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Calhoun: The NPS Institutional Archive
Theses and Dissertations Thesis Collection
1971-09
An algorithm for the solution of
concave-convex games
Gunn, Lee Fredric
Monterey, California ; Naval Postgraduate School
http://hdl.handle.net/10945/15562
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United Statesostaraduate School
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AN ALGORITHM FOR THE SOLUTIONOF CONCAVE -CONVEX GAMES
by
Lee Fredric Gunn
and
Peter Tocha
Thesis Advisor: J. M. Danskin, Jr.
September 1971
AppA.ovQ.ci faon. pubtic n.o.to.ai>2.; du>tsu,biutton Luitimltzd.
,0AM SCHOOL
An Algorithm for the Solution
of Concave- Convex Games
by
Lee Fredric GunnLieutenant, United States Navy
B.A., University of California, Los Angeles, 1965
and
Peter TochaKapitanleutnant , German Navy
Submitted in partial fulfillment of therequirements for the degree of
MASTER OF SCIENCE IN OPERATIONS RESEARCH
from the
NAVAL POSTGRADUATE SCHOOL
I
ABSTRACT
A master's thesis which discusses the solution of con-
cave-convex games. An algorithm is developed, a computer
program written and applied to an anti-submarine warfare
force allocation problem as an illustration. Techniques
for handling concave -convex problems in high dimensions
are included.
TABLE OF CONTENTS
I. INTRODUCTION 5
A. A GAME THEORY APPROACH TO RESOURCEALLOCATIONS 5
B. CONTENT OF THE PAPER, BY CHAPTER 6
II. MATHEMATICAL CONCEPTS 7
A. GENERAL 7
B. THE ALGORITHM FOR THE SOLUTION OF CONCAVE-CONVEX GAMES 8
1. The Derivative Game 8
2. The Lemma of the Alternative 9
a. The Brown-Robinson Iterative Processin the "Auxiliary Game" 9
b. Statement of the Lemma 11
3. The Corollary of the Alternative 12
4. The Algorithm 13
III. PROGRAMMING ASPECTS ON AN ILLUSTRATIVE EXAMPLE - 18
A. FORMULATION OF THE PROBLEM 18
1. The Space X 20
2. The Space Y 20
3. The Function F(x,y) 21
B. BASIC COMPUTATIONS 2 2
1. Computation of 6, . 22r him
2. Partial Derivatives with Respect to x, .
-
23
3. The Value of F(x,y) 23
4. Partial Derivatives with Respect to Y-v" 23
5. Finding Directions of Increase(Decrease) 23
C. PROGRAMMING NOTES 24
1. Presentation of Input - The MissionDescription 24
2. The Contraction with Respect to V, . 27r him
3. Working in Subspaces 27
4. Machine Accuracy Problems 29
5. Testing the Program 30
6. General Comments 31
a. Initial Allocations 31
b. Distance Policy 32
c. Upper and Lower Bounds 32
d. Modification of the ObjectiveFunction 32
e. Assigning Values V, . 33& & himr r«1 • Z 1i i , i , 1 1 » w i > > ' ^ — —— — — — — — — — — — --~ — - - .... -\ •
IV. SUMMARY 35
V. SUGGESTIONS FOR FURTHER STUDY 36
APPENDIX: PROGRAMMING FLOWCHART 38
COMPUTER PROGRAM LISTING 51
BIBLIOGRAPHY 74
INITIAL DISTRIBUTION LIST 76
FORM DD 14 73 7 7
I. INTRODUCTION
A. A GAME THEORY APPROACH TO RESOURCE ALLOCATIONS
Problems in the allocation of resources can be divided
into two descriptive categories: Single-agent problems and
adversary problems. Single-agent problems have only one
participant optimizing without intelligent opposition. In
the solution of adversary problems, opponents work at cross
purposes. Each choice of an allocation of resources by one
participant must be made in light of those of his opponent (s).
Of course, games are adversary problems.
The great interest in game theory as a technique of
modelling which followed von Neumann's statement of the
fundamental concepts in 19Z7 [ISj ^oiilijiues today. The
fascination of the game as a model for conflicts of almost
any sort is enhanced by the fact that the solution to a
game is entirely independent of assumptions regarding the
actual behavior of the antagonist (s)
.
Matrix games and differential games are extensively
treated in the literature; some references are noted here.
Matrix games or games over the square are discussed in
basic form by Williams [21] and a thorough study is done by
Karlin [8]. For a first essay in the subject of differen-
tial games, see Isaacs [6]. Taylor [16,17] gives additional
examples of the modelling of combat operations including
search using differential games.
The development of the derivative game is due to
Danskin [4]. A very large class of concave-convex problems
yield to solution when the algorithm based on this game is
applied. It is with the derivative game and the algorithm
evolved from it that this paper is primarily concerned.
B. THE CONTENT OF THE PAPER, BY CHAPTER
Chapter II surveys some of the more important mathe-
matical ideas necessary to the development of the algo-
rithm for the solution of concave-convex games.
Chapter III is concerned with the programming of an
anti-submarine warfare force allocation example. The ex-
ample serves to illustrate both the use of the algorithm
and the theory on which it is founded. The problem is
formulated and the basic computational steps toward the
solution are enumerated. If any fears stem from the
formidable appearance of the matrices used in the example,
they hopefully will be dispelled by the description of the
form for input in the programming notes. Technical matters
regarding programming are considered in some detail.
Chapter IV draws together this presentation with some
concluding remarks
.
Chapter V contains suggestions for further study.
The Appendix consists of the programming flowchart.
The Computer Program Listing is included as well.
II. MATHEMATICAL CONCEPTS
A. GENERAL
The fundamental theorem of the theory of games in the
form appropriate here is the following:
Suppose F(x,y) is a continuous function on X x J,
where X and Y are compact and convex. Suppose that the set
of points X(y) yielding the maximum to F for fixed y is
convex for each such y, and that the set of points Y(x)
yielding the minimum to F for fixed x is convex for each
such x.
Then there exist pure strategy solutions x° and y°
satisfying
A 1 .'V - V I ' A I A * r! T7 v <, J.
(1)F(x,y°) < F(x°,y°), V xeZ.
A complete proof of the theorem in this form can be found
in [7]. Assuming that the conditions for the existence of
a solution satisfying (1) are met, the problem can be stated
in the form
Max Min F(x,y)
,
x y
which is equivalent to
Max cj)(x)
x
where <KX ) - Mi-n F( x >y)-
y
The theorem applies to two -person zero-sum games. Thus,
Max Min F(x,y) = Min Max F(x,y).x y y x
One important difficulty arises from the fact that,
although F may be smooth, cf> (x) is not in general differ-
entiate in the ordinary sense. Danskin, in [3], has shown
that under general conditions on X and Y, there exists a
directional derivative in every direction. It is this fact
that has provided the key to the solution of problems of
the type described.
B. THE ALGORITHM FOR THE SOLUTION OF CONCAVE -CONVEX GAMES
The algorithm to solve games concave in the maximizing
player and convex in the minimizing player is developed and
presented in great detail in [4]. The following gives a
brief survey of those results which are most important for
the design of the algorithm; to fill the apparent gaps, a
thorough reading of [4] remains necessary.
1 . The Derivative Game
Suppose x° cX . Associate with x° a non-empty set
of admissible directions y> r(x°) . W is defined as the
convex hull (e.g., see [19]) of the set of points
W(y) = {F (x°,y), -••, F (x°,y)},
xl yk
where yel(x°) .
The derivative game then is
H ( y , w ) = y * w
defined over r(x°) x W. The maximizing player maximizes H
by choice of yeT(x°), the minimizing player minimizes H by
choice of weW.
THEOREM I
A necessary and sufficient condition for the exis-
tence of a direction of increase for <J>(x) is that
the value of the derivative game defined by II be
positive at x° . The y° which yields the value of
the derivative game is a pure strategy.
If the value is positive one can find and use this direction.
If the value is non-positive, a direction of increase does
not exist, i.e., the solution has been reached.
The application of the derivative game in practice is greatly
complicated by the necessity for approximations.
2 . The Lemma of the Alternative
The Lemma takes into exact account the approxima-
te ons involved in the application of the derivative game.
It states that a certain process (to be explained below)
must either yield a sufficient increase to cj)(x) at a point
x° or determine that the point x° is nearly optimal. Be-
fore the lemma can be formulated in mathematical terms,
some further difficulties and the tools with which to over-
come them have to be outlined.
a. The Brown-Robinson Iterative Process in the"Auxiliary Game"
The Brown- Robinson (B-R) process employs the
following idea: Let G be the pay-off function. At stage
N=0 both players choose arbitrary strategies x° and y° . At
stage N=l the maximizer chooses x 1 such that G is maximized
against y° ; then the minimizer chooses y1 to minimize G
against x 1, and so forth. At stage N the maximizer chooses
x as if the minimizer ' s strategy were an evenly weighted
mixture of strategies y°,
••• • ,y]
; the minimizer chooses
y as if the maximizer' s strategy were an evenly weighted
mixture of strategies x° ,••• ,x .
IFor matrix games, Julia Robinson [12] proved
thatIN " 1 JN
lim supN-*-°°
r N-l N
m n=0 n=0
Danskin [1] has generalized the proof to hold
for two -person zero -sum games with continuous pay-off de-
fined over X x y, x and Y arbitrary compact spaces. It
should be noted here that the B-R process is very slow in
convergence when applied directly ^o finding an approxims
tion to the value of the game defined by (1). However, it
is not applied to the basic game in this algorithm but rather
to an "auxiliary game" for which an accurate solution is not
required.
The derivative game mentioned above cannot be
solved directly because the set Y(x°) is not known. All one
has is a single element ycY which approximately minimizes
F( x °>y)- The place of the derivative game, therefore, is
taken by the "auxiliary game" employing a modified version
of the B-R process described below. This process makes it
possible to keep track of the approximations involved and
their consequences. The "auxiliary game" is defined as
follows
:
10
For any £>0, denote by Y (x) the set of ycY
such that F(x,y) = <f>(x) + e- Let Y (5) = U Y (x) ,yeT^x ),
yeY(E).E(x°+d Y ,y) - <Kx°)
Then H( Y ,y) = ^ , for dQ
> 0,o
a minimum step size, is a game over r(x°) x Y (5) with y
the maximizing and y the minimizing player. This game has
optimal mixed strategies for both players. Applying the
idea of approximate optimization to the convergence proof in
[1] leads to
THEOREM II
N N-l nLet y be chosen such that ^ E n
H(y,y ) is
maximized to accuracy c, and y be chosen such
that rr Z^ H(y ,y) is minimized to accuracy n
.
i lien
lim sup \\ H( Y) yn
) - I En
H(Yn,y)
n=0 n=0< 2(c+n)
This holds for continuous H
Let 6 be the maximum oscillation of VF(x,y) over a distance
d . Theno
F(x°+do Y
N,y
n) - F(x°,yn ) > y
N F(x°,yn ) - e
_
o
The lemma of the alternative now can be formulated,
b. Statement of the Lemma
Suppose 0<a<3, y°eY (x°).
Then the generalized B-R process will, at some stage N, de
termine that one of the two following statements is true:
1. The maximum over F(x°) of the directional
derivative does not exceed 3.
11
N-N
2. The point x = x° + d y » where
N-N_l £ n
and the point y eY (x ) , where y minimizes
F(x ,y) to accuracy £, satisfy
F(xN,yN
) - F(x°,y°) 3
3> a bo ^- .
o o
3. The Corollary of the Alternative
Suppose that F(x,y) is concave in x and convex in
y. Then the modified B-R process applied at x° will, at
some stage N, determine that one of the two following state-
ments is true
:
1. The pair x = x°, y = y , where
-N _i_ <£ ny N+l L y
'
n=0
are approximate optimal strategies for the game
defined by F.
2. The point x zX yields an increase to <J>(x) by at
least a specified amount.
For details and proof see [4], pp. 36 ff.
Reference [4] continues with a detailed discussion
of delicate problems which can only be listed here: The
choice of the minimal step size d ; the problem of acces-
sibility of a point x from a point x° ; the problem of ob-
struction; the choice of a, 3, c, their interaction with each
other, and the choice of p where p is the accuracy to which
Max <t>(x) is to approximate the value of the game defined by
12
(1) . It must be noted that the conditions derived for the
selection of these parameters are sufficient.
4 . The Algorithm
The algorithm as a consequence of the foregoing
mathematical considerations is presented in section 10 and
11 of [4] and will not be reproduced here in detail. A ver-
bal description of its basic structure - depicted in Figure
1 -, however, may be useful:
The maximizing player, called Max, having arrived
at a point xx , has a direction of maximal increase y, ob-
tained either from the derivative game D cf>(xx) or from
the B-R process in the auxiliary game, and a distance d > d .
The minimizing player, called Min , is at a point yy. F(xx,yy)
is known. Max makes a proposal uo move to a point x = xx + dy
Of course, F(x,yy) > F(xx,yy). Min accepts Max's proposal
and starts minimizing against x, looking for a direction
of maximal decrease g. If there is none, yy is a minimum
against x as well as against xx in which case Max will move
to the point x. If there is a direction of decrease Min
forms a point y = yy + Dg such that F(x,y) < F(x,yy). A test
is performed to determine whether Min has already "beaten"
Max: if F(xx,yy) > F(x,y) Min stops the minimization pro-
cess, and Max discards his proposal x because moving to x
will not increase <K X ) • Max halves the distance d and, with
the same y, forms a new trial point x. If F(xx,yy) < F(x,y)
Min continues to minimize until either F(xx,yy) > F(x,y) or
Min can no longer find a direction of decrease. If now
13
Figure 1. Flowchart of Algorithm.
TRIAL X ANP YALLOCATIONS
YES
FAILURE.FOR X
->SUCCESS
FOR TRIAL X
TRIAL X bECO/AESMEW 5A5E POINT
FIND THE APPARENTDIRECTION OF FASTEST
INCREASE FOR X
YES
FORM A NEWTRIAL X
ADJUST XDISTANCE
YES
ENTER THE5-R PROCESS
PREFER TO FLOW CHARTOF b-R PROCESS)
THE PR06LEM|
i .» in i V i ii
CO TO STOP
|STOP
14
F(xx,yy) < F(x,y) , indicating that Min, even after a complete
minimization, was not able to "beat" Max, Max moves to the
proposed point x realizing a gain for <K X ) • Max then looks
for a new direction of increase. The process terminates when
such a direction does not exist.
The situation that leads into the Brown-Robinson
process is the following one: The proposed points x = xx+dy
have been "beaten" by Min until d gets cut down to d , in
spite of the fact that y, obtained from the derivative game,
is an apparently good direction. If the trial point x = xx
+ d y does not result in a move for Max, the B-R process,
Figure 2, is used.
Denote the present y - the one that so far has
lead to a failure rc-r r.:nx by y" , s^d liit; associated VF
' '' x
by to° . Minimize F completely against x = xx + d y . The
resulting y then leads to a new VF which is averaged with
the previous to,giving to
1. Suppose that to
1 as input to
the derivative game D 4>(x) produces a new y° such that the
value of the derivative game is positive as required. (If
such a y° does not exist the problem is solved). This y,
averaged with y1
,gives y
2 which in turn creates a new
trial point x 2 = xx + d y2
. x 2, or x
1
in general, then is
exposed to Min ' s reaction as described previously. Once Max
finds a direction and an associated trial point that cannot
be "beaten" by Min, Max moves and leaves the B-R routine.
The y-strategies are also averaged and saved although their
average is never used during the computation. In case the
15
Figure 2. The Brown-Robinson Process.
Conditions for entering the. E>-R
2At pt x° there was a direction t of
APPARENT INCREASE, HOWEVER, FAILURE AT
X1
- x°+ d y\ Have cj
START
NE>R-0
compute: y 1
= y 1
— O oCO =COY° =
NE>R=NbR+ 1 <
REFORM AVERAGE, Y, I e (THE SUPERSCRIPT N
CORRESPONDS TONE>R)
TRY XN= X + d T"
FINDMINIMIZING
CALCULATECO*
YES
REFORM OMEGA bAR
m (m-j)Cjw"'+lo
m
U)
GO TO D.F.
ALGORITHM
GET Yu + i
MOVE X
TO X*
D. F.
ALGORITHM
16
game defined by F(x,y) is terminated while in the B-R pro
cess, this average of the y-strategies represents the op-
timal solution for the minimizing player.
17
Ill . PROGRAMMING ASPECTS OF AN ILLUSTRATIVE EXAMPLE
A. FORMULATION OF THE PROBLEM
An algorithm for the solution of a wide class of con-
cave-convex games over polyhedra has been presented in
abbreviated form above. In this chapter that algorithm is
applied to a particular game, an anti-submarine warfare
force allocation problem.
For this problem, five indices are employed: h, i, j,
k, and m. In a meaningful example, the maximum numbers
corresponding to these indices might be, respectively:
5, 25, 10, 500, and 5. The indices have the following
meanings
:
): ; Submarine type
.1
k
Submarine mission
Type of antisubmarine weapon (or vehicle)
Place in which submarine and weapon encounter one
another
m: Stage of the submarine mission.
An additional index is used. &(k) is the "kind of place."
A "kind of place" might be defined by a particular set of
weapon employment parameters. These include the tactical
and the natural environment. The natural environment con-
sists of oceanographic and meteorological conditions. Ex-
amples of the tactical environment are destroyers in an ASW
screen and patrol aircraft in barrier patrol. The "kind of
place" in which an encounter occurs impacts on the outcome
18
of an encounter between submarine and weapon. A reasonable
number of "kinds of places" in the present context might be
25.
A submarine mission is described by a matrix |EU . ., I
.
' '
' hijkm '
'
This matrix has as its elements real numbers denoting the
extent to which a submarine of type h at stage m of mission
i is exposed to a weapon of type j at the kth place. The
effects of these weapons and thus of the encounters are
characterized by a "technical" matrix l^hi£fk")l ^n t ^ie ^°^"
lowing meaning: exp[-C, .
{,, >. ^'V^ ^- s tne probability that
a submarine of type h survives one exposure to y units of
weapons of type j at a "kind of place" £(k). These en-
counters are assumed to be mutually independent. Note that
the probability of survival to the m^n mission stage is
conditioned on the completion of the previous stages. Sup-
pose that there are y., units of force of type j at the ktn3 K
place. Then the probability of a submarine's completing
stage m of mission i is
7| exp[-Ehijkm , ChjA(k) yjk ],
m' <m
which, due to independence of the events, equals exp [-0, . ]
where
6him V *gEhijkm' Chj£(k) yjk*
m ' <m
Now, by carrying out a premultiplication
,
A = E Chijkm "hijkm hj£(k)'
19
the exponent becomes
6him= Z £
Ahijkm' y
jk'
in' <m
In this example, the vast majority of the E, . . , , and there
fore of the A, ..-, , are zero. If 3000 non-zero A,.., arehi j km hi j km
allowed, each type of submarine can be employed on ten dif-
ferent missions and undergo up to 60 encounters with anti-
submarine weapon systems.
1 . The Space X
Let x, . be the proportion of submarines of type h
assigned to the i"th mission. Make X =|
|x, .
|satisfy the
conditions
;OT pvprv h , x, ..hi ' ' hi -
1
and
ahi -xhi - 3hi'
£or ever >' P air h'i
'
where the sets {a, •}, (Bv-1 are supposed to satisfy
L, a,. < 1 < L, 3, • for every h
and
< a, - < 3^ • for every pair h,i.
2 . The Space Y
Let y . , be the proportion of antisubmarine forces
of type j sent to the k™ place. Make Y = l>'-vl satisfy
the conditions
20
and
L y-k
= 1, for every j, y^ k>
JK
ajk - yjk -
bjk
£or ever>r P air J> k
where the sets {a.-, }, {b., } are supposed to satisfy
L a., < 1 , < L b .,
k J k k J k
and
< a.-, < b . , for every pair j ,k
3 . The Function F(x,y)
V, • is the value of accomplishing stage m of
mission i for a submarine of type h. The character of
F(x,y) can be examined.
06 u
w, - = v, . exp r-e, .]him him * L him J
where 8, . is as before. Puthim
hi *-* himm
Thenm
F(x,y) = 7 xhi
Thi
.
h.,i
This function is linear in x and exponential in y and is
therefore a concave -convex game of the type treated in [4],
defined over X x Y. The quantity F(x,y) represents the
total expected payoff to the submarine player. Re-expressing
F(x,y) in its explicit form gives:
21
F(x,y) = L, l, x, . A, V, exp [- /. ), E, . .-, C, .. n sV., 1,v ,7 ^
i - hi ^ him v l v T hiikm h;i£(kyik J
111 m J k J j *, ; j
The remainder of this chapter gives details of the application
of the algorithm to the game defined above.
The principal result is the flow-chart (Appendix).
Since this flow chart is constructed around the algorithm
from [4], it is helpful, though not essential, to have [4]
available. The complete program listing is included following
the Appendix. The program is written in FORTRAN IV and was
run on the IBM 360/67 computer at the Naval Postgraduate
School, Monterey, California.
B. BASIC COMPUTATIONS
1 . C \_> nip u L a 1 1 o ii v f mm
For fixed (h,i), 9, . is non-decreasing in the mis-v'
J' him fa
sion stage m. This reflects the trivial fact that
P [submarine survives stage m]
< P [submarine survives stage m-1].
Equality holds when the "threat" due to the encounters at
stage m is non-existent, i.e., when either no ASW-forces
y., are present or their effectiveness against the submarine,
^h'&fkV ^" s zero - Hence 0, . , for each pair (h,i), is ac-
cumulated over the mission stages as follows:
6, . = G, .r ,. + V V A, . ., y ., , 9, . = .him hi (m-1) h* ^ hijkm 7 jk' hio
22
2. Partial Derivatives with Respect to x, .
hi
Because of the linearity of F in x the partials with
respect to x, . , F , are the coefficients of x, . and do notilJ- Xt z Jii
explicitly contain x. V}iim
exP[-6}lim ] can be represented as
a matrix of dimension (n x m) , where n is the number of pairs
(h,i). Then
Fv = V V, . exp[-0, .]x, . *-> him ^ L him J
is the sum of the elements in the (h,i) row of that matrix.
3. The Value of F(x,y)
F(x,y) is obtained by pre-multiplying F by x, . andXhi
ni
summing the products
F(x,y) - £ x,. Fv_ .
n i
4. Partial Derivatives with Respect to v.,* 7
-} K
Because of the cumulative property of 6, . , a change
in y-k
during some mission stage m ' will affect the following
stages as well. For each pair (h,i), premultiply x, . by
the corresponding Ahi -j km ,> where m' is the mission stage in
which the y., of interest occurs. Sum V, . expf-G, . 1 over1
K
him 'L him J
the mission stages for which m ' <m , multiply the result with
Xhi
Ahiikm'
and sum tne P roducts over h and i:
'\k~~ I ?Xhi
Ahi Jkm .
ra^ Vhim .
exp[-9him ,].
5• Finding Directions of Increase (Decrease )
F and F are inputs to the derivative games forhi } ik
x and y. For x, the direction y° is sought such that D <j>(x) =
VFx«y is maximized; for y, g° is sought such that D Max F(x,y)
g x
23
VF »g is minimized (equivalently , -VF «g is maximized). The
side condition is that y° and g° be unit vectors:
i k J 1 k J
A method of finding such directions - called THE DIRECTION
FINDING ALGORITHM - is derived from the Kuhn-Tucker conditions
and the Schwartz Inequality. It is contained in [4] and has
been applied to a vector - valued function by Zmuida in [22].
C. PROGRAMMING NOTES
The flow chart (Appendix) and the associated program may
not be optimal with respect to machine time and memory space
required, and may provide opportunities for improvement. For
computer routines like this, intended to solve high-dimensional
problems, a trade-off between time and space is generally ap-
parent. The user, considering the particular facilities
available to him, must decide on his optimal trade-off, and
modify the program accordingly. The following discusses some
of the techniques that have been implemented; it points out
the major difficulties that have been encountered and the ways
presently used to deal with these, and offers some remarks
about the impact of the underlying mathematics on the use of
the algorithm for realistic problems. The numbers referenced
are statement numbers.
1 . Presentation of Input - The Mission Description
In a realistic application, most of the A, • • •,, will
be zero because the effectiveness C, .
„
r , ^ contained in A willhj £(k)
be zero. Example: suppose k denotes a place in the western
Baltic and &(k) classifies this place as shallow with extreme-
ly poor sonar conditions; j denotes a nuclear killer-submarine,
1 A
h represents a conventional attack submarine. Then C, . . ,,.
will be zero for all practical purposes.
The method of presenting the matrix E which avoids
computing of and with A when C is zero, is one which is ex-
tremely easy for the user to understand and employ. He gives
his description of a submarine mission by plotting it on a
chart, marks off in order the places the submarine must go,
and notes the forces it might meet at those places. He
will give the exposure E required by the particular mission
in terms of a standard which he will have set. He will mark
off the points in the mission where the various stages of
the mission will have been accomplished. Such a description
might run as follows:
m=0
1% — 1 A — 1 1/-— "1 A — "B — ' Ma - _L 1-1 i\ ~ X J~ ^ J-. — X . U
j=5 E=1.5k=7 j = l E=2.0
j = 3 E=3.0
m=l (the first stage of the first mission is completed)
k=8 j=4 E=1.2j=7 E=1.7
m=2
k=15 j = 3 E=0.3j=2 E=2.0j=4 E=0.5
m=3
The above mission (mission 1 for submarines of type
1) has three stages and nine encounters, in four different
places. Then the listing starts for the next mission with
h=l, i=2, and continues until all missions for all types of
submarines have been described in this manner. Note that
this listing has already taken into account the classification
25
of the places k by £(k). This is done in such a way that
for each encounter on this mission listing the associated
C, •
ps, s is positive. In other words, a place k, and hence
an encounter, will appear on the mission listing only if it
seems possible that the opponent will allocate forces to that
place and the corresponding effectiveness against the maxi-
mized s submarine operating in that place is positive.
There are various possible ways of storing the
information contained in the mission description in the
machine. One way which is very economical in terms of
storage space requirements is to read the list encounter by
encounter, i.e., card by card, starting with an N=l. Then
A(N) = E'C, .„,n and an indicator arrayh3&(k) ;
P(N) = I-J-K-M- (h-1) + J-K-M-(i-l) + K-M-(j-l)
+ M« (k-1) + m + 1
contain the complete information. I, J, K, M denote the
maximum values of the indices i, j, k, m. The reason for
using (m+1) is that the correct m appears after the mth
stage is completed. The disadvantage of this method is
that during computations the individual indices must be re-
computed from P(N):
[•] means "the largestinteger in •"
h = 1 +P(N)
I • J • K • M
a = p(n; - I-J-K:-M(h-l)
i = 1 +A
J • K • M
AA = A J-K-M- (i
26
1 + _MK-M
AAA = AA - K«M(j -1)
k = 1 +AAAM
m = AAA - M(k-l)
In the sample program contained in this paper, the indices
in their original form are stored in vector arrays and are
directly accessible throughout the computations.
2. The Contraction with Respect to Vu .
^_him
This contraction takes into account the possibility
that the stage m of a mission may have been noted, but that
the associated V may have been declared to be zero.nim J
C5j. C'dlatS r.nythlii; IiVO i ViU^'him " '
the corresponding elements in the index arrays of the mission
are eliminated. This is done prior to the execution of the
game and may therefore be referred to as the basic contrac-
tion .
3. Working in Sub spaces
It can be expected that most of the x, . and y.-.
will be on their boundaries at any stage, including the
approximate optimal solution. This suggests the idea of
eliminating computations involving variables which have
fixed values either temporarily or throughout the process.
To do this, one reduces the dimensions of the
spaces, thus saving machine time. One can redefine the
space so as to "freeze" the variables on the boundaries
leaving the remainder free to move.
27
Where in the program should redefinition of the
Y-space take place? The minimizer can hold the subspace
fixed and continue to move in that subspace until an ap-
parent minimum has been reached. At this point the al-
location corresponding to this minimum has to be checked for
optimality in the full space. This will require at least
one iteration through the minimization routine in the full
space. After a minimum valid in the full space has been
obtained, the Y-space is reduced to a new subspace. An
alternative approach is to redefine the space after each
change in the Y-allocation , thus maintaining a continously
changing subspace. The authors feel that the latter will
enable the minimizing variable to stay in the subspace
icnger finding directions oi decicdse , before the ;iccd
arises to employ the full space. This idea. was implemented
in the program.
The contractions for x (statement 1871) and y (125)
eliminate computations involving elements that are on the
lower boundaries. It must be noted that the contraction
itself and the complications caused by its use take machine
time. Worthwhile time savings in computation will not be
realized until this technique is applied to relatively large
scale problems.
INDX(N), N = 1,...,NNX, contains the indices of
the xhi
> 0, INDY(N) , N = 1,...,NNY, contains the indices of
the y.-, > 0. NNX and NNY are the dimensions of the subspaces3 K
for X and Y. These index arrays are used to control which
28
variables are to be involved in the computations. The way
this is done in practice can be seen in the program list-
ing.
One remark concerning the DIRECTION FINDING ALGO-
RITHMS, (400) and (1400), may be in order: these algorithms
have to be applied row by row to the matrices | x.. | and
|y.,|
j. In subspaces, the number of elements belonging to3 K
a particular row varies. The outer do-loop runs over the
number of rows. To find the numbers of elements per row
(these numbers serve as the termination values of the inner
do-loops) a test on the indices stored in INDX and INDY is
conducted (402 f f ) , (1402 f f) . The test value, NTEST, is
the index of the last element in the rows of |x, . I andii hi ''
• y • - i i - i esDect ivelv . Wr> & r< the t-~s' rv<ass&s 313 c 1 ctjigti t sI .
j K . , . I > r - >
in the row at hand have been collected and the algorithm
begins. Before the next row is picked, NSTART is incre-
mented by the number of elements found in the previous row
so that the search through INDX or INDY always starts in the
correct position.
4 . Machine Accuracy Problems
The program calls for frequent testing of floating
point numbers. Throughout the development of the program
these tests have been sources of trouble. Testing for
equality must be strictly avoided even after - as has been
done here in various places - variables close to a fixed
quantity have been reset to that quantity (e.g., (1050 ff)
or (1060 f f ) ) . Although the program specified double-pre-
cision, it took extensive experimentation to maintain
29
feasibility, i.e., satisfy the side conditions
fxhi £ y jk - l
to at least the 13 tn decimal place.
Of critical importance is the accuracy in the
computation of the direction matrices y (GAX) and g (GAY)
.
In the neighborhood of a maximum or minimum, the partials
F and F are close to zero, as are the Lagrange multi-x y ' & &
pliers (XMU and YMU ) , and the differences F -XMU,F -YMU
(SD) . In order to determine the value of the derivative
game, the sum of the squares of these differences has to be
formed, an operation that may very likely lead to erroneous
results which, in turn, carry over into the computation of
y and g.
The countermeasure taken is to premultiply the SD
when they are small by a large number, (505 f f) , (1510 f f) .
5 . Testing the Program
As the calling of subroutines is exceedingly time
consuming, the present program does not use them. This made
debugging tedious. The "standard" routine, i.e., the pro-
gram employing the derivative games in the original form,
was tested running a small scale example where |x| was
(2 X 2) and |y| was (2 X 4), making it possible to hand-
check the computations.
The "auxiliary game" routine using the B-R process
was debugged using the following objective function:
30
. ^ ~ ~ „ ,1
-*- >> s w a
F(x,y) = 2x1
exp[ -2y1-y
2] + 2x
2exp[ -y
1"2y
2]
+ x3
exP [-y1-y
2-y
3 ]
Subject to: L, x. = Zj y, = 1
i k
xi> ^k *
°
with initial allocations x = (0,0,1), y=(0,l,0). For a
discussion of this example in light of the algorithm see
[4]. The reason that this can only be solved through the
B-R routine lies in the fact that, against the initial x,
any y represents an exact minimum, however, the value of
the derivative game for x is positive, yielded by y° =
(0, /2/2, -/2/2).
Pi ri 3 1 1 v pti <=>y -jiiit->1o i.i i f n ^ ^ Q £" COUnt" T2 WHS
up where |x| was (4x4), |y| was (5x10). While this does
not yet represent a problem in high-dimensional space, the
authors are confident that this example provided a suffi-
ciently severe test to demonstrate the validity of the pro-
gram as written.
6 . General Comments
The following remarks are intended to facilitate
the use and modification of the program.
a. Initial Allocation
The initial allocations are generated as corner
points in the stationary part (99). It can be expected that,
in the optimal solution, most of the variables will be on
the boundaries. An initial point on the boundaries insures
that the number of "absurd" allocations is minimal.
31
If one were to use an interior point as a
starting solution, it possibly would involve large numbers
of absurd allocations (e.g., a submarine in an aircraft
barrier patrol) . The machine would spend an enormous
amount of time reducing such allocations to zero.
b. Distance Policy
The initial and re-set values for the dis-
tances are determined from the dimensions of the spaces;
the user may employ his own rules. A compromise between
re-set values too large causing "overshooting" and values
to small causing "creeping" should be considered. In gen-
eral, "creeping" is much costlier in terms of machine time.
The policy for halving distances and its rationale is out-
lined in [ 4 ] .
c. Upper and Lower Bounds
For simplicity, and 1 have been used in the
program. Specifying individual bounds a, •, 3i • and a.-,
,
b., does not introduce additional problems but increases the
storage space required considerably (e.g., for y, two addi-
tional arrays of the same dimension as y)
.
,
d. Modifying the Objective Function
The algorithm as stated is valid for concave-
convex functions under quite general conditions. Though the
present program has been designed to handle a particular lin
ear-exponential function, it is quite flexible. The objec-
tive function mentioned in subsection 5 may serve to illus-
trate this point:
32
F(x,y) = 2x1exp[-2y
1-y
2]
+ 2x^ exp [-y1"2y
2]
+ x3
exp [-yr y 2-y
3]
is produced by a very simple adjustment of the mission
description
:
h=l i=l k=l j=l E=2.0k=2 j=l E=1.0k=3 j=l E=0.0 END OF MISSION 1
m=l
h=l i=2 k=l j=l E=1.0k=2 j=l E=2.0k=3 j=l E=0.0 END OF MISSION 2
m=l
h=l i=3 k=l j=l E=1.0k=2 j=2 E=1.0k=3 j=3 E=1.0 END OF MISSION 3
m=l
/•* r\
Vln = 2.0
v131
- 1.0.
The associated C, . n ,-. ^ are : C nin =Ci 1o = C 11 _=1.0hj£(k) 111 112 113
e. Assigning Values V, .& & him
The values (of completing stage m of mission i
for submarine of type h) are relative measures. When as-
signing V, . , the user should consider that a submarine mayfo ^ him' '
have spent part of its weapons (resources) during stage m-1
This may reduce its operational capabilities for stage m,
and the value for stage m should reflect this fact.
f. Choosing p (RHO)
The parameter p, mentioned in Chapter II,
section B 2.b, has to be chosen by the user. It specifies
33
the accuracy to which Max <}>(x) is to approximate the value
of the game F(x,y) , V. p affects the y-minimization by
controlling the accuracy, e(d), to which the derivative game
D Max F(x,y) = -VF -g has to approximate its mathematicalg x Y
value. The y-minimization is the most time consuming por-
tion of the process.
where 6 is the diameter of the X-space. Considering the
"worst" case, when d = d = p/36-L, (L is the maximum os-
cillation of F ) , it becomes apparent that
2
e ( do^=
36 2 -6-L
is of order of magnitude p2 *10
Recall that the conditions established for
the valididty of the algorithm are intended to guarantee
that, at the approximate optimal solution,|§ (x) -V| <p
.
Test runs of the program seem to indicate that the accuracy
actually obtained is much higher.
Although generally valid conclusions cannot be
derived from this observation the user must be aware of this
feature because he will pay heavily in terms of machine time
when p is unreasonably small.
34
IV. SUMMARY
The devlopment of the mathematical theory underlying
the derivative game and the concave -convex game algorithm
has only been sketched out in this paper. Here, that al-
gorithm has been employed in the formation of a potentially
useful example. With programming techniques designed to
provide economical running in high dimensions, large-scale
problems should be amenable to the application of the con-
cave-convex game algorithm.
35
V. SUGGESTIONS FOR FURTHER STUDY
A. MATHEMATICS
In Section III.C.6.e., the question of the proper choice
of p was addressed. Recall that
Max <f>(x) " V|< p.
x
Even when the user has based his choice of a value for p
on extensive experimentation with a program, he still will
not know exactly how close Max <{>(x) is to the value of thex
game. The desirable state of affairs would be
Max <Kx) _ V| = px
This would require the formulation of necessary conditions
on the functions a 3 and e
.
5
B. PROGRAMMING
The fact that, for the class of games at issue, MaxMinFx y
= MinMaxF can be exploited to arrive at the neighborhood ofy x
the optimal solution faster than the present program will:
Start the problem as MinMaxF. The course of action then is
reversed with considerable advantages. The maximize r sees
the present y-allocation ; the maximization is trivial, as-
signing as much weight as possible to the x, . with the
largest coefficients (the F,(x,y )), which results in a
corner solution for x. Then the B-R technique is employed
directly to F(x,y) which will bring x off the boundaries
36
again, and, after a pre-set number of iterations, will pro-
duce an allocation not far from the solution. At this stage
the problem is reinterpreted as MaxMinF and solved in the
manner of the present program.
Another feasible refinement particularly useful in the
application of the algorithm to objective functions linear
in both x and y, i.e., F = X x.a..y., is the idea of13 J J
"doubling", outlined in [4]. The problem is treated as
MaxMin and MinMax at the same time. Here the value of the
game is approached from below and above simultaneously which
provides a stopping rule when the difference MaxF(x,y) -cj) (x)x
arrives at a pre-specif ied value.
37
APPENDIX FLOWCHART OF THE PROGRAMMED EXAMPLE
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50
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BIBLIOGRAPHY
1. Danskin, J. M. , "Fictitious Play for Continuous Games",Naval Research Logistics Quarterly, 1, 1955
, pp. 313-3TCT
2. Danskin, J. M. , "The Theory of Max-Min, With Applica-tions", Journal of the Society of Industrial and Ap-plied Mathematics , v. 14, I960, pp. 641-664.
3. Danskin, J. M. , The Theory of Max-Min, Springer-Verlag, New YorF] inc
., 19 6 7.
4. Danskin, J. M. , "The Derivative Game in Max-Min andthe Solution of Concave-Convex Games", to appearJournal of the Society of Industrial and AppliedMathematics
.
5. Dresher', M., Games of Strategy, Theory and Applications,Prentice-Hall^ Englewood Cliifs, New Jersey, 1961.
6. Isaacs, R. , Differential Games, John Wiley and Sons,Inc., New York, I96S.
7. Kakutani, 5., "A Generalization of Brou"pr r
s FixedPoint Theorem", Duke Mathematical Journal, v. 8,No. 3, 1941, pp.~T57-4b9.
8. Karlin, S., Mathematical Methods and Theory in Games,Programming and Economics, Vols . I and 1 1 , Addison-Wesley , Reading, Massachus e 1 1 s , 19 59.
9. Koopman , B. 0., Search and Screening, OperationsEvaluation Group Report No. 56 , Washington , D. C.,1946.
10. Kuhn, H. W. , and Tucker, A. IV., (eds.), Contributionsto the Theory of Games , Vol. I, Annals o± MathematicsStudy No. 24 , Princeton University Press, Princeton,New Jersey , 1950
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11. Morse, P. M. , and Kimball, G. W., Methods of OperationsResearch , John Wiley and Sons, Inc
., New York , 1951
.
12. Robinson, J'., "An Iterative Method of Solving a Game",Annals of Mathematics , v. 54, 1951, pp. 296-301.
13. Schroeder, R. G., Mathematical Models of AntisubmarineTactics , Ph.D. Thesis, Northwestern University, Evanston
,
I llinois , 1966
.
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14. Schroeder, R. G., Search and Evasion Games , Naval Post-graduate School Technical Report/Research Paper No. 68,Naval Postgraduate School, Monterey, California, 1966.
15. Suppes , P., and Atkinson, R. C., Markov Learning Modelsfor Multiperson Interactions , Stanford University Press,Stanford, California, 1960.
16. Taylor, J. G., Application of Differential Games toProblems of Military Conflict: Tactical AllocationProblems - Part I , Naval Postgraduate School, Monterey,California, 1970.
17. Taylor, J. G., Application of Differential Games toProblems of Naval Warfare: Surveillance - EvasionPart I , Naval Postgraduate School, Monterey, California
,
1970.
18. von Neumann, J., "Zur Theorie der Gesellschaftsspiele"
,
Mathematische Annalen , Vol. 100, 1928, pp. 295-320.
19. Valentine, F., Convex Sets , McGraw-Hill, New York,1964.
20. Weyl, H. , "Elementary Proof of a Minimax Theorem Dueto von Neumann" , Contributions to the Theory ofG.-imijb , Vol. T
;;\nr:ai" or Mathematics Study No . 24,
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21. Williams, J. D., The Compleat Strategyst , McGraw-Hill,New York, 19 54.
22. Zmuida, P. T., An Algorithm for Optimization of CertainAllocation Models
,M.S. Thesis, Naval Postgraduate
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75
INITIAL DISTRIBUTION LIST
No. Copies
1. Defense Documentation CenterCameron StationAlexandria, Virginia 22314
2. Library, Code 0212Naval Postgraduate SchoolMonterey, California 93940
3. Chief of Naval PersonnelPers-llbDepartment of the NavyWashington, D. C. 20370
4. Department of Operations Researchand Administrative Sciences
Naval Postgraduate SchoolMonterey, California 93940
5. Professor W. P. Cunningham, Code 55Cmm:"isv"r;,,-;.? of OueratiOiiS Research ar,.i
i
Administrative SciencesNaval Postgraduate SchoolMonterey, California 93940
6. Professor J. M. Danskin, Code 55DkDepartment of Operations Research andAdministrative Sciences
Naval Postgraduate SchoolMonterey, California 93940
7. LT Lee Fredric Gunn , USN10002 Enford CourtFairfax, Virginia 22030
8. Kapitanleutant Peter Tocha, German Navyc/o Alo Terhoeven415 KrefeldMarienstr. 43Germany
76
UNCLASSIFIEDSecurity Classification
DOCUMENT CONTROL DATA -R&D(Security clas si fie ation of title, body of ahstract and indexing annotation must be entered when the overall report is classified)
1 ORIGINATING ACTIVITY (Corporate author)
Naval Postgraduate SchoolMonterey, California 93940
2a. REPORT SECURITY CLASSIFICATION
UNCLASSIFIED?b. GROUP
3 REPOR T TITLE
AN ALGORITHM FOR THE SOLUTION OF CONCAVE -CONVEX GAMES
4 DESCRIPTIVE NOTES (Type of report and, inc Ius i vr dates)
Master's Thesis; September 19715 AUTHORISI ffifst name, middle initial, last name)
Lee F. GunnPeter Tocha
6. REPOR T D A TE
September 197170. TOTAL NO. OF PAGES
78
7b. NO. OF REFS
226a. CONTRACT OR GRANT NO.
b. PROJEC T NO
9a. ORIGINATOR'S REPORT NUMBER(S)
9b. OTHER REPORT NO(S) (Any other numbers that may be assignedthis report)
10 DISTRIBUTION STATEMENT
Approved for public release; distribution unlimited.
11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Naval Postgraduate SchoolMonterey, California 93940
13. ABSTRACT
A master's thesis which discusses the solution of concave-convex games. An algorithm is developed, a computerprogram written and applied to an anti-submarine warfareforce allocation problem as an illustration. Techniquesfor handling concave- convex problems in high dimensionsare included.
DD I NOV «!l*» / OS/N 0101 -807-681 1
(PAGE 1
)
UNCLASSIFIED77 Security Classification
A-31408
UNCLASSIFIEDSecurity classification
KEY WORDS
Concave-Convex Games
Games
ASW Forces
ASW Force Allocations
Game Theory
Resource Allocations
Allocations
Allocation Problems
Concave-Convex Functions
Derivative Games
Brown-Robinson Process
MaxMin
Directional Derivative
Algorithms
ROLE
DD .'•"."..1473 < BA«
LINK C
ROLE
B/N 0101-807-61JNCLASSIFIED
78 Security Classification A- U 409
TL-e <*7 hot l^tJU!
2 AUG72 DiJJDERY20906
ThesisG8647 Gunn
131297
c.l Anthe !
algorithm forsolution of con-
cave--convex games.
2 AU072D 1 HUSKY209 06
ThG8c:
Thesis
G8647c.l
Gunn 131297An algorithm for
the solution of con-cave-convex games.
3 2768 002 13596 4
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