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10-1959
Equilibrium Points in Finite Games
Harlan D. Mills
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Mills, Harlan D., "Equilibrium Points in Finite Games" (1959). The Harlan D. Mills Collection.
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EQUILIBRIUM POINTS IN FINITE GAMES
Harlan D. Mills
MATH E MAT I C A
92,A Nau4v. Street
Su.bsidi4r, of
Princeton, New Jersey
COR P 0 RAT ION
0 F
A MER I C A
EQUILIBRIUM POINTS IN FINI T E GAMES
Harlan Mi ll s
Oc t obe r 19 5 9
101 - R MO I
EQUILIBRIUM POINTS IN FINITE GAMES
Harlan Mills
Introduction
The existence of equilibriu:m points in finite ga:me s, as formulated by
von Neu:mann and Morgenstern [11], [6], [7], has been established by Nash [8], [91.
The :method of proof is analytic, e:mploying a fixed point argu:ment.
We give an algebraic characterization of e quilibriu:m points in Theore:ms
1 and In.
This raises the possibility (so far unresolved) of an algebraic proof of
Nash's existence theore:m.
We also show that equilibriu:m points can be described
in ter:ms of sol utions to certain nonlinear progra :ms in Theorems 2. and 2n.
The se
generalize well-known equivalences between :matrix ga:mes a nd linear progra:ms [1],
[2], [3], [10].
In Theore:m 3, equilibriu:m points f o r a t wo-person (nonzero-sum) ga:me
are further characterized in solutions of a syste:m
certain variables :must be integers.
f linear inequalities in which
This c h a r acte r i zation, i n c onjunction with
technique s of integer linear progra:m:ming [4], a llo ws t h e pos sibi lity of syste:matic
co:mputation of equilibrium points for t he t wo-pe rson c a se.
Finally, in Theore:m 4, we note a
u rio u s relation between the Lagrange
:multipliers and the original variables o f the non lin e a r p r ogra:ms of Theore:ms 2
and 2n; a solution to these progra:ms plays a dua l r o l e , serving si:multaneously
as values for the original variables a nd as L a gra nge :multi pliers.
Since our general sche:me can be e a sil y i nfe r red f ro:m the treat:ment of
two-person ga:mes, this case is treated f irst .
notation;
e
We use the conveniences of :matrix
will denote a vector (row or colu:mn) with each entry
or :matrix inequalitie s hold entry by e ntry.
I , while vector
2.
Two-person Ga m s
Let G be a two-person game in normali zed f orm [11], characterized
by two real m by n payoff matrices, A :: (a . .), B = (b
IJ
IJ
i £ {L ... , m} and player 2 (simultaneously) choos e s j
if player 1 chooses
.):
o
{ L ... , n}, the n 1 is
£
paid a .. and 2 is paid b
Mixed strategies for 1 and. 2 a re probability ve c t ors of
IJ
IJ
dimensions m and n; for convenience, we regards l's mixed strategies as row
.'
o
vectors and 2' s as column vectors.
Let X and Y deno t e the sets of mixed strategies
for players 1 and 2; i. e.
X
= {x :;
I
(x..) xe
1
If 1 uses mixed strategy x
E
= 1,
x
~ o}, Y
X and 2 uses y
:=
E
{y ~ (y.)
J
I
ey ~ 1, y> O}.
-
Y, their expected payoffs are
9
respectively
=
xAy
An equilibriwn point
X
E
X, Y
E
L;
i, j
x ,a .. y.,
1 1J J
xBy
(x~
is a pair of mixed strategies,
yL
such that for all
Y,
xAy
A theorem due to Nash
2
xAy,
xBy > xBy.
[9] as serts the existe c
of an equilibrium point for every
We note, for later reference, that A and B Dlay be taken so 0 < a,. < 1,
- IJo -< b 1J, . < .l (all i and j), by a. change in s cale a nd origin of the payoff measurements
without loss of generality in regard to equilib rium points.
game G.
Theorem 1.
(x, y) i s a n equilibriu m point of game G,
if and onl y if there exi s t scalar s
Ay
(1( ,
13
s uch that
< ae, xB < l3e, xAy + xBy :: a + .[3'9
XE
X, Y
E
Y •
Proof. Let (x, y) be an e q ilibr i u m p oint of ga me G and define a == xAy,
13 := xBy.
Then, for all x
E
X, Y
xAy < a,
Since xAy
+ xBy
is satisifed.
;:: a
+ 13,
E
Y,
or Ay < £te ,
by definition, a nd x
E
xBy~
X, Y
~
13,
or xB < l3e.
Y, the system of the Theorem
On the other hand, let (x, y,
QI ,~)
s atis fy 'he syste m of t he Theorem.
Then, multiplying the first relation by x, and the s e c o nd by y, w e have
and these with
xBy <
xAy
~
xAy
+ xBy
a,
=a
xAy :; a ,
Thus , for any
x~
y
X,
£
;
~
+~
impl y
= 13.
xBy
Y,
xAy .::
=
QI
xBy <
xAy,
~
- xBy
and (x, y) is an equilibrium point of game G.
Theorem 2 0 (X9y) is an e quilibrium point of game G
i f and only if (x, y, a , 13) is a sol u i on 'o r program
P:
max xAy
Ay~
when
Proof.
+ xBy
ae,
~
- a
f3
xB~ ~ e ~
We note, to begin with, th a
xe;X , yg Y
by
o mbining the constraint s of
program P we obtain
xAy
hence
9
+ xBy
J3
- a -
~
0
the value of program P, i f it exis t s , is nonp o s i tive.
Let
(x ~
y) be an equilibrium point of game G.
Theorem 1 is satis f ied; 1. e. t here exi s ' a and {:3
Ay .:: a e ,
xB
~
J3e,
x e:
X~
y
The n the syste m o f
h that
BU
y
£:
so (x s y p a, 13) is feasible in program p , a nd i n additi on
xAy
so (x, y
-&' ,9
13)
+ xBy
- ~ = 0
~
is ~ i ndeed, a solut i on f or pr o gram P.
On the other hand . let
(x ~
y,ag
~)
theorem of Nash guarantees the exi s ten ce
b e a solu ti on 'o r p r og r am. P.
f an e qw lib r i u rn point
game G and thereby , by the r e asoning a b v e , t h
e xis t ne e
(x ~
The
y) f or
f a solu i on
4.
~)
(x, y, a?
Then we have, for (x, y, a, I3),
for program P with value zero.
+ xBy
xA Y
Ay ~ ae,
=
- a - {3
xB ~ {3e,
x
X, Y
£
0
E
Y
.
This is the system of Theorem 1; hence (x, y) is an equilibrium point of game G.
Theorem 30
(x, y) is an equilibrium point of game G
scalars a, {3 and vectors
if and only if there exist
u, v, such that
o
< ae - A y
o<
e - u < e - x
~
- v < e - y
{3e - xB < e
X £
X,
Y
£
Y
,
and the components of u, v are integers.
Proof.
of Theorem 1.
We show the system of Theorem 3 is equivalent to the system
The result desired then follows immediately.
Let (x, y,
a,~,
u, v,) satisfy the system of Theorem 3.
O<x~u~e,
Then
O~y~v<e
and components of u and v are 0 or 1.
Now, if u.
1
if
=1,
u.
~
a -
a . . y.
IJ
J
j
_. 0,
x.
1
=
0;
c:::
0;
1
and
~
x.(a -
i
~
Similarly,
1
~
a .. y.)
IJ
J
= 0,
or a ::.: xAy.
= xBy, and it can be yerified that (x, y , a, {3) satisfies the system of
Theorem 1.
On the other hand , let (x, y ,
,(3) satisfy the s y s t e m
and defil).e
1 if x . > 0
_
Ui
{
1
0 if x.. = 0
1
v.
J
= {
1 if y.
<
J
o if y.
J
>0
=0
of Theorem 1,
5.
o <
We also note, since 0 < a .. < 1,
IJ ~
o < a
Then
1
= xAy ~
<
b
that
1,
ij
o
1.,
it can be readily verified that (x, y, a:
< 13
13,
= xBy <
1.
u, v) satisfies the s y stem of
Theorem 3.
n-Person Garnes
An n-per son game
G
in normalized form is characterized by an
n
n-vector of multilinear payoff forms in n vectors, t h e mixed strate gi es o f
the players - 1. e. ,
••
0
,
X
n
£
X
n
x. = (x. ) and each
J
Jk
vector x. is a probability vector of the set of probabilit y vectors X ..
J
J
An equilibrium point is a v ec tor o f mixe d s t r a t egies x
(xl' ... ~ xn)
where each A, is linear (homogeneous) in each vector
1
=
such that for all
x. EX.
1
1
A . (x) > A . (x
Ix.),
I I I
all i
,
the notation on the right side meaning that the
replaced by
X.
every fi n i te
n,
~ th
1
. d
mlxe strategy x .
The theorem of Nash as s er t s that every game
1
.
IS
1
G , for
n
h a s an equilibrium poi n t .
Co r respondi ngl y, we consider the nonlinear program, in vector
variables J
xl~
... , xn
p
' and scalar va.riable s, al' ... , an
maximize
when
A i (x :i) < a, e,
-
1
where Ak (x :i) is t h e vector of c o e ffi c i en s of
the multi linear f orm --k
A (x).
in Ak(x) by Ak(x:ij)
x . £ X. ,
1
1
a ll i
o mpo n e nts o f the v e cto r
xi in
We a . so denote the m a tr i x of c oef f icient s of x.x.
1 J
; note the produ t xiAk (x:i j ) = Ak(x: j ). The generalization
6.
of the two-person _case follows.
Since the proof s a r e direct extensions of
Theorems above, they are omitted.
x = (x.., ••. , x) is an equilibrium
1.
n
point of game G
if and only if there exists a vector
Theorem In.
n
= (aI' ."., an)
tl
A. (x:i) <
1
tl.
-
1
e,
Theorem 2n.
such that
~
k
tl
k
, x
i
E
X.,
1
all i .
x is an equilibrium point of game
G
n
if and only if (x, a) is a solution for program P .
n
The equivalence of Theorem 2n between game G
and program P
n
n
suggests the possibility of further characterization through t he introduction
and interpretation of Lagrange multipliers from nonlin ear programming theory
[5}, [12].
Curiously enough, no further information is f orthcoming as the
following statement shows.
We leave its demonst rat ion for the reader.
Theorem 4.
The Kuhn-Tu cke
ne c essary condition
[5] that a feasible (x, 0') be a soluti on f o r program P
is that there exists Lagrange m
.tip liers ( , 'Y) s u ch
that, for all i
n
This condition is s a tis f ied when (
(x~
a), th e solution for program P
t
n
,,)
is t.aken to b e
, itself .
n
7.
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Op rations