Some comments on Harsanyi's postulates for rational behavior in game situations

behavior in game situations

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Damme, van, E. E. C. (1979). Some comments on Harsanyi's postulates for rational behavior in game situations. (Memorandum COSOR; Vol. 7916). Technische Hogeschool Eindhoven.

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Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 79-16

Some comments on Harsanyi's Postulates for Rational Behavior in Game Situations

by

E.E.C. van Damme

Eindhoven, November 1979 The Netherlands

Some Comments on Harsanyi's postulates for Rational Behavior in Game SLtt.lA.tions

by

E.E.C. van Damme

Abstract: A critical analysis of Harsanyi's postulates for rational behavior in game situations is presented. It is argued that to obtain a good theory the set of postulates should be modified. In the last section it is investigate~whatsolutions a modified set of postulates yields for a simple class of games.

1. INTRODUCTION

In [2], [3] and [6] Professor John C. Harsanyi has tried to develop a general theory of rational decision making. He introduces a set of 8 rationality postulates, and with these postulates he defines a solu-tion for all games. So when the players act according to the postulates, they will agree on the payoffs and strategies, that the theory pres-cribes..

However, there still remained some problems with the postulates. The precise status of them has given rise to misunderstandings ([1], [4] and [7] ), as a consequence of an unclear definition of some notions in the postulates. Moreove~for a certain class of noncooperative games the postulates yield a solution, which is not an equilibriumpoint (but a maximinpoint). Therefore in this paper we will thoroughly analyse the postulates and investigate why they yield a maximinpoint, rather than an equilibriumpoint, as a solution for certain games.

The main purpose of the rationality postulates in Harsanyi's theory, is

to derive the so called "Zeuthen's Principle". HarsanY1 derives the prin-ciple from the rationality postulates in case of a cooperative game, but also applies it to noncooperative games: it is his most important tool to discriminate between equilibriumpoints, and to explain why cer-tain equilibriumpoints cannot be the solution of the game. In this pa-per we will show that for noncoopa-perative games the principle is essen-tially based on a very questionable postulate. Therefore the whole non-cooperative theory is essentially based on this questionable postulate, which is very unsatisfactory.

Therefore a slightly modified set of rationality postulates (excluding_I the dubious one) is proposed, and for this set a modified version of "Zeuthen's Principle" is derived. Finally, we look what solutions the new set of axioms y!elds for a simple class of noncooperative games. It turns out that for this class the axioms yield a solution that coincides with the solution as prescribed by the new theory of Harsanyi and Selten

([5]).

This theory is not an axiomatic theory, but an extension of Bayesian one-person decision theory.

2. PRELIMINARIES

An

niPerson game

is an ordered 2n-tuple

G

=

(Sl'S2' ••• 'Sn'U₁'U₂' ••• 'U_n), where for each i E {l, ••• ,n}: S. is a finite set, the set of pure strategies of player i,

1

-U is a function, U : S XS x •••xS -+ JR, the payoff function of-playeri.

i i 1 2 n .

Although Harsanyi's rationality postulates apply to all classes of games, we will restrict ourselves in this paper to

nonoooperative games.

A

nonoooperative game

is a game, in which no agreements are binding

([6J p. 110) and this implies that the players can't use jointly random-isad strategies. Furthermore we will assume that the game is played

as a

vooaZ

game, which means that the players can communicate with each other. With respect to a given n-person game G the following concepts are introduced:

A mi~ed

strategy

of player i is a probability distribution on S .• This

. 1

set of all mixed strategies will be denoted by

L..

We use

L

1 as an

1 n

abbreviation of Llx .•. xL. lxL. lX .•• xL . If a

=

(0

1,02, .•• ,0 ) E IT

L.,

i ~- 1+ n n i=l 1

then a = (0

1, .•• ,0. 1,0. 1, ••. ,0 ) and we also denote a by

. 1.- ~+ n

(0

1,01.). The m~minpayoffof player i is defined as

'V U. := max 1. min i , i o El i U.(o.,o) 1. 1.

Am~min

strategy

for player i is a strategy

O.

E

L.

such that

1 1

min

i , i

i u.(o.,a) = 1. 1. i U. (T,O' ) 1.

2.,

such that 0'. is a best reply against ai, for all i E {l, ...

,ro.

~ 1.

(=e.p.) ([8J) of G is a strategy combination

S.

L

i can be viewed as : convex subspace of IR ~, the space of functions -from S. to IR. Let

I.

be a convex subset of

I..

Since S. is finite,

2~

.1. J. . ~ 1 J._m .

h~s only a finite number of e~treme points, say 0' , •••• ,0' • The centro1.d iltrateqy of

r

is the strategy

1. An

equitibriumpoint

n C1 E

n

i=1 1 m k

l

0' • mk=1 i i i

Let 0' E

L .

The

oentroid best reply

to 0 is the centroid strategy of tne set of all best replies against oi (which is indeed a convex set). SLmilarly we define the notion of oent~id

max£minstrategy.

Suppose player i thinks that the other players are going to choose strategies corresponding to O'i E

r

i . Then a best reply against O'i will be called

a-subjeotive best repLy

of player i.

3. THE RATIONALITY POSTULATES

In this section Harsanyi's rationality postulate~as far as they relate to noncooperative games, will be reviewed. Before doing this, we first have something to say about the philosophy behind the postulates. The postulates deal with rational behavior in game situations. This means that they apply to individuals who pursue their self interests against individuals who do the same. Now assume a vocal noncooperative game is played. Each player tries to obtain as much as possible. By playing a maximinstrategy a player is absolutely certain of obtaining at least his maximinpayoff. But each player would like to obtain more than this maximinpayoff. To obtain more a player needs cooperation with the other players, but since the game is a noncooperative one, stable cooperation can only occur at an e.p. ([2]J p.53). In a bargaining game

B(G), prior to the noncooperative game G, the players bargain for these e.p.-s, until one e.p. is selected. In this bargaining game the players use the rationality postulates. When the players reach an agreement, then in G they play strategies according to the selected e.p. When

they don't reach an agreement, then in G their behavior will be governed by postulate A

l .

~et us call an e.p.

unreachable

if there exists at least one player, who has rational grounds for not accepting this e.p. Let us call an e.p.

reachable

if it is not unreachable. The purpose of the rationality postulates is to show that for each noncooperative game there is at most one reachable e.p.

The postulates are devided into two classes:

class A: This class contains postulates, that tell a rational player how to choose his strategy.

class B: This class contains postulates, that tell a rational player, what he can expect from his opponents.

A

_{1:Maximin postulate.}

In a game

unprofitable

to you, always use a maximinstrategy.

Remark. The work unprofitable is a key-word in Harsanyi's theory. He defines it as

"you

cannot expect more than your maximinpayoff".

But

since it is not clear what you can expect, this definition is not very useful, and indeed it has given rise to misunderstandings. I will return to this definition later on. For the moment it suffices to remark, that Harsanyi qualifies games without reachable e.p. as unprofitable to all players. So for such a game the theory prescribes a maximinpoint as the solution.

~:

Best-reply postulate.

In a game profitable to you, use a strategy, that is a best reply to the strategy combination used by the other players.

~: Subj~ctive-best-reply

postulate.

In the bargaining game prior to a game profitable to you, use a (bargaining) strategy, that is a

subjec-tive best reply to the strategy combination, that you expect'the others to follow.

A

4:

Aaceptance-of-higher-payoffs postulate.

Part I: If you are willing to agree to some joint strategy

a,

then you

*

are also willing to agree to some joint strategy

a

yielding you more. Part II: Suppose you and the other players are in the bargaining game

*

barqaininq corresponding to the bargaining n-tuple

B.

Suppose

B

is a barqaining n-tuple yielding you more. Then you must be willing to enter

*

an aqreement in which all players shift to

a .

AS:

Centroid postulate.

Let

l~

be a subset of

li

such that

!) all strategies in r~: -are

equally consistent

with the other

ratio-~

nality postulates.

i,i) _{player i} e~ects all strategies in l~ to give him the same payoff.

~

Then player i will be equally likely to choose any strategy in this set (so his behavior will be as if he used the centroid strategy of this set).

(We will return to the italicized phra$es later on> •

B1: MUtuaZly-e~ected-rationality

postulate.

A rational player expects and acts on the expectation that other ratio-nal players likewise follow the ratioratio-nality postulates.

B₂: Sy~tric-e~ectations

postulate.

You cannot expect that other rational players choose a more concessive barqa1ning strategy, than you do.

B3: Independence-of-irreZevant-vaPiabZes postulate.

You cannot expect that other rational players select bargaining

strate-gie~ that depend on variables, whose relevance cannot be established on the basis of the present rationality postulates.

Now let us return to the phrases in the rationality postulates, which have caused misunderstandings. First postulate AS. To what sets

I;

does this postulate apply? In his reply to Mc Clennen ([4J) Harsany1 argues that the set of all best replies of player i against a given strategy combination of the other players qualifies as such a set. This has as a consequence that if 0 is a reachable e.p. of G, then 0 must have the

following property: for all i E {l, ...,n}: 0. is the centroid best reply

~

to oi. E.p's having this property are called

stabLe.

So we see: If an e.p. is unstable, then it is unreachable. In particular: If a game G does not have stable e.p.~s, then each player should play his maximin-strategy.

Now let us look at postulate A

1• Let i be a player in game G. To decide

whether the game is unprofitable to him or not, we have to know what he can expect. We have seen above that, if the game does not have stable e.p.'s, then he can expect only his maximinpayoff. But what about the case when there exists a stable e.p.? Can he expect the payoff corres-ponding to this e.p.? The answer is no, as we shall see in the following example. EXAMPLE 1. (3,1) (0,0) (0,0) (1,3) In this example (X

1'Y1) is a stable e.p., but player 1 cannot expect the payoff of 3. Because when he expects 3, this means that he expects play-er 2 to play Y

1, which is in contradiction with postulate B2. We can

say that, since e.p. (X

1,Y1) is unreachable, player 1 cannot expect the payoff corresponding to this e.p.

We have seen that, in order to decide whether a player has to play his maximinstrategy or not, we first have to know whether there is a reach-able e.p. or not. But we can only get to know this, by applying the postulates. So actually we have an ordering on the postulates, and they must be used in the following way:

The players start bargaining for an e.p. In this bargaining process they make use of all postulates, except A

1• There are two possibilities:

i) The players cannot reach an agreement. Then we use postulate A 1: the players should play their maximinstrategies.

ii) The players can reach an agreement, or equivalently: there exists a reachable e.p. cr. Can we conclude that in this case cr is the so-lution of the game? No, because it can happen that

u.

(cr)

=

u. for

1. 1.

some i. In this case player i can only expect his maximinpayoff, so he should play a maximinstrategy (which is not necessarily equal to cr.). So we have to consider two subcases:

1.

iia) There exists a reachable e.p. cr such that

u. (cr) > u. for all i E {l, ...,n}.

1. 1.

Then cr is the solution of the game.

iib) There is a reachable e.p. cr such that

u.

(cr)

=

u. for all i E N 1

1. 1.

U

i (cr) > ui for all i E N2, where N1 U N2

=

{l, ...,n}.

In this case the players belonging to N

1 should play a maximin-strategy. According to postulate AS they should play their centroid maximinstrategy. Since now the choices of the players from N

1 are known, the players belonging to N

2 are faced with a "smaller" pro-blem: They playa subgame, regarding the choices of the players from N₁ as given. For the subgame we again apply the described procedure, and in an inductive way we obtain a solution for the game.

But is the solution, we obtain by applying postulate A

1 satisfactory? Harsanyi admits that in this case the solution is not a "true solution", but rather a "quasisolutionll

, and says that one may even argue that it

is just another way of saying that such games have no true solutions ([6J, p. 138), But is not there another candidate for the true solution of such games?

EXAMPLE 2. (C,O) (1,2) (2,1) (l , 1)

* *

In this game there are three e.p.~s: (X

1,Y2), (X2,Y1) and (X ,Y ), where

*

1 1

*

1 1

X

=

2'X₁ + 2' X₂' Y

="2

Y

1 + 2' Y2• All these three e.p. ~s are stable.

Be-cause of postulate B

2 the e.p.~s (X1,Y2) and (X2,Y1) are unreachable.

*

*

E.p. (X ,Y ) is reachable. Since i t only yields the maximinpayoffs for the players, they should play their maximinstrategies. So the prescribed solution is (X

2,Y2). Now this point is highly unstable: If player 1 has rational grounds for expecting player 2 to play Y

2, then he will be

mo-tivated to play Xl' and something similar is true for player 2. So, for this game the theory prescribes an unstable solution, while there exists a natural and stable solution. This is of course unsatis-factory. EXAMPLE 3. (0,0) (l,3) (3,1) ( 1 , 1)

* *

In this game the three e.p. ~s are (X

1,Y2), (X2,Y1) and (X ,Y ), where

*

2 1

*

2 1

X =

3

xl +

3

X₂ and y

=

3

Y

1 +

3

Y2 . Because of postulate B2 the

*

*

e.p.~s (X

1'Y2) and (X2,Y1) are unreachable.E.p. (X ,Y ) is unreachable, too, because it is unstable. So the theory prescribes the maximinpoint

(X₂'Y2) as the solution. But this point is, as in example 2, unstable, too. Why does Harsanyi consider to have (X

2,Y2) greater stability than

*

*

(X ,Y ). He says that if player 1 plays X

2 then he is absolutely certain of obtaining 1, while if he plays

x*

his payoff can fall to

I

(cf. [6J, p. 125, example 1). But does player 1 expect his payoff to be } if he

*

plays X 7 The answer is no. If during the bargaining process player 1

*

announces that he is going to play X, then he expects player 2 to choose his centroid best reply againstX*, which is

t

Y₁ +

t

Y₂• This means that

4

he expects his payoff to be

3.

So it is not irrational for ptayer 1 to

*

play X •

A similar reasoning applies to player 2, so we can conclude that

*

*

(X , Y ) is in a sense self-enforcing: both players will be motivated

* *

to bargain for it. The only conclusion must be that (X ,Y ) has greater stability than (X

2,Y2). Again the prescribed solution is

unsatisfac-tory.

In the examples 2 and 3 we have seen, that although the theory yields solutions for all games, it yields unsatisfactory solutions in some cases. We have seen that especially postulate A

1 is unsatisfactory. Now

postulate A

1 is essential in Harsanyi's theory: it provides solutions

for all noncooperative games, not having an e.p. as a solution. In the following section we shall see, that this postulate also plays an essential role in another basic concept in Harsanyi's theory, to wit "Zeuthen's Principle". This principle is an important tool in the theory for deciding whether an e.p. is reachable or not.

4. ZEUTHEN'S PRINCIPLE

Consider the following bargaining situation:

Two players have to agree on one of two alternatives A and B. Player 1

prefers A, while player 2 prefers B. If player 1 sticks to A, and player 2 sticks to B, then a conflict situation C will develop. We will inter-prete the case that 1 chooses Band 2 chooses A, as an agreement to

ob-tain the payoffs U

1(B) and U2(A). We will assume that the players in fact obtain these payoffs. We assume U

1(A) > U1(C) and U2(B) > U2(C).

Furthermore we assume that the players cannot make binding agreements. ObViously the situation can be modelled as the following noncooperative game:

A

A (U₁(A) 'U 2(A»

We want to answer the question, which one of the players sho~ld accept his less desired alternative. Suppose the players are

Bayesian expected

utiZity maximizers.

This means they assign subjective probabilities to the possibilities the other player has and maximize the subjective expected utility. Let q be the subjective probability player 1 assigns to the event "player 2 chooses B". Then if player 1 chooses A he ex-. peets: (l-q) • U

1(a)

+

q. U1(C). If he chooses B he is certain of U1(B). Therefore he should choose A if (l-q) • U

1(A) + q • U1(C) > U1(B), which means U 1(A) - U1(B) q < _{r 1} := U 1(A) - U1(C)

Likewise player 2 should choose B if

U

2(B) - U2(A) p < r₂ := U

2(B) - U2(C)

where p is the subjective probability player 2 assigns to the event "1 chooses A". The quantities r

1 and r2 are called the

risk Umits

of player 1 and player 2, because they measure the highest risk a player is willing to face in order to obtain an agreement on his most prefered alternative, rather than on his opponent's most prefe~ed alternative. Zeuthen argues that the player who is less willing to risk a conflict should yield, while in the situation where they are equally afraid of a conflict, they both should yield.

So "Z~uthen'8

PrineipZe"

says: if r

1 < r2, then player 1 must yield, if r₁ > r

2, then player 2 must yield, and if r₁

=

r₂, then both must yield.

Now, can we derive this principle from Harsanyi'srationality postulates? Since Harsanyi's derivation is only valid for cooperative games, we

cannot apply it. Instead of presen~ing an alternative derivation, we will only briefly sketch the role postulate A

1 plays in such a deri-vation (cf. theorem 1, section 5 and [6J, p. 157). By postulate B

3 we can restrict our attention to relevant variables. Now it is easily seen that the only relevant variables for this game are p,q,r

1, and r2• Since these values don't change if we apply positive linear trans-formations to the utility functions of the players, we can write the game in the following form:

A B A B (l,b) (a,b) (0,0) (a,l)

where a,b E [O,lJ

We have r

1 = 1 - a, r2 = 1 - b. Let us consider the case r

1

=

r2, which means a

=

b. As an example we

1

take a =

3' .

We have the game:

A B A 1 (1,

3)

1 1

(3'

3')

B (0,0) 1

(3'

.1)

which is essentially the game of example 3.

To say that in this case both players should yield is equivalent with saying that the players should play their maximinstrategies. But this means we have to apply postulate A

1• So "Zeuthen's Principle" and with it the whole theory, is essentially based on a very questionable pos-tulate. The only conclusion can be that the theory is unsatisfactory.

5. A MODIFIED SET OF AXIOMS

The question remains whether an axiomatic theory of rational decision making can be set up. After the analysis of the preceding sections it will be clear that such a theory must yield e.p.-solutions for all noncooperative games. So in a good set of axioms postulate A₁ cannot occur. Since not all noncooperative games possess stable e.p.~s, this also implies that postulate AS' in its present form, cannot occur in the set. In this section we will look how ,far we can get if we take the axioms A2,A3,A4,B1,B2 and B

3 as our set of rationality postulates. So in this section a rational player will be a player who obeys these postulates. The rational solution will mean: the unique solution pres-cribed by these 6 postulates. We will see that these 6 postulates do not select from each game a unique strategy n-tuple to be the solution. But, for a certain class of games these postulates select the same strategy n-tuple as the new theory of Harsanyi and Selten ([sJ) does (theorem 2) • First however, we will state a modified version of "Zeuthen's principle", corresponding to this new set of rationality postulates.

THEOREM 1. (Modified version of "ZeuthenIs Principle")

Let G(a,b) be the game

Xl (0,0) (l,b)

X

2 (a,l) (a,b)

where a,b E [O,lJ

If a < _{b, then (X}

1 'Y2) is the rational solution of G(a,b) • If a > b, then (X

2,Y1) is the rational solution of G(a,b) •

* *

*

*

I f a = b, then (X , Y ), where X = (l - a)X

1 + a.x2, Y = (1- a) Y1 + a. Y2 is the rational solution of G(a,b) •

(So if r

1

=

r2, then the players should stick to their most prefered alternative with probability r

1, and yield with probability 1 - r1).

PROOF. The reader can quickly verify that the solution prescribed by the theoram is not in contradiction with the rationality postulates A

A

4,B1,B2,B3, So, indeed i t is a rational solution, We only aave to proof that i t is the only one. Assume a rational player 1 plays Xl with probability p(a,b) in game G(a,b) and a rational player 2 plays

Y1 with probability q(a,b) in game G(a,b). We have to prove that these probabilities conform to the assertion of the theorem.

By postulate A

2 we have:

(1)

( 2)

p(a,b) < 1 - b implies q(a,b)

₌

1 p(a,b) > 1 - b implies q(a,b) 0

q(a,b) < 1 - a implies p(a,b)

₌

1 q(a,b) > 1 - a implies p(a,b)

=

O.

If a

=

b then the game is symmetric, so by postulate B

2 we have:

(3) a = b implies p(a,b) = q(a,b).

Let us first assume a,b E (0,1).

There are three possibilities:

(4) i) p(a,b)

₌

0 and q(a,b)

₌

1 ii) p(a,b)

₌

1 and q(a,b)

₌

0

iii) p(a,b)

₌

1 - b and q(a,b)

=

1 - a.

By combining (3) and (4) we get: a

=

b E (0,1) implies p(a,b)

=

= q(a,b)

=

1 - a.

By postulate A

4 we have:

(5) Ya,a',b E [O,l][a > a' implies p(a,b) S p(a',b)]

V_{b,b',a E} [O,l][b > b' implies q(a,b) S q(a,b')]

(6)

v

I (O,l)[a> a' implies q(a,b) ~ q(a',b)J

a,a ,b E

V b

_{a, ,b}I E (O,l)[b > b ' implies p(a,b) ~ p(a,b') J.

Let a > b. By (6) we have: p(a,b) ~ p (a,a) = 1 - a < 1 - b < 1. So, by (4), p(a,b) =

°

and q(a,b) =

1-Let a < b. By (6) we have: p(a,b) ~ p(a,a) = 1 - a > 1 - b > 0. So (by (4) ) p(a,b) = 1 and q(a,b) = 0.

Thus the theorem is proved for a,b E (0,1). 1

If b > 0, then p(O,b) ~ P(2 b,b) = 1.

By (3~ p(O,O) = q(O,O). Combining this with (1) and (2) we obtain: p(O,O) = q(O,O) = 1.

Applying similar reasonings we can prove the assertion of the theorem

for all remaining a,b E [O,lJ. 0

THEOREM 2. For a

i > 0, bi > 0, i = 1,2, let G(a1,a2,b1,b2) be the following game:

(0,0)

Let p(a

1,a2,b1,b2) (q(a1,a2,b1,b2» be the probability with which a rational player chooses strategy Xl (Y1).

Unless a

1 = a2 = b1 = b2 we have: a

1a2 > b1b2 implies p(a1,a2,b1,b2) = 1 and q(a1,a2,b1,b2) = 0 a

1a2 < b1b2 implies p(a1,a2,b1,b2) =

°

and q(a1,a2,b1,b2) = 1

b a₁a₂ = b₁b₂ implies p(a 1,a2,b1,b2) = a 2

~

b 2 a 1 and q (a 1, a2,b1,b2) = - b

at

₁ PROOF. It is not hard to see that the solution prescribed by the theo-rem is a rational solution (not in contradiction with A2,A3,A4,B1,B2' B

3). We only have to prove that i t is the only one. By applying pos-tulate A

2 we see that the only possible combinations for p(.) and q(.) are those mentioned in the theorem. We just have to decide when each

of these cases will apply. No~ by postulate A

4 we have p(.,a2,b1,b2) and p(a

1,.,b1,b2) are nondecreasing functions, while p(a1,a2,.,b2) and p(a

1,a2,bl,.) are nonincreasing functions. We first consider the case a

l ~ b1 and b2 ~ a2•

By transforming the utility functions, we see that we actually have the game: (0,0) (b, 1) (1,a) (0,0) , where a,b E (0,1], ab < 1.

For this game we can derive the desired result in the same way as in the proof of theorem 1.

In case a

1 ~ b1 and a2 ~ b2, the game is:

(0,0)

(a,b)

(1,1)

(0,0)

, where a,b E (O,lJ, ab < 1.

Assume a < 1. Let ab < a < b. Because of monotonocity

a b

pO,l,a,b) ~ pO,a,a,b)

=

pO, i) , a, i)

a

=

p(l, i) , a, 1)

=

(because of the just proved result)

=

1. So p(1,1,a,b)

=

1 and q(1,1,a,b)

=

O.

Using the monotonocity of q we can also prove the desired result in case b < 1.

Remarks.

1. In theorem 2 the case a₁

=

a₂

=

b_{1 -}- b₂ is excluded. This case reduces to the following game:

--'._--'---Y1 Y₂

Xl (0,0) (l , l)

~

(1,l) (0,0)

Obviously if this game is played as a vocal game, there is no problem. The players will select (X_l,Y₂) or (X2'Yl~ they don't care which. But what if this game is played as a tacit game? Do the rationality

postu-lates yield a solution? The answer is no, they don't discrimate between

1 1 1 1

(X_l'Y2)' _{(X 2},Y_{l ) and ('2Xl} + '2Xl' '2'll + '2 'l2)' although the latter is

intuitively more acceptable. So another rationality postulate is needed. A possibility is, to strenghten A

2 in such a way that only perfect e.p.'s ([9J) are allowed. In this case the e.p.

1 1 1 1

(2

Xl + '2 X₂,

2

'l₁ + '2 Y₂) will be the solution.

2. Finally we look at the case in theorem 2 where some of the a. and

~

b. may be zero. The reader can quickly verify that the assertion of the

:J.

theorem is true as long as a

1 + b1

t=

0 and a2+ b2

t=

O. What happens if

e.q. a

1 + b1 = O? Then we have the following game:

'l 2 (0,0) (O,b) (0,a) (0,0)

For this game the present set of rationality postulates don't yield a solution. This is so because the postulates don't tell player 1 which strategy he must choose. But if a version of postulate AS is valid, then it is applicable to the set of all strategies of player 1. If we accept an additional postulate that says that in such a game player 1 chooses

1 1

2

Xl + _{'2 X2' then for this game we also obtain a rational solution. So,} to obtain a sufficient and consistant set of axioms, we need a weaker version of postulate A .

REFERENCES.

[1]

F~eeman~

R.E.:

Mc Clennen, Harsanyi and the general theory of games: some philosophical questions

in

the construction and justification of a theory of decision-making, Philosophical Studies 31 (1977), 123-131.

[2]

H~8anyi~

J.C.:

A general solution for finite non-cooperative games, based on risk-dominance,

in M.

D~e8he~~

L.S. ShapLey

and

A.W.

Tueke~ (ed): Advances

in

Game Theory. Princeton, N.J. (1964).

[3J - - - : A general theory of rational behavior

in

game situations, Econometrica ~ (1969), 613-634.

[4J - - - : Notes on the so-called incompleteness problem and on the proposed alternative concept of rational behavior, Theory and Decision ~ (1972), 342-352.

[5J - - - : A solution concept for n-person noncooperative games, International Journal of Game Theory

2

(1976), 211-225.

[6J - - - : Rational behavior and bargaining equilibrium in games and social situations, Cambridge University Press, Cambridge (1977).

[7J

Me

CLennen~

E.F. :

An incompleteness problem in Harsanyi's general theory of games and certain related theories of

non-cooperative games, Theory and Decision ~ (1972), 314-341.

[8J Na8h~

J.P.:

Noncooperative games, Annals of Mathematics

~ (1951), 286-295.

[9J SeLten~

R.:

Reexamination of the perfectness concept for equili-brium points in extensive games, International Journal of Game Theory! (1975) 25-55.