An Experimental Study of the Centipede Game
Author(s): Richard D. McKelvey and Thomas R. Palfrey
Source: Econometrica, Vol. 60, No. 4 (Jul., 1992), pp. 803-836
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/2951567
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Econometrica, Vol. 60, No. 4 (July, 1992), 803-836
AN EXPERIMENTALSTUDY OF THE CENTIPEDEGAME
BY RICHARD D.
MCKELVEY AND THOMASR. PALFREY 1
We reporton an experimentin whichindividualsplaya versionof the centipedegame.
In this game, two players alternatelyget a chance to take the larger portion of a
continuallyescalatingpile of money. As soon as one person takes, the game ends with
that playergettingthe largerportionof the pile, and the other playergettingthe smaller
portion.If one views the experimentas a completeinformationgame, all standardgame
theoretic equilibriumconcepts predictthe first movershould take the large pile on the
firstround.The experimentalresultsshow that this does not occur.
An alternativeexplanationfor the data can be given if we reconsiderthe game as a
game of incomplete informationin which there is some uncertaintyover the payoff
functions of the players. In particular,if the subjects believe there is some small
likelihoodthat the opponent is an altruist,then in the equilibriumof this incomplete
informationgame, playersadopt mixedstrategiesin the early roundsof the experiment,
with the probabilityof takingincreasingas the pile gets larger.We investigatehowwell a
versionof this model explainsthe data observedin the centipedeexperiments.
KEYWORDS:Game theory,experiments,rationality,altruism.
AND THERESULTS
1. OVERVIEW
OFTHEEXPERIMENT
THIS PAPERREPORTSTHE RESULTSof
several experimentalgames for which the
predictions of Nash equilibrium are widely acknowledgedto be intuitively
unsatisfactory.We explainthe deviationsfromthe standardpredictionsusing an
approachthat combinesrecent developmentsin game theorywith a parametric
specificationof the errors individualsmight make. We construct a structural
econometricmodel and estimatethe extent to which the behavioris explainable
by game-theoreticconsiderations.
In the games we investigate, the use of backwardinduction and/or the
eliminationof dominatedstrategiesleads to a uniqueNash prediction,but there
are clear benefits to the playersif, for some reason, some playersfail to behave
in this fashion.Thus, we have intentionallychosen an environmentin which we
expect Nash equilibriumto performat its worst. The best known exampleof a
game in this class is the finitely repeated prisoners'dilemma.We focus on an
even simpler and, we believe more compelling, example of such a game, the
closely related alternating-movegame that has come to be known as the
"centipedegame"(see Binmore(1987)).
The centipede game is a finite move extensiveform two persongame in which
each playeralternatelygets a turn to either terminatethe game with a favorable
payoff to itself, or continue the game, resultingin social gains for the pair. As
'Supportfor this researchwas providedin partby NSF Grants#IST-8513679and #SES-878650
to the CaliforniaInstituteof Technology.We thank MahmoudEl-Gamalfor valuablediscussions
concerningthe econometricestimation,andwe thankRichardBoylan,MarkFey, ArthurLupia,and
David Schmidtfor able researchassistance.We thankthe JPL-Caltechjoint computingprojectfor
grantingus time on the CRAY X-MP at the Jet PropulsionLaboratory.We also are gratefulfor
commentsand suggestionsfrom many seminar participants,from an editor, and from two very
thoroughreferees.
803
804
R. D. MCKELVEY AND T. R. PALFREY
far as we are aware, the centipede game was first introducedby Rosenthal
(1982), and has subsequentlybeen studiedby Binmore(1987),Kreps(1990),and
Reny (1988). The originalversions of the game consisted of a sequence of a
hundredmoves (hence the name "centipede")with linearlyincreasingpayoffs.
A concise version of the centipede game with exponentiallyincreasingpayoffs,
called the "Share or Quit" game, is studied by Megiddo (1986), and a slightly
modifiedversionof this game is analyzedby Aumann(1988). It is this exponential version that we study here.
In Aumann'sversion of the centipede game, two piles of money are on the
table. One pile is larger than the other. There are two players,each of whom
alternatelygets a turn in which it can choose either to take the largerof the two
piles of money or to pass. When one player takes, the game ends, with the
player whose turn it is getting the large pile and the other player getting the
small pile. On the other hand, whenever a player passes, both piles are
multiplied by some fixed amount, and the play proceeds to the next player.
There are a finite numberof moves to the game, and the numberis known in
advance to both players. In Aumann'sversion of the game, the pot starts at
$10.50,whichis dividedinto a large pile of $10.00and a smallpile of $.50. Each
time a playerpasses, both piles are multipliedby 10. The game proceeds a total
of six moves, i.e., three moves for each player.
It is easy to show that any Nash equilibriumto the centipede game involves
the first playertakingthe large pile on the first move-in spite of the fact that
in an eight move versionof the game, both playerscould be multi-millionairesif
they were to pass everyround.Since all Nash equilibriamake the same outcome
prediction,clearly any of the usual refinementsof Nash equilibriumalso make
the same prediction.We thus have a situationwhere there is an unambiguous
predictionmade by game theory.
Despite the unambiguousprediction, game theorists have not seemed too
comfortablewith the above analysisof the game, wonderingwhether it really
reflects the way in which anyone would play such a game (Binmore (1987),
Aumann (1988)). Yet, there has been no previous experimentalstudy of this
game.2
In the simple versions of the centipede game we study, the experimental
outcomes are quite different from the Nash predictions.To give an idea how
badly the Nash equilibrium(or iterated elimination of dominated strategies)
predictsoutcomes,we find only 37 of 662 games end with the firstplayertaking
the large pile on the first move, while 23 of the games end with both players
passing at everymove. The rest of the outcomes are scatteredin between.
One class of explanationsfor how such apparentlyirrationalbehaviorcould
arise is based on reputation effects and incomplete information.3This is the
approachwe adopt. The idea is that players believe there is some possibility
2There is related experimentalwork on the prisoner'sdilemmagame by Selten and Stoecker
(1986)and on an ultimatumbargaininggamewith an increasingcake by Guth et al. (1991).
3See Kreps and Wilson (1982a),Kreps et al. (1982), Fudenbergand Maskin(1986), and Kreps
(1990, pp. 536-543).
CENTIPEDE
GAME
805
that their opponenthas payoffsdifferentfrom the ones we tried to induce in the
laboratory.In our game, if a playerplaces sufficientweight in its utilityfunction
on the payoff to the opponent, the rational strategyis to alwayspass. Such a
playeris labeled an altruist.4 If it is believed that there is some likelihoodthat
each player may be an altruist,then it can pay a selfish player to try to mimic
the behaviorof an altruist in an attempt to develop a reputationfor passing.
These incentivesto mimicare very powerful,in the sense that a very smallbelief
that altruistsare in the subjectpool can generatea lot of mimicking,even with a
very short horizon.
The structureof the centipede game we run is sufficientlysimple that we can
solve for the equilibriumof a parameterizedversion of this reputationalmodel.
Using standard maximumlikelihood techniques we can then fit this model.
Using this assumptionof only a single kind of deviation from the "selfish"
payoffsnormallyassumed in induced-valuetheory5 we are able to fit the data
well, and obtain an estimate of the proportionof altruisticplayerson the order
of 5 percent of the subject pool. In addition to estimatingthe proportionof
altruistsin the subject pool, we also estimate the beliefs of the players about
this proportion.We find that subjects' beliefs are, on average, equal to the
estimated "true" proportionof altruists,thus providingevidence in favor of a
versionof rationalexpectations.We also estimate a decision errorrate to be on
the order of 5%-10% for inexperiencedsubjectsand roughlytwo-thirdsthat for
experiencedsubjects,indicatingtwo things:(i) a significantamountof learning
is takingplace, and (ii) even with inexperiencedsubjects,only a smallfractionof
their behavioris unaccountedfor by a simple game-theoreticequilibriummodel
in which beliefs are accurate.
Our experimentcan be comparedto that of Camererand Weigelt (1988) (see
also Neral and Ochs (1989) and Jung et al. (1989)). In our experiment,we find
that many features of the data can be explained if we assume that there is a
belief that a certain percentageof the subjectsin the populationare altruists.
This is equivalent to asserting that subjects did not believe that the utility
functionswe attemptedto induce are the same as the utility functionsthat all
subjects really use for making their decisions. I.e., subjects have their own
personal beliefs about parametersof the experimentaldesign that are at odds
with those of the experimentaldesign. This is similarto the point in Camerer
and Weigelt, that one way to account for some features of behavior in their
experimentswas to introduce"homemadepriors"-i.e., beliefs that there were
more subjectswho alwaysact cooperatively(similarto our altruists)than were
actually induced to be so in their experimentaldesign. (They used a rule-ofthumb procedure to obtain a homemade prior point estimate of 17%.) Our
analysisdiffers from Camererand Weigelt partlyin that we integrateit into a
structural econometric model, which we then estimate using classical tech4We called them "irrationals"in an earlierversionof the paper.The equilibriumimplicationsof
this kind of incomplete informationand altruism has been explored in a different kind of
experimentalgame by Palfreyand Rosenthal(1988).See also Cooperet al. (1990).
5See Smith(1976).
806
R. D. MCKELVEY AND T. R. PALFREY
niques. This enables us to estimate the numberof subjectsthat actuallybehave
in such a fashion, and to address the question as to whether the beliefs of
subjectsare on averagecorrect.
Our experimentcan also be comparedto the literatureon repeatedprisoner's
dilemmas. This literature (see e.g., Selten and Stoecker (1986) for a review)
finds that experienced subjects exhibit a pattern of "tacit cooperation"until
shortlybefore the end of the game, when they start to adopt noncooperative
behavior.Such behaviorwould be predictedby incompleteinformationmodels
like that of Kreps et al. (1982). However, Selten and Stoecker also find that
inexperiencedsubjectsdo not immediatelyadopt this patternof play, but that it
takes them some time to "learn to cooperate."Selten and Stoecker develop a
learningtheory model that is not based on optimizingbehaviorto account for
such a learningphase. One could alternativelydevelop a model similarto the
one used here, where in additionto incompleteinformationaboutthe payoffsof
others, all subjects have some chance of making errors,which decreases over
time. If some other subjects might be making errors, then it could be in the
interest of all subjectsto take some time to learn to cooperate, since they can
masqueradeas slow learners. Thus, a natural analog of the model used here
might offer an alternativeexplanationfor the data in Selten and Stoecker.
2. EXPERIMENTAL
DESIGN
Our budget is too constrainedto use the payoffsproposedby Aumann.So we
run a rather more modest version of the centipede game. In our laboratory
games, we start with a total pot of $.50 divided into a large pile of $.40 and a
small pile of $.10. Each time a playerchooses to pass, both piles are multiplied
by two. We considerboth a two round(four move) and a three round(six move)
version of the game. This leads to the extensive forms illustratedin Figures 1
and 2. In addition,we consider a version of the four move game in which all
payoffs are quadrupled.This "high payoff" condition therefore produced a
payoffstructureequivalentto the last four moves of the six move game.
1
1T
fT
0.40
0.10
1
2
0.20
0.80
FIGURE 1.-The
2
I
.
fT
0.40
0.10
P
P
14
2
6.40
fT
fT
1.60
0.40
0.80
3.20
P
1.60
four movecentipedegame.
1
2
P
P0
2
25.60
P
P
fT
fT
fT
fT
fT
0.20
0.80
1.60
0.40
0.80
3.20
6.40
1.60
3.20
I2.80
2.-The six movecentipedegame.
FIGURE
6.40
26
807
CENTIPEDE GAME
TABLE I
EXPERIMENTAL DESIGN
Session
1
2
3
4
5
6
7
Subject
pool
subjects
PCC
PCC
CIT
CIT
CIT
PCC
PCC
20
18
20
20
20
18
20
Games/
subject
10
9
10
10
10
9
10
Total #
games
moves
High
Payoffs
100
81
100
100
100
81
100
4
4
4
4
6
6
6
No
No
No
Yes
No
No
No
In each experimentalsession we used a total of twenty subjects, none of
whom had previouslyplayed a centipede game. The subjectswere divided into
two groups at the beginningof the session, which we called the Red and the
Blue groups. In each game, the Red player was the first mover, and the Blue
playerwas the second mover.Each subjectthen participatedin ten games, one
with each of the subjectsin the other group.6The sessions were all conducted
throughcomputerterminalsat the Caltech Laboratoryfor ExperimentalEconomics and PoliticalScience. Subjectsdid not communicatewith other subjects
except throughthe strategychoices they made. Before each game, each subject
was matchedwith another subject,of the opposite color, with whom they had
not been previouslymatched, and then the subjects who were matched with
each other played the game in either Figure 1 and Figure 2 dependingon the
session.
All details describedabove were made commonknowledgeto the players,at
least as much as is possible in a laboratory setting. In other words, the
instructionswere read to the subjects with everyone in the same room (see
AppendixB for the exact instructionsread to the subjects).Thus it was common
knowledgethat no subjectwas ever matchedwith any other subjectmore than
once. In fact we used a rotatingmatchingscheme which insuresthat no player i
ever plays against a playerwho has previouslyplayed someone who has played
someone that i has already played. (Further, for any positive integer n, the
sentence which replaces the phrase "who has previouslyplayed someone who
has played someone"in the previoussentence with n copies of the same phrase
is also true.) In principle, this matching scheme should eliminate potential
supergameor cooperativebehavior, yet at the same time allow us to obtain
multipleobservationson each individual'sbehavior.
We conducted a total of seven sessions (see Table I). Our subjects were
students from Pasadena CommunityCollege (PCC) and from the California
Institute of Technology(CIT). No subjectwas used in more than one session.
Sessions 1-3 involved the regular four move version of the game, session 4
6Only one of the three versions of the game was played in a given session. In sessions 2 and 6, not
all subjects showed up, so there were only 18 subjects, with 9 in each group, and consequently each
subject played only 9 games.
808
R. D. MCKELVEYAND T. R. PALFREY
TABLE IIA
PROPORTION OF OBSERVATIONS
AT EACH TERMINAL NODE
f
f7
.20
.35
.23
.01
.11
.12
.01
.02
.01
.253
.078
.014
N
fi
f2
f3
f4
f5
1
2
3
(PCC)
(PCC)
(CIT)
100
81
100
.06
.10
.06
.26
.38
.43
.44
.40
.28
.20
.11
.14
.04
.01
.09
Total
1-3
281
.071
.356
.370
.153
.049
High Payoff
4
(High-CIT)
100
.150
.370
.320
.110
.050
Six
Move
5
6
7
(CIT)
(PCC)
(PCC)
100
81
100
.02
.00
.00
.09
.02
.07
.39
.04
.14
.28
.46
.43
Total
5-7
281
.007
.064
.199
.384
Session
Four
Move
involved the high payoff four move game, and sessions 5-7 involved the six
move version of the game. This gives us a total of 58 subjectsand 281 plays of
the four move game, and 58 subjectswith 281 plays of the six move game, and
20 subjectswith 100 plays of the high payoff game. Subjectswere paid in cash
the cumulativeamount that they earned in the session plus a fixed amountfor
showingup ($3.00 for CIT students and $5.00 for PCC students).7
3. DESCRIPTIVE SUMMARY OF DATA
The complete data from the experimentis given in AppendixC. In Table II,
we present some simple descriptivestatistics summarizingthe behaviorof the
subjects in our experiment. Table IIA gives the frequencies of each of the
terminal outcomes. Thus fi is the proportion of games ending at the ith
terminalnode. Table IIB gives the implied probabilities,pi of takingat the ith
decisionnode of the game. In other words,pi is the proportionof games among
those that reached decision node i, in which the subjectwho moves at node i
chose TAKE. Thus, in a game with n decision nodes, pi =fi/ElEf1f.
All standardgame theoretic solutions(Nash equilibrium,iterated elimination
of dominatedstrategies,maximin,rationalizability,etc.) would predict fi = 1 if
i
=
1, fI
=
0 otherwise. The requirement of rationality that subjects not adopt
dominatedstrategieswould predictthat fn+i = 0 and Pn= 1. As is evidentfrom
Table II, we can reject out of hand either of these hypothesesof rationality.In
only 7% of the four move games, 1% of the six move games, and 15% of the
high payoffgames does the firstmoverchoose TAKE on the firstround.So the
subjectsclearlydo not iterativelyeliminatedominatedstrategies.Further,when
a game reaches the last move, Table IIB shows that the player with the last
move adopts the dominatedstrategyof choosingPASS roughly25% of the time
7The stakes in these games were large by usual standards. Students earned from a low of $7.00 to
a high of $75.00, in sessions that averaged less than 1 hour-average earnings were $20.50 ($13.40 in
the four move, $30.77 in the six move, and $41.50 in the high payoff four move version).
CENTIPEDE
809
GAME
TABLE IBa
IMPLIED
FORTHECENTIPEDE
GAME
TAKEPROBABILITIES
Four
Move
Session
Pi
P2
P3
P4
1 (PCC)
.06
(100)
.10
(81)
.06
(100)
.28
(94)
.42
(73)
.46
(94)
.65
(68)
.76
(42)
.55
(51)
.83
(24)
.90
(10)
.61
(23)
Total 1-3
.07
(281)
.38
(261)
.65
(161)
.75
(57)
4 (CIT)
.15
(100)
.44
(85)
.67
(48)
.69
(16)
S (CIT)
.02
(100)
.00
(81)
.00
(100)
.09
(98)
.02
(81)
.07
(100)
.44
(89)
.04
(79)
.15
(93)
.01
(281)
.06
(279)
.21
(261)
2 (PCC)
3 (CIT)
High
Payoff
Six
Move
6 (PCC)
7 (PCC)
Total 5-7
P5
P6
.56
(50)
.49
(76)
.54
(79)
.91
(22)
.72
(39)
.64
(36)
.50
(2)
.82
(11)
.92
(13)
.53
(205)
.73
(97)
.85
(26)
aThe number in parentheses is the number of observations in the game at that node.
in the four move games, 15%in the six move games,and 31% in the high payoff
games.8
The most obvious and consistent pattern in the data is that in all of the
sessions, the probabilityof TAKE increases as we get closer to the last move
(see Table IIB). The only exception to this pattern is in session 5 (CIT) in the
last two moves, where the probabilitiesdrop from .91 to .50. But the figure at
the last move (.50) is based on only two observations.Thus any model to explain
the data should capturethis basic feature. In additionto this dominantfeature,
there are some less obviouspatternsof the data, which we now discuss.
Table III indicates that there are some differencesbetween the earlier and
later plays of the game in a given treatment which are supportive of the
propositionthat as subjectsgain more experiencewith the game, their behavior
appears"morerational."Recall that with the matchingscheme we use, there is
no game-theoreticreason to expect players to play any differentlyin earlier
games than in later games. Table IIIA shows the cumulative probabilities,
Fj = _j=1fiof stoppingby the jth node. We see that the cumulativedistribution
in the first five games stochasticallydominatesthe distributionin the last five
gamesboth in the four and six move experiments.This indicatesthat the games
end earlier in later matches. Table IIIB shows that in both the four and six
move sessions,in later games subjectschose TAKEwith higherprobabilityat all
8For sessions 1-3, 7 of the 14 cases in this category are attributable to 2 of the 29 subjects. In the
high payoff condition, 4 of the 5 events are attributable to 1 subject.
810
R. D. MCKELVEY AND T. R. PALFREY
TABLE IIIA
CUMULATIVEOUTCOMEFREQUENCIES
(Fj=
E Ifi)
Treatment
Game
N
F1
F2
F3
F4
F5
Four
Move
1-5
6-10
145
136
.062
.081
.365
.493
.724
.875
.924
.978
1.00
1.00
Six
Move
1-5
6-10
145
136
.000
.015
.055
.089
.227
.317
.558
.758
.889
.927
F6
F7
.979
.993
1.000
1.000
TABLE IIIB
IMPLIEDTAKE PROBABILITIES
COMPARISONOF EARLY VERSUS LATE PLAYS IN THE Low PAYOFF CENTIPEDEGAMES
Treatment
Game
Pi
P2
P3
P4
Four
1-5
.06
.32
6-10
(145)
.08
(136)
(136)
.49
(125)
.57
(92)
.75
(69)
.75
(40)
.82
(17)
.00
(145)
.01
(136)
.06
(145)
.07
(134)
.18
(137)
.25
(124)
.43
(112)
.65
(93)
Move
Four
1-5
Move
6-10
P5
P6
.75
(64)
.70
(33)
.81
(16)
.90
(10)
stages of the game (with the exception of node 5 of the six move games).
Further,the numberof subjectsthat adopt the dominatedstrategyof passingon
the last move drops from 14 of 56, or 25%, to 4 of 27, or 15%.
A third pattern emerges in comparingthe four move games to the six move
games(in Table IIB). We see that at everymove, there is a higherprobabilityof
taking in the four move game than in the correspondingmove of the six move
game (.07 vs .01 in the firstmove; .38 vs .06 in the second move, etc.). The same
relationholds between the high payoffgames and the six move games.However,
if we compare the four move games to the last four moves of the six move
games, there is more taking in the six move games (.75 vs .85 in the last move;
.65 vs .73 in the next to last move, etc.). This same relationshipholds between
the high payoff games and the six move games even though the payoffsin the
high payoffgames are identical to the payoffsin the last four moves of the six
move games.
There is at least one other interestingpattern in the data. Specifically,if we
look at individuallevel data, there are several subjects who PASS at every
opportunity they have.9 We call such subjects altruists, because an obvious way
to rationalizetheir behavioris to assumethat they have a utilityfunctionthat is
9Some of these subjects had as many as 24 opportunities
See Appendix C.
to TAKE in the 10 games they played.
CENTIPEDE
GAME
811
monotonicallyincreasingin the sum of the red and blue payoffs,ratherthan a
selfish utility function that only depends on that players'own payoff. Overall,
there were a total of 9 playerswho chose PASS at every opportunity.Roughly
half (5) of these were red playersand half (5) were in four move games. At the
other extreme (i.e., the Nash equilibriumprediction), only 1 out of all 138
subjectschose TAKE at every opportunity.This indicatesthe strongpossibility
that playerswho will alwayschoose PASS do exist in our subjectpool, and also
suggests that a theory which successfully accounts for the data will almost
certainlyhave to admitthe existenceof at least a smallfractionof such subjects.
Finally, there are interesting non-patterns in the data. Specifically, unlike the
ten cases cited above, the preponderanceof the subjectbehavioris inconsistent
with the use of a single pure strategythroughoutall games they played. For
example, subject #8 in session #1 (a red player) chooses TAKE at the first
chance in the second game it participatesin, then PASS at both opportunitiesin
the next game, PASS at both opportunitiesin the fourth game, TAKE at the
first chance in the fifth game, and PASS at the first chance in the sixth game.
Fairlycommonirregularitiesof this sort, which appearratherhaphazardfrom a
casual glance, would seem to require some degree of randomnessto explain.
While some of this behavior may indicate evidence of the use of mixed
strategies,some such behavioris impossibleto rationalize,even by resortingto
the possibilityof altruisticindividualsor Bayesianupdatingacross games. For
example, subject #6 in session #1 (a blue player), chooses PASS at the last
node of the first game, but takes at the first opportunitya few games later.
Rationalization of this subject's behavior as altruistic in the first game is
contradictedby the subject'sbehaviorin the later game. Rational play cannot
accountfor some sequencesof playswe observein the data, even with a model
that admitsthe possibilityof altruisticplayers.
4. THE MODEL
In what follows, we constructa structuraleconometricmodel based on the
theory of games of incomplete informationthat is simultaneouslyconsistent
with the experimentaldesign and the underlyingtheory. Standardmaximum
likelihoodtechniquescan then be applied to estimate the underlyingstructural
parameters.
The model we constructconsistsof an incompleteinformationgame together
with a specificationof two sources of errors-errors in actions and errors in
beliefs. The model is constructedto accountfor both the time-seriesnature of
our data and for the dependence across observations,features of the data set
that derive from a design in which every subject plays a sequence of games
against different opponents. The model is able to account for the broad
descriptivefindingssummarizedin the previoussection. By parameterizingthe
structureof the errors,we can also addressissues of whether there is learning
going on over time, whether there is heterogeneity in beliefs, and whether
individuals'beliefs are on averagecorrect.
812
R. D. MCKELVEY AND T. R. PALFREY
We first describe the basic model, and then describe the two sources of
errors.
4.1. TheBasicModel
If, as appearsto be the case, there are a substantialnumberof altruistsin our
subjectpool, it seems reasonableto assume that the possible existence of such
individualsis commonlyknownby all subjects.Our basic model is thus a game
of two sided incomplete informationwhere each individualcan be one of two
types (selfish or altruistic),and there is incomplete informationabout whether
one's opponent is selfish or altruistic.
In our model, a selfish individualis defined as an individualwho derives
utility only from its own payoff,and acts to maximizethis utility. In analogyto
our definitionof a selfish individual,a naturaldefinitionof an altruistwould be
as an individualwho derives utility not only from its own payoff,but also from
the payoff of the other player. For our purposes, to avoid having to make
parametric assumptions about the form of the utility functions, it is more
convenient to define an altruistin terms of the strategychoice rather than in
terms of the utility function. Thus, we define an altruistas an individualwho
alwayschooses PASS. However,it is importantto note that we could obtain an
equivalentmodel by makingparametricassumptionson the form of the utility
functions. For example, if we were to assume that the utility to player i is a
convex combination of its own payoff and that of its opponent, then any
individualwho places a weight of at least 2 on the payoffof the opponent has a
dominant strategy to choose PASS in every round of the experiment.Thus,
defining altruists to be individualswho satisfy this condition would lead to
equivalentbehaviorfor the altruists.
The extensiveform of the basic model for the case when the probabilityof a
selfish individualequals q is shown in Figure 8 in Appendix A. Hence, the
probabilityof an altruist is 1 - q. This is a standard game of incomplete
information.There is an initial move by nature in which the types of both
playersare drawn.If a playeris altruistic,then the player has a trivialstrategy
choice (namely, it can PASS). If a player is selfish, then it can choose either
PASS or TAKE.
4.1.1. Equilibriumof the Basic Model
The formal analysis of the equilibriumappears in Appendix A, but it is
instructiveto provide a brief overview of the equilibriumstrategies, and to
summarizehow equilibriumstrategiesvaryover the familyof games indexed by
q. We analyticallyderive in AppendixA the solution to the n-move game, for
arbitraryvalues of q (the common knowledgebelief that a randomlyselected
playeris selfish) rangingfrom 0 to 1.
For any given q, a strategyfor the first playerin a six move game is a vector,
(P1, P3, p5), where pi specifiesthe probabilitythat Red chooses TAKE on move
i conditionalon Red being a selfish player. Similarly,a strategyfor Blue is a
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vector (P2, P4, P6) giving the probabilitythat Blue chooses TAKE on the
correspondingmove conditionalthat Blue is selfish. Thus a strategypair is a
vector p = (P1, P2, ... , P6), where the odd componentsare moves by Red and
the even componentsare moves by Blue. Similarnotation is used for the four
move game.
In Appendix A, we prove that for generic q there is a unique sequential
equilibriumto the game, and we solve for this equilibriumas a function of q.
Let us write p(q) for the solution as a function of q. It is easy to verify the
followingpropertiesof p(q), which are true in both the four move and six move
games.
Property 1: For any q, Blue chooses TAKE with probability1 on its last
move.
Property2: If 1 - q > 4,both Red and Blue alwayschoose PASS, except on
the last move, when Blue chooses TAKE.
Property3. If 1 - q E (0, 7) the equilibriuminvolvesmixed strategies.
Property4: If q = 1, then both Red and Blue alwayschoose TAKE.
From the solution, we can compute the implied probabilitiesof choosing
TAKE at each move, accountingfor the altruistsas well as the selfish players.
We can also compute the probabilitys(q) = (sl(q), .. ., s7(q)) of observingeach
of the possible outcomes, T, PT,PPT,PPPT,PPPPT,PPPPPT,PPPPPP. Thus,
s1(q) = qpl(q), s2(q) = q2(1 - pj(q))p2(q) + q(l - q)p2(q), etc. Figures 3 and 4
graph the probabilitiesof the outcomes as a function of the level of altruism,
1 -q.
It is evident from the properties of the solution and from the outcome
probabilitiesin Figures 3 and 4 that the equilibriumpredictionsare extremely
sensitiveto the beliefs that playershave about the proportionof altruistsin the
population.The intuition of why this is so is well-summarizedin the literature
on signallingand reputationbuilding(for example, Kreps and Wilson (1982a),
Kreps et al. (1982)) and is exposited very nicely for a one-sided incomplete
informationversion of the centipede game more recentlyin Kreps (1990). The
guidingprincipleis easy to understand,even if one cannot follow the technical
details of the Appendix.Because of the uncertaintyin the game when it is not
common knowledgethat everyoneis self-interested,it will generallybe worthwhile for a selfish playerto mimic altruisticbehavior.This is not very different
from the fact that in poker it may be a good idea to bluff some of the time in
order to confuseyour opponent aboutwhetheror not you have a good hand. In
our games, for any amount of uncertaintyof this sort, equilibriumwill involve
some degree of imitation.The form of the imitationin our setting is obvious:
selfishplayerssometimespass, to mimic an altruist.By imitatingan altruistone
mightlure an opponentinto passingat the next move, therebyraisingone's final
payoffin the game. The amountof imitationin equilibriumdepends directlyon
the beliefs about the likelihood(1 - q) of a randomlyselected playerbeing an
altruist.The more likelyplayersbelieve there are altruistsin the population,the
more imitationthere is. In fact, if these beliefs are sufficientlyhigh (at least 4,in
our versions of the centipede game), then selfish players will always imitate
altruists,therebycompletelyreversingthe predictionsof game theorywhen it is
814
R. D. MCKELVEY AND T. R. PALFREY
0.8
01
0.2005
0.2
015
I
D
0.05
FIGURE 4.-Equilibrium
0. I
T~~~-
0.2
0.15
outcomeprobabilitiesfor basicsix movegame.
common knowledge that there are no altruists.Between 0 and
predictsthe use of mixed strategiesby selfish players.
7,
the theory
4.1.2. Limitationsof theBasic Model
The primaryobservationto make from the solutionto the basic model is that
this model can account for the main feature of the data noted in the previous
section-namely that probabilitiesof taking increase as the game progresses.
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815
For any level of altruismabove 1/72 in the four move game, and for any value
above 1/73 in the six move game, the solution satisfiesthe propertythat pi >p
whenever i >j.
Despite the fact that the basic model accounts for the main pattern in the
data, it is just as obviousthat the basic model cannot accountfor the remaining
features of the data. It is apparentfrom Figures3 and 4 that for anyvalue of q,
there is at least one outcomewith a 0 or close to 0 probabilityof occurrence.So
the model will fit poorly data in which all of the possible outcomes occur. Nor
can it accountfor any consistentpatternsof learningin the data, for substantial
variationsacross individuals,or for some of the irregularitiesdescribedearlier.
To account for these features of the data, we introduce two additional
elements to the model-the possibilityof errorsin actions,and the possibilityof
errorsin beliefs.
4.2. ErrorsinActions-Noisy Play
One explanationof the apparentlybizarreirregularitiesthat we noted in the
previous section is that players may "experiment"with different strategies in
order to see what happens. This may reflect the fact that, early on, a subject
may not have settled down on any particularapproachabout how to play the
game. Alternatively,subjectsmay simply "goof," either by pressing the wrong
key, or by accidentallyconfusingwhich color player they are, or by failing to
notice that is the last round, or some other random event. Lacking a good
theory for how and why this experimentationor goofing takes place, a natural
way to model it is simplyas noise. So we refer to it as noisyplay.
We model noisy play in the followingway. In game t, at node s, if p* is the
equilibriumprobabilityof TAKE that the player at that node attempts to
implement,we assume that the player actuallychooses TAKE with probability
(1 - e,)p*, and makes a randommove (i.e. TAKE or PASS with probability.5)
with probabilityst. Therefore,we can view st/2 as the probabilitythat a player
experiments,or, alternatively,goofs in the tth game played.We call St the error
rate in game t. We assume that both types (selfish and altruistic)of players
make errorsat this rate, independentlyat all nodes of game t, and that this is
commonknowledgeamong the players.
4.2.1. Learning
If the reasons for noisy play are along the lines just suggested, then it is
natural to believe that the incidence of such noisy play will decline with
experience. For one thing, as experience accumulates,the informationalvalue
of experimentingwith alternativestrategiesdeclines, as subjectsgatherinformation about how other subjectsare likely to behave. Perhapsmore to the point,
the informationalvalue will decline over the course of the 10 games a subject
plays simplybecause, as the horizonbecomes nearer,there are fewer and fewer
games where the informationaccumulatedby experimentationcan be capital-
816
R. D. MCKELVEY AND T. R. PALFREY
ized on. For different,but perhapsmore obvious reasons,the likelihoodthat a
subjectwill goof is likely to decline with experience.Such a decline is indicated
in a wide range of experimentaldata in economics and psychology,spanning
many differentkinds of tasks and environments.We call this decline learning.
We assumea particularparametricform for the errorrate as a functionof t.
Specifically,we assume that individualsfollow an exponentiallearning curve.
The initial error rate is denoted by e and the learning parameter is 8.
Therefore,
Notice that, while according to this specification the error rate may be
differentfor differentt, it is assumedto be the same for all individuals,and the
same at all nodes of the game. More complicatedspecificationsare possible,
such as estimatingdifferent E'sfor altruisticand selfish players,but we suspect
that such parameterproliferationwould be unlikelyto shed muchmore light on
the data. When solvingfor the equilibriumof the game, we assumethat players
are aware that they make errors and learn, and are aware that other players
make errors and learn too.'0 Formally,when solving for the Bayesianequilibrium TAKE probabilities,we assume that e and 8 are commonknowledge.
4.2.2. Equilibrium with Errors in Actions
For e > 0, we do not have an analyticalsolution for the equilibrium.The
solutionswere numericallycalculatedusing GAMBIT,a computeralgorithmfor
calculatingequilibriumstrategiesto incomplete informationgames, developed
by McKelvey(1990).For comparison,the equilibriumoutcomeprobabilitiesas a
functionof q, for , = .2, are illustratedgraphicallyin Figures5 and 6.
4.3. Errors in Beliefs-Heterogeneous Beliefs
In addition to assumingthat individualscan make errors in their strategies,
we also assume that there can be errorsin their beliefs. Thus, we assume that
there is a true probabilityQ that individualsare selfish (yielding probability
1 - Q of altruists),but that each individualhas a belief, qi, of the likelihoodof
selfishplayers,whichmaybe differentfrom the true Q." In particular,individuals' beliefs can differ from each other, givingrise to heterogeneousbeliefs.
10Analternative specification would have it common knowledge that subjects believe others make
errors, but believe they do not commit these "errors" themselves. Such a model is analytically more
tractable and leads to similar conclusions, but seems less appealing on theoretical grounds.
1tThis is related to the idea proposed independently by Camerer and Weigelt (1988) and Palfrey
and Rosenthal (1988), where they posit that subjects' beliefs about the distribution of types may
differ from the induced-value distribution of types announced in the instructions.
CENTIPEDE
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817
GAME
PA
0.8
~~~~4
0.4
*
~~i14~~~~~.:
'
**.**
5
w.
?..
0i2
PPPT
pppp
.
. . .
.........................
FIGURE 5
0
. .
.~.
...............
.
.
Equilibrumoutcome.probailities.for.fourmove.game.....2)
0.05
0.1
0.2
0.15
FIGURE
6.-Equilibriumoutcomeprobabilitiesfor fourmovegame(et = .2).
t1-..+-...
0.4
F~~~~~~~~~~~~~~~~~~~
0~~~~.
a
2
T
PT
,.,~~~~~~~~~~~~~~~~~~7PPPPt
0
0.05
0.1
0.15
0.2
FIGURE6.-Equilibrium outcome probabilities for six move game (Et = .2).
For individuali, denote by qi the belief individuali holds that a randomly
selected opponent is selfish. (We assume that each individualmaintains its
belief throughoutall 10 games that it plays.)Because this convertsthe complete
informationcentipede game into a Bayesiangame, it is necessaryto make some
kind of assumptionabout the beliefs a player has about its opponent'sbeliefs,
etc. etc. If there were no heterogeneityin beliefs, so that qi = q for all i, then
one possibility is that a player's beliefs are correct-that is, q is common
knowledge,and q = Q. We call this rational expectations. One can then solve for
818
R. D. MCKELVEY AND T. R. PALFREY
the Bayesianequilibriumof the game played in a session (whichis unique),as a
function of e, 8, t, and q. An analyticalsolution is derivedin AppendixA for
the case of e = 0.
To allow for heterogeneity,we make a parametricassumptionthat the beliefs
of the individuals are independently drawn from a Beta distributionwith
parameters(a,/3), where the mean of the distribution,q, is simply equal to
a/(a + /3).There are severalways to specifyhigherorder beliefs. One possibility is to assume it is common knowledge among the players that beliefs are
independentlydrawnfrom a Beta distributionwith parameters(a, /) and that
the pair (a, ,3) is also common knowledgeamong the players.This version of
higherorder beliefs leads to serious computationalproblemswhen numerically
solvingfor the equilibriumstrategies.Instead,we use a simpler12 versionof the
higher order beliefs, which might be called an egocentricmodel. Each player
playsthe game as if it were commonknowledgethat the opponenthad the same
belief. In other words, while we, the econometricians,assume there is heterogeneity in beliefs, we solve the game in which the players do have heterogeneous beliefs, but believe that everyone'sbeliefs are alike. This enables us to
use the same basic techniquesin solvingfor the Bayesianequilibriumstrategies
for playerswith differentbeliefs as one would use if there were homogeneous
beliefs. We can then investigatea weaker form of rationalexpectations:is the
averagebelief (a/(a + ,B))equal to the true proportion(Q) of selfish players?
Given the assumptionsmade regardingthe form of the heterogeneity in
beliefs, the introductionof errorsin beliefs does not change the computationof
the equilibriumfor a given individual.It only changesthe aggregatebehaviorwe
will expect to see over a group of individuals.For example,at an error rate of
Et= .2, and parameters a, / for the Beta distribution,we will expect to see
aggregatebehaviorin period t of the six move gameswhichis the averageof the
behaviorgeneratedby the solutions in Figure 6, when we integrateout q with
respect to the Beta distributionB(a, f).
5. MAXIMUM LIKELIHOOD ESTIMATION
5.1. Derivationof the LikelihoodFunction
Considerthe version of the game where a playerdrawsbelief q. For every t,
and for every Et, and for each of that player'sdecisionnodes, v, the equilibrium
solution derivedin the previoussection yields a probabilitythat the decision at
that node will be TAKE, conditionalon the player at that decision node being
selfish, and conditional on that player not making an error. Denote that
probabilityp(e, q, v). Therefore, the probabilitythat a selfish type of that
player would TAKE at v is equal to P,(et, q, v) = (st/2) + (10- st)ps(e, q, v),
and the probability that an altruistic type of this player would take is
Pa(6tgq, v) = Et/2. For each individual, we observe a collection of decisions that
12Whileit is simpler, it is no less arbitrary.It is a version of beliefs that does not assume
"commonpriors"(Aumann(1987)), but is consistentwith the standardformulationof games of
incompleteinformation(Harsanyi(1967-68)).
CENTIPEDE
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GAME
are made at all nodes reachedin all gamesplayedby that player.Let N,i denote
the set of decision nodes visited by player i in the tth game playedby i, and let
D,i denote the correspondingset of decisionsmade by i at each of those nodes.
Then, for any given
(?, 8,
q, v) with vE Ni. we can compute Ps(Et, q, v) from
above, by setting et= e-8(t'). From this we compute irti(D ;e, 8, q), the
probabilitythat a selfish i would have made decisions Dti in game t, with
beliefs q, and noise/learning parameters(8, 8), and it equals the product of
Ps(8, q, v) over all v reached by in game t. Letting Di denote the set of all
decisions by player i, we define rg(Di; s, 8, q) to be the product of the
irti(Dti; 8,8, q) taken over all t. One can similarlyderive mr(Di;E, 8, q), the
probability that an altruistic i would have made that same collection of
decisions. Therefore, if Q is the true populationparameterfor the fractionof
selfish players,then the likelihoodof observingDi, without conditioningon i's
type is given by:
-i(Di; Q, E, 8, q)
=
Qin!(Di; E, 8, q) + (1- Q)wa(Di; E, 8, q).
Finally,if q is drawnfrom the Beta distributionwith parameters(a, /3), and
densityB(q; a, /3), then the likelihoodof observingDi without conditioningon
q is given by:
si(Di; Q, 8,8, a, 13)-= f1(Di;
Q, 8,8, q)B(q; a, () dq.
Therefore, the log of the likelihood function for a sample of observations,
D = (Dl, .. ., D1), is just
L(D; Q,e , 8, a,3)
= E log [si(Di; Q, E, A, a,
)].
i=1
For any sample of observations,D, we then find the set of parametervalues
that maximizeL. This was,done by a global grid searchusing the CrayX-MP at
the Jet PropulsionLaboratory.
5.2. Treatmentsand Hypotheses
We are interested in testing four hypotheses about the parametersof the
theoreticalmodel.
(1) Errorsin action:Is 8 significantlydifferentfrom 0?
(2) Heterogeneity(errorsin beliefs):Is the varianceof the estimateddistribution of priorssignificantlydifferentfrom O?13
(3) Rational Expectations:Is the estimatedvalue of Q equal to the mean of
the estimateddistributionof priors,a/(a +/3)?
(4) Learning:Is the estimatedvalue of 8 positive and significantlydifferent
from 0?
13Whilehomogeneityof beliefs is not strictlynested in the Beta distributionmodel (since the
Beta family does not include degeneratedistributions),the homogeneousmodel can be approximated by the Beta distributionmodel by constraining(a + ,) to be greaterthan or equal to some
large number.
820
R. D. MCKELVEY AND T. R. PALFREY
The firsttwo hypothesesaddressthe questionof whetherthe two components
of error in our model (namely errors in action and errors in belief) are
significantlydifferentfrom 0. The second two hypothesesaddressissues of what
propertiesthe errorshave, if they exist.
In our experimentaldesign, there were three treatmentvariables:
(1) The length of the game (either four move or six move).
(2) The size of the two piles at the beginningof the game (either high payoff,
($1.60,$.40), or low payoff,($.40,$.10)).
(3) The subjectpool (either Caltech undergraduates(CIT) or PasadenaCity
College students(PCC)).
We also test whether any of these treatmentvariableswere significant.
5.3. Estimation Results
Table IV reports the results from the estimations. Before reporting any
statisticaltests, we summarizeseveral key features of the parameterestimates.
First, the mean of the distributionof beliefs about the proportionof altruists
in the populationin all the estimationswas in the neighborhoodof 5%. Figure7
graphs the density function of the estimated Beta distributionfor the pooled
sample of all experimentaltreatmentsfor the four and six move experiments,
respectively.Second, if one looks at the rationalexpectationsestimates(which
constrain a, = /a + 3) to equal Q), the constrained estimate of the Beta
TABLE IV
RESULTS
FROM
Treatment
MAXIMUM
LIKELIHOOD
i3
,
Q
2.75
2.75
2.50
.939
.941
.965
.972
.956
.941
.950
.850
Four
Move
unconstrained
A= Q
S=0
a=0
42
44
68
unconstrained
40
2.00
.952
Six
A= Q
38
2.00
.950
S= 0
=0
34
1.75
unconstrained
42
=Q
a=0
40
unconstrained
42
1.25
.971
28
1.00
.966
.994
.966
.955
.941
.952
Move
PCC
CIT
=
Q
64
48
28
40
-InL
.18
.18
.21
.23
.045
.045
.000
.020
327.35
327.41
345.08
371.04
.904
.06
.030
352.07
.950
.06
.030
352.76
.951
.976
.908
.850
.05
.22
.000
.030
371.01
442.96
2.75
.939
.974
.14
.030
464.14
2.75
.936
.952
.936
.882
.11
.18
.040
.050
464.57
508.60
.880
.22
.040
340.27
.966
.22
.040
342.57
.750
.27
.010
424.83
.900
.938
.938
.930
.22
.22
.14
.18
.050
.050
.050
.050
107.11
435.73
702.80
813.38
a=0
High Payoff
All Four Move
All Low
All Sessions
ESTIMATIONa
2.25
2.25
1.75
2.00
aRows marked g = Q report parameter estimates under the rational expectations restriction that 6/(6 + ,l) = Q. Rows
marked 8 = 0 are parameter estimates under the hypothesis of no learning. Rows marked a = 0 are parameter estimates
under the assumption of no heterogeneity.
CENTIPEDE
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GAME
Density
Ai
125- /
/
Siz
2.5 -
Moveo
F
FourMove
10
7.5
5
2.5
0
0.05
0.1
0.15
0.2
Iq
FiG,URE7.-Estimated distributionof beliefsfor fourand six movegames.
distribution is nearly identical to the unconstrained estimate of the Beta
distribution.
Furthermore,the rational expectations estimates of u are nearly identical
acrossall treatments.Therefore,if we are unable to rejectrationalexpectations,
then it would seem that these beliefs, as well as the true distributionof altruists
are to a large extent independent of the treatments.The greatest difference
acrossthe treatmentsis in the estimatesof the amountof noisy play. While the
estimates of 8 are quite stable across treatments,the estimates of e are not.
This is most apparent in the comparisonof the four move and the six move
estimates of e. We discuss this below after reportingstatisticaltests. Finally,
observethat in the o,= 0 estimates(no heterogeneityof beliefs), the estimateof
A is consistentlymuch largerthan the estimate of Q, and the recoverederror
rates are higher.
It mightseem paradoxicalthat we estimate a level of altruismon the order of
5%,while the proportionof subjectswho choose pass on the last move is on the
order of 25% for the four move experiments and 15% for the six move
experiments.One reason for the low estimate of the level of altruismis that in
the theoreticalmodel, part of this behavioris attributedto errorsin action. But
less obviously, it should be noted that because our equilibriumis a mixed
strategy,there is a sampleselection bias in the set of subjectswho get to the last
move:altruistsare more likelyto make it to this stage, since they pass on earlier
moveswhereasselfish subjectsmix. Thus, even with no errors,we would expect
to see a higher proportionof passing on the last move than the proportionof
altruistsin the subjectpool.
5.4. StatisticalTests
Table V reports likelihood ratio x2 tests for comparisonsof the various
treatments,and for testing the theoreticalhypothesesof rationalexpectations,
learning,and heterogeneity.
822
R. D. MCKELVEY AND T. R. PALFREY
TABLEV
LIKELIHOODRATIO TESTS
Theoretical
Hypotheses
Hypothesis
Treatment
d.f.
-2 log likelihood
ratio
Heterogeneity(a = 0)
4-move
6-move
4-move
6-move
4-move
6-move
1
1
1
1
1
1
87.38*
181.78*
.12
1.38
35.46*
37.88*
4-movevs 6-move
5
51.16*
4-highvs 4-low(PayoffTreatment)
PCCvs CIT
All
4-move
6-move
5
5
5
5
2.54
17.94*
9.63
8.42
Rationalexpectations(
=
Q)
Learning(3 = 0)
Treatment
Effects
*Significantat 1% level.
Our first hypothesis-that E = 0, can be dispensedwith immediately.In both
the four and six move experiments,setting e = 0 yields a likelihoodof zero. So
this hypothesiscan be rejectedat any level of significance,and is not includedin
Table V.
To test for heterogeneity,we estimate a model in which a single belief
parameter,q, is estimated instead of estimatingthe two parametermodel, a
and , (see Table IV, rows marked a = 0). As noted earlier, homogeneityis
approximatelynested in our heterogeneity model. Therefore, we treat the
homogeneousmodel as if it is nested in the heterogeneousmodel, and report a
standard x2 test based on likelihood ratios. All statistical tests were highly
significant(see Table V). Note that by setting Q and all qi to one, then one
obtains a pure random model where individualsPASS with probability?/2.
Hence for any probabilityof passing less than or equal to 2, the pure random
model is a special case of the homogeneous model. Thus the above findings
mean we also reject the pure randommodel (e = 1).
In the test for learning,the null hypothesisthat 3 = 0 is clearlyrejectedfor all
treatments,at essentially any level of significance.We conclude that learning
effects are clearlyidentifiedin the data. Subjectsare experimentingless and/or
makingfewer errors as they gain experience.The magnitudeof the parameter
estimatesof 3 indicate that subjectsmake roughlytwo-thirdsas manyerrorsin
the tenth game, comparedto the first game.
The test for rational expectations is unrejectablein most cases. The one
exception is for the CIT subjectpool, where the differenceis significantat the
5% level, but not the 1% level.
The payofflevel treatmentvariableis not significant.This is reassuring,as it
indicatesthat the results are relativelyrobust.
The other treatment effects, CIT/PCC and four move/six move, are both
significant.One source of the statisticaldifferencebetween the PCC and CIT
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GAME
TABLE VI
CIT-PCC
ESTIMATES, BROKEN DOWN INTO FOUR MOVE AND
PCC 4
CIT 4
PCC 6
CIT 6
a
l3
u
Q
74.0
36.0
40.0
80.0
3.75
1.50
2.50
2.25
.95
.96
.94
.97
.996
.866
.906
.902
Six MovETREATMENTS
-InL
.210
.220
.060
.060
.04
.05
.04
.03
216.47
214.35
231.28
116.58
TABLE VII
CHI-SQUARED
TESTS FOR DIFFERENCES
(UNDER
Parameter
?
a
aSignificant at p =.01
IN E AND a ACROSS TREATMENTS
ASSUMPTION THAT /, = Q)
Treatment
d.f.
-2 log L ratio
4-move vs. 6-move
CIT vs. PCC
1
1
39.28a
4-move vs. 6-move
CIT vs. PCC
1
1
5.76
.02
3.90
level.
estimatesapparentlyderivesfrom the fact that two-thirdsof the CIT data were
for four move games, while only half of the PCC data were for four move
games. Consequently,we break down the statisticalcomparisonbetween PCC
and CIT into the four and six move game treatments (see Table VI). The
subject pool effect is not significantin the six move treatment and is barely
significantat the 10%level in the four move treatment(see Table V).
In orderto pin downthe sourceof the treatmenteffectswe performedseveral
tests. The first one was to test for differencesin learning effects across treatments. This test is done by simultaneouslyreestimatingthe parametersfor each
of the differenttreatments,subjectto the constraintthat 8 is the same for each
treatment, and then conductinga likelihood ratio test. Second, we tested for
differencesin the noise parameter,E. The x2 tests are reported in Table VII.
The results reflect the estimates in Tables IV and V. The only significant(1%
level) differenceis the estimatedinitial errorrate E in the four move versus six
move games. The CIT/PCC differencein E is significantat the 5% level, but
this is due to reasonsgiven in the previousparagraph.The differencebetween 8
in the six move game and four move game is significantat the 5% level.
5.5. Fit
In order to get a rough measureof how well our model fits the data, we use
the unconstrainedparameter estimates from Table IV to obtain predicted
aggregateoutcomes. Table VIII displaysthe predicted frequenciesof each of
the five possible outcomesof the four move game, and comparesthese frequencies to the observedfrequencies.This comparisonis done for each of the ten
periods, t, t = 1,..., 10, and for the time-aggregateddata. Table IX displays
824
R. D. MCKELVEY AND T. R. PALFREY
TABLE VIII
COMPARISON OF PREDICTED (FIRST) VS.
ACTUAL (SECOND) FREQUENCIES, FOUR MOVE
(CELLS 4 AND 5 COMBINED FOR COMPUTATION OF X2)
Predicted
Actual
Period
n
e
fI
f2
f3
f4
f5
fl
f2
f3
f4
f5
x
1
2
3
4
5
6
7
8
9
10
58
58
58
58
58
58
58
58
58
40
.18
.17
.16
.16
.15
.14
.14
.13
.13
.12
.106
.101
.097
.096
.100
.095
.095
.090
.090
.095
.269
.292
.314
.314
.329
.349
.349
.369
.369
.380
.291
.298
.316
.316
.320
.335
.335
.338
.338
.348
.291
.271
.241
.241
.223
.197
.197
.181
.182
.159
.042
.037
.032
.032
.028
.024
.024
.021
.021
.017
.000
.103
.034
.103
.069
.103
.069
.069
.103
.050
.276
.276
.310
.276
.379
.310
.414
.414
.483
.450
.379
.345
.379
.379
.310
.414
.448
.345
.310
.400
.241
.241
.172
.172
.172
.172
.069
.138
.069
.050
.103
.034
.103
.069
.069
.000
.000
.034
.034
.050
7.72
.69
3.11
1.24
1.04
2.00
9.39*
.87
5.10
2.99
Total
562
-
.097
.333
.324
.218
.028
.071
.356
.370
.153
.050
*Significant at .05 level.
similarnumbersfor the six move games. The last columnof the table displaysa
x2 for each period, comparingthe predictedto the actualvalues.14
It is evident from Tables VIII and IX that the fit to the four move games is
better than that of the six move games. For the four move games,in only one of
the periods (period 7), are the predicted and actual frequencies significantly
differentat the .05 level. In the six move games, there are six periods in which
there are differencessignificantat the .05 level, and four of these differencesare
significantat the .01 level. The predictedfrequenciesfor the four move games
also pick up reasonablywell the trends in the data between periods. Thus f2
and f3 increaseover time, while f4 decreases,all in accordancewith the actual
data. In the six move data, on the other hand, the predictedfrequenciesshow
very small trends in comparisonwith those in the actual data. The tables can
also be used to help identifywhere the model seems to be doing badly. In the
four move games, the model overestimatesf1 and f4 and underestimatesf2
and f3. In the six move games, the model underestimates f3.
The differencesin fit between the four and six move games can be accounted
for by the limitationsof the model we fit. As noted earlier,the main difference
between the estimatesfor the four and six move games is in the estimate of S.
In the four move games, we find a high value of e = .18, while in the six move
game we find a lower value of e =.06. The reason for the differencesin the
estimatesof e between the four and six move games is fairlyclear. In order to
obtain a high value of the likelihoodfunctionin the six move games, the model
is forced to estimate a low error rate, simply because there are almost no
observationsat the first terminalnode.15
14The x2 statistic is not reported for the time-aggregated data, since the assumption of
independence is violated.
15In fact, there were no such observations in the first 9 periods and only 2 in the last period.
CENTIPEDE
109 8 7 6 5 4 3 2 1
Total
**Significant
*Significant
562
at at
825
GAME
40585858585858585858
Period
n
.01.05
-
.05.05.05.05.05.05.05.06.06.06
E
level.
level.
f,
.035
.035
.035
.031.030
.030
.030
.030
.030
.030
.030
f2
.076
.076
.077
.077
.077
.076
.076
.076
.076
.076.076
(CELLS
1
f
.113
.113
.113
.113
.113
.113
.096
.096
.096
.108.113
.435
.43514
.435
.432
.433.432
.432
.432
.432
.432
.432
5
.275
.275
.275
.273
.273
.273
.273
.273
.273
.274.273
.068
.072f6
.068
.072
.068
.068
.068
.068
.072
.069.068
.009
.009
.009
.010
.010
.010f7
.009 .009
.009
.009
.009
COMPARISON
AND
2
OF
Predicted
COMBINED
PREDICrED
AND
(FIRST)
CELLS
6 VS.
AND
7
TABLE
IX
ACrUAL
fi
.000
.000
.000
.000
.000
.000
.000
.007 .100
.000
.000
COMBINED
(SECOND)
f2
.103
.069
.069
.069
.069
.069
.069
.000
.034
.064.100
FOR
.241
.241
.103
.103f3
.172
.199.250
.276
.276
.172
.172
FREQUENCIES,
.345
.345
.483
.276
.448
.310
.310h4
.552
.414
.384.350
COMPUTATION
SIX
Actual OF
X2)MOVE
.241
.345
.345fs
.379
.253.050
.138
.379
.207
.172
.207
f6
.103
.103
.103
.069
.069
.069
.078.150
.069
.034
.034
.069
.000
.000
.034
.014.000
.000
.034
.000
.000
.000
5.33 2
3.86 8.39 12.72*
5.32
10.41*
20.66**
15.68**
17.98**
18.99**
826
R. D. MCKELVEY AND T. R. PALFREY
The differencein the estimatesof e also accountsfor the failureof the model
to explainthe trendsin the six move data. The way that our model explainstime
trends is throughthe learningparameter,c, which representsthe rate at which
the errorrate, ?, declines over time. As et declines, the model implies that the
game will end sooner, by predicting more frequent outcomes at the earlier
terminalnodes. In the four move game, the initial error rate is 8 =.18. With
this high initial error rate, a decay of 8 = .045 is able to lead to substantial
changes in predictedbehavior.However,in the six move games, we estimate a
significantlylower e = .06. With this lower initial error rate, a similar rate of
learningwill lead to less dramaticchanges in behavior.
Our model of errors assumes that in any given game, the error rate is a
constant. In other words the likelihoodof makingan error is the same at each
node regardless of whether the equilibriumrecommends a pure or mixed
strategyat that node. It mightbe reasonableto assume instead that individuals
make more errorswhen they are indifferentbetween altematives(i.e., when the
equilibriumrecommendsa mixed strategy)than when they have preferences
over alternatives(i.e., when the equilibriumrecommendsa pure strategy).In
other words,the errorrate may be a functionof the utilitydifferencesbetween
the choices. Such a model mightbe able to explainthe behaviorof the six move
games with error rates on the same order of magnitudeas those of the four
move games. This in turn might lead to a better fit, and a better explanationof
the trends in the six move games. We have not investigatedsuch a model here
because of computationaldifficulties.
One final point regardsthe comparisonsbetween the take probabilitiesin the
four move games and the six move games. As noted in Section 3, the aggregate
data from the four move games exhibit higher take probabilitiesthan the six
move games at each of the decision nodes 1, 2, 3, and 4. But in contrastto this,
when one comparesthe take probabilitiesof the last four moves of the six move
game with the four move game (wherethe terminalpayoffsare exactlycomparable), this relationshipis reversed;the six move data exhibit the higher take
probabilities.Both of these features of the data are picked up in the estimates
from our model. From Tables VIII and IX, the predictedtake probabilitiesfor
the four move and six move games are p4 = (.097,.368,.568,.886), and P6 =
(.031,.078,.120,.552,.778,.885),respectively.
6. CONCLUSIONS
We concludethat an incompleteinformationgame that assumesthe existence
of a small proportionof altruistsin the populationcan accountfor manyof the
salient featuresof our data. We estimatea level of altruismon the order of 5%.
In the version of the model we estimate, we allow for errors in actions and
errorsin beliefs. Both sources of errorsare found to be significant.Regarding
errorsin action,we find that there are significantlevels of learningin the data,
in the sense that subjectslearn to make fewer errorsover time. Subjectsmake
roughlytwo thirds as many mistakes at the end of a session as they make in
CENTIPEDE
827
GAME
early play. Regardingerrorsin beliefs, we reject homogeneityof beliefs. However, we find that rationalexpectations,or on-averagecorrectbeliefs, cannot be
rejected.
While we observesome subjectpool differences,they are small in magnitude
and barelysignificant.The payoff treatmenthad no significanteffect. The only
significantdifferencebetween the parameterestimates of the four and the six
move games was that the estimatedinitial errorrate was lower in the six move
game. A model in which the error rate is a function of the expected utility
difference between the available action choices might well account for the
observed behavior in both the four and the six move games with similar
estimates of the error rate. This might be a promisingdirection for future
research.
Divisionof the Humanitiesand Social Sciences,CaliforniaInstituteof Technology, Pasadena,CA 91125, U.SA.
receivedMay,1990;final revisionreceivedFebruary,1992.
Manuscript
APPENDIXA
In this appendix,we provethat there is a uniquesequentialequilibriumto the n movecentipede
game with two sided incompleteinformationover the level of altruism(see Figure8).
There are n nodes, numberedi = 1,2,...,n. Player 1 moves at the odd nodes, and Player 2
moves at the even nodes. We use the terminology"playeri" to refer to the playerwho moves at
node i. Let the payoffif playeri takes at node i be (ai, bi) where ai is payoffto playeri and bi is
1
Player
1
2
2
7
(l-q)
/
I
1
q(I-q)
q2~
b,
I
Ip
II
I
I!
I
I
I
p
ol p|
I
I? 02
q(l-q)
-
~~~b2
|
bp
b3
I
-
I
IX
b1-2
P ?S p
I101,
p b4 Pb-
b4
*
*
*
b9
P
b+
Ib
P
I
b5,b
Eri0 0T0
1
0 P0
b1
i
p
I
I
,.2
0,
b/
I
I,
Is
t
b
/
I
I
I
p
/
,
I
Player
n-I
n
n-2
p oP1
p
P/
i
I
,-
pP
..
0
ptb \P p_r"P pob.+1
t
0
20 _
b--2
b,..1
0,
b,
8.-Centipede game with incompleteinformation(dashed lines representinformation
sets and open circlesare startingnodes, with probabilitiesindicated).
FIGURE
828
R. D. MCKELVEY AND T. R. PALFREY
payoffto playeri - 1 (or i + 1). Also, if i = n + 1, then (ai, bi) refersto the payoffif playern passes
at the last node. Define
'Iii
=-
ai -bi+I
Ia.+2- bi+l
We assume that ai+2 > ai > bi+,
and that
i7i
is the same for all i. We write 77= 1i (A similar
solutioncan be derivedwhen the 71'are different.)
Now a strategyfor the game can be characterizedby a pair of vectors p = (p.
pn) and
r = (r1,..., rn),wherefor any node i, pi is the probabilitythat a selfishtype takes at that node, and
ri is the conditionalprobability,as assessedby playeri at node i, of the other playerbeing selfish.
Let q be the initialprobabilityof selfishness.So r1= q.
LEMMA1: If p E R' and r e R'
are a sequential equilibrium, then:
(a) forall i, pi=O =*piO for allj < i;
(b) Pn= 1. Also, pi = 1 i=n or i =n-1.
PROOF: (a) Assume pi = 0. Clearlypi-1 = 0, because at node i - 1, the value to playeri - 1 of
passingis at least ai+1, whichis by assumptiongreaterthan ai-1, the payofffrom taking.
(b) By dominance, pn = 1. Suppose pi = 1 for i < n - 1. Then by Bayes rule ri+ 1 = 0, so Pi+ 1 = 0
is the best response by player i + 1, since i's type has been revealed. But pi+, = 0 p1 = 0 by part
(a), whichis a contradiction.
Q.E.D.
Define k to be the firstnode at which0 <pi, and k to be the firstnode for which pi = 1. Clearly
k < k. Then k and k partitionthe nodes into at most three sets, whichwe refer to as the passing
stage(i < k), the mixingstage(k < i < k), and the takingstage(k < i). FromLemma1 it followsthat
there are no pure strategiesin the mixingstage.FromLemma1(b),it followsthat the takingstage is
at least one and at most two moves.
LEMMA 2: In any sequentialequilibrium
(p, r), for 1 < i S n - 1,
=
(a) O<Pi < 1 ,ripi+1
1 -77;
(b) riPi+ 1 < 1 -77 ==Pi = ;
(c) ripi+1 > 1-u =*1=Pi 1.
PROOF:(a) Assume 0 <Pi < 1. Let vi be the value to player i at node i given equilibriumplay
from node i on. Write vn+1 = an+1. Now if i = n - 1, then vi+2 = vn+1 = an+1 = ai+2- If i <n - 1,
then by Lemma1, 0 <Pi ? 0 <Pi+2, which implies Vi+2= ai+2. Now in both cases 0 <pi implies
vi = ai = ripi+1bi+1 + (1 - ripi+1)ai+2. Solving for ripi+1, we get
ai+2 - ai
riPi+1 =
ai+2 - bi+l
1
77.
(b) If rip1+1 < 1 - q, then at node i, v, > ripi+1b,+l + (1 - ripi+i)ai+2 = ai+2 - ripi+,1(ai+2 ai+2 - (ai+2 -ai)=
ai. So vi >ai =ip= 0.
(c) If ripi+ I> 1 - u7,then Pi+1 > 0
=OPi+2> 0 (by Lemma 1). Hence, Vi+2= ai+2. By similar
argument to (b), vi < a5 i = 1.
Q.E.D.
bi+1)>
LEMMA
3: For genericq, for any sequentialequilibrium
thereare an even numberof nodesin the
mixingstage. I.e., k = k + 2K for some integer0 < K < n/2. For any k < K,
(a) rk+2k= 1-(1-q)/nqk;
k+1
1
(b) rk-12k=
PROOF:
We firstshow(a) and (b), and then show k = k + 2K.
(a) For any node i, Bayes rule implies
()
(1 -p,+)ri
ri+2 =
(1-pi+
Ori + (1-ri)
ri-Pi+
1-Pi+lri
CENTIPEDEGAME
829
By assumptionr1= q. And since in the passingstage pi = 0, it followsthat rk = q. Now if both i and
i + 2 are in the mixingstage,it followsfromLemma1 that i + 1 is also, implying0 <pi+ < 1. So by
Lemma 3, ripi+,1 = 1 - q. Hence, (1) becomes
_
ri -_1
(2)
(l -rr)
__
By induction,it followsthat as long as k < '(1 - k), then k + 2k < k is in the mixingstage. So
rr k+2k
1-q
k
1
k
k
1
(b) As above,as long as both i and i - 2 are in the mixingstage,we get
ri-2)
(1
Solvingfor ri-2, we get
ri-2
(l
= 1-
-
ri).
Now from Lemma2, it follows,since PT,= 1,
1rl-
=7
1 -,.
=-1=-=
Pi~
Hence, by induction,as long as k < 2(k - k), we have
rk-1-2k
=
1-7k(l-r
k-1)
= 1-nk+l.
Finally,to show that there are an even numberof nodes in the mixingstage, assume,to the
contrarythat there are an odd number.Then we can write k = k + 2k + 1 for some k > 0. Thus
k = k - 1 - 2k. So by part (b) we have r I = 1 - 7k+ 1. But by (a) we have r k = 1 - (1 - q) = q,
implyingthat q = 1 - nk+ 1. For generic q, this is a contradiction,implyingthat k = k + 2K for
some K > 0. If k > n/2, then k > k + n > n, whichcontradictsLemmal(b). Hence k = k + 2K for
Q.E.D.
some 0 6 K < n/2.
THEOREM4: For generic q, there is a unique sequential equilibrium(p, r) which is characterized as
follows: Let I be the smallest integer greater than or equal to n/2. If 1 - q < ', set K = I - 1, k = 1,
and k = 2I - 1. If 1 - q > t7, let K be the largest integer with 1 - q <K, k= n, and k =k -2K.
The solution then satisfies:
(a) if i<k, then ri+1 = qandpi=0;
(b) if i>k, then ri+1=O andpi 1=;
K
(ii) if i = k + 2k, with
- 1)/qK;
(c) if k S i < k: (i) if i = k, then ri+ I- 1 _
, and pi = (q +
1 S k <K, then rj+j = 1 - 77K-k, and pi = (1 - q)/(1 _ +1 ); (iii) if i= k + 2k + 1, with 0 6
k < K, thenr1+j = 1-(1-q)/rqk
+1, and pi = nk(l _ 7)/(nk_(1-q)).
PROOF:
The formulaefor ri and pi in parts(a), (b), and(c) followby applicationof the previous
lemmastogetherwith Bayes rule. In particularin (a), pi = 0 followsfrom the definitionof k, and
ri+1 = q follows from pi = 0 for i < i together with Bayes rule. In (b), pi = 1 follows from the
definitionof k, and ri1I = 0 then followsby Bayesrule. In (c), all the formulaefor ri1 followfrom
Lemma3. In (c) part (i), we set k = i + 1 in (1) and solve for Pk to get
- rk+1
rk-1
rk=
-rk-lrk+1
But rk - 1 = q and rk+1 =1
q -1
Pk=
+ 77K
K
q?7
-
K. So
830
R. D. MCKELVEY AND T. R. PALFREY
In parts(ii) and (iii) of (c), we applyLemma2a to get that p = (1 - r)/(ri11). Substitutingin for
the valuesof ri -I gives the requiredformulae.
Thus, it only remainsto prove the assertionsabout k and k. We first prove two preliminary
inequalities.First,note, k > 1 implies,by Lemma2,
Pk-1=0
1l-f
rk-lPk
(
i
q+nK
K
=*q + K17
=> 1 q>fK+l
K _ 1K+ I
Hence,
(3)
1-
I=f
q <q+
Second, note k
=
k = 1.
n - 1 implies, by Lemma 2,
r
> i-X
=*r7kp+I
p;=
q+nK_
1
S
,1-Xl
1-q<flK+1
Hence,
(4)
Let
1_q > 77K+1=,k=n.
n
I=[21.
There are two cases.
Case I: 1 - q < i7 FromLemma3, we have
n
n
- -1=oI>K+1.
=K?
K<
2
Thus we have 1 - q <<
121
I
SK+l.
But from (3), this implies k = 1. Now since k > n - 1, it follows
that K = I - 1, and k = k + 2K = 2I - 1.
O > =* (q O
Case II: 1-q > 71. Now Pk
+ qK - 1)/q7K >
1-q < qK. Suppose 1 q < qK+ 1.
Then, from(1), we have k = 1, and by the same argumentas Case I, K = I - 1=>1 q <l K+l1=7 I,
a contradiction.Hence we must have
K+1 < 1-q
<yK.
So K is the largestintegerwith 1 - q < rK.
But now,from(4), it followsthat k = n.
Q.E.D.
In the centipede games describedin the text, the piles grow at an exponentialrate: There are
real numbersc > d > 1 with ai = cbl and ai+ 1 = dai for all i. So f7= (c - d)/(cd2 - d). In our
experimentsc = 4, and d = 2, so Y7= 4. The figuresin the text correspondto the solutionfor the
two and three roundgames(n = 4 and n = 6) for these parameters.
It is interestingto note that since the solutiondependsonly on Y7,the abovesolutionalso applies
if there are linearly increasing payoffs of the form ai + 1 = ai + c, and bi+ 1 = bi + c (with c > 0), as
1 = bi + c. Hence pickingai, bi, and c so that
long as ai > bi+
a,-b2
a3-b2
aI-b1-c
1
ai-bi+c
7
(e.g., a1 = 60, b1= 20, c = 30) one can obtaina gamewith linearlyincreasingpayoffswhose solution
is exactlythe same as the solutionof the gamewith exponentiallyincreasingpayoffstreatedin this
paper.
CENTIPEDE
831
GAME
APPENDIX B
ExperimentInstructions
This is an experimentin group decision making,and you will be paid for your participationin
cash, at the end of the experiment.Differentsubjectsmay earn differentamounts.Whatyou earn
dependspartlyon your decisions,partlyon the decisionsof others,and partlyon chance.
The entire experimentwill take place throughcomputerterminals,and all interactionbetween
you will take place throughthe computers.It is importantthat you not talk or in any way try to
communicatewith other subjectsduringthe experiments.If you disobeythe rules,we will have to
ask you to leave the experiment.
We will startwith a brief instructionperiod. Duringthe instructionperiod,you will be given a
completedescriptionof the experimentandwill be shownhow to use the computers.You musttake
a quiz after the instructionperiod. So it is importantthat you listen carefully.If you have any
questions duringthe instructionperiod, raise your hand and your questionwill be answeredso
everyonecan hear. If any difficultiesarise after the experimenthas begun, raise yourhand, and an
experimenterwill come and assistyou.
The subjectswill be divided into two groups,containing10 subjectseach. The groupswill be
labeled the RED group and the BLUE group.To determinewhich color you are, will you each
please select an envelopeas the experimenterpasses by you.
[EXPERIMENTER PASS OUT ENVELOPES]
If you chose BLUE,you will be BLUE for the entire experiment.If you chose RED, you will be
RED for the entire experiment.Please rememberyour color, because the instructionsare slightly
differentfor the BLUE and the RED subjects.
In this experiment,you will be playingthe followinggame, for real money.
First,you are matchedwith an opponentof the oppositecolor. There are two piles of money:a
LargePile and a SmallPile. At the beginningof the gamethe LargePile has 40 cents and the Small
Pile has 10 cents.
RED has the firstmoveand can either "Pass"or "Take."If RED chooses"Take,"RED gets the
LargePile of 40 cents, BLUE gets the smallpile of 10 cents, and the game is over. If RED chooses
"Pass,"both piles double and it is BLUE'sturn.
The LargePile now contains80 cents and the Small Pile 20 cents. BLUE can take or pass. If
BLUE takes, BLUE ends up with the LargePile of 80 cents and RED ends up with the SmallPile
of 20 cents and the game is over. If BLUE passes, both piles double and it is RED's turn again.
This continuesfor a total of six turns,or three turnsfor each player.On each move, if a player
takes,he or she gets the LargePile, his or her opponentgets the SmallPile, and the gameis over. If
the playerpasses,both piles double againand it is the other player'sturn.
The last moveof the gameis move six, and is BLUE'smove(if the game even gets this far).The
LargePile now contains$12.80and the Small Pile contains$3.20. If BLUE takes, BLUE gets the
LargePile of $12.80and RED gets the SmallPile of $3.20. If BLUE passes,then the piles double
again.RED then gets the LargePile, containing$25.60,and BLUE gets the SmallPile, containing
$6.40.This is summarizedin the followingtable.
PAYOFF CHART FOR DECISION EXPERIMENT
1
2
Move #
3
4
5
6
T
P
T
P
P
T
P
P
P
P
P
P
P
P
P
P
P
P
T
P
P
P
T
P
P
T
P
Large
Pile
Small
Pile
RED's
Payoff
BLUE's
Payoff
.10
.40
.10
.40
.80
.20
.20
.80
1.60
.40
1.60
.40
3.20
6.40
12.80
25.60
.80
1.60
3.20
6.40
.80
6.40
3.20
25.60
3.20
1.60
12.80
6.40
[EXPERIMENTER HAND OUT PAYOFF TABLE]
Go over table to explainwhat is in each columnand row.
The experimentconsistsof 10 games. In each game,you are matchedwith a differentplayerof
the opposite color fromyours.Thus, if you are a BLUE player,in each game,you will be matched
with a RED player.If you are a RED player,in each game you are matchedwith a BLUE player.
832
R. D. MCKELVEY
AND T. R. PALFREY
Since there are ten subjectsof each color, this means that you will be matchedwith each of the
subjectsof the other color exactlyonce. So if your label is RED, you will be matchedwith each of
the BLUE subjectsexactlyonce. If you are BLUE, you will be matchedwith each of the RED
subjectsexactlyonce.
We will now begin the computerinstructionsession.Will all the BLUE subjectsplease move to
the terminalson the left side of the room, and all the RED subjectsmove to the terminalson the
rightside of the room.
[SUBJECrS MOVE TO CORRECT TERMINALS]
Duringthe instructionsession,we will teachyou how to use the computerby goingthrougha few
practice games. During the instruction session, do not hit any keys until you are told to do so, and
when you are told to enter information,typeexactlywhatyou are told to type.You are not paid for
these practicegames.
Please turnon yourcomputernow by pushingthe buttonlabeled"MASTER"on the righthand
side of the panel underneaththe screen.
[WAIT FOR SUBJECTS TO TURN ON COMPUTERS]
When the computerpromptsyou for your name,type yourfull name.Then hit the ENTER key.
[WAIT FOR SUBJECTS TO ENTER NAMES]
When you are asked to enter your color, type R if your color is RED, and B if your color is
BLUE. Then hit ENTER.
[WAIT FOR SUBJECTS TO ENTER COLORS]
You now see the experiment screen. Throughout the experiment, the bottom of the screen will
tell you what is currently happening, and the top will tell you the history of what happened in the
previous games. Since the experiment has not begun yet, the top part of the screen is currently
empty. The bottom part of the screen tells you your subject number and your color. It also tells you
the subject number of the player you are matched against in the first game. Is there anyone whose
color is not correct?
[WAIT FOR RESPONSE]
Please recordyour color and subjectnumberon the top left hand cornerof your recordsheet.
Also recordthe numberof the subjectyou are matchedagainstin the firstgame.
Each game is representedby a row in the upperscreen,and the playeryou will be matchedwith
in each of the ten gamesappearsin the columnlabeled "OPP"(whichstandsfor "opponent")on
the right side of the screen. It is importantto note that you will never be pairedwith the same
playertwice.
We will now start the first practicegame. Remember,do not hit any keys until you are told to
do so.
[MASTER HIT KEY TO START FIRST GAME]
You now see on the bottompart of the screen that the first game has begun, and you are told
who you are matchedagainst.If you are a RED player,you are told that it is your move, and are
givena descriptionof the choices availableto you. If you are a BLUE player,you are told that it is
youropponent'smove, and are told the choices availableto youropponent.
Will all the RED playersnow choose PASS by typingin P on your terminalsnow.
[WAIT FOR SUBJECTS TO CHOOSE]
Since RED chose P, this is recordedon the top part of the screen with a P in the first RED
column,and the cursorhas moved on to the second column,which is BLUE, indicatingthat it is
BLUE'smove.
On the bottompart of the screen,the BLUE playersare now told that it is their turnto choose,
and are told the choicesthey can make.The RED playersare told that it is theiropponent'sturnto
choose, and are told the choices that their opponentcan make. Notice, that there is now a Large
Pile of $.80 and a SmallPile of $.20.
Will all the BLUE playersnow please choose TAKEby typingT at your terminalnow.
[WAIT FOR SUBJECTS TO CHOOSE]
CENTIPEDE
GAME
833
Since BLUE chose T, the firstgame has ended. On the bottompart of the screen,you are told
that the game is over, and that the next game will begin shortly.On the top part of the screen,
BLUE'smove is recordedwith a T in the second column.The payoffsfrom the firstgame for both
yourselfandyouropponentare recordedon the righthandside of the screenin the columnslabeled
"Payoff."Your own payoffis in yourcolor. That of youropponentis in the opponent'scolor.
Please recordyourown payoffon the recordsheet that is provided.
[WAIT FOR SUBJECTS TO RECORD PAYOFFS]
You are not being paid for the practicesession, but if this were the real experiment,then the
payoffyou have recordedwould be moneyyou have earnedfrom the firstgame, and you would be
paid this amountfor that game at the end of the experiment.The total you earn over all ten real
games is what you will be paid for your participationin the experiment.
We will now proceedto the second practicegame.
[MASTER HIT KEY TO START SECOND GAME]
You now see that you have been matchedwith a new playerof the oppositecolor, and that the
second game has begun. Does everyonesee this?
[WAIT FOR RESPONSE]
The rules for the second game are exactlylike the first.The RED playergets the firstmove.
[DO RED-P, BLUE-P, RED-P]
Now notice that it is BLUE'smove.It is the last moveof the game.The LargePile now contains
$3.20, and the Small Pile contains$.80. If the BLUE playerchooses TAKE, then the game ends.
The BLUE playerreceivesthe LargePile and the RED playerreceivesthe SmallPile. If the BLUE
player chooses PASS, both piles double, and then the game ends. The RED playerreceives the
Large Pile, which now contains $6.40, and the BLUE player receives the Small Pile, containing
$1.60.
Will the BLUE playerplease choose PASS by typingP at your terminalnow.
[WAIT FOR SUBJECTS TO CHOOSE]
The second practicegame is now over. Please record your payoffon the second line of your
recordsheet.
[WAIT FOR PLAYERS TO RECORD PAYOFFS]
[MASTER HIT KEY TO START THIRD GAME]
We now go to the thirdpracticegame. Notice again that you have a new opponent.Will all the
RED playersplease choose TAKE by typingT at your terminalnow.
[WAIT FOR PLAYERS TO CHOOSE]
Since the RED playerchose TAKEon the firstmove,the gameis over,and we proceedon to the
next game. Since RED chose TAKE on the firstmove, BLUE did not get any chance to move.
Please recordyour payofffor the thirdgame on the thirdline of yourrecordsheet.
[WAIT FOR PLAYERS TO RECORD PAYOFFS]
This concludesthe practicesession. In the actualexperimentthere will be ten games insteadof
three, and, of course,it will be up to you to makeyourown decisions.At the end of game ten, the
experimentends and we will pay each of you privately,in cash, the TOTAL amount you have
accumulatedduring all ten games, plus your guaranteedfive dollar participationfee. No other
person will be told how much cash you earned in the experiment.You need not tell any other
participantshow muchyou earned.
Are there any questionsbefore we pass out the quiz?
[EXPERIMENTER TAKE QUESTIONS]
O.K., then we will now have you take the quiz.
[PASS OUT QUIZ]
[COLLECT AND MARK QUIZZES]
[HAND QUIZZES BACK AND GO THRU CORRECT ANSWERS]
834
R. D. MCKELVEY AND T. R. PALFREY
We will now beginwith the actualexperiment.If there are anyproblemsfromthis point on, raise
your hand and an experimenterwill come and assistyou. When the computerasks for your name,
please start as before by typingin your name. Wait for the computerto ask for your color, then
respondwith the correctcolor.
[START EXPERIMENT]
[CHECK THAT COLORS ARE OK BEFORE BEGINNING EXPERIMENT]
APPENDIXC
Data
Experimental
The following tables give the data for our experiment.Each row representsa subject. The
columnsare
Col 1: Session number
Col 2: Subjectnumberof Red Player
Col 2 +j: Outcomeof gamej. Letting k be the entryin this column,and n be the numberof
moves in the game(n = 4 for Exp. 1-4, n = 6 for Exp. 5-7), then
k
gameendedwith T on move k,
gameendedwith P on move n.
< n -=>
= n + 1=
The matchingscheme was: In game j, Red subject i is matched with Blue subject [(i +j 1)modmi],where m is the numberof subjectsof each color in the session.Thus,with ten subjectsof
each color, in the firstgame, Red i is matchedwith Blue i. In the second game, Red i is matched
with Blue 1 + i, except Red 10, who is matchedwith Blue 1.
1
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
1
1
1
3
2
3
3
1
3
2
1
3
2
3
4
3
4
2
5
4
2
3
4
3
4
2
3
3
4
2
5
4
4
2
4
3
3
3
5
4
1
2
3
3
4
2
4
1
3
3
4
3
2
2
4
4
3
3
3
3
4
1
3
2
2
3
3
3
4
2
3
3
3
3
4
3
2
3
4
3
3
2
1
2
4
3
3
2
3
2
3
3
2
3
2
2
2
2
3
3
3
2
5
SESSION 1
(Four move, PCC)
2
2
2
2
2
2
2
2
2
1
2
3
4
5
6
7
8
9
4
2
3
3
3
3
2
3
2
4
1
3
3
4
3
4
2
4
3
4
2
3
2
3
3
3
2
2
3
1
3
2
3
5
2
4
3
1
2
3
3
2
4
3
3
SESSION
2
(Four move, PCC)
4
2
1
2
3
3
2
3
2
2
2
1
3
3
2
3
2
3
2
2
1
2
2
3
2
3
2
2
1
1
3
3
2
2
2
2
835
CENTIPEDE GAME
2
4
2
2
3
4
2
3
2
2
3
SESSION
3
2
1
2
2
5
4
3
5
2
2
2
1
3
2
2
5
3
5
3
4
2
2
2
5
2
3
3
4
4
3
4
2
3
3
5
2
2
5
4
1
2
3
4
5
6
7
8
9
10
3
3
3
3
3
3
3
3
3
3
3
2
1
3
4
4
3
2
2
3
2
2
1
3
3
4
2
2
2
3
2
5
3
3
3
2
2
2
4
2
1
4
3
3
2
2
2
5
2
2
3
3
2
2
3
2
1
2
4
3
4
1
2
3
4
2
1
3
2
1
4
1
3
3
2
2
3
2
2
1
2
1
3
2
2
2
2
2
3
1
4
3
2
2
2
2
1
2
3
1
4
2
3
2
1
2
1
3
2
1
4
2
3
3
4
3
5
3
4
2
2
3
3
3
5
3
4
3
2
4
4
2
3
3
5
3
4
2
4
4
3
2
5
3
1
3
2
1
4
6
(Four move, CIT)
4
3
2
3
3
3
3
5
3
2
2
3
4
2
3
3
3
3
3
3
1
2
3
4
5
6
7
8
9
10
4
4
4
4
4
4
4
4
4
4
2
2
3
2
5
4
2
3
3
1
4
SESSION
4
2
2
3
2
5
4
2
3
1
5
2
3
2
3
2
5
4
3
1
(Four move, High payoff, CIT)
5
3
5
3
5
4
3
5
5
5
1
2
3
4
5
6
7
8
9
10
5
5
5
5
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
1
2
3
4
5
6
7
8
9
7
3
4
3
4
4
3
4
5
5
5
3
4
3
4
4
3
5
5
4
5
6
4
5
4
4
4
2
5
6
5
5
5
5
2
6
5
4
4
3
4
3
5
4
3
3
5
4
5
SESSION
(Sixmove,CIT)
4
3
3
3
3
3
3
4
5
5
4
7
5
5
6
5
4
4
5
5
4
4
4
6
4
4
5
4
5
6
5
5
4
4
3
4
5
6
SESSION
(Sixmove,PCC)
2
4
3
3
4
3
3
3
4
5
5
4
4
4
4
4
4
5
4
6
4
4
4
5
4
3
4
4
5
4
4
5
4
7
5
6
3
4
4
5
5
5
6
4
5
4
836
R. D. MCKELVEY AND T. R. PALFREY
7
7
7
7
7
7
7
7
7
7
4
2
5
4
2
7
6
4
4
6
1
2
3
4
5
6
7
8
9
10
4
5
3
2
5
6
4
5
3
4
4
4
2
5
5
6
4
5
5
2
3
2
5
4
4
6
6
4
5
4
3
5
5
6
5
5
4
4
5
3
SESSION7
4
4
4
6
6
4
4
5
4
2
3
3
5
5
4
4
4
5
4
3
3
5
5
4
4
4
4
4
5
3
3
3
4
4
4
4
6
5
4
4
3
4
4
4
4
6
6
4
4
3
(Six move, PCC)
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