Experiments on two person games with incredible

threats�

Peter CoughlanCalifornia Institute of Technology

Richard D. McKelveyCalifornia Institute of Technology

Thomas R. PalfreyCalifornia Institute of Technology

March 8, 1999

�We thank Heather Crawford and Roberto Weber for help in running some of the experiments, andEugene Grayver for writing the computer program for the experiments. The �nancial support of theNational Science Foundation (Grants #SBR-9223701 and #SBR-9631627) is gratefully acknowledged.

Experiments on two person games with incredible

threats

Peter Coughlan Richard D. McKelvey Thomas R. Palfrey

Abstract

We report results of experiments on two person games in which one of the playershas an incredible threat. The basic game has two equilibria, one in which the threat isnot made, and one in which it is. We study three di�erent treatment e�ects: sequentialversus simultaneous play, e�ects of changing payo�s, and the e�ects of di�erent matchingprotocols.

Experiments on two person games with incredible

threats

Peter Coughlan Richard D. McKelvey Thomas R. Palfrey

1 Introduction

Consider a two person game whose extensive form is given in Figure 1(a), and whosecorresponding normal form is in Figure 1(b). Player 1, the row player, can choose eitherU (Up) or D (Down), and Player 2, the column player, can choose either L (Left) or R(Right). The payo�s are assumed to satisfy the constraints:

0 < a1 < b1

and0 < c2 < b2 < a2

This game is similar to a single stage of the well known chainstore paradox game. Therow player can guarantee a secure outcome by choosing Up. The column player has an(incredible) threat in the choice of Right. There are two Nash equilibria to the game,(D;L) and (U;R), but only the �rst of these is perfect.

����

������J

JJJJJJJJJ

1

a1; a2

DU

b1; b2 0; c2

2L R

2L R

U a1,a2 a1,a21

D b1,b2 0, c2

(a) Extensive Form (b) Normal Form

Figure 1: A 2� 2 game of perfect information

In the case when the game is played with error, McKelvey and Palfrey (1998) usethe logistic Quantal Response Equilibrium (QRE) to characterize behavior in this game.They show that the QRE predicts di�erent behavior in the extensive and normal formsof the game. The equilibrium conditions for the logit-AQRE of the extensive form gamein Figure 1(a) are, for any given precision, � of error:

pD =e�qLb1

e�qLb1 + e�a1=

1

1 + e�(a1�qLb1)(1)

qL =e�b2

e�b2 + e�c2=

1

1 + e�(c2�b2)(2)

where pD and qL are the probabilities that players 1 and 2 move D and L respectively inthe extensive form game.

The equilibrium conditions for the normal form game in Figure 1(b) are:

p̂D =e�q̂Lb1

e�q̂Lb1 + e�a1=

1

1 + e�(a1�q̂Lb1)(3)

q̂L =e�[p̂Ua2+p̂Db2]

e�[p̂Ua2+p̂Db2] + e�[p̂Ua2+p̂Db2]=

1

1 + e�p̂D(c2�b2)(4)

where p̂U , p̂D and q̂L are the probabilities that players 1 and 2 move U , D and L respec-tively in the normal form game.

Based on the above formulae, we can establish several properties of the QRE. First,for any value of � > 0, e�(c2�b2) < 1, since c2 < b2. So from equation (2), it follows thatqL > 1

2. A similar argument shows that q̂L > 1

2, since 0 < p̂D < 1. Thus, we have:

Property 1: For all � > 0, qL > 12and q̂L > 1

2.

McKelvey and Palfrey (1998) establish the following relation between the solution forthe extensive and normal form games.

Property 2: For all � > 0, q̂L < qL and p̂D < pD.

It is easy to see why this is true directly from equations (1){(4). For any value ofp̂D 2 (0; 1), since c2 < b2, we get that p̂D(c2 � b2) > (c2 � b2). Therefore, comparingequations (2) and (4), for any value of � > 0, q̂L < qL. Now, comparing equations (1)and (3) using the relation between qL and q̂L, it follows that p̂D < pD.

For the extensive form version of the game, we can also establish the e�ect of a changein the payo�s of the game on the QRE correspondence. For any �, we get

@

@a1qL = 0 (5)

@

@a1pD = �

�e�(a1�qLb1)

(1 + e�(a1�qLb1))2< 0 (6)

2

Thus,

Property 3: For all � > 0, @@a1

qL = 0, and @@a1

pD < 0.

Thus, independent of the amount of error, we obtain predictions about the e�ect ofsequential (extensive form) versus simultaneous (normal form) play, and about the e�ectof changes in payo�s.

The qualitative features of the AQRE correspondence for (pD; qL) and (p̂D; q̂L) as afunction of � in (1) and (2) are shown in Figures 2 and 3. The curves in these �guresshow how the AQRE mixing probabilities vary with �. Note that in the extensive formgame there is a unique AQRE for every value of �, since qL does not depend on pD.

1 Forthe normal form version, there is an additional QRE component for large values of �,which converges to the subgame imperfect equilibrium pD = 0, qL = :5.

2 Experimental design

The goal of the experiment was to study the equilibrium predictions of the QRE. Theprevious section shows that the QRE makes systematically di�erent predictions betweenthe extensive and normal form of the game, and also makes predictions as to the e�ects ofvarying the payo�s. Thus we used these two variables as the primary treatment variables.

We chose two di�erent game matrices for study. The only di�erence between the twogames was in the value of a1, which was 25 for Matrix 1, and 35 for matrix 2. Theremaining parameters were the same in both games: a2 = 20, b1 = 45; b2 = 15, andc2 = 10. This results in the normal form game matrices of Table 1 (a) and (b).

2L R

U 25,20 25,201

D 45,15 0, 10

(a) Matrix 1

2L R

U 35,20 35,201

D 45,15 0, 10

(b) Matrix 2

Table 1:Experiment Payo�s

For each of the above matrices, we considered both the extensive and normal form. Inour experiments, we did not present the game to the subjects in extensive form. Rather,

1In fact any game of perfect information has a unique AQRE (McKelvey and Palfrey, 1988).

3

the game was presented by displaying its payo� table in both treatments. In the normalform version, the subjects chose their moves simultaneously, and in the extensive formversion, the subjects chose their moves sequentially (see next section for more details).This was done to minimize the presentation di�erences between the two treatments, andto focus on the e�ects of simultaneous versus sequential play. We thus refer to the exten-sive form treatment as sequential play and the normal form treatment as simultaneous

play.The above yields two treatment variables, each with two categories. Another variable,

the matching protocol, emerged as an additional important factor during the course ofrunning the experiment. Thus the three treatment variables are:

� Presentation e�ect: sequential (extensive form) versus simultaneous (normal form)play,

� Payo� e�ects: a1 = 25 versus a1 = 35,

� Matching protocol: random versus zipper.

Based on the analysis from the previous section, we have the following predictionsthat are made by the QRE, regardless of the value of �:

� qL > 12and q̂L > 1

2

� q̂L < qL and p̂D < pD

� p1D > p2D and q1L = q2L

Here, the superscript indicates the game matrix. If there is no superscript, it indicatesthat the prediction holds for both matrices.

In Figures 2-3, we illustrate the QRE correspondence for each of the four treatmentsused in our experiments. Each of the �gures plots the QRE of two di�erent treatments,so that the di�erence in the predictions between the di�erent treatments can be easilyseen.

In addition to the predictions speci�ed above, it can be seen from Figure 3(b) thatin Matrix 1, the principal branch of the QRE selects the subgame perfect equilibrium,whereas in Matrix 2, the principal branch selects the equilibrium that involves an incred-ible threat. So this design also allows us to test the equilibrium selection made by theQRE. In addition, on the principal branch of the QRE, we see that for all � the proba-bility the row player moves Down is less in matrix 2 than matrix 1, and the probabilitythe column player moves Left is less in matrix 2 than matrix 1.

Thus, conditional on the principal branch of the QRE being the model of behavior,we obtain the following additional prediction of the QRE:

� p̂1D > p̂2D and q̂1L > q̂2L

4

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qL

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�

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pD

qL

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Figure 2: QRE Sequential (Extensive) vs Simultaneous (Normal)

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qL

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Matrix 2

�

�

p̂D

q̂L

(b) Simultaneous

Figure 3: QRE Matrix 1 vs Matrix 2

5

3 Experiments

Our experiments consisted of sessions in which subjects repeatedly played (with di�erentopponents) one of the games in Table 1 as a one shot game. All of our experiments wererun over a computer network. At the beginning of the experiment, each subject wasassigned to be either a row or column player. Then they were repeatedly matched withsubjects of the opposite role. In each match, subjects chose a strategy for the single shotgame. They then learned the outcome of their match, were re-matched, and repeatedthe process. This continued for a prespeci�ed, and commonly known number of matches.In the zipper matching protocol, the experiment then ended. In the random matchingprotocol, subjects repeated the whole process with a second, and di�erent treatment, afterwhich the experiment ended. After the experiment, the subjects were paid in cash fortheir participation, an amount based on the payo�s accumulated during the experiment.

In the simultaneous play (normal form) version of the game, for each match, thesubjects were presented with a payo� table on their computer screen. They could movetheir mouse over the payo� table. As they did so, the corresponding row (if they were arow player) or column (if they were a column player) would highlight. They could selecttheir strategy by clicking the mouse, and then con�rming. Once all subjects had made achoice, then for each pair of subjects, the outcome determined by the choices of the twosubjects was highlighted on the screen of each subject.2

A match for the sequential play (extensive form) version of the game was run similarlyto the simultaneous play version, except that the column player could not move untilafter the row player had moved. The row that had been selected by the row player washighlighted on the column players screen before they made their move.

We used two di�erent matching protocols to run the experiments. The �rst was arandom matching protocol, and the second was a zipper design. In the random matchingprotocol, subjects were randomly re-matched with another subject of the opposite rolebetween each match. Under this protocol, subjects could be matched with the sameopponent more than once. The opponent was not identi�ed on the players computerscreen. So a subject could not determine who their opponent was. Under the zipperprotocol, subjects were matched together only once.3 This matching protocol guaranteesthat no subject ever plays a subject who they have played before. Also they never playanyone who has played someone they have played before, and so on.

The zipper protocol only allows for n2matches between n subjects, where there is no

limit under the random matching protocol. Our hope was that the matching protocolwould not be a signi�cant variable. In that case, we would be able to collect most ofour data under the random matching protocol, since it allows more repetition. We found

2In the case that the row player chose up, both cells in the upper row give the same payo�, and bothwere highlighted. This was done to avoid any di�erences in the information about previous moves thatsubjects would have about their opponents in the simultaneous move and sequential move versions ofthe game.

3In this matching protocol, the row and column subjects are each numbered from one to six. Inmatch t, row player i is matched with column player i+ t� 1(mod 6).

6

instead that this variable was signi�cant. Hence we report our results separately for eachmatching protocol.

We ran experiments using Caltech undergraduate students as subjects. Each experi-ment used only subjects who had not participated in any of the previous experiments.

In the experiments that were run with the random matching protocol, we had twelvesubjects in each experiment. Half (six) of the subjects were row players and the other halfwere column players. The subjects then participated in two sessions, where each sessionconsisted of one of the treatments discussed above. In each session, the subjects played 30rounds of the same game. After the �rst session, the treatment was changed, and subjectsplayed 30 rounds of the second treatment. The design, showing which treatments wererun with each group, is given in the following table. In this table, N and E refer to thenormal (simultaneous) and extensive (sequential) versions of the game, and the numberrefers to the game matrix used. Thus, N2 is the simultaneous play version of Matrix 2,E1 is the sequential version of matrix 1, etc.

# SessionGroup Subjects 1 2

1 12 N1 N22 12 N2 N13 12 E1 E24 12 E2 E15 12 N1 E16 12 N2 E27 12 E1 N18 12 E2 N2

Table 2: Experimental design for random matching protocol

With the above design, we obtained four sessions of each of the four basic treatments.Each of the four basic treatments was run in the �rst session in half of the experimentsand in the second session the other half. This allowed us to check for learning or sequencee�ects. With this design, we obtained a total of 6 � 30 � 4 = 720 observations for therow players under each of the four treatments. In the simultaneous move experiments,we have the same number of observations of the column players choices. However, inthe sequential version, we only observed the column players choice when the row playermoved down. So the number of observations for the column players was 720� �p, where�p indicates the percentage of row players choosing D.

In the experiments using the zipper matching protocol, we only ran one session witheach group of subjects, since we wanted to be careful not to re-match subjects who hadbeen previously matched. As mentioned above, under the zipper protocol, it is onlypossible to run n

2matches with n subjects. We ran a total of 14 experiments under the

7

zipper design. Of these, there were four of each of the sequential move (extensive form)treatments, and three of each of the simultaneous move (normal form) treatments (seeTable 3). This yielded a total of 6 � 6 � 3 = 108 observations for both the row andthe column players in the simultaneous move games. In the sequential move games, weobtained 6 � 6 � 4 = 144 observations for the row players, and 144 � �p for the columnplayers.

# #Experiments Subjects Treatment

4 12 E14 12 E23 12 N13 12 N2

Table 3: Experimental design for zipper matching protocol

4 Data

Tables 4 and 5 present the data from the experiments, along with the estimates of theQRE and the \noisy Nash" (NNM) model of errors.4

The data are also presented graphically in Figures 4-5 for the random matching, andFigures 6-7 for the Zipper matching protocols. These �gures display the QRE correspon-dence projected into the pD � qL space of possible mixed strategies for the two players.The QRE appears as a solid curved line starting at the centroid of the game (where� = 0 and each strategy is adopted with probability 1

2), and then proceeding to a Nash

equilibrium of the game (when � = 1). The sequential move games have only onebranch to the QRE correspondence. An additional branch of the correspondence appearsin the simultaneous move versions of the game. For the �rst three treatments (all exceptSimultaneous, Matrix 2) the equilibrium selected by the QRE is the perfect equilibrium.For the last treatment, the equilibrium selected is the imperfect equilibrium.

A summary of the QRE predictions, and their evaluation in light of the data appearsin Table 6.

It is immediately evident that in the random matching design, there is very little,if any support for the QRE predictions. From Table 4, we �nd that for both of theMatrix 1 treatments, the con�dence interval for � includes 0. Hence we cannot reject

4The noisy Nash model assumes that players make errors, but that the errors are not taken intoaccount by other subjects. For any , it is a convex combination c+ (1� )n, where c is the centroidof the game, and n is a Nash equilibrium. For the estimates presented here, we use the selection of thesubgame perfect Nash equilibrium for n.

8

random behavior in either of these treatments. From Table 6 we �nd that seven of thetwelve QRE predictions go in the wrong direction, and if we take the prediction as thenull hypothesis, we can reject the hypothesis at the .05 level in six of these seven cases.In particular, we �nd that the prediction that qL > 1

2fails in all four treatments, and

signi�cantly so in three of these.Under the zipper design, the results are more mixed. Here, we �nd that the direction

of the predictions is correct in all cases except for the predictions relating to the com-parison the column players' behavior between the simultaneous and sequential versionsof the game. And in no case can we reject any of the QRE predictions at the .05 level.

In fact, the most striking di�erences we found in our study was the systematic dif-ferences between the random and zipper matching protocols. Table 7 summarizes thesedi�erences. In this table we see that in all four treatments, both the probability that therow player moves Down and the probability that the column player moves Left is lowerunder the Random matching protocol than under the Zipper matching protocol. In allbut one case, these di�erences are signi�cant at the :01 level. Thus, there is a systematictendency towards the imperfect equilibrium in the random matching protocol.

We attribute the di�erences in the two matching protocols to the fact that in the ran-dom matching protocol, subjects play against the same partner more than once. Thus,even though they do not know who their partner is in any match, there is a small prob-ability (1

6in our experiments) that the partner is a particular subject. Hence, subjects

can make (probabilistic) inferences about the behavior of their partner based on pasthistory. Because of that, it is a good strategy to try and establish a reputation. Sincethe preferred equilibrium for the column player is the imperfect equilibrium, we wouldexpect the column players to try and deter the row players from choosing D by choosingL less frequently than they would in a one shot game.

9

Matrix 1 Matrix 2n fi QRE NNM n fi QRE NNM

U 454 0.631 0.504 0.500 572 0.794 0.668 0.500Sequential D 266 0.369 0.496 0.500 148 0.206 0.332 0.500Version L 105 0.395 0.509 0.500 64 0.432 0.624 0.500

R 161 0.605 0.491 0.500 84 0.568 0.377 0.500�j 0.007 0.001 0.101 0.001�loj lo 0.000 0.000 0.083 0.000�hij hi 0.025 0.008 0.121 0.005�L� -683.10 -683.69 -506.15 -602.10U 405 0.563 0.501 0.500 537 0.746 0.742 0.500

Simultaneous D 315 0.438 0.499 0.500 183 0.254 0.253 0.500Version L 219 0.302 0.501 0.500 310 0.431 0.531 0.500

R 501 0.696 0.499 0.500 410 0.569 0.469 0.500�j 0.001 0.001 0.095 0.001�loj lo 0.000 0.000 0.080 0.000�hij lo 0.008 0.006 0.111 0.005�L� -998.37 -998.51 -914.71 -998.59

Table 4: Experiment results (random design)

Matrix 1 Matrix 2n fi QRE NNM n fi QRE NNM

U 32 0.222 0.250 0.293 78 0.542 0.545 0.500Sequential D 112 0.778 0.750 0.707 66 0.458 0.455 0.500Version L 69 0.616 0.700 0.707 35 0.530 0.519 0.500

R 43 0.384 0.300 0.293 31 0.470 0.481 0.500�j 0.169 0.414 0.016 0.001�loj lo 0.146 0.299 0.011 0.000�hij hi 0.187 0.521 0.050 0.102�L� -152.95 -154.83 -144.96 -145.57U 28 0.259 0.261 0.301 59 0.546 0.557 0.491

Simultaneous D 80 0.741 0.739 0.699 49 0.454 0.444 0.510Version L 71 0.657 0.674 0.699 61 0.565 0.511 0.510

R 37 0.343 0.326 0.301 47 0.435 0.490 0.491�j 0.196 0.398 0.0190 0.019�loj lo 0.178 0.272 0.011 0.002�hij lo 0.206 0.515 0.050 0.151�L� -131.29 -132.12 -149.01 -149.68

Table 5: Experiment results (zipper design)

10

Random matching Zipper matchingPrediction Results Z value Evaluation Results Z value EvaluationqL > 1

21 :395 > :500 -3.425 False� :616 > :500 2.455 True

2 :432 > :500 -1.655 False :530 > :500 0.487 True

q̂L > 12

1 :302 > :500 -10.626 False� :657 > :500 3.263 True

2 :431 > :500 -3.703 False� :565 > :500 1.351 True

q̂L < qL 1 :302 < :395 2.763 True :657 < :616 -0.632 False

2 :431 < :432 0.022 True :565 < :530 -0.450 False

p̂D < pD 1 :438 < :369 -2.669 False� :741 < :778 0.683 True

2 :254 < :206 -2.164 False� :454 < :458 0.063 True

p1D > p2D :369 > :206 6.833 True :778 > :458 5.588 True

q1L = q2L :395 = :432 -0.734 True :616 = :530 1.124 True

p̂1D > p̂2D :438 > :254 7.339 True :741 > :454 4.301 True

q̂1L > q̂2L :302 > :431 -5.080 False� :657 > :565 1.387 True

Table 6: Evaluation of predictions (asterisk indicates rejection at :05 level)

Random Zipperprob n prob n Z value

Matrix 1 pD .369 720 .778 144 9.032Sequential qL .395 266 .616 112 3.936

Matrix 2 pD .206 720 .458 144 6.392qL .432 148 .530 66 1.328

Matrix 1 p̂D .438 720 .741 108 5.879Simultaneous q̂L .302 720 .657 108 7.221

Matrix 2 p̂D .254 720 .454 108 4.316q̂L .431 720 .565 108 2.611

Table 7: Comparison of Random versus Zipper matching protocols

11

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Figure 4: Random matching, Matrix 1

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0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

4

2

1

3

pD

qL

(a) Sequential

0.00

0.00

0.10

0.10

0.20

0.20

0.30

0.30

0.40

0.40

0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

222222222

1

4

3

p̂D

q̂L

(b) Simultaneous

Figure 5: Random matching, Matrix 2

12

0.00

0.00

0.10

0.10

0.20

0.20

0.30

0.30

0.40

0.40

0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

333333333

4

1

2

pD

qL

(a) Sequential

0.00

0.00

0.10

0.10

0.20

0.20

0.30

0.30

0.40

0.40

0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

333333333

2 1

p̂D

q̂L

(b) Simultaneous

Figure 6: Zipper matching, Matrix 1

0.00

0.00

0.10

0.10

0.20

0.20

0.30

0.30

0.40

0.40

0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

444

1

2

3

pD

qL

(a) Sequential

0.00

0.00

0.10

0.10

0.20

0.20

0.30

0.30

0.40

0.40

0.50

0.50

0.60

0.60

0.70

0.70

0.80

0.80

0.90

0.90

1.00

1.00

1.00 1.00

0.90 0.90

0.80 0.80

0.70 0.70

0.60 0.60

0.50 0.50

0.40 0.40

0.30 0.30

0.20 0.20

0.10 0.10

−0.00 −0.00

222222222

1

33

p̂D

q̂L

(b) Simultaneous

Figure 7: Zipper matching, Matrix 2

13

REFERENCES

Gale, J., K. Binmore, and L. Samuelson, \Learning to be imperfect: The ultimatumgame," Games and Economic Behavior, 8 (1995):56{90.

Kohlberg, E. and J.-F. Mertens, \On the strategic stability of equilibria," Econometrica,54 (1986): 1003{37.

McKelvey, R. D., and T. R. Palfrey, \Quantal Response Equilibria in Normal FormGames," Games and Economic Behavior, 10 (1995): 6-38.

McKelvey, R. D., and T. R. Palfrey, \Quantal Response Equilibria in Extensive FormGames," Experimental Economics, 1 (1998): 9-41.

Schotter, A., K. Weigelt, and C. Wilson, \A Laboratory Investigation of MultipersonRationality and Presentation E�ects," Games and Economic Behavior, 6 (1994):445{68.

14

5 APPENDIX: Experiment instructions

NORMAL FORM GAME { ZIPPER MATCHINGThis is an experiment in decision making, and you will be paid for your participation incash, at the end of the experiment. Di�erent subjects may earn di�erent amounts. Whatyou earn depends partly on your decisions and partly on the decisions of others.

The entire experiment will take place through computer terminals, and all interactionbetween you will take place through the computers. It is important that you not talkor in any way try to communicate with other subjects during the experiment. If youdisobey the rules, we will have to ask you to leave the experiment.

We will start with a brief instruction period. During the instruction period, you willbe given a complete description of the experiment and will be shown how to use thecomputers. If you have any questions during the instruction period, raise your hand andyour question will be answered so everyone can hear. If any di�culties arise after theexperiment has begun, raise your hand, and an experimenter will come and assist you.

The subjects will be divided into two groups, each containing an equal number ofsubjects. The groups will be labeled the RED group and the BLUE group. To determinewhich color you are, will you each please select an envelope as the experimenter passesby you.

[EXPERIMENTER PASS OUT ENVELOPES]Inside each envelope is an index card labeled either BLUE or RED. If you chose

BLUE, you will be BLUE for the entire experiment. If you chose RED, you will beRED for the entire experiment. Please remember your color, because the instructionsare slightly di�erent for the BLUE and RED subjects.

This experiment will consists of several periods or matches. I will now describe whatoccurs in each match. First, you will be randomly paired with a subject of the oppositecolor. Thus, if you are a BLUE subject, you will be paired with a RED subject. If youare a RED subject, you will be paired with a BLUE subject.

[TURN ON OVERHEAD PROJECTOR]After you have been paired, each subject will simultaneously be asked to make a

choice. The RED subject in each pair will be asked to choose one of the two rows in thematrix which will appear on the computer screen and which is also now shown on thescreen at the front of the room. The RED subject can choose either "Up" or "Down".The BLUE subject in each pair will be asked to choose one of the two columns in thematrix, either "Left" or "Right". Neither subject will be informed of what choice theother subject has made until after all choices have been made.

After each subject has made his or her choice, payo�s for the match are determinedbased on the choices made. Payo�s to each subject are indicated by the numbers in thematrix. The payo� to the RED subject is in red and appears in the lower left of eachcompartment, while the payo� to the BLUE subject is in blue and appears in the upperright of each compartment. The units are in dimes.

Thus, if the RED subject chooses "Up" and the BLUE subject chooses either "Left"

15

or "Right", the RED subject receives a payo� of dimes, or $ , while the BLUEsubject receives a payo� of 20 dimes, or $2.00. If the RED subject chooses "Down"and the BLUE subject chooses "Left", the RED subject receives a payo� of 45 dimes,or $4.50, while the BLUE subject receives a payo� of 15 dimes, or $1.50. Lastly, if theRED subject chooses "Down" and the BLUE subject chooses "Right", the RED subjectreceives a payo� of 0 dimes while the BLUE subject receives a payo� of 10 dimes or$1.00.

[SWITCH OVERHEADS ON PROJECTOR]The experiment will consist of 6 matches. In each match, you will be paired with a

di�erent subject of the opposite color. Thus, if you are a BLUE subject, in each matchyou will be paired with a RED subject. If you are a RED subject, in each match youwill be paired with a BLUE subject.

Since there are 6 subjects of each color, this means that you will be paired with eachof the subjects of the other color once and only once. You will be paired with everysubject of the opposite color and you will never be paired with any subject twice. Thus,if your label is BLUE, you will be paired with each of the RED subjects exactly once. Ifyou are RED, you will be paired with each of the BLUE subjects exactly once.

The exact experiment pairings are now shown on the screen at the front of the room.[DESCRIBE PAIRINGS AS SHOWN ON SCREEN]

Your �nal earnings for the experiment will be the sum of your payo�s from all 6matches plus a $3.00 participation bonus.

[BEGIN COMPUTER INSTRUCTION SESSION]We will now begin the computer instruction session. Will all RED subjects please

move to the computers to my left, near the window, and will all BLUE subjects pleasemove to the computers to my right, near the door to the hallway.

[WAIT FOR SUBJECTS TO MOVE TO APPROPRIATE COMPUTERS][TURN OFF OVERHEAD PROJECTOR]

During the computer instruction session, we will teach you how to use the computerby going through a few practice matches. Do not hit any keys until you are told to doso, and when you are told to enter information, type exactly what you are told to type.You are not paid for these practice matches.

Please turn on your computer now by pushing the button labeled "MASTER" on theright hand side of the panel underneath the screen.

[WAIT FOR SUBJECTS TO TURN ON COMPUTER]When the computer prompts you for your name, type your full name. Then hit the

ENTER key. Con�rm your entry by pressing the Y key when prompted, or press the Nkey to correct your entry.

[WAIT FOR SUBJECTS TO ENTER NAMES]When you are asked to enter your color, type R if your color is RED, and B if

your color is BLUE. Then hit ENTER. Con�rm your entry by pressing the Y key whenprompted, or press the N key to correct your entry.

[WAIT FOR SUBJECTS TO ENTER COLORS]

16

You now see the experiment screen. Throughout the experiment, you will be toldwhat is currently happening at the left or very bottom of the screen. The strip along thebottom of the screen tells the history of what happened in your previous matches. Sincethe experiment has not yet begun, this strip along the bottom is currently empty. In themiddle of the screen is the matrix which you have previously seen up on the screen atthe front of the room. At the top left of the screen you see your color, your subject IDnumber, and your name. Is there anyone whose color is not correct?

[WAIT FOR RESPONSE]We will now pass out the experiment record sheet, on which you will record all of

the results from this experiment. When you receive an experiment record sheet, pleaserecord your name, color, and today's date on the top of the sheet. Do not record yoursubject ID number at this time.[EXPERIMENTER PASS OUT EXPERIMENT RECORD SHEETS AND PENCILS]

[WAIT FOR SUBJECTS TO RECORD INFORMATION]We will now start the �rst practice match. Remember, do not hit any keys or click

the mouse button until you are told to do so.If you are a RED subject, on the left of the screen you are asked to please choose

a row. If you are a BLUE subject, you are asked to please choose a column. You willchoose a row or column by moving the mouse to the appropriate choice and clicking themouse button.

Will all RED subjects now move the mouse so that the arrow on the screen is pointingto the bottom row labeled "D" and will all BLUE subjects now move the mouse so thatthe arrow on the screen is pointing to the left column labeled "L".

[WAIT FOR SUBJECTS TO MOVE MOUSE TO APPROPRIATE ROW OR COLUMN]Note that the row or column to which you are pointing with the mouse is now sur-

rounded by a ashing rectangle. Will all RED subjects please choose "Down" and allBLUE subjects please choose "Left" by clicking the mouse button now while the arrowis pointing to the appropriate row or column. After choosing the row or column, con�rmyour choice by clicking on the "Yes" icon at the bottom of the screen or click on the "No"icon to correct your choice.[WAIT FOR SUBJECTS TO CHOOSE ROW OR COLUMN AND CONFIRM CHOICE]

After all subjects have con�rmed their choices, the match is over. The outcome ofthis match, Down-Left, is now highlighted on everybody's screen. Also note that themoves and payo�s of the match are recorded in the experiment history at the bottom ofthe screen. The outcomes of all of your previous matches will be recorded at the bottomof the screen throughout the experiment so that you can refer back to previous outcomeswhenever you like. The payo� to the RED subject for this match is 45 dimes and thepayo� to the BLUE subject is 15 dimes. Please record the outcome of this match on yourexperiment record sheet in the �rst row labeled "PRACTICE". After you have �nishedrecording the outcome of this match, use the mouse to click on the "OK" icon at thebottom of the screen to indicate that you are ready to continue.

[WAIT FOR SUBJECTS TO RECORD OUTCOME AND CLICK "OK"]

17

You are not being paid for the practice session, but if this were the real experiment,then the payo� you have recorded would be money you have earned from the �rst match,and you would be paid this amount for that match at the end of the experiment. Thetotal you earn over all 6 real matches, in addition to the $3.00 bonus, is what you willbe paid for your participation in the experiment.

We will now proceed to the second practice match.[EXPERIMENTER HIT KEY TO START SECOND MATCH]

For the second match, each subject has been paired with a di�erent subject of theopposite color. You are not paired with the same subject with whom you were paired inthe �rst match. After you have been paired with a subject once, you will never be pairedwith that subject again. The rules for the second match are exactly like the �rst. Willall RED subjects again choose "Down" by clicking on the bottom row and con�rmingthe choice. Also, will all BLUE subjects choose "Right" by clicking on the right columnand con�rming the choice.[WAIT FOR SUBJECTS TO CHOOSE ROW OR COLUMN AND CONFIRM CHOICE]

The outcome of this match, Down-Right, is now highlighted on everybody's screen.The payo� to the RED subject for this match is 0 dimes and the payo� to the BLUEsubject is 10 dimes. Please record the outcome of this match on your experiment recordsheet in the second row labeled "PRACTICE". After you have �nished recording theoutcome of this match, use the mouse to click on the "OK" icon at the bottom of thescreen to indicate that you are ready to continue.

[WAIT FOR SUBJECTS TO RECORD OUTCOME AND CLICK "OK"]We will now proceed to the third practice match.

[EXPERIMENTER HIT KEY TO START THIRD MATCH]Each subject has once again been paired with a di�erent subject of the opposite color.

You are not paired with either of the two subjects with whom you were paired in theprevious matches. Will all RED subjects choose "Up" by clicking on the top row andcon�rming the choice, and will all BLUE subjects choose "Left" by clicking on the leftcolumn and con�rming the choice.[WAIT FOR SUBJECTS TO CHOOSE ROW OR COLUMN AND CONFIRM CHOICE]

The outcome of this match is now highlighted on everybody's screen. Note that theentire top row is highlighted and thus that the column which the BLUE subject choseis not revealed. Also observe in the history at the bottom of the screen that only theRED subject's choice is recorded. If the RED subject chooses "Up", the RED subjectwill not be told whether the BLUE subject chose "Left" or "Right" as the payo�s arethe same in either case. The payo� to the RED subject for this match is dimes andthe payo� to the BLUE subject is 20 dimes. Please record the outcome of this matchon your experiment record sheet in the third row labeled "PRACTICE". If you are aRED subject, you should put a hyphen or dash in the column labeled "BLUE CHOICE",since your screen does not indicate whether the BLUE subject chose "Left" or "Right".If you are a BLUE subject, you should record your choice as usual by putting a "L" inthe column labeled "BLUE CHOICE", since you should know which column you chose.

18

After you have �nished recording the outcome of this match, use the mouse to click onthe "OK" icon at the bottom of the screen to indicate that you are ready to continue.

[WAIT FOR SUBJECTS TO RECORD OUTCOME AND CLICK "OK"]We will now proceed to the fourth practice match.

[EXPERIMENTER HIT KEY TO START FOURTH MATCH]Each subject has once again been paired with a di�erent subject of the opposite color.

You are not paired with the any subject with whom you were paired in the �rst, second,or third practice match. Will all RED subjects choose "Up" by clicking on the top rowand con�rming the choice, and will all BLUE subjects choose "Right" by clicking on theright column and con�rming the choice.[WAIT FOR SUBJECTS TO CHOOSE ROW OR COLUMN AND CONFIRM CHOICE]

The outcome of this match is now highlighted on everybody's screen. Note once againthat the entire top row is highlighted and thus that the column which the BLUE subjectchose is not revealed. The payo� to the RED subject for this match is dimes andthe payo� to the BLUE subject is 20 dimes. Please record the outcome of this match onyour experiment record sheet in the fourth row labeled "PRACTICE". After you have�nished recording the outcome of this match, use the mouse to click on the "OK" iconat the bottom of the screen to indicate that you are ready to continue.

[WAIT FOR SUBJECTS TO RECORD OUTCOME AND CLICK "OK"][EXPERIMENTER HIT KEY TO END PRACTICE SESSION]

This concludes the practice matches. The computer screen now indicates your totalpayo� for the four practice matches. This is the amount you would have earned for thesematches if these were matches in the actual experiment. You do not need to record thistotal.

In the actual experiment there will be 6 matches and, of course, it will be up to youto make your own decisions. At the end of the experiment, we will pay each of youprivately, in cash, the total amount you have accumulated during all 6 matches, plusyour guaranteed $3.00 participation bonus. No other person will be told how much cashyou earned in the experiment. You need not tell any other participants how much youearned.

Are there any questions before we begin the experiment?[EXPERIMENTER TAKE QUESTIONS]

O.K., then we will now begin with the actual experiment. Please press the spacebaronce and wait a moment for the current screen to clear.

[WAIT FOR SUBJECTS TO PRESS SPACEBAR AND CLEAR SCREEN]After the screen has changed, please type "DL" and hit the "Enter" Key.

[EXPERIMENTER START EXPERIMENT PROGRAM]If there are any problems from this point on, raise your hand and an experimenter

will come and assist you. When the computer asks for your name, please start as beforeby typing your name. Wait for the computer to ask for your color, then respond withthe correct color.

[WAIT FOR SUBJECTS TO INPUT NAME AND COLOR]

19

[EXPERIMENTER HIT KEY TO START MATCHING]At the top left of your screen you once again see your color and subject ID number.

This ID number may be di�erent than the ID number you were given during the practicematches. Will you now please record this ID number at the top of your experiment recordsheet where it says "Subject ID#". Also, please make sure that the color indicated onthe screen is correct.

[WAIT FOR SUBJECTS TO RECORD ID NUMBERS]Okay, we will now begin match number 1.

[START EXPERIMENT][AFTER 1ST MATCH, REMIND SUBJECTS THEY ARE PAIRED WITH A NEW PERSON]

20