Beyond Bayes's Theorem: Effect of Base-Rate Information in … · 2017-05-16 · 84 GILAT, MEYER,...

Journal of Experimental Psychology: Applied1997, Vol. 3, No. 2, 83-104

Copyright 1997 by the American Psychological Association, Inc.1076-898X/97/$3.00

Beyond Bayes's Theorem: Effect of Base-Rate Informationin Consensus Games

Sharon Gilat, Joachim Meyer, Ido Erev, and Daniel GopherTechnion-Israel Institute of Technology

Many of the practical implications of behavioral decision-making researchare based on the assumption that behavioral trends have to be comparedwith normative prescriptions. The present article demonstrates that incertain settings this approach is both inapplicable because there is no"normative" prescription and unnecessary because robust quantitativepredictions can be made without reference to normative prescriptions.

Experiment 1 demonstrates that a simple learning rule can be used to predictthe base-rate effect in consensus games with multiple equilibria. Experi-ment 2 shows that information about the payoff rule affects participants'initial propensities but does not affect the learning process. Some implica-tions of these results for the understanding of decision groups in socialcontexts, such as employment decisions in organizations, are pointed out.

Most attempts to derive practical implicationsfrom research on judgment and decision makingare based on comparisons of observed descriptivetendencies to normative prescriptions of rationaldecision theory. Much of this research focuses ona debiasing technique that reduces the differencebetween observed decisions and the optimalprescription (e.g., see Von Winterfeldt & Ed-wards, 1986). Whereas this normative-based ap-proach has led to useful results, its applicability is

Sharon Gilat, Joachim Meyer, Ido Erev, and DanielGopher, Research Center for Work Safety and HumanEngineering, Technion-Israel Institute of Technology,

Haifa, Israel.This study was supported by the Committee for

Research and Prevention in Occupational Safety andHealth, Israel Ministry of Labor and Social Affairs.Parts of this study were presented at the 15th BiannualConference on Subjective Probability, Utility andDecision Making (SPUDM-15), Jerusalem, Israel.

Correspondence concerning this article should beaddressed to Ido Erev, Department of Industrial Engi-neering and Management, Technion—Israel Institute ofTechnology, Haifa 32000, Israel, or to Joachim Meyer,who is now at the Department of Industrial Engineer-ing and Management, Ben Gurion University of theNegev, P.O. Box 653, Beer Sheva 84105, Israel.Electronic mail may be sent via Internet [email protected].

limited. The present article demonstrates this

limitation and provides an evaluation of analternative approach by focusing on one example:the effect of base-rate information in situations in

which decision makers are motivated to reach aconsensus. We chose this example because

Bayes's theorem, which implies a specific utiliza-

tion of base-rate information, is one of the bestunderstood and most useful tools of rationaldecision theory.

Previous studies that explored the effect ofbase-rate information on subjective judgment anddecision making focused on simple situations in

which the optimal utilization of the base rates canbe derived from Bayes's theorem. Generally, ithas been found that people tend to use base-rate

information less than optimally (e.g., Bar Hillel,1980; Beyth-Marom & Fischhoff, 1983; Fis-chhoff & Bar-Hillel, 1984; Kahneman, Slovic, &Tversky, 1982; Kahneman & Tversky, 1973;Lyon & Slovic, 1976; Tversky & Kahneman,1974, 1980). This finding is quite robust, al-though numerous variables were shown to affectthe utilization of base-rate information (e.g.,Ajzen, 1977; Gigerenzer, Hell, & Blank, 1988;Nisbett & Ross, 1980; Tversky & Kahneman,1982). The robustness of the insufficient usage ofbase-rate information is particularly clear inbinary decisions that can be considered as signal

83

84 GILAT, MEYER, EREV, AND GOPHER

detection tasks. Experimental studies reveal thatindividuals tend to underweight the base rates,even after long practice periods (Healy & Kubovy,1981; but see Birnbaum, 1983, for a criticalassessment of Bayesian inference as the basis forassessing optimal behavior with base-rate infor-mation).

Whereas the comparison of base-rate utiliza-tion to the prescription of Bayes's theorem isinstructive, it is important to recall that thesituations under which Bayes's theorem de-scribes rational behavior are quite limited. Bayes'stheorem can be used to find the alternative thatmaximizes utility when decisions are made underuncertainty, but it is often insufficient for comput-ing the best course of action in social interactionsinvolving numerous actors who may be interde-pendent in the outcomes of their actions.

The case we present here are situations whereactors not only attempt to be accurate in theirdecisions but also strive to reach agreement withanother person. For instance, when deciding inwhat restaurant to reserve a table for dinner witha guest whose preferences are not known, restau-rant goers will probably choose a place that theynot only like themselves but also one they hopethe guest will enjoy; they will make the decisionby guessing from what they think most otherpeople like. Base rates are likely to have a majoreffect on behavior in situations of this type. Werefer to situations in which consensus with an-other person is reinforcing as consensus games.In the present article, we demonstrate that inthese situations there often exists more than oneoptimal way to use a base rate and that the effectof base rate on choice behavior can be substan-tial.

The basic type of decisions that we model hereare binary decisions where an actor has to decideto which of two mutually exclusive categories anevent belongs. A large number, in fact, perhapsmost decisions in social or organizational settingsare of this type. For instance, a personnel man-ager may have to decide whether a candidateshould be hired for a certain job. Other examplesinclude the bank official who has to decidewhether to grant a loan to an applicant, theequipment safety supervisor who has to decidewhether the system is safe to operate, or thereviewer of a scientific journal who has to decidewhether a submitted paper should be accepted for

publication. Decisions of this type are naturallymodeled in terms of signal detection theory (SDT),as presented by Green and Swets (1966/1988).

Classic One-Person SignalDetection Theory

It is convenient to distinguish between threesubmodels that are implicit in classical SDT: themodel of the strategies, the payoff matrix, and thechoice rule. The first model assumes that theobserver considers cutoff strategies along thecontinuum of a single variable (e.g., the candi-dates' qualifications for the job) in which eachinstance (i.e., candidate) is summarized by acertain value. Assuming that the two categoriesof candidates (those who should be hired andthose who should not) each have a distribution ofvalues, one can specify the possibility of distin-guishing between members of the two categoriesthrough the standardized distance between themeans of the two distributions termed sensitivityor d' (see Figure 1). Usually one distribution isNoise (N), whereas the other is referred to asSignal (S) or Signal + Noise. The decisionmaker, or personnel manager, is assumed toconsider cutoff strategies. Each strategy implies acutoff point above which the decision maker willcategorize an event as belonging to the S distribu-tion and below which the event is categorized asbelonging to the N distribution. This point isusually denoted as response criterion or (3.

Noise Slgnal+NolM

FA

"N"cutoff "S"

Figure 1. The Signal (S) and Noise (N) distributionsin one-person signal detection. The distance betweenthe means of the two distributions (d1) is the sensitiv-ity, and the cutoff point above which the decisionmaker indicates that this is a signal specifies theresponse criterion (P). CR = correct rejection;FA = false alarm.

BASE RATE IN CONSENSUS GAMES 85

The second model, the payoff matrix, is anabstraction of the possible outcomes. Because theperceived signal includes noise, the observercannot be completely accurate; therefore, thereare four possible outcomes (see Figure 2a):correct detection of a signal (hit in the commonSDT terminology), correctly say N (correct rejec-tion or CR), say N when a signal was presented(miss), or say S when there was no signal (falsealarm or FA).

The third model, the choice rule, is oftenreferred to as the ideal observer model. It impliesthjit the detector is a utility maximizer who setsan optimal response criterion. For a single deci-sion maker, the optimal response is computed byusing Bayes's theorem. We should note that thisis a strong interpretation of the ideal observerassumption. According to a weaker definition, anideal observer uses a likelihood ratio responserule that may not be optimal.

a

State of

Nature

Signal

Noise

Player's Response

•Signal'

Hit

FalseAlarm

•Noise-

Miss

CorrectRejection

b

State of

Nature

Player 1's

Response

Player 2's

Response

Signal

Noise

•Signal1

•Signal'

HH1

HH2

FF1

FF2

•Noise'

HM1

HM2

FC1

FC2

•Noise'

'Signal-

MH1

MH2

CF1

CF2

•Noise'

MM1

MM2

CC1

CC2

Figure 2. Notations for one-person (a) and two-person (b) signal detection. For die two-person matrix(b), the upper left combination of letters indicates theutility for Player 1, and the lower right combination ofletters indicates the utility for Player 2. The letterscorrespond to the four possible outcomes in signaldetection theory, hit (H), miss (M), false alarm (F),and correct rejection (C), so that, for instance, thecombination MH2 indicates the utility for Player 2when Player 1's response was a miss and Player 2'sresponse was a hit.

Two-Person SDT

Assume that the decision maker is not workingin isolation; rather, the decision maker has toreach agreement with another person. For in-stance, the personnel manager mentioned abovehas to reach the same decision as another officialin the company in order for this decision to be

approved. It is natural to assume that not having adecision approved may be unpleasant. Under thisassumption, the two actors are involved in aconsensus game, with each trying to make deci-

sions that are most likely to correspond to theother's decision. Analyzing situations like thisrequires an extension of SDT to two-personsituations. Two complementary extensions have

been suggested in the literature. Sorkin and Dai(1994) extended SDT to situations in which ateam of observers who can communicate have toreach common decisions. &ev, Gopher, Itkin,and Greenshpan (1995) extended the theory tononcooperative games, situations in which eachobserver has to make an independent decision.The current article focuses on problems that arebest modeled by Fjev et al.'s extension.

Erev et al. (1995) noted that only the payoffmatrix has to be modified in order to addresstwo-person noncooperative signal detectiongames. Because there are only two players andtwo possible responses, the interaction betweenthe players can be summarized in a payoff matrix(see Figure 2b). It presents the players' utility,given the state of nature and the two responses.The top left and bottom right entries in each cellpresent Player 1 's and Player 2's utilities, respec-

tively.

The models of the strategies and the decisionrule do not have to be modified to considertwo-person games. Yet the technical extension ofthe ideal observer model to the two-person case isnot trivial. The calculation of the strategy thatmaximizes utility in the two-person game isdifferent from the calculation in the case of asingle detector. A utility maximizing player wouldhave to behave differently when playing alonethan when playing with other players. Accordingto classical game theory, rational players choosetheir best response, which is based on the assump-tion that the other players will choose the bestresponse to their own strategy. This logic leads toequilibrium predictions. Following Erev et al.


(1995), Appendix A presents the computation ofthe equilibria for two-person signal detectiongames when the two players are independent (seeindependent signals).

Formal Definition and Equilibriain Consensus Games

The notation of two-person games uses letterpairs to denote the payoffs for the various combi-nations. In each letter pair in the two-personcondition, the left letter refers to Player 1 and theright letter refers to Player 2, where H = hit, C =correct rejection, F = false alarm, and M = miss.The numerals 1 and 2 indicate the payoff of eitherPlayer 1 or Player 2. For example, HH1 refers tothe payoff for Player 1 when both Player 1 and 2made a hit, and CF2 refers to the payoff for Player2 when Player 1 made a correct rejection andPlayer 2 made a false alarm. In these notations, atwo-person binary consensus game can formallybe defined by two constraints on the possiblepayoffs: First, Player ;'s payoff, given a certainchoice and a state of nature, is maximal whenPlayer j makes the same decision; for example,for Player 1: HH1 > HM1 and MM1 2 MH1 andCC1 £= CF1 and FF1 > FC1; second, in at leastone case, Player i benefits from consensus (forPlayer 1: HH1 > HM1 or MM1 > MH1 orCC1 > CF1 or FF1 > FC1. Note that in naturallyoccurring consensus games, the decision makersare likely to be in conflict between the attempt tofind the best decision and the strive to reachconsensus. Such a conflict can be modeled by thedifferent payoffs.

In consensus games when the consensus issufficiently important (for Player 1: MM1 > HM1and FF1 == CF1), the extreme cutoffs (p —» 0 orIn P —* —o° and P —* °° or In |3 —* oo) are alwaysequilibria. That is, when reaching a consensus ismore important than being correct, there areequilibria in which players ignore the signal andalways use one of the two responses; in this way,they can achieve 100% agreement rate. In addi-tion, in some games additional equilibria pointsmay exist. Moreover, in an extreme condition(considered below) where the two observers seethe same signal in each trial (or a dependentsignals case) every symmetrical pair of cutoffs isan equilibrium. Thus, generally there is no singlerational cutoff in consensus games.

The case of dependent signals is probablyextremely rare in actual social settings, and mostsituations that resemble consensus games shouldprobably be considered independent games, thatis, games in which the two players form theirjudgments, which are based on different stimuli.This is obvious when the two decision makersreceive different information, as when two peopleinterview an applicant independently, each obtain-ing somewhat different information from theinterview. Yet even given that the two interview-ers may have conducted the interview jointly,they will still be involved in an independentgame. A candidate's evaluation is the result of thecombination of the values in a number of dimen-sions (e.g., her or his past experience, communi-cation skills, or demeanor), each of which issubject to some random error and may be weighteddifferently by different evaluators. Hence, theevaluators' scores on the combined dimension onwhich judgments are based (e.g., the applicant'ssuitability for the position) will most likely differto some extent.

An Alternative Choice Model:A Reinforcement-Based Learning Rule

The observation that consensus games can havemultiple equilibria does not imply that behaviorin these situations is unpredictable. Rather, itimplies that behavior is likely to be sensitive tothe exact learning process. An understanding ofthe learning process is necessary in order topredict which equilibrium is an attraction point.

Roth and Erev (1995) suggested a simplereinforcement-based learning model that has beenfound to provide a useful approximation ofbehavior in a wide set of experimental games (seeBornstein, Erev, & Goren, 1994; Erev, Maital, &Or-Hof, in press; Erev & Roth, 1995; Ochs, 1995;Rapoport, Seale, Erev, & Sundali, in press; Roth& Erev, 1995). The model is based on the law ofeffect (Thorndike, 1898), which states that theprobability that a certain strategy will be adoptedincreases when this strategy is positively rein-forced. Similar models have been suggested byHarley (1981) to describe animal learning pro-cesses and by Bush and Mosteller (1955). Erev etal. (1995) adapted the model to detection gamesunder the assumption that the set of strategiesavailable to the players is a set of possible cutoff


points (i.e., each value of p is considered astrategy). The adapted model's basic assumptionsare

1. Each player considers a finite set of cutoff

strategies.

2. The player has an initial propensity to select

each of the strategies.

3. Reinforcements (payoffs relative to a refer-

ence point) affect the propensity to choose the

selected strategy again in line with the law of

effect with the addition of a generalization and a

forgetting processes.

4. The choice probabilities are determined by

relative propensities.

A quantification of these assumptions is re-quired to allow a derivation of the model'spredictions by a computer simulation. Erev etal.'s (1995) quantification is presented in Appen-dix B. This quantification provides a good ap-proximation of Erev et al.'s results and repro-duces the main experimental results obtained intraditional one-person signal detection studies(for a review, see Erev, 1995). In one-persontasks, the model typically (see exceptions inErev) predicts a slow convergence toward theideal observer's predictions. Thus, it reproducesthe base-rate underutilization phenomenon in thiscase; the predicted cutoffs are less extreme thanthe cutoffs predicted under the assumption ofefficient base-rate utilization.

The model's predictions for consensus gamesdepend on the dependency between the twoobservers; it predicts an extreme base-rate effectwhen the two observers are independent, but aweak effect, as in the one-person case, when thetwo players see dependent signals.1 In addition,the model predicts more variability in cutoffselection in the two-person task with independentobservers. This variability is expected to lead tolower estimated d' scores in this condition. Themodel's quantitative predictions for the gamesstudied here are presented in the Results andDiscussion sections below. These sections alsoshow that the model's qualitative predictions,which are stated above, are relatively insensitiveto the exact choice of parameters.

The set of experiments described below weredesigned to examine how people perform inexperimental consensus games. Specifically, thedegree to which the equilibrium predictions andthe predictions gained from the reinforcement-

based learning model correspond to participants'behavior in these games was assessed. The experi-ments utilize an external signal detection para-digm (see Kubovy, Rapoport, & Tversky, 1971;Lee, 1963) in which participants are asked tocategorize stimuli that are sampled from one oftwo normal distributions.

We decided to conduct an entire game ratherthan have participants play against the computerin order to avoid arbitrary decisions about thecomputer strategy. In games with multiple equilib-ria, like the game we describe below, minorchanges in the opponent's strategy can lead tocompletely different behavior.

Experiment 1

Experiment 1 focused on the consensus gamepresented in Figure 3b, as it was played by twosymmetrical players with administrated d' (thedistance between the two experimentally deter-mined normal distributions, referred to as D') of1.5 when the prior probability (base rate) of asignal, which is denoted p(S), is .6. Given theseparameters and assuming that the functions S,(.)and N,-(.) are symmetrical normal distributions,with variance 1 and with the means —D'I2and D'/2, respectively. Appendix A implies thatthe game has three distinguishable sets of equi-librium points: Two low-equilibria cutoffs atp —• 0 (In p — -oo) and P = .02 (In p = -4), amoderately low-equilibrium cutoff at p = .42(In (J = —.87), and two high-equilibria cutoffs atP = 163 (In P = 5.8) and P —•« (In p — »).

In choice tasks, the base-rate effect can be

1 Whereas the model's predictions are not trivial

(we were surprised by them), they can be understood

by examination of the reinforcement structure of a

simplified situation in which Player i sets a cutoff at

the center (In p; = 0) and only ; learns from experi-

ence. Given this cutoff, Player i will choose the

frequent responses more than 50% of the time (assum-

ing d( > 0). Consider the independent case first. From

fs point of view, the simplified game is similar to a

one-person game in which the frequent response is

more likely to be reinforced because i chooses it in

more than 50% of the trials. Thus, an extreme

base-rate effect is predicted. In the dependent game,

on the other hand,./ maximizes reinforcement probabil-

ity by selecting i's cutoff (the center). Some learning is

still possible due to the generalization process.

GILAT, MEYER, EREV, AND GOPHER

a

Slate of

Nature

Signal

Noise

Player's Response

•Signal*

10

•10

•Noise'

-10

10

b

State of

Nature

Player 1's

Response

Player 2's

Response

Signal

Noise

"Signal"

•Signal*

10

10

-10

-10

•Noise'

-10

-10

-10

-10

•Noise1

•Signal-

-10

-10

-10

-10

"Noise"

-10

-10

10

10

Figure 3. Payoff matrices for the one-person (a) andthe two-person (b) conditions in the experiments. Forthe two-person matrix (b), the upper left number is thepayoff for Player 1, and the lower right number is thepayoff for Player 2.

evaluated by the proportion of choices of themore frequent category. At the three distinctequilibria, the relevant proportions are expectedto be 1.0, .72 and 0, respectively. Thus, rationaldecision theory cannot be used to predict theoptimal base-rate effects.

As noted above, the rational behavior is evenless clear when players are dependent (i.e., theysee the same stimulus). In this case, everysymmetrical cutoff set (pi = p2) is an equilib-rium because Figure 3 implies that Player ishould try to make the same decision as Player./.In the dependent game, the probability for consen-sus reaches 1 when both players choose the samestrategy. Thus, Player i cannot improve his or hergains by selecting a cutoff that differs from Player/s cutoff. Clearly, in the dependent games, it isimpossible to predict a specific optimal strategythat may be adopted by the players.

In comparison to the multiple equilibria thatcharacterize the two-person games, there is typi-cally one optimal cutoff in traditional one-personsignal detection tasks. For example, in a one-person simplification of the present game (seeFigure 3a), the optimal cutoff is P = .67(In p = -.4). The expected proportion of thefrequent category in this case is .63.

Method

Participants

Fifty Technion students served as paid partici-pants in the experiment. They performed theexperiment in unmixed male or female pairs andwere randomly assigned to the experimentalconditions. One pair had biased prior informationregarding the experiment and was omitted fromthe experiment. The exact payoffs were contin-gent on performance and ranged from 18 to 26shekels ($6.00-$8.50).

Apparatus

The experiment was programmed and con-ducted with Visual Basic 3 for Windows 3.1. Thissystem was installed on a 486 PC with a SuperVGA screen approximately 35.5 cm diagonal. Inthe display, there were two gray rectangles (6.6cm wide X 13.0 cm high on a blue background asshown in Figure 4). Above each rectangle weretwo white fields (2.2 cm wide X 0.8 cm high) inwhich participants' scores were displayed. Thelower field showed the last trial's score, and theupper field showed the cumulative score.

Participants in all experimental conditions satside by side facing the screen at a distance ofapproximately 50.0 cm from the screen. Theyresponded by pressing keys on the keyboard

Figure 4. Schematic depiction of the experimentalscreen as seen by the participants. Each participantsaw only one half of the screen, and the other half washidden by the partition. The distributions are notshown during the experiment and are only added herefor demonstration purposes.

BASE RATE IN CONSENSUS GAMES

positioned in front of them. A partition wasplaced between the participants in order to pre-vent each from seeing the displays or responsesof the other participant.

In each trial, the participants saw for 1 s awhite square, 0.2 cm wide and 0.2 cm high at 1 of66 possible locations along the right border of thegray rectangle. After both participants in a pairresponded, the feedback was shown, and after a3-s interval the next stimulus was displayed. Thestimuli were sampled from one of two normaldistributions with equal variances, one of whichwas 1.5 standard deviations (2.8 cm) higher thanthe other. (The distributions are shown in Figure4 but were not displayed to the participants.) Thebase rate was set so that 60% of the stimuli weredrawn from one distribution and 40% from theother distribution. The location where the stimuliappeared was determined randomly for eachparticipant in the pair in the independent condi-tion in each trial. That is, the participants saw thestimulus at different locations, although the stimu-lus was always sampled from the same distribu-tion, the higher or the lower, for both of them,keeping the state of nature the same for bothparticipants. In the dependent condition, bothparticipants saw the stimuli at exactly the samelocation in each trial, thus the state of nature wasidentical for both players. In the single-playercondition, half of the pairs saw the stimuli atexactly the same location in each trial, as in thedependent condition, and half of the pairs saw thestimuli at different locations at each trial, as in theindependent condition.

Participants responded to the stimuli by press-ing one of two keys. A response that the stimuluswas sampled from the higher distribution wasgiven by pressing the (6) key by the right playerand (A) by the left player. A response that thestimulus was sampled from the lower distributionwas given by pressing the (3} and (Z) keys,respectively.

Procedure

Participants received written instructions stat-ing that they would see the heights of men andwomen from a sample of students and that theirtask was to decide for each stimulus whether itwas a man or a woman from that sample (i.e.,whether it belonged to the higher or the lower

distribution). The participants were also told thatthere might be more or fewer men than women inthe sample and they should not assume that thenumber of men and the number of women wouldbe equal. The payoff matrix was not explained.Instead, participants were told that correct an-swers were rewarded in a probabilistic fashion;that is, incorrect answers were always costly, butcorrect answers were also costly in some of thetrials.

The experiment consisted of five blocks with100 trials each. At the beginning of each block,the participants received 2,000 points. In eachtrial, they could gain or lose 10 points accordingto the payoff matrix, and the cumulative score foreach block could range between 1,000 and 3,000points. Blocks were separated by a short break.At the end of the experiment for each participant,one of the blocks was randomly selected, and theparticipant was paid according to his or her scorein this block. The value of a point was .01 shekel.The experiment lasted approximately 1 hr.

Experimental Design

There were 24 pairs who participated in theexperiment, 8 in the single-player condition, 8 inthe independent condition, and 8 in the dependentcondition. In the two-person conditions, partici-pants were rewarded only when both were correctand agreed with each other. In the single-playercondition, participants were rewarded accordingto the correctness of their own decisions. Thesingle-player condition was the control conditionfor both two-person games. Therefore, 4 pairs inthis condition saw dependent stimuli (i.e., thestimuli appeared at the same location in thedisplay for both players), and the other 4 sawindependent stimuli. In order to balance the baserate, for half of the pairs in each condition 60% ofthe stimuli were drawn from the higher distribu-tion and 40% from the lower; for the other half,40% were drawn from the higher distribution and60% from the lower.

Results and Discussion

Descriptive Statistics

Table 1 presents the mean raw results for theexperimental conditions and blocks. To derive the


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signal detection statistics, we arbitrarily desig-nated the more frequent distribution as signal andthe less frequent distribution as noise. Accordingto this distinction, P(hit) and P(FA) are theproportion of frequent responses for stimuli thatwere sampled from the more and less frequentdistribution, respectively. The profit (P) is thenumber of points accumulated during an experi-mental block beginning with initial 2,000 points.Table 1 presents the means and standard devia-tions of these averages for the participants.

In line with traditional signal detection re-search, the analysis reported below focuses onstatistics derived from P(FA) and P(hit): In P andestimated d'. In addition, we analyzed the propor-tion of frequent responses, P(frequent), that pro-vides a direct assessment of the base-rate effect.Because players in a pair cannot be considered tobe independent, we used pairs as the unit forstatistical hypothesis testing. Hence, we firstcomputed the variables separately for each partici-pant and block, and then the mean for each pairwas computed. These means were analyzed withtwo- way analyses of variance (ANOVAs)Condition X Block with repeated measures on theexperimental block. We present these analyses follow-ing the derivation of the model's predictions.

The Learning Model's Predictions

Basic simulations. We conducted computersimulations in order to derive the model's predic-tions in a virtual replication of the experiment.There were 800 virtual pairs who participated hithe simulation of each experimental condition(100 virtual replications of each of the eight pairsin each condition) with Erev et al.'s (1995)original parameters. Like the actual participants,each pair of virtual subjects participated in 500trials. The following steps were taken in each trial.

1. The state of the world was randomly deter-mined in accordance with the assumed priorprobabilities.2. The players' cutoffs (ct and c2, which impliesp! and p2 values) were randomly determined inaccordance with A4i (see Appendix B).3. The perceived signals (*i and *2) were se-lected from the assumed normal distribution,given the state of the world.4. Players' decisions were determined (Playerj"s response was S if and only if xt > c,).


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With variability

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1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1-Person2-Pereon, Ind. Big.

2-Pereon dep. slg.90% Cl

Figure 5. Results predicted from the learning model without variability (left) andwith variability (middle) and the results obtained in Experiment 1 (right) for the threeexperimental conditions in the proportion of frequent responses, P(frequent), In f5and d'. ind. sig. = independent signal detection; dep. = dependent signal detection;CI = confidence interval.

5. Profits were calculated using the relevant

payoff table.

6. Propensities were updated in accordance with

A2i (see Appendix B).

7. The reference point for the next trial was

calculated (see Appendix B).

To evaluate the model's predictive power, we

calculated the same statistics that were calculated

for the actual pairs for the virtual pairs. Figure 5

(left column) presents the three statistics that we

chose to analyze. Each curve shows the average

of 800 simulations summarized in five blocks of

100 rounds in one of the three experimental

conditions. Three qualitative predictions can be

observed: Teams with independent observers are

expected to have more extreme In (J and P(fre-


quent) and lower estimated d'. The first twoeffects are indications of faster learning in thedirection of base-rate utilization in the two-person independent condition. The third effect isan indication of a larger within-subject cutoffplacement variation in this condition.

The fact that the model is probabilistic meansthat it not only predicts a central tendency but italso makes a prediction of the complete distribu-tion of possible outcomes. The 90% confidenceintervals (CIs) predicted by the model are repre-sented by the vertical lines in Figure 5. Recallthat 100 virtual replications of the experimentwere run and the CI were derived by setting thelower bound of the interval (for each statistic) bythe 5th percentile's simulation and the upperbound by the 95th percentile. The CIs were alsoestimated under the assumption of normal distri-butions by estimating the virtual pairs' standarddeviation, deriving the expected standard error ina sample of eight dyads, and by either adding orsubtracting 1.645 time this value from the means.The two sets of estimated CI practically coin-cided. This finding implies that the observedmeans in the two-person condition do not reflecta prediction that each dyad will converge to oneof the possible equilibria. Rather, there appears tobe one attraction point in each condition.2

Sensitivity analyses. Two types of sensitivityanalyses were conducted to evaluate the robust-ness of the model's predictions. The first setincludes simulations in which one of the valuesof the nonzero original parameters (m, C^x, S(l),<|>, D, w~, w+, a,, <rg) was either increased ordecreased by 50%. There were 100 virtual dyadsincluded in each of 18 additional simulations(nine parameters x two variations). Table 2presents the average results over blocks of theoriginal and the altered simulations. It shows thatthe three qualitative predictions stated above holdin all cases. This table also shows that the para-meters affect the model's quantitative predictions.

A second set of sensitivity analysis simulationsexamined the effect of between-subjects variabil-ity in the learning parameters. In this analysis, thevalue of the seven nonzero learning parameters(all the parameters considered above with theexception of m and Cmax, the cognitive strategiesparameters) were randomly selected for eachvirtual subject. Following Erev and Roth (1995),the values were drawn from a uniform distribu-

tion with a mean at the value of Erev et al.'s(1995) original parameter. The values were se-lected from the interval, 0,2 (original parameter).Thus, for example, the values for <j>, .001 in theoriginal model, were randomly selected fromthe interval (0, .002). Figure 5 (center column)presents the predicted learning curves under thisvariant of Erev et al.'s model. It shows that thebetween-subjects variability slightly improvesthe model's predictions.

Hypothesis Testing

Figure 5 (right column) presents the experimen-tal statistics for which the model's predictionwhere derived. The data's 90% CI were calcu-lated by either adding or subtracting 1.645 SEunits from the data means.

Proportion of frequent responses. The propor-tion of frequent responses given by participantsindicates the degree to which their responsescorrespond with the base rate. Figure 5 (upperright panel) presents the proportion of frequentresponses observed in the three experimentalconditions as a function of time. The main effectof the time was significant, F(4, 21) = 9.8, p =.0001, because of an increase of the proportion offrequent responses in all conditions. The overallproportion of frequent responses in the indepen-dent condition was significantly higher than inthe other conditions, F(2, 21) = 8.48, p = .002.The last two blocks of the different conditionsalso revealed a significant difference, F(2, 21) =13.07, p < .01, between the conditions thatresulted from a higher proportion of frequentresponses for the independent condition (.77)than the dependent (.62) and the single-player(.61) conditions. The linear trend between thefirst and last block was significantly stronger inthe independent condition, F(2, 21) = 4.1, p <.04, than in the other two conditions. This pattern

2 The overlap of the three confidence intervals (CIs)in each block implies that the power of the current testof the model is limited. Only over blocks does theprobability of incorrect acceptance of the null hypoth-esis fall below .2. We have derived the CI predictiononly after the completion of the first experiment, andwe increased the design power by reducing d' andincreasing the number of dyads in each condition inthe second experiment.


Table 2Sensitivity Analysis: Mean Predictions of One-Parameter Variants With Original Parameters

Original game parameters

P(frequent) d'

2-person 2-person 2-person 2-person 2-person 2-person1-person independent dependent 1-person independent dependent 1-person independent dependent

Measure .62 .65 .62 -.29 -.42 -.31 1.24 1.20 1.26

1.54.5

8.0125.0375

'.752.25

w~.01.03

w+/w~

.25

.75V

.00005

.00015,1<P

.0005

.0015m

51151

Cmax

2.57.5

.61

.613

.64

.63

.60

.66

.64

.62

.624

.65

.61

.62

.63

.63

.59

.62

.63

.63

.68

.633

.62

.69

.61

.69

.67

.67

.633

.69

.65

.66

.67

.66

.61

.68

.68

.67

.61

.626

.66

.61

.59

.65

.65

.62

.626

.66

.64

.62

.63

.60

.58

.62

.62

.61

-.37-.25

-.31-.32

-.20-.40

-.35-.30

-.31-.39

-.26-.29

-.30-.34

-.18-.31

-.3-.35

-.54-.35

-.43-.53

-.29-.53

-.49-.47

-.34-.54

-.43-.43

-.50-.45

-.28-.49

-.51-.52

-.26-.34

-.40-.27

-.19-.38

-.44-.31

-.31-.46

-.36-.28

-.33-.20

-.12

-.29

-.31-.27

1.261.23

1.241.23

1.371.13

1.201.23

1.231.21

1.251.26

1.211.24

1.381.23

1.231.24

1.24

1.18

1.191.15

1.361.09

1.181.18

1.181.19

1.231.21

1.181.21

1.351.15

1.151.19

1.321.23

1.271.28

1.421.15

1.291.25

1.24

1.26

1.281.22

1.241.27

1.391.27

1.271.29

Note. The original values of the parameters were S(l) = 3, crg = .025, CT, = 1.5, p(l) = 0, w~ = .02, w+/w~ = .5, u = .0001,<|> = .001, m = 101, Cm, = 5. In each of the variations, the value of one of the nonzero parameter was either increased ordecreased by 50%.

of results corresponds with the predictions of thelearning model.

Response criterion In ft. The In p statistic isan estimate of the players' response criterion, thatis, the point above which all stimuli will beperceived as drawn from the signal distribution,and below it, from the noise distribution. Thevariable is computed as follows: In p = cd',where c is c = -0.5[Z(hit) + Z(FA)]. Figure 5presents In (3 values in the three experimentalconditions as a function of time. The main effectof time was again significant, F(4, 21) = 9.15,p = .0001, as was the difference between theindependent condition and the dependent and

single-player conditions, F(2, 21) = 6.33, p <.008. The values of In (J were significantly lowerin the independent condition. For the last twoblocks of the different conditions, a significantdifference between the conditions, F(2, 21) =7.51, p < .01, was found. The value of In |3 waslower for the independent condition (—.75) thanfor the dependent (—.26) and single-player (—.28)conditions. The linear trend between the first andlast block was again stronger in the independentcondition, F(2, 21) = 3.94, p < .04. These resultsalso correspond with our predictions.

The sensitivity d'. Figure 5 (bottom row)shows the average estimated d' scores in the


three conditions. The observed condition effect issignificant, F(2, 21) = 7.3, p < .004. In line withthe model's predictions, lower d' values wereobserved in the independent condition. Recallthat under the model this effect is a result of anincrease in cutoff variability; our virtual subjectshad a fixed perceptual ability. Indeed, the partici-pants in the experiment exhibited more violationsof the assumption of static cutoff in the indepen-dent conditions.3

Cumulative profits. Table 1 presents the cu-mulative profits in Israeli shekels in the threeexperimental conditions and blocks. The maineffect of time was significant, F(4, 21) = 3.35,p < .02, because of an increase of the cumulativeprofits in all three conditions. The differencebetween the independent condition and the depen-dent and single-player conditions, F(2, 21) =68.92, p < .0001, was significant. The cumula-tive profits were significantly lower in the indepen-dent condition.

Quantitative Assessment of the Model'sPredictions

The experimental results support the robustqualitative predictions made by the model. Asnoted above, these predictions are relativelyinsensitive to the exact values of the model'sparameters. The current section examines thequantitative predictions made by the model withErev et al.'s (1995) original parameters (with andwithout between-subjects variability). To derivequantitative fitness scores, we created a data setfrom each row in Figure 5. Each data set includes15 observations (5 blocks X 3 conditions) and itsvariables are Figure 5's statistics.

Table 3 presents two types of fitness scoresderived from these data sets: the proportion ofobservations outside the 90% CI, and the correla-tions between the data and the models. This tableand Figure 5 reveal that the experimental statis-tics fall outside the original model's CI in 10 ofthe 45 cases (3 statistics X 3 conditions X 5blocks). The central tendencies predicted by theoriginal model fall outside the data's CI in 12 ofthe 45 cases.

Between-subjects variability improves the mod-el's calibration. Only in 5 of the 45 cases did theobserved statistics fall outside the model's CIwhen variability was assumed. This value is as

Table 3Model Fitness Scores for Experiment 1

Model P(frequent) In 3 d' Total

Original parameters

Exp outModel outr(model, exp)

3/154/15

.93*

1/153/15

.89*

6/155/15.37

10/4512/45

With variability

Exp outModel outr(model, exp)

2/154/15.94*

0/15 3/152/15 2/15

.91* .53*

5/458/45

Note. Frequency of experimental observations (exp) thatfell outside the model's 90% confidence interval (CI, exp.out), the frequency of average predictions that fell outsidethe data's 90% CI, model out, and Pearson's correlations,/•(model, exp). P(frequent) = proportion of frequent re-sponses.*p < .05.

close as possible to the predicted 10% under theassumption of perfect calibration. The model felloutside the data's CI in 8 of the 45 cases.

Positive correlations between the model's meanpredictions and the mean results were observed inall 6 (3 statistics X 2 versions of the model)cases. Only one of the six correlations, d' with novariability, did not reach significance. The corre-lations for the response criteria and P(frequent)are above .89.

These results suggest that in addition to theaccurate qualitative prediction, the current modelwith Erev et al.'s (1995) parameters provides auseful quantitative approximation of the data. Infact, with variability, the model's quantitativepredictions cannot be rejected, even if the 45statistics are assumed to be independent. Theprobability that of a sample of 45 independentobservations 8 or more will fall outside the 90%CI is above .05. Of course, this does not meanthat the model is accurate. Rather, the quantita-tive finding is important: The model provides a

3 To assess cutoff variability, we assessed the propor-tion of static cutoff violations (SCV), which aredefined as choices that are inconsistent with the staticcutoff that best describes the participants' decisions.Over blocks, the proportion of SCV was 14.85 in theindependent condition compared with 9.26 in thedependent condition and 10.21 in the single-playercondition.


90% CI that is not too wide (over the threestatistics, the geometric average is only 25%wider than the CI estimated from the data) and iswell calibrated.4

Experiment 2

In Experiment 1, we demonstrated that partici-pants adjust their response criteria in accordancewith the predictions of the reinforcement-basedlearning model. This occurred even though partici-pants were unaware of the game's payoff struc-ture and their interdependence. One purpose ofExperiment 2 was to understand the effect ofprior knowledge regarding the exact payoff struc-ture on. participants' behavior. The question ofhow prior knowledge of being engaged in aconsensus game affects behavior is of majorimportance for applying the results to actualsocial situations where people usually are awarethat they are participating in a consensus game.Participants' prior knowledge regarding the pay-off structure has obviously no effect on theoptimal cutoffs. However, the knowledge mayaffect the actual learning process. In particular, asnoted by Roth and Erev (1995), instructions arelikely to affect the player's initial propensities.The sensitivity analysis (see Table 2) suggeststhat whereas such an effect is not likely to affectthe qualitative trends, it can affect the magnitudeof the differences.

A second purpose of Experiment 2. was toexamine the learning process for games in whichplayers have a lower sensitivity. Therefore, weused the same consensus game as in Experiment1, which is presented in Figure 3, but the twosymmetrical players' D' (administrated d') was1.0 instead of 1.5. The prior probability of asignal was .6, as in Experiment 1. Given theseparameters, Appendix A implies that the gamehas only two distinguishable equilibria: a lowequilibrium at (J —» 0 (In P —> — oo) and twoexperimentally indistinguishable high equilibriaat p = 5.1 (In p = 1.63) and p — » (In p — «).The proportion of choices of the more frequentcategory at the two distinguishable sets of equilib-ria are expected to be 1 and 0, respectively. In thesingle-player conditions, the optimal cutoff isP = .67 (In p = — .4). The expected proportionof the frequent category in this case is .67.

Method

Participants

There were 96 Technion students who servedas paid participants in the experiment. Theyperformed in unmixed pairs of men and womenwho were randomly assigned to one of fourexperimental conditions. The exact payoffs werecontingent on performance and ranged from 16 to24 shekels ($5-$8). None of the participants inExperiment 2 had taken part in Experiment 1.

Apparatus

The experiment was conducted with the samecomputer system that was used in Experiment 1.Again, the base rates were 60% for one distribu-tion and 40% for the other distribution. Onemajor difference was the use of D' = 1.0 insteadof 1.5, as in Experiment 1. The screen distancebetween the means of the two distributionsremained the same (2.8 cm); thus the D' wasreduced to 1.0 by increasing the variance of thedistributions. The result was a greater difficulty todiscriminate among the distributions.

Procedure

The general procedure was the same as inExperiment 1, but the experiments differed in theinstructions the participants received. In the with-out-information conditions, participants receivedthe same instructions as in Experiment 1. In thewith-information conditions, participants re-ceived basically the same instructions, but theywere also informed about the exact payoff struc-ture and the game's rules according to the condi-tion to which they were assigned (two person orone-person). Participants in the two-player condi-tion were told that whenever they were correctand in agreement with each other, they wouldreceive 10 points; in all other situations (when

4 Note, however, that there appears to be an increasewith time in the number of observations outside themodel's confidence intervals (CIs). This observationsuggests that the model does not capture all the factorsthat affect the experimental participants. For example,our virtual subjects are not getting tired even after 500rounds.


they disagreed or when they agreed on an incor-rect judgment) they were told they would lose 10points. In the single-person condition with infor-mation, players were told that they would receive10 points when they were correct and lose 10points when they were wrong.

Experimental Design

There were 48 pairs who participated in theexperiment: 12 in the single-player conditionwithout information, 12 in the single-player con-dition with information, 12 in the independentcondition without information, and 12 in theindependent condition with information. In thetwo-person conditions, players were rewardedonly when both were correct and agreed witheach other. In the single-player conditions, play-ers were rewarded according to the correctness oftheir own decisions. In order to balance the baserate, for half of the pairs in each condition, wedrew 60% of the stimuli from the higher distribu-tion and 40% from the lower. For the other half,40% were drawn from the higher distribution and60% from the lower.

Results and Discussion

Descriptive Statistics

Table 4 presents mean raw results. As inExperiment 1, we analyzed the three statisticsderived from P(hit) and P(FA): In (3, d', andP(frequent). The values of these statistics arepresented in Figure 6 (see Columns 3 and 4).They are discussed following the derivation ofthe predictions of the learning model.

The Learning Model's Predictions

Again, computer simulations were conductedin order to derive the model's predictions with theparameters of the experiment. As in Experiment1, 100 virtual replications of each of the condi-tions (1,200 virtual pairs) were conducted inwhich each simulated subject participated in 500rounds summarized in five blocks of 100 rounds.The model's predictions with and without variabil-ity are presented in Figure 4 (see the lefthandcolumns). Comparison of these columns with theleft-hand columns of Figure 5 shows that the

decrease in D' (from 1.5 to 1.0) did not affect thetrends predicted by the model. This assertion wasreinforced in an additional sensitivity analysisthat examined the effect of each of the param-eters, which is in line with Table 4's analysis. Asin Experiment 1, neither addition nor subtractionof 50% from any of the parameters affected thequalitative predictions.

Hypothesis Testing

Proportion of frequent responses. Figure 6(top panel, columns 3 and 4) presents the propor-tion of frequent responses in the four experimen-tal conditions as a function of time. The maineffect of time was significant, F(4, 44) = 8.65,p = .0001, because of an increase of the propor-tion of frequent responses in all conditions,except for the single-player condition with infor-mation. The proportion of frequent responsesover all five blocks in the two-person conditionswas not significantly different from the single-player conditions, F(l, 44) = 1.62, ns. However,the proportion of frequent responses in the lasttwo blocks combined was significantly higher inthe two-person conditions, F(l,44) = 5.43,p < .03, than in the single-player conditions. Thelinear trend between the first and last block wasstronger in the two-person conditions, F(l, 44) =7.96, p < .001. This pattern of results corre-sponds with the predictions of the learning model(see Figure 4).

In the overall analysis, neither the main effectof information nor any of the interactions involv-ing information approached significance. In theseparate analysis of the last two blocks, there wasa significant main effect of the information,F(l, 44) = 4.44, p < .05. With information, theproportion of frequent responses was lower thanwithout information.

Response criterion In J3. Figure 6 (see middlepanel, columns 3 and 4) presents In (3 values inthe four experimental conditions as a function oftime. The main effect of time was again signifi-cant, F(4,44) = 7.31, p = .0001. The values ofIn (3 over all five blocks in the two-personconditions were not significantly different fromthose in the single-player conditions, F(l, 44) =.49, ns. For the analysis of the combined In (3values in the last two blocks, there was a margin-ally significant difference between the single- and


•5

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'g-sQ.§

C <wa-s

00— VD — VD fNdo, do, vb oo

; ^H wi ^H

S 04 — •*— tf) t—i

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S

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o o o o 06 od^^ " « T*1

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Original Org. model Exp. data: Exp. data: Model withwlthvar. No inf. With inf. 0|=.75

P(frequent)

1.0-

0.8

0.6-

0.4-

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

0.2-

o.o--0.2-

-0.4-

-0.6-

-0.8-

-1.0-

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1.3-

1.1 -

0.9-

0.7-

0.5-

0.3-

1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5

1-Peraon2-Pereon ind. sig.

90% Cl

Figure 6. Results of the simulations and the empirical data for the proportion offrequent responses (P(frequent)), hip and d' for Experiment 2. From left to right areshown predictions of the learning model without variability (var.), predictions of thelearning model with variability; experimental results for the condition in whichparticipants had no information (inf.) about the game; experimental results whenparticipants received information about the game; and predictions of the learningmodel with variability and a smaller variance of the distribution of the initialpropensities, ind. sig. = independent signal detection; CI = confidence interval.


the two-person conditions, F(l, 44) = 3.01, p <.09. The linear trend between the first and lastblock for the two-person conditions was signifi-cantly stronger, F(l, 44) = 5.62,p < .03, than forthe single-player conditions. These trends werepredicted by the learning model.

Information had no main effect, and none ofthe interactions involving information approachedsignificance in the overall analysis. For the lasttwo blocks, there was a significant main effect ofthe information, F(l, 44) = 4.72, p < .04, whichresulted from higher values of In p in the condi-tions with information compared with the condi-tions without information.

The sensitivity d'. Replicating the findings ofExperiment 1, the overall analysis revealed asignificant main effect of the condition on thevalues of d', F(\, 44) = 12.68, p < .001. As inExperiment 1, the sensitivities were lower in thetwo-person conditions compared with the single-player conditions. Neither the main effect nor theinteractions of information had a significant ef-fect on the values of d'. Again, comparing theresults of the experiment with the predictions ofthe learning model shows that the results corre-spond with the predictions (see Figure 6).

Cumulative profits. The main effect of timewas significant, F(4,44) = 2.80, p < .03, be-cause of a slight increase of the cumulativeprofits in all three conditions. The differencebetween the two-person conditions and the single-player condition was significant, F(l,44) =

446.71, p < .0001. The cumulative profits weresignificantly lower in the two-person conditionscompared with the single-player conditions. Theinformation had no main effect, and none of theinteractions involving information approachedsignificance.

Quantitative Assessment of the Model'sPredictions

Table 5 presents Table 3's fitness scores forExperiment 2. It shows that whereas Erev et al.'s(1995) parameters with variability provide a goodfit for the conditions without information, theyhave to be modified in order to account for theslower learning process observed with informa-tion. For example, the frequency of experimentalstatistics that fall outside the model with variabil-ity CI are 5 of 30 in the no-information condition,but 14 of 30 in the condition with payoff-matrixinformation.

Table 4 results are in line with Roth and Erev's(1995) assertion that manipulation of the initialpropensities parameter can account for the ob-served information effect. Slower learning ispredicted when the players have tighter distribu-tions of initial propensities (smaller CT,-). Note thatsuch an effect is consistent with the intuition thatwith information the players start the experimentwith a clearer notion of their preferred strategies.

Figure 6 (right side) presents the prediction ofa variant of the model with variability and <r,- =

Table 5Model Fitness Scores for Experiment 2 in Table 5's Format

Without information With information

4/10 3/10 5/10 12/305/10 5/10 5/10 15/30.75* .78* .081/10 1/10 3/10 5/30

8/1010/10.89*8/10

9/10 2/10 19/309/10 2/10 22/30.78* .266/10 0/10 14/30

Model and fitness score P(freq) In (3 d' Total P(freq) In £ d' Total

Original parametersExp. outModel out/•(model, exp)Exp. out

With variabilityModel out/•(model, exp)

With variability and CTJExp. outModel outKmodel, exp)

2/10 3/10 3/10 8/30 10/10 9/10 1/10 20/30

.77* .68* .32 .90* .74* .42= .75

0/10 0/10 4/10 4/30 2/10 1/10 3/10 6/300/10 1/10 5/10 6/30 3/10 5/10 4/10 12/30

.89*.76* .66* .39 .77* .43

Note. P(freq) =*p < .05.

proportion of frequent responses; exp. = experimental observation.


.75, which is half the original value. This columnand Table 7 (bottom columns) show that with thesmaller a,- the virtual subjects behave similarly tothe players who know the payoff rule. Mostimportant, the lower a,, like the information inthe experiment, slowed the learning process orled to higher In p. In addition, it increased theestimated d'.,The information in the experimenthad a similar effect that did not reach signifi-cance.

In line with the results of Experiment 1, Table5 shows positive correlations between the mod-el's predictions and the observed statistics. Onlythe correlations for d' did not reach significance.5

Finally, again in line with the results of Experi-ment 1, Table 5 shows that between-subjectparameter variability tend to improve the model'sfitness scores. A similar conclusion was reachedby Erev and Roth (1995).

General Discussion

The present research demonstrates that theeffect of base-rate information on decision mak-ing cannot always be described by consideringthe optimal response rule (by means of Bayes'theorem) and the experimentally observed devia-tion from it. In natural situations in which interde-pendence between decision makers is likely,Bayes's theorem cannot be used to compute therational decision. Moreover, in these situations,multiple equilibria may exist. For example, undercertain conditions, all consensus games have atleast two equilibria.

However, the fact that rational decision theorydoes not allow unique predictions of behaviordoes not imply that decisions are random. Rather,the present article demonstrates that behavior intwo-person consensus games can be predictedfrom an understanding of the reinforcement struc-ture and the learning process. In line with thepredictions derived from Erev et al.'s (1995)learning model, an extreme base-rate effect wasobserved in consensus games that involve twodecision makers who evaluate independent infor-mation.

The attempt to go beyond Bayes's theorem,which distinguishes the present research frommost previous base-rate studies, is only oneindication of a deeper difference between the

approach taken here and the traditional judgmentand decision-making approach. The present re-search was designed within the framework ofcognitive game theory (Erev & Roth, 1995; Roth

& Erev, 1995). Cognitive game theory relies onprevious results in judgment and decision-making research but replaces some of the basic

implicit assumptions of the traditional frame-work.

Most importantly, traditional experimental re-search implicitly accepts the framework of ratio-nal (high) game or decision theory. Under thisframework, it is possible to distinguish between

the economic and the psychological determinantsof behaviors. The economic factors are modeledby a game, and the psychological factors aremodeled by expected utility theory with certainadditions. According to Savage's (1954) influen-tial version of expected utility theory, the psychol-ogy of decision making can be summarized bytwo processes: probability assessments and choiceamong gambles. Moreover, the probability assess-ment process is expected to be consistent withBayes's theorem. Whereas previous experimentalstudies demonstrate that the specific assumptionsof rational decision theory are often not descrip-tive, the general framework has been kept. Theutilization of Bayes's theorem as a benchmark forthe study of base-rate effects is an example of theimplicit acceptance of the rational framework.

Under cognitive game theory, psychologicalprocesses play a major role in decision making,and psychology is not limited to the choiceamong gambles and the assessment of subjectiveprobabilities. Rather, in a cognitive game theoreti-cal analysis, three factors have to be consideredin order to predict choice behavior:

First, the set of strategies considered by theplayers has to be abstracted. The identification ofthe relevant strategies and the initial propensitiesis usually based on experimental results. In the

5 The relative inaccuracy of the predicted d' scores

appears to characterize the two experiments. Param-

eter fitting might improve the model's predictions

even for the no-information condition. Yet, given the

high predicted and the observed variance (large CIs),

we believe that a larger data set is needed to perform

this task.


present research, following results from signaldetection research, we assumed that players uti-lize cutoff strategies.

A second abstraction involves the payoff rule.The specific payoff structure that applies to asituation has to be defined. Payoff here maydepend on the state of the world and the actionsof other people (e.g., the consensus with them inour study). The combination of strategies andassociated payoffs gives rise to the abstraction ofthe "cognitive game," that is, the expected payofffor each strategy set. In the context of two-personsignal detection games, it is the normal formpresentation of the game. (For an explicit presen-tation of the game, see Erev et al., 1995.)

Finally, the way by which players select amongthe different strategies in the cognitive game hasto be modeled. Traditional approaches assumethat people try to select the strategy that maxi-mizes expected utility. In contrast, the presentapproach distinguishes between long-term (possi-bly equilibrium) predictions and intermediate(learning-curve) predictions. In the present set-ting, the learning model predicts convergencetoward one of the game's equilibria, but indifferent settings there may not be such a conver-gence.

Our results demonstrate the potential of cogni-tive game theory for the analysis of consensusgames as created in an experiment. However, thisapproach also has much wider implications. Weare all constantly engaged in consensus gameswhere the outcome of our actions depend at leastin part on consensus with other people. Forinstance, most decisions in organizational set-tings have to be approved by a number of people,and participants are motivated to reach consensuswith others who are involved in the decisionprocess. This situation can be modeled as aconsensus game; from the results of our study, wemay expect a strong weighting of base-rateinformation. For example, assume that there aretwo applicants, one belonging to a group whosemembers are usually hired when they apply for ajob and another person who belongs to a groupwhose members have been less likely to be hired.In this case, we expect to find a strong tendencyto chose the person belonging to the more readilyaccepted group. Hence, considering hiring deci-sions as consensus games leads to the prediction

of discrimination against certain groups in the

population.Another set of situations for which the current

model applies involves detection of malfunction-ing in a system by a team of workers. The modelpredicts that an incentive to reach consensus willlead teams to ignore (miss) rare signals.

In addition to the identification of a consensuseffect that can bias decisions, two specific non-trivial implications are directly derived from thecurrent experimental results. First, the problem isexpected to increase as decision makers gainexperience. Thus, practice will not have a posi-tive effect in this setting, and the severity of theproblem cannot be assessed by examining thebehavior of inexperienced decision makers.

A second direct implication is related to theeffect of explicit information. Our results suggestthat trying to hide the problematic game (e.g., bynot informing the decision makers that they areevaluated based on between-judge agreements)might increase the bias.

Other implications can be derived from thequantitative model supported in the current re-search. This model can be used to derive quantita-tive predictions of the effect of the incentive toreach consensus in different settings. Given anabstraction of the incentive structure and theplayer's sensitivity, the model predicts the ex-pected decision criteria as a function of experi-ence. Most important, the model can be used toidentify the conditions in which the bias is likelyto be particularly strong and to suggest possibledebiasing techniques. For example, the modelsuggests (in simulations that have not beenreported and studied here) that two major factorsare likely to increase the bias: a decrease in theobservers sensitivity and an increase in the ratioof consensus-to-accuracy payoffs.

The examples considered above are only asmall set of social situations that can be analyzedin terms of consensus games specifically orcognitive game theory generally. Specifying thestrategies, payoff structure, and learning modelthat apply in a social interaction allows theresearcher to predict intermediate term outcomesin this interaction through applying a minimumof basic principles and may be a valuable tool forthe understanding of decision making undervarious circumstances.

102 G1LAT, MEYER, EREV, AND GOPHER

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Appendix A

Two-Person Signal Detection Model

The two-person signal detection model is anextension of classic signal detection theory (SDT)for the case of two independent detectors. Liketraditional SDT, the model assumes that in eachtrial, Player i observes a stimulus (*,-) that wasrandomly selected from one of two distributions:N/(.) or S,-(.). Player i is assumed to set a cutoff cf)

which implies p; = S,-(c;)/N, (c,-) value, and toconsistently respond "S" if *,• is larger than c,.Erev et al. (1995) showed that when the twoplayers are independent, the equilibrium cutoffs(for °= > P > 0) satisfy the following set ofequations:

CC, - FQ

+ (CF, - CC, - F?! + Fd) J" N2(jc) dx

HHj - MH,

+ (HMj - ! - MM! + MHO

and

S2(c2)

N2(c2)

+ (CF2 - CC2 - FF2 + FC2) N,

HH2 - MH2 '

+ (HM2 - HH2 - MM2 + MH2) J^ S,(*) dx

where C = correct rejection, F = false alarm,H = hit, and M = miss. In addition, the game can

also have extreme equilibria ((} — » 0, P — > °°).The equilibrium calculation can be greatly simpli-fied when symmetrical consensus games of thetype studied here (where HH1 = CC1 > HM1 =MH1 = MM1 = FF1 = CF1 = FC1) areconsidered. The following hold for these games:(a) All equilibria are symmetrical (see Gilat,1995), and (b) At all nonextreme equilibria(0 < 3 < co), Player i is indifferent between thetwo responses ("N" and "S"), thus,

P,(N|(C; = JC,-))P,<"N/' \(X,, Cj = Xff)

= P,<S|(c, = xdW'Srl , Cj = x,)).

where P,(Z|(c/ = *,-)) is the probability that iassigns to the event that the state is Z (S or N)given a signal xt that equals to her or his cutoff,and Pj(Zj\(Xi, Cj = xi)) is the probability that iassigns to the event that j will respond Z giveni's signal and the assumption that j uses thesame cutoff (when y's signal is not known to i).because

P,<S|(C,. = *,-))=! - P,<N | (C; = *;)),

and

P,.("S," |(*;, cj = *,.)) = 1 - P,<"N/' !(*,., cj = *,.)).

Simple algebra implies that

P,(N|(Cj. = *,.)) = Pf."S"\<*, Cj = *;))•


Appendix B

A Reinforcement-Based Learning Rule

Erev et al.'s (1995) adaptation of Roth andErev's (1995) learning rule can be summarizedby the following four assumptions (A).

A Finite Number of Uniformly andSymmetrically Distributed Cutoffs

Al«: Player / considers a finite set of m cutoffs.The location of cutoff/ (1 <y < m) is c/ = cmin +A(; - 1). Erev et al. defined A = 2(cmax)/(m - 1) and set the two strategy-set parametersto m = 101 and cma): = 5.

Initial Propensities

A2").: At Time t = 1 (before any experience).Player i has an initial propensity to choose the/thcutoff.

Two initial propensities parameters were set:S(l) and cr,-. To set these parameters, Erev et al.(1995) defined $<,-(!) = p*(l)S(l), where S(l) =2f=1 qj(t) and pt(l) is the probability that cutoffk will be chosen at the first round. They assumedthat S(l) = 3 * (the average absolute profit in thegame, given randomly selected cutoffs) and deter-mined p*(l) by the area above cutoff k under anormal distribution with a mean at the center ofthe two distributions and SD u-, = 1.5.

Reinforcement, Generalization, and Forgetting

The learning process is the result of theupdating of the propensities through reinforce-ment, generalization, and forgetting.

A3q: If cutoff k was chosen by Player i at Timet and the received payoff was v, then the propen-sity to set cutoff/ is updated by setting

qj(t + 1) = max [v, (1 - tp)<?,O) + Gk(j,R, (v))],

where D is a technical parameter that ensures thatall propensities are positive, tp is a forgetting

parameter, Gk(.,.) is a generalization function,and R(.) is a reinforcement function.

Erev et al. (1995) setu = .0001 and cp = .001.The reinforcement function was set to R,(v) =v — p(t), where p is a reference point that isdetermined by the following contingent weightedaverge adjustment rule

p(0(l -

p(t

+), ifv(t) >

v(/)(vO,ifv(0<p«,

where w+ — .01 and w~ = .02 are the weights bywhich positive and negative reinforcements af-fect the reference point.

The generalization function was set to

Gk(j, R(v)) = R(v)(F{[Cj + cj+l]/2\

~ F\[c,

where F\.} is a cumulative normal distributionwith mean ck and SD = o-g. Erev et al. set crg =.25.

The Relative Propensities Sum

The final assumption states the choice rule.Mfl: The probability that the observer sets strat-egy k at time f is determined by the relativepropensities sum

Received September 11,1995Revision received October 15, 1996

Accepted October 22, 1996

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