There are probably no concepts which currently occupy the reflective man as much as cooperation, conflict, and competition. Nevertheless the vast literature on these concepts is almost devoid of precise definitions. For example, more often than not "conflict" and "competition" are used interchangably. Furthermore, little attention has been given to measuring these relations. This note is directed toward reducing these deficiencies.

The definitions to be developed are based on the concept of the purposeful state of a decision maker (individual or group). I have used the "purposeful state" in other places as a basis for defining (1) communication and related concepts¹, and (2) best decisions in the context of decision theory2. It is not surprising that communication theory, decision theory, and a theory of cooperation-conflict can be built on a common conceptual foundation.

The definition of a purposeful state itself requires use of the following

/ = an individual or group whose behavior is observable.

N = the individual's environment.

d(1 i m) = the courses of action available to the individual,
defined so as to be exclusice and exhaustive.

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* Director, Operations Research Group, Case Institute of Technology, Cleveland, Ohio.

Oj (1 ;' n) = the possible outcomes of the courses of action,
defined so as to be exclusive and exhaustive.

P; = P (Ci\l,N) = the probability that / will select & in N.
(Note that 2 Pi= 1.0.)
! = 1

Eij = P(Oj\&,I,N) = the probability that Oj will occur if / selects
d in N: the efficiency of /'s use of Ci for Oj
uxN.
(Note that 2 Eij = 1.0.)
7 = 1

Fj = the relative value of Oj to / in N.

An individual (or group) can be said to be in a purposeful state if the
following conditions hold:

(1) There are at least two courses of action, Ci and Cz, for which
Pi and P2 are greater than zero: f has at least two potential
courses of action in N.

(2) There is at least one outcome, Oi, for which V\ > 0 for f.

(3) Relative to at least one outcome, Oj, for which Vj > 0, Eij > 0,
E2l > 0, and E\j 4= £27; that is, f's choice can "make a difference".

In ordinary English these conditions state that f is in a purposeful
state if he wants something and if he can pursue it by alternative means
which have some, but unequal, efficiency with respect to what he wants.

The expected value of a purposeful state (S), then, is

if the Vj's are independent. If they not, the O/s can be combined by a
Boolean expansion into outcome-complexes whose values are independent.

If 2 Vj = a (a is usually equal to 1.0), then
j

since 2 P{ = 1.0 and 2 En = 1.0.

The concepts cooperation, conflict, and competition, involve interactions
between individuals and for groups. Therefore, we need the
following expressions:

EVi (S\h) = The expected value of S to h if h is present in JV.

EVi (S\h') = The expected value of S to h if h is not present in N.

The degree of cooperation of h with h (DC2I) can now be defined as

and the degree of cooperation of h with h as

These quantities measure the difference in the expected value of the state to one party with and without the other party present. DC2I and DCI2 are not necessarily equal, a fact that I shall use below. If max EV (S) = a, the degree of cooperation also has a maximum value of a. Its minimum value is —a. Negative values of the degree of cooperation represent degrees of conflict. If this measure is equal to zero, say DC2I — 0, this means the value of the state to h is independent of h.

Now let us consider the significance of DCn + DC2I. Tris means that one of the parties is exploiting the other. If DCI 2 > D2l, then h is exploiting h, if DC2I < D2l, then h is exploiting h. The degree to which f1 exploits h is

If this negative, then f1 is being exploited by h. It is apparent that

If DCI2 and DCI2 are both positive quantities, but unequal, then the exploitation is called benevolent, since both parties benefit, though unequally. If DCI2 and DC2I are both negative quantities, but unequal, then the exploitation is called malevolent, since both parties suffer. If one is positive and the other is negative we have what I suppose might be called normal exploitation.

If the minimum and maximum of the degree of cooperation are —a
and +a, respectively, then the minimum and maximum degree of exploitation
are —2a and +2a.

Now where does competition come in? The most useful suggestion I have found in the literature is that competition is conflict in accordance with rules; that is, regulated conflict3. On this basis, for example, we can distinguish between a prize fight (as competition) from a street brawl

(as conflict). What function do the rules have? Clearly, they must be intended to constrain the conflict to a type which serves some purpose. This suggests that competition involves both conflict and cooperation, but how?

Consider three individuals or groups - h, h, and Is - of whom two - 7i and h - are in conflict with each other. Now if this state of conflict increases f3's expected value of his state, then h and h are competing relative to h. h and h may be "competing" business firms and h their consumers; or h and h may be two prize fighters and h the audience. Rules or laws control such conflicts to assure their service to the "third" party.

But clearly two people on a tennis court or on opposite sides of a chess board can compete without an audience. They can, but to see how they can we must look inside their states. Suppose 7i and h are in conflict with respect to two objectives (Oi = 7i wins, and O2 = h wins). Suppose further that both 7i and h pursue a third obpective (O3 = recreation) which is efficiently served by the conflict relative to O\ and 02. Then h and h can be said to be competing intensively. Competition with respect to a "third" party is extensive. Of course, f1 and h may be competing both intensively and extensively.

This concept of competition cannot be represented by a single measure. The degree of competition between f1 and h clearly depends on DCn and DC2I and would increase as these terms decrease (since negative values represent conflict), and hence as their sum decreases. It also depends on how "even" the conflict is, that is, the competition would be more "intense" as the difference between DCI2 and DC2I decreases, and, hence, as the degree of exploitation decreases. Finally it also depends on how efficiently the conflict serves the "third" party or objective; that is, on the degree of cooperation with respect to this party or objective. I can see no way at present of conveniently combining these considerations into a single measure.

REFERENCES:

'■ Russell L. Ackoff: Towards a Behavioral Theory of Communication. Management
Science, Vol. 4, No. 3, April, 1958.

2. Russell L. Ackoff: Scientific Method: Optimizing Applied Research Decisions. John
Wiley & Sons, New York, 1962.

3- Daniel Katz and R. L. Schanck: Social Psychology. John Wiley & Sons, New York.
1938.