Unraveling Games

8/4/2019 Unraveling Games

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American Economic Association

Unraveling in Guessing Games: An Experimental StudyAuthor(s): Rosemarie NagelSource: The American Economic Review, Vol. 85, No. 5 (Dec., 1995), pp. 1313-1326Published by: American Economic AssociationStable URL: http://www.jstor.org/stable/2950991 .

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Unravelingn Guessing Games: An ExperimentalStudyBy ROSEMARIENAGEL*

Consider the following game: a large num-ber of players have to state simultaneously anumber in the closed interval [0, 100]. Thewinner is the person whose chosen number isclosest to the meanof all chosen numbersmul-tiplied by a parameterp, where p is a prede-termined positive parameterof the game; p iscommon knowledge. The payoff to the winneris a fixed amount, which is independentof thestatednumberand p. If thereis a tie, the prizeis divided equally among the winners. Theother players whose chosen numbers are fur-ther away receive nothing.'The game is played for four rounds by thesame group of players. After each round, allchosen numbers,the mean, p times the mean,the winning numbers,and the payoffs are pre-sented to the subjects. For 0 c p < 1, thereexists only one Nash equilibrium:all playersannounce zero. Also for the repeated super-game, all Nash equilibriainduce the same an-nouncements and payoffs as in the one-shotgame. Thus, game theory predicts an unam-biguous outcome.The structureof the game is favorable forinvestigating whether and how a player's men-tal process incorporates the behavior of theother players in conscious reasoning. An ex-planation proposed, for out-of-equilibriumbe-havior involves subjects engaging in a finitedepth of reasoning on players' beliefs about

one another.In the simplest case, a player se-lects a strategyat randomwithout forming be-liefs or picks a number that is salient to him(zero-order belief). A somewhat more so-phisticatedplayer forms first-order beliefs onthe behavior of the other players. He thinksthatothers select a numberat random, and hechooses his best response to this belief. Or heforms second-order beliefs on the first-orderbeliefs of the others and maybe nth order be-liefs about the (n - I )th order beliefs of theothers, but only up to a finite n, called the n-depth of reasoning.The idea that players employ finite depthsof reasoning has been studied by varioustheorists (see e.g., Kenneth Binmore, 1987,1988;ReinhardSelten, 1991;RobertAumann,1992; Michael Bacharach, 1992; CristinaBicchieri, 1993; Dale 0. Stahl, 1993). Thereis also the famous discussion of newspapercompetitions by John M. Keynes (1936 p.156) who describes the mental process ofcompetitors confronted with picking the facethat is closest to the mean preference of allcompetitors.2Keynes's game, which he con-sidered a Gedankenexperiment, has p = 1.However, with p = 1, one cannot distinguishbetween different steps of reasoning by actualsubjects in an experiment.There are some experimental studies inwhich reasoning processes have been analyzedin ways similar to the analysis in this paper.Judith Mehta et al. (1994), who studied be-haviorin two-personcoordination games, sug-gest thatplayerscoordinateby either applyingdepth of reasoning of order I or by picking afocal point (Thomas C. Schelling, 1964),which they call "Schelling salience." StahlandPaul W. Wilson (1994) analyzedbehaviorin symmetric3 x 3 games and concluded thatsubjectswere using depthsof reasoning of or-ders 1 or 2 or a Nash-equilibrium strategy.

* Department of Economics, Universitat Pompeu Fa-bra, Balmes 132, Barcelona08008, Spain. Financial sup-port from Deutsche Forschungsgemeinschaft (DFG)throughSonderforschungsbereich303 and a postdoctoralfellowship from the University of Pittsburgh are grate-fully acknowledged. I thank Reinhard Selten, DieterBalkenborg,Ken Binmore,JohnDuffy, MichaelMitzkewitz,Alvin Roth, Karim Sadrieh,Chris Starmer,and two anon-ymous referees for helpful discussions and comments. Ilearned about the guessing game in a game-theory classgiven by Roger Guesnerie,who used the game as a dem-onstration experiment.'The game is mentioned, for example, by HerveMoulin (1986), as an example to explain rationalizability,and by Mario H. Simonsen (1988).

2This metaphor is frequently mentioned in the mac-roeconomic literature see e.g., Roman Frydman, 1982).1313


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1314 THEAMERICANECONOMICREVIEW DECEMBER1995Both of these papersconcentratedon severalone-shot games. In my experiments, the deci-sions in first period indicate that depths ofreasoning of order 1 and 2 may be playing asignificant role. In periods 2-4, for p < 1, Ifind that the modaldepth of reasoning does notincrease, although the median choice de-creases over time.3 A simple qualitative earn-ing theory based on individual experience isproposed as a better explanation of behaviorover time than a model of increasing depth ofreasoning. This is the kind of theory thatSelten and Joachim Buchta (1994) call a"learning direction theory," which has beensuccessfully applied in several other studies.Other games with unique subgame-perfectequilibria that have been explored in the ex-perimental literature include Robert Rosen-thal's (1981) "centipede game," a marketgame with ten buyers and one seller studiedexperimentallyby Roth et al. (1991), a public-goods-provision game studied by VesnaPrasnikar and Roth (1992), and the finitelyrepeated prisoner's dilemma studied experi-mentally by Selten and Rolf Stoecker (1986).In the experimental work on the centipedegame by Richard McKelvey and ThomasPalfrey (1992) and on the prisoner'sdilemmasupergame, the outcomes are quite differentfrom the Nash equilibriumpoint in the open-ing rounds, as well as over time. While theoutcomes in Roth et al. (1991), PrasnikarandRoth (1992), and my experiments are also farfrom the equilibrium in the opening round,they approachthe equilibrium in subsequentrounds. Learningmodels have been proposedto explain such phenomena(see e.g., Roth andIdo Erev, 1995).

I. TheGame-TheoreticolutionsFor 0 c p < 1, there exists only one Nashequilibriumat which all playerschoose 0.4 All

announcing 0 is also the only strategy com-bination that survives the procedure of infi-nitely repeated simultaneous elimination ofweakly dominated strategies.5For p = 1 andmore than two players, the game is a coordi-nation game, and there are infinitely manyequilibriumpoints in which all playerschoosethe same number(see Jack Ochs [1995 ] for asurvey). For p > 1 and 2p < M (M is thenumber of players), all choosing 0 and allchoosing 100 are the only equilibriumpoints.Note that for p > 1 there are no dominatedstrategies.6 The subgame-perfectequilibriumplay (Selten, 1975) does not change for thefinitely repeated game.

II. A Model of Boundedly Rational BehaviorIn the firstperiod a player has no informa-tion about the behavior of the other players.He has to form expectations about choices ofthe other players on a different basis than insubsequentperiods. In the subsequent periodshe gains informationabout the actual behaviorof the others and about his success in earlierperiods. Therefore, in the analysis of the dataI make a distinction between the first periodand the remaining periods.

- This kind of unraveling is similarto the naturallyoc-curring phenomena observed by Alvin E. Roth andXiaolinXing (1994) in manymarketsn which it is importantto actjust a littleearlier n time thanthe competition.4 Assume that there is an equilibriumat which at leastone player chooses a positive numberwith positive prob-ability. Let k be the highest numberchosen with positiveprobability, and let m be one of the players who chooses

k with positive probability.Obviously, in this equilibriump times the mean of the numbers chosen is smallerthank.Therefore, player m can improve his chances of winningby replacing k by a smaller number with the same prob-ability. Thereforeno equilibriumexists in which a positivenumberis chosen with positive probability.'Numbers in (lOOp, 100] are weakly dominated bylOOp; n the two-player game, 0 is a weakly dominantstrategy. The interpretation f the infinite iterationprocessmight be: it does not harm a rational player to excludenumbers in the interval (lOOp, 100]. If this player alsobelieves thatall otherplayers are rational,he consequentlybelieves that nobody will choose from (lOOp, 100], andtherefore he excludes (lOOp2, 100]; if he thinks that theothers believe the same, (lOOp3, 100] is excluded, andso on. Thus, 0 remains the only nonexcluded strategybased on common knowledge of rationality. If choiceswere restricted to integers, all choosing I is also anequilibrium.

6 It is straightforward o show that all choosing 0 andall choosing 100 are equilibria:it does not pay to deviatefrom 0 (100) if all other players choose 0 (100) and thenumberof players is sufficiently large. There is no othersymmetric equilibrium since with a unilateral small in-crease a player improves his payoff. Also, other asym-metric equilibriaor equilibria in mixed strategies cannotexist for analogous reasons, as in the case p < 1.


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VOL. 85 NO. 5 NAGEL: UNRAVELINGN GUESSINGGAMES 1315The model of first-periodbehavior is as fol-lows: a player is strategic of degree 0 if hechooses the number 50. (This can be inter-preted as the expected choice of a player whochooses randomly from a symmetric distribu-tion or as a salient number a la Schelling[1960]). A person is strategic of degree n ifhe chooses the number50p", which I will calliteration step n. A person whose behavior isdescribed by n = 1 just makes a naive bestreply to randombehavior.7However, if he be-lieves that the others also employ this reason-ing process, he will choose a numbersmallerthan50p, say 50p2, the best reply to all otherplayers using degree-i behavior. A higher

value of n indicates more strategic behaviorpaired with the belief thatthe otherplayers arealso more strategic; the choice converges tothe equilibriumplay in the limit as n increases.For periods 2-4, the reasoning process ofperiod 1 can be modified by replacingthe ini-tial referencepoint r = 50 by a referencepointbased on the information from the precedingperiod. A naturalcandidate for such a refer-ence point is the mean of the numbersnamedin the previous period. With this initial refer-ence point, iterationstep 1, which is the prod-uct of p and the mean of the previous period,is similar to Cournot behavior (Antoine A.Cournot, 1838) in the sense of giving a bestreply to the strategychoices made by the oth-ers in the previous period (assuming that thebehavior of the others does not change fromone period to the next).8I can also consider "anticipatory earning,"in which an increase in iteration steps is ex-

pected of the other players. Specifically, onecan ask whether, with increasing experience,higher and higher iteration steps will be ob-served. I will show, however, that the modalfrequency,polled over all sessions, remains atiterationstep 2 in all periods. In Section V-Ca quite different adjustmentbehavior is ex-amined, which does not involve anythingsimilar to the computation of a best replyto expected behavior. Instead of this, a behav-ioral parameter- the adjustment factor- ischanged in the direction indicated by the in-dividual experience in the previous period.III. The Experimental Design

I conducted three sessions with the param-eterp = '/2 (sessions 1-3), four sessions withp = 2/3 (4-7), and three sessions with p =4/3(8-10).9 I will refer to these as 1/2, 2/3, or 4/3sessions, respectively. A subject could partic-ipate in only one session.The design was the same for all sessions:15-18 subjectswere seatedfarapart n a largeclassroom so thatcommunicationwas notpos-sible. The same groupplayed for four periods;this design was made known in the written in-structions. At each individual's place were aninstruction sheet, one response card for eachperiod, and an explanation sheet on which thesubjects were invited to give writtenexplana-tions or comments on their choices after eachround. The instructionswere read aloud, andquestions concerning the rules of the gamewere answered.")After each round the response cards werecollected. All chosen numbers, the mean, and

7 If the meanchoice of the others is 50, the numberthatreally comes nearest to p times the mean is a little lowersince this player's choice also influencesp times the mean.My interpretationof iterationstep 1 is comparableto thedefinition of secondary salience introduced by Mehta etal. (1994) or the level-l type in Stahl and Wilson (1994).' Actually, Coumot behavior inresponseto an assumedmean choice x-_ of the other players would not lead to ptimes the mean, but toM-l1p X_jM-p

where M is the number of players. However, there is noindication that subjects try to compute this best reply.

Moreover, for M between 15 and 18, the numberof sub-jects in my experiments, the difference between this bestreply andp times the mean is not large." I use p = '/2, because it reduces calculation difficul-ties. With p = 2/3, I am able to distinguish between thehypothesis that a thoughtprocess startswith the referencepoint 50 and the game-theoretichypothesis that a rationalperson will start the iterated elimination of dominatedstrategies with 100. Forp > 1, p = 4/3 is used to analyzebehavior. There are no sessions with p = 1; this game issimilar to a coordination game with many equilibria,which has already been studiedexperimentally(e.g., JohnVan Huyck et al., 1990).0'A copy of the instructions used in the experimentmay be obtainedfrom the authorupon request.


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1316 THEAMERICANECONOMICREVIEW DECEMBER1995the product of p andthe mean were writtenonthe blackboard (the anonymity of the playerswas maintained). The number closest to theoptimalnumberandthe resultingpayoffs wereannounced. The prize to the winner of eachroundwas 20 DM (about $13). If therewas atie, the prize was split between those who tied.All otherplayersreceived nothing." Afterfourrounds, each player received the sum of hisgains of each period and an additional fixedamount of 5 DM (approximately $3) forshowing up. Each session lasted about 45minutes, including the instructionperiod.

IV. TheExperimentalResultsThe raw data can be found in Nagel (1993)and are also available from the author uponrequest. Whereas I use only nonparametrictests in the following sections, Stahl (1994)applies parametric ests to these dataand con-firms most of the conclusions.

A. First-Period ChoicesFigure 1 displays the relative frequenciesofall first-period choices for each value of p,separately. The means and medians are alsogiven in the figure. All but four choices areintegers. No subject chose 0 in the 2A,and "/2sessions, and only 6 percent chose numbersbelow 10. In the 4/3 sessions, only 10 percentchose 99, 100, or 1. Thus, the sessions withdifferentparametersdo not differ significantlywith respect to frequencies of equilibriumstrategies and choices near the equilibriumstrategies. Weaklydominatedchoices, choiceslarger than lOOp, were also chosen infre-

quently: in the "/2 sessions, 6 percent of the

0.15 A. median 17mean 27.05ao* 0.10-c

0 10 20 30 40 50 60 70 80 90 100ChosenNumbers

01-B. median 3mean 36.73, 0.10-o

IL

X 0.05-

0.00

0 10 20 30 40 50 60 70 80 90 100Chosen Numbers

0.15 C. median 66mean 60.12o 0.10

0.05-

0.00-,,1,,,1,,F11|Sl]BS1|B|X,Jg||

0 10 20 30 40 50 60 70 80 90 100Chosen NumbersFIGURE 1. CH()ICES IN THE FIRST PERIO)D:A) SEssIo)Ns1-3 (P = /2); B) SEssio)Ns 4-7 (P = /3);C) SEssIo)Ns 8-10 (P = 4/3)

subjects chose numbersgreaterthan 50 and 8percent hose50; in the / sessions,10percentof the chosen numbers were greater than 67,and 6 percentwere 66 or 67. Fromthese results

" This all-or-nothing payoff structuremight trigger un-reasonable behavior by some subjects which in turn im-pedes quicker convergence. In John Duffy and Nagel(1995), the behavior in p-times-the-median game wasstudied, in an effort to weaken the influence of outliers.While firstroundbehavior in both the mean- and median-treatments was not significantly different, fourth roundchoices in p-times-the-median game were slightly lowerthanthose inp-times-the-mean game. Changes in the pay-off structure, for example, negative payoffs to losers,might affect the evolution of behavior on the guessinggame in a different way.


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VOL.85 NO. 5 NAGEL: UNRAVELING N GUESSINGGAMES 1317one might infer that either dominatedchoicesare consciously eliminated or referencepointsare chosen that preclude dominated choices.For p > 1, dominated strategies do not exist.Apart from the similarities just mentioned,there are noticeable differences between thedistributions of choices in sessions with dif-ferent values of p. When p was increased, themean of the chosen numberswas higher. I canreject the null hypothesis that the data fromthe'/2 and 2,3 sessions are drawn from the samedistribution, n favor of the alternativehypoth-esis that most of the chosen numbers in the1/2 sessions tend to be smaller, at the 0.001level of statistical significance, according to aMann-Whitney U test. The same holds for atest of the data from the 2A- essions againstthose of the 4/3 sessions: the chosen numbersin the former tend to be smaller than those inthe latter;the null hypothesis is rejected at the0.0001 level. This result immediately suggeststhat many players do not choose numbers atrandombut instead are influenced by the pa-rameterp of the game.I also tested whether the data exhibit thestructuresuggested by the simple model givenin Section III, that is, taking 50 as an initialreference point and considering several itera-tion steps from this point (SOp'). Figure 1shows that the data do not correspondexactlyto these iterationsteps. However, are the dataconcentratedaround those numbers?In orderto test this possibility, I specify neighborhoodintervals of SOp'",or which n is 0, 1, 2, ....Intervalsbetween two neighborhoodintervalsof 50pfl+ ' and SOp' are called interim inter-vals. I use the geometric mean to determinethe boundaries of adjacentintervals. This ap-proachcapturesthe idea that the steps are cal-culated by powers of n. The interimintervalsare on a logarithmic scale approximately aslarge as the neighborhood intervals, if round-ing effects are ignored.'2

0.35-. A.0.30 step 2

0.25-c ~~~~~~~~~~~step

a)~ ~ ~ ~ ~ ~ ~~~~B

: 0.20-0r steps0.15-

0.10-0.05- step 30.00


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1318 THE AMERICANECONOMICREVIEW DECEMBER 1995sessions, almost 50 percent of the choices arein the neighborhood intervalof either iterationstep 1 or 2, and there are few observations inthe interim interval between them. In all ses-sions only 6-10 percent are at step 3 andhigher steps (the aggregation of the two left-hand columns in Figure 2A and Figure 2B[p = '/2 and p = 2/4], respectively, and theright-hand column of Figure 2C [p 4/])*(Choices above 50 in the '/2 and 2, sessionsand choices below 50 in the 4/3 sessions aregraphed only in aggregate.) The choices aremostly below 50 in the '/2 and 2Asessions andmostly above 50 in the 4/3 sessions; thisdifference is statistically significant at the1-percent level, based on the binomial test.To test whether there are significantlymoreobservations within the neighborhood inter-vals than in the interim intervals I consideronly observations between step 0 and step 3.Hence, the expected proportion within theneighborhood interval under the null hypoth-esis (that choices arerandomlydistributedbe-tween interim and neighborhoodintervals) isthen the sum of the neighborhood intervalsdi-vided by the interval between step 0 and step3. Note that this is a strongertest than takingthe entire interval 0-100. The one-sided bi-nomial test, taking into considerationthe pro-portion of observations in the neighborhoodintervals, rejects the null hypothesis in favorof the hypothesis that the pooled observationsare more concentrated n the neighborhoodin-tervals (the null hypothesis is rejected at the1-percent level, both for the '/2 sessions andfor the 2A sessions; it is rejected at the 5-percent level for the 4/, sessions).'3"'4Note that over all '/2 sessions, the optimalchoice (given the data) is about 13.5, which

belongs to iteration step 2, which is also wherewe observe the modal choice, with nearly 30percent of all observations. Over all 2/' ses-sions, the optimal choice is about 25, whichalso belongs to iterationstep 2 with about 25percentof all observations, the second-highestfrequency of a neighborhood interval. Thus,many players are observed to be playing ap-proximately optimally, given the behavior ofthe others.B. The Behavior in Periods 2, 3, and 4

To provide an impression of the behaviorover time, Figures 3-5 show plots of pooleddata from sessions with the same p for eachperiod; the plots show the choices of each sub-ject from round t to t + 1. In the 1/2 and 2A,sessions, 135 out of 144 (3 transitionperiodsx48 subjects) and 163 out of 201 observations(3 x 67 subjects), respectively, are below thediagonal, which indicates that most subjectsdecrease their choices over time. In all ses-sions with p < 1, the medians decrease overtime (see Table 1); this is also true for themeans except in the last period of the I/2ses-sions. In the 4/3 sessions, the reverse is true:133 out of 153 observations(3 x 51 subjects)are above the diagonal and the medians in-crease and are 100 in the third and fourthpe-riods. Thus from roundto round,the observedbehavior moves in a consistent direction, to-wardan equilibrium. (It is this movement thatis reminiscent of the unraveling in time ob-served in many markets by Roth and Xing[1994].)In the 1/2 sessions, more than half of the ob-servations were less than 1in the fourthround.However, only three out of 48 chose 0. In the2A sessions, only one player chose a numberless than 1. On the other hand, in the 4/3 ses-sions, 100 was already the optimal choice inthe second period, being chosen by 16 percentof the subjects;and in the third and fourthpe-riod, it was chosen by 59 percentand 68 per-cent, respectively. Thus, for the 4, sessions Iconclude that the behavior of the majorityofthe subjects can be simply described as thebest reply (100) to the behavior observed inthe previous period. (Some of the subjectswho deviated from this behavior argued thatthey tried to influence the mean [to bring it

'3 Since the iteratedelimination of dominatedstrategiesstarts the reasoning process at 100, I also tested whetherthat initial reference point would structure the data in acoherentway for the different parameters.For p = 4/3 alliteration steps collapse into 100; thus spikes cannot beexplained. For the 2/3 sessions the data are not only con-centratedaround 100 x (2/3)". On the other hand, the pat-tern of the '/A-session ata is similarto the pattern n Figure3A, except that step n becomes step n + 1. Hence, 100 isnot a plausible initial reference point for most subjects.14 The written comments of the subjects also seem tosupportour model. For details see Nagel (1993).


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VOL. 85 NO. 5 NAGEL: UNRAVELINGN GUESSINGGAMES 1319100- 100-A. A.90- 90-

-~80 . 80- +o ~~~~~~~~~~~~~~~~~~~~0ii 70 i 70-

c 60- c 60-o 0o 0a) 50- a) 50-U) U~~~~~~~~~~~~~~~~~C)+40- + -40 ++Si Si + ~~~~~~~~~~~~~~~~~~~~~~~~~~~+530 ~~~~~~~~~~~~~~~~~~~~~~~~~+ 30 ++

300++ + +++ + +~~~~~~~~~~~~~~

+ +~~~~~~~~~~~~~~~~0 10 20 3 40 do d io ~0 40 100 0 10 20 3 40 5 60 70 80 90 100Choices in First Period Choices in First Period

100- 100-B. B.90- 90-80- 80-

.070 c 70-o 60- *2 60-H 50 H 50-

40- 40-o 0+30- 0 30-o) 0+ +20- 20 + *. ++++ +10 .- + 10: +40 10 20 30 40 50 60 70 80 90 100 0 10o 20 30 40 0 60 70 80 90 100Choices in Second Period Choices in Second Period

100-i 100-C. C.90- + 90-80- 80-

o0 0070 c 70Q.c60 . 60-

LO 50- 0 50 +0 0L40- C 40-5 30 s~~~~~~~~~~~~~~~~~~~0-

20- ~~~~~~~~~~~~~~~20-++10- 10- + ++

0 10 20 30 40 50 60 70 80 90 100 0 10o 20 30 40 50 d0 70 80 90 10Choices inThirdPeriod Choices in ThirdPeriodFIGURE 3. OBSERVATIONS OVER TIME FOR SESSIONS 1-3 FIGURE 4. OBSERVATIONS OVER TIME FOR SESSIONs 4-7(p = A/):A) TRANSITION FRom FIRST TO SECOND (p 74): A) TRANSITION FROm FIRST TO SECONDPERIOD; B) TRANSITION FROM SECOND To THIRD PERIOD; PERIOD; B) TRANSITION FROM SECOND To THIRD PERIOD;C) TRANSITION FROm THIRD To FoURTH PERIOD C) TRANSITION FROm THIRD To FoURTH PERIOD

down again] or wrote that the split prize wastoo small to state the obvious right answer.)The adjustmentprocess towardthe equilibri-um in the 1/2 and 2A3essions is quite differentfrom that in the 4/3 sessions. Zero is never the

best reply in the 1/2 and 2Asessions, given theactual strategies. Instead the best reply is amoving target that approaches 0. The adjust-ment process is thus more complicated. Com-paring Figures 3 and 4, one can see that the


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1320 THEAMERICANECONOMICREVIEW DECEMBER1995100 - H - !n 80A. + + + +90- +

c 80- + ,+, 4 /70-

60-0 +50O40-

o 30- /20-

0 10 20 30 40 50 60 70 80 90 100Choices in First Period

8 observations100- H HB.90- +80-

a) 70 +- 60-H 50- +U) 40-

*5 30+20- /10-

0 1 2 3 40 50 60 70 80 90 1i00Choices inSecond Period

27observations100- C. +90a c-o80-0 700-

-C 60- +LL050-*) 40-0 30-

020-

0 10 20 3 4 0 60 7 90 100Choices in ThirdPeriodFIGURE 5. OBSERVATIONS OVER TiME FOR SESSIONS8-10 (p = 44): A) TRANSITION FROm FIRST To SECONDPERIOD; B) TRANSITION FROM SECOND TO THIRD PERIOD;C) TRANSITION FROM THIRD TO FOURTH PERIOD

choices in '2, sessions converge faster towardO than those in '4Asessions. However, thereason might be that the initial distributionisat lower choices in the formersessions thaninthe latter.Therefore,to investigatewhethertheactual choices decrease faster for p = '/2 than

for p = 2/, I define a rate of decrease of themeans and medians from period 1 to period 4within a session by(la) Wmclan (mean), - (mean), =4(mean),=,(1b ) wmcd (median), - (median),=4(median), ,IThe ratesof decrease of the single sessions areshown in Table 1, in the last lines of panels Aand B. The rates of decrease of the sessionmedians in the '/2 sessions are higher thanthose in the 2/ sessions, and the difference isstatistically significant at the 5-percent level(one-tailed), based on a Mann-Whitney Utest. There is no significant difference in therates of decrease of the means. The medianseems more informative than the mean, sincethe mean may be strongly influencedby a sin-gle deviation to a high number. Thus, I con-clude that the rate of decrease depends on theparameterp.Analyzing the behavior in the firstperiod, Ifound some evidence that r = 50 was a plau-sible initial reference point. Below, I classifythe data of each of the subsequentperiods ac-cording to the reference point r (mean of theprevious period) and iteration steps n: rp".Numbers above the mean are aggregated to"above mean,_" (see Table 2).'5As was the case for the first-periodbehav-ior, one cannot expect thatexactly these stepsare chosen. Grouping the data of the subse-quentperiods and sessions in the same way asin the first period, namely, in neighborhoodintervals of the iterationsteps and interim in-

'"The chosen numbers tend to be below the mean ofthe previous period,and the difference is significant atthe5-percent level for all 'A and 2/Asessions and all periodst = 2-4, according to the binomial test. The same testdoes not reject the null hypothesis for p times the meanof the previous period, for periods 2 and 3. In the fourthperiod the chosen numbers are significantly (at the 1-percent level) below p x r, in six out of seven sessions.Note that if I had analyzedthe datastarting romreferencepoint "naive best reply of the previous period" (p x r),instead of startingfrom the mean, step n would becomestep n - 1, and all choices above the naive best replywould be aggregatedto one category.


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VOL.85 NO. S NAGEL: UNRAVELING N GUESSING GAMES 1321TABLE 1-MEANS AND MEDIANS OF PERIODS 1-4, AND RATE OF DECREASE FROM PERIOD 1 TO PERIOD 4

A. Sessions withp '1

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