Spontaneous Order

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School of Economics University of East
Anglia Norwich NR4 7TJ, United Kingdom
Spontaneous Order Lecture 2 Salience Robert
Sugden (based on joint work with Nicholas
Bardsley, Judith Mehta and Chris Starmer) Fudan
University, November 2008
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A core problem for theories of spontaneous order
when a game has two or more ESSs, which
emerges? Recall the crossroads game ...
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ESS (L gives way)
q 1
q 0.8
unstable equilibrium
ESS (R gives way)
p 0.8
p 1
dynamics of asymmetrical crossroads
game ppr(slow down right), q pr(slow down
left).
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A major difference between my ERCW and other
evolutionary theories of spontaneous order I
emphasise the role of salience (focal points) in
determining which ESS emerges.
In this lecture, I discuss -- the concept of
salience -- alternative theoretical explanations
of salience -- relevant evidence.
Most of the analysis in this lecture will be
concerned with one-shot games. In later
lectures, I will explain the significance of
salience for the evolution of conventions.
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A pure coordination game (due to Schelling) Two
players, unable to communicate with one another.
Each has instruction Name a place in New York
city and a time of day at which to meet the other
person. Both are rewarded if and only if they
both give the same answer.
In Schellings unscientific sample, almost
everyone chooses 12 noon, a majority choose Grand
Central Station. (Replication under more
controlled conditions Mehta, Starmer and Sugden,
AER 1994).
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How would this game look in game-theoretic
notation?
Consider the name a time part of the
game player Bs strategy
0000 0001 0002 0003
... player As 0000 1, 1 0, 0
0, 0 0, 0 ... strategy 0001
0, 0 1, 1 0, 0 0, 0 ...
0002 0, 0 0, 0 1,
1 0, 0 ... ...
... ... ... ...
...
i.e. (if answers in whole minutes) a 1440 x 1440
game with 1440 completely symmetrical Nash
equilibria. Labels of strategies treated as
irrelevant. So, no way of explaining why players
coordinate with frequency greater than 0.0007
(but in fact, frequency close to 1!)
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Implication labels matter (even though standard
game theory assumes they dont).
Schelling pointed this out in 1960. He argued
that understanding importance of labels was
crucial for many problems of negotiation,
especially Cold War military strategy (e.g.
distinction between nuclear and conventional
weapons, spheres of influence).
Schellings explanation some labels are more
prominent / salient / obvious than others
when people are trying to coordinate, they
focus their expectations on the most salient
label (the focal point).
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Since 1960, most game theorists have recognised
that Schelling is right about how real people
play real coordination games but make very
little use of salience in their theories. Why?
Because they havent been able to integrate
salience into game theory in (what they see as)
an acceptable way.
But there is a minority-taste literature which
tries to explain what salience is and how it
affects players strategy choices. This is what
this lecture is about (and particularly my own
recent experimental work with Bardsley, Mehta and
Starmer, forthcoming in Economic Journal).
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Theory Consider a pure coordination game for 2
players (i.e. each player chooses from a set of n
distinct labels payoffs (1, 1) if both choose
same label, (0, 0) otherwise). E.g. Heads and
Tails. How do players coordinate with
probability gt 1/n? Some candidate explanations
1. Primary salience. Each player just chooses
the first thing to come to mind, but what comes
to mind is correlated across players (common
psychology, common culture, common experience...)
This explanation tested by Mehta et al 1994,
mostly for open-ended tasks (e.g. Name any
colour). Test is to compare picking and
coordinating treatments. Pickers responses
are positively correlated, but coordinators are
much more likely than pickers to choose same
labels.
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2. Secondary salience. Idea first proposed by
Lewis (Convention, 1969) it can be formulated
more rigorously in terms of cognitive hierarchy
theory (Stahl and Wilson, 1995 Camerer, Ho and
Chong, 2004).
A label has secondary salience to the extent
that players believe it to have primary salience
for co-players in general. (Similarly
third-order salience, etc.) E.g. Choose one of
grapefruit juice, apple juice, orange juice,
mango juice. Primary salient for me
grapefruit (comes to mind as my favourite).
Secondary salient for me orange (I guess, the
most common favourite).
We should expect secondary salience to be more
concentrated than primary salience (because of
information which helps to predict what is
primarily salient for others). In principle,
third-order salience could be more concentrated
still, but it seems unlikely that ordinary people
have access to the relevant information.
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The hypothesis people solve coordination games
by choosing secondarily salient
labels. (Cognitive hierarchy version Level 0
players just pick what is primarily salient for
them Level 1 players assume co-players are at
level 0 Level 2 players assume co-players are
at level 0 or level 1, ... )
In Mehta et als (1994) design, secondary
salience hypothesis implies that coordinators
responses should have same mode as pickers. For
most tasks, this was observed. But some
interesting exceptions, e.g. Write down any
positive number pickers mode is 7,
coordinators is 1 (as predicted by Schelling).
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3. Team reasoning (Hodgson, 1967 as applied to
coordination games Sugden, 1991 Bacharach,
1993, 2006 Beyond Individual Choice, Princeton
UP ed. Natalie Gold and Robert Sugden). Basic
ideas
Players choose options under descriptions
i.e. each player responds to the game as she
describes it to herself, not as the game theorist
describes it for modelling purposes.
Players ask Which rule is best for us? (not
What is best for me, given what I can expect you
to do?) Rules are formulated in terms of
descriptions accessible to the players
themselves.
E.g. Write down any positive number. Some
numbers (e.g. 652) are nondescript. Others have
descriptions (e.g. smallest, luckiest, most
popular) with varying degrees of ambiguity and
accessibility. Choose the smallest is
unambiguous and highly accessible.
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Experimental evidence Our experiment aims to
discriminate between cognitive hierarchy theory
and team reasoning (without requiring a full
specification of either theory, since these dont
yet exist). We use two distinct tests
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1. Pure coordination games (text tasks) Design
extends that of Mehta et al (as suggested by an
anonymous Oxford graduate student in the early
1990s). There are three conditions
Pickers are instructed to pick one of a set of
labels. Each picker is paid provided she picks
something the amount is independent of what is
picked.
Guesers are anonymously paired with previous
pickers, and instructed to guess what the picker
chose. They are rewarded for correct guesses.
Coordinators are anonymously paired with one
another, and instructed to try to choose the same
label as their co-players. They are rewarded for
successful coordinations.
Notice that pickers responses reveal primary
salience while guessers responses reveal
secondary salience.
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With this design, we can test
Are randomly-matched coordinators more likely
to give same responses than randomly- matched
guessers? (i.e. are coordinators more successful
than they would have been, had they used
secondary salience?)
Are the distributions of responses different
for coordinators and guessers? (e.g. are the
modes different?)
This test has the potential to disconfirm
cognitive hierarchy theory. Strictly, it cant
disconfirm team reasoning (because secondary
salience might be the best rule) but we can
design our problems to make the credible best
rule diverge from secondary salience. Our
recipe
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Example choose one of Berlin, Calais, Paris,
Prague, Rome
We guess that pickers will go for favourite
destinations, but each of the capitals will
appeal to some people so pickers responses will
be dispersed among the four capitals. The same
will be true for guessers.
Coordinators will see that the rule Choose the
most popular favourite is not likely to be very
successful.
But Calais is the odd one out (not a capital
city, not a tourist destination). So Choose the
odd one out is a better rule.
So, cognitive hierarchy (CH) coordinators will
choose capitals while team reasoning (TR)
coordinators will choose odd ones out. TR
predicts that coordinators are more likely to
match than guessers.
General principle one label stands out but is
not particularly attractive the other labels are
roughly equal in attractiveness.
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2. Nondescript Hi-Lo games (number tasks)
Consider coordination game in which each player
chooses one of four nondescript labels (e.g.
, , , ). Payoff is (9, 9) if
both choose , (10, 10) if both choose the
same one of the other three labels, (0, 0)
otherwise.
CH predicts that level 0 players randomise among
labels (or favour the 10-point labels). So
level 1 players do best to choose 10-point
labels. Similarly for levels 2, 3, .... So,
most players choose 10-point labels.
TR (and Schelling) predicts that coordinators
choose the 9-point label it stands out, even
though the feature by which it stands out (low
payoff) is intrinsically unattractive.
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The experiment
In fact, two experiments using an (almost) common
design, one at University of Amsterdam
(Bardsley), one at University of Nottingham (the
rest of the team). Common features
Subjects recruited from general student
population.
Subjects randomly allocated to pick/guess and
coordination treatments. Each pick/guess subject
faces a set of picking tasks, then guesses how an
anonymous co-player has picked on (different)
guessing tasks. Each coordinator tries to
coordinate with an anonymous co-players from same
group.
Design balanced so that each task is faced in
picking, guessing and coordinating treatments.
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Two types of task
Text tasks choice from a set of objects,
each with a distinct text label all labels carry
10 points. Labels constructed according to our
recipe one label stands out but is not
particularly attractive, the others are roughly
equally attractive (we hope).
Number tasks choice from a set of distinct
but nondescript objects each carries a number of
points, which may differ between objects.
Nondescript labels are same for paired subjects,
but randomised across pairs (see later).
Pickers score the points for the objects they
choose.
Guessers score the points for the object they
guess, if their guess is correct.
Coordinators score the points for the object
they choose, if it is also chosen by the
co-player.
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Payoff is at a constant money rate per point
scored.
No feedback until the end of the experiment.
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Nottingham text task subject ticks one box
(order of boxes randomised between co-players)
Nottingham number task subject ticks one box
(order of boxes randomised between co-players
displays created randomly but are same for paired
subjects)
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Amsterdam text task discs swim about in the
fish tank subject clicks on one. Each disc
has a label and a number of points (10 in every
case).
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Amsterdam number task each disc has a number of
points (not necessarily the same number), but no
other label. Movements of discs are randomised
but are same for paired subjects.
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Results edited highlights
We use normalised coordination index (NCI) as
our main summary statistic. Consider a group of N
individuals, each of whom chooses one option from
the same set of n objects. Given their
responses Coordination index (CI) probability
that two individuals, selected at random, without
replacement, choose the same object. This is a
measure of success in coordinating responses.
Expected value of CI is 1/n if individuals choose
at random.
NCI CI n. This is a measure of success
which controls for the number of objects in the
set. Expected value of NCI is 1 if individuals
choose at random.
We can calculate similar cross-treatment NCIs,
e.g. the probability that a randomly selected
picker and a randomly selected guesser choose the
same object.
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Amsterdam results NCIs for text tasks
pick guess coord pick 1.12 1.17 1.19 guess 1
.26 1.27 coord 1.82
Notice NCI much higher for coordinators (Cs)
than pickers (Ps), as in Mehta et al 1994.
Implication coordinators do more than follow
primary salience.
NCI much higher for Cs than for guessers (Gs).
Implication coordinators do more than follow
secondary salience (i.e. cognitive hierarchy
theory disconfirmed).
Gs are about as likely as Cs to match with Ps
or with other Gs. Implication coordinators are
not just guessing what guessers do (third-order
salience) they seem to be using some other
principle.
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A typical Amsterdam text task (TA2)
pick guess coord Ford 10 11 31 Ferrari 13 22
11 Jaguar 20 7 5 Porsche 10 12
9 NCI 1.04 1.12 1.48 sig wrt pick ns sig
wrt guess chisq wrt guess
P and G responses are widely distributed, Cs
are concentrated on the odd one out (Ford). NCI
for C is significantly greater than for G
(one-tail bootstrap test). Distribution of
responses for C is significantly different from G
(chi squared test).
This is consistent with team reasoning Cs use
the odd one out rule, and are more successful
than if theyd used secondary salience.
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Amsterdam results number tasks
Type 1 best rule for pair (unless very odd
risk attitudes) is to choose the low payoff)
NA1 pick guess coord 10x3 50 48 8 9x1 3
4 48
Pickers and guessers choose high payoffs
coordinators choose the low payoff. This is
consistent with TR and inconsistent with CH.
(Could Cs have used the labels i.e. swimming
patterns as a coordinating device? Test is to
look at number tasks where all objects have 10
points. NCI 1.34 for Cs surprisingly high,
but nowhere near high enough for a best rule.)
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Nottingham results NCIs for text tasks
pick guess coord pick 1.20 1.42 1.47 guess 1
.98 2.07 coord 2.20
Notice NCI much higher for Cs than Ps, as in
Mehta et al 1994. Implication coordinators do
more than follow primary salience.
NCI only slightly higher for Cs than for Gs.
Generally, Cs and Gs seem similar. Implication
coordinators are following secondary salience
(i.e. cognitive hierarchy theory confirmed).
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A typical Nottingham text task (TN3)
pick guess coord Ford 3 4 2 Mercedes
8 11 13 Pontiac 4 1 0 Porsche 21 29 26 Volksw
agen 9 0 3 NCI 1.43 2.36 2.15 sig wrt
pick sig wrt guess chisq wrt guess ns
P responses are quite concentrated on Porsche
(a credible favourite). Gs are much more
concentrated on Porsche (i.e. guessers are good
at guessing). No apparent difference between Gs
and Cs.
This is consistent with CH. The intended
stander-out (Ford as generic car) was not used
by Cs.
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Nottingham results number tasks
Type 1 best rule for pair (unless very odd
risk attitudes) is to choose the low payoff)
NA1 pick guess coord 10x4 42 44 35 9x1 3
1 9
Pickers and guessers choose high payoffs. A
large majority of coordinators choose high
payoffs (consistently). This is consistent with
CH and inconsistent with TR. Some suggestion
that a small minority of Cs are acting
consistently with TR.
(Could Cs have used the labels as a coordinating
device? Test is to look at number tasks where
all objects have 10 points. NCI 1.40 for Cs
surprisingly high, but nowhere near high enough
for a best rule.)
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So, a puzzle Amsterdam results confirm TR,
Nottingham results confirm CH. Possible
explanations
Differences in subject pools? Seems unlikely,
but needs investigation.
Differences in displays? On number tasks,
Nottingham display perhaps more likely to direct
attention to nondescript patterns rather than
to payoffs but majority of Nottingham
coordinators favour high-payoff objects.
Differences in text tasks? Our ex post hunch was
that the labels we intended as standers-out were
in fact more obviously standing-out in the
Amsterdam tasks. And the Nottingham tasks had
more obvious conventional favourites (for
guessers, NCI 1.26 in Amsterdam, 1.98 in
Nottingham). So Nottingham tasks may tend to
prompt ideas about favouriteness, Amsterdam tasks
to prompt ideas about odd ones out.
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Some supporting evidence
Questionnaire given to Amsterdam and Nottingham
students (a year after the experiments).
Respondents are shown the list of labels for
various text tasks and asked to say which of the
items is your favourite or (a different sample)
for you, which of the items stands out from the
others. We collect responses for all the text
tasks (in both experiments) at both locations.
No systematic differences between Nottingham and
Amsterdam responses for given tasks.
Implication no evidence of subject pool
differences.
But we now have independent evidence of which
labels stand out, and which are favourites, for
our subject pool. We can use this to try to
infer the reasoning of subjects in the original
experiments ...
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NCIs for text tasks comparing experiment and
questionnaire
Amsterdam Nottingham stand out
favourite stand out favourite pick 1.16
1.17 1.19 1.36 guess 1.23 1.20
1.49 1.76 coord 1.52 1.07
1.57 1.84 stand out 1.33
1.19 1.40 1.44 favourite
1.40 1.71
Association between standing-out and favourite
is less in Amsterdam tasks than Nottingham tasks.
(Perhaps Nottingham labels stand out as desirable
while Amsterdam ones stand out as odd ones out?)
Favouriteness is much more concentrated in
Nottingham tasks than in Amsterdam ones. (So,
choose the conventional favourite is a better
rule for coordinating in Nottingham.)
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Provisional interpretation Amsterdam text
tasks prompt players to notice odd ones out they
realise that choose the odd one out is a
team-optimal rule and use it as such they carry
this mode of team reasoning over to the number
tasks.
Nottingham text tasks prompt players to notice
favourites this prompts secondary salience/ CH
reasoning, which is carried over to number tasks.
But more generally....
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Cognitive hierarchy theory and team reasoning
theory describe two different modes of reasoning
which can be used to solve coordination problems.
Our experiments were designed to tease these
apart by presenting subjects with problems in
which CH reasoning pulled in one direction, TR in
another.
Our results suggest that each mode of reasoning
is used in some contexts which is used is
sensitive to details of design and framing. Each
theory captures an element of focal point
reasoning, but not the whole.
And our subjects are surprisingly successful at
using these design details to coordinate on a
mode of reasoning. Implication Schellings
intuitions are right, but existing attempts to
express them in game theoretic terms are failing
to capture the richness of salience-based
reasoning.
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