Conflict, Negotiation, and
General-Sum Game Theory

Joseph George Caldwell

25 August 2001

Copyright © 2001 Vista Research Corporation. All rights reserved.

Table of Contents

II.
Background Information: Nash's Bargaining Solution to a General-Sum Game

III.
The Special and General Bargaining Solutions

"All too often in OR dealing with military problems, war is viewed as a zero-sum two-person game with perfect information. Here I must state as forcibly as I know that war is not a zero-sum two-person game with perfect information. Anybody who sincerely believes it is a fool. Anybody who reaches conclusions based on such an assumption and then tries to peddle these conclusions without revealing the quicksand they are constructed on is a charlatan....

"There is, in short, an urgent need to develop positive-sum game theory and to urge the acceptance of its precepts upon our leaders throughout the world."

-- Joseph H. Engel, Retiring Presidential Address to the Operations Research Society of America, October 1969

This article is a reprint, with minor notational changes, of
the report, *Conflict, Negotiation, and General-Sum Game Theory*, by J. G.
Caldwell, published 19 May 1970 by Lambda Corporation as Paper 45, A Technical
Report Submitted to the Office of Naval Research Under Contract
N00014-69-C-0282.

__Conflict, Negotiation, and
General-Sum Game Theory__

To date, most war gaming, weapons allocation, and force procurement models have been developed either using zero-sum payoffs (one player's loss is the other's gain), or ignoring the relationship of conflict to negotiation. This situation prevails in spite of the fact that two-player general-sum (positive-sum) game theory has been developed to the point where both of these aspects can be taken into account. One reason for this situation is that, although good analytical solutions are theoretically determined, explicit computation of the solution is in general very difficult.

This paper examines the problem of determining a
computationally tractable general-sum game-theoretic solution to war, taking
into account the effect of the threat of war on negotiations. An idealized situation that seems to possess
the essential characteristics of most actual situations is examined. If A and B denote the payoff matrices of the
general-sum game between the players, then it is shown in general that optimal
threat strategies for conducting the war are given by the optimal solutions to
the zero-sum game A-cB, where the constant c is such that this game has value k
(a constant depending on the nature of the particular problem). Furthermore, if all payoffs are negative, if
the losses associated with combat are quite large compared to those associated
with negotiation, and if the losses from negotiation are close to zero, then k
is approximately equal to zero. (In
addition, the optimal strategies correspond approximately to the solution of
the ratio game A/B, and this game has value c.)
If, further, the two players' combat losses tend to be proportional,
then c is approximately equal to V_{A}/V_{B}, where V_{A}
and V_{B} are the values of the zero-sum matrix games A and B. That is, approximately optimal strategies for
conducting the war are given by the solutions to the zero-sum game (V_{B}/V_{A})^{1/2}
A - (V_{A}/V_{B})^{1/2} B. Thus, the difficult mathematical problem of
solving the general-sum game (which is felt to represent war more adequately
than a zero-sum formulation) is reduced to a simpler mathematical problem,
viz., the solving of a particular zero-sum game derived from the general-sum
game.

This paper will summarize current theory relating to the solution of finite two-player general-sum games, and examine certain situations which appear to relate to war and negotiation and which suggest reasonable criteria for conducting a war.

There is a well-established procedure for playing games in which the interests of the players are diametrically opposite (so that one player's loss is the other player's gain). Such a game is called a constant-sum, or zero-sum, game. For "rational" players, optimal strategies are available; these strategies are given by the minimax (von Neumann-Morgenstern) solution to the game.

A game in which the interests of the two players are not necessarily exactly opposite is called a general-sum game. The theory of general-sum games does not, in general, provide optimal strategies for the players. General-sum games are classified into two types: noncooperative and cooperative. In a noncooperative game, any type of collusion, such as side payments and correlated strategies, is not permitted between the players.

If we accept that there is a relationship between the
negotiations conducted by two nations and the wars occurring between those
nations, it is reasonable to view negotiation and conflict in a combined
context. The interests of two nations
are not in general directly opposed, i.e., it is possible for __both__ to
profit as they change their strategies.
It is hence reasonable to examine negotiation and conflict as a
general-sum cooperative game, in the hope of determining some useful
information concerning good criteria for negotiation and waging war.

A finite zero-sum game is characterized by a payoff (or
loss) matrix, A, whose (i,j)-th element a_{ij} indicates the gain to
player 1 (loss to player 2) if player 1 selects his i-th strategy and player 2
selects his j-th strategy. If, in
playing the game, player 1 selects his i-th strategy with probability xi, then we
say that he is employing a mixed strategy; we denote this strategy by the
vector __x__ = (x_{1},x_{2},...,x_{n})'. If player 1 selects strategy __x__ and
player 2 selects strategy __y__, then the gain to player 1 (loss to player
2) is __x__A__y__'.

If we assume that player 1 wishes to maximize his gains, and that player 2 wishes to minimize these gains (his losses), then it can be proved that there are optimal strategies, x*, y* for both players, defined by

__x__*A__y__*' = max(__x__)
min(__y__) __x__A__y__' = min(__y__) max(__x__) __x__A__y__'
,

where max(__x__) denotes maximization with respect to __x__
(of the following function of x) and min(__y__) denotes minimization with
respect to __y__.

A finite general-sum game is characterized by a pair of
payoff (or loss) matrices, A and B. The
element a_{ij} of the matrix A specifies the gain to player 1 if he
selects strategy i and player 2 selects strategy j. The element b_{ij} of the matrix B
specifies the gain to player 2 if player 1 selects strategy i and player 2
selects strategy j.

Unlike the situation in the constant-sum game, there is no
unique "natural" solution to a general-sum game. The most reasonable approach developed to
date, however, seems to be that developed by John Nash. Nash actually developed two solutions, which
he calls the __special bargaining solution__ and the __general bargaining
solution__, respectively. We shall
briefly describe these solutions. See
reference 2 for further details.

It is convenient to describe the solutions in terms of a
graph showing the __payoffs__ (u,v) accruing to the two players,
corresponding to various strategies (__x__,__y__). Both solutions are defined in terms of a
point (u*,v*) -- called the __status quo point__ -- which corresponds to the
least (expected) payoff that a player is willing to accept. How this point is chosen is the difference
between Nash's two solutions. In
general, if we do not specify the nature of the status quo point, we shall
simply refer to the corresponding solution as a __bargaining solution__ to
the game.

Let us denote by S the __feasible set__, or set of all
possible joint payoffs (u,v) realizable by the two players acting
together. If A and B denote the payoff
matrices of the two players, then S is the set of convex combinations of the
set of points (a_{ij},b_{ij}).
Given the status quo point (u*,v*), Nash proved that if there are any
points (u,v) in S, then there exists a unique point (u_{b},v_{b})
(called the bargaining solution) that satisfies the following six axioms:

1. (u_{b},v_{b}) ≥ (u*,v*). (That is simply a statement of the condition
that neither player will accept a payoff less than u* or v*, respectively.)

2. (u_{b},v_{b}) is
in S.

3. If (u,v) is in S and (u,v) ≥ (u_{b},v_{b}),
then (u,v) = (u_{b},v_{b}).
(This axiom is called the condition of __Pareto optimality__.)

4. If (u_{b},v_{b})
is in T, a subset of S, and (u_{b},v_{b}) is the bargaining
solution for the set S when (u*,v*) is the status quo point, then (u_{b},v_{b})
is the bargaining solution for the set T with the same status quo point. (This axiom is called the condition of __independence
of irrelevant alternatives__. It says
that if both players reject feasible alternatives, then those alternatives
cannot affect the solution, so long as the status quo point remains the same.)

5. The solution is linearly invariant. That is, if T is obtained from S by the linear transformation

u' = a_{1} u + b_{1}

v' = a_{2} v + b_{2}
,

then, if (u_{b},v_{b})
is the bargaining solution for S with status quo point (u*,v*), then (a_{1}u
+ b_{1}, a_{2}v + b_{2}) is the bargaining solution for
T with status quo point (a_{1}u* + b_{1}, a_{2}v* + b_{2}).

6. The solution is symmetric with respect to the players. That is, if S is such that

(u,v) is in S if and only if (u,v) is in S ,

and (u_{b},v_{b})
is the bargaining solution for S with status quo point (u*,v*), where u* = v*,
then u_{b} = v_{b}.

Furthermore, Nash proved that if there are any points (u,v)
in S that satisfy u > u* and v > v*, then the bargaining solution is the
point (u_{b},v_{b}) that maximizes the function

g(u,v) = (u - u*) (v - v*)

subject to u ≥
u* (or v ≥ v*) and (u,v)
in S. (If there is no point in S such
that u > u*, then u = u* for (u,v) in S, and the solution is (u_{b},v_{b})
where u_{b} = u* and

v_{b} = max (v such that
(u,v) is in S, v ≥ v*) v .

If there is no
point in S such that v > v*, then v = v* for (u,v) in S, and the solution is
(ub,vb) where vb = v* and

u_{b} = max (u such that (u,v) is in S, u ≥ u*) u .)

The geometric
nature of the solution is indicated in Figure 1. Note that if A and -cB are approximately
equal, then the points (a_{ij},b_{ij}) of S tend to lie on a
line of slope -c, and the payoff possible through negotiation differs little
from that if the players do not negotiate.
If A = -cB, then we have the zero-sum situation, of course, in which
negotiation is of no use (one player's loss equals the other's gain). If A is approximately equal to cB, then the
points (a_{ij},b_{ij}) of S tend to lie along the line u = cv,
and there is a great deal to be gained by negotiation. If A = cB, then the interests of the players
coincide, and there is but one point in the negotiation set.)

We shall now
describe how the status quo point is chosen.
In the special bargaining problem, u* and v* are the payoffs that the
players can obtain by unilateral action, whatever the other player does; i.e.,
they are the minmax values of the respective matrix games, A and B:

u* = max(__x__) min(__y__) __x__A__y__'

v* = max(__y__) min(__x__) __x__B__y__' .

Note that if A =
-cB, then the general-sum game reduces to a zero-sum game.

In the general
bargaining problem, u* and v* (called threat payoffs) are chosen as
follows. First, we describe some
characteristics of the bargaining solution.
Observe from the figure that, from axiom 3, whatever the values of u*
and v*, the solution (u_{b},v_{b}) is restricted to that part
of the boundary of S such that u and v cannot simultaneously be increased. By changing u* and v*, we move the solution
(u_{b},v_{b}) along the Pareto-optimal boundary, and one player
can profit only if the other player loses.
(This tradeoff is not linear unless the Pareto-optimal boundary is
linear.) The objectives of the two
players in choosing their threats (u* and v*) are directly opposed, and it can
be proved that there are in fact optimal choices for u* and v*.

If the
Pareto-optimal boundary of S is linear (au + bv = 1), then the strategies
corresponding to the optimal threats are the maxmin solutions to the (zero-sum)
matrix game

aA - bB .

This is easy to
see. Subject to the constraint au + bv =
1, the values of u and v that maximize (u - u*) (v - v*) are

u_{b} = (1/a + [u* - (b/a)v*])/2

and

v_{b} = (a/b) (1/a - [u* - (b/a)v*)/2 .

Now u* = __x__A__y__'
and v* = __x__B__y__' are the "status-quo" payoffs, or
"threat" payoffs, to the respective players if they choose threat
strategies x and y. Hence, in terms of __x__
and __y__, we have

u_{b} = (1/a + [__x__(A - (b/a)B)__y__'])/2

v_{b} = (a/b) (1/a - [__x__(A - (b/a)B)__y__'])/2 .

Clearly, player 1
wishes to maximize

__x__(A - (b/a)B)__y__'

while player 2
wishes to minimize the same quantity.
Hence the optimal threat strategies x and y are the minmax values __x___{t}
and __y___{t} to the (zero-sum) matrix game A - (b/a)B, or,
equivalently, to the matrix game aA - bB.

The situation is
depicted in Figure 2. We define V_{A}
and V_{B} to be the values of the two players' zero-sum games:

V_{A} = max(__x__) min(__y__) __x__A__y__'

V_{B} = max(__y__) min(__x__) __x__B__y__' .

We define

u_{t}* = __x___{t}A__y___{t}'

and

v_{t}* = __x___{t}B__y___{t}'

where xt and yt are
defined by

__x___{t} (aA -bB) __y___{t}' = max(__x__)
min(__y__) __x__ (aA - bB) __y__' .

For the graph, we
have assumed that V_{A} = V_{B}, and that a is sufficiently
smaller than b so that aA - bB is approximately equal to -bB. Hence v_{t}* is approximately equal
to V_{B}, but u_{t}* could vary considerably from V_{B}. An interpretation of this situation is as
follows. Since, relative to the value V_{B}
of his "threat" game B, player 2 has less to gain or lose through
negotiations than player 1, he concentrates on holding player 1 to his minimum
gain, rather than paying much attention to his own gain or loss. From the graph, we can see that this attitude
is reasonable, since it is clear that fluctuations in the threat payoff u_{t}*
have much less effect on player 1's negotiated payoff u_{b} than comparable
fluctuations in v_{t}* have on v_{b}.

If the
Pareto-optimal boundary of S is nonlinear, then finding the optimal threats is
not as straightforward as above. We
shall investigate a situation which seems analogous to reality; namely, one in
which the losses if the threats (war) are carried out are far greater than the
gains from negotiation.

We assume that the
negotiation set (Pareto-optimal boundary of S) can be approximated by the
equation uv = c. Then we want to find
values for u and v that maximize (u - u*)(v - v*) subject to uv = c. These values are

u_{b} = -(cu*/v*)^{1/2}

v_{b} = -(cv*/u*)^{1/2} .

As before, u* = __x__A__y__'
and v* = __x__B__y__' are the threat payoffs to players 1 and 2 if they
choose threat strategies __x__ and __y__, respectively. Clearly, player 1 wishes to minimize

u*/v* = __x__A__y__'/__x__B__y__'

while player 2 wishes to maximize this quantity. (This curious situation, in which player 1
wishes to minimize the ratio of his gains to those of player 2, occurs because
all payoffs are negative. Were the
negotiation set in the positive quadrant, there would be a more natural
determinant of the optimal threat strategy.)
Hence the optimal threat strategies for the players are the strategies __x___{t}
and __y___{t} corresponding to

min(__x__) max(__y__) __x__A__y__'/__x__B__y__'
.

Schroeder (Reference 2) suggests an iterative procedure for
solving this ratio game. He notes that,
if c denotes the value of the game, then the zero-sum game A - cB will have
value zero; furthermore, optimal strategies for the zero-sum game A - cB are
optimal for the ratio game. He suggests
an iterative procedure for determining c (and hence optimal strategies __x___{t}
and __y___{t} for the ratio game).

As an alternative to the iterative solution, the following
approximate solution is suggested.
Maximizing u*/v* is equivalent to maximizing ℓn u*/v* (where ℓn
denotes the natural logarithm function).
Let us denote the zero-sum solutions to the games A and B by (__x___{A},__y___{A})
and (x_{B},y_{B}), respectively. Then for __x__A__y__' close (in a
percentage sense) to V_{A} = __x___{A}A__y___{A}'
and __x__B__y__' close to V_{B} = __x___{B}B__y___{B}'
we can write

ℓn u* = ℓn
__x__A__y__'

= ℓn (__x___{A}A__y___{A}' + __x__A__y__'
- __x___{A}A__y___{A}')

= ℓn (V_{A} + __x__A__y__' - V_{A})

approximately = ℓn V_{A} + (1/V_{A})(__x__A__y__'
- V_{A})

= ℓn V_{A} + (1/V_{A})(__x__A__y__') - 1

and similarly,

ℓn v* approximately = ℓn V_{B} + (1/V_{B})(__x__B__y__')
- 1

so that

ℓn (u*/v*) approximately = ℓn (V_{A}/V_{B})
+ (1/V_{A})__x__A__y__' - (1/V_{B})(__x__B__y__') ,

and hence, it seems
reasonable to approximate

min(__x__) max(__y__) ℓn (u*/v*) approximately = min(__x__)
max(__y__) (ℓn (V_{A}/V_{B})

+ (1/V_{A})xAy' - (1/V_{B})__x__B__y__')

or

min(__x__) max(__y__) ℓn (u*/v*) approximately = ℓn (V_{A}/V_{B})

+ min(__x__) max(__y__) __x__((1/V_{A})A
- (1/V_{B})B)__y__' .

Hence near-optimal
threat strategies __x___{ta} and __y___{ta} are those
strategies corresponding to

min(__x__) max(__y__) __x__((1/V_{A})A - (1/V_{B})B)__y__'
,

i.e., to the
solution of the (zero-sum) matrix game (1/V_{A})A - (1/V_{B})B. Hence we see that c is approximately equal to
V_{A}/V_{B}, provided that the losses from combat are very
large compared to the losses associated with negotiation.

The situation is
described in Figure 3. As usual, V_{A}
and V_{B} are the values of the zero-sum games A and B. We define

u_{ta}* = __x___{ta}A__y___{ta}'

and

__v___{ta}* = __x___{ta}B__y___{ta}'

where __x___{ta}
and __y___{ta} are defined by

__x___{ta} ((1/V_{A})A - (1/V_{B})B) __y___{ta}'
= min(__x__) max(__y__) __x__ ((1/V_{A})A - (1/V_{B})B)
__y__' .

If we assume that V_{A}
= V_{B}, then __x___{ta} and __y___{ta} are the
zero-sum solutions to the matrix game A - B.

The preceding approximate solution is
noted simply in passing. The really
significant result of this section is that there does exist a constant c such
that the optimal threat strategies are solutions of the zero-sum game A -
cB. As noted earlier, the constant c is
the solution of the ratio game A/B and it therefore has the property that the
zero-sum game A - cB has value zero.

It is interesting
to compare the threat strategies of the general-sum game in the case of a
linear negotiation set to the approximate threat strategies in the case of a
hyperbolic negotiation set. The zero-sum
games that determine the threat strategies in these two cases are

aA - bB

and

(1/V_{A})A - (1/V_{B})B ,

or

(a/b)^{1/2} A - (b/a)^{1/2} B

and

(V_{B}/V_{A})^{1/2} - (V_{A}/V_{B})^{1/2}
B ,

respectively.

It is noted that
the derivations of this section are dependent upon the assumption that the
center of the hyperbola defining the negotiation set is located at the origin,
and that all payoffs are negative. If
this assumption is removed, then c is such that the matrix game A - cB has some
specified value different from zero, and c will no longer be approximately
equal to V_{A}/V_{B}.

In reality, it
would appear that there is a correlation between the losses to one player and
losses to the other. That is, the
feasible set S would tend to be "close" to the line u = cv. We shall now investigate this case.

The situation is
most easily described in terms of Figure 4.
The problem is to find u = u_{b} and v = v_{b} that
maximize

(u - u*)(v - v*)

subject to (u,v)
being on the negotiation set of the boundary

((u cos θ + v sin θ)/a)^{2} - ((-u sin θ + v cos
θ)/b)^{2} = 1
(1)

of S. The values u_{b} and v_{b}
that maximize (u - u*)(v - v*) subject to (1) can be shown to be the solutions
to the equations (1) and

(v - v*/2)^{2} - (u - u*/2)^{2} = (v*^{2} - u*^{2})/2
.
(2)

In general, the solutions u_{b}
and v_{b} will lie on the boundary of S, but not in the negotiation
set. This fact, together with the fact
that the above system of equations is difficult to solve, makes it difficult to
see what the optimal threat payoffs u_{t}* and v_{t}* are. We hence look for an approximate
solution. Note that, since S is
"long and narrow" and the threat payoffs are far from the negotiation
set, the two players will try to move u* and v* in opposite directions along
the line

u = -(1/c)v + k .

Hence player 1 is
trying to minimize

u - cv

while player 2 is
trying to maximize this same quantity.
But

u - cv = __x__ (A - cB) __y__' ,

and so approximate
optimal threat strategies are the solutions __x___{ta} and __y___{ta}
to the zero-sum (matrix) game

A - cB .

In order for S to
be long and narrow, and lie along a line of slope c, the matrix A must be
approximately equal to cB, in which case it is likely that V_{A} = cV_{B},
or c approximately = V_{A}/V_{B}. Hence the approximate optimal threat
strategies are probably close to the solution to the game

A - (V_{A}/V_{B})B

or the game

(1/V_{A})A - (1/V_{B})B ,

the same result as
in the case of the rectangular hyperbolic negotiation set.

The following
alternative method of reasoning reaches the same conclusion. The hyperbola defined by (2) asymptotically
approaches the lines

u/v = u*/v* = k

say, and

u/(v - v*) = - (u*/v*) = -k .

(In fact, if u* =
v*, the hyperbola degenerates into two crossed lines.) Only the first line is of interest, since it
will be the line that intersects (or comes closest to) the negotiation
set. Now player 1 wants the hyperbola
defined by (2) to intersect the negotiation set as far to the right as
possible. By the preceding observation,
this is approximately the same as wanting the line u/v = k to intersect as far
to the right as possible. Hence, since u
and v are both negative, player 1 wishes (approximately) to minimize the ratio
u/v = k, while player 2 wishes to maximize the same quantity. Hence approximately optimal threat strategies
are the solutions to

min(__x__) max(__y__) u/v = min(__x__) max(__y__) (__x__A__y__')/(__x__B__y__')
. (4)

Hence, from the
results in the case of a rectangular hyperbolic negotiation set, approximately
optimal threat strategies correspond to the solution of the zero-sum game A -
cB, where c is the value of the ratio game A/B and hence has the property that
the zero-sum game A - cB has value zero.
As in the case of the rectangular hyperbolic negotiation set,
approximately optimal threat strategies are those strategies xta and yta
corresponding to

min(__x__) max(__y__) __x__ ((1/V_{A})A - (1/V_{B})B)
__y__' ,

i.e., to optimal
solutions of the matrix game

(1/V_{A})A - (1/V_{B})B .

As before, the
derivations of this section assume that the center of the hyperbola defining
the negotiation set is located at the origin, and that all payoffs are
negative.

__Approximate
Solution__

We shall now extend
the results of the preceding section to the situation in which the negotiation
set has neither a hyperbolic negotiation set nor a "long, narrow"
shape. As illustrated in Figure 5, there
are certain regions of the feasible set in which it is unreasonable to expect expected
payoffs (such as the threat payoffs) to occur.
For example, in order for the expected payoffs (u*,v*) to occur in
region R_{1}, player 1 would have to act in such a way that he tends to
both minimize his own gains and those of his opponent. Similarly, in order for the expected payoffs
to occur in region R_{2}, player 2 would have to act in a similar
fashion. Neither of these situations
would correspond to those of "rational" strategies, regardless of the
form of the negotiation set. Hence,
"reasonable" pairs (u*,v*) tend to lie in a much narrower region than
the entire feasible set. As usual,
player 1 is attempting to push (u*,v*) toward R_{2}, while player 2 is
attempting to push (u*,v*) toward R_{1}. Since the "reasonable" region is
likely to be narrow, such efforts by the two players will correspond to their
attempting to push (u*,v*) in opposite directions along lines approximately
perpendicular to the narrow region. If
the "reasonable" region is fairly straight, then these perpendicular
lines are all of about the same slope, and hence the actions of the players
correspond to playing a __particular__ zero-sum game. The "reasonable" region would be
fairly straight if the players' anticipated losses from war tended to be
linearly proportional. Hence, to the
extend that this last condition (proportional losses) holds, it is not
difficult to determine (approximately) the zero-sum game to which the optimal
strategies correspond.

To be perfectly explicit regarding the
result that we have derived, let us denote the "slope" of the
"reasonable" region by c.
Then, the two players are in effect trying to move the threat payoffs u_{t}*
and v_{t}* in opposite directions along a line

u = -(1/c)v + k .

Hence, as we
observed in the preceding section, player 1 is trying to minimize

u - cv

while player 2 is
trying to maximize this same quantity.
But

u - cv = __x__ (A -cB)) __y__' ,

and so the optimal
threat strategies are the optimal solutions __x___{t} and __y___{t}
to the zero-sum (matrix) game A - cB.

If the line through
the optimal threat point and the corresponding bargaining solution passes near
the origin, we can reason as before that

c approximately = V_{A}/V_{B} .

__Exact Solution__

We shall close this
section with the observation that, even though the optimal threat strategies
are in general not the optimal solutions of a zero-sum game involving a known
linear combination of the matrices A and B, there does exist __a__ zero-sum
game whose optimal solutions are the optimal threat strategies. The usefulness of this result is conditioned
by the fact that the matrix of this particular zero-sum game may depend on the
optimal solution to the general-sum game.
We shall first present a heuristic proof of the result, and then a
precise derivation.

Let us refer to
Figure 6. We denote the optimal threat
payoffs by (u_{t}*,v_{t}*), and the corresponding bargaining
solution by (u_{b},v_{b}).
Let us denote the slope of the line joining these two points by c*;
i.e.,

c* = (u_{b} - u_{t}*)/(v_{b} - v_{t}*) .

Let the equation of this line be u -
c*v = k*. At the optimal threat point (u_{t}*,v_{t}*)
the two players are attempting to move u_{t}* and v_{t}* in
opposite directions along the line of slope -1/c* through the point (u_{t}*,v_{t}*),
i.e., along the line

u + (1/c*)v = k' .

Consider the family
of lines

u - c*v =k

parallel to the
line

u - c*v = k*

joining the optimal
threat point and the corresponding bargaining solution. Since (u_{t}*,v_{t}*) is the
threat point optimal for both players, then by the reasoning used earlier in
this section, we must have

k* = min(__x__) max(__y__) k

which holds if

u_{t}* - c*v_{t}* = min(__x__) max(__y__) (u* -
c*v*)

or

__x___{t}A__y___{t}' - c*__x___{t}B__y___{t}'
= min(__x__) max(__y__) (__x__A__y__' - c*__x__B__y__)

or

__x___{t} (A - c*B) __y___{t}' = min(__x__)
max(__y__) __x__ (A - c*B) __y__' .

Thus the optimal
strategies (__x___{t},__y___{t}) for the matrix game

A - c*B

are optimal threat
strategies for the nonzero-sum game.
Thus we have shown that there does exist a zero-sum game whose optimal
strategies are optimal threat strategies.
Unfortunately, this zero-sum game (A - c*B) depends on c*, which is
known only after we in fact have the solution to the problem we are attempting
to solve. Note that the value of the
game A - c*B must be k*, which is also unknown.
As Schroeder (Reference 2) has noted (for the case k* = 0), if we know
the value of k*, then we can determine c* (and hence the optimal strategies (__x___{t},__y___{t}))
iteratively, by choosing a trial value c_{a}* for c*, and adjusting the
trial value if the value of the game A - c_{a}*B differs
"too" much from k*. Note that
if the negotiation set is "near" (relative to the rest of the
feasible set) the origin, then k* is "close" to zero.

We now present a
rigorous derivation of the preceding result.
We assume that u < 0 and v < 0 for all (u,v) in the feasible set;
that the negotiation set is defined by a continuously differentiable curve
f(u,v) = 0; that the family of straight lines intersecting f(u,v) = 0 and
having slope negative to that of f at the point of intersection is defined by a
continuously differentiable function g(u,v) = k, where the various lines are
defined by varying the parameter k. Let
the straight line corresponding to g(u,v) = k be u - c(k)v = h(c). Let (u*,v*) denote the optimal threat
point. Now as we vary u, the parameter c
of the function g also varies, but, by the nature of the negotiation set, we
must have Δc/ Δu ≥ 0.
Similarly, we have Δc/ Δv ≤ 0. Now, by optimality, if

g(u*,v*) = k*

then

g(u* + Δu,v*) ≤ k*

and

g(u*,v* + Δv) ≥ k* .

That is,

(u* + Δu) - (c* + Δc/ Δu)v* ≤ k*

or

(u* + Δu) - c*v* ≤ k* + v*( Δc/ Δu) ≤ k*

since v* < 0 and
Δc/ Δu ≥ 0.

Also,

u* - (c* + Δc/ Δv)(v* + Δv) ≥ k*

or

u* - c*(v* + Δv) ≥ k* + v*( Δc/ Δv) ≥ k*

since v* < 0,
and Δc/ Δv ≤ 0. Hence,
since u* - c*v* = k*, the optimal threat strategies u* and v* are the optimal
solutions to the zero-sum game A - c*B.
Furthermore, this game has value k*.

Note that the
principal assumption made above is that all payoffs are negative. Unless some assumption is made that will
restrict the value of k*, however, the result does not appear to be very useful
from a computational viewpoint. If we
know the value of k*, then we can iteratively determine the value of c*, and
(simultaneously) the optimal threat strategies.
if the negotiation set is "close" (relative to the remainder
of the feasible set) to the origin, then k* is approximately equal to
zero. As we noted earlier, if the
expected losses for the two players tend to be proportional, then c* is the
proportionality constant (and k* is the u-intercept of the line of
proportionality).

In general, it appears
that many negotiation and conflict situations in the real world would have the
essential characteristics of the situations of the previous section:
essentially, expected losses that are large compared to the payoff for
negotiation. Hence it is felt that, in
optimization problems involving war between two parties, a reasonable payoff
function for both parties to examine in conducting the war is the matrix A -
cB, where A and B are the respective parties' payoff matrices and c is chosen
so that the value of the matrix game A - cB is zero. (Note that here we are assuming that the two
players' payoff functions are translated so that the feasible set is in the
lower left quadrant, close to the origin.)
Furthermore, c is approximately equal to V_{A}/V_{B},
where V_{A} and V_{B} denote the values of the matrix games A
and B.

The above result
appears to be a useful one, since it provides the military analyst with a
zero-sum game representation of war that is generally consistent with the
general-sum formulation of negotiation and conflict. The result hence offers the advantage of
retaining computational simplicity (since it deals with zero-sum games) while
appearing to be a more adequate representation of the real world.

A drawback
associated with implementation of the preceding result is that the matrices A
and B must specify payoffs resulting from all possible actions: those resulting
from negotiations and those resulting from conflict. It is noted, however, that the preceding
result is not heavily dependent on the slope of the negotiation set but only on
its general location. We shall now
heuristically derive a useful implication of this situation, for the case in
which the only payoffs we know are those corresponding to conflict (and we
assume that the negotiation set is "far" from the "conflict
set"). We shall here use the
approximation that c is approximately equal to V_{A}/V_{B}.

We suppose, then,
that it is possible to define payoffs corresponding to military actions that
are likely to result from failure of negotiations. Thus we assume knowledge of submatrices A_{s}
and B_{s} of A and B; the matrices A_{s} and B_{s}
define the payoffs corresponding to feasible military actions. It seems reasonable to assume that the values
of the zero-sum games A_{s} and B_{s} would be close to the
values of the zero-sum games A and B, since if player 1's losses are equivalent
to player 2's gains, it is obviously to player 2's advantage in the zero-sum
game A to declare war, and similarly to player 1's advantage in the zero sum
game B to declare war. (The underlying
assumption here is that waging war is costly, whereas negotiation is not.) Thus we assume that

V_{A} approximately = V_{As}

and that

V_{B} approximately = V_{Bs} .

For the same
reason, it appears reasonable to assume that in the zero-sum game

(1/V_{A})A - (1/V_{B})B ,

the strategies of
primary interest correspond to military actions. Hence the optimal strategies for the zero-sum
game

(1/V_{A})A - (1/V_{B})B

are likely to be
similar to those for the zero-sum game

(1/V_{A})A_{s} - (1/V_{B})B_{s} .

But since V_{A}
approximately = V_{As} and V_{B} approximately = V_{Bs},
the game is approximately the same as the game

(1/V_{As})A_{s} - (1/V_{Bs})B_{s} .

Hence, it appears
that military analysts should examine the zero-sum game

(1/V_{As})A_{s} - (1/V_{Bs})B_{s}

in their
analyses. This result appears to be a
useful one in that it does not place on the military analysts the burden of
identifying the negotiation set.

1. Owen. G., Game
Theory, W. B. Saunders Co., Philadelphia, 1968.

2. Schroeder, R.
G., "Linear Programming Solutions to Ratio Games," Operations
Research, April, 1970, Vol. 18, No. 2, pp. 300-305.