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Dictionary of the History of Ideas | ||

*7. Basic Concepts: Game Theory and Social Struc-
ture.* The description of a game of strategy involves

a number of new concepts. Obviously, games are first

classified by virtue of the number of players or partici-

pants: 1, 2,...,

*n.*Second, when the winnings of some

are compensated exactly by the losses of others, the

game is zero-sum. The sum can also be positive (when

all gain), negative (when all lose), constant, or variable.

Games are “essential,” when there is an advantage in

forming coalitions, which can happen even in zero-sum

games, but only when

*n*≧ 3. This expresses advan-

tages in cooperation; it can develop even when there

are only two players, but then the game has to be

non-zero sum. Games are “inessential” when there is

no such advantage, in which case each player proceeds

independently for himself. Note, however, that he still

does not control the outcome for himself by his actions

alone; the “others” are always present, and sometimes

also nature is present as an agent.

Games are played according to *rules* which are

immutable and must be known to the players. A rule

cannot be violated since then the game would cease;

it would be abandoned or go over into another

game—if that is possible. A tacit assumption is that

players agree to play. They do this without doubt when

playing for pleasure. When games are used as models,

it may however happen that one's participation in the

modeled situation is not voluntary. For example, a

country may be forced into a military conflict; or, in

order to survive and to earn a living a person may

have to engage in certain economic activities. Games

come to an end; the rules provide for this. Again in

the modeled situation one play of a certain game—a

play being the concrete, historical occurrence of a

game—may follow another play of the same game, or

the play of one game may follow that of another game

and so on. Sometimes it is possible to view such se-

quences as supergames and to treat them as an entity.

In some games as in chess the players are perfectly

informed about all previous moves, in others they have

only partial information about them. Sometimes the

players are not even fully informed about themselves,

as e.g., in bridge, which is a two-player game, but each

player (e.g., North and South) plays through two rep-

resentatives. In this case information about oneself and

to oneself is only disclosed by the manner of playing.

In addition chance enters, since the cards are dealt at

random. This example gives a first indication of the

great complexity that confronts any attempt at theory

even under simple conditions. In poker, bluffing is

added, as the pretense by some players of having cer-

tain sets of cards can become an element in the play.

of possible bluffs by the other players, how to surmise

bluffs by others, and many more such factors.

The rules normally specify sequences of moves,

countermoves and tell when the game has terminated.

It is possible to view games described in this “*extensive*”

*form* strictly equivalently by introducing the notion of

strategies, which are the complete plans made up by

each player for such series of moves. Games are then

described in the “*normalized*” *form* and it is thus that

they shall be treated in what follows. In choosing a

pure strategy, i.e., by specifying the precise complete

course of action, the player may or may not be at a

disadvantage in expected values if he has to make his

choice openly before the other player makes his choice.

If there is no disadvantage, then the game has a *saddle
point* in the payoff matrix, for if the first player chooses

his optimal strategy, then no matter what the second

player may do, he cannot depress the first one's ex-

pected payoff below a certain value which is the value

of the saddle point. Exactly the same is then true

conversely for the second player. Games having saddle

points in

*pure*strategies are strictly determined. In

these cases there is no value of information flowing,

voluntarily or involuntarily, from one player to the

other. Each behaves rationally if in pursuit of his in-

tended maximum benefit he chooses his pure strategy

so that he is guaranteed at least as much as corresponds

to the value of the saddle point. If the other player

deviates from his optimal strategy, i.e., behaves “irra-

tionally,” the first one can only gain.

However, games usually have no saddle points in

pure strategies. A player forced to disclose his pure

strategy would then be at a disadvantage and the

question arises whether there is at all an optimal way

of playing. The attempt of opponents to outguess each

other by the chain of thought: I think that he thinks

that I think he thinks... will never lead to a resolution

of the dilemma exemplified by the Sherlock Holmes—

Professor Moriarty pursuit case (Morgenstern, 1928),

which corresponds exactly to a qualified game of

matching pennies. How then shall one proceed?

John von Neumann proved in 1928 that for these

games which are not strictly determined a saddle point

*always* exists if the players resort to proper so-called

*mixed strategies:* the now famous minimax theorem.

A mixed strategy means that instead of selecting a

particular pure strategy from the whole set of all

available pure strategies, the player must *assign* a

specific probability to each one of them such that at

least one will be played. A properly chosen chance

device will then determine the strategy actually

chosen. The player himself will not know which strat-

egy he will actually play; hence he cannot be found

out by his adversary and he cannot even accidentally

disclose his choice, which if he did would be disastrous.

The fundamental “Minimax Theorem” assures that

the player, using mixed strategies, can always find a

correctly computed optimal mixed strategy to protect

himself (minimizing the worst in expected values that

can happen to him) precisely as in strictly determined

games he can identify, and even announce, his optimal

*pure* strategy. The original proof of this theorem in-

volved very advanced methods of topology and func-

tional analysis. The theorem is of outstanding impor-

tance and has had wide ramifications: the original

theory of games for any number of players rests on

it. Though the implications of the theorem have often

been found uncomfortable (and were termed “pessi-

mistic”), it stands unchallenged. As is often the case

in mathematics, other simpler proofs have later been

offered, by von Neumann himself as well as by others,

such as using concepts from the theory of convex

bodies, a theory which in turn has greatly benefited

from these developments.

It is necessary to examine the significance of the use

of mixed strategies since they involve probabilities in

situations in which “rational” behavior is looked for.

It seems difficult, at first, to accept the idea that

“rationality”—which appears to demand a clear, defi-

nite plan, a deterministic resolution—should be

achieved by the use of probabilistic devices. Yet pre-

cisely such is the case.

In games of chance the task is to determine and then

to evaluate probabilities inherent in the game; in games

of strategy we *introduce* probability in order to obtain

the optimal choice of strategy. This is philosophically

of some interest. For example, the French mathe-

matician É. Borel asserted that the human mind cannot

produce random sequences of anything; humans need

to invent devices which will do this for them. Borel

did not and could not give a mathematical proof be-

cause his assertion is not a mathematical one. It is

noteworthy, incidentally, that recent studies of the

brain seem to indicate, however, that some uncertainty

and randomness in its operation are essential for its

proper functioning.

The identification of the correct probabilities with

which to use each pure strategy is a mathematical

task—sometimes computationally formidable—and is

accomplished by use of rigorous theory. Putting these

probabilities to use requires then a suitable physical

generating device which always can be constructed.

In practice players may merely approximate such

devices where these would tend to be very compli-

cated. In some cases they will produce them exactly,

as in matching pennies. In this game, on matching

either heads or tails, one unit will be paid to the first

first to the second. This game, clearly zero-sum and

of complete antagonism between the two players, is

not strictly determined. Hence each will protect him-

self against being found out. As is well known the

optimally correct way of playing is for both players

to toss his coin simultaneously with the other player,

which is equivalent to choosing each of the only two

available strategies with probabilities 1/2, 1/2. The coin

itself when tossed will either show heads or tails pre-

cisely with the required probabilities.

The manner in which this game is played makes

it appear to be a game of chance, but in reality it is

one of strategy. This incidentally illustrates a grave

difficulty of giving correct descriptions of social events!

The probabilities of 1/2, 1/2, have to be changed if there

should be a premium, say, on matching on heads over

matching on tails. The new probabilities that secure

the saddle point can no longer be guessed at or be

found intuitively; they have to be computed from the

theory, so quickly does the true, mathematical analysis

which requires the full use of the complex theory have

to be invoked. When the number of strategies goes

beyond two the computational difficulties increase at

any rate; the computations may become impossible

even when the game is strictly determined, as in chess,

where there are about 10120 strategies. The existence

proofs of optimal strategies are valid nevertheless.

The problem now arises how a *social equilibrium*

can be described when there are more than two deci-

sion makers. Here only the most basic concepts can

be indicated as a full description would require much

space and intricate mathematical analysis. The struc-

ture is this: when in a zero-sum game *n* ≧ 3, then the

possibility of cooperation among players arises, and

they will form coalitions wherever possible. In order

to be considered for inclusion in a coalition a player

may offer side payments to other players; some may

be admitted under less favorable terms (when *n* > 3)

than those set by the initial members of the coalition

and the like. When a coalition wins, the proceeds have

to be divided among the partners and these then find

themselves in the same kind of conflict situation which

arises for the players of a zero-sum two-person game.

The totality of all payments to all players is an

“imputation.” In order to determine an equilibrium it

appears to be necessary to find a particular imputation

that is “better,” that is, more acceptable, from among

all possible ones than any other. Such an imputation

then “dominates” all other imputations. But that would

be the case in inessential games. Only for those is there

a *unique* social optimum, a division of the proceeds

of the game played by society which cannot be im-

proved upon and which therefore is imposed or im

poses itself upon society as the best stable arrangement.

But since cooperation is a basic feature of human

organization these games are of little interest. No such

single imputation exists for essential *n*-person games.

Domination is then not transitive, thus reflecting a

well-known condition of social arrangements in which

circularity often occurs (as, for instance, in the relative

values of teams in sports).

Thus the hope of finding a uniquely best solution

for human affairs is in vain: there is no stability for

such arrangements. Political, social, and economic

schemes have been proposed under the tacit, but fre-

quently even open, assumption that this is possible

when men organize themselves freely. Only the iso-

lated individual or a fully centralized (usually dicta-

torial) society can produce a scheme that it considers

better than any other and that it hopes to be able to

enforce.

Thus there is, in general, no “best” all dominating

scheme of distribution or imputation; but there may

be a number of imputations which do *not dominate
each other* and which among them dominate everything

else. Such imputations, therefore, must be considered

by society. They form a special “stable set,” originally

called the “solution set.” Any one of the imputations

belonging to this stable set is a possible, acceptable

social arrangement.

A stable set is precisely a set *S* of imputations, no

one of which dominates any other, and such that every

other possible imputation not in *S* is dominated by

some imputation in *S.* (Technically, the imputations

belonging to each stable or solution set are not even

partially ordered and, *a fortiori,* the elements of this

set are not comparable with one another.)

The stability that such a set possesses is unlike the

more familiar stability of physical equilibria. For no

single imputation can be stable by itself; it can always

be disturbed, not by “forces” (as a physical equilibrium

could be), but by the proposal of a different arrange-

ment by which it is dominated. Such a proposal must

necessarily lie outside of *S.* But for every such proposal,

there is always a counter-proposal which dominates

the proposal, and which lies in *S.* Thus a peculiar,

delicate but effective equilibrium results which has

nothing to do with the usual equilibria of physics; the

process of proposal and counter-proposal always leads

to an imputation in *S.* Indeed the present notion differs

so profoundly from the usual ideas of stability and

equilibrium that one would prefer to avoid even the

use of the words. But no better ones have yet been

found.

There may exist, even simultaneously, different,

conflicting solution sets or standards of behavior, *each*

one with any number of different imputations, always

those within the respective solution sets are merely

*alternative*to each other; they are not in fundamental

conflict as are the different standards.

Clearly, it is difficult to identify solutions, i.e., sets

of imputations with the required properties, even from

the whole set of all possible imputations. In 1968 W. F.

Lucas made the important discovery of a game of

10 players that has no solution (in the so-called charac-

teristic function form). The question is open whether

this is a rare case and what modifications in concepts

and methods may be necessary to assure solvability.

In all other cases so far investigated solutions have been

found.

These admittedly difficult notions emerge from the

rigorous mathematical theory whose empirical basis is

formed by facts that are not questioned even by current

social and economic theory, though these theories have

not rendered a successful account of the nature of

decision-making. The lack of identification of a single

settlement or imputation is not a deficiency of game

theory. Rather there is herein revealed a fundamental

characteristic of social, human organization which

cannot be described adequately by other means.

In the light of these considerations one of the stand-

ard concepts currently used in describing a social op-

timum, the so-called Pareto optimum (formulated by

v. Pareto, 1909) appears at best to be an oversimplifi-

cation. That notion says that the optimal point is

reached when no one can improve his position without

deteriorating that of others. What is lacking in that

formulation, among other things, is to account for

nonuniqueness, uncertainty, deceit, etc., hence a more

comprehensive frame within which individuals make

decisions that guarantee a precisely defined but differ-

ent stability (Morgenstern, 1965).

The appearance of novel and complicated notions

is due to a mathematical analysis that is germane to

the subject matter and has nothing to do with any

ideological or other conception of society. The mathe-

matical analysis unravels implications of some gener-

ally accepted facts and observations, axiomatically

stated, and then leads via the fundamental minimax

theorem to the discovery of relationships in the empir-

ically given social world which without the aid of the

new theory have either escaped notice altogether or

were at best only vaguely and qualitatively described.

Since inventions are possible in the social world this

process is an unending one, which means that new

concepts and theorems have arisen and more are bound

to arise. For example, new concepts of solution struc-

ture have emerged. It may even happen that social

organizations are proposed that have no stable sets;

and that only work in a manner that is quite different

from the original intentions, even though these may

have involved sound philosophical and ideological

principles.

Physics studies given physical facts and is *not* con-

fronted with this type of creation; it faces in this sense

a static world (though it may be expanding!) as far as

we can tell. Not all *given* physical facts are known;

new effects are constantly being discovered but it is

doubtful that they are currently being created, while

it is certain that novel forms of social organization are

being and will be invented. We know that the life

sciences are also, and in fact more clearly, confronted

with the evolutionary creation of new phenomena, not

only with their discovery, as in the case of physics.

But on the other hand, the time spans which are neces-

sary for genetic change are so great as to make this

concern with the creation of new phenomena (other

than breeding of new plants and animals) to have as

yet no practical importance in this context.

This goes to show that the intellectual situation in

the social sciences is disquieting even when one ab-

stracts from the further complication presented by the

existence of frequently changing ideologies.

There is thus no hope to penetrate into the intricate

web of social interdependencies by means of concepts

derived from the physical sciences, although thinking

along such lines still dominates. This is partly due to

the immense success of physics and the slow develop-

ment even of any proper description of the social

world. Where this description has used abstract con-

cepts these were mainly taken from the physical sci-

ences. Thus a recasting of the records of past social

events is necessary. The two movements of description

and theory formation are as inseparably interrelated

as they were in physics and astronomy where the

analysis of simple processes, for instance, that of a

freely falling body, led to mechanics and to the dis-

covery of the appropriate tool of the differential cal-

culus. Fate will not be easier for the social sciences

and in this methodological situation lies the deep phil-

osophical significance of game theory, i.e., of the new

analyses of human decision-making and the interlocking

of such decisions.

To give but one illustration: a formal system of

society may be fully symmetric, i.e., give each member

exactly the same possibility, such as laissez-faire, and

thereby have provisions of complete freedom and

equality. But the possibility of cooperation via coali-

tions, agreements, and the like produces nonsymmetric

arrangements so that the intent of the law-maker can-

not be maintained without forbidding coalitions which

then would run afoul of the principle of freedom.

While this asymmetry is sometimes not very hard to

discover there are other, more elusive cases; but in

first yield results which are also obtainable from com-

mon sense experience. However, theory must in addi-

tion be able to predict the emerging structures and

show how the inner nature of social processes works.

Dictionary of the History of Ideas | ||