Some sufficient conditions are given to show the existence of equilibrium points with finite spectrum for nonzero-sum two-person continuous games on the unit square. We also examine the question of uniqueness of the equilibrium point for such games

Parthasarathy, T.

Raghavan, T. E. S.

English

Publications of the IAS Fellows

PACIFIC JOURNAL OF MATHEMATICS
Vol. 57, No. 1, 1975
EQUILIBRIA OF CONTINUOUS TWO-PERSON GAMES
T. PARTHASARATHY AND T. E. S. RAGHAVAN
Some sufficient conditions are given to show the existence
of equilibrium points with finite spectrum for nonzero-sum
two-person continuous games on the unit square. We also
examine the question of uniqueness of the equilibrium point
for such games.
1* Players I and II choose secretly an x and a y in the closed
interval [0, 1]. Player I receives K
x
{x, y) and player II receives K2(x, y)
where Klf K2 are continuous on the unit square. The following theorem
is classical in game theory: ([3], see page 156).
There exists a pair of probability distributions (F°, G°), called
Nash equilibrium points satisfying
K,(F\ G°) ^ K
λ
{x, G°) for all x in 0 ^ x < 1
and
K2(F°, G°) ^ K2(F°, y) for all y in 0 ^ y ^ 1
where K
λ
{F, G) = \\κix, y)dF(x)dG(y) and
fa G) = ^K,{x, y)dG(y) etc.
Let £f be the set of such pairs (F°, G°).
One can ask the following questions. When does an (F°, G°) e g7
with the spectrums of F° and G° being finite? For what class of
games, g" has a unique point? Here we try to answer these questions
by giving some sufficient conditions.
2* In this section we prove the following results.
THEOREM 1. Let K
λ
(x, y) and K2(x, y) be continuous on 0 ^ x,
y ^ 1. Let K2(x, y) be concave in y for each x. Then there is an
equilibrium {F\ G°) such that G° is a degenerate probability dis-
tribution and F° is concentrated at most at two points.
THEOREM 2. Let K^x, y) and K2(x, y) be continuous on 0 ^ x9
y ^ 1. Let (d^/dy^Kzix, y) ^ 0. Then there is an (F°, G°) e <£ with
the spectrum of F° and G° finite.
265
266 T. PARTHASARATHY AND T. E. S. RAGHAVAN
We need the following theorem of Bohnenblust, Karlin, and
Shapley [1] in the sequel.
PROPOSITION. Let K be compact convex in R%. Let g
a
 be a family
of continuous convex functions on K. Let sup
α
 g
a
(x) > 0 for each
xeK. Then there exists (n + 1) induces alf <x2, -*-,an+1 such that
Σ ? ί ί ^iQctiiώ) > 0 for all x. Here (Xly λ2, •• , λ Λ + 1 ) is a probability
vector.
REMARK. Theorems 1 and 2 are known for zero-sum games [1],
[2]. Our proof follows similar lines.
Proof of Theorem 1. We first prove the result when K2(x, y) is
strictly concave in y for each x.
Let (F°, G°) e gf. Since K2(x, y) is strictly concave in y, σ(G°), the
spectrum of G° contains just one element, say y°. Let C = σ(F°) =
the spectrum of F°. Consider
max {K2(x, y°) - K2(x, y)} = f(y) .
c
We claim ψ(y) ^ 0 for all y, for otherwise
K2{x, y°) — K2(x, y') < 0 for all x in C and for some yf .
Thus K2{F\ y°) - K2(F°, y') < 0 contradicting (F°, G°) e ϊf. Since K2{x,
yQ) — K2(x, y) is convex for each x and continuous in x for each y
over the compact set C it follows from the above proposition that
for some 0 ^ λ ^ 1
XK2(xu y°) + (1 - X)K2{x2i y°) ^ \K2(xu y) + (1 - X)K2(x2, y)
for all y. Define F* = XlXι + (1 - X)IX2. Here Ix stands for the de-
generate distribution at x. Clearly ( F j ° ) e ^ . The general case
is handled by approximating K2{x, y) by a sequence of strictly concave
functions in y. For the proof of Theorem 2 we need the following
lemma of Glicksberg [2].
LEMMA. Let h(y) 2: 0 on the unit interval with hn(y) > 0. Then
there exists a polynomial p(y) ^ 0 of degree at most (n — 1) such
that h(y) — p(y) ^ 0 on [0, 1] and has exactly n roots counting mul-
tiplicities.
For the sake of completeness we reproduce the proof.
Proof. By Rollers theorem we know that h{y) has at most n
roots. If it has exactly n roots p(y) = 0 is a choice. If h(y) has
EQUILIBRIA OF CONTINUOUS TWO-PERSON GAMES 267
fewer than n roots, then p(y) is constructed as follows. Let ylf y2,
• , yk be the roots of h(y). Let q(y) be a polynomial with the same
roots and multiplicities. Then q(y) ^ 0 and of degree at most (n — 1).
Let m4 be the multiplicity of yt. Then
limM.iim =§
ί
y^vi q(y) v-+vi qmi{y)
Hence we have open sets Et around yt with
h in
2Q(V) 2
The complement of U Et is a compact set, hence h/q is the ratio of
two continuous nonvanishing functions and that it achieves its min-
imum δ0. Hence for ε < min(<50, δJ2) h(y) — εq(y) ;> 0. Let ε0 be the
supremum of all ε for which h(y) — εq(y) ^ 0. Then h(y) — εQq(y)
has at least one more root for otherwise h(y) — e
o
q(y) would satisfy
all the conditions that h(y) satisfies and we cound find an ε' > 0 with
[h(y) - ε
o
q(y)] - εfq{y) ^ 0
which contradicts that ε0 is the supremum. Therefore we either have
a new root or at least the multiplicity of a former root is increased.
The function h — ε
o
g ;> 0 satisfies
dn
 (h - εq) > 0 .
dyn
We may continue the process until we arrive at a p(y) satisfying
the conditions of the lemma. Further one easily checks that this
polynomial is unique.
Proof of Theorem 2. First we prove the theorem when K2(x, y)
is of the form Kt(x, y) = ΣJ-oflφOϊ/' w i t h (Snβyn)K2(xf y) < 0. Let
(F°, G°)e gf. Since K2{F\ G°) = maxK2(F°, y) and (dn/dyn)K2(F\ y) <
0 we see that σ(G°) is finite. We will produce an F* with σ(F*)
finite and (ί7*, G°) e %?. Consider S = {a = (a0, alf- , αΛ): α€ = [y*dG, i =
0,1,2, •••,%, for some probability distribution G). Let M(x, a) =
Σ< ai(x)oCi. Clearly S is compact and convex in Rn+1 and M(x, a) is
aίfine in a for each x. Let C = σ(F°) and K2(x, G°) =
Define π/r(α) = max
ϊ e C {M(x, α°) — M(cc, a)}. As in Theorem 1 we
have ψ(a) :> 0 for every a in S and by the proposition of Bohnenblust,
Karlin, and Shapley we have a finite number of points x
u
 x2y -- ,xm
in C such that
268 T. PARTHASARATHY AND T. E. S. RAGHAVAN
Σ \M(xif a0) ^ Σ \M(xif a) for all a e S
where λ = (λ^ •••, λ
w
) is a probability vector. Let F* — Σ i V x f
Clearly (F*9 G°) e g\ This proves Theorem 2 for the special case.
Proof of the theorem in the general case. Let (F°, G°) 6 g* and
h(y) = ^Kt(x, y)dF°(x). Without loss of generality K2{F\ G°) = 0, and
hence h(y) <: 0 for all y. From the above lemma of Glicksberg we
have a polynomial P(y) with h(y) ^ P(y) ^ 0 and P{y) — h(y) has
exactly n roots counting multiplicities. For each x we can find coef-
ficients a,i(x)9 i = 0, 1, , n — 1 with Σ ^ι(^)yi ~ K2{x, y) having the
same roots and multiplicities as that of p(y) — h(y).
Now we will show that the coefficients at{x) are continuous in
x9 for i — 0, 1, 2, , n — 1. Fix x = x0 and consider Σ*=o <^i(^o)yi —
K2(x0f y). The polynomial Σ*=o cbi(^o)yί is unique for otherwise we will
have 2 distinct polynomials of degree less than n whose difference
will have n roots counting multiplicities. This is clearly not possible.
Thus for each fixed x the coefficients at(x) are uniquely determined.
In fact we may write n linear equations for the unknown a
o
(x), ,
α
w
_j(x). The matrix of coefficients in these equations is nonsingular
and since this matrix is independent of x9 it follows from the con-
tinuity of K2{x, y) in x, that the a^xYs are continuous in x. Now
since the roots interior to [0, 1] are of even multiplicity
Σ diίtyy* — K2(x, y) Ξ> 0 for all (x9 y)
or
Σ ^i(^)yi — K2(x9 y) ^ 0 for all (x, y) .
But \ Σ ai(x)yidFfi{x) — K2(F°, y) has roots and multiplicities as that
of P(y) — h(y). Since P(y) is unique P(y) = \ Σ ai(x)yidF°(x). Si
> M /^) f° r some y the first inequality holds.
Since K2(x, y) ^ Σ oΛ&V* = J°(^ »)
0 = ίΓ2(F°, G°) ^ (J Σ ai(x)dF0(x)yidG0(y) -
^ \p(y)dG°(y) ^ 0 .
Thus P(F°, G°) = 0^ P{F\ y) = P(y) for all y. We therefore have
{F\ G°) as an equilibrium point of the auxiliary game. We know
from the first part that this can be replaced by (ί7*, G°) where
σ(jF*) is a finite subset of σ(F°). We claim (F*f G°) is also an equi-
librium point for the original game. To show this it is enough
to prove # 2(F*, G°) = P(F*, G°) for then K2(F*, G°) = P(F*, G°) ^
EQUILIBRIA OF CONTINUOUS TWO-PERSON GAMES 269
P(F\ G) ^ K2(F*, G) for all G. Suppose KΛ(F*, G°) < P(F*, G°) (the
other inequality cannot hold). Then K2(x, y) < P(x, y) for some
x e σ(F*) (Z σ(F°) and yeσ(GQ) and K2(x, y) £ P(x, y) for all (x, y).
Combining these two statements we have 0 = K2{F\ G°) < P(F°, G°) ^
0 a contradiction. Hence the assertion. This completes the proof of
Theorem 2.
3. In this section we give a set of sufficient conditions for the
uniqueness of optimal strategies and equilibrium points for continuous
games.
DEFINITION. Let μlf μ2 be two probability measures on the unit
interval. We say μt ^ μ2 if for some k > 0 μx{E) ^ kμ2(E) for all E.
We say μ
x
 ~ μ2 if μ1 ^ μ2 and μ2 ^ μx.
THEOREM 3. Let K{x, y) he a continuous payoff on 0 <^  x, y ^ 1 for
a zero-sum two-person game. Suppose every optimal strategy for
each player has the spectrum the entire unit interval. Further
assume every pair of optimal strategies for each player be equivalent.
Then there is only one optimal strategy for each player.
THEOREM 4. Let K
x
{x, y), K2(x, y) be continuous payoffs on 0 ^
#,2/^1 for a nonzero-sum two-person game. Let for every (μ
ί9 vλ) e
gf, (μ2, v2) e &σ(μ,) = σ(μ2) = σfa) = σ(v2) = [0, 1] and μx - μ2y v, - v2.
Then g7 has just one element.
Proof of Theorem 3. It suffices to prove that the compact convex
set of optimal strategies for each player has exactly one extreme
point.
Let if possible μlf μ2 be two distinct extreme optimal strategies,
say for player I. Since μ1 ~ μ2 we have a k > 2 such that μλ(E) ^
kμ2(E) for all E. Define
μ\E) = (1 + θ)μ2{E) - θμ^E)
and
μ"(E) = (1 - θ)μ2(E) + θμ^E)
where
0 < ^ < 1 .
k — 1
When the spectrum of an optimal strategy for player II is the
whole unit interval every optimal strategy for player I is an equalizer.
That is
270 T. PARTHASARATHY AND T. E. S. RAGHAVAN
and
\K(x, y)dμ](x) = v for all y
\κ(x, y)dμ2(x) = v for all y .
Thus it is easily seen that μ'y μ" are two distinct optimal strategies
for player I and that μ2 = (μf + μ")/2. This contradicts the fact that
μ2 is an extreme optimal for player I. Hence the theorem.
Proof of Theorem 4. Since for any pair (μl9 vλ) e gf, (μ2, v2) e g7
we have
^x, y)dυ2{y) = c2, \^K2{x, y)dμ1(x) = a,
^α, y)dv
ί
(y) = clf ^K2(x, y)dμ2(x) = a2
(μlf v2) and (μif vt) also belong to i? and that g7 is a convex set. Of
course g7 is compact. The rest of the proof is as in Theorem 3.
REMARK 1. For example if every optimal strategy for each player
possesses a continuous strictly positive density, then one can check
that the conditions of Theorem 3 are satisfied; hence such games will
have unique optimal strategies.
REMARK 2. For matrix games the notion of equivalence coincides
with the notion of completely mixed strategies.
REMARK 3. It would be interesting to know whether Theorem
3 is valid if we just assume that the spectrum of every optimal
strategy for each player is the unit interval.
ACKNOWLEDMENT. We wish to thank the referee for some useful
comments.
REFERENCES
1. H. F. Bohnenblust, S. Karlin, and L. S. Shapley, Games with continuous convex
payoff, Annals of Mathematics Studies, 24 edited by H. W. Huhn and A. W. Tucker,
Princeton University Press, (1950), 181-192.
2. I. Glicksberg, A derivative test for finite solutions of games, Proc. Amer. Math.
Soc, 4 (1953), 895-897.
3. T. Parthasarathy and T. E. S. Raghavan, Some Topics in Two Person Games,
Amer. Elsevier Publ. Co., New York, 1971.
Received February 27, 1974. The first author is on leave from the Indian Statis-
tical Institute, Calcutta.
UNIVERSITY OF ILLINOIS


Equilibria of continuous two-person games

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Equilibria of continuous two-person games

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