Atkinson, Suppes - AN ANALYSIS OF TWO-PERSON GAME SITUATIONS IN.pdf - Psychologia - koni86

Journal o/ Experimental Psychology

Vol. 55, No. 4, 1958

AN ANALYSIS OF TWO-PERSON GAME SITUATIONS IN

TERMS OF STATISTICAL LEARNING THEORY '

RICHARD C. ATKINSON 2 AND PATRICK SUPPES

Applied Mathematics and Statistics Laboratory, Stanford University

This study represents an extension

of statistical learning theory to a class

of two-person, zero-sum game situa-

tions. Because the theory has been

mainly developed in connection with

experiments dealing with individual

learning problems, its predictive suc-

cess in an experimental area involving

interaction between individuals pro-

vides an additional measure of the

scope of its validity. It should be

emphasized that the study reported

here was not conceived as providing

an empirical test of the adequacy of

learning theory as opposed to game

theory; although we use the language

of game theory to describe the study,

the game characteristics of the situa-

tion were not apparent to Ss. This

point is amplified below.

For the purposes of this experiment,

a play of a game is a trial. On a given

trial each of the two players inde-

pendently makes a choice between

one of two alternatives—that is, he

makes one of two possible responses.

After the players have indicated their

choices, the outcome of the trial is

announced to each player.

On all trials, the game is described by

the following pay-off matrix.

The players are designated A and B.

The responses available to A are AI and

AS; similarly, the responses available to

B are BI and B 2 . If A selects AI and B

selects Bi then there is a probability ai

that A is "correct" and B is "incorrect,"

and a probability 1 — ai that A is "in-

correct" and B is "correct." These two

joint events are exhaustive since it is

required that exactly one player is cor-

rect on each trial. The outcomes of the

other three response pairs are identically

specified in terms of az, as and &i.

The interaction of the players is limited

by two factors: (a) neither player is

shown the pay-off matrix, (b) neither

player is directly informed of the re-

sponses of the other player. Thus, from

the standpoint of the general theory of

rational behavior (4), S should not regard

himself as playing a 2 X 2 game with

known pay-off matrix but should view

the situation as a multi-stage decision

problem against an unknown opponent.

However, selection of an optimal strategy

in this multi-stage decision problem is

far from a trivial task mathematically,

and it is scarcely to be expected that any

S would use such a strategy. The virtue

of statistical learning theory is that it

yields a quantitative prediction of how

organisms actually do behave in such

situations.

Our theoretical analysis of the be-

havior of Js in the situation described is

based on two distinct but closely related

models. Since a detailed mathematical

analysis of these models will be presented

elsewhere, the present statement will

concern only the most salient facts and

omit mathematical proofs.

Linear model. —The first model is an

extension of a linear model developed by

Estes and Burke (6). Experimental

tests of this formulation for one-person

learning situations have been reported

1 This research was supported by the Be-

havioral Sciences Division of the Ford Founda-

tion and by the Group Psychology Branch of the

Office of Naval Research. The authors are

indebted to W. K. Estes for several stimulating

discussions of the ideas on which this experiment

is based.

2 Now at the University of California, Los

Angeles.

369

370

RICHARD C. ATKINSON AND PATRICK SUPPES

(2, 9, 13). The basic assumption of the

model is that response probability on a

given trial is a linear function of the

probability on the preceding trial. When

a response is reinforced its probability

increases; the reinforcement of any other

response decreases its probability.

In the present situation, where two

responses are available to each S, if a

response occurs and is designated as

"correct," then the response is rein-

forced; if a response occurs and is desig-

nated as "incorrect," then the alternative

response is reinforced. More specifi-

cally, let a n be the probability of response

AI on Trial «. The rules of change are:

(a) if AI is reinforced on Trial n, then

a n+ 1 = (1 — 0A.)a n + 0A; (b] if Aj is rein-

forced on Trial n then a n+ \ = (1 — 0A.)a n ,

where 0 < BA. < 1. Identical rules are

specified for /3 n , the probability of a B:

response, in terms of (?B-

The following pair of recursive equa-

tions can then be derived for the mean

probabilities a n and j3 n , where •)>„ is the

mean probability of the joint event that

on Trial » Player A will make response

Ai and Player B response BI.

The line determined by this equation

has been labeled the interaction line since

the exact point on the line specifying the

asymptotic probabilities 5 and /3 is a

function of both 8 A and &•&• It is inter-

esting to observe that in the correspond-

ing one-person learning situation, the

interaction line degenerates to a point,

while in the three-person situation an

interaction surface is obtained.

Finite Markov model. —In this model

the simplifying assumption is made that

on all trials a player's response behavior

is determined by a single stimulus—that

is, the event associated with the onset of

a trial. The S is described as being in

one of two possible states: (a) if in State

1, the stimulus is conditioned to Response

1 and, in the presence of the stimulus,

Response 1 will be elicited; (b) if in State

2, the stimulus is conditioned to Re-

sponse 2 and, in the presence of the

stimulus, Response 2 will be elicited.

Thus, on any Trial «, the two players are

described in terms of one of the following

fourstates: <!,!>, <1,2>, <2,1> and

<2,2> where the first member of a

couple indicates the state of Player A

and the second, the state of Player B.

For example, <2,1> means that Player

A will make response Aa and Player B

will make response Bj. It is postulated

that the change of states from one trial

to the next is Markovian, and the follow-

ing analysis is used to derive the transi-

tion matrix (10, 11) of the process.

When one of Player A's responses is

reinforced on Trial n there is (a) a prob-

ability #A that the stimulus governing

Player A's response will be conditioned

to the reinforced response and therefore,

on Trial n + 1 Player A will make the

response reinforced on Trial n and, (b) a

probability 1 — (?A that the conditioned

status of the stimulus will remain un-

changed and therefore, on Trial n + 1

Player A will repeat the response made

on Trial n. Identical rules describe the

process for Player B in terms of 0B- 4

4 The Markov process derived from these as-

sumptions differs in certain respects from that

which can be derived from the Estes and Burke

stimulus sampling model (6). In their formula-

tion the stimulus is conceptualized as being

+ 0A(a 4 -

/§»*,! = (1 - 0 B a s - 0Ba 4 )i3 n

— a 4 )5,,

It may be shown that 3, /3 and 7, the

asymptotic probabilities in the sense of

Cesaro (II), 3 exist and are independent

of the initial probabilities «o, |8o> To-

However, in general these asymptotic

quantities depend on (?A and #B, and no

simple results are obtainable for the

quantities individually. On the other

hand, an interesting linear relation be-

tween a and /3, which is independent of

•y, #A and 0s, can be derived, namely:

[(a s + a 4 — ai — a 2 ) + (aia 2 — aaa 4 )]5

J(a s + a 4 — &\ — a 2 )

— a a ). (1)

3 To be explicit,

_ _

a = lira - S "on,

_ n-»» » i-l

and similarly, for /3 and 7.

1 " "

TWO-PERSON GAME SITUATIONS

371

For this set of assumptions and the

pay-off probabilities ai, aa, a-a and a.*, the

transition matrix describing the learning

process can be derived and is as follows:

<2,1>

<2,2>

ai(0A -

320B

a 2 (0A - SB)

(1 - a s )0A

a(0 A - SB)

+ (1 - fa)

<2,2>

4(0A - fe)

+ (1 - fa)

Rows designate the state on Trial n

and columns the state on Trial n + 1.

Thus (1 — aa)0A, the entry in Row 3,

Column 1, is the conditional probability

of being in State <!,!> on Trial n -f- 1

given that the pair of "Ss was in State

<2,1> on Trial n, because:

taking the limits of a and /3 as the ratio

#A /6s approaches zero or becomes large.

Particular cases of the theoretical

analysis may be illustrated by examining

predictions for the parameter values em-

ployed in this experiment. Three sets

of ai values were used corresponding to

three classical cases of 2 X 2 games in

the theory of zero-sum, two-person

games (12).

The first case is labeled the Mixed

Group, since both players have mixed

minimax strategies. The as values are

given by the pay-off matrix

+ (1-0A)0B-0+(1-0A)(1-0 B )'0.

From these one-stage transition prob-

abilities an explicit solution is obtained

for the Cesaro asymptotic probabilities

of an AI and Bj response; as in the case

of the linear model_ these quantities are

denoted as 5 and & respectively. The

general equations for S. and /3 are too

lengthy to reproduce here but certain

results are noteworthy. It can be shown

that a and /3 are related by the identical

interaction line determined by Equation

1 of the linear model. For the Markov

model, however, it can in addition be

proved that the point on the interaction

line describing a particular pair of Ss'

asymptotic behaviors is uniquely deter-

mined by the ratio of 0A to O-B. Further,

even without a knowledge of the specific

values of BA. and BB one can specify a

fairly narrow interval on the interaction

line within which a and /3 must fall by

The minimax strategy for Player A is to

choose Ai with probability 3, and the

minimax strategy for B is to choose BI

with probability f. In the Markov

model

5= .600

- 35(0 A /fla)+22

P ~ 50(0 A /S B ) + 40'

(2)

(3)

Note that a is independent of 0A/0B-

From Equation 3 one obtains as bounds

on /3:

composed of a large number of stimulus elements

each of which is sampled with probability 6 and,

once sampled, conditioned to the reinforced re-

sponse with probability 1. Further, the prob-

ability of a response is defined as the proportion

of stimulus elements in the sample conditioned

to the response. In the model used in this paper

it is assumed that the single stimulus is sampled

on each trial with probability 1.

.550 < /3 < .700.

(4)

If one assumes 0A = OB, then /3 = .633.

For this case the interaction line is the

line satisfying Equation 2.

The second case is labeled the Pure

Group, since both players have pure mini-

max strategies. The particular values

372

RICHARD C. ATKINSON AND PATRICK SUPPES

are given by the matrix

Bi 62

Here ai = J is a saddle point of the ma-

trix and from the standpoint of game

theory the optimal strategy for Player A

is to play AI with probability 1 and for

B to play BI with probability 1. In the

Markov model

= .667

j3 ^ 6(fl A /fl B ) + 5

(S)

METHOD

Subjects.— The Ss were 120 undergraduates

obtained from introductory courses in psy-

chology and philosophy at Stanford University.

They were randomly assigned to the Mixed,

Pure, and Sure Groups with the restriction that

there were 20 pairs of Ss in each group.

Apparatus. —The Ss, run in pairs, sat at

opposite ends of an 8 X 3-ft. table. Mounted

vertically on the table top facing each S was a

50-in. wide by 30-in. high black panel placed

22 in. from the end of the table. The E sat

between the two panels and was not visible to

either S. The apparatus, as viewed from S's

side, consisted of two silent operating keys

mounted 8 in. apart on the table top and 12 in.

from the end of the table; upon the panel, three

milk-glass panel lights were mounted. One of

these lights, which served as the signal for S to

respond, was centered between the keys at a

height of 17 in. from the table top. Each of the

two remaining lights, the reinforcing signals, was

at a height of 11 in. directly above one of the

keys. The presentation and duration of the

lights were automatically controlled. The Ss

were not visible to one another and could not see

each other's responses or panel lights.

Procedure. —The Ss were read the following

instructions: "We always run Ss in pairs because

this is the way the equipment has been designed

and also because it is the most economical pro-

cedure. Actually, however, you are both work-

ing on two completely different and independent

problems.

"The experiment for each of you consists of a

series of trials. The top center lamp on your

panel will light for about 2 sec. to indicate the

start of each trial. Shortly thereafter one or the

other of the two lower lamps will light up.

Your job is to predict on each trial which one of

the two lower lamps will light and indicate your

prediction by pressing the proper key. That is,

if you expect the left lamp to light press the left

key, if you expect the right lamp to light press

the right key. On each trial press one or the

other of the two keys but never both. If you

are not sure which key to press then guess.

"Be sure to indicate your choice by pressing

the proper key immediately after the onset of

the signal light. That is, when the signal light

goes on press one or the other key down and

release it. Then wait until one of the lower

lights goes on. If the light above the key you

pressed goes on your prediction was correct, if

the light above the key opposite from the one

you pressed goes on you were incorrect, and

should have pressed the other key. At times

(6)

If one assumes

|9 = .611.

The third case is labeled the Sure

Group since both players have sure-thing

strategies (i.e., given the pay-off matrix

one of the two responses available to each

player is at least as good or better than

the other response regardless of what his

opponent does). The parameter values

are given by the matrix

(7)

0B, then

The sure-thing strategies for Players A

and B are AI and BI respectively. In

the Markov model

- = 5(9 A /e B ) + IS

7 //) /fl i i 0 "\

i ^PA/CB/ i* "^

•J6

-23'

(8)

r *"*

(9)

and as bounds one has:

.652 < a < .711 (10)

.696</3<.711. (11)

If one assumes that #A = #B, then

a = .667 and /3 = .700. For this case

the interaction line is determined by the

equation:

35=10/3-5.

(12)

As in the previous case, a is independent

of 6\ /0B and the interaction line is the

line satisfying Equation 5. From Equa-

tion 6 one obtains as bounds on /3:

.555 < 0 < .667.

that

TWO-PERSON GAME SITUATIONS

373

you may feel frustrated or irritated if you cannot

understand what the experiment is all about.

Nevertheless, continue trying to make the very

best prediction you can on each trial."

For each pair of Ss, one was randomly selected

as Player A and the other as Player B. Further,

for each S one of the two response keys was

randomly designated Response 1 and the other

Response 2 with the restriction that the following

possible combinations occurred equally often in

each of the three experimental groups: (a) Ai

and BI on the right, (b) AI on the right and Bi

on the left, (c) AI on the left and BI on the right,

and (</) Ai and BI on the left.

Following the instructions, 200 trials were run

in continuous sequence. For each pair of Ss

sequences of reinforcing lights were generated in

accordance with assigned values of ai and ob-

served responses.

On all trials the signal light was lighted for

3.5 sec.; the time between successive signal ex-

posures was 10 sec. The reinforcing light fol-

lowed the cessation of the signal light by l.S sec.

and remained on for 2 sec.

At the end of the session each S was asked to

describe what he thought was involved in the

experiment. Only one S indicated that he be-

lieved the reinforcing events depended in any

way on a relationship between his responses and

the other player's responses. His record and

that of his partner were eliminated from the

analysis and replaced by another pair.

- MIXED GROUP

.6-

.5-

O .6-

.5-

SURE GROUP

BLOCKS OF 40 TRIALS

FIG. 1. Observed proportions of AI and Bi

responses in blocks of 40 trials for the three

experimental groups.

RESULTS AND DISCUSSION

Mean learning curves and asymptotic

results. —Figure 1 provides a descrip-

tion of behavior over all trials of the

experiment. In this figure the mean

proportions of AI and Bj responses in

successive blocks of 40 trials are given

for the sequence of 200 trials. An in-

spection of this figure indicates that

responses are fairly stable over the

last 100 trials except possibly for BI

responses in the Pure Group. To

check the stability of response be-

havior for individual data, t's for

paired measures were computed be-

tween response proportions for the

first and last halves of the final block

of 60 trials. In all cases the obtained

values of t fall short of significance at

the .05 level.

It appears reasonable to assume

that a constant level of responding has

been reached; consequently the pro-

portions computed over the last 60

trials were used as an estimate of 5.

and /3. Table 1 presents the observed

mean proportions of AI and BI re-

sponses in the last 60 trial block and

the SD's associated with these means.

TABLE 1

PREDICTED AND OBSERVED MEAN PROPORTIONS

OF Aj AND BI RESPONSES OVER THE

LAST BLOCK OF 60 TRIALS

Response Ai

Response Bi

Pred.

Obs.

.60S

.670

.606

.0794

.0832

.1005

Pred.

Obs,

.649

.602

.731

.0874

.0634

.0760

Mixed

Pure

Sure

.600

.667

.633

.611

.700

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