The function that assigns to each matrix game (i.e., the mixed extension of a finite zero-sum two-player game) its value is axiomatized by a number of intuitive properties.game theory;noncooperative games;matrix games;axiomatic methods

Norde, H.W.

Voorneveld, M.

English

Research Papers in Economics

 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
No. 2003–17 
 
 
 
AXIOMATIZATIONS OF THE VALUE OF 
MATRIX GAMES 
 
By Henk Norde and Mark Voorneveld 
 
February 2003 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
ISSN 0924-7815 Axiomatizations of the value of matrix games
Henk Norde
Department of Econometrics and OR
Tilburg University
P.O. Box 90153
5000 LE Tilburg
The Netherlands
H.Norde@uvt.nl
Mark Voorneveld
Department of Economics
Stockholm School of Economics
Box 6501
1 1 38 3S t o c k h o l m
Sweden
Mark.Voorneveld@hhs.se
February 24, 2003
Abstract
The function that assigns to each matrix game (i.e., the mixed extension of a ﬁnite zero-sum
two-player game) its value is axiomatized by a number of intuitive properties.
JEL-code: C72 .
Key-words: matrix game, value.
1I n t r o d u c t i o n
The minimax theorem of Von Neumann (1928) states that every ﬁnite zero-sum two-player
game (or matrix game) has a well-deﬁned value. Formally, an m×n matrix game is represented
by a matrix A ∈ Rm×n,w i t hm,n ∈ N.T h e e n t r y aij ∈ R indicates the payoﬀ to player 1
if he chooses row i ∈{ 1,...,m} andplayer 2 chooses column j ∈{ 1,...,n}.T h e g a m e i s
zero-sum, so the payoﬀ to player 2 then equals −aij. Payoﬀs are extended to mixed strategies
in the usual way. Let G = ∪m,n∈N Rm×n be the set of all matrix games. For k ∈ N,l e t
∆k = {x ∈ Rk
+ |
￿k
i=1 xi =1 } denote the unit simplex in Rk.T h evalue of a game A ∈ Rm×n is
deﬁned by
v(A)=m a x
x∈∆m
min
y∈∆n
xtAy =m i n
y∈∆n
max
x∈∆m
xtAy,
where the superscript t denotes transposition.
The purpose of this paper is to axiomatize the function v : G → R that assigns to each
matrix game A ∈ G its value v(A) ∈ R. Recalling that the value can be interpretedas the
expectedpayoﬀ that player 1 (the row player) can guarantee himself, regard less of the strategy
choice of his opponent, the following properties are intuitive:
Monotonicity: If all payoﬀs in the matrix are weakly increased, this should have a nonnegative
eﬀect on the value.
Symmetry: If the role of the row player andthe column player is exchanged , the value of the
new game equals minus the value of the original game.
1Objectivity: In a trivial game where each of the two players has only one strategy, the value
of the game is the payoﬀ corresponding with this strategy combination.
Subgame property: Removing one of the rows implies a decrease in the strategic possibilities
of the row player. This shouldhave a nonpositive eﬀect on the value.
Independence of irrelevant alternatives: Adding a row that is payoﬀ equivalent with a
mixedstrategy of the row player d oes not change his strategic possibilities. This shouldnot
aﬀect the value.
Lower bound property: The value of the game is at least as large as the smallest payoﬀ in
the matrix.
Strictly dominated action property: If one of the rows is strictly dominated (i.e., if the row
player has a mixedstrategy that is strictly better than that row, regard less of the choice of the
column player), then removing that row leaves the value of the game unaﬀected.
These properties are formalizedin Section 2 andare usedto give several axiomatizations of the
value function. In Section 3, it is shown that the axiomatizations employ logically independent
axioms. Potential relaxations of some axioms andd irections for further research are d iscussed
in Section 4.
2 Axiomatizations
In this section, the properties from the introduction are formally deﬁned and four axiomatiza-
tions of the value function are provided. For a function f : G → R we introduce the following
properties:
Monotonicity: For each m,n ∈ N,a n dA,B ∈ Rm×n,i fA ￿ B (i.e., aij ￿ bij for all entries),
then f(A) ￿ f(B).
Symmetry: For each m,n ∈ N and A ∈ Rm×n,−f(A)=f(−At).
Objectivity: If A =[ a] ∈ R1×1,t h e nf(A)=a.
Subgame property: For each m,n ∈ N with m ￿ 2 and A ∈ Rm×n,i fB ∈ R(m−1)×n is
obtainedfrom A by deleting one of its rows, then f(A) ￿ f(B).
Independence of irrelevant alternatives: For each m,n ∈ N and A ∈ Rm×n,i fB ∈
R(m+1)×n is obtainedfrom A by inserting a row that is a convex combination of the rows of A,
then f(A)=f(B).
Lower bound property: For each m,n ∈ N and A ∈ Rm×n, f(A) ￿ mini,j aij.
Strictly dominated action property: For each m,n ∈ N with m ￿ 2 and A ∈ Rm×n,i f
B ∈ R(m−1)×n is obtainedfrom A by deleting a strictly dominated row1,t h e nf(A)=f(B).
1For k ∈ N and i ∈ {1,...,k},l e tei ∈ R
k denote the i-th standard basis vector (with one as the i-th coordinate
and zero everywhere else). Row i is strictly dominated if there exists a strategy x ∈ ∆m for which e
t
iAy < x
tAy
for all y ∈ ∆n. Similarly, column j is strictly dominated if there exists a strategy y ∈ ∆n for which x
tAej >x
tAy
for all x ∈ ∆m.
2Remark 2.1 A simple argumentation shows that symmetry andind epend ence of irrelevant
alternatives of f imply: for each m,n ∈ N and A ∈ Rm×n,i fB ∈ Rm×(n+1) is obtainedfrom
A by inserting a column that is a convex combination of the columns of A,t h e nf(A)=f(B).
This property is usedin the proof of Theorem 2.2.
Similarly, if f satisﬁes symmetry andthe strictly d ominatedaction property, then d eleting a
strictly dominated strategy of the column player does not aﬀect the function value. In particular,
this implies that in a game A ∈ R1×n where the row player has only one strategy, f(A) depends
exclusively on the smallest entry in the matrix. This property is usedin the proof of Theorem
2.4. ￿
Theorem 2.2 The value function v i st h eu n i q u ef u n c t i o no nG satisfying monotonicity, sym-
metry, objectivity, the subgame property, and independence of irrelevant alternatives.
Proof. The value function v clearly satisﬁes the ﬁve properties. Let f : G → R also satisfy
them. To show: f(A)=v(A) for all A ∈ G.L e tm,n ∈ N and A ∈ Rm×n.L e tz ∈ ∆m be a
maximin strategy of the row player: miny∈∆n ztAy = v(A).T h i si m p l i e s
∀j ∈{ 1,...,n} : ztAej ￿ min
y∈∆n
ztAy = v(A). (1)
Let B ∈ R(m+1)×n be the matrix obtainedfrom A by inserting ztA as a ﬁnal row:
B =
￿
A
ztA
￿
.
Independence of irrelevant alternatives implies
f(A)=f(B). (2)
Deleting all rows of B, except the ﬁnal one, andrepeated ly applying the subgame property
yields
f(B) ￿ f([ztA]). (3)
Property (1) implies that [ztA] ￿ [v(A)···v(A)] ∈ R1×n. By monotonicity:
f([ztA]) ￿ f([v(A)···v(A)]). (4)
Repeatedapplication of symmetry andind epend ence of irrelevant alternatives (see Remark 2.1)
andobjectivity yield
f([v(A)···v(A)]) = f([v(A)]) = v(A). (5)
Combining (2) to (5) yields f(A) ￿ v(A).S i n c eA is an arbitrary game, this implies that f ￿ v.
Conversely, for every A ∈ G: f(A)=−f(−At) ￿ −v(−At)=v(A) by symmetry. So f ￿ v.
Hence f(A)=v(A) for all A ∈ G. ￿
Theorem 2.3 The value function v i st h eu n i q u ef u n c t i o no nG satisfying symmetry, the sub-
game property, independence of irrelevant alternatives, and the lower bound property.
3Proof. The value function v clearly satisﬁes the four properties. Let f : G → R also satisfy
them. To show: f(A)=v(A) for all A ∈ G.L e tm,n ∈ N and A ∈ Rm×n.L e tz ∈ ∆m be a
maximin strategy of the row player: miny∈∆n ztAy = v(A).S i n c eminj ztAej =m i n y∈∆n ztAy =
v(A), the smallest entry of the matrix [ztA] ∈ R1×n equals v(A). By the lower boundproperty:
f([ztA]) ￿ v(A). (6)
Consecutively using independence of irrelevant alternatives, the subgame property (m times),
and(6) yield s
f(A)=f
￿￿
A
ztA
￿￿
￿ f([ztA]) ￿ v(A).
Since A is an arbitrary game, this implies that f ￿ v. Conversely, for every A ∈ G: f(A)=
−f(−At) ￿ −v(−At)=v(A) by symmetry. So f ￿ v.H e n c ef(A)=v(A) for all A ∈ G. ￿
Theorem 2.4 The value function v i st h eu n i q u ef u n c t i o no nG satisfying monotonicity, sym-
metry, objectivity, independence of irrelevant alternatives, and the strictly dominated action
property.
Proof. The value function v clearly satisﬁes the ﬁve properties. Let f : G → R also satisfy
them. To show: f(A)=v(A) for all A ∈ G.L e tm,n ∈ N and A ∈ Rm×n.L e tz ∈ ∆m be a
maximin strategy of the row player: miny∈∆n ztAy = v(A).L e tC ∈ Rm×n be a matrix with all
entries strictly smaller than mini,j aij andconsequently also strictly smaller than v(A).T h e n
f(A)=f
￿￿
A
ztA
￿￿
￿ f
￿￿
C
ztA
￿￿
= f([ztA]) = f([v(A)]) = v(A),
where the ﬁrst equality follows from independence of irrelevant alternatives, the inequality from
monotonicity, the secondequality from repeatedapplication of the strictly d ominatedaction
property. To establish the thirdequality, remember that miny∈∆n ztAy =m i n j ztAej = v(A).
Symmetry andthe strictly d ominatedaction property (see Remark 2.1) allow us to d elete
from [ztA] ∈ R1×n all columns with a number larger than v(A), while independence of ir-
relevant alternatives allows us to delete multiple occurrences of the coordinate v(A) if nec-
essary. This proves the thirdequality. The ﬁnal equality follows from objectivity. Hence
f(A) ￿ v(A).S i n c e A is an arbitrary game, this implies that f ￿ v.C o n v e r s e l y , f o r e v e r y
A ∈ G: f(A)=−f(−At) ￿ −v(−At)=v(A) by symmetry. So f ￿ v. Hence f(A)=v(A) for
all A ∈ G. ￿
Vilkas (1963) was the ﬁrst to provide an axiomatization of the value function v : G → R.I n
ad d ition to monotonicity, symmetry, andobjectivity, he usedthe following axiom:
Dominance: For each m,n ∈ N and A ∈ Rm×n,i fB ∈ R(m+1)×n is obtainedfrom A by
inserting a row [b1···bn] such that [b1 ···bn] ￿ ztA for some z ∈ ∆m,t h e nf(A)=f(B).
It is easy to see that the value function v satisﬁes dominance and that dominance implies both
independence of irrelevant alternatives and the strictly dominated action property. Thus, Vilkas’
characterization is a direct consequence of Theorem 2.4:
4Corollary 2.5 [Vilkas, 1963] The value function v i st h eu n i q u ef u n c t i o no nG satisfying
monotonicity, symmetry, objectivity, and dominance.
Notice that dominance is stronger than the conjunction of independence of irrelevant alternatives
andthe strictly d ominatedaction property: it also makes statements about weakly d ominated
actions. Yet, according to Theorem 2.4, the latter properties, together with symmetry, objectiv-
ity, andmonotonicity, suﬃce to axiomatize the value function. This makes Theorem 2.4 much
more appealing than Corollary 2.5.
3 Logical independence of the axioms
In this section, it is shown that the four axiomatizations use logically independent properties.
Proposition 3.1 The axioms from Theorem 2.2 are logically independent.
The result is proven by ﬁve examples of real-valuedfunctions on G, each of which violates exactly
one of the axioms. In all but one of the examples, it is straightforwardto check that certain
a x i o m sa r es a t i s ﬁ e d.T h i sp a r ti sl e f tt ot h er e a de r .
The construction of a function that satisﬁes all axioms in Theorem 2.2 except monotonicity
is the most troublesome. This example is discussed in detail. For each n ∈ N andeach A =
[a1 ···an] ∈ R1×n,de ﬁ n e
g(A)=2m i n
i
ai − max
i
ai.
For each m ∈ N andeach A =[ a1 ···am]t ∈ Rm×1,de ﬁ n e
h(A)=2m a x
i
ai − min
i
ai.
Moreover, for each m,n ∈ N and A ∈ Rm×n,de ﬁ n e
L(A)=m a x
x∈∆m
g(xtA) and U(A)=m i n
y∈∆n
h(Ay).
Since the functions g and h are continuous andthe unit simplices are compact, these maxima
andminima exist. Notice that for an m × 1 matrix x =[ x1···xm]t:
g(−xt)=g([−x1···−xm]) = 2min
i
(−xi) − max
i
(−xi)
= −2max
i
xi +m i n
i
xi = −h






x1
. . .
xm





 = −h(x)
andconsequently that for an arbitrary m × n matrix A:
L(−At)=m a x
x∈∆n
g(xt(−At)) = max
x∈∆n
g(−(Ax)t)=m a x
x∈∆n
−h(Ax)=− min
x∈∆n
h(Ax)=−U(A). (7)
Finally, deﬁne for each A ∈ G:
f1(A)=
1
2
L(A)+
1
2
U(A).
5By (7), the function f1 satisﬁes symmetry: for every A ∈ G, f1(−At)=1
2L(−At)+1
2U(−At)=
1
2(−U(A)) + 1
2(−L(A)) = −f1(A).
The function f1 satisﬁes objectivity: if A =[ a] ∈ R1×1,t h e nf1(A)=1
2L([a]) + 1
2U([a]) =
1
2a + 1
2a = a.
The function f1 satisﬁes the subgame property: let m,n ∈ N with m ￿ 2,l e tA ∈ Rm×n,
andlet B ∈ R(m−1)×n be obtainedfrom A by deleting one of its rows, say the i-th one. Then
L(A)=m a x
x∈∆m
g(xtA) ￿ max
x∈∆m:xi=0
g(xtA)= m a x
x∈∆m−1
g(xtB)=L(B).
Moreover, for every y ∈ ∆n,t h ev e c t o rBy is obtainedfrom Ay by deleting its i-th coordinate.
Hence maxj(Ay)j ￿ maxj(By)j and minj(Ay)j ￿ minj(By)j, which implies that h(Ay) ￿
h(By). Consequently
U(A)=m a x
y∈∆n
h(Ay) ￿ max
y∈∆n
h(By)=U(B).
Since both L(A) ￿ L(B) and U(A) ￿ U(B), it follows that f1(A) ￿ f1(B).
The function f1 satisﬁes independence of irrelevant alternatives: let m,n ∈ N, A ∈ Rm×n,
andlet B ∈ R(m+1)×n be obtainedfrom A by inserting a row (for notational convenience, we
will take this to be the ﬁnal row) that is a convex combination of the rows of A, i.e., there is a
z ∈ ∆m such that
B =
￿
A
ztA
￿
.
The convex hull of the rows of A is the same as the convex hull of the rows of B,s oL(A)=L(B).
Moreover, for every y ∈ ∆n,
h(By)=h
￿￿
Ay
ztAy
￿￿
= h(Ay),
since ztAy is a convex combination of the numbers in Ay andhence mini(Ay)i ￿ ztAy ￿
maxi(Ay)i. Hence also U(A)=U(B). Conclude that f1(A)=f1(B).
The function f1 does not satisfy monotonicity:
L([0 0]) = U([0 0]) = 0, so f1([0 0]) = 0, but L([0 1]) = −1,U([0 1]) = 0, so f1([0 1]) = −
1
2
.
The function f2 : G → R deﬁned by f2(A)=m a x i,j aij for all A ∈ G satisﬁes all axioms in
Theorem 2.2 except symmetry:
−f2
￿￿
0
1
￿￿
= −1 ￿=0=f2([−10 ] ) .
The function f3 : G → R deﬁned by f3(A)=0for all A ∈ G satisﬁes all axioms in Theorem 2.2
except objectivity. The function f4 : G → R deﬁned by f4(A)=1
2(maxi,j aij +m in i,j aij) for all
A ∈ G satisﬁes all axioms in Theorem 2.2 except the subgame property:
f4
￿￿
0
1
￿￿
=
1
2
< 1=f4([1]).
6The function f5 : G → R deﬁned by f5(A)=1
2(maxi minj aij +m i n j maxi aij) for all A ∈ G
satisﬁes all axioms in Theorem 2.2 except independence of irrelevant alternatives:
f5
￿￿
10
01
￿￿
=
1
2
￿=
3
4
= f5




10
01
1
2
1
2



.
Proposition 3.2 The axioms in Theorem 2.3 are logically independent.
This is easily checkedusing the functions d eﬁnedearlier:
The function satisﬁes all properties in Thm 2.3 except
f2 symmetry
f4 subgame property
f5 independence of irrelevant alternatives
f3 lower boundproperty
Proposition 3.3 The axioms in Theorem 2.4 are logically independent.
For each A ∈ G,l e td(A) ∈ G be the game obtainedfrom A by the iteratedelimination
of strictly d ominatedrows andcolumns. Recall, for instance from Osborne andRubinstein
(1994, Section 4.2.2), that d(A) is well-deﬁned. The function f6 : G → R deﬁned by f6(A)=
1
2(maxi,j d(A)ij +m i n i,j d(A)ij) satisﬁes all axioms in Theorem 2.4, except monotonicity:
f6
￿￿
11
10
￿￿
=
1
2
< 1=f6
￿￿
11
00
￿￿
.
This function, together with those deﬁned earlier, prove the proposition:
The function satisﬁes all properties in Thm 2.4 except
f6 monotonicity
f2 symmetry
f3 objectivity
f5 independence of irrelevant alternatives
f4 strictly dominated action property
Proposition 3.4 The axioms in Corollary 2.5 are logically independent.
Vilkas (1963) does not prove this result: his example violating monotonicity also violates objec-
tivity. The proposition follows from earlier examples:
The function satisﬁes all properties in Cor. 2.5 except
f6 monotonicity
f2 symmetry
f3 objectivity
f5 dominance
74 Concluding remarks
The full strength of the diﬀerent axioms is not always required in the proofs. We brieﬂy discuss
a number of relaxations of the axioms. In the proof of Theorem 2.2, a much weaker form of
monotonicity suﬃces:
Restricted monotonicity: f : G → R is monotonic on R1×2.
To see this, consider [ztA] ∈ R1×n as in (3). Independence of irrelevant alternatives allows us
to eliminate n − 2 elements that are not equal to the maximal andminimal elements of this
matrix. This yields an [y1 y2] ∈ R1×2 whose coordinates are maxj ztAej and minj ztAej = v(A)
with f([ztA]) = f([y1 y2]).S i n c e[y1 y2] ￿ [v(A) v(A)], restrictedmonotonicity andobjectivity
give
f([ztA]) = f([y1 y2]) ￿ f([v(A) v(A)]) = v(A),
just like in (4) and(5).
For the same reason, the lower boundproperty in Theorem 2.3 need s to holdonly for 1 × 2
games. Although it suﬃces to require monotonicity andthe lower boundproperty only on a
small set of matrix games, the original axioms are more elegant: they avoidthe consid eration
of a somewhat artiﬁcial subclass of games.
In conjunction with symmetry, the objectivity axiom can be relaxedas follows:
Restricted objectivity: If A =[ a] ∈ R1×1,t h e nf(A) ￿ a.
Indeed, let A =[ a] ∈ R1×1.T h e nf(A) ￿ a by restrictedobjectivity and f(A)=−f(−At) ￿
−(−a)=a by symmetry andrestrictedobjectivity. Thus f(A)=a, as requiredby objectivity.
Let us conclude by brieﬂy mentioning two potential topics for further research. Tijs (1975,
1981a) extends the characterization of Vilkas (1963) to diﬀerent classes of two-person zero-sum
games in which the players may have an inﬁnite set of pure strategies. Following his line of
proof, we believe that our axiomatizations of the value function can be extended to this context
of inﬁnite games.
A more interesting question is the following: it is well-known that von Neumann’s minimax
theorem and the duality theorem from linear programming are closely related. Is it possible to
use properties similar to the ones presentedin our paper to characterize the value function on
the set of feasible linear programs? The results from Tijs (1981b) suggest a positive answer; this
topic is taken up in further research by the authors.
References
J. ￿￿￿ N￿￿￿￿￿￿ (1928) Zur Theorie der Gesellschaftsspiele. Mathematische Annalen 100,
295-320.
M.J. O￿￿￿￿￿￿ ￿￿￿ A. R￿￿￿￿￿￿￿￿￿ (1994) A course in game theory,C a m b r i dg eM A :M I T
Press.
S.H. T￿￿￿ (1975) Semi-inﬁnite and inﬁnite matrix games and bimatrix games, PhD Disserta-
tion, University of Nijmegen.
S.H. T￿￿￿ (1981a) A characterization of the value of zero-sum two-person games. Naval Re-
search Logistics Quarterly 28, 153-156.
8S.H. T￿￿￿ (1981b) A characterization of the value of linear programs. S i m o nS t e v i nQ u a r t e r l y
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and its Applications 8, 304-307.
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