Adaptive Dynamics in Coordination Games
Author(s): Vincent P. Crawford
Source: Econometrica, Vol. 63, No. 1 (Jan., 1995), pp. 103-143
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/2951699
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Econometrica,
Vol. 63, No. 1 (January, 1995), 103-143
ADAPTIVE DYNAMICS IN COORDINATION GAMES
BY VINCENTP. CRAWFORD1
This paper proposes a model of the process by which players learn to play repeated
coordination games, with the goal of understanding the results of some recent experiments. In those experiments the dynamics of subjects' strategy choices and the resulting
patterns of discrimination among equilibria varied systematically with the rule for determining payoffs and the size of the interacting groups, in ways that are not adequately
explained by available methods of analysis. The model suggests a possible explanation by
showing how the dispersion of subjects' beliefs interacts with the learning process to
determine the probability distribution of its dynamics and limiting outcome.
KEYwORDS:
Equilibrium selection, coordination, learning, strategic uncertainty.
1. INTRODUCTION
IN ECONOMICS, COORDINATION PROBLEMS are usually modeled as noncooperative
games with multiple Nash equilibria in which any Pareto-efficientstrategy
combinationis an equilibrium,but players' strategychoices are optimal only
when they are based on sufficientlysimilarbeliefs about how the game will be
played.Althoughsuch games have no incentiveproblemsas these are normally
characterized,playingthem often involvesreal difficulties.Similardifficultieslie
at the heart of many questions usually analyzed under the assumptionthat
playerscan coordinateon any desired equilibrium.These include how incentive
schemes should be structured;which outcomes can be supportedby implicit
contract in a long-termrelationship;whether, and how, bargainersshare the
surplusfrom makingan agreement;and the role of expectationsin macroeconomics.
Convincinganswersto such questionsmust go beyondthe observationthat if
rationalplayershave commonlyknown,identicalbeliefs, then those beliefs must
be consistent with some equilibriumin the game. However, the traditional
approachto analyzinggames with multiple equilibriarelies on refiningNash's
notion of equilibriumuntil (ideally) only one survives,and traditionalrefinements do not accomplishthis for coordinationgames.This suggeststhat players
are unlikelyto base their decisions entirelyon deductionsfrom rationality,and
highlightsthe importanceof gatheringinformationfrom other sources about
how coordinationproblemsare solved.
1I am grateful to Brian Arthur, Antonio Cabrales, John Conlisk, Robert Engle, Daniel Friedman, Clive Granger, Jerry Hausman, Yong-Gwan Kim, Mark Machina, Robert Porter, Garey
Ramey, Michael Rothschild, Larry Samuelson, Joel Sobel, Maxwell Stinchcombe, Glenn Sueyoshi,
John Van Huyck, Halbert White, Peyton Young, and anonymous referees for helpful suggestions; to
Bruno Broseta and Pu Shen for valuable advice and outstanding research assistance; to Ray
Battalio, Richard Beil, and John Van Huyck for access to their experimental data; to the Santa Fe
Institute and the Department of Economics, University of Canterbury (New Zealand) for their,
hospitality; and to the National Science Foundation for research support.
103
104
VINCENT P. CRAWFORD
Perhapsthe most importantsource of such informationnow availableis the
rapidlygrowingexperimentalliteratureon coordination.A number of recent
-studies-such as Banks, Plott, and Porter (1988); Cooper, DeJong, Forsythe,
and Ross (1990); Isaac, Schmidtz,and Walker (1989); Roth and Schoumaker
(1983); and Van Huyck, Battalio, and Beil (1990, 1991) (henceforth
"VHBB")-report experiments in which subjects repeatedly played simple
coordinationgames, uncertainin most cases only about each other's strategy
choices. The results suggest that players' initial beliefs in such games are
normally widely dispersed; that this dispersion, which I shall call strategic
uncertainty, makes players' experience with analogous games an important
determinantof their decisions; and that interactionsbetween strategicuncertaintyand the processof learningfrom experiencecan exert a strongand lasting
influenceon coordinationoutcomes.
The effectsof strategicuncertaintyshow up especiallyclearlyin VHBB (1990,
1991).The large strategyspaces and the varietyof modes of interactionin their
designs yielded remarkablyrich dynamics,with persistentpatternsof discrimination among equilibriaemergingover time. These patternsvaried systematically with the environment,in ways that are not adequately explained by
availablemethods of analysisbut which can be better understood,as I shall
argue,by takingstrategicuncertaintyfully into account.
This paper presents a simple model of the learningprocess that suggests a
unified explanation of the dynamics and patterns of discriminationVHBB
observed.The model is a repeated game in whichplayersadjusttheir strategies
in the underlyingcoordinationgame in response to their experience.It implies
that playersnormallyconvergeto some equilibriumin the underlyinggame, as
usually happens in coordination experiments. The question remains, which
equilibrium?The model answersthis questionby showinghow strategicuncertainty interacts with the learning process to determine the prior probability
distributionof its outcome. When players'beliefs are identical from the start,
the outcome is completelydeterminedby their initial responsesto the underlying coordinationgame, and does not varywith the environmentexcept as their
initial responsesdo. But for realisticlevels of strategicuncertainty,decliningat
realistic rates as players learn to predict how the game will be played, the
distributionof the dynamicsand limitingoutcome varies with the environment
much as it did in the experiments.
The model's predictionsare influenced(but not completelydetermined)by
the differencesin the sizes of the basins of attractionof the equilibriain these
environmentsdiscussedin Crawford(1991). MaynardSmith's(1982) notion of
evolutionarystability,Harsanyiand Selten's (1988) risk-dominanceaxiom, and
the techniquesused by Kandori,Mailath,and Rob (1993) and Young (1993a)to
study the limiting outcomes of related adjustmentprocesses also respond to
those size differences, discriminatingamong equilibriain ways that in some
respects resemble the patterns VHBB observed. The extent to which these
alternativeapproacheshelp to explaintheir resultsis discussedbelow.
COORDINATION
GAMES
105
The model departs from traditionalnoncooperativegame theory in three
main ways. First, players'behavioris adaptiveratherthan rational,in that they
view their strategies in the underlyingcoordination game as the objects of
choice, adjustingthem over time in ways that are sensible but not necessarily
consistent with equilibriumin the repeated game that describes the entire
adjustmentprocess or the underlyinggame.2 One could conduct a traditional
equilibriumanalysisof the repeated game, but such an analysiswould share the
indeterminacyof the underlyinggame but nonetheless require the assumption
that players'beliefs are coordinatedwhen play begins.This begs the questionof
how coordination comes about, and often leads in practice to dictating a
solution by applyingrefinementsthat are largelyinsensitiveto the difficultyof
coordination.It seems better for my purposesto allow players'beliefs to differ
and studythe dynamicsof the process by which they converge.
Second, the model relies on simplifyingassumptionsabout the structureof
the strategic environmentin the spirit of evolutionarygame theory. These
assumptionsare satisfied by VHBB's designs (see Crawford(1991)) and are
commonto a numberof interestingeconomicmodels (see Woodford(1990) and
Cooper and John (1988)). They allow a more informative analysis of the
implicationsof strategicuncertaintythan now seems possible for more general
games.The resultsare applicableto some importantquestions,and the methods
seem likely to be useful in other settings.
Finally, instead of fully endogenizingplayers' beliefs and strategy choices
(whose differences cannot be traced to differencesin players' informationor
other characteristics)the model characterizesthem statistically,treatingcertain
aspects of the process that describeshow they evolye as exogenousparameters
to be estimatedon a case-by-casebasis. This departurereflects the conviction,
first expressedby Schelling(1960), that it is impossibleto predictfrom rationality alone how people will respond to coordinationproblems. I illustrate the
usefulness of this approachby using the data from VHBB's experimentsto
estimate the model, treatingthe idiosyncraticcomponentsof players'beliefs as
errorterms, and then using the estimatesto infer the priorprobabilitydistributions of outcomesin the variousenvironments.In each case the model provides
an adequatestatisticalsummaryof subjects'behaviorwhile closely reproducing
the dynamicsof their interactions.The distributionsit implies suggest that the
observed dynamicsare not anomalous,but in some cases they strengthenor
modifythe impressionscreated by the raw data.
The paper is organized as follows. Section 2 introduces the model and
summarizesVHBB's experimentaldesigns and results. Sections 3 and 4 carry
out an analysis under the simplifyingassumptionthat players' strategies are
continuouslyvariable.Section 5 extendsthe analysisto the case in whichplayers
2Because players' strategy choices normally converge to an equilibrium in the underlying game,
this "irrationality" does not persist, and is therefore immune to the most common criticism of
adaptive analyses.
106
VINCENT P. CRAWFORD
have discrete, finite strategyspaces, as in the experiments.Section 6 reports
econometricestimatesof the model's parametersbased on VHBB's experimental data, and Section 7 discusses the implicationsof these estimates for the
probabilitydistributionsof coordinationoutcomes.
2. VAN HUYCK, BAII7ALIO, AND BEIL'S EXPERIMENTAL
DESIGNS AND RESULTS
In VHBB's(1990, 1991)experiments,subjectsrepeatedlyplayed coordination
games in which their payoffs were determined by their own strategies and
simple summarystatistics of other players' strategies.Explicit communication
was not allowed, but the relevant summarystatistic was publicly announced
after each play. (In some experimentsentire strategy profiles were also announced; this made little differenceto the results.)With apparentlyunimportant exceptions, all details of the structureof each experimentalenvironment
were publicly announced at the start. There was ample evidence that the
subjectsunderstoodthe rules and were paid well enough to induce the desired
preferences.3
VHBB's designsare best understoodin simplifiedversionsof the games used
in their 1990 experiments.Suppose that a numberof players choose between
two efforts,1 and 2. The minimumof their effortsdeterminestheir total output,
which they share equally.Effort is costly, but sufficientlyproductivethat if all
playerschoose the same effort their output sharesmore than repay their costs.
If anyoneshirks,however,the balanceof the others'effortsis wasted.Assuming
for definitenessthat output per capita is twice the minimumeffort and each
player'sunit effort cost is 1, payoffsare determinedas follows:
Minimum Effort
2
Player's
Effort
1
2
1
2
0
1
A game with this structurewas suggestedby Bryant(1983) (see also Cooper
and John (1988)) as a model of Keynesianeffectivedemandfailures.This game
has a long historyin economics,which can be traced to the stag hunt example
Rousseau (1973 (1775), p. 78) used to discussthe originsof the social contract.
To see the connection, imagine (adding some game-theoretic detail to
Rousseau's discussion)that each of a numberof hunters must independently
decide whetherto join in a stag hunt (effort2) or hunt rabbitsby himself(effort
1). Huntinga stag yields each hunter a payoffof 2 when successful,but success
requiresthe cooperationof every hunter and failure yields 0. Hunting rabbits
yields each hunter a payoff of 1 with or without cooperation, and thereby
determinesthe opportunitycost of effort devoted to the stag hunt.
3VHBB
(1990, 1991) and Crawford (1991) describe the designs in more detail.
COORDINATION
GAMES
107
For any numberof players,stag hunt has two pure-strategyNash equilibria,
one with all choosing effort 2 and one with all choosing effort 1. The effort-2
equilibriumis the best feasible outcome for all players,and all strictlyprefer it
to the effort-1equilibrium.This rationalefor playingeffort 2 does not depend
on game-theoreticsubtleties;it is clearly the "correct"coordinatingprinciple.
However,effort 2's higher payoffwhen all playerschoose it must be traded off
againstits riskof a lower payoffwhen someone does not. For a playerto prefer
effort 2 (treatingthe influenceof his choice on future developmentsas negligible) he must believe that the correctnessof this choice is sufficientlyclear that it
is more likely than not that all of the other players will believe that its
correctnessis sufficientlyclear to all. Informalexperimentssuggest that people
are often uncertainaboutwhetherother people will believe this, and that most
people accordinglybelieve that effort 2 is a good bet in small groupsbut not in
large groups.4
VHBB's experimentsshowedthat this kind of uncertaintycan have profound
and lasting consequences.Their subjectschose among seven efforts instead of
two, with both the summarystatisticused to determinepayoffsand the size of
the groups playing the game varying across treatments. In the "minimum"
experimentsreportedin VHBB (1990)populationsof 14-16 subjectsrepeatedly
played gameslike stag hunt, firstin large groupswith the minimumeffortin the
entire population determiningpayoffs, and then in random pairs with each
subject'spayoff determinedby his currentpair's minimum.5In the "median"
experimentsreported in VHBB (1991) populationsof nine subjectsrepeatedly
played games in which the entire population'smedian effort determinedtheir
payoffs,with variationsin the payoff function across three treatments.In each
case the game had the "same"seven strict, symmetric,pure-strategyequilibria;
this similarityin strategic structureextends to the experimentalenvironments
when the random-pairingtreatmentis viewed as a game played simultaneously
by all members of the population, with players' expected payoffs evaluated
before the uncertaintyof pairingis resolved.
These environmentselicited roughlysimilarinitial distributionsof effort,with
high to moderatemeans and variances.Subjects'subsequentbehavior,however,
differed strikinglyand systematicallyacross treatments,with consequencesfor
equilibriumselection that appearedlikely to persist indefinitely.In the largegroup minimum treatment subjects' choices gravitated strongly toward the
4Note that these beliefs are self-confirming. They are plausible because if players choose
independently, with probabilities independent of the number of players, the clarity of the principle
is less likely to be sufficient the larger the group. Because this intuition concerns a choice between
strict equilibria, however, it is not captured by traditional refinements like trembling-hand perfectness or strategic stability.
5 In the random-pairing treatment each subject was told only his current pair's minimum. There
was also a treatment in which the minimum game was repeatedly played by fixed pairs, with subjects
informed that their pairings were fixed. Those subjects were evidently aware of the importance of
repeated-game strategies, and usually achieved efficient coordination. Because repeated-game
strategies raise difficult new issues-the horizon over which players evaluate their strategies, for one
-that treatment is not discussed here. It may be possible to explain its results along the lines
suggested by Crawford and Haller (1990) or Kim (1990).
108
VINCENT P. CRAWFORD
lowest effort,even thoughthis led to the least efficientequilibrium.By contrast,
in the random-pairingminimum treatment subjects' efforts converged very
slowlywith little or no trend; and in the median treatmentssubjectsinvariably
convergedto the equilibriumdeterminedby the initial median, even though it
varied across runs in each treatment and was usually inefficient.6Thus, the
dynamics were highly sensitive to group size, with very different rates of
convergenceand patterns of discriminationamong equilibriain the randompairingand large-groupminimumtreatments.They were also highlysensitiveto
the summarystatisticused to determinepayoffs,with no trends and very strong
history-dependencein the median treatmentsbut strongtrends and no historydependence in the large-groupminimumtreatment.
The similarityof the distributionsof subjects'initial responses across treatments suggests that the differences in their subsequent behavior can be attributed to the dynamicsof the learning process. ExplainingVHBB's results
requiresimposingstructureon this process.Manystructuresare consistentwith
some aspects of the results, and the dynamicswill be at least as importantas
their limitingoutcomes in discriminatingamong explanations.It is instructive,
however,to considerthe extent to which the outcomesthat emerged(or seemed
to be emerging)in the experimentscan be explainedby applyingequilibrium
refinementsto the underlyingcoordinationgames.7
Traditionalrefinementslike trembling-handperfectnessand strategicstability
do not addressthe strategicissues raisedby VHBB's games. Those refinements
are motivated by asking whether a given equilibriumis self-enforcingif all
playersexpect it to governplay. They therefore do not discriminateamong the
strict equilibriain these games, which all pose essentially the same strategic
question in the absence of strategicuncertainty.
More promisingare refinementslike Harsanyiand Selten's (1988) risk-dominance axiom or the "general theory of equilibriumselection" of which it is a
part, which do discriminatebetween strict equilibria.Although Harsanyiand
Selten assume that players'beliefs alwaysconvergeto a particularequilibrium
before play begins, the mental tatonnements that model their convergence
process are sensitive to strategicuncertainty,with implicationsfor equilibrium
selection that are similar in some respects to the patterns VHBB observed.
However,neither Harsanyiand Selten's theorynor a variantthat eliminatesthe
precedencethey give payoff-dominance(allowingrisk-dominance,whichembodies most of their ideas about the effects of strategicuncertainty,to determine
the outcome in most of VHBB'streatments)correspondsverycloselyto VHBB's
6
Isaac, Schmidtz, and Walker (1989) and several more recent studies report similar results.
Comparing the results across VHBB's (1990) treatments that differed only in subjects' prior
experience suggests that the patterns they observed are not eliminated when subjects learn to
anticipate them; if anything, they are reinforced.
7Applying refinements to the repeated game that describes the entire learning process does not
increase their explanatory power, and seems less plausible in this setting.
COORDINATION
CGAMES
109
subjects'strategychoices.8As will be seen, this is due mainly to the influence
strategicuncertaintyexerts on the learningdynamics,an influencetheir theory
rules out by assumption.
Also more promising is Maynard Smith's (1982) notion of evolutionary
stability.Formallya static equilibriumrefinement,it is motivatedby considering
conditions for local stability of evolutionarydynamics that resemble simple
models of the learning dynamics(see Crawford(1989)). Crawford(1991) explored the possibility of an "evolutionary"explanation of VHBB's results,
verifyingthat VHBB's environmentssatisfythe structuralassumptionsof evolutionary game theory, so that the notion of evolutionary stability, suitably
generalized to allow finite populations and interactions other than random
pairing,can be applied. Like risk-dominance,evolutionarystabilityrespondsto
the large differences in the sizes of basins of attraction in the large-group
minimum treatment by discriminatingamong strict equilibria in a way that
resembles VHBB's results. Although suggestive,an analysis along these lines
does not determine players' initial strategychoices, as would be required to
determine the patterns of equilibriumselection in VHBB's other treatments,
and the analogy between evolution and learning may well be unreliable in
environmentswith strategic uncertainty.The structureof evolutionarygames
nevertheless has important advantages in modeling the learning dynamics.
Section 3 introducesa model that combinesthis structurewith a flexiblemodel
of individualstrategy adjustment,in which players learn from experience in
comparativelysophisticatedways.
3. A MODEL WITH CONTINUOUSLY
VARIABLE STRATEGIES.
In this section and the next it is assumed that players' pure strategies are
continuouslyvariable.The analysisresolves most of the issues that arise when
the model is extended to the discretestrategyspaces of VHBB's experimentsin
Sections 5-7.
8As explained in Crawford (1991), Harsanyi and Selten's theory selects the equilibrium with
effort 7 (in response to payoff-dominance) in all of VHBB's treatments but median treatment P,
where it selects the equilibrium with effort 4 (in response to symmetry). These selections correspond
to 7% of subjects' last-period choices in the large-group minimum treatment, 43% in the randompairing minimum treatment, and 0%, 67%, and 33% in median treatments F, Q2, and '. These
success rates are mostly better than random, which with seven effort levels would imply a 14%
success rate, but low. Harsanyi and Selten's theory without payoff-dominance selects the equilibrium
with effort 1 in the large-group minimum treatment (in response to risk-dominance); the equilibrium
with effort 4 in the random-pairing minimum treatment (because risk-dominance is neutral in
VHBB's two-person minimum game and the theory therefore applies the tracing procedure to a
uniform prior over the undominated strategies); the equilibrium with effort 7 in median treatments
F and 12 (in response to risk-dominance); and the equilibrium with effort 4 in median treatment P
(in response to risk-dominance, after imposing symmetry). This raises the success rate to 72% in the
large-group minimum treatment, lowers it to 17% in the random-pairing minimum treatment, and
leaves it unchanged in the median treatments. These success rates are again better than random,
but low. The patterns differ for first-period choices, but the success rates are no higher on average.
110
VINCENT P. CRAWFORD
The model is defined for a class of environmentswith the structure of
evolutionarygames,which generalizesVHBB's experimentaldesigns.There is a
finite numberof indistinguishableplayers,who repeatedlyplay a game in which
their roles are symmetric.This game should be thought of as describingthe
interactionsof the entire population of players, with their payoffs evaluated
takinginto accountany uncertaintyabout how they interact.Players'strategies
are identified,so that it is meaningfulto say that differentplayers choose the
same strategy,or that a player chooses the same strategyin differentperiods.
Playersplay only pure strategies,and their strategyspaces are one-dimensional.9
I focus on coordinationgames in which any symmetriccombinationof pure
strategies is an equilibrium,and they are the only pure-strategyequilibria.
Because players'roles are symmetric,these equilibriaare Pareto-rankedunless
players are indifferentbetween them. This makes the coordinationproblem
particularlysimple, with a one-dimensional continuum of possible limiting
outcomesand an unambiguousmeasureof efficiency.I furtherrestrictattention
to games in which each player'sbest replies are given by a summarystatisticof
the strategieschosen by all players in the populationwheneverthe statisticin
question is unaffectedby his choice. The summarystatistic can be written in
general as y, =f(x,), where xt = (x1t, ..., xnt) and xit denotes player i's strategy choice at time t. I assume that f( ) is continuousand, for any xt and any
constants a and b > 0, f(a + bx t, ***, a + bXnt) =a + bf(Xlt*
Xnt). These
assumptionsare satisfiedwhen f(*) is an orderstatistic,as in VHBB'sminimum
and median treatments,or a convexcombinationof order statisticssuch as the
arithmeticmean.10To see what they entail, note that because players'roles are
symmetricf( ) must be a symmetricfunction of the xit; its value is therefore
completelydeterminedby the order statisticsof their empiricaldistribution.My
9The results of Crawford (1989) suggest that equilibria in which individuals play mixed strategies
would not emerge if they were allowed. The polymorphic configurations of pure strategies that
mixed-strategy equilibria often represent in evolutionary game theory are allowed, but do not
emerge in the games studied here.
10 When players are risk-neutral their best replies in VHBB's random-pairing minimum treatment are actually given by the median effort in the population. A risk-neutral player i would like,
ideally, to choose the xi that maximizes
E[ p min {xi, xj}
-
qxi] =p[1 - G(xi)]xi
= (p
- q)xi
-pJ
+pf
xG(xj)
- 00
xj dG(xj) - qxi
dxj,
where p > q > 0, G(*) is the empirical distribution function of the other players' efforts, and the
second equality is obtained by integrating the Stieltjes integral by parts. Computing the left- and
right-hand derivatives of this objective function makes it clear that xi* solves this problem if and
only if G(x) < (>) 1 - q/p when x < (>) x*, so that x* is an order statistic of the distribution
G(-). In the random-pairing minimum experiments p = 2q, so that xi* is the median when it is
well-defined. VHBB also conducted some large-group minimum experiments in which the cost of
effort was lowered to zero, making all efforts at or above the current minimum best replies and the
highest effort a weakly dominant strategy. The effect of dominance on players' strategy choices is
easily handled within the model, but the generic multiplicity of players' best replies raises new
issues, not addressed here.
COORDINATION
GAMES
ll
assumptionsrule out most nonlinearfunctionsof these order statistics,which is
certainlyrestrictivebut probablynot unrepresentativeof symmetricgames.
The structureof the environmentis made public before play begins, and the
value of the relevantsummarystatisticis publiclyannouncedafter each play."
Players'beliefs and strategychoices evolve as follows. Their beliefs at the start
of play determine their initial choices. They then observe the value of the
summarystatistic that results, update their beliefs and make new choices, and
the process continues. Thus, players face uncertaintyonly about each other's
strategychoices, and both the effects of those choices and the informationthey
receive about them are filtered throughthe summarystatistic.It follows that in
choosing their strategies, they need form beliefs only about the summary
statistic.
In this learningprocess, even if playersform and revise their beliefs sensibly
and choose their strategies optimally given their beliefs, their beliefs and
strategychoices may differ in unpredictableways. I imagine that each player
beginswith a prior about the process that generates yt, and that each period he
updates his prior in response to his new informationand then chooses the
strategythat is optimal given his beliefs about yt. Playerswhose priors differ
may then have differentbeliefs, even if they observe the same historyand use
the same procedures to interpret it. I suppose that each player treats the
probabilitythat his decisionsinfluencethe {ytj process as negligible,so that his
optimal choice each period is determinedby his beliefs about the currentvalue
of yt; this is not strictlynecessary,but it is a natural simplificationgiven the
unimportanceof such influences in VHBB's experiments.12 Finally, I assume
that each player'sprioris sufficientlyundogmaticthat he will eventuallylearn to
predict Ytcorrectlyif it converges.
The importanceof the differencesin players'beliefs and the need to supplement the theory with empiricalinformationabout them make it essential to
representbeliefs accuratelyand flexibly,and to describetheir evolutionin terms
of observablevariables so that it is possible to estimate the parametersleft
undeterminedby the theory. It also seems advisableto avoid undulyrestricting
the form of players'priors about the {ytj process, given the lack of theory on
this issue. These considerationssuggest a model in the style of the adaptive
control literature.'3The key insight of the control literatureis that describing
the evolution of agents' beliefs does not require that they be explicitlyrepresented as probabilitydistributionsor their moments:It is enough to model the
"lAlthough subjects were told only their pair minimum in the random-pairing minimum treatment, this can be viewed as a noisy estimate of the population median that determined their best
replies (see footnote 10), so that the limitation on their information can be treated as an increase in
the dispersion of their beliefs.
12 In a large-group minimum game a player can expect to influence the minimum only if he
reduces his effort below his estimate of the minimum of the others' choices, but the model implies
that he will never do this. In more than 90% of the periods in the other treatments, no single
sub)ect's choice could have altered the summary statistic.
3See for example Ljung and Soderstrom (1983) and Nevel'son and Has'minskii (1973). My
discussion follows Woodford (1990, Section 2), who gives an excellent overview.
112
VINCENT P. CRAWFORD
dynamicsof the estimates of the optimal decisions they imply, which are the
only aspects of agents' beliefs that affect the outcome. This simplificationwill
accommodatethe wide range of learningbehaviorVHBB's subjects exhibited
and make it possible to describe the dynamicsin a simple frameworkthat
satisfiesthe desideratamentionedabove.
The logic of this approachand the need for accuracyand flexibilitysuggest
that beliefs are best representedby continuousvariables.When strategiesare
also continuouslyvariableit is possible,if playersare risk-averseor risk-neutral,
to represent their beliefs directly by their optimal decisions.14 Thus, in this
section and the next, xi, will represent both beliefs and strategychoices. (In
later sections xi, will continue to represent beliefs and remain continuously
variable, with strategy choices determined by a standard model of discrete
choice in which the xit are latent variables.)Note that this way of representing
players'beliefs requiresno specific restrictionson their priors or risk preferences. There is also no need for a player'sbeliefs, as representedby his strategy
choices, to be related in any simpleway to the momentsof the distributionthat
describeshis beliefs about yt. If, however,playerscome to expect a particular
value of yt the xit will approachthat value; thus, in the limit, their choices can
be viewed as estimatesof the mean of yt.
I follow the control literaturein assumingthat players'beliefs and strategy
choices evolve accordingto linear adjustmentrules of the followingform:
(1)
xi=
aj?
and
(t=1,...).
-1
xit= ait + bityt-1 + (1
The ait and bit in (1) and (2) are exogenouscoefficients,which represent any
trends in players' beliefs and how their beliefs respond to new information,
thereby reflectingtheir precision.These coefficientsare allowed to varywith i
and t, as describedbelow, to accommodatedifferencesin players'beliefs and
learningbehavior.
Although (2) resembles partial adjustmentto the strategychoice the latest
observationof yt suggests is optimal, it is taken here to representfull adjustment to his currentestimate of his optimal decision, which normallyresponds
less than fullyto each new observationbecause it is only part of the information
he has about the {ytj process.'5Suppose, for instance,that player i's decisions
are certainty-equivalent,so that his optimal strategychoice equals his current
estimateof the mean of yt. Then if he is convincedthat the yt are independent
drawsfrom a fixeddistribution,and if he puts as muchweighton his prioras he
(2)
14 When players'optimal decisions do not preserveenough informationabout their beliefs to
representthem accurately,it is necessaryto keep trackof beliefs and decisionsseparately,as in the
analysisof discretestrategychoice in Section5.
1 Note that these learning rules differ from those discussed in the psychologicallearning
literature(see Roth and Erev (1995) for an interestingapplicationin economics)in that they
incorporateinformationaboutthe game'sbest-replystructure.This seems appropriatefor VHBB's
experiments,but mightbe unrealisticin other settings.
COORDINATION
GAMES
113
would on X such draws, he will set ait 0 and bit 1/G(+t+ l), as in
"fictitiousplay."If he believes insteadthat the {yt}processis a driftlessrandom
walk, he will set ait 0 and bit 1, as in a "best-reply"process. Thus (2)
includes as special cases two of the learning rules most often studied in the
game theoryliterature.
In generalplayers'priorsare more complexthan this, and their decisionsmay
not be well approximatedby certaintyequivalence.With or without certainty
equivalenceor strong restrictionson priors, playerswhose adjustmentsfollow
(2) will learn to predict yt if it convergesand choose xit that are optimal,given
their predictions, as long as a -- 0 as t -- ooand 0 < bit < 1 for sufficiently large
t. More generally,learning rules of this kind have been shown in the control
literatureto provide a robust, effective approachto the estimation problems
faced by agents who understandthe forecastingproblems they face but are
unwillingto make the specificassumptionsabout the process or unable to store
and process the large amountsof informationan explicitlyBayesianapproach
would require.
It is plain that the learningrules in (2) are usuallyless than fully rationalin
the sense used in game theory,because players'priorsaboutthe structureof the
{yt} process are not requiredto be correct,and because the form of (2) may be
inconsistentwith the adjustmentrules that are optimalgiven their priors(which
in general may depend nonlinearly,and nonseparably,on the observablehistory).It mightbe possible to find repeated-gameequilibria,perhapswith payoff
perturbationsas in McKelveyand Palfrey's(1992) analysisof experimentswith
the centipede game, that are statisticallyconsistent with VHBB's subjects'
behavior.But for this applicationit is both simplerand, giventhe unimportance
of individualsubjects'influenceson yt noted above,more plausibleto adopt the
workinghypothesisthat playersfocus on stage-gamestrategies.'6Under simple
restrictionson the learningprocessthis approachyields specificpredictionsthat
correspond to the dynamics VHBB observed. Although the possibility that
players choose optimallygiven correct beliefs is not ruled out, it is strongly
rejectedin the empiricalanalysis.
To sum up, although the specificationof players' learning rules used here
cannot be defended by an appeal to rationality,it accommodatesa wide range
of sensiblerules,whichlocate players'best repliesquicklyand reliably.The case
for this specificationis most compellingwhen it is difficultto defend specific
restrictionson the formof players'Bayesianpriors.However,the adjustmentsit
16
One might still try to impose rationality in the weaker sense of requiring players' rules to be
statistically optimal for the process implied by the model, on the assumption that their influences on
Yt are negligible. If players' choices are certainty-equivalent, (2) includes rules (exponential smoothing with time-varying coefficients) that are optimal in this sense for {yj} processes close to the one
implied by the model (which is shown below to resemble a random walk plus noise, with drift and
declining variances); see Harvey (1989). However, the logic of this approach also requires using the
underlying variances of the process to determine the optimal coefficients of the learning rule. Even
without the restrictive form of (2), attempts to close the loop in this way ultimately founder on the
multiplicity of equilibria in the learning process, and in any case do not help to explain the
differences in players' beliefs.
114
VINCENT P. CRAWFORD
implies are qualitativelysimilarto those of Bayesianplayersand coincide with
them in leading cases. Thus, using (2) mightnot be a significantsource of error
even if the populationconsisted entirelyof Bayesianswith knownpriors.More
to the point, it appears to be a reasonableway to describe players'behavior
when, as here, precise knowledge about their priors and decision rules is
lacking.
When players'learningrules are describedby (2), differencesin their beliefs
appear,to an outside observeror the playersthemselves,as randomvariations
in the ait and bit, In keeping with their unpredictability,I characterizethem
statistically,treatingcertainparametersof their distributionsas behavioraldata.
Thus, let
(3)
sit ait - at
and
it -bit -8t
(t = 1,...)
where at and t are the common expected values of the ait and bit, The
restrictionson the ait and bit discussedabove suggestthat at -O as t --oo (but
that the at may have either sign before they converge) and that 0 <,bt < 1. By
construction, EEit = E-1it = 0 for all i and t, where Ez denotes the expected
value of the randomvariablez ex ante (that is, before randomvariablesdated 0
are drawn).I assumethat the Eit and nit are seriallyindependent.This amounts
to assumingthat xiti, fully captures any future effects of idiosyncraticinfluences on player i's beliefs throughperiod t - 1, which is restrictive,but seems a
reasonablesimplification.It is then a naturalextension of my assumptionthat
players are indistinguishableex ante to assume that the Eit and nit are jointly
independentlyand identicallydistributed(henceforth,"i.i.d.")across i for any
given t, with exogenous common variancesand covariances,denoted o-, at
and Kt respectively.17
It is convenientto rewritethe model by substituting(3) into (1) and (2) and
letting
(4)
V iO =io
and
tit
8it+
(yt_l-xitJ)qit
(t=
1,...),
which yields
(5)
xio=
ao + Vio
and
t
(t= 1,...).
xit = tt +f tyt-l + (l-t)Xit-1+
The vit are idiosyncraticrandom variables that represent the differences in
players'initial beliefs and in how they respondto new observations.It is easily
verifiedthat, like the Eit and nits they are ex ante identicallydistributedacross i
for any given t. Further,letting E,(z I ) representthe conditionalexpectation
of the randomvariablez given the informationavailableat time s (that is, after
randomvariablesdated s are drawn)and usingthe law of iteratedexpectations,
(6)
17
Note that it is not assumed that Eit and nit are independent of each other. This would be
inappropriate because ait and bit are different aspects of the same learning rule.
COORDINATION
GAMES
115
for any i#j and any t; and E;it=?,
E[;it;jt]=E[Et-,(;it;jtlxt-,)]=O
O for any
E itl ) = O, and E[;i,vjt] = E[E,(;i jtlx,, i)] = E[;iE(jtlx,)]=
i, j and any s, t with s < t. Thus, although the distributionof Vit depends on
xt-1 and therefore on
-it_l
and the other -jt_,, the Vitare uncorrelated across i
for any given t and seriallyuncorrelated.The common ex ante varianceof the
cit' which is endogenousbut well defined once ot- 171t, and Kt are specified,is
denoted o-J, and the conditionalvarianceof Vitgiven xt1 is denoted o2tIx_1.
Under these distributionalassumptions,(5) and (6) define a Markovprocess
with state vector xt, in whichplayers'beliefs and strategychoices are identically
distributed ex ante, but not in general otherwise. The recursive structure,
together with the conditional independence of players' deviations from the
average learning rule, captures the requirementthat players must form their
beliefs and choose their strategies independentlywhich is the essence of the
coordinationproblem.
The dynamicsare drivenby the dispersionof players'beliefs, as represented
by the otlxt 1. This is true even though the model is formallyconsistentwith
any historyof the yt for any n and ffQ), with the at varyingas necessaryover
time and players'beliefs constrainedto be identicalthroughout.(If o Ix_1 = 0
for all i and t, so that Vit= 0 with probabilityone, then (5) and (6) imply that
=yt= Et=oaa.) Solutionsof this kind, in which playersjump simultaneously
it
from one stage-gameequilibriumto the next followingsome commonlyunderstood pattern,correspondto equilibriaof the repeatedgame that are difficultto
rule out using traditionalrefinements.Yet it is clear that they provideneither a
meaningful explanation of the dynamics of yt nor an adequate statistical
summaryof subjects'strategychoices. I shall thereforeask that the model meet
both of these goals simultaneously,without such ad hoc variations in its
parameters.
This requires that players differ significantlyin their responses to new
informationas well as in their initial beliefs. To see this, assume for simplicity
but ot Ixtl1=0 for all i and
If ;2=E2>0
that at=0 for all t=1.
t = 1,..., then (6) (with Vit= a-t= 0 and 0 <J3t < 1) implies that (xit - yt- 1) and
(xit -1 - yt- 1) alwayshave the same sign, with xit closer to yt- 1 than xit -1 was.
It follows that players'strategychoices converge to yt monotonically,without
overshooting,and therefore,when ff() is an order statistic,that yt Y0for all
t, independent of n and f( ). This extreme history-dependenceis consistent
with the results for VHBB's median treatments but not for their minimum
treatments. 8
Given that t- Ixt 1 > 0, the model is closest to other work on adaptive
dynamicswhen at = 0 for all t = 1,... and 8t and ;it xt- remainconstantat
18
In the large-group minimum treatment, for instance, subjects whose efforts were above the
minimum adjusted their efforts only part of the way toward it on average. Thus, without persistent
differences in subjects' responses to new information the minimum would never have fallen. In fact,
however, there was enough variation in subjects' responses, given the size of the population, to
make the minimum fall in 9 out of the 13 instances in which it was not already at the lowest possible
level.
116
VINCENT P. CRAWFORD
positivelevels or convergeto positivelimits,so that the processhas stationaryor
asymptoticallystationarytransitionprobabilitiesand a unique, globally stable
ergodicdistribution.Then, the methodsof Freidlinand Wentzell(1984),as first
applied in this area by Foster and Young (1990) and subsequentlyadapted to
discrete state spaces by Kandori,Mailath,and Rob (1993) and Young (1993a),
can be appliedto the model with discretestrategychoice analyzedin Section 5.
In the long run the process cycles perpetuallyamongthe pure-strategyequilibria of the underlyinggame, with their prior probabilitiesat any given time
determinedby the ergodicdistribution.As the o-t Ixt_ are allowedto approach
zero (remainingconstantover time) the ergodic distributionassignsprobability
approachingone to the equilibriumwith the lowest effort whenever the summary statistic is below the median, and approaches a limit with positive
probabilityon every equilibriumwhen the summarystatistic is the median, in
each case independentof the numberof players.19
Thus, an analysisis the style of Kandori,Mailath,and Rob (1993) and Young
(1993a)discriminatesbetween VHBB's large-groupminimumtreatmenton one
hand and their median treatmentsand random-pairingminimumtreatment(in
whichthe relevantsummarystatisticis also the median,as explainedin footnote
10) on the other, but otherwise does not distinguishbetween them. These
conclusionsare not logicallyinconsistentwith the systematic,persistenteffects
of the numberof playersand the summarystatisticwithin the finite horizonsof
VHBB's experiments,but they suggest that an analysisbased entirely on the
limitingdistributionsof an ergodiclearningprocesswith infrequent"mutations"
is of limited use in understandingthose effects.
The methods developed below characterizethe probabilitydistributionsof
the entire time paths of yt and the xit, whether or not the process is ergodic
and the o-J Ixt_ are small. As will become clear, the model's dynamicsare
closest to VHBB's resultswhen the o-JlIxt_ decline steadilyover time-eventually to zero, given subjects'tendencyto "lock in" on an equilibrium-just as
one would expect as playersgain experienceforecasting yt.
4. ANALYSIS
The model's "evolutionary"structuremakes it analyticallytractabledespite
its nonlinearityand the nonstationarityof its transition probabilities.When
players'interactionsare filteredthrougha summarystatisticf(x1,,..., x"t) with
the property that f(a + bxlt, ... , a + bXnt) a + bf(Xlt, ..., xnt), the outcome
can be expressed as a function of the idiosyncraticshocks that represent the
differences between their beliefs. In what follows, sums with no terms (like
-1 Bs+ 1fs for t = 0) should be understoodto equal 0, and productswith no
terms should be understood to equal 1. All limits are taken as t -- oo.
19
This follows because fewer changes in individual players' efforts are required to get from a
high-effort equilibrium to the basin of attraction of a low-effort equilibrium than vice versa if, and
only if, the summary statistic is below the median. Robles (1994) analyzes a closely related model for
the games studied here, showing that this conclusion changes only slightly when players do not
ignore their influences on y,.
COORDINATION
117
1: The unique solution of (5) and (6) is given, for all i and t, by
PROPOSITION
t
(7)
GAMES
Xit=
t-1
acs + E Ps+l s + Zit
s=O
s=O
and
(8)
t
t-1
s=O
s=O
Yt=5 ats+ E 1s+lfs +fttS
where
t
(9)
t -s
Zit
S=0 _J=1
PROOF:
1-ot-j+ 1)
vjs
and ft=f (zj,..,znt)
The solution follows immediatelyby induction on t, recalling that
f(a + bx1t,.*.*, a + bxt)n a +bf(Xlt..... xIn) and notingthat zit = (1 -/3dZit+ Vitfor all t. Uniqueness also follows immediately by induction.
Q.E.D.
Proposition1 representsthe outcome of the learningdynamicsas the cumulative effect of trend and shock terms from each period, each of whose effects
persist indefinitely.(The remainingterms, zit in (7) and ft in (8), are subsumed
in the shock terms after the period in which they first appear.) Thus the
coordinationprocess resemblesa randomwalk,but with decliningvariancesand
possibly nonzero drift. Although Proposition l's solution remains valid no
matter how the Vjtare generated, much of its usefulness in the analysisbelow
stems from the fact that the shock terms are knQwnfunctionsof the zit, which
under my assumptions are ex ante identically distributed and uncorrelated
across i for all t.
The next propositionprovidesconditionsunderwhichthe dynamicsconverge,
with probability1, to one of the symmetricNash equilibriaof the underlying
coordinationgame. In this proposition,and sometimesbelow, it is necessaryto
bound players' strategies. This is accomplishedby increasing xit to its lower
bound, denoted x, or reducingit to its upper bound, denoted x, whenever it
would otherwisefall outside the interval[x, x]. This is equivalentto increasing
Vioto x - ao (or reducing it to x - ao) when it would otherwise fall below
- (1 -8)xjt_1
(or
(above) that value and increasing Vjt to x - a -tyt
reducing it to X - ao - tyt-l- (1)xit-) when it would otherwise fall
below (above) that value. The resultingtruncationof the conditionaldistributions of the Vjtis consistent with the distributionalassumptionsintroducedin
Section 3, except that it may bias the E;io or the Et_1(;itjxt_j)
away from zero
and induce serial or contemporaneouscorrelationin the Vit;I indicate below
when this affects the results.
2: Assume that the distributions of the Vit are truncated so that
the xit always remain in the interval [x, x]. Then if 8t is bounded above 0, with
PROPOSITION
118
VINCENT P. CRAWFORD
0 < 3 < 13 < 1 for all t, and E,=oa, and E,=o j2 are finite, y, and the xit
converge, with probability 1, to a common, finite limit.
PROOF: The proof followsthe martingaleconvergenceargumentsof Nevel'son
and Has'minskii (1973, Theorem 2.7.3).2? Consider the Lyapunov function
n. Clearly,
where the summation is taken over all i, j=1,...,
if
for
all
i
and
all
=
0
if
and
0
for
with
only
Substituting
>
j.
xit=xjt
Vt
Vt
xt,
Vt- Li j(x-xit)2,
from (6) and simplifying,
(10)
Vt =
E [(1-8t
)(xit-1
+
x-xt-)
vit-jt]
i,j
-
[(1
13t)2(Xit-l-xit_l)2
i, J
+2(1 -It)(xit-1
+ (-it
-xt-1)(;it-jt)
jt)]
Takingexpectationsin (10) yields
(11)
= (1-_,t)2
Et_1(VtlXt_l)
( V~~~~~~~~,
(Xit,_-xjtJ
i,
l)
j
+ 2(1 -8t) E (xit-,
-xjt-,)Et-,
[;it - jt] |xt-1)
i,iJ
+ J:Et_1([Ri jt-
2
]
Xt_J)
i, J
The first term on the right-handside of (11) is plainlyboundedbelow Vt-> for
all xt-1 outside any given neighborhoodof the set for which Vt 1 = 0. Without
truncationthe second term equals0, and the thirdterm eventuallyapproaches0
to be finite and this
with probability1 because o2 must approach0 for E=0
cannot occur unless Et_ 1(;2tIxt 1) approaches 0 with probability 1. Thus,
without truncation{Vt}eventuallybecomes a nonnegativesupermartingale,and
therefore convergeswith probability1, under the stated variancecondition, to
its lowerbound,0. Becausetruncationat time t can neverincreaseEt 1(VtIx_
t)
or Et- 1(2tIxt-1), this conclusion extends to the truncatedversion of the {Vt}
process. It follows that (xit - xjt) converges to 0, with probability1, for all
i, j = 1, ..., n, and therefore (because yt =f(xt), ff() is continuous, and
f(a .. ., a) a) that (yt -xit) also converges to 0 with probability 1. The
convergenceof the xit (and not just their differences)then follows, given that
they are bounded, from two observations:(i) xt cannot (with positive probability) returninfinitelyoften to a given point at which xit =A
yt for some i, because
20
Nevel'son and Has'minskii's(1973) generalizationof Lyapunov'sglobal-stabilitycriterionto
stochastic dynamicsystems is a useful alternativeto the techniques of Ljung (see Ljung and
Soderstrom(1983))used by Woodford(1990).Nevel'sonand Has'minskiirequirethat the Lyapunov
functionapproachinfinitywith llxtIl,but they use this conditiononly to ensurethe boundednessof
solution paths; in Proposition2 boundednessis assumed directly.The variancecondition they
imposeis just what one would expectby analogywith the stronglaw of large numbers.
COORDINATION
GAMES
119
the (y, -xi) convergeto 0 with probability1; and (ii) x, cannot (with positive
probability)oscillate infinitelyoften between distinctpoints at which xi, = y, for
all i, because E'=oas and the Es=O;is also converge(the latter with probability
1) underthe stated conditions.
Q.E.D.
Because the xit, and therefore the (xit - Yt), are bounded, (4) implies that for
E7=o
07to be finite it is sufficientthat Eos=oa2sand Es=0 7 be finite, and
necessary that ES=o-2, be finite. Although it is plainly not necessary that
S
be finite, I have not been able to use this possibility to weaken these
sufficientconditions for convergencein any useful way. I have also not been
able, withouttruncatingthe distributionsof the jit,to rule out the possibilityof
solutionsin which Yvand the xit divergetogetherto ? oo,but the conclusionof
Proposition2 seems likelyto remainvalid for well-behavedunboundeddistributions.
Propositions 1 and 2 show that the model determines a prior probability
distributionover limitingequilibriathat is normallynondegenerate,ratherthan
singling out a particular equilibrium.This is consistent with the variations
observed across different runs in some of VHBB's treatments, and follows
naturallyfrom characterizingplayers' beliefs statisticallyinstead of assuming
that they can be determinedentirelywithin the model.
I now considerthe model'scomparativedynamicsproperties,focusingon how
changes in VHBB's treatment variables, the number of players, n, and the
summarystatistic, ff(), affect the prior probabilitydistributionsof the yt and
xit. 21 Propositions 3 and 4 characterizethese effects qualitativelyfor given
values of the behavioralparametersat, 3t, oet a7t, and Kt, showingthat they
are fully consistent with the patterns of variation in the dynamics VHBB
observedacrosstreatments.Propositions5 and 6 then show how the means and
variancesof the yt and xit are determinedby the treatmentvariablesand the
parameters,to assess the quantitativemagnitudesof these effects.
These results are then used to explain the patterns of variation in the
dynamics.As with any model whose predictionsdepend on empiricalparameters, the explanationrests on the assumptionthat the parametersare stable
across runs in any given treatment.(Sample-splittingtests and inspectingthe
data suggestthat this is not a bad assumption.)But somethingmore is involved
here, because each treatment is a different game, and there is no good
theoretical reason to expect players' responses to be the same in different
games. Thus, in principle, the effects of changes in the treatment variables
identified in Propositions3 and 4 might be swampedby large, unpredictable
changes in the behavioralparameters.Such changes would leave the model's
ability to predict coordinationoutcomes unaffected,as long as the behavioral
21
Broseta(1993a) studies the comparativedynamicsof a closely related model that allows for
autoregressiveconditionalheteroskedasticityin the idiosyncraticcomponentsof players'beliefs,
obtainingsome results like those given here and others concerningthe effects of changesin the
initialmean and dispersionof players'beliefs.
120
VINCENT P. CRAWFORD
parametersremained stable across runs in each treatment.They would, however, reduce the patternsof variationin the dynamicsto a primarilyempirical
question, limitingthe theory'susefulnessbeyond those environmentsfor which
the parametershave been estimated and leaving it unable to explainwhy the
dynamicsvaried across treatmentsas they did in VHBB's experiments.There
would then be no reason to expect similarresultsor patternsof variationin the
dynamicsfor nearbyvalues of the treatmentvariables.
This question is addressedbelow by showingthat the most importantdifferences VHBB observed across treatments were mainly the product of the
interactionsbetween the numberof players,the summarystatistic,and strategic
uncertaintycharacterizedin Propositions3-6. Unreported simulationsof the
present model and the sensitivityanalysisof a closely related model in Broseta
(1993a) reveal that the patternsin the dynamicsare highlyrobust to moderate
changes in the behavioral parameters,which are shown in the econometric
analysis in Section 6 to vary surprisinglylittle across treatments. Estimates
based on Propositions5 and 6 suggestthat the effects of the importantchanges
in treatmentvariableswere systematicand large enoughto swampthe effects of
the changes in the behavioralparametersthey induced; this is confirmedin
Section 7 when the model is simulatedusing Section 6's estimates.Thus, there
is good reason to expect similarresults in nearbyenvironments,and to believe
that the patternsVHBB observedare replicable.
In what follows,the jth order statisticof an empiricaldistribution(l,.... ., n)
is defined to be the jth smallestof the ei; thus, the minimumis the first order
statistic and the median (for odd n) is the ((n + 1)/2)th. If ffQ) is an order
statistic,the index j identifieswhich one; and if ffQ) is a convexcombinationof
order statistics, j indexes the order statistics from which it is computed. An
"increasein j" refers, in general,to a shift in the weightsused to computeffQ)
that increases j in the sense of first-orderstochastic dominance,viewing the
weights as a probabilitydistributionfor the purpose of applyingthe definition.
A randomvariableis said to "stochasticallyincrease"if its probabilitydistribution shifts upwardin the sense of first-orderstochasticdominance.
PROPOSITION 3: Supposethat bit> 0 for all i and t, so that axi,/ayt1 > 0 with
probability one. Then, holding n constant, increasing i stochastically increases y,
and the xi, for any t.
PROOF:
The proof is immediate from (2) by induction, noting that for
any given value of x,-
1' y-1
for the largervalue of j.
and therefore x,, is weakly larger when computed
Q.E.D.
PROPOSITION 4: Supposethat bi, > 0 for all i and t, so that axi,/ay,- 1 > 0 with
probability one. Then, holding j (or the weights on alternative values of j)
constant, increasing n stochastically decreases yt and the xit for any t.
COORDINATION
121
GAMES
PROOF: The proof is immediate from (2) by induction, noting that for any
given value of xt -1 (with i runningfrom 1 to the largervalue of n), yt -1 and
thereforext, is weaklysmallerwhen computedfor the largervalue of n. Q.E.D.
Propositions3 and 4 hold whether or not the distributionsof the vit are
truncated. They make precise, in the probabilisticsense appropriateto the
model, the commonintuitionsthat coordinationis more difficult,the smallerthe
subsets of the population that can adversely affect the outcome, and that
coordinationis more difficultin larger groups because it requires coherence
among a largernumberof independentdecisions.
Let a-2tdenote the common ex ante varianceof the zit. Because the vit are
serially uncorrelated, it follows from (9) that o,2t- Et = o[Ht s(l 1)]2- _j+
Define
Because the random variables
**... Zntl/ozt)
Ef(Z1/t-zt
/-t
zitl/zt
are
standardized,with common mean 0 and commonvariance 1, A-'t is completely
determined by n, f0 ), and the joint distribution of the Zitl/zt
-tt is subscripted
only because the distribution of the
is generally time-dependent; its
zitl/zt
dependenceon n and fQ ) is suppressedfor clarity.It is easily shown,using the
propertiesof order statistics,that 1-tt is decreasingin n and increasingin j and
that, if n > 2 and the zit are distributedsymmetricallyabout 0, then A-'t < 0 if
1 <i <(n + 1)/2, A-'t = 0 if j=(n + 1)/2, and At > 0 if (n + 1)/2 <j < n.
PROPOSITION
5: The ex ante means of
Yt
and the xit are given, for all i and t,
by
t
(12)
t-1
Exit= E a5 + Efs+ 1o5zs,s
5=0
5=0
and
t-1
t
(13)
Eyt= Eas+
s=0
ESs+lCzsAs+ztAt
s=0
PROOF: Takingexpectationsin (7) and (8), using (9), and noting that
( 14)
Ef (Z l s
**
Zn,) )E
[ 0rZSA
Z ls/ozs
,**
Znslozs
)
]-Czs/s
Q.E.D.
immediatelyyields (12) and (13).
Let Varz and Cov(z, z') denote the ex ante varianceof the randomvariable
z and the covariance of the random variables z and z'. Let at-2
Vart(Zlt/?"zts
* * * l Znt/zt),
Zt)f(Zlt/ztl
I * *nt/zt)]
(for any i),
and 8st = Cov[f(zs/zS...,
Zns/zs), A Z1t1?/zt- Izntazt)]. Like /tt the parameters yt-, Vt' and 8st are completely determinedby n, f(), and the joint
distribution of the zisl/zs and Zitl/zt; they are subscripted only because that
distributionis generallytime-dependent.
122
VINCENT P. CRAWFORD
PROPOSITION 6:22
The ex ante variances of yt and the xi, are given, for all i and
t, by
t-1
(15)
t-s
t-1
p
2+13 2or2+ E 3S+jo2
Varxi,=
s=O
s=O
(1(1 8-1+i)v5
j=l
t-1 s-1
+2E
s=1
E
r=O
r+i13s+izr'0zs
8rs +
zt
and
t-1
t-1
(16)
Var yt=
E o82+1 2r2
+
E
>S+313oztU
s=O
s=O
t-1
+2E
s=1
s-1
E 1r+i13s+iO'zrOTzs
r=O
rs +
t(7t
PROOF:Taking variances in (7) and (8) yields
t-1
t-1
(17)
Varxit=
E1s+1Cov(fS,z
E 32+1VarfS+
t)
s=O
s=O
t-1 s-1
+2
E
s=1
E,
r+13s+ 1COV( fr, fs) + Varzit
r=O
and
t-1
t-1
(18)
Vary=
yt
E82+1Varf5+
E8s5+1Cov(fS,ft)
s=O
s=O
t-1 s-1
+2 E
s=1
E,
r+13s+C
Cov(fr,fs)
+ Varft.
r=O
Because the zisl/ozs are standardized,with mean 0 and variance1,
Varfs Varf (zls,...,* z,s)
(19)
For any t >
-Var [o-zsf ( Z15/o *.*,zns/o5)]
...s, zit in (9) can be expressedas
t-s
t-s
(20)J_
Zi
1=1
1-8j+l)i
Because Ezit = O and E(zisit)
is
=
-0
rt-s-k
E FL
k=1 -Lj=1J (1
-8t-j+l)1;sk
E[Es(zisitlxs, zis)] = E[zisEs(;itxs)]
=
O for
22 Because the proof of Proposition 6 uses the properties of the untruncated distributions of the
vit, its formulas would need to be modified with truncation.
COORDINATION
123
GAMES
any s < t, it follows from (20) and my assumptionson ffQ) that, for all s < t,
(21)
Cov(fs,
Zit)
= E[zij
=
(zls,
zns)]
z
f (
z2sE[(zis1qzs/)
ls/lzs*
Znslzs),
t-s
xFl
j=l
(1 -8t
j+l)
t-s
=ozsys17
j=1
(1-t_j+l)
and
(22)
Cov
fs,
ft
=Cov[f(Zls
-ZS(7zt
***
Cov
[f ( z
Zns)
***
Znt)I
Znslo/zso),
s/(Zs
f ( Z ltl'7zt,
o-(zs(Jzt
f(Zlt
-
*Znt/Orzt)
I
st'
Substituting(19), (21), and (22) into (17) and (18) yields (15) and (16).
Q.E.D.
Techniqueslike those used to prove Propositions5 and 6 make it possible to
compute the ex ante variances o-t and zt recursivelyfrom the fundamental
parameters.Given (4) and (9),
(23)
and
z2-vo
(24)
(J = (1-
EO
1)2z2t
+ Jt.
Combining (4), (7), (8), (14), and (21) and recalling that
= or)2t+
1r+2t
(25)
2 ooztILL
t Kt+zt(t+
Kt - Cov(Eit,
nit)
yields
2
Intuition suggests that both O2t and ozt should decline over time as players
Yt. It is clear from (23) and (24) that (with 0 <J8t S 1) the o2
are decliningwheneverthe az2tare, so that requiringthe ozt to decline is more
stringentthan requiringthe a2t to decline.
The formulasin Propositions5 and 6 show with considerablegeneralityhow
the dispersion of players' beliefs interacts with the strategic environmentto
determine the mean and variance of the outcome. But before those formulas
can be put to practicaluse, numericalvalues mustbe assignedto the parameters
learn to forecast
that appear in them. As noted above, the behavioral parameters at' Pt' and o-t,
171t,and Kt (which determinethe azt) must be estimatedfor each environment;
this is done for VHBB's experimentsin Section 6. The remainingparameters
and 8st are completely determined by n, fQ ), and the distributions of
V7t2
A ot2,
the
zit
(which are determined by the distributions of the
Eit
and Tit), but are
difficult to evaluate due to the complexity of their dependence on those
distributions.
This problem can be overcome in two ways. Evaluating l
a t2
7t,
and
8st
can
be sidesteppedby estimatingthe probabilitydistributionsof outcomes implied
124
VINCENT P. CRAWFORD
by the model directlyby repeatedsimulation;this is done in Section 7, using the
parameterestimatesreportedin Section 6. Alternatively,the outcome distributions can be approximatedanalyticallyunder simplifyingassumptionsabout the
distributionsof the zi,
Because the analyticalresultsprovideinsightinto the workingsof the model,
I outline the approximationmethod here.23The approximationsare based on
the assumptionthat the zit are jointly normallydistributedfor any given t.
Normalityis a reasonable approximation,at least when truncationeffects are
negligible, because the zit are weighted sums of the cit, which are weakly
dependent and likely to be approximatelyconditionallynormal for familiar
reasons.24Normalitysimplifiesmatters by makingthe common distributionof
the zit/o?t independentof t, so that At - X
and yt -y. Given that
,y2the zit/azt are uncorrelated for any given t, the parameters ,u and o-2 are
tabulatedin Teichroew(1956) for any order statisticof the normaldistribution
and any n < 20; and it can be shown that y = l/n.25 Finally,for jointlynormal
zis and zit ,st is completely determinedby the common correlationbetween
the standardized zis/o-zs and the corresponding Zit?zt.
Because E(zisit) = 0
whenevers < t, (20) implies that this correlationthen equals
t-s
(26)
Cov( Zis
=
Zit) /?zsozt
H1 (1 - 38t-j 1)(Var zis) /o-zso-zt
j=1
t-s
=
171(1-
j=1
3t-j+0)zs1?z,
23AAn earlier version, Crawford (1992), provides more detail about the approximations and
com2paresthem with the probabilities estimated directly by repeated simulation.
2 It does not seem possible to give an exact justification for this assumption. Normality of the
zit
would follow from joint normality of the vit, even though they are dependent, because the normal
distributions form a stable class. But even if the eit and nit were normal the vit could not be jointly
normal, because then their lack of serial correlation would imply that they were serially independent, a contradiction.
25 Let f'( ) denote the jth order statistic. If (.
are any i.i.d. random variables, it follows
from symmetry, the linearity of the expectations operator, and the fact that the sum of an empirical
distribution's elements equals the sum of its order statistics that
(ib YJE[f f
=E[n)
n j=
-E f(( ,
1
n~~~~~
.
fO((*E[f(
n)E f
nj=
n)Eij
n)]
j=1
It is well known (see for example Jones (1948)) that when the (i are standard normal, the sum in the
last term equals 1 for any order statistic f( ); this conclusion extends immediately to the case where
f(*) is a convex combination of order statistics.
COORDINATION
125
GAMES
Thus, under normality 8st can be written as 8(pst), where pst-- FlI-s(l It is clear that 0 pst < 1, and that the function 8( ) is differenf3t-j+ )?s/?t
tiable and monotonicallyincreasing on the interval [0, 1], with 8(0) = 0 and
8(1) = -2. The function 5( ) has not been tabulated, to my knowledge, but
could be tabulatedwith some effort.
An explanationfor the patterns of variation in the dynamicscan now be
discerned.(This explanationis made precise in Section 7, taking the discreteness of players' strategychoices into account.) For simplicity,ignore the discreteness and boundednessof yt and the xit, assume normalityof the zit, and
suppose that as 0 and Bs =,1 for all s = 1.... and that Et0=o-s -* S. Proposition 5 then implies that Eyt and Exit approachthe approximatecommonlimit
a0 + A,u,S.Given that ,u is determinedby n and ff(), this formulashows how
the means and dispersionsof players'beliefs and the averagerate at which they
respond to new information interact with the number of players and the
summarystatistic to determine the ex ante mean of the limitingcoordination
outcome.
For VHBB's random-pairingminimumtreatment(viewed as a median treatment, as in footnote 10) and their median treatments, A,= 0 because of
symmetry.Thus, in these treatmentsthe approximatecommonlimit of Eyt and
Eyit is a0. The estimatesof a0 in Section 6 rangefrom 4.30 in the random-pairing minimumtreatmentto 4.71 and 4.75 in median treatments ' and F and
6.26 in median treatment n (whose payoff structure,describedin footnote 34
below, made the efficientequilibriumwith effort 7 more prominentthan in the
other treatments). For VHBB's large-group minimum treatment,
At=
-1.74.26
The estimates of a0 and p are 5.45 and 0.25 respectively,and S (which is
difficultto estimate for this treatment,due to boundaryeffects) appearshighly
unlikely to be less than 10. Thus, the approximatecommon limit of Eyt and
Exit, ao + At1S, is at most 1.10.
Even the simplifyingassumptionsmade above in approximatingthe means do
not yield simple expressionsfor Var yt and Varxit, but Proposition6 can be
used to get some idea of how they vary with the environment.Because 5(pst)
increases smoothlyfrom 0 to o-2 as Pst ranges from 0 to 1, its value can be
roughlyapproximatedby psto2. It then follows from (15) and (16) that Var yt
and Varxit are approximatelyproportionalto o-2 for each treatment, with
factorsof proportionalitydeterminedby n, 13,and the z2t.
Like A, oJ2 iS determinedby n and f(-). o-2 = 0.17 in the mediantreatments,
0.10 in the random-pairingminimumtreatment, and 0.30 in the large-group
minimumtreatment.The estimates of o-t and O-1treported in Section 6 are
small for the median treatments,significantlylarger for the large-groupminimum treatment,and much largerfor the random-pairingminimumtreatment.
This suggeststhat the ex ante varianceof the limitingoutcome is small in the
26
This value for ,u and the values for 0,2 given below are taken from Teichroew's (1956) Tables I
and II, setting n = 15 in the large-group minimum treatment for simplicity.
126
VINCENT P. CRAWFORD
median treatments, larger in the large-groupminimum treatment, and still
largerin the random-pairingminimumtreatment.
These approximationssuggest that the limitingprior probabilitydistribution
of y, is relatively concentrated,with means 4.71, 4.75, and 6.26 in median
treatments ', F, and Q respectively;somewhat more diffuse, with mean no
greaterthan 1.10, in the large-groupminimumtreatment;and still more diffuse,
with mean 4.30, in the random-pairingminimumtreatment. These estimated
means are very close to those implied by the precise estimates of the limiting
prior probabilitydistributionsimplied by the model, and almost as close to
those impliedby the last-periodfrequencydistributionsin VHBB's experiments
(both of these distributionsare reported in Tables VI-Xa). The suggested
ordering of the variances is also qualitativelyconsistent with the precisely
estimated limiting distributionsand VHBB's limiting frequency distributions,
once the effect of the lower boundaryin the large-groupminimumtreatmentis
taken into account.I do not estimate the varianceshere because this adds little
intuition,and the distributionsare estimatedpreciselybelow.
The most important changes in the dynamics across treatments VHBB
observed were between the random-pairingand large-groupminimumtreatments, and between the median treatments and the large-groupminimum
treatment.Viewing the random-pairingminimumtreatmentas a median treatment, as explained in footnotes 10 and 11, the model treats the differences
between these treatmentsprimarilyas changes in the summarystatistic (even
though the former differenceis "really"a change in group size and the latter
also involves a change in the size of the group, from 9 to about 15). The
estimates of the limiting means of y, given above suggest that each of these
changes altered the drift of the process by much more than the accompanying
changes in the behavioralparameters.
VHBB also observeddifferencesin the dynamicsbetween the median treatminimumtreatment.These were generallysmaller
ments and the random-pairing
and of a differentcharacter,havingto do with the rate of convergenceand, in
one case, the mean outcome. The model treats the differencesbetween these
treatmentsmainlyas changesin groupsize, which are relativelysmall and in any
case have no effect on drift. The above estimates of the limitingmeans of y,
suggest that the effects of these changes in treatment variables are mainly
determinedby changesin the behavioralparameters.
5. DISCRETE STRATEGY CHOICE
I now extend the analysisto the discrete strategyspaces of VHBB's experimental environments.The xi, will continue to represent players'beliefs, and
remain continuously variable, but instead of representing players' strategy
choices directly they will now determine them as the latent variables in a
discrete-choicemodel. The fact that players' strategychoices and best replies
are naturallyorderedby their payoffimplicationssuggestsmodelingthem as an
orderedprobit(see for exampleMcFadden(1984)).I thereforeassumethat the
COORDINATION
127
GAMES
are conditionallynormallydistributedgiven x-'1, and that players'choices
among the feasible discrete alternativesare determinedby roundingthe latent
variables that represent their beliefs as described below. These assumptions
determineplayers'choice probabilitiesas functionsof the nonrandomcomponents of their latent variables.
Let playerschoose, as in VHBB's experiments,amongseven pure strategies,
numbered 1, ... ,7. I assume that player i's choice at time t is determinedby
roundingxi, to the nearest feasible strategy,so that he chooses strategy1 when
X., < 3/2; strategyx, for x = 2,... ,6, when x - 1/2 <xi, < x + 1/2; and stratvit
egy 7 when 13/2 <xi.
27
Conditional on
xt_l,
eit
is normal with mean 0 and
variance(computedfrom (4))
(27)
OtIxt1
Thus, letting c[
=-1+2(yt1-xit1)Kt
+ (Yt-1
xit-1)
O2t2
denote the standardnormal distributionfunction, at time t
player i chooses strategy 1 with probability P[(3/2 - at - it Yt_-1- (1 8t)xit - 1)1(of;itIxt- 1)]; strategy x, for x = 2, ... ,6, with probability
P[(X + 1/2 -at -,t yty1 -(1 -t)xit
-[(x
-1/2
-
at -,8tyt-
1-
-/(0f0t xt_1)]
(1 -8t)xit
x
and strategy 7 with probability 1 - P[(13/2 - at - ,tyt-1 - (1 - Bt)xit-,)l
(0ot Ixt- 1)1.
The evolution of players'beliefs can be describedby (5) and (6) as before,
with players'strategychoices determinedby roundingthe xit, yt computedby
evaluatingf(*) at those roundedvalues, and the Vitsatisfyingthe distributional
assumptionsin Section 3. The dynamicsare still a Markovprocesswith the state
vector xt representingplayers' beliefs; the only difference is in how players'
beliefs determinetheir strategychoices and the summarystatistic.To see how
this differenceaffects the results, let h(x1t,..., xnt) _
where xit
* nt,x),
denotes the rounded value of xit, so that h(-) determines the value of the
summarystatistic that players observe as a function of their beliefs. Because
h(a + bxlt,.. ., a + bXnt)= a + bh(x10 ... Xxnt) only by coincidence,Proposition
1 does not carryover directlyto the present model. I now argue,however,that
Propositions2, 3, and 4-which do not depend on Proposition1-continue to
hold exactlyas stated with discretestrategychoice, and that Propositions5 and
6-which do depend on Proposition1-hold approximatelywheneverthe grid
of feasible strategiesis sufficientlyfine.
The proof of Proposition2 goes throughbecause the xit remaincontinuously
variableand yt-1, the only variablein (6) that is affectedby rounding,cancels
27It would be easy to allow other values for the boundaries of these choice regions, or to estimate
them. But rounding this way is optimal for the payoffs in most of VHBB's treatments when players
are risk-neutral and xit is viewed as player i's estimate of Et -.1(ytIxt .1), and allowing other values
is unlikely to change the results very much.
128
VINCENT P. CRAWFORD
out of the Lyapunovfunctionin (10). Thus, underthe statedvarianceconditions
and bounds on the It,, players'beliefs and strategychoices still convergewith
probabilityone to one of the pure-strategyequilibriaof the game. The proofsof
Propositions3 and 4 also go through,because h(*) inheritsthe weak monotonicity propertiesof f( ). Thus, the qualitativecomparativedynamicsare unaltered
by discretechoice.
Discrete choice does affect the quantitativecomparativedynamics.Proposition 1, and thus Propositions5 and 6, remainformallyvalid,with the conditional
distributionsof the vi, adjusted for discreteness and , ot-72, Yet,and 8st
evaluated accordingly.The difficultyof evaluatingthose parameterslimits the
usefulness of these results. It is more helpful to identify settings in which the
means and variances for the model with continuouslyvariable strategies are
good approximationsto their analogswith discretechoice.
Plainly, the errors in approximatingthe means and variances in this way
cannot be negligible unless the probabilityof significantboundaryeffects is
negligible.28Given this, the errors are negligiblewheneverthe grid of feasible
strategiesis sufficientlyfine. To see this, note that roundingthe xit can alter
their order only by creatingties, which does not affectorder statistics.It follows
that if f(Q) is an order statistic, then h(x1t,..., xnt) equals the rounded value of
f(xlt,....
xnt);
and if f(-) is a convex combination of order statistics, then
h(x1t,..., xnt) equals that combinationof the rounded values of those order
statistics,rounded as necessarywhen yt is also requiredto be discrete. Thus,
the effects of roundingin (5) and (6) are filtered through yt, bounded by half
the size of the grid each period, and additive across periods. It can then be
shown,usingthe fact that the convergenceof the xit is asymptoticallygeometric
in both models, that the cumulativeeffect of roundingon yt remainsfinite with
probabilityone and approacheszero with the size of the grid.The qualityof the
approximationsreportedin Crawford(1992) suggeststhat the grids in VHBB's
experimentswere fine enoughto make the effects of the discretenessnegligible.
This section's analysisshows that most of the theoreticalproblemscaused by
discrete strategy spaces can be overcome for the present model. It would,
however, be significantlyeasier to analyze the results of experimentswith
approximatelycontinuousstrategyspaces.
6. ESTIMATION
Althoughthe theoreticalanalysisof Sections4 and 5 sheds considerablelight
on the patterns of discriminationamong equilibriain VHBB's experiments,a
full explanationdepends on the values of the parametersleft undeterminedby
the theory. The model provides a simple specificationof the distributionof
28Boundary effects were a significant problem in both of VHBB's minimum treatments and, to a
lesser extent, one of their median treatments, Q2.
COORDINATION
GAMES
129
players'beliefs within which those parameterscan be estimated. This section
reportsestimatescomputedusingVHBB'sexperimentaldata, in preparationfor
the analysisof coordinationoutcomes in Section 7.29
The model was estimated separately for each of VHBB's experimental
treatmentsthat came firstin its sequence,pooling the data from all such runsof
each treatment.30The estimates were obtained by maximum-likelihoodtechniques, taking the discreteness of strategy choices into account using the
orderedprobitmodel of Section 5, with the latentvariablesdeterminedas in (5)
and (6) and the error terms taken to be heteroskedasticas in (4), conditionally
independentacrossplayers,and conditionallynormallydistributed.3'
The model was first estimatedfor each treatmentwith no restrictiorison the
behavioralparameters.32
Allowingthe estimates to vary freely over time in this
way maximizesthe usefulnessof their plausibilityas an informaldiagnosticand
minimizesthe risk of specificationbias. However,as explainedin Section 3, the
large numberof parameters(5 per period minus 3) also leaves room for doubt
about whether the model can explain the dynamics only through ad hoc
parametervariations.To resolve this doubt and to providethe structureneeded
for beyond-sampleprediction, the model was reestimatedfor each treatment
under the simplestintertemporalconstraintsthat appearedto have a chance of
not being rejected. These required that at = 0 for t = 1,... in the median
treatments; that t = a for t = 1,... in the minimum treatments, in which at =0
is implausible;that f3t= f3 for t = 1,... in all treatments;and that the parameters of the variance-covariance
matrix, -t, O2t,and Kt, all decline over time like
1/tA for t = 1, ... in all treatments.33Includinga0 and o_-O,
which are allowedto
vary independently to reflect the mean of players' initial beliefs and their
differentstochasticspecification,these constraintsreduce the numberof inde29
Broseta (1993b) estimates a related model that allows for autoregressive conditional heteroskedasticity in the idiosyncratic components of subjects' beliefs. This richer intertemporal
stochastic structure describes the data better in some respects.
30An exception was made in including the random-pairing minimum treatment, which was always
run following other treatments. Only the longer of the two runs in this treatment was used, to avoid
problems with missing observations. All other treatments were sometimes run first in a series and
sometimes later on; prior experience clearly affected subjects' initial beliefs, making pooling
inadvisable. The two (out of nine) runs of the large-group minimum treatment in which entire
strategy profiles were announced were omitted for similar reasons.
31 To make the computations manageable, the likelihood function was constructed period by
period, using the xit- 1, subjects' (rounded) strategy choices from the previous period, as proxies for
the xit-1, their unobservable (unrounded) beliefs. Without this substitution, this procedure would
yield consistent and asymptotically efficient parameter estimates under my assumptions. The grid
ap ears to have been sufficiently fine that the substitution made little difference to the estimates.
2 In each case the reported estimates are unrestricted maximum likelihood, except that /81 in
treatment Q2was constrained to lie between 0 and 1 because the unrestricted estimate, 1.28, was
implausible. In this and a few other cases, indicated by dashes in parentheses in Tables I-V, it
proved impossible to compute standard errors.
33 No cross-treatment restrictions were imposed, because the unconstrained estimates suggest
that most simple restrictions of this kind would be rejected, and such restrictions are neither
suggested by the theory nor needed for the analysis.
VINCENT P. CRAWFORD
130
TABLE I
F
ESTIMATES
FORMEDIANTREATMENT
(54 observations each period; asymptotic standard errors in parentheses)
Unconstrained Model
t
at
0
4.75
(0.18)
-0.07
(0.13)
0.15
(0.11)
0.02
(0.07)
0.10
(0.10)
-0.10
(0.12)
0.09
(0.09)
1
2
3
4
5
6
t
0
at
1
4.75
(0.18)
0.00
2
0.00
3
0.00
4
0.00
5
0.00
6
0.00
lo
0.56
(0.12)
0.80
(0.14)
0.63
(0.12)
0.57
(0.20)
0.48
(0.23)
0.41
(-)
1.62
(0.37)
0.57
(0.18)
0.41
(0.11)
0.14
(0.04)
0.05
(0.02)
0.04
(0.02)
0.04
(0.02)
0.19
(0.14)
0.11
(0.10)
0.05
(0.08)
0.55
(0.27)
0.50
(0.27)
0.43
(0.02)
0.21
(0.11)
-0.12
(0.09)
-0.01
(0.07)
-0.16
(0.05)
0.14
(0.05)
-0.04
(0.02)
Constrained Model
,
t
Kt
l
0.58
(0.07)
0.58
(0.07)
0.58
(0.07)
0.58
(0.07)
0.58
(0.07)
0.58
(0.07)
7,Ct
Log L
1.62
1.62
-86.7
1.17
0.86
- 67.9
0.58
0.53
-59.4
0.25
0.17
-37.9
0.22
0.18
-24.2
0.20
0.14
-15.6
0.12
0.05
-8.7
eKt
1.62
(0.37)
0.43
(0.20)
0.23
(0.06)
0.17
(0.03)
0.13
(0.02)
0.11
(0.02)
0.09
(0.02)
0.78
(0.31)
0.43
(0.11)
0.30
(0.08)
0.24
(0.07)
0.20
(0.07)
0.17
(0.07)
-0.05
(0.13)
-0.03
(0.07)
-0.02
(0.05)
-0.01
(0.04)
-0.01
(0.03)
-0.01
(0.02)
't
Ct
1.62
1.62
0.82
0.54
0.61
0.47
0.44
0.33
0.30
0.22
0.21
0.16
0.16
0.12
Et Log Lt
-334.3
pendent behavioralparametersto 8 in the minimumtreatmentsand 7 in the
median treatments,independentof the numberof periods.
Tables I-V report the unconstrainedand constrainedparameterestimates
for the three median treatments, F, Q, and ', and the large-groupand
minimumtreatments,A and C, with asymptoticstandarderrors
random-pairing
in parentheses. Each treatment lasted 10 periods except for treatment C, in
which the run for which the estimateswere computedlasted 5 periods, but no
estimatesare reportedfor some periodsnear the ends of the mediantreatments
in which there was no longer enough sample variationto identifythe parameters. To avoid specificationbias, the constrained estimates for treatment A
were computed omitting the data from period 9, in which the unconstrained
estimates revealed large end-of-treatmenteffects, described below. The estimates of the JZ and o-2 reportedfor the mediantreatmentswere approximated
131
GAMES
COORDINATION
TABLE II
ESTIMATES FOR MEDIAN TREATMENT Q2
(27 observations each period; asymptotic standard errors in parentheses)
0,,2t
t
oat
0
6.26
(0.36)
0.34
(0.30)
-0.16
(0.22)
-0.01
(0.36)
0.95
(0.18)
0.97
(0.44)
aot
lot
1
2
3
l
1.00
(-)
Unconstrained Model
0on2t
Kt
2.37
(1.04)
1.47
(0.87)
0.04
(0.06)
0.02
(0.05)
-
,2t
0,2
a;,
Log L
2.37
2.37
-35.9
7.00
- 25.8
1.06
-12.0
0.05
-0.2
-
2.47
(1.73)
0.15
(0.14)
0.03
(0.05)
-1.90
(1.16)
0.04
(0.15)
-0.03
(0.88)
7.00
1.08
-
0.05
-
Constrained Model
t
0
1
2
3
6.26
(0.36)
0.00
0.00
0.00
0.97
(0.13)
0.97
(0.13)
0.97
(0.13)
I
2.37
(1.04)
0.89
(0.41)
0.09
(0.04)
0.02
(0.16)
t
Kt
K,tt
2.37
Et
2.37
Log Lt
-76.5
-
0.90
(0.64)
0.09
(0.06)
0.02
(0.02)
0.54
(0.60)
0.06
(0.05)
0.01
(0.01)
2.91
2.91
-
0.35
0.35
-
0.03
0.03
-
TABLE III
'
(27 observations each period; asymptotic standard errors in parentheses)
ESTIMATES FOR MEDIAN TREATMENT
Unconstrained Model
t
0
1
2
o
4.71
(0.20)
-0.30
(0.12)
0.11
(-)
3
0.16
(0.18)
lo
0.95
(0.16)
0.40
(-)
0.92
(0.27)
2,
0a
0.97
(0.30)
0.22
(0.10)
0.00
(1.13)
0.04
(0.04)
2
0t
,2t
UZ
Kt
-
0.97
2t
Log L
0.97
-38.5
0.29
- 23.6
0.00
-4.5
0.05
-5.1
-
0.07
(0.11)
0.00
(0.94)
0.10
(-)
0.09
(0.10)
0.00
(0.57)
0.00
(
0.29
-
0.11
-
0.05
-
Constrained Model
t
0
1
2
3
ozt
4.71
(0.20)
0.00
0.00
0.00
Pt
lot
0.74
(0.11)
0.74
(0.11)
0.74
(0.11)
Kt
cet
0.97
(0.30)
0.22
(0.10)
0.08
(0.02)
0.04
(0.02)
-
0.16
(0.14)
0.06
(0.06)
0.03
(0.03)
-0.01
(0.08)
-0.00
(0.03)
-0.00
(0.01)
ut2
a;C
0.97
0.97
0.43
0.36
-
0.13
0.10
-
0.05
-
0.04
Log Lt
-79.9
VINCENT P. CRAWFORD
132
TABLE IV
ESTIMATES FOR LARGE-GROUP
MINIMUM TREATMENT
A
(107 observations each period; asymptotic standard errors in parentheses)
Unconstrained Model
t
aot
0
5.45
(0.19)
0.51
(0.23)
0.10
(0.41)
0.27
(0.25)
-0.63
(0.37)
-0.54
(0.58)
0.11
(0.22)
-0.35
(0.35)
-0.49
(0.25)
-5.19
(1.97)
0.52
(0.09)
0.34
(0.14)
0.54
(0.11)
0.15
(0.13)
0.20
(0.25)
0.46
(0.22)
0.40
(0.18)
0.11
(0.17)
-4.47
(2.24)
t
at
Pt
0
5.45
(0.19)
-0.27
(0.09)
- 0.27
(0.09)
- 027
(0.09)
-0.27
(0,09)
-0.27
(0.09)
-0.27
(0.09)
-0.27
(0.09)
-0.27
(0.09)
1
2
3
4
5
6
7
8
9
P
It
3.47
(0.64)
1.00
(0.41)
6,85
(2.69)
2.96
(0.98)
6.00
(2.06)
1.76
(1.28)
1.32
(0.51)
0.55
(0.42)
1.17
(0.40)
26.0
(1.73)
Kt
Log L
-180.5
0.04
(0.03)
0.75
(0.32)
0.48
(0.20)
0.52
(0.22)
0.76
(0.27)
1.14
(0.45)
0.24
(0.11)
0.39
(0.19)
37.7
(1.04)
-0.21
-184.1
(-)
2.08
(0.93)
1.01
(0.45)
1.64
(0.67)
1.06
(0.54)
1.06
(0.40)
0.19
(0.19)
0.59
(0.24)
30.5
(1.30)
- 170.4
- 154.8
- 140.7
- 106.3
- 115.5
- 78.3
- 80.7
-88.0
Constrained Model
1
2
3
4
5
6
7
8
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
0.25
(0.04)
It
3.47
(0.64)
3.05
(0.61)
2.34
(0.35)
2.00
(0.27)
1.79
(0.23)
1.64
(0.21)
1.53
(0.21)
1.44
(0.20)
1.37
(0.20)
Kt
Et Log Lt
-1265.0
0.59
(0.14)
0.45
(0.09)
0.38
(0.08)
0.34
(0.07)
0.32
(0.06)
0.29
(0.06)
0.28
(0.06)
0.26
(0.06)
0.91
(0.25)
0.69
(0.17)
0.59
(0.14)
0.53
(0.12)
0.49
(0.11)
0.45
(0.10)
0.43
(0.10)
0.41
(0.09)
from the estimated parametersusing (23)-(25) and assumingnormalityof the
Zit; large boundary effects made it impractical to compute the analogous
estimatesfor the minimumtreatments.
Despite its simplicity,the model gives a reasonablyaccuratestatisticalsummary of subjects'behaviorin all three median treatments.The unconstrained
parameterestimates are generallyplausible and reasonablystable across periods. With minor exceptions the trend parameters, at for t = 1,...,are not
COORDINATION
133
GAMES
TABLE V
ESTIMATES FOR RANDOM-PAIRING
MINIMUM TREATMENT C
(16 observations each period; asymptotic standard errors in parentheses)
Unconstrained Model
t
rat
0
4.30
(1.13)
1.59
(0.71)
0.35
(0.61)
1.50
(0.01)
0.80
(0.91)
0.67
(0.28)
0.51
(0.65)
1.00
(0.01)
0.42
(0.37)
t
aot
P
lo
0
4.30
(1.13)
1.21
(0.44)
1.21
(0.44)
1.21
(0.44)
1.21
(0.44)
1
2
3
4
1
2
3
4
l
0.54
(0.17)
0.54
(0.17)
0.54
(0.17)
0.54
(0.17)
t
ret
17.29
(10.47)
5.26
(3.15)
4.74
(2.24)
9.30
(6.48)
6.51
(4.16)
Kt
Log L
-28.2
0.96
(0.49)
3.33
(2.87)
9.30
(6.48)
0.72
(0.44)
2.24
(1.15)
3.97
(2.30)
9.30
(6.48)
2.16
(1.32)
- 22.6
-24.2
- 17.0
-18.8
Constrained Model
Kt
17.29
(11.02)
8.30
(4.89)
8.04
(2.56)
7.89
(2.64)
7.79
(3.37)
Et LogLt
- 119.5
1.01
(0.57)
0.98
(0.35)
0.96
(0.39)
0.95
(0.49)
2.86
(1.68)
2.77
(0.94)
2.72
(1.00)
2.68
(1.26)
significantlydifferentfrom 0. The adjustmentparameters,f3t, are significantly
differentfrom 0, rangingfrom around0.5 in treatment F to just below 1.0 in
treatmentQ. The initialmean, a0, is noticeablyhigherin treatmentQ2,in which
(unlike in treatment F) the cost of failing to coordinateon the highest effort is
no greaterthan for lower efforts, and the simplicityof the payofffunctionmay
make the argumentfor the highest effort easier to apprehend.34However,ao is
not significantlyhigherin treatmentF, where highereffortsare also associated
with greaterefficiency,than in treatment P, where they are not; and the other
at are no higher in treatmentsF and Q than in treatment P. In treatmentQ
the at and
,t
are close to the values that would correspond to best-reply
dynamicsif there were no dispersion,but in the other treatmentsthe at and f3t
are significantlydifferentfrom both fictitiousplay and best-replydynamics.
With minor exceptions,the unconstrainedestimatesof the variancesot and
u2 decline smoothlytoward0 over time, except for an upwardjumpin ay2t from
34 MediantreatmentF combineda commonpreferencefor a highermedian,other thingsequal,
with increasinglysevere penalties for being furtherand further away from the median. Median
treatment' maintainedthese penaltieswhile eliminatingthe preferencefor a highermedian;and
median treatment Q2maintainedthe preference for a higher median while imposing different
penaltiesfor being awayfrom the median,usuallyhigherthan in treatmentF but independentof
the distance.See VHBB (1991)or Crawford(1991,Section2) for details.
134
VINCENT P. CRAWFORD
period 3 to period 4 in treatmentF. The estimatedcovariances,Kt, are almost
all insignificantlydifferentfrom 0. The implied values of the ; and o-2talso
decline smoothlytoward 0, except for a sharp upwardjump from period 0 to
period 1 in treatment 12. Inspectingthe data for the periods beyond those for
which estimatesare reportedreveals that maximumlikelihoodestimatesof o-t
and 2 would differ at most slightlyfrom 0, continuingtheir generallydownward trends and implyinga continuingdownwardtrend in the zot and ait.
The standarderrorsof both the unconstrainedand the constrainedestimates
indicate that constraintsthat rule out strategicuncertainty, t -7t - Kt -,
would be stronglyrejected in each case. The impressionthat the dispersionof
subjects'beliefs diminished steadily over time created by the unconstrained
estimatesis confirmedby the estimatesof A, the commonrate of decline of a,,,t,
2, and Kt in the constrainedspecification,which are significantlypositivein all
three mediantreatments:0.86 (0.34) in F, 3.27 (0.77) in Q, and 1.52 (0.53) in 1
(standarderrorsin parentheses).These estimatesdo not fully explainthe strong
convergenceVHBB observedin the median treatments,because in treatments
F and ' they are not significantlygreaterthan one as Proposition2's variance
condition requiresin this case. This might reflect the fact that Proposition2's
condition is sufficient,but not necessary,or that convergencewith probability
one is an unrealisticallystringentcriterion.
Likelihoodratio tests show that the constraintscannot be rejected in treatments Q and 0, but are stronglyrejectedin treatmentF. The X2 statisticsare
5.3 with 10 degrees of freedomfor treatmentQ, with p-value 0.87; 16.6 with 10
degrees of freedom for treatment 1, with p-value 0.08; and 67.8 with 25
degrees of freedom for treatment F. Otherwisethe constrainedestimates are
quite plausible,and similarto the unconstrainedestimates.
The rejectionin treatment F appearsto be due to the upwardjump in the
dispersionof players'responses from period 3 to period 4, which is likely to
violate any simple intertemporalrestrictions.As the rejected constraintsare
neither an implicationnor a necessarypart of the theory, the rejectionhas no
bearingon its validity.Because they can still be used to show that the model can
explain the dynamicswithout ad hoc variationsin the parameters,I have not
tried to find alternativeconstraintsthat would not be rejected.
The model does less well summarizingsubjects'behavior in the minimum
treatments.The unconstrainedmodel yields implausibleestimatesof the trend
and adjustmentparametersin period 9 of the large-grouptreatment, A, in
which significant end-of-treatmenteffects were visible, with some subjects
jumpingall the way from effort 1 to effort 7.35 The unconstrainedestimatesare
35These end effectsseemed to be due to subjects'perceptionthat the coordinationof the timing
of adjustmentsrequiredto breakout of an inefficientequilibriumwas unlikelyto be achievedin any
periodother than the last;see Crawford(1991,p. 57). Theywere strongestin the firstthree runsof
treatmentA, in whichthe endpointwas announced;but theywere visibleeven in the last four runs,
in which it was not announced,perhapsbecause subjectsbelieved that the experimentersor their
partnerswere habituatedto the decimalsystem.Estimatescomputedseparatelyfor each of:these
regimesdid not differenoughto justifyreportingthem separately.
COORDINATION
GAMES
135
otherwise plausible and reasonablystable over time. The initial mean, a0, is
higher in treatment A than in the random-pairingtreatment,C, and roughly
comparableto the initialmeans estimatedfor the mediantreatments.The trend
parameters, at for t = 1, .. ., are usually insignificantlydifferent from 0 in
treatment A and positive, often significantlyso, in treatmentC, showingsome
tendencyto decline over time in treatmentA but no clear trend in treatmentC.
The trend parametersare systematicallyhigherin treatmentC than in all other
treatments,suggestingthat subjectsmay have correctedfor the fact that their
currentpair minimumtends to underestimatethe median of the other subjects'
efforts that determined their best replies (see footnote 10). The adjustment
parameters,Pt, are usually significantlypositive in treatment A and positive,
often significantlyso, in treatment C. In each case the at and Pt differ
significantlyfrom the values impliedby fictitiousplay and best-replydynamics.36
In both minimumtreatmentsthe unconstrainedmodel yields estimatesof the
varianceparametersthat show only a weak tendencyto decline over time, with
several upwardjumps in the estimatedvalues of the a2t and 2 . This impression is confirmed by the constrained estimates of A, the rate of decline
(standarderrorsin parentheses):0.39 (0.11) in treatment A, significantlypositive but smallerthan in the median treatments;and 0.05 (0.59) in treatmentC,
insignificantlypositive. The convergenceobservedin treatment A is explained
in Section 7 by considerationsother than those addressedby Proposition2, so
the fact that its estimated A is significantlyless than 1 is no cause for concern.
The much smaller rate of decline in treatment C reflects its extremelyslow
convergence,which was to be expected, given the very noisy informationits
subjects received about the population median that determined their best
replies (see footnote 11).
The unconstrainedestimates of the Kt are generallypositive, usuallysignificantly so, in treatment A and positive, but generally insignificantlyso, in
treatment C, as in the median treatments.The fact that the estimatedcovariances are significantlydifferent from zero only in the large-groupminimum
treatmenthas a plausibleexplanation.Because all subjectsare tryingto forecast
the same summarystatistic based on the same experience, high values of sit
tend to be associatedwith high values of wit in treatmentA, where the relevant
summarystatisticis the minimum,so that the typicalsubject'seffort is above it,
and yt-1 - xit-1 < 0 in (4). In the other treatments,by contrast, the relevant
summarystatisticis the median, so that as manysubjects'effortsare above it as
below it.
Likelihood ratio tests show that the constraints are strongly rejected in
treatment A even when the last period is omitted, but that they cannot be
rejected in treatmentC. The relevant x2 statisticsare 107.4with 34 degrees of
freedomfor treatmentA; and 17.4 with 14 degreesof freedomfor treatmentC,
36Boylanand El-Gamal(1993) analyzedgrouped data from differentrandom-pairingexperiments, comparingrandomlyperturbedversionsof fictitiousplay and best-replydynamics.On the
assumptionthat subjects'unperturbedresponsesall followedeither one or the other of these rules,
they found that the evidencefavorsfictitiousplay.
136
VINCENT P. CRAWFORD
with p-value 0.24. The constrainedestimatesotherwiseseem plausible,and are
generallyquite close to the unconstrainedestimates.
Inspectingthe unconstrainedestimates suggests that the rejection in treatment A is again due to upwardjumps in the dispersionof subjects'responses,
which are likelyto violate any simple intertemporalrestrictions.As in treatment
F, this has no bearingon the validityof the theory,and I shall use the rejected
constraintsas simplifyingrestrictionsto show that the model can explain the
dynamicswithout ad hoc parametervariations.
7. COORDINATION
OUTCOMES
The parameterestimatescan be used to infer the probabilitydistributionsof
coordinationoutcomes from the results of a limited number of experimental
trials.Those distributionscan be approximatedanalyticallyas in Section 4, but
they can be estimated precisely only by repeated simulation. This section
describessimulationsbased on the constrainedparameterestimatesreportedin
Section 6 and uses the resultingestimateddistributionsto flesh out the explanation of VHBB's resultsoutlined in Section 4.
The estimateddistributionsplay an essentialrole in evaluatingthe model. As
explained in Section 3, the model can explain the dynamicsVHBB observed
only if there are significantdifferencesin players'beliefs. Because it does not
seem useful to try to explain such differences when players are otherwise
identical, the model treats them as error terms within a given stochastic
structure,and therefore determines a probabilitydistributionover outcomes
rather than predictinga particularoutcome. This inevitablylimits the model's
goodness of fit, so that it should be evaluated by comparingthe frequency
distributionsfrom the experimentswith the probabilitydistributionsit implies.
Such comparisonscan be made either conditionallyor unconditionally;both
kinds of test are reportedhere. The simulationstake into accountthe discreteness of the rounded y, and xi,, denoted 9, and xi. Because 9, and the x'
normallyapproacha commonlimit except in treatmentC, I simplifyby focusing
on 9t in treatmentsother than C.
I begin by describingthe results of the conditionalsimulationsand comparisons. These were designed to estimate the conditionalprobabilitydistribution
of 9t in the last period for which estimates are reported, given the realized
value of x .37 Because the realizedvalue of x_ differedacrossruns in each
treatment,a separateestimatebased on 500 simulationruns was computedfor
each. In order to test the model's ability to "predict"beyond sample, these
simulationswere based on parameterestimates computedunder the intertemporal constraintsdiscussedin Section 6 but omittingthe last period of data in
each treatmentbut A, and the last two periods of data in treatmentA. (These
estimatesare not reported,but are close to the constrainedestimatesin Tables
I-V.)
37To make the computationsmanageable,the x iti1 were used as proxiesfor the unobservable
xit_i. This appearsto have made little differenceto the results.
COORDINATION
GAMES
137
In all twelve experimentswith median games and all seven experimentswith
treatment A, the model assigns conditionalprobabilityof at least 0.998, given
the realizedvalue of x 1', to the value of 9, realizedin the last period.38These
fits are so close that formal tests are unnecessary.In the experimentwith
treatmentC consideredhere the model assignsconditionalprobability0.37 to 6,
the realized last-period median (the relevant summary statistic here) and
conditionalprobabilitiesof 0.53, 0.09, and 0.01 to the alternativevalues 7, 5, and
4.39 The x2 statisticfor a test of goodnessof fit is 1.7 with 6 degreesof freedom,
with p-value 0.94. (The analogoustest for the xi, is not independentof this test.
However,its x2 statisticis 3.8 with 6 degrees of freedom,with p-value 0.70.)
I conclude that the model conditionallypredicts the 9t well in every treatment. These are weak tests, because it is not difficult to make conditional
predictions of variables that vary as little over time as the 9t do. But this
weaknessis inherentin the data set, which simplydoes not allowvery powerful
tests of this kind.40
I now turn to the unconditionalsimulations,whichwere designedto estimate
the prior probabilitydistributionsof the 't. These simulationswere carriedout
usingboth the unconstrainedand the constrainedparameterestimates,with 500
runs for each treatment. I report here only the results for the constrained
estimates, despite the fact that the constraintswere rejected in treatments F
and A, because they provide a more stringent test of the model's ability to
explainthe dynamicsand help to answerthe criticismthat it can explainthem
only throughad hoc variationin its parameters.(The simulationresults for the
unconstrainedestimates, which are very similar, are reported in Crawford
(1992).)
Because the entire time paths of 9t are importantin discriminatingamong
alternativeexplanations,the results of these simulationsare reported for as
manyperiodsas the estimatespermittedin each treatment.The firstand second
lines of the cells in Tables VI-Xa reportthe actualfrequencydistributionsand
the estimatedprior probabilitydistributionsof the 't. The distributionfor the
last period should approximatethe commonlimitingdistributionof 9t and the
xit implied by the model in each treatmentbut C, in which convergencewas
slow. Table Xb supplementsthe results for the 9t in treatment C reported in
Table Xa with the analogousdistributionsfor the xit.
38All500 runsyieldedthe realizedmedianor minimumin all but one r and one Q2experiment,
in which all but one of 500 runsyielded the realizedmedian.
39Ambiguitiesof the mediandue to the evennessof n in treatmentC were resolvedby splitting
the weightof ambiguousobservationsbetweenthe two possiblevalues.
40 Breakingthe sample earlier and tryingto predictlonger historiesof the Yt would yield little
additionalinformationbecause the Yt do not change much even in the early periods, and would
make it even more difficultto estimatethe intertemporalrelationshipsrequiredfor beyond-sample
prediction.Comparingconditionalpredictionsof the xit with the experimentalresults in the
treatmentsother than C wouldamountto a joint test of the model'sabilityto conditionallypredict
the jt, which has alreadybeen tested, and the assumptionmaintainedin the simulationsthat the
(unrounded)xit are conditionallynormallydistributed,whichis a workinghypothesisratherthan a
substantiveimplicationof the model. (Inspectingthe data neverthelesssuggeststhat conditional
satisfied.)
normalityis approximately
138
VINCENT P. CRAWFORD
TABLE VI
'
DYNAMICS OF
IN MEDIAN TREATMENT F
(actual frequency distributions in first lines,
simulated frequency distributions in second lines)
7
6
5
4
3
2
1
Po
P,
92
93
94
95
96
0.00
0.00
0.00
0.06
0.50
0.62
0.50
0.31
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.09
0.50
0.56
0.50
0.31
0.00
0.03
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.10
0.50
0.53
0.50
0.33
0.00
0.03
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.11
0.50
0.52
0.50
0.32
0.00
0.04
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.11
0.50
0.53
0.50
0.32
0.00
0.04
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.11
0.50
0.52
0.50
0.32
0.00
0.04
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.11
0.50
0.52
0.50
0.32
0.00
0.04
0.00
0.01
0.00
0.00
The coarseness of experimentalfrequencydistributionsbased on 1-7 trials
makes them unlikely to resemble the probabilitydistributionsimplied by the
model closely. Formal tests suggest, however, that the model replicates the
dynamicsin every treatment.The most informativetests appearto be x2 tests
of goodnessof fit comparingthe estimatedpriordistributionsfor the last-period
At with the correspondingexperimentalfrequencydistributions,because it is
harder to track the prior distributionsfor the last period than for earlier
TABLE VII
DYNAMICS OF
A
IN MEDIAN TREATMENT Q2
(actual frequency distributions in first lines,
simulated frequency distributions in second lines)
7
6
5
4
3
2
1
PO
Pi
92
93
0.67
0.34
0.00
0.55
0.33
0.10
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.67
0.37
0.00
0.47
0.33
0.15
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.67
0.36
0.00
0.47
0.33
0.15
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
0.67
0.36
0.00
0.47
0.33
0.15
0.00
0.02
0.00
0.00
0.00
0.00
0.00
0.00
139
GAMES
COORDINATION
TABLE VIII
DYNAMICS OF Y IN MEDIAN TREATMENT 'P
(actual frequency distributions in first lines,
simulated frequency distributions in second lines)
7
6
5
4
3
2
1
Yo
Yi
Y2
Y3
0.00
0.00
0.00
0.04
0.67
0.68
0.33
0.28
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.67
0.66
0.33
0.29
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.67
0.66
0.33
0.29
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.05
0.67
0.66
0.33
0.29
0.00
0.00
0.00
0.00
0.00
0.00
TABLE IX
DYNAMICS OF Y IN LARGE-GROUP
MINIMUM TREATMENT
A
(actual frequency distributions in first lines,
simulated frequency distributions in second lines)
7
6
5
4
3
2
1
Y0
pi
Y2
P3
Y4
Y5
0.00
0.00
0.00
0.00
0.00
0.00
0.29
0.09
0.14
0.37
0.29
0.33
0.29
0.21
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
0.00
0.04
0.57
0.18
0.43
0.78
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.14
0.01
0.29
0.03
0.57
0.96
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.01
1.00
0.99
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
1.00
1.00
periods.41 These tests never reject in any treatment.The relevantx2 statistics,
each with six degrees of freedom, are 2.54 in treatment C, with p-value 0.86;
0.00 in treatment A; 1.55 in treatment F, with p-value 0.95; 2.95 in treatment
Q, with p-value 0.81; and 0.16 in treatment 1, with p-value 0.99. (The
41 More powerful tests involve entire histories of Yt and/or
xi,. These are complicated by
statistical dependence, and the results suggest that rejection is unlikely.
140
VINCENT P. CRAWFORD
TABLEXa
DYNAMICS OF Yt IN RANDOM-PAIRING
MINIMUM TREATMENT C
(actualfrequencydistributionsin firstlines,
simulatedfrequencydistributionsin secondlines)
7
6
5
4
3
2
1
Y0
Y1
92
93
Y4
0.00
0.06
0.00
0.14
0.00
0.09
0.00
0.21
0.00
0.18
0.00
0.26
1.00
0.30
0.00
0.27
0.00
0.38
1.00
0.28
0.00
1.00
1.00
0.00
0.00
0.25
1.00
0.29
0.00
0.19
0.00
0.08
0.29
0.00
0.27
0.00
0.11
0.00
0.02
0.28
0.00
0.21
0.00
0.06
0.00
0.01
0.26
0.00
0.14
0.00
0.04
0.00
0.00
0.22
0.00
0.10
0.00
0.01
0.00
0.00
0.00
0.01
0.00
0.00
0.00
0.00
0.00
0.00
0.00
0.00
analogoustests for earlier periods and/or the unconstrainedestimates,which
are not independent,would yield the same conclusions,with the exception of
the constrainedestimates for period 2 in treatment A.) The model replicates
the unconditionalfrequencydistributionsfrom VHBB's experiments.
The statisticalanalysissuggeststhat VHBB's resultswere not anomalous.In
some cases, however,the model strengthensor modifiesthe impressionscreated
by the raw data. The inefficientoutcomes they observedin treatment A now
appear inevitable, appearingin all 500 simulationruns. It appears, however,
TABLEXb
DYNAMICS OF THE Xit IN RANDOM-PAIRING
MINIMUM TREATMENT C
(actualfrequencydistributionsin firstlines,
simulatedfrequencydistributionsin secondlines)
iio
7
6
0.31
0.30
0.00
iXj
Xi2
Xi3
ii4
0.31
0.33
0.06
0.25
0.37
0.19
0.63
0.42
0.00
0.50
0.46
0.00
0.09
0.09
0.10
0.11
0.11
5
0.13
0.31
0.19
0.19
0.25
0.09
0.11
0.11
0.10
0.10
4
0.19
0.06
0.06
0.06
0.06
0.09
0.11
0.11
0.10
0.09
3
0.06
0.09
0.06
0.08
0.25
0.25
0.06
0.10
0.06
0.07
0.13
0.20
0.06
0.10
0.13
0.06
0.13
0.16
0.00
0.08
0.13
0.06
0.00
0.13
0.00
0.07
0.13
0.05
0.06
0.12
2
1
COORDINATION
GAMES
141
that playersmight do better than VHBB's subjectsin treatmentF, and worse in
treatment Q. The model also suggests somewhatmore stronglythan the data
that treatmentC is likely to yield a high-effortequilibrium.
The model attributesthe completelack of history-dependencein treatmentA
to the stronglynegative drift in this treatment,which overwhelmedits variance
so that the lower bound on players' strategies determined the final outcome
with high probability.The stronghistory-dependencein the median treatments
and the moderatehistory-dependencein treatmentC can be traced to the low
to moderate drifts in these treatments,the large conditionalvariancesof y, in
treatment C, and the small conditionalvariances of y, in the median treatments. The analysis suggests, however, that the perfect history-dependence
VHBB observedin all twelve of their experimentswith mediangamesoverstates
the importanceof historicalaccidents in such environments.In the simulation
results the correlation between yO and the last yt reported is only 0.68 in
treatment r, 0.76 in treatment (2, and 0.90 in treatment 1, so that the initial
median "explains"only 46%, 58%, and 81% of the variancesof the limiting
outcomes in these treatments.
8. CONCLUSION
The model providesa simple,unified explanationfor the complexpatternsof
history-dependenceand discriminationamong equilibria in VHBB's experiments. Three directionsfor furtherresearchseem promising.
It would be desirable to conduct similar experiments,in order to test the
theory'spredictionsbeyond the samplesthat influencedits specification,and to
learn more about how the strategic environmentinfluences the dispersionof
players' beliefs and the rate at which it is eliminated as they accumulate
forecastingexperience.
It would also be of interest to relax the assumptionthat players'beliefs and
strategychoices are statisticallyidenticalex ante. The substanceof this assumption is the absence of externallyobservabledifferencesbetween players.It may
be tractable to allow, instead, a fixed population made up of two or more
observable "types" of players, with each player's payoffs and best replies
determined in each play by his type, his strategy choice, and the current
populationdistributionof strategychoices by type. A random-pairingmodel in
game,
whichplayers'types are identifiedwith their roles in a "divide-the-dollar"
as in Young (1993b), seems of particularinterest. Such a model, in which
players'preferencesover equilibriaare opposed, is a naturalcomplementto the
present analysisand may help in interpretingexperimentalevidenceon bargaining outcomes (see Roth (1987)).
Finally,it would be of interest to extend the analysisfrom tacit coordination,
in whichplayerscommunicateonly by playingthe game, to explicitcoordination,
in which playerscan send signalsthat are not directlyrelated to the game (and
which may not be costly). Tacit coordinationraises many issues that must be
resolved to understandexplicit coordination,and generalizingthe techniques
developed here may yield useful models of explicit coordination.A natural
142
VINCENT P. CRAWFORD
place to begin is games with one or more rounds of costless, simultaneous
pre-playcommunication(see Farrell(1987), Palfreyand Rosenthal (1991), and
Crawford(1990)) or the games with costly pre-play communicationused in
VHBB's (1993) experiments(see Crawfordand Broseta (1994)). Considering
how the meaningsof players'messages and their decisions evolve may resolve
the ambiguitythat plagues traditionalanalyses of preplaycommunicationand
shed new light on proposedequilibriumrefinements.
Dept. of Economics,Universityof California,San Diego, 9500 GilmanDr., La
Jolla, CA 92093-0508,U.S.A.
ManuscriptreceivedJanuary, 1992; final revision received May, 1994.
REFERENCES
BANKS, JEFFREY, CHARLES PLOTr, AND DAVID PORTER
(1988): "An ExperimentalAnalysis of
Unanimity in Public Goods Provision Mechanisms," Review of Economic Studies, 55, 301-322.
BOYLAN,
RICHARD,
AND
MAHMOUD
EL-GAMAL
(1993): "Fictitious Play: A StatisticalStudy of
Multiple Economic Experiments," Games and Economic Behavior, 5, 205-222.
BROSETA, BRUNO (1993a): "Strategic Uncertainty and Learning in Coordination Games," Discussion Paper 93-34, University of California, San Diego.
(1993b): "Estimation of a Game-Theoretic Model of Learning: An Autoregressive Conditional Heteroskedasticity Approach," Discussion Paper 93-35, University of California, San
Diego.
BRYANT, JOHN (1983): "A Simple Rational Expectations K6ynes-Type Model," QuarterlyJournal of
Economics, 98, 525-528.
COOPER, RUSSELL, DOUGLAS DEJONG, ROBERT FORSYTHE, AND THOMAS Ross (1990): "Selection
Criteria in Coordination Games: Some Experimental Results," American Economic Review, 80,
218-233.
COOPER,
RUSSELL, AND ANDREW
JOHN
(1988):"CoordinatingCoordinationFailuresin Keynesian
Models," QuarterlyJournal of Economics, 103, 441-463.
Equilibriain EvolutionaryGames,"
CRAWFORD, VINCENT P. (1989):"Learningand Mixed-Strategy
Journal of Theoretical Biology, 140, 537-550; to be reprinted in Proceedings of the Second
Workshop on Knowledge, Belief, and Strategic Interaction, ed. by Cristina Bicchieri and Brian
Skyrms. New York: Cambridge University Press.
(1990): "Explicit Communication and Bargaining Outcomes," American Economic Review
Papers and Proceedings, 80, 213-219.
(1991): "An 'Evolutionary' Interpretation of Van Huyck, Battalio, and Beil's Experimental
Results on Coordination," Games and Economic Behavior, 3, 25-59.
(1992): "Adaptive Dynamics in Coordination Games," Discussion Paper 92-02R, University
of California, San Diego.
Auctioningthe
CRAWFORD, VINCENT P., AND BRUNO BROSETA (1994):"WhatPrice Coordination?
Right to Play as a Form of Preplay Communication," manuscript in preparation, University of
California, San Diego.
CRAWFORD,VINCENT P., AND HANS HALLER (1990): "LearningHow to Cooperate:OptimalPlay in
Repeated Coordination Games," Econometrica, 58, 571-595.
FARRELL, JOSEPH (1987): "Cheap Talk, Coordination, and Entry," Rand Journal of Economics, 18,
34-39.
GameDynamics,"TheoretFOSTER, DEAN, AND H. PEYTONYOUNG (1990): "StochasticEvolutionary
ical Population Biology, 38, 219-232.
FREIDLIN, MARK, AND ALEXANDER WENTZELL (1984): Random Perturbationsof Dynamical Systems.
New York: Springer-Verlag.
HARSANYI, JOHN, AND REINHARD SELTEN (1988): A General Theory of Equilibrium Selection in
Games. Cambridge, Massachusetts: MIT Press.
COORDINATION
HARVEY,
ANDREW
GAMES
143
(1989): Forecasting, Structural Times Series Models and the Kalman Filter.
Cambridge,U.K., and New York:CambridgeUniversityPress.
ISAAc,R. MARK, D. SCHMIDTZ, AND J. WALKER (1989):"The AssuranceProblemin a Laboratory
Market," Public Choice, 62, 217-236.
JONES,
HOWARD
(1948): "Exact Lower Moments of Order Statisticsin Small Samples from a
Normal Distribution," Annals of Mathematical Statistics, 19, 270-273.
MICHIHIRO, GEORGE MAILATH, AND RAFAEL ROB (1993):"Learning,Mutation,and Long
Run Equilibriain Games,"Econometrica,61, 29-56.
KIM YONG-GwAN (1990): "EvolutionaryAnalysisof Two-PersonFinitely Repeated Coordination
Games,"manuscript,Universityof California,San Diego.
KANDORI,
LJUNG, LENNART, AND TORSTEN SODERSTROM'(1983):
Theoryand Practice of Recursive Identification.
Cambridge,Massachusetts:M.I.T.Press.
MAYNARD SMITH, JOHN (1982):Evolution and the Theory of Games. Cambridge,U.K.: Cambridge
University Press.
DANIEL (1984): "EconometricAnalysisof QualitativeResponse Models,"Ch. 24 in
Handbookof Econometrics,VolumeII, ed. by Zvi Grilichesand MichaelIntriligator.Amsterdam
and New York:ElsevierSciencePublishing.
RICHARD, AND THOMAS PALFREY (1992): "An ExperimentalStudy of the Centipede
McKELVEY,
McFADDEN,
Game," Econometrica, 60, 803-836.
M. B., AND R. Z. HAS'MINSKII (1973):Stochastic Approximation and Recursive Estimation, Vol. 47, Translationsof MathematicalMonographs.Providence,Rhode Island:American
MathematicalSociety.
PALFREY, THOMAS, AND HOWARD ROSENTHAL (1991): "Testingfor Effectsof CheapTalkin a Public
Goods Gamewith PrivateInformation,"Gamesand EconomicBehavior,3, 183-220.
ROBLES, JACK (1994):"Evolutionand Long Run Equilibriain CoordinationGameswith Summary
StatisticPayoffTechnologies,"DiscussionPaper94-12,Universityof California,San Diego.
ROTH, ALVIN (1987):"BargainingPhenomenaand BargainingTheory,"Chapter2 of Laboratory
in Economics,ed. by Alvin Roth. New York:CambridgeUniversityPress.
Experimentation
Games:ExperimentalData and
ROTH, ALVIN, AND IDO EREV (1995): "Learningin Extensive-Form
SimpleDynamicModelsin the IntermediateTerm,"GamesandEconomicBehavior,8, forthcoming.
ROTH, ALVIN, AND FRANCOISE SCHOUMAKER (1983):"Expectationsand Reputationsin Bargaining:
NEVEL'SON,
An Experimental Study," American Economic Review, 73, 362-372.
in The Social Contract
JEAN-JACQUES (1973):"A Discourseon the Originof Inequality,"
and Discourses(translatedby G. D. H. Cole). London:J. M. Dent & Sons, Ltd., 27-113.
SCHELLING, THOMAS (1960): The Strategy of Conflict, First Edition. Cambridge,Massachusetts:
ROUSSEAU,
Harvard University Press.
D. (1956): "Tables of Expected Values of Order Statisticsand Productsof Order
Statistics for Samples of Size Twenty and Less from the Normal Distribution,"Annals of
TEICHROEW,
Mathematical Statistics, 27, 410-426.
HUYCK, JOHN, RAY BATTALIO, AND RICHARD BEIL (1990): "Tacit CoordinationGames,
StrategicUncertainty,and CoordinationFailure,"AmericanEconomicReview,80, 234-248.
-~
(1991):"StrategicUncertainty,EquilibriumSelectionPrinciples,and CoordinationFailurein
VAN
-~
Average Opinion Games," QuarterlyJournal of Economics, 106, 885-910.
(1993): "Asset Markets as an Equilibrium Selection Mechanism: Coordination Failure,
Gamesand EconomicBehavior,5, 485-504.
Game FormAuctions,and Tacit Communication,"
"Learningto Believe in Sunspots,"Econometrica, 58, 277-307.
Econometrica, 61, 57-84.
YOUNG, H. PEYTON (1993a):"The Evolutionof Conventions,"
-~
(1993b):"An EvolutionaryModel of Bargaining,"Journalof EconomicTheory,59, 145-168.
WOODFORD, MICHAEL(1990):