Kaufman, Becker - THE EMPIRICAL DETERMINATION OF GAMETHEORETICAL.pdf - Teoria gier - Piffony

Journal of Experimental Psychology

1961, Vol. 61, No. 6, 462-468

THE EMPIRICAL DETERMINATION OF GAME-

THEORETICAL STRATEGIES *

HERBERT KAUFMAN AND GORDON M. BECKER 2

Electric Boat Division, General Dynamics Corporation

The present paper reports a study

of the behavior of naive subjects in

a strategy-making situation where a

normative solution, i.e., game theory,

can be stated. In simplest game

theory terms, 5s were required to

find optimal strategies in a two person

zero sum game under conditions to be

described. The game theory solution

provides a logical reference or ideal

to which the actual game playing is

compared. The procedure provides

a dynamic behavioral analysis of

decision problems usually handled by

the game theory model.

Atkinson and Suppes (1958, 1959),

Lieberman (1959), and Simon (1956)

have previously reported empirical

studies involving game theory con-

cepts. These studies, almost uni-

versally, are concerned with behavior

in game situations where individual

moves, rather than strategies, are

made at each trial. The present

paper represents a radical departure

from this usual procedure in that the

individual playing a matrix game is

required to indicate a game strategy

at every trial, i.e., to select 100 moves

at each trial, rather than simply one

move.

In order to explain the experimental

procedure, some characteristics of the

game theory model will be stated:

(a) In the two person, two choice,

1 This research was done as part of the

Basic Research Program of General Dynamics

Corporation. The authors wish to thank

Mortimer H. Applezweig for his help in

providing Connecticut College psychology

laboratory facilities and students.

2 Now at the University of California,

Los Angeles.

zero sum game represented by a

2X2 matrix there exist optimal

strategies for each player, (b) If

either player plays his optimal strat-

egy the play of the other has no

effect on the average payoff. (c)

Playing an iterated game involves,

for each player, both a decision about

the relative frequencies of play for

the two choices, the game strategy,

and a decision about the sequence of

play, the game tactics. In the absence

of any predictive criteria on the play

of the opponent, a random sequence

of the choices is to be preferred in

order to guarantee the expected

payoff for a given strategy.

In the experiment to be described,

the strategy-making aspect of the

game decisions was isolated by re-

moving the burden of tactical play

from 5s. The major questions to be

answered were: (a) Do naive 5s learn,

through the experience of playing a

game, to find and maintain a strategy,

optimal or otherwise? (b) Do games

having the same values (average

payoff for optimal strategy), but

different optimal strategies, differ

in ease of learning ?

METHOD

Strategy decisions were emphasized by

forcing 5s to choose relative frequencies of

play, and not allowing them to decide upon

the sequence of play. At each trial S divided

100 choices between his two alternatives.

Each block of 100 choices amounts to a

strategy. Thus, at each trial 5 announced

the strategy he wanted on that trial, e.g.,

42 of one alternative and 58 of the other.

The 5s strategy was paired with a strategy

chosen by E to be punishing to 5 (see below).

That is to say, in every game S's opponent

462

GAME-THEORETICAL STRATEGIES

463

was in fact E. The payoff to S for this pair

of opposing strategies was calculated under

an assumption of random play by both

players, equivalent to arranging the two

sequences of choices each in random order

and summing the 100 payoffs thus determined.

Figure 1 is the graphical representation of

Game D shown in Table 1. T and B are the

alternative choices for the row player, L and

R the alternatives for the column player,

and the cell entries represent the payoffs

to the row player for the given combinations

of row and column choices. In this figure

the strategies of the row player are shown

along the abscissa, labeled as the proportion

of top row (T choices). For example, the

point 92 on the abscissa corresponds to the

row player's choice of 92T and 8B. The

ordinate is payoff points to the row player.

The four ordinate values, a, b, c, and d, are

the cell entries of the matrix in the top row

(a, b) and the bottom row (c, d), from left to

right. For example, in Game D (Table 1),

a is 1.12, b is 0.78, c is 0.4S, and d is 1.47.

The column strategies are represented para-

metrica'lly by the lines cutting across from

the line X = 0 to the line X = 100, labeled

as left column proportion. For example,

the line running from (100, 85) to (0, 126)

corresponds to the column player's strategy

of 20L and 80R. The payoff to the row

player for 92T-8B against a column strategy

of 20L-80R is found on the graph to be 88

points.

TABLE 1

FIVE EXPERIMENTAL GAMES

Game

5 Chooses

Rows

T & B

Payoffs to 5

When E Chooses

Column

Optima!

Strategy for S

(100 Moves)

0.95

0.28

0.68

1.35

1.02

0.35

1.12

0.45

0.60

1.30

0.95

1.65

1.23

0.53

0.88

1.58

100

0.78

1.47

1.28

0.62

5 50

All the lines intersect at the point (X',

Y'). X' is the row player's optimal strategy;

in this case 75T-25B. Y' is the value, V,

of the game; in this case 95 points.

The S was always the row player. His

opponent (E) was the column player. 'The

JE's choices were obtained as follows: (See

Fig. 1). Each strategy of 5 defines an inter-

val (containing V) of possible payoffs to S.

Each point in this interval corresponds to a

strategy for E. All points with payoff

greater than V (shaded area in Fig. 1) were

eliminated so that S could never do better

than V. The remaining interval shown in

the unshaded portion of the graph was divided

into 10 equal parts, each part corresponding

to the E strategies 100-0, 95-5, . . ., 55-45.

One of the 10 strategies in the interval

determined by S's strategy was selected

randomly to give 5 his payoff on that play.

The five 2X2 matrix games shown in

Table 1 were used. These games were selected

according to the following specifications:

(a) All the games have the same value

(V = .95). (b) The optimal S strategies

for the games are: 100-0, 90-10, 75-25,

60-40, 50-50, these numbers referring to the

proportions of T and B, respectively.

deviating from the optimum are approxi-

mately equal for equal absolute deviations

and E's most punishing choices (always

either 100-0 or 0-100). Graphically this

means that the lines representing E's 0-100

and 100-0 strategies have slopes of approxi-

mately equal magnitude, but opposite in sign.

FIG. 1. Graphical illustration of Game D.

464

HERBERT KAUFMAN AND GORDON M. BECKER

The payoffs for the five games were ob-

tained graphically, by drawing the lines ac

and bd, as in Fig. 1, and reading off the

intercepts at pairs of perpendiculars lOOx

units apart; the left perpendicular at x'

(100-0 game), 10 units to the left of x' (90-10

game), etc. The values were rounded to the

nearest hundredth. The slopes of the lines

ac and bd were chosen with two ideas in

mind.

In the first place, if the slopes were too

small, differences in players' responses would

not be differentially rewarded and it would

be difficult to locate the optimal strategy

by trial and error. On the other hand, if the

slopes were large, the range of payoffs would

be large, leading both to easier response

differentiation and to awkward point-to-

money conversion rates.

In the second place, the two lines were

chosen with slightly different absolute slopes

to avoid obvious symmetries in the SO—50

game and to avoid the easily learned char-

acteristic of equal slopes, i.e., probability

of playing a given row is simply the difference

between the payoffs in the other row divided

by twice the slope. Furthermore, the slope

for any line is simply the difference between

the column payoffs. Therefore, for equal

slopes, the optimal strategy would always be

the difference between the payoffs in the

opposite row, divided by twice the difference

between the payoffs in either column. The

values actually used in the experiment reflect

a compromise among these conflicting con-

siderations. The slopes of the lines ac and

bd were the same for all five games. Since

these slopes determine a reward gradient

for responses deviating from optimal strate-

gies, it is clear that the reward gradients were

the same for all games. This means that the

average reward and range of rewards for

responses deviating by a given absolute

amount from optimal strategies was the same

for all games. The games were obtained

empirically rather than analytically partly

because of the ease with which the graphic

method provided a solution to these problems

and partly because two place accuracy was

considered more than sufficient for the

purposes of this experiment.

The 5s were told that their point total

would be converted to money after the experi-

ment—the more points, the more money—

with a conversion factor of about 100 points

equal to 1 cent. They could receive up to

$1.00 more than the minimum guaranteed

($1.50) for participating in the experiment.

They were urged to make all reasonable

efforts to maximize their point total for

each trial and for the entire series of tasks.

The 5s were, in fact, paid on a linear scale

where the lowest point total equaled $1.50

and the highest equaled $2.50.

The same game value was used for all five

games to avoid confounding the effect of

value with that of the major variable, optimal

strategy. The value of 95 points was chosen

so that the point-to-money conversion could

be made at reasonable cost, i.e., it was felt

that a ratio of approximately 100 points to

1 cent would give a return satisfactory to 5s

and tolerable to the experimental budget.

The value of the games was not made a

rounded figure, say 100 points, in order to

avoid making this particular aspect of the

procedure too transparent to 5s.

Instructions. —In the instructions given 5

the following points were emphasized: (a)

Your ability to make decisions is being tested.

The better your decisions, the higher will be

your payoff, (b) The E is single-minded in

his determination to keep your winnings as

low as possible. He will be informed of your

choices but his information will not be

complete. He will be told enough, however,

so that unwise choices on your part can be

punished severely, (c) A given payoff will

be determined entirely by two factors: your

strategy and your opponent's strategy.

The procedure for obtaining the payoffs

from the two strategies at each trial was

explained carefully to 5s. One reason was

to avoid the misconception that if, for ex-

ample, the two strategies were T90-B10 and

L90-R10, there would be 90TL combinations,

10BR combinations and no BL or TR com-

binations, when in fact there would be 81TL,

9TR, 9BL, and 1BR combinations.

The procedure was represented to 5s as

an analog of real life situations involving two

choices each for two antagonists. The pay-

off rules of the game as given by the matrix

were explained, but no instructions were

given involving game theory concepts.

Specifically, 5s were not told that all games

had the same value, nor what the value of

any game was, nor indeed did they have

any explanation of what a game value was.

They were told nothing of the possible payoff

beyond the information contained in the

matrix and the important facts labeled (6)

immediately above. All this was aimed at

creating a situation in which 5s were con-

fronted by a hostile and knowledgeable

opponent. By choosing their optimal strategy

5s could guarantee themselves a certain

number of points each trial. They could

never do better than the optimal strategy

payoff but they could do considerably worse.

GAME-THEORETICAL STRATEGIES

465

Procedure. —The procedure for each game

was as follows: 5 was given a three-page

answer sheet with the matrix game repro-

duced on each page. The alternatives were

denoted by capital letters and were changed

from game to game and from 5 to 5, The 5

announced his strategy (frequencies of the T

and B alternatives adding to 100). The E

chose a strategy punishing to 5 as described

above. The E then announced the strategy

he had chosen together with 5's payoff. The

5 recorded both strategies and his payoff.

This procedure was continued for 50 trials

or until criterion was reached. (See Results

and Analysis).

Experimental design. —The design, shown

in Table 2, was a S X 5 latin square replicated

four times; each replication had the 5 games,

5 orders of presentation (each with a different

S), and 5 temporal positions.

TABLE 3

FREQUENCY DISTRIBUTIONS FOR ADs FROM

OPTIMAL STRATEGY BY POSITIONS

Position

19.5

1 8

12.0

11.5

13.5

40 +

30-39.9

20-29.9

10-19.9

0-9.9

Mean

19.0

Tables 3 and 4. The other measure

was trials to achieve a steady-state

response. Steady-state was defined

as repetition, without deviation, of

the same response up to and including

the last trial (Trial 50), or 5"s an-

nounced intention to continue in this

way. The percentage of 5s reaching

criterion in each of the five positions

was 10, 20, 40, 50, and 65, and in each

of the five games was 50, 40, 30, 35,

and 30 for Games A through E,

respectively. All steady-state solu-

tions to the game problems were the

"correct" game theory (maximin) so-

lutions, but a steady-state was not

always achieved.

Because a steady-state solution

was not achieved in many games,

the analyses of the criterion scores

were performed using a nonpara-

TABLE 2

EXPERIMENTAL DESIGN

Group

Order of Presentation

of Games

III

I V

The order of the rows in each game matrix

was varied randomly from one S to another.

Since each game in each of the five order of

presentation groups was presented to four

different 5s, two of the four 5s in each group

had the matrix as given in Table 1, and the

other two had the rows inverted.

Subjects. —Twenty paid volunteers, under-

graduate women students from Connecticut

College, served as 5s.

TABLE 4

FREQUENCY DISTRIBUTIONS FOR ADs FROM

OPTIMAL STRATEGY BY GAMES

RESULTS

Two measures were used, each

corresponding to a somewhat dif-

ferent aspect of performance. To

measure the overall correspondence

of the response to the optimal strategy

an "integrated error" score was used.

This was the average deviation (AD)

from optimal strategy over all trials

for each game. These are shown in

Game

40 +

30-39.9

20-29.9

10-19.9

0-9.9

Mean

20.5

10.0

16.0

14.5

1 6

14.5

466

HERBERT KAUFMAN AND GORDON M. BECKER

TABLE 5

ANALYSIS OF VARIANCE OF AD SCORES

received by 5s. Although the main

concern was centered on what 5s

did (responses given as strategies)

rather than what they received (rein-

forcement given as number of points

received) there is a possibility that

the lack of response differentiation

might have been mediated by reward

constancy, or in simpler terms, that

5s failed to seek and find better

strategies because the extra effort

did not lead to any particular gain

in payoff. The analysis of variance

on the point totals reveals the same

effects as those demonstrated in the

response measure. The tau correla-

tion between earning (points) and

deviations from optimal strategy,

computed over 5s, was —0.947

(P < .001).

The "how-much" scores of average

error and points, and the "how-long"

score of trials-to-criterion lead to

similar conclusions concerning effects

of 5s and temporal position. A

major discrepancy is found for the

five games which differ in their

optimal strategies for 5. Here it is

found that 5s tend on the average

to be further from the optimal

strategies on the 100-0 and 90-10

games than on the other three games

in the presolution portions of the

game, but do not differ significantly

in the number of trials taken to

achieve a solution.

Source

Between 5s

Order of presentation

Replicates (Groups)

Residual among 5s

Within 5s

Games

Position

Residual from L.S.

Residual within 5s

109.86

375.65

231.84

268.59

388.14

58.91

1.62

3.94*

4.56*

6.59*

12\ 7 2

*P <.01.

metric ranking test, the Friedman

two-way analysis of variance on ranks.

Average deviation scores.- —The anal-

ysis of variance performed on the AD

scores is shown in Table 5. The

results show that performance, using

this measure, varied with the person

playing the game, the game being

played, and the point in the sequence

at which the game was played.

Trials-to-criterion scores. —In per-

forming the analysis of variance on

ranks, parallel analyses were done

on rankings with ties, and on rankings

with ties broken by appeal to the

AD scores. The results, shown in

Table 6, indicate that performance,

using this measure, varied with the

person playing the game and the

point in the sequence at which the

game was played, but did not vary

with the game.

Of the 100 games played by the 20

5s, 37 were solved by 13 5s. Among

the 13 5s who solved at least one

game the mean number of solutions

was 2.8 (median of 3) out of a possible

five. In only one case did an 5 fail

to solve a game following a solution

to an earlier game.

In addition to the above two re-

sponse measures the point totals

collected by each 5 for each game were

analyzed to see what effects were

reflected by the amount of points

TABLE 6

FRIEDMAN TWO-WAY ANALYSIS OF VARIANCE

ON RANKS OF TRIALS-TO-CRITERION

SCORES

. —

•t

Soincc

Games

Position

Tics

2.4

15.0*

36.7*

No Tics

5.0

27.7*

45.1*

60/ 7 2

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