some formal problems with the von neumann

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EDWARD F. MCCLENNEN

SO ME FO R M A L PR O B L E M S WIT H T H E VO N N E U M A N N

A N D M O R G E N ST E R N T H E O R Y O F T WO -PE R SO N ,

ZERO -SU M GAMES, I: THE DIRE CT PROO F

1 I N T R O D U C T I O N

In Chapter 111, Section 14 of T h e o r y o f G a m e s a n d E c o n o m i c B e h a v i o r 1,

Von Neumann and Morgenstern offer a theory concerning rational

choice in zero-sum, two-person games for which there exist saddle-pointsin pure strategies strictly determined games). The analysis in Section 14

stands in an intimate relation to the analysis of Section 17, which has

as its objective the construction of a general theory of rational solutions

for all zero-sum, two-person games. Serious students of game theory are,

of course, drawn to the discussion in Section 17, since it is there that a

proof is offered of the so-called main theorem, the Minimax Theorem.

The Minimax Theorem is essentially an existence theorem, which states

that all zero-sum, two-person games with finite sets of pure strategies for

each player have at least one saddle-point in mixed strategies. The line of

reasoning developed in Section 14 deserves careful attention, however,

for the reasoning of Section 17 consists primarily of an extension of the

reasoning of Section 14, from the case of pure, to the case of mixed,

strategies. This extension is major in terms of mathematical interest and

complexity, for it now becomes necessary to prove the existence of

saddle-points, whereas in the earlier analysis their existence was not at

issue. In other respects, however, the arguments of the two sections are

strictly parallel.

Section 14 can thus be said to lay the basis for a significant portion of

the construction of the more general theory in Section 17. As such, it is

worth considering what exactly is attempted in Section 14, and to what

extent that attempt is successful.

Stated somewhat informally, the theory offered with respect to zero-

sum, two-person games with a saddle-point in pure strategies strictly

determined games), is that if both players choose rationally, then the

pair of strategies chosen will be a maximin-minimax pair constituting asaddle-point of the game, and the payoff to each player in the game will

Theory and D ecision 7 1976) 1-28. All Rights ReservedCopyright 9 1976by D . R eidel Publishing Comp any, Dordrecht-Holland



2 EDW ARD F MCCLENNEN

b e t h e s a d d l e - p o in t p a y o f f . T h e a n a ly s is o f f e r e d i n s u p p o r t o f th i s t h e o r y

d iv ides in to th ree main pa r t s . In the f i r s t pa r t (14 .1 th rough 14 .3 ) , a

de ta i l ed ana lys i s i s o f f e red o f tw o va r ian t s o f an a rb i t r a r i ly s e lec ted

gam e G , - va r ian t s fo r wh ich , i t i s a rgued , the con cep t o f r a t iona l cho ice

c a n b e i n t e r p r e t e d i n a p r e c i s e a n d u n a m b i g u o u s m a n n e r . I n t h e s e c o n d

pa r t (14 .4 ), the ana lys i s o f the tw o va r ian t s o f G i s app l ied f o r heu r i s t i c

purposes to G i t s e l f . In the th i rd and l a s t pa r t (14 .5 ) , wha t i s va r ious ly

d e s c r i b e d a s a f in a l a n d a d i r e c t p r o o f o f t h e t h e o r y f o r m u l a t e d a b o v e

i s p resen ted . T he p r oo f i s s a id to be f ina l in the sense tha t i t cons t i tu tes

a f o r m a l a r g u m e n t f o r t h e t h e o r y w h i c h i s l o g ic a ll y i n d e p e n d e n t o f t h e

ana lyses con ta ine d in the f ir s t tw o pa r t s , an d , in pa r t i cu la r , o f the heur -i s ti c s t eps to b e fou nd the re . I t i s s a id to be a d i r ec t p r oo f to d i s t ingu i sh

i t f r o m a q u i t e d i f fe r e n t p r o o f o f t h e t h e o r y w h i c h i s o f fe r e d s u b s e q u e n t l y

in Sec t ion 17 , - a p r oo f w h ich i s, in the log ica l sense o f the t e rm, in -

d i re c t , a n d w h i c h b u i l d s u p o n a n d c o m p l e t e s t h e h e u r i st ic r e a s o n i n g o f

the f i rs t two pa r t s o f the ana lys i s in S ec t ion 14. 2

I n o r d e r t o fa c il it a te a n a p p r a i sa l o f t h e V o n N e u m a n n a n d M o r g e n -

s t er n a r g u m e n t I s h al l b e g in b y g iv i ng a s u m m a r y o f w h a t I t a k e t o b e

the m ain s t eps in the th ree p a r t s o f the ana lys i s d is t ingu ished a bove .A s i d e f r o m t h e a d o p t i o n o f a s o m e w h a t s i m p l i f i e d n u m b e r i n g s y s t e m ,

I h a v e s o u g h t t o r e m a i n a s f a i t h f u l a s p o s s i b l e t o t h e i r a r g u m e n t , b o t h

w i t h r e g a r d t o i t s s t r u c t u r e a n d t h e t e r m i n o l o g y w h i c h t h e y e m p l o y ,

e x c e p t f o r t w o m a t t e rs . F i r s t, a n u m b e r o f s te p s i n t h ei r a r g u m e n t i n v o l v e

c l a im s c o n c e r n i n g w h a t i s v a r i o u s l y d e s c r ib e d a s t h e g o o d w a y t o p l a y ,

a g o o d w a y t o p l a y , p l a y i n g w e l l , e tc . I t is c le a r f r o m t h e i r o w n d i s-

c u s s i o n , h o w e v e r , t h a t t h e y t a k e s u c h e x p r e s s i o n s a s s y n o n y m o u s w i t h

expres s ions o f the fo rm, r espec t ive ly , the r a t iona l w ay to p lay , a

r a t i o n a l w a y t o p l a y , p l a y i n g r a t i o n a l ly , e tc . 3 I n t h e f o l lo w i n g s u m m a r y

I h a v e a d o p t e d t h e l a t te r t e r m i n o l o g y a n d e m p l o y e d i t c o n s is t e n tl y

th rou gho u t . Second , a s ign i fi can t po r t ion o f the i r ana lys is i s fo rm ula te d

in mathemat ica l t e rms . The s teps , however , wh ich I th ink a re e spec ia l ly

d e s e rv i n g o f c lo s e e x a m i n a t i o n a ll t u r n o n p o i n t s w h i c h c a n n o t b e e s ta b -

l is h e d s im p l y b y a p p e a l t o t h e v a r i o u s m a t h e m a t i c a l p r o p e r t i e s o f th e

m a t r ix o f t h e g a m e . I n o r d e r t o b r i n g t h e s e n o n m a t h e m a t i c a l a s p e c ts o f

the a rgumen t c lea r ly in to focus , I have sough t , wherever pos s ib le , to

s u b s t i tu t e f o r t h e v a r i o u s m a t h e m a t i c a l e x p r e ss io n s w h i c h t h e y e m p l o y as o m e w h a t s i m p l e r a n d l oss m a t h e m a t i c a l t e r m i n o l o g y . W h e r e a p a r t i c u l a r



V O N N E U M A N N M O R G E N S T E R N T W O P E R S O N G A M E S

step turns on the mathematical properties of the matrix, however, I have

explicitly indicated this.

In subsequent sections of this paper I argue for two related theses

which, briefly summarized, are as follows: (1) The formal or direct proof

of their theory concerning the rational solution to G is logically defective

in the sense that most of the conclusions which they arrive at do not

follow in any rigorous sense from the premises which they employ

(Section 3). (2) The problem with the formal or direct proof turns out

to be the problem of deriving maximin and minimax criteria of rational

choice from the expected-utility maximization principle (EUX) in game-

decision contexts, and there are strong reasons for supposing that nosuch direct derivation of these criteria is possible (Section 4). In the

final section I shall also briefly discuss the relation betweexl the direct

proof attempted in Section 14 of their work, and the indirect proof

which is offered in Section 17. The remarks in this final section of the

paper are primarily anticipatory, since Part II of this present work will

be devoted to a full discussion and critique of such an indirect argument

for the theory of zero-sum games.

2. A S U M M A R Y O F T H E V O N N E U M A N N A N D

M O R G E N S T E R N A R G U M E N T

2 1 Preliminaries: the Definition of G and its two Variants

Consider any zero-sum, two-person game G for which each player has a

finite set of pure strategies, and there exists a saddle-point in pure

strategies. Such games can be characterized as strictly determined two-

person, zero-sum games. Following the usual procedure, the essential

features of such a game can be expressed in matrix (normal) form: n

rows for the n strategies available to one player, m columns for the m

strategies available to the other player, with the matrix elements specifying

the payoff o be associated with each row-column pair of strategies. Since

the game is zero-sum, the payoff values can be expressed by a single

number, in each matrix element, which is, by convention, taken to give

the payoff value for the row-chooser. The payoff values for the column-

chooser are then simply the negatives of these numbers (since these two

values must sum to zero). Let player 1 be the row-player and player 2 bethe column-player. Player 1 can be said to be interested in maximizing the



E DW AR D F MC C L E NNE N

payoff from the game, and player 2 can be said to be interested in

minimizing the payoff from the game. Finally, it is assumed that the

usual conditions of deliberation and choice obtain: (1) both players are

assumed to know the strategy and payoff structure of the game; (2) each

player is assumed to know that he is playing against a fully rational

opponent who also knows the strategy and payoff structure of the game;

(3) the game is to be played only once; and (4) strategy choices by the

two players are to be made simultaneously.4

Consider now two variants of G, G~ and G2. Let G~, the minorant of

G, be a game which agrees with G in every detail except that in G~

player 1 must choose a strategy first, and player 2 is informed as to howplayer 1 has chosen, before he is required to make a choice. Let G2, the

majorant of G, be a game which agrees with G in every detail except that

in G2 player 2 must choose a strategy first, and player 1 is informed as

to how player 2 has chosen, before he is required to make a choice.

Thus G~ and G2 differ from G in respect that they each specify a sequen-

tial, as distinct from a simultaneous, order of choice, and the player who

goes second is informed as to how the player who goes first has chosen.

Given that the interests of the two players are in strict opposition, it canbe noted that, intuitively, player 1 is at a disadvarttage in G1, as compared

with G (and G2), since he must play first and reveal his strategy choice

to his opponent; and player 2 is at a disadvantage in G2, as compared

with G (and G1), for a similar reason. 5

What is to be gained by an analysis of Gx and G2 ? Von Neumann and

Morgenstern argue that for G~ and G2 the best way of playing , i.e., the

concept of rational choice, has a clear and precise meaning, and further

that G itself- for which there is not yet a clear and precise concept of

rational choice - lies between G~ and G2, in terms of an ordering of

relative advantage for each player, as noted above. Thus it is to be

hoped that G1 and G2 will provide lower and upper limits for the signif-

icant quantities concerning G, i.e., for the rational outcome or payoff

for G. 6

2.2. The nalysis of Gt

In G~ player 1 chooses a strategy first, and then player 2 responds,

knowing how player 1 has chosen. Player 2 s concern, however, is tominimize the payoff. Thus, whatever choice player 1 makes, player 2 can



V O N N E U M A N N - M O R G E N S T E R N T W O P E R S O N G A M E S 5

be expec ted to r espond by s e lec t ing a s t r a tegy fo r wh ich the r esu l t ing

p a y o f f , g i v e n t h a t c h o i c e b y p l a y e r 1 , i s m i n i m u m . T h e r e m a y b e , o f

cour se , more than one s t r a tegy ava i l ab le to p layer 2 wh ich s a t i s f i e s th i s

c o n d i t i o n . L e t e a c h s u c h s t r a t e g y b e t e r m e d ' a minimizing response-

strategy ( to a spec i fi ed cho ice by p layer 1 ) ' J Th us i t can be in fe r r ed tha t

p lay er 1 w i l l expec t p laye r 2 to s e lec t a m in imiz ing r esponse s t r a tegy to

w h a t e v e r h e h i m s e l f c h o o s e s. A n d s in c e p l a y e r l ' s c o n c e r n i s t o m a x i m i z e

t h e p a y o f f , p l a y e r 1 c a n b e e x p e c t e d t o c h o o s e a s t ra t e g y w h o s e p a y o f f ,

g iven a min imiz ing r esponse - s t r a tegy by p layer 2 , w i l l be max imum.

Aga in , the re ma y be m ore than one s t r a tegy ava i l ab le to p laye r 1 s a t is fy ing

th i s cond i t ion . Le t each such s t r a tegy be t e rm ed a maximin s t r a tegy ' fo rp layer 1 .s Re fo rm ula t ing th e a bo ve reason ing exp l ic i tly in t e rms o f the

c o n c e p t o f ra t i o n a l b e h a v i o r , i t c a n b e c o n c l u d e d t h a t :

( A : a )

a n d

T h e r a t i o n a l w a y f o r p l a y e r 1 t o p l a y t h e m i n o r a n t g a m e G 1

i s to choo se a maximin s t r a tegy ;

( A : b ) T h e r a t io n a l w a y f o r p l a y e r 2 t o p l a y t h e m i n o r a n t g a m e G 1

is to choose a minimizing response=strategy ( t o p l a y e r l ' s

max imin cho ice ) .

9 N o w , i f p l a y e r 1 c h o o s e s a m a x i m i n s t ra t e g y , a n d p l a y e r 2 c h o o s e s a

min imiz ing r esponse - s t r a tegy ( to p layer l ' s max imin cho ice ) , i t ca r t be

s h o w n , b y a p p e a l t o t h e m a t h e m a t i c a l p r o p e r t i e s o f t h e p a y o f f m a t r i x ,

tha t the p ay of f to p layer 1 w i ll be a ce r ta in f ixed quan t i ty , nam ely , the

m a x i m u m o f th e m i n i m u m p a y o ff s a s so c i at e d w i th e a c h row. S t a te d m o r e

fo rma l ly , l e t :

( D 1 ) v ~ = d f t h e m a x i m u m o f t h e m i ni m u m p a y o f f v al u e s a ss o ci at ed

w i t h e a c h r o w ( f o r s h o r t , t h e maximin p a y o f f f o r

6 1 1 0 .

G i v e n v l s o d e f i ne d , i t c a n b e s h o w n t h a t :

A : c I f b o t h p l a y e r 1 a n d p l a y e r 2 c h o o s e r a ti o n a l ly i n th e m i n o r a n t

g a m e G ~, t h e p a y o f f t o p l a y e r 1 w ill b e v~ ( a n d t h e p a y o f f t oplay er 2 wi l l be - v l ) .



E DW AR D F Mc C L E NNE N

T h e a u t h o r s n o w p r o c e e d t o a r g u e t h a t : i l

. .. t he en t i re d i s cus s ion sho u ld m ake i t c lea r tha t eve ry p layo f the gam e Gx has a de f in it e value or each p layer . T h i s va lue is

the abo ve v l fo r the p layer 1 and the re fo re - v l fo r the p layer 2 .

T h e c o n c e p t o f t h e v a l u e o f a p l a y o f th e g a m e , w h i c h is h e r e i n t ro d u c e d

fo r the f i r s t t ime , p lays an impor tan t ro le in the subsequen t ana lys i s .

A t t h i s p o i n t , h o w e v e r , t h e y c h o o s e t o l e a v e t h e c o n c e p t i n q u e s t i o n

u n d e f i n e d i n a n y e x p l i c i t s e n s e , a n d p r o c e e d i n s t e a d t o e x p l o r e i n m o r e

deta i l th e s ignificance o f the valu e v 1 . Specifica lly, the y argu e, by ap pe al

a g a i n t o t h e m a t h e m a t i c a l p r o p e r t i e s o f t h e m a t ri x , t h a t :

( A : d ) P l a y e r 1 c a n , b y p l a y i n g a p p r o p r i a t e l y , s e c u re f o r h i m s e l f a

ga in t> vx i r respec t ive o f w ha t p laye r 2 does . P layer 2 can , b y

p l a y i n g a p p r o p r ia t e l y , s e c u r e f o r h i m s e l f a g a i n t > - v ~

i r respec t ive o f w ha t p lay er 1 does .

M o r e o v e r , ( A : d ) c a n b e s h o w n t o b e e q u iv a l en t t o :

( A : e ) P l a y e r 2 c a n , b y p l a y i n g a p p r o p r i a t e l y , m a k e i t s u r e t h a t t h e

ga in to p layer 1 i s ~ v 1 , i.e ., p reven t h im f rom ga in ing > v 1

i r respec t ive o f wha t p layer 1 does . P lay er 1 car t, by p lay ing

a p p r o p r i a t e l y , m a k e i t s u r e t h a t t h e g a i n t o p l a y e r 2 i s ~ - v a ,

i .e ., p r e v e n t h i m f r o m g a i n in g > - v ~ i rr e sp e c ti v e o f w h a t

p laye r 2 does .

T h e p r o o f o f b o t h ( A : d ) a n d ( A : e ) is, th e y n o t e , t ri v ia l : t h e a p p r o p r i a t e

cho ice fo r p lay er 1 i s s im ply any maximin s t ra t e g y ; t h e a p p r o p r i a t e

cho ice fo r p layer 2 i s s imply any min imiz ing r esponse - s t r a tegy to wha t -

ever cho ice p layer 1 m akes . T h i s com ple tes the ana lys i s o f G1.

2.3. The Analysis of G

Th e ana lys i s o f G 2 i s s tr i c tly pa ra l l e l to tha t o f G 1. Reca l l tha t in G2 i t

i s p laye r 2 who ch ooses fi rs t, and p layer 1 w ho c hoose s on ly a f t e r be ing

i n f o r m e d a s t o h o w p l a y e r 2 h a s c h o se n . N o w p l a y e r l s c o n c e r n i s t o

max imize the payof f . Thus , wha tever cho ice p layer 2 makes , i t i s to be

expe c ted tha t p layer 1 w i l l r e spo nd by s e lec t ing a s t r a tegy fo r wh ich the

resu l t an t payof f , g iven tha t in i t i a l cho ice by p layer 2 , i s max imum. Le tthe s e t o f s t r a teg ies ava i l ab le to p layer 1 wh ich s a t i s fy th i s cond i t ion be



V O N N E U M A N N - M O R G E N S T E R N T W O P E R S O N G A M E S

t e r m e d t h e s e t o f 'maximizing response-strategies ( to a speci f ied choice

by p lay er 2) ' . t~' I t can be infer red , then , tha t p lay er 2 wi ll ex pec t that

wh a tever c ho ice he mak es , p layer 1 w i ll s e lec t a m ax imiz ing r espo nse -

s t r a tegy to h i s cho ice . And , s ince h i s concern i s to min imize the payof f ,

he can be expec ted to s e lect a s t r a t egy who se payof f , g iven a max imiz ing

response - s t r a tegy to i t , is min imum . Le t the s e t o f s tr a t eg ies ava i l ab le to

p l a y e r 2 w h i c h s a t i s f y t h i s c o n d i t i o n b e t e r m e d t h e s e t o f minimax

s t r a teg ies ' f o r p layer 2 . lz Re fo rm ula t ing these con c lus ions exp l i c i tly in

t e r m s o f t h e c o n c e p t o f r a ti o n a l b e h a v i o r , i t f o ll o w s t h a t :

B : a )

a n d

( B : b )

T h e r a t i o n a l w a y f o r p la y e r 2 t o p l a y t h e m a j o r a n t g a m e G 2

is to choo se a minimax s t r a tegy ;

T h e r a t io n a l w a y f o r p l a y e r 1 t o p l a y t h e m a j o r a n t g a m e G 2

i s to choose a maximizing response-strategy ( to p layer 2 ' s

min imax cho ice ) .

A s i n t h e c a s e o f G t , i t c a n b e s h o w n t h a t i f b o t h p l a y e r s c h o o s e i n t h e

m a n n e r s p ec if ie d i n ( B : a ) a n d ( B : b ) , t h e p a y o f f t o p l a y e r 1 ( a n d h e n c e

a l so to p layer 2 ) w i l l be a f ixed quan t i ty , / )2 - 1)2 fo r p lay er 2) , w here

( D 2 ) / ) 2 = d f t h e m i n i m u m o f t h e m a x i m u m p a y o f fs a s so c ia te d

w i t h e a c h c o l u m n i n G 2 f o r shor t , the minimax

pa yo f f fo r G2).

Th a t is , i t can b e shown tha t :

( B : c ) I f b o t h p l a y e r 1 a n d p l a y e r 2 c h o o s e r a ti o n a l ly in t h e m a j o r a n t

9 g a m e G 2 , t h e p a y o f f t o p l a y e r 1 w i ll b e v2 ( a n d t h e p a y o f f t o

player 2 wi l l be - / )2) .

O n c e a g a i n , t h e a u t h o r s n o t e t h a t t h e d i s c u s s i o n u p t o t h i s p o i n t m a k e s

i t c lea r tha t each p lay o f the gam e G2 has a de f in it e va lue , and th i s va lue

i s v2 fo r p laye r 1 and - / )2 fo r p layer 2 . M oreov er , v2 has the fo l lowing

signif icance:

( B : d ) P l a y e r 1 c a n , b y p l a y i n g a p p r o p r i a t e l y , s e c u re f o r h i m s e l f a

pa yo f f > t v2 i r re spec t ive o f wh a t p layer 2 does . P lay er 2 can ,

by p lay ing approp r ia te ly , s ecu re fo r h im se l f a p ay of f ~> - v 2i r respec t ive o f w ha t p layer 1 does .



8 EDWA RD F MCCLENNEN

A n d , once aga in , ( B : d ) can b e s how n t o be equ i va l en t t o :

( B : e ) P l aye r 2 can , by p l ay i ng app r op r i a te l y , m ake i t s u r e t ha t thepa yo ff to pla ye r 1 i s ~<v2, i . e . , prevent him f rom gaining

> v 2 ir r espec t ive of wha t p la yer 1 does . P layer 1 can , by

p l ay i ng app r op r i a te l y , m ake i t s u r e tha t t he p ayo f f t o p l aye r 2

is ~< -v 2 , i.e ,, p rev ent h im f ro m ga in ing > -v 2 i r respec t ive

o f w ha t p l aye r 2 does.

2.4. The Value o f a Play of ~

H o w is th e c o n c e p t o f t h e v a l ue o f a p l a y o f G~ t o b e u n d e r s t o o d ? W h il e

t he au t ho r s do no t exp l ic i tl y de f ine i t, t hey do no t e t ha t ( A : c ) , ( A : d ) ,

(A : e ) an d (B : c ), (B : d ) , (B : e ) ca n b e t ake n as characterizing t he va l ue o f

a pla y o f G 1 an d G2, respect ively.14 An d, in a second re ference to the

sam e not ion , the y re fe r to (A : a ) , (A : b ) and (B :a ) , (B :b ) as the r easonings

w h i ch l ead t o t he concep t o f t he va l ue o f a p l ay o f a game , w h i le t he

con cep t i t s e lf is as soc ia ted , th i s t ime, w i th jus t (A : d ) , (A : e ) a nd (B : d ) ,

(B :e) . 15 Th e om ission , in th i s s econ d ins tance , o f r e fe ren ce to (A :c) and

(B: c ) is s ign if ican t , fo r th e as soc ia t ion o f the con cep t w i th th a t pa i r o f

propo s i t ions suggest s the fo l lowing as the ap prop r ia te def in i tion :

( D 3) T he va lue o f a p l ay o f G i = d f t he am ou n t w h i ch each p l aye r

will r ece ive i f bo th p layer s choo se in a rational m an ne r in G~.

I n con t r a s t, t he a s s oc i a ti on o f t he conc ep t w i t h j u s t ( A : d ) , ( A : e ) and

(B: d) , (B: e) suggests inste ad :

( D 4 ) T h e v a lu e o f a p l a y o f G i = d f t h e m a x i m u m w h i c h e a c h

p l a y e r c a n guarantee him self in Gi .

Subs equen t r e f e r ences t o t he co ncep t o f t he va l ue o f a p l ay o f a gam e

s imply tend to r e inforce the am bigui ty jus t n o ted . In Sec t ion 14 .4 .1 e i ther

(D3) or (D4) w ou ld seem to f it w i th w ha t i s sa id , whi le in Se c t ion 14 .5 .1 ,

to be d i scussed shor t ly , on ly (D4) w ould seem acceptab le . In co nt ras t , in

Sec t ion 14 .3 .1 , wh ere the co ncep t i s f i rs t in t roduc ed , (D3) i s the m os t

na t u r a l i n t e r p r e t a t ion , s ince the concep t is i n t r oduced t he r e p r i o r t o t he

f o r m u l a t i o n a n d p r o o f o f ( A : d ) a n d ( A : e), a n d r ig h t a f te r th e f o r m u l a t i o n

o f ( A : c ) . T o be s u r e , w i th i n t he con t ex t o f G 1 and G 2, t he t w o i n te r -p r e t a t i ons a r e ex t ensi ona ll y equ i va len t , bu t t h i s cann o t be t ake n a s



V O N N E U M A N N M O R G E N S T E R N T W O P E R S O N G A M E S 9

s u f f i c i e n t g r o u n d s f o r t r e a t i n g t h e m a s e x t e n s i o n a l l y e q u i v a l e n t i n t h e

c a s e o f G i ts e lf . T o d o s o w o u l d b e t o b e g p r e c i se l y w h a t i s a t i s su e w i t h

r e g a r d t o G , n a m e l y , w h a t c o n s t i t u t e s r a t i o n a l c h o i c e b e h a v i o r f o r G . 16

F o r t h e p r e s e n t i t w i l l s u ff ic e , h o p e f u l l y , t o k e e p b o t h d e f i n i ti o n s i n m i n d ,

t a k i n g c a r e f u l n o t e w h e r e n e c e s s a r y o f w h i c h i n t e r p r e t a t i o n i s m o r e

a p p r o p r i a t e .

2.5. The Relation Between Gx and G

T h e p a r a l le l a n a l y se s s u m m a r i z e d a b o v e m a k e i t c l e ar t h a t e v e r y p l a y o f

G1 , and o f G2 , has a de f in i t e value ( i n e i t h e r s e n s e D 3 o r D 4 ) f o r e a c h

p l a y e r . F o r G x , t h i s v a l u e i s v ~ f o r p l a y e r 1 a n d - v ~ f o r p l a y e r 2 . F o r

G 2 , t h i s v a l u e is v 2 f o r p l a y e r 1 a n d - v2 f o r p l a y e r 2 . W h a t i s t h e r e l a t io n -

s h i p b e t w e e n t h e v a l u e o f a p l a y o f G~ a n d t h e v a l u e o f a p l a y o f G 2 ( i n

e i t h e r s e n se D 3 o r D 4 ) ? I n t u i t iv e l y , s in c e p l a y e r 1 i s a t a d i s a d v a n t a g e i n

G 1 a s c o m p a r e d w i t h G 2, i t is t o b e e x p e c te d t h a t t h e v a l u e o f a p l a y o f

G ~ ( i n e it h e r s e n s e ) f o r p l a y e r 1 is le s s t h a n , o r e q u a l t o , b u t n o t g r e a t e r

t h a n , t h e v a l u e o f a p l a y o f G 2 ( i n e i th e r s e n s e ) f o r p l a y e r 1 . T h a t i s , i t is

to be ex pec ted th a t v~ ~< v2 .

2.6. The Heuristic Extension of the Analysis of GI and G to G

T h e a u t h o r s n o w p r o c e e d t o a r g u e ( i n S e c t io n 1 4 .4 .1 ) t h a t g i v e n c e r ta i n

a s s u m p t i o n s c o n c e r n i n g t h e i n te l le c t o f e a c h p l a y e r , th e v a l u es v 1 a n d

v2 c a n b e c o r r e l a t e d w i t h G i ts e lf . T h e r u l e s o f G r e q u i r e t h a t c h o i c e s b e

m a d e s i m u l t a n e o u s l y , b u t i t i s c o n c e i v a b l e t h a t o n e o f t h e p l a y e r s , e .g .

p l a y e r 2, f i n d s o u t h i s o p p o n e n t , i. e., t h a t h e h a s s o m e h o w a c q u i r e d

k n o w l e d g e a s t o w h a t s t ra t e g y hi s o p p o n e n t w i ll c h o o s e . T h e b a s i s o f

t h i s k n o w l e d g e n e e d n o t b e o f c o n c e r n h e r e : f o r t h e p r e s e n t i t s uf fic es t oa s s u m e t h a t o n e p l a y e r h a s a c q u i r e d s u c h k n o w l e d g e , 7 S u p p o s e , n o w ,

t h a t p l a y e r 1 is a w a r e t h a t p l a y e r 2 h a s f o u n d h i m o u t . I n t h i s ca se , t h e

c o n d i t i o n s o f G i t s e l f b e c o m e e x a c t l y t h e s a m e a s i f t h e g a m e w e r e G x .

S i m i la r ly , i f i t is a ss u m e d t h a t p l a y e r 1 h a s f o u n d o u t p l a y e r 2, a n d t h a t

p l a y e r 2 i s a w a r e o f th i s , t h e n t h e c o n d i t i o n s o f G it s e l f b e c o m e e x a c t l y

t h e s a m e a s i f t h e g a m e w e r e G 2 . I n t h e l i g h t o f t h e s e c o n s i d e r a t i o n s , i t

c a rt b e c o n c l u d e d t h a t :

( H 1 ) T h e v a l u e o f a p l a y o f t h e g a m e G ( i n e it h e r s en s e D 3 o r D 4 ,a l t h o u g h D 3 i s p e r h a p s m o r e a p p r o p r i a t e h e r e ) i s a w e l l -



1 E D W A R D F . M C C L E N N E N

d e f in e d q u a n t i t y i f o n e o r t h e o t h e r o f th e f o U o w in g e x tr e m e

a s s u m p t i o n s i s m a d e : E i t h e r t h a t p l a y e r 2 f i n d s o u t h i s

a d v e r s a r y , o r t h a t p l a y e r 1 f i n d s o u t h i s a d v e r s a r y . I n th e

f ir st c as e t h e v a l u e o f a p l a y o f G i s v 1 f o r p l a y e r 1, a n d - v l

f o r p l a y e r 2 ; i n t h e s e c o n d c a s e t h e v a l u e o f a p l a y o f G i s v 2

f o r p l a y e r 1 a n d - v 2 f o r p l a y e r 2 .

N o w , l e t v b e t h e v a l u e o f a p l a y o f G i ts e l f ( in e i t h e r s e ns e D 3 o r D 4 ,

a l t h o u g h , a g a in , D 3 is p e r h a p s m o r e a p p r o p r i a t e ) - w i t h o u t a n y fu r t h e r

q u a l i fi c a ti o n s o r m o d i f i c a ti o n s i n t e r m s o f t h e a b o v e - m e n t i o n e d e x t r e m e

a s s u m p t i o n s : s O n t h e b a si s o f ( H I ) a n d t h e i n t u it iv e n o t i o n s o f r e la t iv e

a d v a n t a g e w h i c h h o l d b e t w e e n G , G 1, a n d G 2 , f o r e a c h p l a y e r , it c a n b e

i n f e r r e d t h a t :

( H 2 ) I f v i s a w e l l - d e f i n e d q u a n t i t y a t a ll , t h e n it m u s t li e b e t w e e n

t h e v a l u e s v l a n d v 2, i .e . , i t m u s t b e t h a t

v~ <~v<~vz.

T h e l e n g t h o f t h e i n t e rv a l , ,4 = v z - v ~ / > 0 c a n b e t a k e n t o e x p r e s s t h e

r e la t iv e a d v a n t a g e w h i c h is g a i n e d in t h e g a m e G b y f i n d i n g o u t o n e so p p o n e n t in s te a d o f b e i n g f o u n d o u t b y h i m .

C o n s i d e r n o w a g a m e i n w h i c h i t d o e s n o t m a t t e r w h i c h p l a y er a n d s

o u t h i s o p p o n e n t . I n t e r m s o f t h e a b o v e , t h i s w ill b e t h e c a s e i f a n d o n l y

i f A = 0 , o r e q u i v a l e n t l y , v~ = vz. B y a p p e a l t o t h e m a t h e m a t i c a l p ro p e r t ie s

o f z e r o - s u m , t w o - p e r s o n g a m e s , i t c a n b e e s t a b l i s h e d t h a t f o r a g i v e n

g a m e , v~ = v z i f a n d o n l y i f th e g a m e h a s a s a d d l e -p o i n t. B u t G h a s b e e n

d e f i n e d a s a g a m e f o r w h i c h t h e r e e x i s ts a s a d d l e - p o i n t i n p u r e s t ra t e g ie s .

T h u s i t c a n b e i n fe r r e d t h a t f o r G v =vz. A n d t h is , t a k e n t o g e t h e r w i t h( H 2 ) i m p l i e s t h a t :

( H 3 ) I f v , t h e v a l u e o f a p l a y o f G , is a w e l l- d e f in e d q u a n t i t y a t all,

t h e n

C ~ - - - / 1 ~ / ' ) 2

T h i s c o m p l e t e s w h a t V o n N e u m a n n a n d M o r g e n s t e rn r e g a rd a s t h e

h e u r i s ti c e x t e n s io n o f t h e r e a s o n i n g o f t a n d G 2 t o G it se lf . T h e y d o n o t

t a k e t h i s r e a s o n i n g a s e s t a b l i s h i n g i n a n y f o r m a l s e n s e t h a t f o r G ,v = v l = v 2 . T h a t i s , t h e a r g u m e n t s o f a r i s t o b e t a k e n m e r e l y a s s u g g e s t in g



V O N N EU M A N N - M O RG EN S TERN TWO P ERS O N G A M ES I

a c e r t a i n c o n c l u s i o n . W h a t r e m a i n s i s t o c o n s t r u c t a f o r m a l a n d d i r e c t

a r g u m e n t w i t h r e g a r d t o G i ts e lf .

2.7. The Formal nalysis of G

T h e f o r m a l a n a l y s i s o f G , w h i c h c o m m e n c e s i n S e c t i o n 1 4 . 5 . 1 , i s c o m -

p l ic a t e d b y a n o t h e r a m b i g u i ty . I n th e p r e c e e d i n g s e c ti o n , a s s u m m a r i z e d

a b o v e , v w a s i n t r o d u c e d a s t h e h o p e d - f o r v a l u e o f a p l a y o f G i ts e l f

( f o r p l a y e r 1 ) . A t t h e o u t s e t o f 14 .5 .1 , h o w e v e r , v i s t a k e n a s d e s i g n a t i n g

b o t h t h e m a x i m i n a n d m i n i m a x v a l u e s a s so c ia te d w i t h t h e p a y o f f m a t r ix ,

w h i c h a r e , o f c o u rs e , e q u i v a l e n t f o r s t r ic t ly d e t e r m i n e d g a m e s . T h u s t h e

a u t h o r s b e g in b y n o t i n g t h a t

. .. i t i s t o b e h o p e d . . , t h a t i t w il l b e p o s s i b l e t o i n t e r p r e t t h e

q u a n t i t y

/ .-~- V 1 ~.~-V2

a s t h e v a l u e o f a p l a y o f G ( f o r t h e p l a y e r 1 ). 19

I n e f fe c t , t h e n , v h a s b e e n r e c h a r a c t e r i z e d a s a d e f i n it e q u a n t i t y , a n d

w h a t i s t o b e s o u g h t f o r i s a n a r g u m e n t w h i c h e s t a b l is h e s t h a t v - --v I = v2i s t h e v a l u e o f a p l a y o f G ( fo r p l a y e r 1 ). S t a t e d m o r e f o r m a l l y , t h e f o l l o w -

i n g d e f i n i t i o n i s im p l i c it l y a d o p t e d :

(D5 ) v = d f v l = v2.

A g a i n s t t h e b a c k g r o u n d o f (D 5 ) , th e a u t h o r s n o w p r o c e ed t o f o r m u l a t e

a p r o p o s i t io n w h i c h c o r re s p o n d s to ( A : d ) a n d ( B : d ) :

( C : d ) P l a y e r 1 c a n , b y p l a y i n g a p p r o p r i a t e ly , s ec u r e f o r h i m s e l fa g a i n > /v , ir r e s p e c ti v e o f w h a t p l a y e r 2 d o e s . P l a y e r 2 c a n ,

b y p l a y i n g a p p r o p r i a te l y , s e c u r e f o r h i m s e l f a g a i n > i - v ,

i r re s p e c t iv e o f w h a t p l a y e r 1 d o e s .

T h e p r o o f o f ( C : d ) is st r a ig h t f o r w a r d . T h e a p p r o p r i a t e c h o i c e f o r

p l a y e r 1 is a n y m a x i m i n s t ra t e g y . T h e a p p r i o p r i a t e c h o i c e f o r p l a y e r 2

is a n y m i n i m a x s t ra t e g y . I f p l a y e r 1 c h o o s e s a m a x i m i n s t ra t e g y , i t c a n b e

s h o w n t h a t t h e p a y o f f t o h i m w i ll be a t l e a s t v, a n d i f p l a y e r 2 c h o o s e s a

m i n i m a x s t r a te g y , it c a n b e s h o w n t h a t t h e p a y o f f t o h i m w i ll b e a t l e a s tv . M o r e o v e r , ( C : d ) c a n b e s h o w n t o b e e q u i v a le n t t o :



I ~ E D W A R D i~ M C C L E N N I ~ N

C : e ) H a y e r 2 c a n , b y p l a y i n g a p p r o p r i a t e l y , m a k e i t s u r e t h a t t h e

g a i n o f p l a y e r 1 i s ~ < v , i .e ., p r e v e n t h i m f r o m g a i n i n g > v

i rr e sp e c t iv e o f w h a t p l a y e r 1 d o e s . P l a y e r 1 c a n , b y p l a y i n g

a p p r o p r i a t e l y , m a k e i t s u r e th a t t h e g a i n o f p l a y e r 2 i s

~ < - v , i .e ., p r e v e n t h i m f r o m g a i n in g > - v i rr e sp e c t iv e o f

w h a t p l a y e r 2 d o e s .

H a v i n g e s ta b l is h e d b o t h C : d ) a n d C : e ), t h e a u t h o r s n o w a r g u e th a t

t h e s e t w o p r o p o s i t i o n s s u ff ic e t o e s t a b l i s h t h a t :

N ) t h e v a l u e o f a p l a y o f G i s v f o r p l a y e r 1 , a n d - v f o r

p l a ye r 2 . 2o

T h i s is t h e f ir st p l a c e a t w h i c h t h e a m b i g u i t y i n th e c o n c e p t o f t h e v a l u e

o f a p l a y o f a g a m e b e c o m e s c ri ti ca l. I f D 4 ) g iv e s t h e p r o p e r i n t e rp r e t a -

t i o n , t h e n t h e i n fe r e n c e f r o m C : d ) a n d C : e ) t o N ) i s w a r r a n t e d -

i n d e e d , t r iv i a ll y s o , si n ce D 4 ) t a k e s t h e v a l u e o f a p l a y o f G t o b e w h a t

e a c h p l a y e r c a n g u a r a n t e e h i m s e lf , a n d C : d ) a n d C : e ) e s t a b li s h th a t

t h is a m o u n t i s v f o r p l a y e r 1 a n d - v f o r p l a y e r 2. O n t h e o t h e r h a n d , i f

D 3 ) i s t h e p r o p e r i n t e r p r e t a ti o n - i f t h e v a l u e o f a p l a y o f a g a m e i s w h a t

e a c h p l a y e r w i l l r e c e iv e i f b o t h c h o o s e r a t i o n a l l y - t h e i n f e r e n c e is d e a r l y

n o t w a r r a n t e d . 21 T h u s , i f t h is s t e p o f t h e a r g u m e n t i s t o s u c c e ed , i t m u s t

b e a s s u m e d t h a t D 4 ) , a n d n o t D 3 ) i s c o r r ec t .

J u s t h o w c r u c ia l it is t h a t N ) b e i n t e r p r e te d i n s u c h a m a n n e r t h a t i t

c a n b e s h o w n t o fo l l o w f r o m C : d ) a n d C : e ) b e c o m e s q u i t e c l ea r i n t h e

l ig h t o f t h e n e x t s t ep o f t h e a r g u m e n t , i n w h i c h t h e a u t h o r s r e m a r k t h a t :

O w i n g to C : d ) . . . i t is r e a s o n a b l e t o d e fi n e a [ s ic ] g o o d w a y

f o r 1 t o p l a y th e g a m e G a s o n e w h i c h g u a r a n t e e s h i m a g a in

w h i c h i s g r e a t e r t h a n o r e q u a l t o t h e v a l u e o f a p l a y f o r 1,

i r re s p e c t i v e o f w h a t 2 d o e s ; i .e ., a c h o i c e [ f o r w h i c h t h e r e s u l-

t a n t p a y o f f i s ] ~> v f o r a l l [ p o s s i b l e c o u n t e r - c h o i c e s b y

p l a y e r 2 ] .

a n d

. .. b y C : d ) i t i s r e a s o n a b l e t o d e f in e t h e g o o d w a y f o r 2 t o

p l a y th e g a m e G a s o n e w h i c h g u a r a n t e e s h i m a g a i n w h i c h

i s g r e a t e r t h a n o r e q u a l t o t h e v a l u e o f a p l a y f o r 2 , i r re s p e c -

t i v e o f w h a t 1 d o e s ; i .e ., a c h o i c e [ f o r w h i c h t h e r e s u l t a n tp a y o f f i s ] i> - v f o r a ll [ p o s s i b l e c o u n t e r - c h o i c e s b y p l a y e r 1 ].z ~



VON NE UM AN N MORGENSTERN TWO PERSON GAMES 3

T a k e n t o g e t h e r , th e s e tw o r e m a r k s c o n s t i t u t e a n a r g u m e n t f o r a c r i te r i o n

o f r a t i o n a l c h o i c e w h i c h n o t o n l y m a k e s r e f e re n c e t o t h e c o n c e p t o f t h e

v a l u e o f a p l a y o f t h e g a m e , b u t w h i c h p r es u p p o s e s t h a t t h e v a l u e o f a

p l a y o f G is t h e d e t e r m i n a t e q u a n t i t y v , f o r p l a y e r 1, a n d - v f o r p l a y e r 2.

I n s h o r t , t h e c r i t e r i o n p r e s u p p o s e s ( N ) . N o w , g i v e n ( D 4 ) , t h i s p r e -

s u p p o s i t i o n i s w a r r e n t e d , a n d , c o n s e q u e n t l y , t h e c r i t e r i o n i s r e n d e r e d

d e t e r m i n a t e . O n t h e o t h e r h a n d , g i v en ( D 3 ) , th e p r e s u p p o s i t io n i s n o t

w a r r e n t e d , a n d t h e c r i te r i o n im p l i c i tl y f o r m u l a t e d i n t h e a b o v e - q u o t e d

r e m a r k s r e m a i n s i n d e t e r m i n a t e . 23

T h e a b o v e - m e n t i o n e d c o n s i d e r a t io n s w o u l d s e e m t o s e tt le c o n c lu s i v el y

t h a t ( D 4 ) , a n d n o t ( D 3 ) , g iv e s t h e p r o p e r i n t e r p r e t a t i o n o f t h e c o n c e p t o ft h e v a l u e o f a p l a y o f G . I t r e m a i n s t o b e d e t e r m i n e d ju s t h o w t h e c r i te r i o n

o f r a t i o n a l c h o i c e im p l i c it ly f o r m u l a t e d i n t h e a b o v e r e m a r k s is t o b e

e x p li c it ly f o r m u l a t e d . I n t h e f i rs t o f th e t w o r e m a r k s q u o t e d a b o v e , t h e

a u t h o r s s p e a k o f a g o o d w a y t o p l a y , w h i l e i n t h e s e c o n d r e m a r k , t h e y

s p e a k o f t h e g o o d w a y t o p la y . T h e u s e o f a i n t h e f o r m e r ca s e m u s t b e

a s s u m e d t o b e a m o m e n t a r y l a p s e , f o r , a s w i l l b e c o m e e v i d e n t s h o r t l y ,

w h a t i s r e q u i r e d f o r th e r e m a i n d e r o f t h e a r g u m e n t t o g o t h r o u g h , i s a

n e c e s s a r y , a s w e l l a s a s u f fi c ie n t, c o n d i t i o n o n r a t i o n a l c h o i c e . T h a t i s ,w h a t is r e q u i r e d i s t h a t :

( D 6 ) x i s a r a t i o n a l c h o i c e f o r p l a y e r i i n i f and on ly i f x g u a r a n -

t e es a p a y o f f t o p l a y e r i a t le a s t a s g r e a t a s t h e v a l u e o f a p l a y

o f G fo r i ( i. e. v fo r p l ay er 1 an d - v fo r p l aye r 2 ). 34

G i v e n ( D 6 ) t h e r e m a i n d e r o f t h e a r g u m e n t g o e s t h r o u g h i n s t ra i g h t-

f o r w a r d f a s h i o n . I t c a n b e s h o w n t h a t p l a y e r l s m a x i m i n s tr a te g ie s a r e

t h e o n l y o n e s w h o s e g u a r a n t e e d p a y o f f is a t le a s t a s g r e a t a s v, a n d t h a tp l a y e r 2 s m i n i m a x s t ra t eg i es a r e t h e o n l y o n e s w h o s e g u a r a n t e e d p a y o f f

i s a t l e a s t a s g r e a t a s - v . T h u s i t f o l l o w s t h a t :

t h e r a t i o n a l w a y f o r p la y e r 1 t o p l a y G is to c h o o s e a m a x i m i n

s t r a t e g y ;

C:a)

a n d

( C : b ) t h e r a t i o n a l w a y f o r p l a y e r 2 t o p l a y G is to c h o o s e a m i n i m a x

s t ra t egy .

M o r e o v e r , ( C : a ) a n d ( C : b ) i m p l y , in t u r n , t h a t :



14 EDWARD F MCCLENNEN

C:c) i f bo t h p l aye r s 1 and 2 p l ay G r a t iona l l y the p ayo f f t o p l aye r 1

will be v and the p ay of f to p layer 2 will be - v .

The au thors conc lude the i r ana lys i s by no t ing tha t , g iven the def in i t ion

of a sadd le-poin t , and the def in i tions of m axim in and m in im ax s tr a teg ies,

( C : c ) i m p l ie s :

c : f ) bo t h p l aye r s 1 and 2 p l ay G r a t i ona l ly i f and on l y i f t he ou t -

co m e o f the i r r espec t ive choices i s a saddle-poin t fo r G .

I mm ed i a t e l y f o l low i ng t he p r oo f o f ( C : f ) i t is r em ar ked t ha t t he s e t o f

propos i t ions (C :a) th rou gh (C : f ) s e t tl e every th ing as f a r as st r ic t lyde t e r m i ned tw o- pe r s on gam es a r e con ce r ned .25 Th e au th ors a re qu ick

t o no t e , how eve r , t ha t t he r ea s on i ng in s uppor t o f t he ana ly s is o f G

di ffe rs s ign i fi can tly f ro m tha t em ploy ed in the ana lys is o f G1 and G2.

In th e case, fo r exam ple , o f G1, the a rg um en t p roceed s f ir s t by es tab l i sh ing

( A : a ) and ( A : b ) . ( A : c ) is t hen de r i ved f r om ( A : a ) an d ( A : b) . F i na ll y ,

(A : d) an d (A: e ) a re der ived q u i te indepe nden t ly . In the case o f G i tse lf ,

h o w e v e r, t h e a r g u m e n t c o m m e n c e s w i t h a p r o o f o f ( C : d ) a n d ( C : e ),

wh ich a r e then , in tu rn , u t il ized to der ive (C : a ) , (C :b ) , an d f ina l ly ,(C :c) , v ia the in te rme diary steps (N) a nd (D6) .

3. A CRITIQUE OF THE ANALYSIS OF V

D oe s t he i n f e rence cha i n f r om ( C : d ) t o ( C : a ) , ( C : b ) and ( C : c ) , v ia the

impl ic i tly form ula te d steps (N) an d (D6) , rea l ly se t t le every th ing wi th

r ega r d t o G ? I have a l r eady r em ar ked t ha t t he s t ep f r om ( C : d ) t o ( N ) is

r ende r ed p r ob l em a t i c i f ( D 3) r a t he r t h an ( D 4) i s t aken t o g ive t he p r op e r

i n t e rp r e t a ti o n o f t h e c o n c e p t o f t h e v a l u e o f a p l a y o f a g a m e . R o u g h l y

s peak ing , t he p r ob l em is t h is : ( C : d ) does no t s eem to s ay any t h i ng i n

an d o f it s e l f ab ou t r a t iona l choice , whi le (N) , i f i t is in te rpre ted in t e rm s

o f ( D 3), does s ay s om e t h i ng abou t r a t iona l cho i ce . T h i s p r ob l em w as

avo i ded by t ak i ng ( D 4) t o g ive t he p r o pe r i n t e r p r e t a ti on . B u t t he p r o b l em

can no l onge r be avo i ded , f o r t he au t ho r s make i t qu i t e dea r , i n t he

r em ar ks quo t ed above , t ha t t hey a l so t ake ( C : d ) t o p r ov i de t he g r ound s

f o r t he c r i te r i on o f r a ti ona l cho i ce w h i ch I have exp li ci tl y f o r m u l a t ed a s

(D6) . In sh or t , they d o suppose th a t (C : d ) impl ies som eth ing qu i te specif icabo u t w ha t cou l d con s t it u t e r a ti ona l cho i ce w i th i n t he con t ex t o f G .




The cen t ra l ques t ion , then , con cern ing the i r ana lys is o f G i s wh ethe r

(C: d ) p rov ides any supp or t fo r (D6) . Two th ings a re , in th i s r egard , qu i t e

c lear. F i rs t , the au th ors p resen t no a rgu m ent for th i s st ep. Th ey conte n t

themselves w i th r em ark ing tha t i t is ow ing to (C :d) tha t i t is r easonable

t o i n t r oduce t he c r i te r i on o f r a ti ona l i ty f o r mu l a t ed i n ( D 6) . Second , t he

inference f rom (C: d ) to (D6) co uld hard ly be sa id to be so o bvious as to

requi re n o fur the r d i scuss ion . (C : d ) es tab li shes tha t there i s a w ay fo r

each p l aye r t o choos e w h i ch gua r an tees h i m a ce r t a in am oun t . M or eove r ,

i t c an be s how n t ha t t he am ou n t in ques t ion (v , f o r p l aye r 1 and - v , f o r

p l a y e r 2 ) i s t h e m a x i m u m a m o u n t w h i c h e a c h p l a y e r c a n g u a r a n t e e

h imsel f. But (D6) goes wel l be yo nd th i s to r equi re , in e f fec t , tha t jus t andon l y w ha t gua r an t ees a t l e a s t t h i s amoun t i s r a t i ona l . W i t hou t any

f u r t he r qua l if ica ti on , t he i n f e r ence f r om ( C : d ) t o ( D 6) t akes on a mo s t

ques t ionable cha rac te r . I t appear s , in e f fec t, to be a n in ference f rom the

ex i st ence o f a ce r t a i n w ay o f p l ay i ng t o t he conc l u s ion t ha t s uch a w ay

of p lay ing , and on ly such a wa y o f p lay ing , is r a t iona l . A t th e ve ry l east ,

i t mus t be s uppos ed t ha t t he au t ho r s have i n mi nd no t t ha t ( C : d ) by

i t se lf wa r ren t s (D6) , bu t tha t (C : d ) , t aken in co njun c t ion wi th cer t a in

o the r p remises , wa r ren t s (D6) .No w, in the case o f G1 and G2, i t w il l be r eca l l ed , the a rgum ents for

( A : a ) , ( A : b ) a nd ( B : a ) , ( B : b ) , t u r ned on t he a s s umpt i on t ha t p l aye r l s

conce r n w as t o max i mi ze t he payo f f, w h i le p l aye r 2 s conce r n w as t o

mi n i mi ze t he payo f f. A nd , w h i le t he a r gum en t s w er e deve l oped qu i t e

i n f o rma l l y , they a r e capab l e o f be ing f o r mu l a t ed r igo r ous l y i n t e r ms o f

the pr inc ip le tha t r a t iona l choice i s cho ice which maximizes expec ted

ut i li ty (EUX ) . 26 Th at is , g iven E U X and the spec if ica t ion of the sequent ia l

ru les o f p lay for G1 and G2, (A :a ) , (A :b) an d (B : a ) , (B : b ) can be der ived

i n a s t r a igh t f o r w ar d man ne r . P r e s um ab l y w ha t d i st ingu is hes G f r om G 1

an d Gz is p rec i se ly tha t no such d i rec t der iva t ion of (C :a) an d (C :b) is

poss ib le . The l ine of a t t ack wh ich the au th ors em ploy sugges ts , h owe ver ,

t ha t t hey s uppos e t ha t E U X , w hen t aken t oge t he r w i t h ( C : d ) , w i l l

suf f i ce for the der iva t ion of the c r i t e r i a fo rmula ted in (C :a) and (C :b) .

I t i s t o be admi t t ed t ha t ( C : d ) and E U X t oge t he r do i mp l y ce r t a i n

th ings conc ern ing wh at could cons t i tu te a r a t iona l choice in G . But an

exp l o r a t i on o f t he s e i mp l i ca t i ons s e r ves on l y t o r ende r t he pu t a t i ve

i n f e rence f r om ( C : d ) and E U X t o ( D 6) m or e , no t l ess , quest ionab l e .Cons ider , f irs t, a gam e in w hich p layer 1 , fo r exam ple , has a s t r a tegy



16 E DW AR D F MC C L E NNE N

a v a il ab l e t o h i m w h o s e m a x i m u m p o ss ib le p a y o f f is le ss t h a n v. I f E U X

a n d C : d ) h o l d , c h o i c e o f s u c h a n a l t e r n a t iv e c a n n o t b e ra t i o n a l . T h e v e r y

l e a s t h e c a n e x p e c t t o g e t b y s e l e c t i n g a m a x i m i n c h o i c e w i l l b e v , w h i l e ,

ex h yp o t h es i t h e m o s t h e c a n e x p e c t t o g e t b y c h o o s i n g t h e s t a t e d

a l t e rn a t i v e is le ss t h a n v . E U X t h u s d i c ta t e s t h a t a n y m a x i m i n c h o i c e w i ll

b e m o r e r a t i o n a l t h a n s u c h a n a l t e r n a t i v e . T h u s i t c a n b e i n f e r r e d t h a t :

D 7 ) x i s a r a t i o n a l c h o i c e f o r p l a y e r i i n G o n l y / f i t i s n o t t h e c a s e

t h a t t h e m a x i m u m p o s s i b le p a y o f f f o r x is < t h e v a lu e o f a

p l a y o f G f o r p l a y e r i .

N o t e , h o w e v e r , h o w w e a k t h i s c o n c l u s i o n is i n c o n t r a s t to t h e h o p e d - f o rc o n c l n si o rt D 6 ) . D T ) r e q u ir e s o n l y t h a t t h e m a x i m u m n o t t h e g u a r a n -

t e e d ) p a y o f f b e a t l e a s t a s g r e a t a s t h e v a l u e o f a p l a y o f t h e g a m e .

C o n s i d e r , s e c o n d , a g a m e i n w h i c h o n e o f t h e p l a y e r s , sa y p l a y e r 1,

c o m e s t o f o r m a n e x p e c t a t i o n c o n c e r n i n g h o w t h e o t h e r p l a y e r w i l l

c h o o s e . T h a t i s, s u p p o s e p l a y e r 1 c o m e s t o f o r m c e r t ai n e s t im a t e s o f t h e

l i k e l ih o o d o f p l a y e r 2 c h o o s i n g a n y o n e , r a t h e r t h a n a n y o t h e r , o f t h e

s t ra t e g ie s a v a i la b l e to h i m . A n d s u p p o s e , f u r t h e r , t h a t p l a y e r 1 h a s

a v a i la b l e to h i m a s t r a te g y w h o s e e x p e c t e d p a y o f f a s c a lc u l a t e d i n t e rm so f t h o s e e s t im a t e s ) is le ss t h a n v . T h e n a g a i n , E U X t o g e t h e r w i t h C : d )

i m p l ie s t h a t c h o i c e o f s u c h a n a l t e r n a t iv e w o u l d b e i r r a ti o n a l . F o r C : d )

i m p l ie s t h a t t h e e x p e c t e d u t il i ty o f a n y m a x i m i n c h o ic e m u s t b e a t l e a s t

v , a n d , e x h y p o t h e s i , t h e e x p e c t e d u t i l it y o f th e a l t e r n a t i v e i s l es s t h a n v .

T h u s i t c a n b e i n f e rr e d t h a t :

D 8 ) x i s a r a t i o n a l c h o i c e f o r p l a y e r i i n G o n l y / f i t i s n o t t h e c a s e

t h a t t h e e x p e c t e d u t i l it y o f x is < t h e v a l u e o f a p l a y o f G fo r

p l ay er i .

O n c e a g a i n , h o w e v e r , th i s c o n c l u s io n is c o n s i d e r a b l y w e a k e r t h a n D 6 ) ,

w h i c h r e q u ir e s t h a t t h e g u a r a n t e e d a n d n o t s im p l y t h e e x p e c te d ) p a y o f f

m u s t n o t b e l es s t h a n t h e v a l u e o f a p l a y o f t h e g a m e f o r t h a t p l a y e r.

W h a t is e v e n m o r e d i s t u r b i n g i s t h a t a f u r t h e r e x p l o r a t i o n o f t h e

i m p l ic a t io n s o f E U X w o u l d s e e m to d e s t ro y a n y h o p e w h a t s o e v e r o f

d e r i v in g D 6 ) f r o m E U X a n d C : d ) . S u p p o s e , f o r e x a m p l e , t h a t p l a y e r 1

h a s f o r m e d c e r t a i n e s ti m a t e s o f th e l i k e li h o o d o f p l a y e r 2 c h o o s i n g o n e ,

r a t h e r t h a n a n y o t h e r , s tr a te g y . A n d s u p p o s e f u r t h e r t h a t p l a y e r 1 h a s as t r a t e g y x w h o s e g u a r a n t e e d p a y o f f is le ss t h a n v b u t w h o s e e x p e c t e d




p a y o f f a s c a l c u la t e d in t e r m s o f t h o s e e s t im a t e s ) is g r e a te r t h a n t h e

e x p e c t e d p a y o f f o f a n y o f h i s m a x i m i n c h o ic e s. 27 G i v e n E U X , x is a m o r e

r a t i o n a l c h o i c e t h a n a n y o f h is m a x i m i n s t ra te g ie s . B u t th i s is t o s a y t h a t

u n d e r t h e c o n d i t i o n s s p e c i f i e d t h e r a t i o n a l c h o i c e f o r h i m w i l l b e o n e

w h i c h v i o la t e s D 6 ) . T h u s , s o f a r f r o m i ts b e i n g t h e c a se t h a t E U X a n d

C : d ) i m p l y D 6 ) , E U X a c t u a l ly i m p li e s c e r t a i n c r it e ri a o f c h o i c e w h i c h

a r e i n c o m p a t i b l e w i t h D 6 ) .

N o w i t m i g h t b e a r g u e d a g a i n s t t h e a b o v e , t h a t t h e i m p l i c a t i o n s o f

E U X w h i c h c o nf li ct w i t h D 6 ) c a n b e o b t a i n e d o n l y o n t h e a s s u m p t i o n

t h a t p l a y e r 1 c o m e s t o e x p e c t p la y e r 2 t o c h o o s e s o m e t h i n g o t h e r t h a n a

m i n i m a x s t r a t eg y , a n d t h a t n o s u c h e x p e c t a t i o n o n t h e p a r t o f p l a y e r 1c o u l d b e r a t i o n a l o r r e a s o n a b le . T h e p r e s u p p o s i t i o n in q u e s t i o n m u s t b e

a c k n o w l e d g e d . I f p l a y e r 1 w e r e t o e x p e c t t h a t p l a y e r 2 w o u l d s el ec t a

m i n i m a x s t ra t e g y , h e c o u l d n o t r a ti o n a ll y ) e x p e c t t o g e t m o r e t h a n v ,

n o m a t t e r h o w h e c h o o s e s. 2a H e n c e t h e r e c o u l d b e n o a l t e rn a t i v e x

a v a i la b l e t o h i m w h o s e e x p e c te d p a y o f f w a s g r e a t e r t h a n v . O f c o u rs e ,

t h i s b y i t s e lf d o e s n o t s uf fic e t o d i sp o s e o f th e u n w a n t e d i m p l i c a t io n o f

E U X . I t m u s t a ls o b e s h o w n t h a t t h e o n l y r a t io n a l ) e x p e c t a t i o n w h i c h

p l a y e r 1 c o u l d h a v e w o u l d b e t h a t p l a y e r 2 w o u l d c h o o s e a m i n i m a xs t r a t e g y . I s h a l l r e t u r n t o t h i s q u e s t i o n i n S e c t i o n 4 . F o r t h e p r e s e n t ,

h o w e v e r , i t is i n te r e s ti n g to n o t e t h a t e v e n i f i t c o u l d b e s h o w n t h a t

p l a y e r 1 c a n n o t r a t i o n a l l y ex p e c t p l a y e r 2 to c h o o s e o t h e r t h a n a m i n i m a x

s t ra t e g y , s ti ll D 6 ) c a n n o t b e d e r iv e d f r o m C : d ) a n d E U X , a n d t h is

s i m p l y b e c a u se , e v e n u n d e r t h e c o n s t r a i n t i n q u e s ti o n , E U X h a s i m -

p l i c a t i o n s w h i c h c o n f l ic t w i t h D 6 ) . S u p p o s e , f o r t h e s a k e o f t h e a r g u m e n t ,

t h a t p l a y e r 1 d o e s e x p e c t p l a y e r 2 t o c h o o s e a m i n i m a x s tr a te g y . I t m a y

s ti ll h a p p e n t h a t p l a y e r 1 h a s o t h e r s t r a te g i e s, i n a d d i t i o n t o h i s m a x i m i n

s tr a te g ie s , w h o s e e x p e c te d p a y o f f is v . T h e e x p e c t e d p a y o f f o f a n y m a x -

im in s t r a t egy , i n such a s i t ua t i on , w i l l be , o f cou rse , v . B u t i n such a case ,

E U X i m p l ie s t h a t c h o i c e o f a rt y s u c h a l t e r n a t i v e is j u s t a s r a t i o n a l a s t h e

c h o i c e o f a m a x i m i n s t ra t e g y . 29 I n c o n t r a s t , D 6 ) r u le s a g a i n s t t h e r a t i o n -

a l i t y o f s u c h a r t a l t e r n a ti v e , n o n - m a x i m i n s t r a t e g y , s in c e i f i t is n o t a

m a x i m i n s t r a t e g y i ts g u a r a n t e e d p a y o f f m u s t b e le ss t h a n v .

T h u s , w h e t h e r i t i s a s s u m e d t h a t a g i v e n p l a y e r s c o n s t r a i n e d w i t h

r e g a r d t o w h a t h e c a n e x p e c t t h e o t h e r p l a y e r t o d o , o r s n o t s o c o n -

s t r a i n e d , E U X w i l l h a v e , f o r c e r t a i n g a m e s , i m p l i c a t i o n s w h i c h c o n f l i c tw i t h D 6 ) . A n d s in c e t h e a u t h o r s m a k e i t c l ea r t h a t E U X is t o b e t a k e n



18 EDW ARD F. MCCLENNEN

a s a f r a m e w o r k a s s u m p t i o n f o r t h e w h o l e a n a ly s is o f G , t h is c a s t s s e ri o u s

d o u b t u p o n t h ei r c l a im t o h a v e s e t tl e d e v e r y t h i n g c o n c e r n i n g G . T h e

i n f er e n c e c h a i n f r o m ( C : d ) t o ( C : a ) , ( C : b ) a n d ( C : c ) b r e a k s d o w n a t t h e

c r it ic a l s te p f r o m ( C : d ) t o ( D 6 ) . N o t o n l y d o t h e y f a il t o s h o w t h a t ( D 6 )

f o l lo w s i n a n y s e n se f r o m ( C : d ) , b u t t h e c o n s i d e ra t i o n s a d d u c e d a b o v e

sugges t tha t no such de r iva t ion i s pos s ib le , a t l eas t a s long as the p r inc ip le

o f e x p e c t e d - u ti l it y m a x i m i z a t i o n ( E U X ) i s t a k e n a s r e g u la t iv e f o r t h e

analysis .

4. THE POSSIBIL ITY OF A DIRE CT PROOF OF MAXIMIN AND

M I NI M AX C R I TER I A FOR THE ZER O- S UM TWO- P E R S ON GAM E

I n S e c t i o n 3 I w a s c o n c e r n e d p r i m a r i l y w i t h t h e q u e s t i o n o f w h e t h e r V o n

N e u m a n n a n d M o r g e n s t e rn s u c c ee d e d i n d e ri vi n g ( C : a ) a n d ( C : b ) f r o m

( C : d ). T h e o b j e c t i o n s ra i se d , h o w e v e r , s e e m to p o i n t b e y o n d t h e s i m p l e

conc lus ion tha t the de r iva t ion , a s they choose to fo rmula te i t , f a i l s ; i t

w o u l d a l s o s e e m t h a t t h e r e a r e r e a s o n s f o r s u p p o s i n g t h a t n o s u c h d e ri v a -

t ion c ou ld be pos s ib le . In th i s f ina l s ec t ion o f the pap er , I w an t to con -

s ide r th i s s t ronger conc lus ion i ll m ore de ta i l, a l though , fo r r eason s wh ichwi l l be co m e ev iden t sho r t ly , a f ina l ad ju d ica t ion o f th i s ma t te r w i ll

h a v e t o a w a i t t h e r e s u lt s o f a n e x a m i n a t i o n o f i n d ir e c t a r g u m e n t s f o r

such c r i te r i a , wh ich exam ina t ion w i l l be the top ic o f the s eque l to th i s

p a p e r .

L e t u s a s s u m e t h a t b o t h E U X a n d ( C : d ) h o l d . T h e p r o b l e m i s h o w t o

a v o i d t h e i m p l i c a t io n s o f E U X w h i c h c o n f l ic t w i t h ( D 6 ) , a n d h e n c e w i t h

( C : a ) a n d ( C : b ) .

(1 ) Cons ide r the f i r s t imp l ica t ion , i . e . , t he pos s ib i l i ty tha t a p layer

m i g h t c o m e t o e x p e c t h is o p p o n e n t t o p l a y o t h e r th a n a s e c u r it y

s t r a te g y ( a m a x i m i n o r m i n i m a x s t ra t e g y ), a n d , h e n c e , i n te r m s o f th a t

e x p e c t a t i o n , h a v e a n e x p e c t e d u t i l i t y m a x i m i z i n g r e a s o n f o r c h o o s i n g

h i m s e l f o t h e r t h a n a s e c u r i ty s t ra t eg y . I t c o u l d b e a r g u e d t h a t s u c h a

p o s s i b il i ty i s ru l e d o u t w i th i n t h e c o n t e x t o f th e k i n d o f t h e o r y w h i c h w e

s e e k t o d e v e l o p . W h a t w e a re l o o k i n g f o r is n o t a t h e o r y c o n c e r n i n g

w h a t a p l a y e r s h o u l d d o u n d e r a n y a n d a l l c i r c u m s t a n c e s , b u t s i m p l y

w h a t h e s h o u l d d o o n t h e a s s u m p t i o n t h a t h e i s p l a y i n g a g a i n s t a f u l l y

r a t i o n a l o p p o n e n t . P r e s u m a b l y , t h e n , i n s o f a r a s i t c a n b e e s t a b l i s h e dt h a t t h e r e a r e v a l id c r it e ri a o f r a t io n a l c h o i c e a p p l ic a b l e t o p l a y e r j , t o




that extent we are entitled to assume that player i (who is taken to be

rational and to know that player j is rational) will be aware of these

criteria, and will take them as regulative of his expectations concerning

how player j will choose. More specifically, if there were some way in

which security criteria could be derived from EUX and (C:d) - and

whatever additional assumptions proved necessary - the derivations in

question would establish for each player principles which were regulative

not merely of each player s choice, but each player s expectations con-

cerning how the other player would choose. And given the consequent

narrowing of the range of expectations which each could rationally

entertain, hypothetical arguments concerning what it w o u l d be rationalfor a given player to do, wer e he (counterfactually) to expect his opponent

to behave in a certain manner, would have no weight against the theory

in question.

(2) It might be objected that the above argument begs the question,

since what is at issue here is whether such a satisfactory derivation of

security criteria is possible. Until such a derivation is detailed, how could

hypothetical considerations be ruled out? This objection can be got

around by formulating the above argument as a condition on what is tocount as an adequate derivation of criteria of rational choice against a

rational opponent. Let me state such a condition, and then argue for it

by appeal to certain cases in which, intuitively at least, there would seem

to be a successful derivation of criteria. The condition runs as follows:

(C1) An assumption that player i expects player j to choose a

particular (kind of) strategy is w r r n t e d as a step in the

derivation of criteria of rational choice for player i against

a rational opponent j i f n d o n l y i f choice of that (kind of)strategy by player j is requ i red by valid criteria of rational

choice for player j against a rational opponent i.

(a) Consider now the analysis of the variant game G 1. Player 1 must

make a choice and player 2 is informed of what player 1 has selected

before making his own choice. Since player 2 is concerned to minimize

the payoff, and chooses in response to a known choice by player 1, one

can establish, directly by appeal to EUX, that the only rational way for

player 2 to choose is to select a minimizing response-strategy (to thechoice which player 1 makes). This amounts to satisfaction of the if



2 E D W A R D F M C C L E N N E N

c l a u s e o f ( C 1 ) : c h o i c e o f a m i n im i z i n g r e s p o n s e - s tr a t e g y b y p l a y e r 2 i s

r e q u i r e d b y E U X . B u t n o w i t is a s s u m e d ( b e c a u s e p l a y e r 1 is ra t io n a l ,

k n o w s p l a y e r 2 t o b e r a t i o n a l , a n d k n o w s w h a t r a t i o n a l i t y r e q u i r e s )

tha t p layer 1 w i l l expec t p laye r 2 to ch oose a m in imiz ing r espon se -

s t r a tegy . And i t i s p rec i s e ly th i s a s sumpt ion , when t aken toge the r w i th

E U X a g a in , t h a t g e n e r a te s t h e c o n c l u s i o n t h a t p l a y e r l s o n l y ra t i o n a l

c h o i c e i s a m a x i m i n s t ra t e g y . T h e i f p o r t i o n o f ( C 1 ) t h u s c a p t u r e s th e

l in e o f r e a so n i n g e m p l o y e d i n th e a n a l y s is o f G a. M o r e o v e r , t h e o n l y i f

p o r t i o n o f (C 1 ) c a p t u r e s t h e n o t i o n t h a t sin ce , u n d e r t h e c o n d i t io n s

s p ec if ie d , a n y o t h e r c h o i c e b y p l a y e r 2 w o u l d n o t b e r a t io n a l , i t w o u l d b e

i n a d m i s si b le t o r e a s o n f r o m a n a s s u m p t i o n t h a t p l a y e r 1 e x p e c ts p l a y e r 2t o c h o o s e i n s o m e o t h e r m a n n e r , t o a c o n c l u s i o n c o n f l i c t i n g w i t h t h a t

a r r iv e d a t a b o v e .

( b ) S u c h a m o d e o f r e a s o n i n g c a n a l s o b e a p p l i e d t o c e r ta i n s i m u l -

t a n e o u s c h o i c e g a m e s . C o n s i d e r , f o r e x a m p l e , a g a m e i n w h i c h p l a y e r 2

has a s t r a t egy which s t rong ly domina tes eve ry o the r s t r a t egy ava i l ab le to

h i m . B y a p p e a l t o E U X i t c a n b e s h o w n t h a t t h e o n l y r a t io n a l c h o i c e f o r

h i m i s t h e d o m i n a n t s t ra t e g y . B u t t h e n i t c a n b e a r g u e d , i n t h e s p i ri t o f

(C1) , tha t w e a re en t it l ed to a s sum e tha t p laye r 1 w i ll expec t tha t p laye r 2w i l l c h o o s e t h e d o m i n a n t s t r a t e g y . A n d t h i s a s s u m p t i o n , t a k e n t o g e t h e r

w i t h E U X , s uffic es f o r t h e d e r i v a t i o n o f a c o n c l u s i o n a b o u t w h a t i t is

r a t io n a l f o r p l a y e r 1 t o d o . M o r e o v e r , o n c e a g a in , th e o n l y i f p o r t i o n o f

( C 1 ) g i ve s e x p r e ss i o n to t h e n o t i o n t h a t a n y o t h e r s u p p o s i t i o n a s t o w h a t

p l a y e r 1 e x p e c t e d p l a y e r 2 t o d o w o u l d b e c o u n t e r f a c t u a l , a n d n o t

adm is s ib le in to the a rgum en t . 3~

S o fa r , t h e n , ( C 1 ) w o u l d s e e m t o f o r m u l a t e a n i n t u i ti v e ly a c c e p t a b l e

c o n d i t i o n o n r e a s o n i n g a b o u t r a ti o n a l c h o i c e in g a m e s i tu a t io n s , a n d o n e

w h i c h s a t is f a c to r i ly d i s p o s e s o f t h e f ir st o f t h e u n w a n t e d i m p l i c a ti o n s o f

E U X .

( 3) U n f o r t u n a t e l y , ( C 1 ) p r o v e s t o b e o f n o u s e a t a l l i n d e a l in g w i t h ,

a n d i n d e ed e x a c e r ba t e s, t h e p r o b l e m o f t h e s e c o n d o f th e u n w a n t e d

imp l ica t ions o f EU X . T he p r ob lem here , i t w i ll be reca l led , i s tha t i f ( a s

w e h o p e t o b e a b l e t o p r o v e ) t h e o n l y r a t io n a l w a y f o r a n y p l a y e r t o

choose in a ze ro - sum, two-per son game i s to s e lec t a s ecu r i ty s t r a t egy ,

t h e n p l a y e r i , e x p e c t in g p l a y e r j t o c h o o s e i n s u c h a m a n n e r , m a y h a v e n o

r a t io n a l g r o u n d s , i n te r m s o f E U X , f o r c h o o s i n g h i s s ec u r it y st r at e g y inp re fe rence to ce r t a in o the r s t r a t eg ies ava i l ab le to h im. The d i f f i cu l ty




ar i ses p rec i s e ly in v i r tue o f ou r r ea son ing , in the sp i r it o f (C1) , tha t

p laye r i m us t exp ec t p layer j to cho ose a s ecu r i ty s t ra t egy . F or i t is th i s

w a r r a n t e d e x p e c t a t io n o n p l a y e r i s p a r t w h i c h , w h e n c o u p l e d w i t h E U X

leads inexorab ly to the conc lus ion tha t i t i s not the c se t h a t t h e o n l y

ga t iona l cho ice ava i l ab le to p layer i is a s ecu r i ty s t r a tegy , the re by un der -

min ing wha tever pu ta t ive de r iva t ion o f s ecu r i ty c r i t e r i a was employed .

T h e p r o b l e m h e r e i s b o t h s y s t e m a t i c a n d s e r i o u s . G i v e n E U X a s V o n

N e u m a n n a n d M o r g e n s t e r n a x i o m a t i z e it , ( C 1 ) c art b e e m p l o y e d t o

u n d e r m i n e a n y p u t a t i v e d e r iv a t i o n o f m a x i m i n a n d m i n i m a x c r it e ri a o f

r a t iona l cho ice fo r ze ro - sum, two-per son games aga ins t a r a t iona l

op po ne nt . 31( 4) C a n ( C 1 ) b e m o d i f i e d i n s o m e m a n n e r o r o t h e r s o a s t o a v o i d th e

p r o b l e m p o s e d a b o v e ? C e r t a i n p o s s ib i li ti e s c a n b e s u g g e s te d , b u t i n e a c h

case the re a r e s e r ious coun te r -ob jec t ions .

( a ) I t can be no ted , f i rs t o f al l, t ha t the on ly i f po r t ion o f (C1) i s a l l

t h a t i s r e q u i r e d t o d e a l w i t h t h e f ir st o f th e u n w a n t e d i m p l i ca t io n s o f

E U X . D r o p p i n g t h e i f p o r t i o n w o u l d p e r m i t u s t o c h al le n g e t h e l in e o f

r eason ing ske tched in (3 ) . Bu t th i s wou ld s e rve on ly to de fe r the who le

p r o b l e m , f o r i t w o u l d r e m a i n t o d e t e r m i n e w h y o n e s h o u l d c o n t i n u e t oregard the r eason ing o f (2a) and (2b ) a s accep tab le , and ye t re j ec t the

reaso ning o f (3) , wh ich is s t r ic tly para l le l to th at o f (2a) and (2b) .

( b ) A s o m e w h a t m o r e r a d ic a l a p p r o a c h w o u l d b e t o a r g u e a s f o ll o w s .

(C1) i s r ea l ly qu i t e accep tab le on the on ly i f fo rm ula t ion : i t s imp ly does

n o t , i n th a t r e g a rd , g o f a r e n o u g h . W h a t t h a t p o r t i o n o f (C 1 ) p o i n t s t o i s

t h e n o t i o n t h a t p e r m i s s ib l e e x p e c t a ti o n s a b o u t w h a t a r a t io n a l o p p o n e n t

wi l l do a re a func t ion o f wha tev er c r it e r ia o f r a t iona l cho ice can be

d e v e l o p e d , n o t v i c e v e rs a . B u t i f e x p e c t a ti o n s a r e t o b e t a k e n a s d e p e n d e n t

up on the app l i cab le c r i t e ri a o f ra t iona l cho ice , then i t m igh t be a rgued

t h a t s u c h e x p e c t a ti o n s c a n n o t b e f e d b a c k i n t o t h e re a s o n i n g in a n y

w a y w h i c h w o u l d u p s e t w h a t c o u l d b e i n d e p e n d e n t l y a n d d i r e c t l y

e s t a b li s h e d t o b e a r a t i o n a l w a y t o p l a y . T h e p o i n t w o u l d b e t o d i st in g u i sh

b e t w e e n w h a t c o u ld b e t e rm e d p r i m a r y a n d s e c o n d a r y i nf er en c es f r o m

E U X t o c r it e ri a o f r a t io n a l c h o i c e f o r g a m e s i t ua t io n s . I f t h e r e w e r e s o m e

w a y t o e s t a b l i s h s e c u r i t y c r i t e r i a b y a p p e a l d i r e c t l y t o E U X a n d , f o r

exam ple , (C : d ) , then , on the v iew in ques t ion , such de r iva t ions cou ld no t

b e u n d e r m i n e d b y t a k i n g th e e x p e c ta t io n s w h i c h t h e s e p r i m a r y d e r iv a -t i o n s w o u l d g e n e r a t e a n d r e a p p l y i n g E U X t o t h e m . T h e e x p e c t a t i o n s i n



EDWARD F MCCLENNEN

q u e s t i o n w o u l d h a v e t o b e r e g a r d e d a s p a r a s i ti c u p o n t h e cr i te r ia so

d e r i v e d , a n d , a s s u c h , n o t regul tive fo r suc h c r i t e r ia . 82

S u c h a w a y o f d e a l in g w i t h t h e p r o b l e m , i t c o u l d b e a r g u e d , w o u l d n o t

e a s t d o u b t u p o n t h e r e a s o n i n g o f ( 2a ) a n d ( 2 b) . I n fe r e n c e s m a d e i n a

r o u n d a b o u t f a s h i o n , f r o m w h a t i t i s r a t i o n a l f o r p l a y e r i t o d o , t o w h a t

p l a y e r j c a n e x p e c t p la y e r i t o d o , a n d f in a l ly f r o m t h e r e t o w h a t i t is

r a t i o n a l f o r p l a y e r j t o d o , - s u c h i n f e re n c e s a re p e r f e c t l y a c c e p t a b l e s o

l o n g a s t h e y s q u a r e w i t h w h a t c a rt, i f a n y t h i n g , b e o b t a i n e d d i r e c t ly b y

a p p l i c a t i o n o f E U X t o p l a y e r j . I n t h e c a se o f a c o nf li c t, h o w e v e r , t h e

c r i t e ri a w h i c h c a n b e e s t a b l i s h e d d i r e c t ly m u s t p r e v a i l .

( c ) T h e r e a r e , u n f o r t u n a t e l y , t w o s e r i o u s o b j e c t i o n s w h i c h c a n b er a i s e d t o t h i s a p p r o a c h . ( i) T h e p o s s i b i li t y o f a d e r i v a t i o n o f s e c u r i ty

c r it e ri a w o u l d n o w h i n g e o n t h e p o s s i b il it y o f s h o w i n g t h a t t h e r e i s a w a y

t o r e a s o n f r o m E U X a n d ( C : d ) t o s e c u r it y c r it er ia without a n y a p p e a l t o

a s s u m p t i o n s a b o u t w h a t e a c h p l a y e r c a r t e x p e c t i n t h e w a y o f s p e c i f i c

b e h a v i o r o n t h e p a r t o f t h e o t h e r p l a y e r. 33 B u t i f s u c h a d e r i v a t i o n w e r e

p o ss ib le , th i s w o u l d b e t a n t a m o u n t t o e s t a b li sh i ng t h a t th e V o n N e u m a n n

a n d M o r g e n s t e r n s o l u t i o n t o t h e z e r o - s u m , t w o - p e r s o n g a m e i s a l s o a

s o l u t i o n t o t h e c la s si ca l p r o b l e m o f i n d iv i d u a l d e c i s io n m a k i n g a g a i n s tn a t u r e u n d e r c o n d i t io n s o f ( c o m p l e te ) u n c e r t a i n t y . T h e w o r k o f t h e l a s t

t w e n t y o r s o y e a r s o n t h e l a t t e r p r o b l e m s t r o n g l y s u g g e s t s t h a t n o s u c h

s t r a i g h t f o r w a r d d e r i v a t i o n o f s e c u r i t y c r i t e r ia i s p o s s ib l e . ~4 T h e p o i n t

h e r e i s t h a t i f w e d o n o t n e e d t o i n t r o d u c e c o n s i d e r a t io n s o f re c i p r o c al

e x p e c t a t i o n s i n t o t h e a n a l y s i s , i n o r d e r t o d e r i v e s e c u r i t y c r i t e r i a , t h e

d i s t i n c t io n b e tw e e n t h e t w o k i n d s o f p r o b l e m w o u l d s e e m t o c o ll ap s e .

E U X i s p r e s u m e d t o h o l d i n b o t h c a s e s . ( C : d ) c a r t b e s h o w n t o h o l d i n

b o t h c a se s . A n d t h e a s s u m p t i o n , i n t h e g a m e c a se , t h a t a g i ve n p l a y e r s

e x p e c t a t io n s a b o u t h o w t h e o t h e r p l a y e r w i ll c h o o s e i s i r re l e v a n t t o t h e

d e r i v a t i o n o f s e c u r i t y c r i te r i a , is m a t c h e d , i n t h e c a s e o f i n d i v i d u a l

d e c i s i o n - m a k i n g a g a i n s t n a t u r e , b y t h e a s s u m p t i o n t h a t t h e d e c i si o n -

m a k e r h a s n o e x p e c t a ti o n s a b o u t h o w n a t u r e w i ll r e s p o n d .

( ii) P e r h a p s e v e n m o r e s e r io u s ly , t h e v i e w s k e t c h e d a b o v e w o u l d c a s t

c o n s id e r a b le d o u b t u p o n t h e v a r i o u s indirect a r g u m e n t s w h i c h h a v e b e e n

e m p l o y e d t o g r e a t a d v a n t a g e i n g a m e - t h e o r y c o n s t r u c t io n . S u c h a r g u m e n t s

e m p l o y t h e s t r a te g y o f s u p p o s i n g t h a t w h a t e v e r t h e o r y o f r a t i o n a l c h o i ce

c a n b e d e r i v e d f o r a g a m e s i t u a t i o n , s u c h a t h e o r y n o t o n l y d i c t a t e s ar a t i o n a l c o u r s e o f a c t i o n f o r e a c h p l a y e r , i t a ls o d e t e r m i n e s w h a t e a c h



VON NEUMANN - MOR GENS TERN TW O P ER SON GAMES 23

p l a y e r m u s t e x p e c t t h e o t h e r p l a y e r to d o , a n d t h a t t h e s e e x p e c t a t i o n s ,

t a k e n t o g e t h e r w i th E U X , m u s t b e s u p p o s e d t o b e re g u l a ti v e o f t h e

p l a y e r s b e h a v i o r , a n d , h e n c e , r e g u l a ti v e o f t h a t t h e o r y i ts e lf . B u t t h is

s o r t o f f e e d i n g b a c k i n t o t h e r e a s o n i n g f o r th e t h e o r y it s e lf w h a t e v e r

e x p e c t a t i o n s t h e t h e o r y g e n e r a te s , i s p r e c is e l y w h a t is r u l e d o u t b y t h e

p r o p o s a l i n q u es t io n . V o n N e u m a n n a n d M o r g e n s t e r n t h em s e l v es o ff e r

a v e r s i o n o f s u c h a n i n d i r e c t a r g u m e n t f o r t h e i r t h e o r y , i n S e c t i o n 1 7,

a n d L u c e a n d R a i f f a o f f e r a d i f f e r e n t v e r s i o n i n t h e i r w o r k . 35

( d ) T h e r e a r e , n o d o u b t , o t h e r a n d l e s s d r a s t i c w a y s i n w h i c h o n e

c o u l d a v o i d t h e u n w a n t e d i m p l ic a ti o n s o f E U X . W h a t w o u l d s e e m t o

e m e r g e , h o w e v e r , is t h a t t h e q u e s t i o n o f t h e p o s s i b i li ty o f a d i r e c t p r o o fo f m a x i m i n a n d m i n i m a x c r i te r ia f o r th e z e ro - s u m , t w o - p e r s o n g a m e i s

i n t er tw i n e d w i t h q u e s ti o n s c o n c e r n i n g b o t h r e a s o n in g a b o u t r a t i o n a l

c h o i c e u n d e r c o n d i t io n s o f u n c e r t a in t y a n d i n d ir e c t m o d e s o f r e a s o n i n g

a b o u t g a m e p r o b l e m s . M o r e o v e r , o n e b e g in s t o s u s p e c t t h a t t h e r e a l

p r o b l e m l ie s n o t s o m u c h w i t h ( C 1 ) a s w i t h w h a t l ie s b e h i n d i t, a s it w e r e,

n a m e ly , t h e v er si o n o f E U X w h i c h V o n N e u m a n n a n d M o r g e n s t e rn

a x i o m a t i z e i n t h e i r w o r k , a n d w h i c h h a s b e e n e m p l o y e d b y v i r t u a l l y

e v e r y o t h e r g a m e t h e o r i s t. I f th e p r o b l e m s k e t c h e d i n (3 ) a b o v e i s t o b er e s o lv e d , e i t h e r s o m e r e s t ri c t io n m u s t b e p l a c e d O ll t h e w a y i n w h i c h t h i s

p r i n c i p le c a n b e e m p l o y e d , o r t h e p r i n c i p le i ts e l f m u s t b e r e f o r m u l a t e d :

i n e i th e r c as e , w h a t c a n f a i rl y b e s a id t o b e t h e s t a n d a r d o r a c c e p t e d v i e w

o f r a t i o n a l i t y m u s t b e r e v is e d . 36

I n P a r t I I o f t h i s p a p e r I s h a ll a t t e m p t t o u n d e r s c o r e th i s c o n c l u s i o n b y

s h o w i n g t h a t t h e p r o b l e m s w h i c h a r e t o b e e n c o u n t e r e d i n c o n s tr u c t in g

a d i r e c t p r o o f o f m a x i m i n a n d m i n i m a x c r i te r ia c a rr y o v e r to a n y a t t e m p t

t o c o n s t r u c t a n i n d i r e c t a r g u m e n t , a n d t h a t , o n c e a g a i n , t h e s e p r o b l e m s

s te m f r o m th e v er si o n o f E U X w h i c h V o n N e u m a n n a n d M o r g e n s t er n

a x i o m a t i z e in t h e i r w o r k .

Washington niversity

N O T E S

1 Th ird Edition (John W iley and Son s, New York, 1953). Hereinafter referred to asT .

T., pp. 147-8.8 T ., p. 10 0; eft, also pp. 31-2.4 (1), (2), and (3) are no t explicitly formulated b y Von Neum arm and M orgenstern in



2 4 E D W A R D F. M C C L E N N E N

S e c t i o n ! 4 , b u t a r e d e a r l y i m p l i c it in , f o r e x a m p l e , t h e a n a l y s is o f t h e v a r i a n t g a m e s

( p p . 1 0 0 - 1 0 5 ) . ( 4 ) , o r r a t h e r , f r o m a s t r a t e g i c p o i n t o f v ie w , i ts e q u i v a l e n t - v i z ., a

r e q u i r e m e n t t h a t p l a y e r s m u s t m a k e t h e i r c h o i c es i n t o ta l i g n o r a n c e o f w h a t t h e i r o p -p o n e n t s h a v e c h o s e n - i s e x p l i c it l y f o r m u l a t e d i n 1 4.1 .1 ( 1) . 9 8 ) . C f. p . 1 4 7 f o r a n e x -

p l i c i t r e f e r e n c e t o ( 3 ) .

5 T h e q u a l i f i c a t i o n t h a t i n t e r e s t s m u s t b e i n s t r i c t o p p o s i t i o n i s c r i t i c a l h e r e . T h e s e

r e l a ti o n s o f r e la t iv e a d v a n t a g e d o n o t n e c e s s a r il y o b t a i n i n t h e n o n - z e r o - s u m g a m e ,

w h e r e b e i n g a b l e t o g o f i r st m a y p r o v e t o b e a d v a n t a g e o u s .6 T . p . 100 .

7 T h e s e t o f s u c h s tr a te g ie s is V o n N e u m a n n a n d M o r g e n s t e r n 's s e t B ~1 - t h e s e t o f

t h o s e v ~ f o r w h i c h :,Y f(zl, v 2) a s s u m e s i ts m i n i m u m v a l u e M i n ~ f ' ( ~ l , v a) .

s T h e ' s e t o f s u c h s t r at e g ie s i s V o n N e u m a n n a n d M o r g e n s t e r n ' s s et A - t h e se t o f th o s e

T1 f o r w h i c h M i n ~ ( v l , v 2) a s s u m e s i ts m a x i m u m v a l u e M ax r~ M i n ~ 2 f ' ( r l , ~ ) .

a H e r e a n d s u b s e q u e n t l y I h a v e si m p l if ie d t h e a u t h o r ' s n u m b e r i n g s y s te m : ( A : a ) c o r r e -s p o n d s t o t h e i r ( 1 4 : A : a ) , e tc . F o r p r o p o s i t i o n s w h i c h t h e y d o n o t n u m b e r , b u t w h i c h

p l a y a c r u t i a l r o l e i n t h e i r a n a l y s is , I h a v e e m p l o y e d s i n g l e l e t te r , e .g . ( N ) , o r l e t t e r -

a n d - n u m b e r , e . g . , ( H 1 ) , ( D 4 ) , e t c . , d e s i g n a t i o n s .x0 T. , p . 101.

11 T. , p . 102.

12 T h e s e t o f s u c h s t r a t e g ie s is th e a u t h o r ' s s e t A r~ - t h e s e t o f t h o s e z l f o r w h i c h

,,q '( ~1 , z~ ) a s s u m e s i t s m a x i m u m v a l u e M a x r l , .~ g ' ( rl , ~ 2 ).

l a T h e s e t o f s u c h s t ra t e g ie s is th e a u t h o r ' s s e t B - t h e s e t o f t h o s e ~ f o r w h i c h M a x r z

9 ( ~ 1 , ~ 2) a s s u m e s i ts m i n i m u m v a l u e M i n r~ M a x r l ~ g ( z l , T ~).14 T. , p . 104.

15 T. , p . 104.

16 T o a s s u m e t h a t ( D 3 ) a n d ( D 4 ) a r e e x t e n s i o n a l l y e q u i v a l e n t f o r G i s t o a s s u m e t h a t

t h e r a t i o n a l p a y o f f t o e a c h p l a y e r i s e q u i v a l e n t t o t h e s a d d l e - p o i n t p a y o f f. B u t t h i s i s

o n e o f t h e r e s u l t s w h i c h t h e a n a l y s i s i s s u p p o s e d t o e s t a b l i s h , t _ f . P r o p o s i t i o n ( C : f ) .

17 T h i s is p r e c is e ly t h e p o i n t u p o n w h i c h t h e ' i n d i r e c t ' a r g u m e n t ( T ., p p . 1 4 7 - 8 ) t u r n s .

T h e n o t i o n i s t h a t i f t h e r e e x is t s a f u l ly a d e q u a t e a n d c o n v i n c i n g t h e o r y w h i c h is k n o w n

t o b o t h p l ay e r s, t h e n e a c h p l a y er m u s t a s s u m e t h a t h i s c h o i c e h a s b e e n ' f o u n d o u t ' ; f o r

h i s o p p o n e n t w i ll b e a b le , b y r e f e r e n c e to t h a t t h e o r y , t o a n t i c i p a t e w h a t h e w i ll d o .

i s ( D 3 ) , r a t h e r t h a n ( D 4 ) , i s m o r e a p p r o p r i a t e h e r e , s i n c e t h e a m o u n t w h i c h e a c h

p l a y e r c a n g u a r a n t e e h im s e l f i n G c a n b e s h o w n t o b e a d e t e rm i n a t e a m o u n t w i t h o u t

a n y o f t h e q u a l i fy i n g a s s u m p t i o n s , w h i le t h e a m o u n t w h i c h e a c h g e ts i f b o t h p l a y

r a t i o n a l l y i s s t il l p r o b l e m a t i c u n l e s s t h e q u a l i f y i n g a s s u m p t i o n s a r e i n t r o d u c e d .1 9 T . p . 107 .

s0 T . , p . 108 .

21 ( C : d ) a n d ( C : e ) c o u l d b e s a i d t o w a r r e n t ( N ) o n ( D 3 ) o n l y i f i t c a n b e i n d e p e n d e n t ly

e s t a b l i sh e d t h a t ( D 3 ) a n d ( D 4 ) a r e e x t e n s i o n a l l y e q u i v a le n t . B u t i t tu r n s o u t t h a t t h i s

c a n b e e s t a b l is h e d o n l y b y a p p e a l , a m o n g o t h e r t h i n g s , to ( N ) i t se lf .

~ T. , p . 108.

2a T h e s u g g e s t i o n h e r e i s t h a t ( D 4 ) g i v es t h e p r o p e r i n t e r p r e t a t i o n o f t h e c o n c e p t o f t h e

v a l u e o f a p l a y o f G b o t h i n ( N ) a n d i n t h e c r i t er i o n . T h i s s av e s th e s t ep f r o m ( C : d) a n d

( C : e ) t o ( N ) a n d a l s o e n s u r e s t h a t t h e c r i t e ri o n is d e te r m i n a t e . I t m i g h t b e s u p p o s e d ,

a l t e r n a t i v e l y , t h a t t h e a u t h o r s i n t e n d ( D 3 ) i n b o t h i n s t a n c e s . B u t t h e n ( N ) d o e s n o t

f o l l o w f r o m ( C : d ) a n d ( C : e ) , a n d t h e c r i te r i o n m a k e s r e f e r e n c e t o a n a s y e t u n d e t e r -

m i n e d q u a n t i t y . T h e o n l y o t h e r p o s s ib i l it y w o u l d b e t o s u p p o s e t h a t t h e y i n t e n d ( D 4 )i n t h e c a s e o f ( N ) b u t ( D 3 ) i n t h e e a s e o f t h e c r it e r io n . T h i s w o u l d r e n d e r t h e c r i te r i o n



6 E D W A R D F . M C C L E N N E N

M u t a t i s M u t a n d i s t h i s s t r i c t u r e a p p l i e s t o p r e f e r r i n g a s e c u r i t y s t r a t e g y t o a n o n -

s e c u r i t y s t r a te g y w h e n t h e e x p e c t e d u t i l i t y o f e a c h i s t h e s a m e .

a0 C f . , f o r e x a m p l e , J o h n C . H a r s a n y i , ' O n t h e R a t i o n a l i t y P o s t u l a t e s U n d e r l y i n g t h eT h e o r y o f C o o p e r a t i v e G a m e s ' , J . Confl ic t Resolut ion 5 (196 ) , pp . 179-80 . (C1) i s

f o r m u l a t e d b y H a r s a n y i a s h i s P o s t u l a t e o f M u t u a l l y E x p e c t e d R a t i o n a l i t y , w h i c h

t o g e t h e r w i t h E U X h e t a k e s a s t h e ' w e a k p o s t u l a t e s o f r a ti o n a l i t y '. I t i s h i s c o n t e n t i o n

t h a t t h e s e t w o p o s t u l a t e s t o g e t h e r s u tt ic e , in t h e c a s e o f z e r o - s u m g a m e s , f o r t h e

d e r iv a t i o n o f s e c u r i ty c r i t e r i a : i t i s , o f c o u r s e , p r e c i s e ly t h i s c o n t e n t i o n w h ic h I a m

c h a l l e n g in g i n t h i s p a p e r .

a l T h e p o i n t h e r e is t h a t E U X , a s a x i om a t i z e d b y V o n N e u m a n n a n d M o r g e n s t er n ,

r e q u i r es o n e t o r e g a r d a n y t w o o r m o r e s t r a t e gi e s a s e q u a l ly r a t io n a l i f t h e e x p e c t e d

u t i li t ie s a re , i n e a c h c a s e, m a x i m a l . G i v e n ( C I ) , i t m a k e s a b s o l u te l y n o d i ff e re n c e h o w

t h e s e c u r it y c r i te r i a h a v e b e e n ' d e r i v e d ' : b y a p p e a l t o i t , o n e c a n u n d e r m i n e t h e c l a i m

t h a t t h eonly

r a t i o n a l c h o i c e f o r a g iv e n p l a y e r w i l l b e h i s s e c u r i t y s t r a te g i e s .

32 I s h a l l e x p lo r e t h is s o r t o f p o s s ib i l i t y i n c o n s id e r a b l e m o r e d e t a i l i n t h e s e q u e l t o t h i s

p a p e r . I n t h e l ig h t o f t h e c o m m e n t s i n f o o tn o t e 2 9 , i t q u i t e c le a r l y i n v o lv e s a s i g n i f i c a n t

r e c o n c e p t u a l i z a t i o n o f u t i l it y t h e o r y as tha t theory i s to app ly to gam e deci sion prob lems .

( H o w e v e r , t h e s o - c a ll e d ' E l ls b e r g P r o b l e m ' s u g g es t s t h a t s u c h a r e c o n c e p t u a l i z a t i o n

m a y b e c a l l e d f o r e v e n i n t h e c a s e o f in d i v i d u a l d e c i si o n p r o b l e m s a g a i n s t n a t u r e .

C f . D a n i e l E l ls b e r g, ' R i s k , A m b i g u i t y , a n d t h e S a v a g e A x i o m s ' , Quar t . J . Econ . 75( 19 61 ), 6 4 3 - 6 6 9 ; a n d D a v id H . K r a n t z e t a l. F o u n d a ti o n s o f M e a s u r e m e n t ( A c a d e m i c

P r e ss , N e w Y o r k , 1 9 7 ) , p p . 4 1 7- 4 2 0 .) I n t e r e s ti n g l y e n o u g h , V o n N e u m a n n a n d

M o r g e n s t e r n t h e m s e l v e s p a r t i a l l y a n t i c i p a t e s u c h a v i e w p o i n t w h e n t h e y r e m a r k ,

e a r l y o n i n th e w o r k , t h a t :

E v e r y p a r t i c i p a n t c a n d e t e r m i n e t h e v a r i a b l e s w h i c h d e s c r i b e h i s o w na c t i o n s b u t n o t t h o s e o f t h e o t h e rs . N e v e r th e l e s s th o s e ' a l i e n ' v a r i ab l e s

c a n n o t , f r o m h i s p o i n t o f v i e w , b e d e s c r i b ed b y s t a ti s ti c a l a ss u m p t i o n s .

T h i s i s b e c a u s e t h e o t h e r s a r e g u i d e d , j u s t a s h e h i m s e l f , b y r a t i o n a l

p r i n c ip l e s - w h a t e v e r t h a t m a y m e a n - a n d n o m o d u s p r o c e d e n d i c a n b e

c o r r e c t w h i c h d o e s n o t a t t e m p t t o u n d e r s t a n d t h o s e p r i n c i p l e s a n d t h e

in t e r a c t i o n s o f t h e c o n f l i c t i n g i n t e r e s ts o f a l l p a r t i c i p a n t s . ( T . , p . 1 1)

T h e a u t h o r s t h e n p r o c e e d , o n t h e n e x t p a g e ( p. 1 2), t o s k e t c h a d i s t i n c ti o n b e t w e e n

w h a t t h e y t e r m ' d e a d ' v a r i a b l e s , w h i c h r e f e r t o t h e ' u n a l t e r a b l e p h y s i c a l b a c k g r o u n d

o f t h e s i t u a t i o n ' , a n d v a r i ab l e s w h i c h ' r e f le c t a n o t h e r p e r s o n ' s w i l l o r i n t e n t i o n o f a n

e c o n o m i c k i n d - b a s e d o n m o t i v e s o f t h e s a m e n a t u r e a s h i s o w n ' . B u t t h e y d o n o t , a sf a r a s I c a n d e t e r m i n e , e x p l o i t t h is d i s t in c t i o n i n t h e r e m a i n d e r o f t h e b o o k . A n o t h e r

e x a m p l e in w h i c h o n e c a n d i s c e rn a n a t t e m p t t o b r e a k o u t f r o m t h e c la s s ic a l e x p e c t e d -

u t i l i t y p a r a d i g m i s T . C . S c h e l l i n g ' s T h e S t r a t e g y o f C o n f l i c t ( H a r v a r d U n i v e r s i t yP r e s s , Ca m b r id g e , 1 96 0) , o f . , i n p a r t i c u l a r , C h a p t e r 4 , ' T o w a r d a T h e o r y o f I n t e r -

d e p e n d e n t D e c i s i o n ' . C o n c e r n i n g t h e c r i t i c a l q u e s t i o n o f t h e s t a t u s o f e x p e c ta t i o n s

w h i c h o n e p l a y e r h a s c o n c e r n i n g t h e b e h a v i o r o f t h e o t h e r p l a y e r , S c h e l l i n g q u o t e s

f i r s t C a r l K a y s e n ( f r o m t h e la t t e r' s r e vi e w o f V o n N e u m a n n a n d M o r g e n s t er n ) a n d

t h e n L u c e a n d R a i f f a a s f o l l o w s :

T h e t h e o r y o f s u c h g a m e s o f s t r a te g y d e a l s p r ec i s e ly w i t h t h e a c t i o n s o f

s e v e r a l a g e n t s , i n a s i t u a t i o n i n w h ic h a l l a c t i o n s a r e i n t e r d e p e n d e n t ,

a n d w h e r e , i n g e n e r a l , t h e r e i s n o poss ib i l it y o f w ha t we ca l led para-metr i za t ion t h a t w o u l d e n a b l e e a c h a g e n t ( p l a y e r ) t o b e h a v e a s i f t h e



V O N N E U M A N N - M O R G E N S T E R N T W O P E R S O N G A M E S 7

ac t ions o f the o thers were g iven . I n f a c t , i t i s th i s very lack o f para-

metr i za t ion wh ich is the essence o f a game . ( e m p h a s i s a d d e d ) ( Ka y s e n )

I n tu i t i v e ly , t h e p r o b l e m o f c o n f l i c t o f i n t e r e s t i s, f o r e a c h p a r t i c i p a n t ,a p r o b l e m o f individual decision making u n d e r a m i x t u r e o f r is k a n d

uncertainty t h e u n c e r t a i n t y a r i s i n g f r o m h i s i g n o r a n c e a s t o w h a t t h e

o th e r s w i l l d o . ( e m p h a s i s a d d e d ) ( L u c e a n d Ra i f f a )

O n t h e c l a s si c al e x p e c te d - u t il i ty p a r a d i g m , h o w e v e r , s u c h a l a c k o f p a r a m e t r i z a t i o n

m u s t t r a n s la t e i t s e lf i n t o t h e a s s u m p t i o n ( a n t ic i p a t e d i n th e q u o t a t i o n f r o m L u c e a n d

R a i f f a ) t h a t t h e g a m e d e c i s i o n p r o b l e m i s a p r o b l e m o f i n d i v i d u a l d e c i s i o n m a k i n g

u n d e r c o n d i t i o n s o f ( c o m p l e te ) u n c e rt a i nt y . T h e o n l y a l te r n a t i v e w o u l d s e e m t o b e t o

e m p l o y t h e so r t o f a r g u m e n t w h i c h V o n N e u m a n n a n d M o r g e n s t e rn c a ll a n i n d ir e c t

p r o o f , a c c o r d i n g t o w h i c h it m u s t b e s u p p o s e d t h a t w e r e t h e r e a f u l ly a d e q u a t e t h e o r y

o f r a t i o n a l c h o i c e f o r g a m e p r o b l e m s , r a t i o n a l p l a y e r s w o u l d b e a w a r e o f i t , a n d ,

h e n c e , b e in a p o s i t i o n t o e x p e c t c e r ta i n t h i n g s a b o u t h o w t h e i r o p p o n e n t s w o u l d p l a y .T h e n o t i o n h e r e i s , i n e f f e c t , t h a t w e r e t h e r e a n a d e q u a t e t h e o r y k n o w n t o b o t h

p l a y er s , t h e n e a c h p l a y e r i s i n a p o s i t io n , i n v i r tu e o f k n o w i n g t h a t t h e o r y a n d w h a t i t

i m p l i e s a b o u t h o w h i s o p p o n e n t w i l l c h o o s e , t o t a k e h i s o p p o n e n t s b e h a v i o r a s

p a r a m e t r i z e d , i . e . , given. Bu t , a g a in , i n t e r m s o f t h e c l a ss i c a l e x p e c t e d u t i l i t y p a r a d ig m ,

t h is m u s t t r a n s la t e i t s e l f i n t o a n a s s u m p t i o n t h a t e a c h p l a y e r i s c o n f r o n t e d w i t h a

d e c i s io n p r o b l e m u n d e r c o n d i t i o n s o f c e r t ai n t y o r r i s k ( i f m i x e d s t ra t e gi e s a r e p o s s i bl e )

a n d t h e E U X p r in c ip l e, i f i t is n ot quali fied g e n e r at e s t h e u n w a n t e d c o n c l u s i o n t h a t,

f o r c e r t a i n g a m e s , a r a t i o n a l p l a y e r w h o k n o w s t h a t h i s o p p o n e n t i s ra t i o n a l w i ll h a v e

n o r a t i o n a l g r o u n d s f o r p r e f e r r i n g h i s s e c u r it y le v e l s t r a t e g y t o c e r t a in o th e r s t r a t e g i e s

a v a i l a b le t o h i m . F o r s o m e t h i r ty y e a r s n o w g a m e t h e o r y h a s b e e n u n s u cc e s sf u l i n i ts

a t t e m p t t o a v o i d e i t h e r t h e S c y l l a w h i c h r e s u l t s f r o m c o m b i n i n g t h e c l a s s i c a l E U Xp r i n c i p l e w i t h t h e a s s u m p t i o n o f n o n p a r a m e t r i z a t i o n ( w h i c h le a d s, u n p r o m i s i n g l y , t o

a c o l l a p s in g o f t h e d i s t i n c t i o n b e tw e e n g a m e d e c i s io n s a n d i n d iv id u a l d e c i s i o n s a g a in s t

n a t u r e u n d e r c o n d i t i o n s o f u n c e r ta i n t y ) o r t h e Charybdis w h i c h r e s u lt s f r o m r e a s o n i n g

v i a s o m e v e rs i o n o f t h e i n d i r e c t a r g u m e n t t o t h e c o n c lu s i o n t h a t e a c h p l a y e r c a n

t a k e h i s o p p o n e n t s b e h a v i o r a s p a r a m e t r i z e d ( w h i c h l e a d s to t h e u n w a n t e d c o n c l u s i o n

th a t i t is n o t a lwa y s t h e c a s e t h a t a p l a y e r s o n ly r a t i o n a l s t r a t e g y i s o n e wh ic h m a x -

im iz e s h is s e c u r i t y le v e l) . T h e s u g g e s t i o n i n t h e t e x t i s t h a t t h e r e i s a wa y o u t o f t h i s

d i l e m m a , b u t i t i n v o l v es , a m o n g o t h e r t h i n g s, i n t r o d u c i n g t h e n o t i o n o f e x p e c t a ti o n s

w h i c h a r e p a r a s i t i c u p o n , a n d , h e n c e , n o t r e g u l a ti v e f o r , a t h e o r y o f r a t i o n a l c h o i c e

f o r g a m e s it u a ti o n s . U n f o r t u n a t e l y , a s I t r y t o a n t i c i p a t e i n th e c l o s i n g p a r a g r a p h s o f

t h i s p a p e r , a n d a s I s h a l l t r y t o s h o w in g r e a t e r d e t a i l i n t h e s e q u e l , s u c h a m o v e n o to n ly ( a s s u g g e s t e d a b o v e ) r e q u i r e s a s u b s t a n t i a l r e f o r m u la t i o n o f t h e c l a s s i c al v e r s io n

o f E U X , i t al s o c a st s s e r io u s d o u b t u p o n t h e v a l i d it y o f v a r i o u s v e r si o n s o f t h e q n -

d i r e c t a r g u m e n t w h i c h V o n N e u m a n n a n d M o r g e n s t e r n , a s w e l l a s L u e e a n d R a i f f a ,

e m p l o y . I f I a m c o r r e c t, t h e w h o l e o f t h e t h e o r y o f z e r o - s u m ( a s w e l l, in c i d e n ta l l y , as

n o n - z e r o - s u m ) g a m e s w i ll h a v e t o b e r e t h o u g h t f r o m s t a r t t o f i n is h .

a a S u c h a d e r iv a t i o n m ig h t b e t e r m e d a d i r e c t , expectat ion-independent a r g u m e n t , t o

d i s t i n g u i s h i t f r o m wh a t wa s e m p lo y e d , e . g . , i n t h e a n a ly s i s o f GI a n d G~ , i . e . , wh a tc o u ld b e c h a r a c t e r i z e d a s a d i r e c t , expec ta t ion-dependen t a r g u m e n t . T h e p o i n t h e r e i s

t h a t i f t h e o n l y w a y s u c h c r i t e r i a c a n b e d e r i v e d i s b y m e a n s o f s o m e e x p e c t a t i o n -

d e p e n d e n t a r g u m e n t , i t i s di ff ic u lt t o s e e h o w o n e c a n a v o i d t h e u n w a n t e d i m p l i c a t i o n

o f E U X .3~ Cf . , f o r e x a m p le , L u c e a n d Ra i f f a , op. c i t . C h a p t e r 1 3 ; a n d K . J . A r r o w a n d

L . H u r w i c z , A n O p t i m a l i t y C r i t e ri o n f o r D e c i s i o n - M a k i n g u n d e r I g n o r a n c e , i n



2 8 E D W A R D F . M C C L E N N E N

C. F . Ca r t e r a n d J . L . F o r d ( e d s . ) . Uncer ta in ty and Expec ta t ions in Economics ( B l a c k -

wel l , Oxfo rd , 1972).

35 L u c e a n d Ra i f f a p r e s e n t a p a r t i c u l a r l y s u c c in c t v e r s io n o f s u c h a l i n e o f r e a s o n in g ,w h i c h r u n s a s f o l l o w s :

I t seems p laus ib le tha t , i f a the or y o f fe rs c an d fl~o as su i tab le s t ra teg ies ,

t h e m e r e k n o w l e d g e o f t h e t h e o r y s h o u l d n o t c a u s e e i t h e r o f t h e p la y e r s

to c h a n g e h i s c h o i c e : j u s t b e c a u s e t h e t h e o r y s u g g e s t s P~0 t o p l a y e r 2

s h o u l d n o t b e g r o u n d s f o r p l a y e r 1 t o c h o o s e a s t r a t e g y d i f f e r e n t f r o m

a q ; s i m i la r ly , t h e t h e o r e ti c a l p r e s c r i p ti o n o f a s h o u l d n o t l e a d p l a y e r 2

to s e l e c t a s t r a t e g y d i f f e r e n t f r o m BJ o. P u t i n t e r m s o f o u t c o m e s , i f t h e

theo ry s ing les ou t (~xq , PJo) , the n :

( i ) N o o u t c o m e O~jo s h o u l d b e m o r e p r e f e r re d b y 1 t o O q i o

( ii ) N o o u t c o m e O j s h o u l d b e m o r e p r e f e r r e d b y 2 t o O q t 0

A n y c a n d Bf0 s a t i s fy in g c o n d i t i o n s ( i ) a n d ( i i ) a r e s a id t o b e i nequilibrium a n d t h e a pr ior i d e m a n d m a d e o n t h e t h e o r y i s t h a t t h e p a i rs

o f s t r a te g i e s i t s i n g le s o u t s h a l l b e i n e q u i l i b r i u m . ( L u c e a n d Ra i f f a ,

op. cir. t5. 63)

T h e ' g r o u n d s ' h e r e , a re , o f c o u r se , w h a t E U X d i c ta t e s, g i v e n t h e e x p e c t a ti o n s w h i c h

t h e t h e o r y o f r a t i o n a l c h o i c e f o r g a m e s g e n e ra t es . T h u s t h e s e e x p e c t at i o n s, c o u p l e d

w i t h E U X i ts e lf , i m p l y s o m e t h i n g a b o u t w h a t i t w o u l d b e ra t i o n a l f o r e a c h p l a y e r t o d o ,

a n d , h e n c e , w h a t t h e c o n t e n t o f t h e t h e o r y w e a r e s e a rc b i n g f o r m u s t b e li k e . T o a r g u e ,

a s I s u g g e s t i n t h e t e x t , t h a t s u c h e x p e c t a t i o n s a r e s e c o n o a r y a n d p a r a s i t i c u p o n t h e

t h e o r y d e v e l o p e d , a n d , a s s u c h , c a n n o t , i n c o n j u n c t i o n w i t h E U X , b e t a k e n a s c o n -

t r o l l i n g f o r t h e t h e o r y , i s t o r u l e o u t p r e c i s e ly t h i s s o r t o f l i n e o f r e a s o n in g .86 I t i s w o r th r e m a r k in g h e r e t h a t I f i n d i t d if f ic u l t t o s u p p o s e t h a t a n y o t h e r w a y o f

p l a y i n g e x c ep t c h o o s i n g a s e c u r i t y st r a te g y c o u l d c o u n t a s r a t i o n a l i n t h e c o n t e x t o f

z e r o - s u m g a m e s a g a i n s t a r a t i o n a l o p p o n e n t . W h a t o n e m u s t o b j e c t t o a b o u t t h e w a y

t h a t E U X h a s b e e n t r a d i t i o n a ll y f o r m u l a t e d a n d e m p l o y e d is p r e c is e l y t h a t i t m u s t

f r u s t ra t e a n y a t t e m p t o n o u r p a r t t o e s t a b li s h t h is c o n c l u si o n . T h e t r a d i t i o n a l v e r s i o n

o f E U X i s a l s o re s p o n s i b l e f o r w h a t I t a k e t o b e s o m e e x t re m e l y c o u n t e r -i n t u i t iv e

c o n c l u s i o n s i n t h e c a s e o f n o n - z e r o - s u m t h e o r y , e . g . , t h e ' d o u b l e - c r o s s ' s o l u t i o n t o

' p r i s o n e r ' s d i l e m m a ' g am e s . O n e m i g h t h o p e t h a t a r e c o n c e p t u a li z a t i o n o f E U X f o r

t h e z e r o - s u m c a s e w o u l d a l s o s u gg e s t n e w w a y s t o l o o k a t t h e p r o b l e m o f r a ti o n a l

c h o i c e i n n o n - z e r o - s u m g a m e s .

some formal problems with the von neumann

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