American Economic Association
Tacit Coordination Games, Strategic Uncertainty, and Coordination Failure
Author(s): John B. Van Huyck, Raymond C. Battalio, Richard O. Beil
Source: The American Economic Review, Vol. 80, No. 1 (Mar., 1990), pp. 234-248
Published by: American Economic Association
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Tacit Coordination Games, Strategic Uncertainty,
and Coordination Failure
By JOHN B. VAN HUYCK, RAYMOND C. BATTALIO, AND RICHARD 0. BEIL*
Deductive equilibriummethods-such as
Rational Expectations or Bayesian Nash
Equilibrium-are powerfultools for analyzing economies that exhibit strategicinterdependence. Typically, deductive equilibrium
analysis does not explain the process by
which decision makers acquire equilibrium
beliefs. The presumption is that actual
economies have achieveda steady state. In
economies with stable and unique equilibrium points, the influence of inconsistent
beliefs and, hence, actions would disappear
over time, see Robert Lucas (1987). The
power of the equilibriummethod derives
from its ability to abstractfrom the complicated dynamic process that inducesequilibrium and to abstractfromthe historicalaccident that initiatedthe process.
Unfortunately, deductive equilibrium
analysis often fails to determinea unique
equilibriumsolutionin manyeconomiesand,
hence, often fails to prescribeor predictrational behavior.In economieswith multiple
equilibria, the rational decision maker formulatingbeliefs using deductiveequilibrium
*Department of Economics, Texas A&M University,
College Station, TX 77843; Department of Economics,
Texas A&M University, College Station, TX 77843; and
Department of Economics, Auburn University, Auburn,
AL 36849, respectively. Mike Baye, John Bryant, Cohn
Camerer, Russell Cooper, Vincent Crawford, John
Haltiwanger, Pat Kehoe, Tom Saving, Steve Wiggins,
Casper de Vries, a referee, and seminar participants at
the NBER/FMME 1987 Summer Institute, the 1988
meetings of the Economic Science Association, the University of New Mexico, and Texas A&M University
made constructive comments on earlier versions of this
paper. Sophon Khanti-Akom, Kirsten Madsen, and Andreas Ortmann provided research assistance. The National Science Foundation, under grants no. SES8420240 and SES-8911032, the Texas A&M University
Center for Mineral and Energy Research, and the Lynde
and Harry Bradley Foundation provided financial support. A Texas A&M University TEES Fellowship has
supported Battalio.
234
concepts is uncertain which equilibrium
strategyother decision makerswill use and,
when the equilibriaare not interchangeable,
this uncertaintywill influence the rational
decision-maker'sbehavior. Strategicuncertainty arises even in situationswhereobjectives, feasible strategies, institutions, and
equilibriumconventionsarecompletelyspecified and are common knowledge. While
multiple equilibriaare common in theoretical analysis, consideration of specific
economies suggests that many equilibrium
points are implausible and unlikely to be
observedin actualeconomies.
One response to multipleequilibriais to
arguethat some Nash equilibriumpoints are
not self-enforcingand, hence, are implausible, because they fail to satisfy one or more
of the following refinements:eliminationof
individuallyunreasonableactions,sequential
rationality, and stability against perturbations of the game-see Elon Kohlburgand
Jean-FrancoisMertens(1986) for examples
and references.Equilibriumrefinementsdetermine when an outcome that is already
expected would be implementedby rational
decision makers.
In general,many outcomeswill satisfythe
conditionsof a givenequilibriumrefinement.
The equilibriumselectionliteratureattempts
to determine which, if any, self-enforcing
equilibriumpoint will be expected.A satisfactory theory of interdependentdecisions
must not only identifythe outcomesthat are
self-enforcingwhen expectedbut also must
identify the expectedoutcomes.Consequently, a theory of equilibriumselection would
be a useful complement to the theory of
equilibriumpoints.
The experimental method provides a
tractable and constructiveapproachto the
equilibrium selection problem. This paper
studies a class of tacit pure coordination
games with multiple equilibria,which are
VOL. 80 NO.1
strictly Paretoranked,and it reportsexperiments that provide evidenceon how human
subjectsmake decisionsunderconditionsof
strategicuncertainty.
Game
I. A Pure Coordination
To focus the analysisconsiderthe following tacit coordination game, which is a
strategic form representation of John
Bryant's (1983) Keynesian coordination
game. The baseline game is defined as follows: Let el,..., en denote the actions taken
by n players.The period gameA is defined
by the following payoff function and strategy space for each of n players:
(1)
235
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES
a(ei, e1) = a [min(ei, e1)] - bei,
a > b > O,
where ei equalsmin(el,..., ei_1,ei+,,..., en).
Actions are restrictedto the set of integers
from 1 to J. The players have complete
information about the payoff function and
strategy space and know that the payoff
function and strategy space are common
knowledge.'
If the players could explicitlycoordinate
their actions, the-real or imagined
planner'sdecision problemwould be trivial.
Given a - b greater than 0, each player
should choose the maximumfeasibleaction,
e. Moreover, a negotiated "pregame" agree-
ment to choose e- would be self-enforcin*.
Unlike games with incentiveproblems,here
'Apparently, this game is similar to Rousseau's "stag
hunt" parable, which he used to motivate his analysis of
the social contract, see Crawford (1989, p. 4). In the
stag hunt game, each hunter in a group must allocate
effort between hunting a stag with the group and hunting rabbits by himself. Let e, denote effort expended on
the stag hunt. Since stag hunting in that era required the
coordinated effort of all the hunters, the probability of
successfully hunting a stag depends on the smallest e.
The parameter a in equation (1) reflects the benefits of
participating in the stag hunt: eating well should the
hunt succeed. Hunting rabbits does not require coordination with the other hunters. The parameter b in
equation (1) reflects the opportunity cost of effort allocated to the stag hunt that could have been allocated to
rabbit hunting: a meal-however, meager.
the firstbest outcomeis an equilibriumpoint.
However,when the playerscannotengagein
"pregame"negotiationthey face a nontrivial
coordinationproblem.
Suppose that the players attempt to use
the Nash equilibriumconceptto informtheir
strategic behavior in the tacit coordination
game A. A player'sbest responseto ei is to
choose ei equal to ei. By symmetryit follows
that any n-tuple (e,...,
e)
with e E {1,
2,..., e-} satisfies the mutual best response
property of a Nash equilibriumpoint. All
feasible actions are potentialNash equilibrium outcomes. The Nash concept neither
prescribesnor predictsthe outcome of this
tacit coordinationgame.
(Standardequilibriumrefinementsdo not
reducethe set of equilibria.For example,the
equilibria are strict-each player has a
unique best response-and, hence, trembling-handperfect.)
II. Coordination
ProblemsandEquilibrium
SelectionPrinciples
The analysis in Section I follows convention and abstractsfrom the equilibriumselection problem. However,a rationalplayer
using deductive equilibriumconcepts confronts two nontrivialcoordinationproblems
in period game A. First, playersmay fail to
correctly forecast the minimum, ei, and,
hence, regret their individualchoice, that is,
ei = ei. This type of coordinationfailureresults in disequilibrium:outcomesthat do not
satisfy the mutual best-responsepropertyof
an equilibrium.
An equilibriumselectionprincipleidentifies a subset of equilibriumpoints according
to some distinctivecharacteristic.An interesting conjectureis that decisionmakersuse
some selectionprincipleto identifya specific
equilibrium point in situations involving
multiple equilibria.This selection principle
would solve the problemof coordinatingon
a specificequilibriumpoint. Hence, the outcome will satisfy the mutual best-response
propertyof an equilibrium.
A second coordination problem arises
when the equilibriacan be Pareto ranked.
In such situations, all players may give a
best response,but, nevertheless,implementa
236
THE AMERICAN ECONOMIC REVIEW
Pareto dominated equilibrium, that is,
en) # J. Whilenot regrettingtheir
individualchoice, they regretthe equilibrium
implemented by these individual choices.
Consequently,the outcomeresultsin coordination failure. What equilibriumselection
principlescould a playeruse to resolvethese
two coordinationproblems?2
Deductive selectionprinciplesselect equilibrium points based on the descriptionof
the game. Deductiveselectionprinciplespreserve the equilibrium method's desirable
propertyof independencefromhistoricalaccidents and from complicateddynamicprocesses. Inductive selection principles select
equilibriumpoints based on the history of
some pregame.3Hence, inductive selection
principles are not independentof accident
and process.
When multiple equilibriumpoints can be
Pareto ranked,it is possible to use concepts
of efficiency to select a subset of selfenforcing equilibriumpoints: examples include R. Duncan Luce and HowardRaiffa's
(1956, p. 106) concept of joint-admissibility,
Tamer Basar and Geert Olsder's(1982, p.
72) concept of admissibility, and John
Harsanyiand ReinhardSelten's(1988,p. 81)
concept of payoff-dominance.An equilibif it
riumpoint is said to be payoff-dominant
is not strictlyParetodominatedby any other
equilibriumpoint. When unique,considerations of efficiency may induce players to
focus on and, hence, select the payoffdominant equilibrium point, see Thomas
Schelling(1980, p. 291).
In period game A, the equilibriumpoints
are strictly Pareto ranked. Each player
prefers a larger minimum.The only equilibrium point not Pareto dominated by
any other equilibriumpoint is the n-tuple
min(el,...,
(e,...,
e):
the payoff-dominant equilibrium
point. Consequently,payoff-dominanceselects the n-tuple(e,..., e-) in gameA.
2The "disequilibrium
Keynesians"
emphasizethefirst
coordinationproblem. The "equilibriumKeynesians"
emphasizethe secondcoordinationproblem,see Cooper
and John (1988) for examplesand references.
3We use the term inductionin the logical, rather
than mathematical,sense of reasoningfrom observed
facts-history-to a conclusion.
MARCH 1990
Selecting the unique payoff-dominant
equilibriumpoint not only allows playersto
coordinateon an equilibriumpoint but also
ensures that they will not coordinateon an
inefficientone. Payoffdominancesolvesboth
the individual and the collective coordination problemsof disequilibriumand coordination failure and, as Harsanyiand Selten
suggest, should take precedenceover alternative selectionprinciples.
The tacit coordinationgameA providesa
severe test of payoffdominance,becausethe
minimum rule exacerbatesthe influenceof
uncertaintyabout the strategiesof the other
n -1 players. Define the cumulativedistribution functionfor a player'sactionas F(e.).
In the payoff-dominantequilibrium,F&e)
equals 1 and F(e1) equals0 for ej less than
e. A well-known theorem is that if el,..., en
are independent and identicallydistributed
with common cumulativedistributionfunction F(ej), then the cumulativedistribution
function for the minimum, F.,n(e), equals
1- [1 -F(e)];
see A. M. Mood, F. A.
Graybill, and D. C. Boes (1974). In the
payoff-dominantequilibrium,Fmin(e-)
equals
1 and Fmin(e) equals 0 for e less than e-.But
suppose that a player is uncertainthat the
n -1 playerswill select the payoff-dominant
action, J. Specifically, let F(1) be small but
greater than 0, then as n goes to infinity
Fmin(l) goes to 1. Consequently,when the
number of players is large it only takes a
remote possibility that an individualplayer
will not select the payoff-dominantaction eto motivate defection from the payoffdominantequilibrium.
Several deductive selection principles
based on the "riskiness"of an equilibrium
point have been identifiedand formalized.A
maximin action, which is an action (pure
strategy)with the largestpayoffin the worst
possible outcome, is secure, see John Von
Neumann and Oskar Morgenstern(1944,
1972). Given existence, security selects the
equilibrium point supported by player's
maximin actions. Securitymay select very
inefficientequilibriumpointsin nonzerosum
games.
In period game A, a player can ensurea
payoff of a - b by choosing ei equal to 1,
which is the largestpayoffin the worstpossi-
VOL. 80 NO. I
VAN HUYCKETAL: TACIT COORDINATION GAMES
237
TABLE1-EXPERIMENTAL DESIGN
Experiment
No.
Date
Size
1
2
3
4
5
6
7
June
June
June
Sept
Sept
Sept
Sept
16
16
14
15
16
16
14
A
Payoff A
Fullsize
B
Payoff B
Fullsize
1P,2,... 10
1P,2,..., 10P
1P,2,... 10P
1P,2P,..., 10P
1P,2P".. ,1OP
lP,2P,..,10lP
1P,2P,..0. lop
-
11,...,15
11,...115
11P,"...15
11p...,15
11"..., 15
1ip,1... 15
A'
Payoff A
Fullsize
C
Payoff A
Size Twoa
-
-
16P,..., 20
16P, ... ,20
16,...,20
16,.,20
20
16,
16,...,22
21, ,27
21,.,27
21,.,25
23,...,25
P- Denotes a period in which subjects made predictions.
- - In experiment 4 and 5 pairings were fixed, while in experiments 6 and 7 pairings were random.
Notice that the principlesof efficiencyand
security can be defined independentlyof
equilibrium.The experimentsin this paper,
(1,...,1).
Since payoff-dominance and security select differentequilibriumpoints in this which are designedto study the conflictbetacit coordinationgame (equilibriumpoints tween efficiency and security, are not dewith the highest and lowest payoffs,respec- signed to study how repeatedplay of period
tively), an importantand tractableempirical game A influences the set of equilibrium
questionis which, if any, deductiveselection points for A(T).
principleorganizesthe experimentaldata.
Having t periods of experiencein A(T)
It is often possibleto applymorethanone
provides a player with observed facts, in
deductive selection principle to a game. addition to the descriptionof the game,that
Hence, subjects may choose disequilibrium can be used to reasonabout the equilibrium
selection problem in the continuationgame
outcomes unless they behaveas if thereis a
hierarchyof selection principles.When de- A(T - t). This experience may influence the
ductive selectionprinciplesfail to coordinate outcome of the continuationgame A(T- t)
beliefs and actions, inductiveselectionprin- by focusing expectationson a specificequiciples based on repeated interaction may librium point. For example, one adaptive
hypothesis is that players will give a best
allow players to learn to coordinate.
response to the minimumobserved in the
Consider a finitely repeatedgame A(T),
which involves the n playersplayingperiod previous period. This adaptive behavior
game A for T periods.The payoff-dominant would immediatelyconvergeto an equilibequilibrium of A(T) is just the repeated rium in A(T-1). The selected equilibrium
implementation of the payoff-dominant involves all players choosing the period 1
minimum for the T- 1 periods of the conequilibriumof period game A, because the
tinuationgame A(T -1).
first-best outcome for period game A is
ble outcome. Consequently,in this tacit coordinationgame, securityselects the n-tuple
(e,..., e). Similarly, the secure equilibrium
of A(T) is just the repeatedimplementation
of the secureequilibriumof periodgameA.4
4Crawford (1989)emphasizesthat the secureequilibrium is also the only equilibriumthat is evolutionary
stable.In repeatedplay,playersusingadaptivebehavior
may be led to implementthe secureequilibrium.Hence,
while the experimentsreportedin this paper discriminate sharplybetween strategicstabilityand evolutionary stability,they do not discriminatebetweenlearning
to use securityand certainkinds of adaptivebehavior.
III. Experimental
Design
Table 1 outlines the design of the seven
experimentsreportedin this paper. The instructionswere read aloud to ensurethat the
descriptionof the game was commoninformation,if not, commonknowledge.'No pre5The originalworkingpaper,Van Huyck, Battalio,
and Beil (1987), which includesthe actualinstructions,
MARCH 1990
THE AMERICAN ECONOMIC REVIEW
238
PAYOFFTABLE A
Smallest Value of X Chosen
Your
Choice
of
X
7
6
5
4
3
2
1
7
6
5
4
3
2
1
1.30
-
1.10
1.20
-
0.90
1.00
1.10
-
0.70
0.80
0.90
1.00
-
0.50
0.60
0.70
0.80
0.90
-
0.30
0.40
0.50
0.60
0.70
0.80
-
0.10
0.20
0.30
0.40
0.50
0.60
0.70
-
-
PAYOFFTABLE B
Smallest Value of X Chosen
Your
Choice
of
X
X
7
6
5
4
3
2
1
7
6
5
4
3
2
1
1.30
-
1.20
1.20
-
1.10
1.10
1.10
-
1.00
1.00
1.00
1.00
-
0.90
0.90
0.90
0.90
0.90
-
0.80
0.80
0.80
0.80
0.80
0.80
-
0.70
0.70
0.70
0.70
0.70
0.70
0.70
play negotiation was allowed. After each
repetitionof the periodgame, the minimum
action was publicly announcedand the subjects calculated their earningsfor that period. The only commonhistoricaldata available to the subjectswas the minimum.
During the courseof an experimentsome
design parameterswerealteredresultingin a
sequence of treatmentslabeled A, B, A',
and C. Instructionsfor continuationtreatments were given to the subjectsafterearlier
treatmentshad been completed.The feasible
actions in all treatmentsof all experiments
were the integers 1 through 7: hence, e
equaled7.
In treatmentA and A', the followingvalues were assignedto the parametersin equation (1): parametera was set equal to $0.20,
parameterb was set equal to $0.10, and a
payoff tables, questionnaire,extra instructions and
recordsheet used in the experimentsand a moreextensive analysis of the experimentalresults, is available
from the authorsupon request.
-
-
-
constant of $0.60 was added to ensurethat
all payoffs were strictly positive. Consequently, the payoff-dominantequilibrium,
(7, ... , 7), paid $1.30 while the secure equilibrium, (1,...,1),
paid $0.70 per subject per
period.6Subjectsweregiventhis information
in the form of a payoff table, see payoff
Table A. In treatmentA, the periodgameA
was repeatedten times.The numberof players, n, varied between 14 and 16 subjects.
(In treatment C, the numberof players, n,
was reduced to two.) TreatmentA' designates the resumptionof these conditionsafter treatmentB.
In treatment B, parameterb in equation
(1) was set equal to zero,see payoffTable B.
This gives the subjectsa dominatingstrategy
(play 7 regardlessof the minimum),which
eliminates the coordination problem. The
numberof players, n, remainedthe same as
in treatmentA.
6For the remainderof this paper, an equilibrium
denotes a mutual best-responseoutcome in the period
game.
VOL. 80 NO. 1
VAN HUYCK ETAL: TACIT COORDINATION GAMES
239
Occasionally,subjectswere asked to pre- riod one was never greaterthan 4. Hence,
dict the actions of all the subjects in the
the largest payoff in period one was $1.00
treatment.7 For each prediction in the
and some payoffs were $0.10. (The payoffSeptemberexperiments,a subjectwas paid
dominant equilibriumwould have paid ev$0.70 less 0.02 times the sum of the absolute eryone $1.30). All of these outcomes are
value of the differencebetween the actual inefficient.The subjectswere unable to use
and predictedactions.(The rule used in the
any deductive selectionprincipleto coordiJune experimentswas less sensitive to pre- nate on an equilibriumpoint.
diction errors).At the end of the experiment,
Only 10 percentof the subjectspredicted
the subjectswere told the actualdistribution an equilibriumoutcome in period one. Inof actionsand werepaid.
stead, most subjects(95 of 106) predicteda
The subjectswere undergraduatestudents disequilibriumoutcome.8Moreover,the subattending Texas A&M Universityand were jects' predictionswere dispersed:one third
recruited form sophomoreand junior ecoof the subjectspredictedat least one 1 and
nomics courses.A total of 107 studentspar- one 7-a rangeof 6-and the averagerange
ticipated in the seven experiments.After of the predictionswas 4.0. The subjects'disreading the instructions,but before the ex- persedpredictionssuggestthat they expected
periment began, the students filled out a
other subjectsto respondto the payoff table
questionnaireto determinethat they under- differently than they did. These data are
stood how to read the payofftable for treat- inconsistent with any theoryof equilibrium
ment A, that is, map actions into money selection that assumesthat, becausea player
payoffs. The instructionswould have been will derive his prior probabilitydistribution
re-read if needed, but all 107 students re- over other players' pure strategies strictly
from the parametersof the game,all players
sponded correctly.
will have the samepriorprobabilitydistribuIV. Experimental
Results
tion. Instead, some subjectsmade optimistic
predictions and some subjects made pesTable 2 reports the experimentalresults simisticpredictions.
for treatmentA. The data in periodone are
While the subjectswere unable to coordiparticularlyinterestingbecause the subjects nate beliefs and actions,in almost all cases
can only use deductiveselectionprinciplesto
their individualpredictionsand actionswere
inform their behavior.
consistent. Of the 107 subjects,106 subjects
In periodone, the payoff-dominant
action, predicted that at least one other subject
7, was chosen by 31 percentof the subjects would choose an actionequalto or less than
(33 of 107) and the secure action, 1, was their choice. (Only one subjectpredictedhe
chosen by 2 percent of the subjects (2 of
would determinethe minimum.)Most sub107). Neither deductive selection principle jects mapped predictionsinto actions in a
succeeds in organizingmuch of the data, reasonable way. Those subjects who made
althoughpayoff-dominanceis more success- pessimisticpredictionsabout what the other
ful than security.The popularityof actions4
subjects would do chose small values for
and 5-chosen by 18 and 34 subjects,re- their action and subjects who made optispectively-is consistentwith many subjects mistic predictionsaboutwhat the other subhaving nearly diffusepriorbeliefs about the jects would do chose large values for their
outcome of the periodgame.
action.9
The initial play of all seven experiments
exhibitboth individualand collectivecoordination failure.The minimumaction for pe8One subjectis excludeddue to predictingonly 15
choices.
7In two earlier pilot experimentspredictionswere
not made in any period. The substantiveresultswere
the same as those reportedhere.
9Ananomalyis that,of the 95 subjectswhopredicted
a minimumless than 7, 87 subjectschose an action
greaterthan the minimumthey predicted.Van Huyck,
Battalio, and Beil (1987) provide an expected value
model to explainthis anomaly.
240
THE AMERICAN ECONOMIC REVIEW
MARCH 1990
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENTA
Period
1
2
3
4
5
6
7
8
9
10
No. of 7's
No. of 6's
No. of 5's
No.of4's
No. of 3's
No. of 2's
No.of l's
8
3
2
1
1
1
0
1
2
3
6
2
2
0
1
1
2
5
5
2
0
0
0
1
4
5
4
2
0
0
0
1
4
8
3
0
0
0
1
1
7
7
0
0
1
1
1
8
5
0
0
0
0
1
6
9
0
0
0
0
0
4
12
1
0
0
0
1
1
13
Minimum
Experiment 2
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
2
2
2
1
1
1
1
1
1
1
4
1
3
4
1
3
0
0
0
3
6
4
2
1
1
1
2
2
2
6
2
0
0
1
3
5
5
2
0
0
0
3
0
5
8
0
1
0
0
1
9
5
0
0
1
0
1
3
11
0
0
1
0
0
4
11
0
0
0
0
1
3
12
1
0
1
0
0
1
13
2
1
1
1
1
1
1
1
1
1
4
2
5
3
0
0
0
4
0
6
3
0
1
0
1
2
1
2
7
1
0
0
0
1
1
6
4
2
1
0
1
2
0
5
5
1
0
0
1
2
3
7
1
0
0
0
3
6
4
0
0
0
0
0
3
11
0
0
0
0
0
2
12
2
0
0
1
0
2
9
Minimum
Experiment 4
No. of 7's
No. of 6's
No.of5's
No. of 4's
No. of 3's
No. of 2's
No.of l's
4
2
2
1
1
1
1
1
1
1
6
0
8
1
0
0
0
0
6
5
1
2
1
0
1
2
5
4
3
0
0
1
0
5
6
2
0
1
0
0
0
7
4
2
2
0
1
1
1
3
3
6
1
0
0
2
2
7
3
0
0
0
1
2
4
8
0
0
0
1
1
2
11
0
0
0
0
0
2
13
Minimum
4
2
3
1
1
1
1
1
1
1
Experiment1
Minimum
Experiment 3
No. of 7's
No. of 6's
No.of 5's
No. of 4's
No.of 3's
No. of 2's
No. of l's
An interestingquestionis whetherthe subjects' predictions correspondto the actual
distribution of actions more closely than
predictions based on payoff-dominanceor
security. Using the numberof actions correctly predictedas a statistic,the data reveal
that 95 percentof the subjectspredictedthe
actions of the other n -1 subjects more
accurately than did payoff-dominance.
This statistic is used to measureprediction
accuracybecause the subjectspayoff were a
linear transformationof the predictionaccuracy score. The differenceof the mean prediction accuracy score was always positive
and in most cases significantlydifferentfrom
zero at the 1 percentlevel: a result that is
robust to non-parametricstatistical procedures. Obviously,securitydoes even worse.
Subjects predicted the observed heterogenous responseto the descriptionof the game
and the resultingcoordinationfailurein period one.
Repeated play of the period game allows
the subjectsto use inductiveselectionprinciples or to learn to use deductiveselection
principles. Hence, repeated play makes it
more likely that subjects will be able to
obtain mutualbest-responseoutcomesin the
continuation game. Repeating the period
game does cause actions to converge to a
VOL. 80 NO. I
VAN HUYCK ETAL: TACIT COORDINA TION GAMES
241
TABLE2-EXPERIMENTAL RESULTSFOR TREATMENT
A, Continued
Period
Experiment 5
No. of 7's
No. of 6's
No.of5's
No. of 4's
No.of3's
No. of 2's
No. of l's
Minimum
Experiment 6
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No.of3's
No.of2's
No. of l's
Minimum
Experiment 7
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No.of3's
No. of 2's
No. of l's
Minimum
1
2
3
4
5
6
7
8
9
10
2
1
9
3
1
0
0
2
3
3
4
2
2
0
3
1
0
6
2
2
2
1
0
4
2
4
3
2
1
0
1
1
6
4
3
1
0
0
2
0
6
7
1
0
2
0
0
5
8
0
0
0
2
0
2
12
0
0
0
1
0
5
10
0
0
0
1
1
3
11
3
2
1
1
1
1
1
1
1
1
5
2
5
2
1
0
1
3
0
1
3
5
2
2
1
0
0
4
4
4
3
1
0
0
0
2
5
8
1
1
0
0
2
3
9
1
0
1
0
2
3
9
2
0
0
0
1
6
7
2
0
0
0
0
4
10
2
0
0
0
2
5
7
3
0
0
0
0
5
8
1
1
1
1
1
1
1
1
1
1
4
1
2
4
1
1
1
1
3
0
3
0
3
3
2
1
1
0
0
1
2
2
8
1
1
0
0
2
1
2
8
1
1
0
0
1
1
4
7
1
1
0
0
0
0
4
9
1
1
0
0
0
0
4
9
1
1
0
0
0
0
4
9
1
1
0
0
0
0
5
8
1
1
0
0
0
0
3
10
1
stable outcome, see Table2. But ratherthan
converging to the payoff-dominantequilibrium or to the initial outcome of the treatment, the most inefficientoutcome obtains
in all seven experiments.
The change in a subject'saction between
period one and period two providesinsight
into the subjects'dynamicbehavior.Of the
eleven subjects who determinedthe minimum in period one the averagechange in
action betweenperiodone and two was 0.73:t
seven subjects increasedtheir action, three
did not change, and one decreasedhis action. In every experimentsomeonewho had
not determinedthe minimumin period one
determines the minimum in period two.
Moreover, in experimentsone throughfive
the intersection of the set of subjectswho
determinethe minimumin period one with
the set of subjectswho determinethe minimum in period two is empty. Since a subject's payoff is increasing in ei when he
(she) uniquelydeterminesthe minimum,this
adaptivebehaviorcan be rationalized.
However, a subject'spayoff is decreasing
in ei when he (she) played above the minimum and those subjectsthat played above
the minimumreducedtheirchoice of action.
The observed mean reductionis increasing
in the differencebetween a subject'saction
and the reported minimum,and the mean
reductionis smallerthan this difference.The
observed correlation between the current
choice of these optimisticsubjects and the
minimumreportedin period 1 suggeststhat
behavior in the continuationgame A(9) is
not independentof the historyleadingup to
continuation game A(9). However,only 14
of 107 subjects give a best-responseto the
period one minimumin periodtwo.
Some subjectsplay below the minimumof
the preceding period. This observed"overshooting" cannot be reconciledwith adaptive theories that predict the currentaction
242
THE AMERICAN ECONOMIC REVIEW
will be a convex combinationof last periods
action and last periodsoutcome.Apparently,
some subjects learn how "risky" it is to
choose an action other than the secure action, 1, underthe minimumrule and learnto
use security to inform their behaviorin the
continuationgame.
Although it failed to predict the initial
outcome, security predicts the stable outcome of period game A. By period ten 72
percentof the subjects(77 out of 107) adopt
their secure action, 1, and the minimumfor
all seven experimentswas a 1. The observed
coordinationfailureappearsto resultfrom a
few subjects concludingit is too "risky"to
choose an action other than the secure action and from most subjectsfocusingon the
minimum reportedin earlierperiod games.
The minimumrule interactingwith this dynamic behaviorcausesthis treatmentto converge to the most inefficientoutcome.
In treatment B, parameterb of equation
(1) was set equal to zero. Becausea player's
action is no longer penalized, the payoffdominant action, 7, is a best responseto all
feasible minimums.Action 7 is a dominating
strategy. Hence, treatmentB tests equilibrium refinementsbasedon the eliminationof
individually unreasonableactions. For example, a simple dominanceargumenteliminates all of the equilibriumpoints except
one: (7,.. ., 7). Any strategic uncertainty
would cause an individuallyrationalplayer
to choose the payoff-dominantaction,7.
Table 3 reports the experimentalresults
for treatmentB and treatmentA'. In period
eleven, the payoff-dominantaction, 7, was
chosen by 84 percentof the subjects(76 of
91). However,the minimumin periodeleven
was never more than 4 and in experiments
four, five, six, and sevenit was a 1.10
Of course, a subjectthat adopts action 7
need not worry about what actions other
10At least one subject did not understand how the
payoff table had changed. Subject 3 in experiment 5,
who plays a 1 in every period of the B treatment,
predicts that all 16 players will choose 1 but only he
does so. When the actual distribution was revealed,
subject 3 appeared genuinely amazed and confessed
that he had not understood how the payoffs had
changed.
MARCH 1990
subjects take and, apparently,most subjects
did not. This propertyof dominatingstrategies resulted in the B treatmentexhibiting
different dynamics than the A treatment.
Like the A treatment, those players who
determine the minimum increase their action, but, unlike the A treatment,those players who were above the minimumdo not
decreasetheir action.11This dynamicbehavior converges to the efficientoutcome-the
payoff-dominantequilibrium-in four of the
six experiments.By periodfifteen,96 percent
of the subjects chose the payoff-dominant
action, 7.
Even in the experimentsthat obtainedthe
efficient outcome, the B treatmentwas not
sufficientto induce the groupsto implement
the payoff-dominantequilibriumin treatment A': parameter b equals $0.10 once
again. Returningto the originalpayoff table
in period sixteen, 25 percentof the subjects
chose the payoff-dominantaction,7.12 However, 37 percent chose the secure action, 1.
Period sixteen predictionswere peakedwith
subjects choosing a 7 predictingmost subjects would choose 7 and subjectschoosinga
1 predictingmost subjectswould choose 1.
This bi-modal distributionof actions and
predictionssuggeststhat play priorto period
sixteen influenced subjects'behavior.However, the subjectsexhibit a heterogenousresponse to this history.
Security predicts the stable outcome of
treatmentA'. In treatmentA', the minimum
in all periods of all six experimentswas 1.
By period twenty, 84 percentof the subjects
chose the secure action, 1, and 94 percent
chose an action less than or equal to 2.
(Experimentstwo and four even satisfy the
"1The two exceptions were due to subject 3 in experiment five, see fn. 10, and subject 12 in experiment six.
Subject 12 predicts that he will uniquely determine the
minimum in period 11, verifies this in period 12 by
choosing a 3, and then chooses a one for the remainder
of the B treatment. Perhaps, subject 12 became vindictive. He had chosen a 7 in period 1.
12The large fluctuations in behavior resulting from
changes in the parameter b -between treatment A and
B and between treatment B and A'-suggest that subjects are influenced by the description of the game. In
our view, these data are inconsistent with backwardlooking theories of adaptive behavior.
VOL. 80 NO. 1
TABLE 3-EXPERIMENTAL
Experiment 2
No.of 7's
No. of 6's
No. of 5's
No.of 4's
No. of 3's
No. of 2's
No.of l's
Minimum
Experiment 3
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
Minimum
Experiment 4
No. of 7's
No. of 6's
No.of5's
No. of 4's
No. of 3's
No. of 2's
No.of l's
Minimum
Experiment 5
No. of 7's
No. of 6's
No.of5's
No. of 4's
No. of 3's
No. of 2's
No.ofl's
Minimum
Experiment 6
No.of 7's
No. of 6's
No.of S's
No. of 4's
No. of 3's
No. of 2's
No. of l's
243
VAN HUYCK ETAL.: TACIT COORDINATION GAMES
RESULTS FOR TREATMENT
Treatment B
14
13
B AND
TREATMENT
A'
Treatment A'
19
18
15
16
17
16
0
0
0
0
0
0
16
0
0
0
0
0
0
8
0
1
1
1
3
2
2
0
0
2
1
3
8
0
0
0
0
1
4
11
0
0
0
0
1
2
13
7*
7*
7*
1
1
1
1
13
0
0
0
1
0
0
12
1
1
0
0
0
0
13
1
0
0
0
0
0
14
0
0
0
0
0
0
6
1
0
1
0
2
4
2
0
2
0
0
4
6
2
0
1
0
0
2
9
1
0
0
0
0
3
10
1
0
0
1
0
0
12
4
3
5
6
7*
1
1
1
1
1
12
0
1
0
0
0
2
13
0
0
1
1
0
0
14
0
0
1
0
0
0
14
0
1
0
0
0
0
15
0
0
0
0
0
0
3
0
0
2
2
2
6
1
0
0
0
0
1
13
0
0
0
0
0
2
13
0
0
0
0
0
0
15
0
0
0
0
0
0
15
1
3
4
5
7*
1
1
1
1*
13
0
1
1
0
0
1
13
0
1
1
0
0
1
15
0
0
0
0
0
1
15
0
0
0
0
0
1
15
0
0
0
0
0
1
1
0
0
0
1
3
11
0
0
0
0
1
4
11
0
0
0
0
0
2
14
0
0
O
0
0
2
14
0
0
0
0
0
3
13
1
1
1
1
1
1
1
1
1
1
13
0
0
1
0
1
1
13
1
1
0
1
0
0
12
1
1
1
0
0
1
12
1
0
1
1
0
1
13
0
1
0
0
1
1
2
0
0
1
1
5
7
2
0
0
0
0
6
8
2
0
0
0
0
7
7
2
0
0
0
0
6
8
2
0
0
0
0
5
9
11
12
13
1
0
1
1
0
0
15
0
1
0
0
0
0
16
0
0
0
0
0
0
3
5
13
0
0
1
0
0
0
20
0
0
0
0
0
0
16
1*
1*
Minimum
Experiment 7
No. of 7's
No. of 6's
No.of S's
No.of 4's
No.of 3's
No. of 2's
No.of l's
1
3
1
1
1
1
1
1
1
1
12
0
0
1
0
0
1
14
0
0
0
0
0
0
13
1
0
0
0
0
0
13
0
0
0
1
0
0
14
0
0
0
0
0
0
3
0
1
2
2
2
4
4
0
0
0
0
4
6
2
0
0
0
0
2
10
2
0
0
0
0
2
10
2
0
0
0
0
1
11
Minimum
1
7*
6
3
7*
1
1
1
1
1
*
Denotes a mutual best-response outcome.
244
THE AMERICAN ECONOMIC REVIEW
mutual best-responsepropertyof an equilibrium by period twenty.) Obtainingthe efficient outcome in treatmentB failed to reverse the observedcoordinationfailure.Like
the A treatment, the most inefficientoutcome obtained.
V. ExperimentalResultsfor TreatmentC:
GroupSize Two
TreatmentC was addedto the September
experiments to determineif subjects were
influencedby groupsize whenchoosingtheir
actions. In theory,any uncertaintyabout the
actions of an individualplayerin the gameis
exacerbated by the minimum rule as the
number of players increases,see Section II.
The C treatmentreducesgroup size to two.
Table 4 reports the experimentalresults
for the C treatmentof experimentsfour and
five, which permanentlypairedsubjectswith
an unknown partner.In period twenty-one,
42 percent of the subjectsplay theirpayoffdominant action, 7, and 74 percent of the
subjectsincreasetheiraction.This resultoccurred even though the minimum for the
precedingfive periodshad been a 1 and all
31 subjects had played either a 1 (28 subjects) or a 2 (3 subjects)in period twenty.
Clearly,eitherthe subjectsthoughtthat their
partner in treatment C would change his
(her) action in response to reduced group
size or the subjectsexpectedalternativedynamicsin repeatedplay.
The subjectsin experimentsfour and five
used an adaptive behaviorin the C treatment similarto the adaptivebehaviorexhibited in the A treatment.Subjectsthat played
the minimumincreasedeffortby an average
of +2.0 and subjectsthat played above the
minimum rediced effort by an averageof
- 1.9. However,unlikethe A treatment,there
was no ".overshooting" to the secureaction,
1. Occasionally, both subjects simultaneously chose the payoff-dominantaction, 7V13
13Recall that subjects only observe the minimum and
their own action. Hence, it is not possible to unilaterally
"signal" a willingness to implement the payoff-dominant equilibrium, that is, subjects could not use Os-
MARCH 1990
This dynamic behaviorconvergedto the efficient outcome-the payoff-dominantequilibrium-in 12 of 14 trails.Hence,even with
an extremely negative history payoff-dominance predicts the stable outcome of the
tacit coordinationgamewith fixedpairs.
Experiments six and seven randomly
paired subjects with an unknownpartner.'4
Hence, experiments six and seven test
whetherthe resultsobtainedin Experiments
four and five were due to subjectsrepeating
the period game with the same opponent.In
the first period of these experiments,37 percent of the subjectschose the payoff-dominant action, 7, and 73 percentof the subjects
increasedtheir choice of action,see Table 5.
Moreover, the subjects' dynamic behavior
was similarto that found in the fixedpair C
treatment.While the resultsfor the random
pair C treatment are influencedby group
size, no stable outcomeobtains.
The C treatmentconfirmsthat there are
two consequencesof the minimumrule.First,
groupsize interactingwith the minimumrule
alters the subjects' initial choice of action.
Second, group size interactingwith the minimum rule alters the convergenceof the subjects' dynamicbehaviorin disequilibrium.
borne's (1987) refinement of a "convincing deviation"
to inform their behavior, see also van Damme (1987).
14Experiments by Cooper, DeJong, Forsythe, and
Ross (1987) report that after eleven repetitions randomly paired groups of size two almost always obtain
the payoff-dominant equilibrium. However, Cooper
et al. also report coordination failure when subjects can
choose from a strategy space that includes certain kinds
of dominated cooperative strategies. Their game illustrates an interesting distinction between Luce and
Raiffa's "solution in the strict sense," which depends on
joint-admissibility and Harsanyi and Selten's solution,
which depends on payoff-dominance. Because the
first-best outcome requires using a strictly dominated
strategy-as in the prisoner's dilemma game-and because joint-admissibility admits efficiency comparisons
with disequilibrium outcomes, the Cooper et al. game
with dominated cooperative strategies has no "solution
in the strict sense" of Luce and Raiffa. Because the
first-best outcome is an equilibrium in period game A,
joint-admissibility-appropriately defined for n person
games-and payoff-dominance select the same equilibrium point in period game A.
VOL. 80 NO. 1
VAN HUYCK ETAL.: TACIT COORDINATION GAMES
245
TABLE4-EXPERIMENTAL RESULTSFOR TREATMENTC:
FIXED PAIRINGS
Period
21
22
23
24
25
26
7
7
7
7
7
7
7
7
7
7
7
7
7
7
Minimum
Pair 2
Subject 2
Subject 15
7*
7*
7*
7*
7*
7*
7*
7
1
2
7
7
3
7
6
7
7
7
7
7
7
Minimum
Pair 3
Subject 3
Subject 14
1
2
7
7
7
7
7
1
1
1
1
1
7
1
1
1
1
1
1
1
7
Minimum
Pair 4
Subject 4
Subject 13
1*
1*
1
1*
1*
1*
1
1
7
7
2
7
5
7
7
7
7
7
7
7
7
Minimum
Pair 5
Subject 5
Subject 12
1
2
5
7*
7*
7*
7*
1
1
7
4
4
7
7
7
7
7
7
7
7
7
Minimum
Pair 6
Subject6
Subject 11
Minimum
Pair 7
Subject 7
Subject 10
1
4
4
7*
7*
7*
7*
5
7
5
7
7
7*
7
7
7*
7
7
7*
7
7
7*
7
7
7*
7
7
7*
1
5
7
3
6
6
7
7
7
7
7
7
7
7
Minimum
Pair 8
Subject 8
Subject 9
1
3
6*
7*
7*
7*
7*
7
3
6
5
6
7
7
7
7
7
7
7
7
7
3
5
6
7*
7*
7*
7*
7
2
7
3
4
6
5
6
6
7
6
7
7
7
Minimum
Pair 2
Subject 3
Subject 14
2
3
4
5
6
6
7*
5
7
7
7
7
7
7
7
7
7
7
7
7
7
Minimum
Pair 3
Subject 4
Subject 13
5
7*
7*
7*
7*
7*
7*
1
7
1
1
1
1
1
3
4
1
4
1
1
2
Minimum
Pair 4
Subject 5
Subject 12
Minimum
1
1*
1*
1
1
1
1
5
7
5
7
7
7*
7
7
7*
7
7
7*
7
7
7*
7
7
7*
7
7
7*
Experiment 5
Pair 1
Subject 1
Subject l6
Minimum
Experiment 6
Pair 1
Subject 2
Subject 15
27
246
THE AMERICAN ECONOMIC REVIEW
TABLE4-FIXED
MARCH 1990
PAIRINGS,Continued
Period
21
22
23
24
25
26
4
4
5
5
7
7
7
7
7
7
7
7
7
7
4*
5*
7*
7*
7*
7*
7*
SubjectlO
5
5
7
7
7
7
7
7
7
7
7
7
7
7
Minimum
5*
7*
7*
7*
7*
7*
7*
Pair5
Subject6
Subject11
Minimum
Pair6
Subject7
*
27
Denotes a mutualbest-responseoutcome.
TABLE 5-DISTRIBUTION OF ACTIONSFOR TREATMENTC:
RANDOMPAIRINGS
Period
21
22
23
24
25
5
0
2
3
1
1
4
5
1
5
1
1
1
2
4
3
3
1
1
2
2
10
0
3
1
0
2
0
8
0
4
1
0
2
1
-
-
6
1
0
2
2
0
3
5
0
3
1
0
0
5
5
1
0
4
0
1
3
Experiment6
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
Experiment 7
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
VI. Treatment
A withMonitoring
As a refereepoints out, a reasonableconjecture is that revealingthe distributionof
actions each period-in additionto the minimum-might influencethe reporteddynamics. For example, subjects could signal a
willingness to coordinate on the payoffdominant equilibriumand optimistic subjects might delay reducing their action if
they knew the minimumwas determinedby
just one subject.Two experiments,each using payoff Table A and 16 naive subjects,
were conductedin whichthe entiredistribution of actionswas recordedon a blackboard
at the end of each periodand was left there
for the entire experiment.
The initial distributionof actions and the
dynamicsof the two monitoringexperiments
were similar to those reported above, see
Table 6. If anything, the convergenceof
actions to the secure action, 1, was more
rapid under the monitoring treatment.In
fact, a mutual best-responseoutcome was
obtained in one experiment:without monitoring mutual best-responseoutcomes are
not observedfor period game A until treatment A'. Apparently,monitoringhelps solve
the individualcoordinationproblem-more
subjects give a best-responsesooner-but
VOL. 80 NO. 1
247
VAN HUYCK ETAL.: TACIT COORDINA TION GAMES
TABLE6-DISTRIBUTION OF ACTIONSFOR TREATMENT
A WITH MONITORING
Period
Experiment 8
No.of7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
Minimum
Experiment 9
No. of 7's
No. of 6's
No. of 5's
No. of 4's
No. of 3's
No. of 2's
No. of l's
Minimum
1P
2P
3
4
5
6
7
8
9
lop
4
1
4
5
1
0
1
0
1
0
4
4
2
5
0
0
1
2
1
1
11
0
0
0
1
0
2
13
2
0
0
0
0
3
11
0
0
1
0
2
2
11
0
0
0
1
0
1
14
1
0
0
0
0
1
14
1
0
0
0
0
2
13
1
0
1
0
0
1
13
1
1
1
1
1
1
1
1
1
1
6
1
2
4
1
0
2
2
2
2
2
5
1
2
0
0
1
3
3
0
9
1
0
0
1
1
5
8
0
0
0
0
0
4
12
0
0
0
0
0
1
15
0
0
0
0
0
0
16
0
0
1
0
0
0
15
0
0
0
0
1
1
14
0
0
0
1
0
2
13
1
1
1
1
1
1
1
1
1
1*
P- Denotes a period in which subjects made predictions.
* - Denotes a mutual best-response outcome.
not the collectivecoordinationproblem-the
minimumwas a 1 in all ten periodsof both
experiments.
VII. ConcludingComments
These experimentsprovide an interesting
example of coordinationfailure. The minimum was never above four in period one
and all seven experimentsconvergedto a
minimum of one within four periods. Since
the payoff-dominantequilibriumwouldhave
paid all subjects $19.50 in the A and A'
treatments-excluding predictions-and the
average earnings were only $8.80, the observed behavior cost the average subject
$10.70 in lost earnings.
This inefficientoutcomeis not due to conflictingobjectivesas in "prisoner'sdilemma"
games or to asymmetricinformationas in
"moralhazard"games.Rather,coordination
failure results from strategic uncertainty:
some subjectsconcludethat it is too "risky"
to choose the payoff-dominantaction and
most subjects focus on outcomesin earlier
period games.The minimumrule interacting
with this dynamicbehaviorcausesthe A and
A' treatmentsto convergeto the most inef-
ficientoutcome.
Deductive methodsimply that all feasible
actions are consistentwith some equilibrium
point in this experimental coordination
game. However, the experimentalresults
suggest that the first-bestoutcome,which is
the payoff-dominantequilibrium,is an extremely unlikely outcome either initially or
in repeatedplay. Instead,the resultssuggest
that the initial outcomewill not be an equilibriumpoint and only the secure-but very
inefficient-equilibrium describes behavior
that actual subjectsare likely to coordinate
on in repeatedplay of periodgame A when
the numberof playersis not small.
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