pitchik

Int. Journal of Game Theory, Vol. 10, Issue 3/4, page 207-221. �9 Physica. Verlag, Vienna.

Equilibria of a Two-Person Non-Zerosum Noisy Game of Timing 1 )

By C Pitchik, New York 2)

Abstract: Necessary and sufficient conditions are obtained for the existence of an equilibrium point (as well as for the existence of a dominating equilibrium point) in a two-person non-zerosum game of timing.

1. Introduction

Two toothpaste manufacturers are competing for a larger share o f the dentifrice market. Each is in the process of developing a new and bet ter toothpaste. The longer

one company waits to introduce its new toothpaste, the better its chances are of successfully capturing a share o f the market, if its product hits the stores first. (This is as- suming that the toothpaste is being technologically improved as time goes on.) Alter- natively, if a company waits too long to introduce its product , then it might be too late to successfully capture any of the market. (Everyone might already be quite hap- py brushing with the other company's toothpaste introduced just last week!) Essen- tially, the problem for each company is one of choosing a time at which to introduce their particular brand of toothpaste to the public.

Two researchers are working independently on a particular problem. When to pub- lish one's results is a big question. By publishing one's results first, one has some ad-

vantage over the other. Alternatively, by waiting until later, one can capitalize on weaknesses in the other 's results.

1) The first version of this paper was written in September 1979 while the author was a Ph.D. student at the University of Toronto and was revised in January 1981. This research was supported by a Natural Sciences and Engineering Research Council Canada Scholarship. The author is very grateful to Anatol Rapoport and Marc Kilgour for encouragement, suggestions and support, to Martin Osborne and two referees for constructive criticisms and suggestions and to both the Insti- tute for Advanced Studies at the Hebrew University of Jerusalem and Cowles Foundation for Re- search in Economics at Yale University for providing a stimulating working atmosphere. This research was also partially supported by the Office of Naval Research, Contract Number N00014-77-C-0518.

2) Prof. Carolyn Pitchik, Dept. of Economics, New York University, 269 Mercer Street, 7th floor, New York, N.Y. 10003, U.S.A.

208 C. Pitchik

The above examples illustrate some characteristics of a 2-person noisy game of timing which may or may not be zerosum. Mathematically, a 2-person noisy game of timing has the following structure. The player set is {P1, P2 ). The pure strategy set for Pi consists of all choices of times of action in [0, 1 ], the closed unit interval. The strategy set for Pi then consists of all cumulative distribution functions on the closed unit interval. Let the strategy set for Pi be denoted by F. Thus F ~ F if F is a right-continuous, non-negative, non-decreasing real-valued function defined on the real line R such that F (t) = 0 for t < 0 and F (t) = 1 for t ~> 1. Let the degenerate distribution with a jump of 1 at a point T E [0, 1 ] be denoted by ~ T" Thus

~ T ( t ) = { i fort~>Tf~

or, alternatively, we may write 8 T (T) -- 8 T (T--) = 1. The payoff to Pi, if each Pk acts according to a pure strategy 8 tk, tg E [0, 1 ] for

k = 1,2, is denoted by K i (~ti, ~t/) and is equal to

L i(ti) if t i< t /

Ki(6t i ' ~ ) = r if t i= t / (t:) if t i > t i

where Li, ~)i and 34i are real-valued functions defined on [0, 1 ]. Thus P/receives (i) L i (ti), if P/acts first at time ti, (ii) 49 i (ti), if both Pi and P~ act simultaneously at time ti, or (iii) M i (t/), if P/acts first at time t/. The above game is zerosum if K1 + K2 = 0 at all times.

If P1 and P2 choose mixed strategies F1 and F2 in F, then the payoff to Pi' denoted by K i (F i, F]), is equal to the Lebesgue-Stieltjes integral of the kernel

K i (~ti , ~t] ) with respect to the measures F~ and F2, i.e.

Ki(Fi, F] ) = f Ki(St, F])dFi(t) [0,1]

= f ( f Mi ( s )dF / ( s )+ i~ i ( t )%( t )+Li ( t ) (1 - -F / ( t ) ) )dF i ( t ) [0,11 [0,t]

where a] (t) = F] (t) -- F] (t --) is the size of the ]ump at t ofF]. The above 2-person game of timing will be denoted by ( F, K1, K~). A strategy pair (F1, F2) is an equilibrium point (hereafter denoted by EP) of

( F, Ka, K2 ) if and only if K i (F i, F]) >1 K i (F, F]) for all F e F, i = 1,2, {i, ]} = {1,2}. An equivalent definition is that a strategy pair (F1, F2) is an EP of ( F, K1, K2 ) if and only if K i (F i, F/) >~ K i (5 T' F/) for all T E [0, 1]. They are equivalent since F E F is a right-continuous, non-negati{,e, non-decreasing function on [0, 1 ] such that F (1) = 1.

The early literature concentrates on EP's of 2-person zerosum games of timing with various restrictions on the kernels of each player. See BlackweIl [ 1948], Blackwell

Equilibria of a Two-Person Non-Zerosum Noisy Game 209

[ 1949], Glicksberg [ 1950], Blackwell/Girshick [ 1954], Karlin [ 1959], F o x / K i m e l d o r f

[1969], Owen [1976]. Sudkzute [ 1969] initiated the study of non-zerosum silent games of timing. In a

silent game of timing, Li, r and 34i are functions of both t i and t] (signifying that each player does not know if the other has acted or not). More recently, Kilgour [ 1973], has obtained sufficient conditions for the existence of an EP in a 2-person non-zerosum noisy game of timing (with differentiability conditions on the kernel which imply conditions (i), (ii) below).

This paper is concerned with obtaining necessary and sufficient conditions for the existence of an EP in the (not necessarily zerosum) 2-person noisy game of timing in which Pi's kernel satisfies the following for i = 1,2:

Let a i maximize Min ~L i (t), M i (t)} in [0, 1 ].

(i) Li' r and M i are continuous real-valued functions on [0, 1 ] such that L i is a strictly increasing function while M i is a strictly decreasing function.

(ii) either lira [L i (t) -- L i (ai)]/[L i (t) --&I t. (t)] exists and is strictly positive (here- t ~ a i

after, this condition will be known as Condition I), or a i = 0 and either L i (0) >

3~/i (0) (which implies that lim [L i (t) - L i (ai)]/[L i (t) - - M i (t)] = 0) or t ~ a i

L i (0) = M i (0) and 3 e > 0 such that L i is differentiable in (0, e] and L~ (t)/[L i (t) - M i (t)] is bounded for t E (0, e] (hereafter, this condition will be known as Condition II).

Condition I is used solely in the "only if" part of Lemma 7A while Condition II is used solely in the "only if" part of Lemma 7B.

The first main result of this paper, Theorem 8, gives necessary and sufficient conditions for the existence of an EP in a game (F, KI, K2) which satisfies conditions (i) and (ii) above. (Hereafter, (F, K1, K2) will denote a game of timing (F, K1, K2) de- scribed in the Introduction which satisfies conditions (i) and (ii) above.)

A strategy pair (F1, F2) is a domina t ingEP of (F , K1, K~) if and only if (F1, F2) is an EP such that K i (F i, 1~) >~K i (G i, G/) for i = 1, 2, {i, ]} = {1,2) for any EP (G~, G2) of (F, K1, K2), i.e., a dominating EP is an EP at which the payoff to each player is larger than or equal to the payoff received at any other EP. A second result of this paper, Theorem 10, gives necessary and sufficient conditions for the existence of a dominating EP in (F, K1, K2) satisfying, in addition to (i), (ii) above, (iii) below:

(iii) L i (0) <<,M i (0) for i = 1,2.

2. Preliminary Notation and Definitions

Alternate proofs of Lemmas 2, 3 and 4 in Section 3 and of the " if" part of Lemma 7 in Section 4 can be found in Kilgour [1973] and Kilgour [1979]. For completeness,

210 C. Pitchik

the author offers these proofs (some of which use Lemma 1 very efficiently). In order to begin, the following notation and definitions will be needed.

Let Supp (F) denote the support of F E F, i.e., Supp (F) is the complement of the set of all points which have a neighborhood on which F is constant.

Let J/denote the set of jump points o fF i E F, i.e., J / = {t E Supp (Fi): F i (0 -- Fi ( t - - )>O).

Recall that a i (t) denotes the size of the jump at a jump point ofF/ , i.e., ai (t) = F i (t) - F i (t - ) for t E J/. If t q3 Ji' then ~i (t) = O.

3. Preliminary Lemmas

This section gives shape to the supports of strategy pairs which are possible EP's of ( ~:, K1, K2). The first simple yet useful lemma (see the proofs of Lemmas 3, 4, 5 and 6) is an elaboration of Lemma 2.2.1 in Karlin [1959]. Basically, Lemma 1 states that, if (F1, F2) is an EP of (F, K1, K2) and T E Supp (Fi), then, either T contributes to K i (Fi, F/) as much as the whole Supp (/7/) does or there exist points S n E Supp (Fi), S n q: T, S n converging to T that do the job. Let 3 denote "there exist"; V, "for each"; and let • denote, for U C [0, 1], the function defined by

= / 0 if t ~ U

Xu (t)

L 1 if tEU.

Given a strategy 17i E F we define a new function: H/: [0, 1] • t: ~ R as follows

(r, w = ; M t (t) a w (t) + (73 (73 • (73) + L; (73 (1 (73). [O,T)

Note the difference between this function H i and the restriction of the payoff func- t ionK i of Player i, K i : [0, 1] • F ->R defined previously as

Ki (S T, 1)) = f Mi (t) dFj (t) + aj (T) r (73 + Li (T) ( 1 - 1 ~ (73). [0,T)

These functions may differ whenever the first variable T belongs to CJ i ~ ~ (where C denotes the complement of Ji in [0, 1 ]) since, in that case, r (7) does not appear in the computation of H i (T, Fj) but does appear in the computation of K z. (6 T' Fj).

The following facts are almost immediate

(A) H i (T, FI) =K i (6 T, Fi) whenever T ~ J / U CJj.

Since f K i (5 T' Fi) dFi (T) = f ( f M i (t) dFj (t) + o~ (73 r (73 U U [0,T)

+ L i ( T ) ( 1 - - F / ( 7 3 ) ) d F / ( T ) = f ( f Mi (t) dFl (t) + a/ (T) r (T) X(o,1 ] (ai (T)) U [0,T)

+ L i (73 (1 --/7/(T))) dF/(T) = U f H/(T, Fi) dF/(T), it is true that

Equilibria of a Two-Person Non-Zerosum Noisy Game 2 1 1

(B) u f K i (~T' I~) dF i (7) = f H i (T, 1~) dF i (7) for any closed set U_C [0, 1].

Suppose that (F1, F2)is an EP of (F ,KI , K2). If ever Ki (F/, F]) <Hi (T, F]) on a set U of positive F/measure, then one could define a new distribution G i (by translat- ing F i and multiplying by a normalizing constant) on a closed set V of positive F i measure such that K i (Gi, F]) = (by (B)) : H i (T, 1~) dG i (7") > K i (Fi, 5 . ). This

would contradict the hypothesis that (F1, F2) is an EP of ([:, KI, K2). Thus, it is true that

(C) If (F,, F2) is an EP of ( F, K,, K2), then K i (F i, 1~) >1 H i (T, 1~) almost everywhere with respect to F i.

By facts (B) and (C), it is true that Ki (Fi, F/) = [0:1 ]Hi (T, F]) dF/(7) and

K i (F i, F/) >1 H i (T, t~) almost everywhere with respect to F i whenever (F,, F2) is an EP of (F,K1, K2). Thus,

(D) K i (F i, 1~) = Hi (T, F/) almost everywhere with respect to F i whenever (F1, F2 ) is an EP of ([:, KI, K2 ).

Supp (Fi) is a closed set in [0, 1 ] whose only possible isolated points must be jumps of the distribution F i. Thus,

(v,) If TE Supp (Fi) and T~ ]/, then 3 a sequence {Tn} C Supp (F/) 63 ( T - e , 7) for some e > 0 (and/or 3 a sequence {T n } C Supp (Fi) 63 (T, T + e) for some e > 0) such that T n converges to T (to be denoted by T n -+ 7).

Finally, Lemma 1 can be stated as follows:

Lemma 1. Suppose that (F1, F2) is an EP of([ : ,K1, K2). I f T E Supp (Fi) for some i = 1 or 2, then

(1) If r e ] / , then K i (F i, F/) = K i (ST, F]) = H (T, I~),

(2) If ~ a sequence {S n } C Supp (Fi) 63 ( T - el, 7) for some el > 0 such that S n ~ T, then 3 {T n } C Supp (Fi) 63 (T-- el, 7) such that T n -+ Tand such

that K i (F i, t~) = H i (Tn, 121. ) V n,

(3) If 3 a sequence (S n } C Supp (Fi) A (T, T + e2) for some e~ > 0 such that S n ~ T, then 3 {T n } C Supp (Fi) N (T, T + e2) such that T n -+ T and such

that K i (F i, t~) = H i (Tn, F/) V n.

Proof. Let (Fj, F2) be an EP of([: ,K1, K2) and let TE Supp (Fi) for some i = 1 or 2.

(1) is true by facts (A) and (D) since (T) is a set of positive F/measure if T E ]/. Suppose that 3 {Sn} C Supp (F/) N Vwhere V= ( T - - e l , 7) or V= (T, T + e2)

212 C. Pitchik

for some el , e2 > 0. Let V m = ( T - - (e l /m) , T) or let V m = (T, T + (e2/m)) depend- ing on whether V = ( T - - el, T) or V = (T, T + e2) respectively. For each m, 3 n such

that S n 6 Vm,SO that f dF i (t) > 0 (since S n E Supp (Fi)). Thus 3Tin E V m such Vm

that H i (T m , F]) = K i (F i, F1. ) by (D), i.e., 3 {T m } C Supp (Fi) Cl V such that T m -+ T

and K i (F i, F]) = H i (T m , F]) V m. []

Since the limit of a constant sequence exists, lim H i (T m , F])

({T m } c Supp (F/) (3 V as in (2) or (3) of Lemma 1) exists and is equal to

K i (Fi, 5 ) = lim H i (T m , F]) = f M i (t) dl~ (t) m-+~ [0,T)

+ %. (7) [L i (7) X(T.el,T ) (T1) + M i (T) X(T,T+e2 ) (T1)]

+ L i (7) (I - Fj (7))

by the Lebesgue Dominated Convergence Theorem and the fact that 2) a. (Tin) exists m I

implies that ~] (Tin) -+ 0 which implies that ct] (T m ) 0 i (T m) X[ 0,11 (ai (Tin)) -~ 0 as T m -+ T since r is bounded.

Thus, one can conclude that, if (F1, F2) is an EP of ( F, K1, K~ ) and T E Supp (Fi), i = 1 or 2, then (i) if T E J1 ;3 J2 and 3 sequence {S n } satisfying (2) in Lemma 1, then

cp i (T) = L i (T), (ii) if T E J 1 ('lJ 2 and 3 sequence {Sn} satisfying (3) in Lemma 1,

then r (7) =34i (i 0 and (iii) if T E J] and 3 sequences satisfying both (2) and (3) in

Lemma 1, then L i (7) = M i (7).

Lemma 2: The pure timing strategy pair (F1, F2) with F k = 5Tk for k = 1, 2, is an EP of (F ,K1 , K2) if and only if T1 = 7"2 = Tand for i = 1, 2

L i (1) if T = 1

r (T)/> Max {L i (T), M i (T)} if 0 < T < 1

M/(0) if r = O.

Proof." If T i <2 T/, then by the definition of an EP, L i (Ti) = K i (F i, F]) must be greater than or equal to K i (St, F]) = L i (t) for each t E (T., T].); but, this contradicts the assumption that L i is a strictly increasing function. Thus T1 = 7"2 = T. By the definition of an EP,

L i (t) if t < T

~i (T) = K i (F i, F]) >~ K i (fit' 5 ) = ~ Oi (T) if t = T

( M i ( T ) if t > T

for all t E [0, 1 ]. The Lemma now follows from the continuity and monotonicity of

L r []

Equilibria of a Two-Person Non-Zerosum Noisy Game 2l 3

Lemma 3 [for an alternate proof, see Kilgour, 1973, 1979] indicates that, if (F1, F2) is an EP of (F, K1, K2), then the supports ofF1 and F2 are identical until the probability of at least one player's having acted is one. A precise statement of this idea requires the following definitions.

Let e (F) = Max (t C [0, 1 ] : t E Supp (F)}. Thus e (F) is the earliest time of certain action corresponding to F.

Let Supp (F, G) = Supp (F) ~ Supp (G), i.e., Supp (F, G) denotes the common support of F and G.

Lemma 3." If (F1, F2)is an EP of (F,K1, K2) such that e (Fi) ~<e (E]), then

Supp (Fi) = Supp (E]) ~ [0, e (Ft.)].

Proof." Suppose that there exists a point T E Supp (Ft.) such that T ~ Supp (E]). Since Supp (F]) is closed and e (Ft.) ~< e (F]), there must exist points tl < t2 such that TE [tl, t2), [tl, t2] N Supp (E]) = ~, andE] (t2) < 1. By Lemma 1 and facts (A) and (E), 3 S E [h, t2) such that

K i (F i, F.) = K i (SS' F])

f M. (t) dE] (t) + f [0,S] (S, t2]

L i(S) dE](t)+ f L i(S) dE](t) (t2,1]

< f M/(t) dE/(t) + [0,s]

f M i(t) dF](t)+ f L i( t2)dF](t) (S, t21 (t2,1 ]

= Ki (6t2' El),

since F] (h ) -- F] (S) = 0, F. (t2) < 1, and L i (S) < L i (h) . This contradicts the hypothesis that (F1, F2 ) is an EP. []

Thus, Supp (F1,/;'2) = Supp (Ft.) whenever (F1, F2) is an EP of (F,K1, K2) and e (Ft.) ~< e (F]). Hereafter, the term initial support ofF1 and F2 naturally describes, and is sYnonymous with, the common support ofF1 and F2 whenever (F1, F2) is an EP of (F, K1, K2 ). This result contrasts with one of S~d~i~t~'s results in [ 1969] which states that if (F1, F2) is an EP of a silent non-zerosum game of timing, then Supp (F1) f3 (0, 1) = Supp (F2) r (0, 1). Lemmas 7A and 7B show that this is certain- ly not true for our noisy game of timing, (F, K1, K2).

Lemma 4 [for an alternate proof, see Kilgour, 1973, 1979] gives us more information about the possible behavior of an EP of (F, K1, K2). Recall that a i maximizes Min (L i (t), M i (t)) for t E [0, 1].

Lemma 4: Suppose that (F1, F2) is an EP of (F, K1, K2) such that T C- Supp (F1, F2). If T < e (F/.) and T < a i for i :/:/, then Tis a common jump of the EP 071, F2), i.e., T E J 1 A J 2 .

Proof." Suppose, to the contrary, that (El, F2) is an EP of (F, K1, K2) such that

214 C. Pitchik

3 T ~ Supp (F1, F2) satisfying (i) T < e (F]), ai, for i r and (ii) T ~ J l A J2- Since T<e(Fi.) ,ai , 3 e ~>0 such that Be1 > e such that T + el ~ J] and T + e l < e (/~), a i and such that B a sequence {T n) C (T - e, T + e) satisfying the conclusions in (2) or (3) of Lemma 1 (such a sequence exists by Lemma 3 and fact (E)) so that

Ki (Fi, t~) =lim H i ( T n , F / ) = f Mi (t) dl~ (t) + al (T) [Mi (T) x ( r r+e) (r l ) n~** [0,T)

+ (T) x(r.,, T) ( r , ) ] + Le (13 (1 - F/(T))

< f M i ( t ) d l ~ ( t ) + f Mi( t )dF]( t ) [0,T) [T,T+el)

+ f L i (T + el ) dF/(t) (T+el,ll

= ci e?, since T + el < e (F]), a i, and T + el ~ l], i.e., since F] (T + el ) < 1, L i (T) <

Li (T + e 1), /t4/ (t), for t, T "< a i and aj (T + e 1 ) = 0. This contradicts the definition

of EP. D

An immediate corollary to Lemma 4 is the following: Suppose that (F1, F2) is an EP of (F,K1, K2) such that TE Supp (Fx, F2) and T < e (F/) for some i. If Tis not a common jump, then T >I- a/for / q: i.

4. Key Lemmas and First Main Theorem

The following lemma provides the key to both theorems. Lemma 5 tells us that the existence of an EP (F1, F2) of (F, K1, K2) with a common jump implies the existence of a pure EP o f ( F , K1, K2). (If, in addition, L i (0) ~<M/(0) for i = 1, 2, then it also implies the existence of a pure EP of (F , K1, K2) which dominates (F1, F2).) The reason this information provides the key is that Section 3 already tells us a lot about the initial support of an EP of (F, K1, K2) which does not have a common jump. Sec- tion 3 also gives us necessary ar.d sufficient conditions for the existence of a pure EP of (F, K1, K2). Thus, after Lemma 5, it only remains to find necessary and sufficient conditions for the existence of an EP of (F, K1, K2) without any common jumps.

Lemma 5: Suppose that (F1, F2) is an EP o f ( F , K1, K2). If T E J 1 f~J2, then

(ST,~T) is an EP of (F,K1, K2). Suppose further that L i (0) <<. M i (0) for i = 1,2. If 3 T ~ J~ A J2, then 3 a pure

EP of ( F, K1, K2) which dominates (F1, F2).

Proof.' Suppose that (F1, F~) is an EP o f (F ,K1 , K2) such that T E J I (~J~. For any sequence (S n } C [0, T) if T > 0 and any sequence (Sn} C (T, 1 ] if T < 1, i f S n converges to T, then

Equilibria of a Two-Person No n-Zerosum Noisy Game 215

Ki(6Sn, l~) = f Mi ( t )dF( t )+a / (Sn ) r )+ f Li(Sn)dF/(t) . [O,S n) (Sn,1 ]

But, by the Lebesgue Convergence Theorem and the fact that ~ (Sn) ~ O,

Lim K i (SSn, t~) = f M i (t) d1~ (t) + M~ (7) ~ (T) XIT,1) ($1) n~** [0,T)

+Li (70 ~ (r)• ) (Sa ) + (r,1 f j Li (70 aFl (t)

> [0,T) f Mi(t) d t ~ ( t ) + ( ~

= K i (Ft., 1~) (by Lemma 1)

if either r (70 <M/(70 for T < 1 (choose (Sn} C (T, 1]) or r (7) < L i (70 for T > 0 (choose (S n) C [0, 70) since a/(T) > 0. This would contradict the hypothesis that (FI, F2) is an EP of (F, Kl, K2). Thus r (70 >~Mi (70if T < 1 and r (7) >1 L i (70 if T > 0, i.e., (6 T, 6T) is an EP of (F ,K1, K2) (by Lemma 2).

Further suppose that L i (0) ~ M i (0) for i = 1,2 and that 3 a common jump of (Yl, V~).

Let T = Inf (S E J1 r J2 ). (6 s, 6 s) is an EP V s E J1 N J2 (by above). Thus (6T, 6T) is an EP of(F ,K1, K2) by the continuity of Li, 4) i and M/. It remains to show that (5 T' 5 T) dominates (F1, Fz).

If T = 0, then r (0)/> M i (0)/> L i (0) (by above and by assumption respectively) implies that

~] (0) r (0) + (1 -- ~ (0)) L i (0) = H i (T, I~)

if T E J 1 N J 2 K i (5 o, 5 o) = 0i (0) >/

a] (0)114/(0) + (1 --a] (O)) L i (0) =Hi (T, F])

if T nEJa (~J2 and T n - + T

=Ki(F ,F? (by Lemma I).

If 0 < T ~< 1, then r (70 ~> Li (I") since (5 T' 5 T) is an EP. Thus, by ]_.emma 1,

[/-/i (T,F]) if r e J 1 fq:2

Ki(Ft"F] )=~l im H . ( T , if Jx tqJz, T n T I n - - - t n El) ThE --->

/I~ (0 aF/(0 + r (~ % (Z) + L~ (~ (1 --w 03) |

[ [0fT)M/(t) dF] (t) + 34/(T) a/(T) + L i (T) (1 -- F] (T))

216 C. Pitchik

since f M i (t) dF] (t) is non-zero only if T does not begin the support ofF] in which [0,T)

case ~i (73 >~Li (T) > M i (t) for all t E Supp (FI) f3 [0, T) (by Lemma 4, since Tis the earliest possible common jump).

Thus, (~ T' ~ T ) is an EP of (F, K 1, K2 ) which dominates (F1, F2 ). []

And so, ( F, K1, K2 ) has an EP with a common jump if and only if ( F, K1, K2) has a pure EP (see Lemma 5) if and only if q~i' i = 1,2, are both large for some TE [0, 1 ] (large in the sense of Lemma 2). Another consequence of Lemma 5 is that, i fL i (0) <~ M i (0) for i = 1,2, then it is only necessary to search among pure EP's of ( F, K1, K2) and EP's of (/:, K1, K2) without any common jumps for the existence of

a dominating EP of ( F, K1, K2). Lemma 2 already gives us necessary and sufficient conditions for the existence of a

pure EP of (F, Kx, K2). It remains to find necessary and sufficient conditions for the existence of an EP of (F, K1, K2) with no jumps in common. Lemmas 6, 7A and 7B provide us with exactly this information.

The next lemma rules out the possibility of an EP among a certain class of strategy pairs with no common jumps.

Lemma 6: Suppose that (F1, F2) is an EP o f (F ,K1 , K2). If J1 N J2 = 0, then Supp (F~, F2) = {e (F])} where e (F]) < e (Fi) i 3] .

Proof." By Lemma 3, Supp (F1, F2) = Supp (F]). Suppose that Supp (F]) contains more than one point, i.e., let T = Min Supp (F1, F2) and suppose that

T<e(l~)<~e(Fi) , {i ,]}= {1,2}. Since T E Supp (F~, F2) is not a common jump, 3 a sequence

{S n } C Supp (F1, F2) C~ (T, T + e) for some e > 0 such that S n -~ T (by fact (E) and Lemma 3). Thus, by Lemma i, 3 a sequence {T n } satisfying the conclusion in (3) of

Lemma 1. ThusK k (F k, Fl) = lim H k (T n, Ft) = M k (T) F t (73 + Llc (73 (1 --Fi(73) n - - ~ oo

for k = 1,2, (k, l} = {1,2}. T' Similarly, since e (F]) E Supp (F1, F2) is not a common jump, 3 a sequence { n }

satisfying the conclusion in (2) of Lemma 1. Let l be such that a t (e (/~)) = 0. Thus T' K k (F k, Ft) = lim H k ( n' Fl) = f M k (t) dF l (t). But then a contradiction

results since n - ~ [0,e(F.)l

K k (Fk, FI) = M k (T )F t (T) +L k (73 (1 - - F 1 (73)

> f M k (t) dF l (t) = K k (F k , El) [T,e(~)l

since T>~ a l (by Lemma 4), F l (73 < 1 and M k is strictly decreasing.

Among the strategy pairs without any common jumps, the only remaining candidates for an EP are those for which the initial support, Supp (F1, F2) is a singleton

[]


set {T} such that T ~ Jt N J2- The next lemma stipulates exactly under what conditions this type of EP can occur.

Let 0_ be the set of strategy pairs of (F, K~, Ks) without any jumps in common but with Supp (F1, F2) = Supp (Fi) = {T } where e (Fi) ~< e (Fj), i.e.,

Q = ((F1,F2)E F X F :Supp(F1 ,F2)=Supp(Fi )= {T}

where e (Fi) ~< e (F]), F/(T) = 0, a i (7) = 1}.

Note that, in the above, T < 1. For the proof of the "if" part of Lemma 7A, see Kilgour [1973, 1979]. Recall that Condition I states that lira [L (t) - -L (ai)]/[L (t) - -M (t)] exists and is

strictly positive, t~ai

Lemma 7A: 3 (FI, F2) E Q such that (F1, F2) is an EP of (F, K1, K2) (satisfying Condition I) if and only if 3 TE [0, 1) such that a i < T<<.~ andM] (T)/> q~/(T).

Proof." "if" Suppose that 3 TE [0, 1) such that Li (T) >M/(T) , M] (T) 1>/,] (T), ~! (7). Let F/(T) -- F i (T - ) = 1. Let F] be any absolutely continuous distribution such that Fj (T) = 0 and

z i (t) (73 F] (t) >~ Li (t) --%. (70 in (T, 1]

(such an F. exists since L i (t) > M i (7) for t E [T, 1]). Then 1

[ (t) = M,. (:o--> j(n

L (Z)

if t < T

if t = T

if t > T

= Kj(6t, 5 ) VtE[O,I],

i.e., F/is a best response for P] against F i.

Also g i (F., 1~) = L i (T) ~ M i (7) 5 (t) q- L i (t) (1 - -5 ' (s (by assumption) >~ f M. (s) dFj (s) + L i (t) (1 --/7] (t)) = K i (6 t, F/.) V t E [T, 1] since M/is de-

IT, t) creasing and K i (F i, F1. ) = L i (7 0 > L i (t) = K i (6 t, ~ ) V t E [0, T), i.e.,F/is a best response for Pi against F/.

Thus, (FI, F2) E Q is an EP of ( F, K1, K2). "only if" Suppose that (F1, F2) E Q is an EP of ( F, K1, K2). By definition of EP,

/,j (t) if t < T

M] (T) = K] (Fi, FI) >~ K] (6 t, Fi) = r (T) if t = T

M/(T) if t > r .

218 C. Pitchik

Also, T ~ Supp (F1, F2), T not a common jump, T < e (~)) implies that T >~ a i (by Lemma 4). Thus a i <~ T <~ ~ and A/i (7)/> ~l (7).

It remains to show that T must be strictly larger than a i. Suppose, to the contrary, that L i (T) = M i (T), Then

L i (7) = K i (Fi, Fj) >t K i (6 t' FI) : f M i (S) dt~ (S) + I~ (t) (1 -- L i (t)) [r,t)

>~ M i (t) Fj (t) + b) (t) (1 - -L i (t)),

for all but at most a countable set of t EJ], implies that/~ (t) ~> [L i (t) - -L i (7)1 /

[L i (t) - - M i (t)] which implies that lira. F] (t) ~> lim [1. i (t) -- L i (7)]/[L i (t) - - t -~ T t ~ a i

M/(t)] > 0 which contradicts the hypothesis that F 1 (7) = 0 since F] must be right- continuous. []

The counterpart of Lemma 7A uses Condition II. Recall that Condition II states that a i = 0 and either L i (0) > M i (0) or L i (0) = 31i (0) and 3 e > 0 such that L i is

�9 " ! " E differentlable in (0, e] and L i (t)/[L i (t) - - M i (t)] is bounded for t (0, e].

Lemma 7B: 3 (F1, F2) E O. such that (F1, F2) is an EP of (F,K1, K2) (satisfying Condition II) if and only if 3 r E [0, 1) such that a i <<. r <~ ~ and MI (7) ~> ~b i (7).

Proof." "only if" 3///. (7)/> L l (T), ~b l (T), as in Lemma 7A. Also r < e (/~) implies that T i> a: (by Lemma 4) Thus a. ~< T ~< a. and M. (7) i> ~b. (7).

" t 1 1 1 "if" if either 0 < T < a j andM, (7) ~> Oj (7) or T = 0 and Li(O) > M i (0),

M/(0) i> # (0), then (6 T' F]) E Q as in the if part of Lemma 7A will be an EP of (F,K1, K2)-

It remains to show that if T = a i = 0 and Mj (7)/> ~j ( t ) then 3 Fj ~ F such that (ST, Fj) E Q is an EP o f ( F,K1, K2).

Choose Fj E F as follows. Let Fj be any absolutely continuous distribution such that Fj (0) = 0, Fj (e) = 1 and such that Fj' (t) > L; (t)/[L i (t) - - M i (t)] in (0, el. Kj (Fj, 6 .) >~ K. (F, ~ o) for all F ~ F by assumptions on Mj. It remains to show that

1 I 8 0 is best for Pi against Fj.

K i (6 t, w = y (S) dFj (S) + (t) (1 --F/(t)). [0,t)

The derivative of K i (5 t, t~) with respect to t i sM i (t) F]' (t) + L~ (t) (1 - -F / ( t ) ) --

L i (t) F: (t) < 0 by assumption on F]' (t) for all t E (0, e]. But K i (8o, F]) = L i (0). 1 Thus K i (6 t, FI) < K i (60, F i) for all t E (0, e]. Also K i (St, Fi) = = f M i (s) dFj (s) < L i (0) for all t E (e, 1 ]. Thus, 8 o is best for Pi against FI.

[O,e(Fl)] Therefore (60, F / )~ Q is an EP of (F,K1, K2). []


And so, by Lemmas 6, 7A and 7B, the only candidates for an EP of (F, K1, K2) without any common jumps are those strategy pairs whose initial common support is a singleton set (T } such that T E [0, 1), M/(7)/> ~1 (T) and a i < T <<. ai if Condition I holds (a i = 0 ~< T ~<a/if Condition II holds).

We are now ready to state necessary and sufficient conditions for the existence of an EP of (/:, K1, K2).

Theorem 8: The game of timing, (/: ,K1, K2), has an EP if and only if there exists a point TE [0, !] such that

either (i) T = 0 and ~i (0) ~>M/(0) for i = 1, 2

or (ii) 0 < T < 1 and r (7) ~> Max (L i (T), M i (7)} for i = 1,2

or(iii) T = l a n d r ) for i = 1 , 2

or(iv) ai<T<<.~andM.(T)>~r for i4:]

or(v) O = a i = T < . a , M ( T ) > ~ (T) for i--l:] andCondi t ionl Iholds( ie , 1 1 1 "" either L i (0) > M i (0) or the derivative of L i exists in some interval (0, e] for e > 0 and L~ (t)/[L i (t) - -M i (t)] is bounded in (0, e]).

Proof." " if" If (i), (ii) or (iii) is true, then (5 T' 8 T) is an EP of (/:, K1, K2) by Lemma 2. If(iv) is true, then 3 (F1, F2) E Q which is an EP of (/: ,K1, K2) by the "if" parts of Lemmas 7A and 7B. If (v) is true, then 3 (Fl, F2) E Q which is an EP of (/:, K1, K2) by the "if" part of Lemma 7B.

"only if" Suppose that (FI,/72 ) is an EP of (/:, K~, K2 ). If there exists a point T E J~ fl J2, then (5 T' 8 T) is an EP (by Lemma 5) which implies, by Lemma 2, that one of (i), (ii) of (iii) is true. If J1 A J2 = 0, then (iv) or (v) is true by Lemmas 6, 7A and 7B. []

5. Dominance Theorem

Let us assume that, in this section, the game of timing, (F, K1, K2) under consid- eration also satisfies condition (iii) in the Introduction, i.e., in addition to the continuity and monotonicity conditions, the kernel also satisfies the condition that L i (0) <<. M i (0) for i = 1,2. Thus far, this condition was assumed only in the second part of Lemma 5 which established that, i fL i (0) <<.M i (0) for i = 1, 2, then the existence of a common jump in an EP (F1, F2) of (F, K~, K2) implies the existence of a pure EP (ST, ST) of( / : , K1, K2) which dominates (F1, F2). [IfLi (0) > M i (0) for some i, then this is not necessarily true, i.e., there may then exist an EP (F1, F2) with a jump in common even though no pure EP dominates (F1,/;'2).]

Theorem 10, in this section, will provide necessary and sufficient conditions for the existence of a dominating EP in (/:, K1, K2). Before we proceed, we need some addi- tional notation. Let

220 C. Pitchik

Q = (S E [0, 1] : ~ EP (F1, F2) E O. with Supp (F1, F2) = {S}}.

Thus, if Condition I holds, then

Q = {SE(0, 1 ) : a i<S<~5 ,M] (S )>~) ] (S )} ;

while, if Condition II holds, then

Q= {SE[O, 1) : ai=O<~S ~,Mi(S)>~qbj(S)}.

Let

P = {SE[0, 1] : (~S,5s)isanEPof(F,K~,K2) }.

Now, Theorem 8 can be restated as follows: An EP of (F, K1, K2) exists if and only i fP U Q ~ 0- The next lemma states that, if 3 EP E Q which is a dominating EP of (F, K1, K~), then Q is a singleton set.

Lemma 9: Suppose that Q contains more than one point. If (Fa, F2) E Q is an EP of (F, K1, K~_), then (F1, F2) is not a dominant EP o f (F ,K1 , K2).

Proof." Let Supp (F1, F2) = {T } for some T E Q. The payoffs to ~ and P] are L i (T), 3///(7) respectively, for i r (see Lemmas 7A, 7B). There exists a point S :# T, S E Q such that either both L i (T) < L i (S) and/14/(T) >/14] (S) or both L i (T) 2> L i (S) and 114 i (T) < 3//] (S). This is due to the monotonicity of L i and/14] and to the assumption that Q contains more than one point. By Lemmas 7A, 7B there exists an EP (of (F,K~, K2)) (G~, G2)E Q such that Ki (G i, ~ ) =L i (S) and/(] (G], Gi) =M i (S). The EP (F1, F2) does not dominate the EP (G1, G2). []

We are now prepared to state the conditions necessary and sufficient for the existence of a dominating EP of (F, K1, K2).

Theorem 10: A dominating EP of (/:, K1, K2) exists if and only if

either (i) 3 p C P such that r (P) >~ r (T) V TEP, i = 1,2 and (~i (P) >~Li (T), ~. (p) ~M. (T) for all T E Q, i :/:] (in which case, (6p 5p) is a dominating 1 1

EP of (F, K,, K2))

or (ii) Q = ( q } a n d L i ( q ) > r i~ ] ( inwhichcase 3 (F1, F2) E Q. which is a dominating EP of (F, K1, K2)).

Proof." "if" Suppose (i) or (ii) is true. Let (F,, F2) be the EP of (F, K,, K2) with respective payoffs to Pi, P'' i --/=], equal to r (P)' 491 (P) if (i) is true (equal to L i (q), 1 M i (q) if (ii) is true). Let (G1, G2) be any EP of (F, gl, K2). If there exists a common jump in Supp (G1, G2) then, by Lemma 5, there exists a pure EP of (F, K1, K2) which dominates (G1, G2). But, by assumption, (F1, F2) dominates all pure EP's of (F, K1, K2). Thus (F1, F2) dominates (G1, G2). If there does not exist a common


jump in Supp (G1, G2) then, by Lemmas 7A and 7B, the payoffs to Pi and P / f r o m

(G1, G2) are L i (T) ,3~ (T) respectively, for some T E a . Thus (F1, F2) dominates

(GI, G2). "only i f" Suppose that (F1, F2) is a dominating EP o f ( F, K, , K2). (1) I f

3 p E J 1 r J2, ((2) if J1 A -/2 = ~) then K i (F i, F,) = r (P) for i = 1,2 by Lemma 5; since, otherwise, K i (/7/, F/) < r (P) for i = 1, 2 / o r a pure EP (~. , 6 . ) contradicts the

dominancy of (FI, F2) (then K i (F i, F/) = L i (q}and K/(F/ , F/) e=/1~ (q) for {q} = a

by Lemmas 6, 7A, 7B and 9). Thus (1) if 3 p EJ~ A J 2 ((2) if J1 f~r2 = ~)) then (i)

((ii)) must be true since V T E P 3 EP payoffs of~ i (70 f o r P / f o r i = 1,2 and V T E Q

3 EP payoffs of L i (70, ~ (70 for P/, P] respectively, i 4= ], by Theorem 8. []

References

Blackwell, D. : The silent duel, one bullet each, arbitrary accuracy. The RAND Corporation, RM-302, 1948.

- : The noisy duel, one buUet each, arbitrary, non-monotone accuracy. The RAND Corporation, D-442, 1949.

Blackwell, D., and M.A. Girshick: Theory of Games and Statistical Decisions. New York 1954. Fox, M., and G.S. Kimeldorf: Noisy Duels. SIAM J. Appl. Math. 17, 1969, 353-361. Glicksberg, L : Noisy duel, one bullet each, with simultaneous fixe and unequal worths. The RAND

Corporation, RM-474, 1950. Karlin, S. : Mathematical Models and Theory in Games, Programing and Economics. Reading, Mass.

1959. Kilgour, D.M. : Duels, truels, and n-uels. Ph.D. Thesis, University of Toronto, Toronto 1973. - : The non-zerosum noisy duel, I. Circulated manuscript, 1979. Owen, G . : Existence of equilibrium pairs in continuous games. Int. J. Game Theory 5 (2/3), 1976,

97-105. Sffcl~ifft~, D.P. : The shapes of the spectra of equilibrium strategies for some non-zerosum two-

person games on the unit square. Lith. J. Math. 9 (3), 1969, 687-694. - : The existence and shape of equilibrium strategies for some two-person non-zerosum games with

the choice of a moment of time. Lith. J. Math. 10 (2), 1970, 375-389.

Received February 1981 (revised version September 1981)

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