-
the environment. Instead, they always need some refer-
duce any contrast effect on the uncooperative games in
ence standard either retrieved from memory, or existing
the second half. Thus, we demonstrated that sequential
in the current environment, with which to compare the
context effects arise, given variations in cooperativeness
current game. Such reference standard in our study is
of PD games, only as local context effects, because peo-
provided by the previous games that have different co-
ple compare the current game with the previous game and
operativeness. This explanation is further motivated by
quickly learn to coordinate their behaviour depending on
a new theory of decision making developed by Stewart,
the game played; but there are no signi?cant global con-
Chater, and Brown (2006), in which, in contrast with tra-
trast effects, when the comparison is between entire se-
ditional models, there are no underlying psychoeconomic
quences of games. Another important result was the fact
scales. Instead, consistent with some interpretations of
that the average cooperation in the positive games in the
psychophysical data (e.g., Laming, 1997), this model as-
Mixed condition was signi?cantly higher than the posi-
sumes that an attribute’s subjective value is based only on
tive games in both the High-Low condition and the Low-
its rank within a small sample of attribute values drawn
High condition. This result could be explained by the
from memory or from the current decision context.
presence of a very powerful contrast (between the two
Another plausible account of the contrast effects
game types) operating on a trial by trial basis, which en-
caused by the games in the sequence could be some
abled the participants to quickly learn cooperate in the
relativistic theory of human judgement like the range-
positive (cooperative) games.
frequency theory proposed by Parducci (1965, 1974),
The average cooperation in the negative games in all
which claims that the subjective value given to an at-
three conditions was around 18%, indicating that the con-
tribute is a function of its position within the overall range
text affected only the positive games. This asymmetry of
of attributes, and its rank. Therefore, this model implies
the context effect suggests that there is some default min-
that games’ attributes (e.g., cooperativeness) are judged
imum cooperation level which cannot be reduced. This
purely in relation to one another and their subjective value
result is an indicator of a ?oor effect (i.e., cooperation
is independent of their absolute value (de?ned by the co-
cannot go lower than certain level), which is most likely
operation index in the case of PD games).
due to the documented existence of so-called uncondi-
Vlaev and Chater (2006) already demonstrated a con-
tional cooperators, who tend to cooperate no matter what
text effects in games, which could be explained by such
(as Kurzban & Houser, 2001, seemed to ?nd). Therefore
relativistic theoretical frameworks. The results presented

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Judgment and Decision Making, Vol. 2, No. 6, December 2007
Context effect in games
388
in this article elaborate these ?ndings by showing for ?rst
plained by the relative judgment model (Stewart, Brown,
time that the source of the context effect is a compari-
& Chater, 2005).
son of the current game with the immediately preced-
Our results also imply that games are not considered
ing games as opposite to comparing the current game
independently of the previously played game. Thus, the
with some amalgam of the games experienced during the
present research demonstrates a new and large anomaly
whole session (the short-lived contrast effect in the Low-
for normative rational theories (like game theory) of
High condition supports this conclusion). Our study was
decision-making under strategic uncertainty. Therefore,
speci?cally motivated by the relative judgment model
the present study demonstrates the extent to which ba-
(Stewart, Brown, & Chater, 2005), which not only as-
sic aspects of human magnitude representation play an
sumes that people have no absolute access to psychophys-
important role in strategic decision making. Our results
ical magnitudes, but that they can only make ordinal
also suggest that any descriptive theory of choice will be
judgments concerning the size of “jumps” between pairs
incomplete without taking into account this fundamental
of magnitudes (this model explains a wide range of abso-
aspect of human cognition.
lute magnitude experiments, including trial-to-trial con-
text effects). Note that other existing psychophysical
models for a wide range of perceptual phenomena like
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