A relation between perfect equilibria in extensive form games and proper equilibria in normal form games

A relation between perfect equilibria in extensive form games

and proper equilibria in normal form games

Citation for published version (APA):

Damme, van, E. E. C. (1980). A relation between perfect equilibria in extensive form games and proper equilibria in normal form games. (Memorandum COSOR; Vol. 8019). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1980

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Department of Mathematics

PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP

Memorandum COSOR 80-19

A relation between perfect equilibria in extensive form games and proper equilibria in normal form games.

by

E.E.C. van Damme

Eindhoven, December 1980 The Netherlands

by

E.E.C. van Damme

Abstract

The concept of almost perfect equilibrium points for games in extensive form is introduced. It is proved, that proper equilibrium points of the nor-mal form of a game induce almost perfect equilibrium points in the exten-sive form of this game.

1. INTRODUCTION

For a game in extensive form with perfect recall Selten [3J introduced the concept of perfect equilibrium points in order to exclude the possibility that disequilibrium behavior is prescribed at unreached parts of the game. Selten achieves this equilibrium behavior in the following way: it is

assumed that any player will make mistakes with a small probability; there-fore every player will make any choice with a positive probability and hence any part of the game will be reached with a positive probability. So, in such a "perturbed game" there are no unreached parts and therefore every player will make an equilibrium choice everywhere. A perfect equilibrium point is then defined as an equilibrium point, which can be obtained as the

limit of a sequence of equilibrium points, associated with a sequence of per-turbed games for which the mistake parameter goes to zero. Selten

[3J

proved that any finite game in extensive form with perfect recall has at least one perfect equilibrium point, so indeed the perfectness concept is a useful refinement of the equilibrium concept.

Selten [3J also introducedtheperfectness concept for finite games in normal form. However, the relation between perfect equilibrium points in extensive form games and perfect equilibrium points in normal form games, is not as nice as one (perhaps) would like i t to be. Namely, let

r

be an n-person ex-tensive game and let G be the normal form associated with

r.

Let b=(b

1, ••• ,bn) be a behavior strategy combination and let q=(q1, ••. ,qn) be a mixed strategy combination such that the behavior strategy induced by q. is b. (section 2).

~ ~

It can be proved that, if b is a perfect equilibrium point of

r,

then q is a perfect equilibrium point of G. However, if q is a perfect equilibrium

point of G, then it is not necessarily true that b is a perfect equilibrium point of

r.

Selten [3J has given 2 examples of this phenomenon, which will be reviewed later on.

Myerson [2J has noted that perfect equilibrium points of normal form games possess another undesirable property. Namely,that adding strictly dominated strategies may change imperfect equilibrium points into perfect ones. There-fore, he introduced a refinement of the perfectness concept: the properness concept. Furthermore, he showed that any n-person game in normal form has at least one proper equilibrium point.

In this paper, we will relate perfect equilibrium points of a game in exten-sive form to proper equilibrium points of the associated normal form. We will prove, that proper equilibrium points of a normal form game induce very sensible strategies in the extensive form of this game.

Namely, if

r

is a finite game in extensive form, G the normal form associa-ted with

r

and q a proper equilibrium point of G, then in a natural (but not standard) way q induces a behavior strategy combination b=(b1, ••• ,b

n), for which it can be proved, that for any player i:

if b. prescribes that a certain choice must be taken with a positive pro-~

bability, then, given that player i will play b. after c, c is a best reply ~

against a sequence of small perturbations of (b1,···,b

i_1, bi+1,···, bn) So, the main result of this paper, which is proved in section 4, says: if a player plays (in the extensive form) in accordance with a proper equi-librium point of the normal form, then he acts optimally against mistakes of the other players, against past mistakes of himself, but not necessarily against future mistakes of himself. In section 3 we will review the discrepancy be-tween perfect equilibrium points of games in extensive form and perfect

equilibrium points of games in normal form. The examples given in this section will illustrate that the result of section 4 is the best we can obtain: we cannot expect that a choice prescribed by a proper equilibrium point is a best reply against mistakes of the player himself. Throughout the paper, the no-tation used will be the same as in Selten [3J. For the convenience of the

B

iu is compact.

A behavior strategy 2. PRELIMINARIES

In this paper we consider noncooperative finite n-person games in extensive form and the normal form games associated with them. A finite game in ex-tensive form is determined by:

(2.1) A game tree K; this is a tree with finitely many vertices and edges, (2.2) for each vertex of K a specification of which player has to move:

at this vertex.

(2.3) a specification of the information each player has, when he has to move; this specification is provided via the information sets, (2.4) a description of the possible choices at each move,

(2.5) a specification of the probabilities that are involved with the chance moves, and

(2.6) a specification of the rewards for each player.

In all extensive games we willencounter,each.player has .perfect recall ([lJ). This means,that at any instant of the game each player knows what he previously has known and what he previously has done.

Below, some definitions with respect to a finite game

r

in extensive form with perfect recall are given.

Let u be an information set of player i in the game

r.

A local strategy biu at the information set u is a probability distribution over A, where A denotes the set of all choices at u. b. (c) denotes the

u u ~

probability that the local strategy b. assigns to c. B

i denotes the set

~u u

of all local strategies at the information set u. Since the game is finite,

b. of player i is a function that assigns to every

in-~

-formation set u of player i a local strategy. U

i denotes the set of all in-formation sets of player i and B. denotes the set of all behavior strategies

~

of this player.

A behavior strategy b

i € Bi is called completely randomized, if biu(C) > 0 for all u € U. and c € A •

k ~k u

Let {(bl, •.•,bn)}k€~ be a sequence of behavior strategy combinations. Since the game is finite and B. is compact for all i and u, there exists a

subse-k k ~u k

quence {(b

1 Q" •••,bn Q,)}nJV€:JN such that lim b.,2;+00 ~u Q, exists for all i and u. The behavior strategy combination b=(b

1, ••• ,bn) is called a limit behavior strategy combination of the sequence

{(b~,

•••

,b~)}k€:JNif

there exists a .. subsequence of this sequence that converges to b.

A pure strategy ~i of player i is a function that assigns an element of A to every u € U.•

We will write ~iu for the choice that ~i assigns to u. IT

i denotes the set of all pure strategies of player i.

distribution ove.r IT .•

1.

to ~. (~. E IT.). A mixed strategy

J. J. 1.

a

for all ~. E IT .• Q. is the set

J. 1. 1.

A mixed strategy q. of player i is a probability

J.

q. (~.) is the probability that q. assigns

1. J. J.

q. is called completely mixed if q. (~.) >

J. J. 1.

of all mixed strategies of player i.

Associated with every endpoint x of K, there is a payoff for each player; h. (x) denotes the payoff for player i when x is reached. Once each player

1.

has fixed his strategy, an expected payoff for each player is determined.

H ).

n

that in

r

the players can H. denotes this eXPected payoff for player i. The normal form associated

1.

with

r

is the normal form G= (IT

1, · · · , ITn, H1, ••• ,

From the assumption of perfect recall it follows,

restrict themselves to behavior strategies ([lJ). In G, the players will use

playing q., if there exists

J.

is played (see (1]).

mixed strategies. In the following we relate these two concepts. Let q. be a

1.

mixed strategy of player i. An information set u E U. is relevant when J.

a vertex x in u, that can be reached, when q.

J.

By Rel(q.) we denote the set of all information sets that are relevant, 1.

when playing qi. Furthermore, we define for a choice c at some information

set U:

Ess(c) :=

_hi

E IT

i ; u E Rel(~.),1. ~.1.U

=

C}, and

P(c) := {~

.

E IT

i; ~. = cL

1. 1.U

We call the elements of Ess (c) : the strategies that are essential for c.

if u E ReI (q. ) J.

b. (c)

=

J.U

(2.7)

P(c) is the set of all pure strategies that choose c at u. The behavior stra-tegy induced by the mixed strastra-tegy q. is the strastra-tegy b., which is (as in [lJ)

J. 1. defined by: (2.8) b. (c)

=

1.U

I

~.EP(C) J. q. (~.) J. J. ifutRel(q.)1.

(we adopt the convention that

I

aE(3

x = 0).

f, then b. ~ ~ n for all

c

We now turn to the definition of perfect equilibrium points in extensive form games. Let f be an n-person game in extensive form with perfect re-call. Let n be a function that assigns to every choice c a strictly posi-tive number n •

c

The perturbed game (f,n) is a game which has the same game tree, informa-tion sets, etc. as f, but in which only particular behavior strategies are allowed. Namely, if b. is a behavior strategy of player i in

~

is admissible as a behavior strategy in (f,n) only if b. (c) ~u information sets u of player i and all choices c at u.

The game (f,n) is best interpretated as being an infinite game; pure stra-tegies are not allowed in (f,n).

A behavior strategy combination (bl, ••• ,b ) is a perfect equilibrium-point

k n

of f, if there exists a sequence {(f,n )}k of perturbed games and a

k k €:JN

sequence {(bl, .•• ,bn)}k€:JN of behavior strategy combinations, such that:

(2.9)

(2.10)

(2.11)

lim nk_c = 0 for all c k+CXl

lim

b~

= b. for all i

k+CXl ~ ~

k k k

(bl, ••• ,b

n) is an equilibrium point of (f,n )

Next, we turn to the definitions of perfect and proper equilibrium points in normal form games. Let G

=

(TI1, ••• ,TIn,Hl, .•• ,H_n) be a finite game in normal form and let € > O. A mixed strategy combination q

=

(ql' •••'~) is an €-perfect equilibrium point of G if:

(2. 12)

(2. 13)

qi is completely mixed (i € {l, •.• ,n})

For all i € {l, ••• ,n} and all ~i' ~i € TI i:

,

H. (q/~.) < H. (q/~.) => q. (~.) ~ €

~ ~ ~ ~ ~ ~

(H. (q/~.) is the expected payoff for player i when he plays ~.

~ ~ ~

A mixed strategy combination q = (q1""'~) is a perfect equilibrium

point of G, if there exists a sequence {(€k' q€k)}kE~sUChthat €k converges to

a

q£k converges to q (k-+00) and q€k is an E:

k-perfect equilibrium point of G (kE:N) •

A mixed strategy combination q = (q1""'~) is an E:-proper equilibrium point of G if:

(2.14)

_qi

is completely mixed (i E {l, . . .,n})

(2.15) _{For all i ~} {l, ••• ,n} and all ~i' ~i E IT i

,

H. (q/~.) < H. (q/~. ) J. J. J. J.

,

=> q. (~.) S; .E: q. (~.) • J. J. J. J.

A mixed strategy combination q = (q1""'~) is a proper equilibrium point of G, if there exists a sequence {(€k' qE:k) }k€:N such that E:_k converges

to 0, qE:k converges to q (k-+00) and qE:k is an E:

k-proper equilibrium point of G(k€JN).

Let

r

be an n-person extensive game with perfect recall and let G be the normal form associated with

r.

Let q be a proper equilibrium point of G and let {q€k}kE:N be a sequence of €k-proper equilibrium points, such that lim E:

k

=

a

and lim qE:k

=

q. Let b£k be the behavior strategy combination

k-+oo k-+oo

induced qy q€k (k€:N). In section 4 we are interested in behavior strategy combinations, which are limits of such a_sequence {bE:k}k€JN'

We will call a limit of such a sequence a limit behavior strategy combina-tion induced by q. Note that there is a difference between behavior stra-tegies induced by q and limit behavior strastra-tegies induced by q (see e.g. example 2).

Remarks:

1. The definition of perfect equilibrium points for normal form games is somewhat different from the definition given in Selten [3] We have chosen for the equivalent definition, given by Myerson [2]

2. If G is a game in normal form,

a

< _€1 < E:₂ and q is an E:₁-perfect equi-librium point of G, then q is also an E:

2-perfect equilibrium point of G. Hence, if

q

is a perfect equilibrium point of G, then for all E: >

a

there exists an €-perfect equilibrium point q~ such that lim q€

=

q.

€+O

In the following it will be convenient to consider such sequences (q€)€>o of €-perfect equilibrium points. The same remark applies if

q

is a proper equilibrium point. 3. EXAMPLES

In this section, we review the examples, by means of which Selten [3J illustrated the discrepancy between perfectness in the extensive form and perfectness in the normal form. The discussion of the examples will ex-hibit the relation between perfectness in the extensive form and properness in the normal form. In example 3 we illustrate what kind of information cannot be regained by considering proper equilibrium points.

EXAMPLE 1

a

a

5 4 4

a

a

o

a

3

a

3 extensive form

r

1 3 2

a

a

a

1 3

_a

a

_a

₅

a a

0

a a

0

a a

a

_a

0 0 1 3

_a

2

a

0 1 3 4 ₄

_a

a

3

o

3 3

a

₃ 3

a

3 3

a

₃

q. € € € € q

=

_{(ql' q2' q3)} € €1·L1Q, + _{€2· Ll}r + €3· RIQ, + €4· R1r q1 € (1-6) .L 2 + O.R2 q2 = € n.L 3 + (1-n) .R3 q3

=

One may prove that the equilibrium point q

=

(R1r, L

2, R3) is a perfect equilibrium point of G, but that it is not a perfect equilibrium point of f. Let us briefly investigate the reason for this fact. First, observe that q induces (in f) disequilibrium behavior for player 2: if player 2 expects that the others will play in accordance with q, then it is better for him to play R

2 (this yields 4, while L2 yields 3). Next, let us in-vestigate, why in G it is not the case that R

2 is better than L2• There-fore, let for € > 0 an €-perfect equilibrium point q€ be given, such that

€ lim q = €-+O Write: € Now, lim q2

=

L 2 €-+O e is sufficiently e € and hence L

2 must be a best reply against (q1' q2) if small.

Therefore,it is necessary that:

(3.1) ~ -(1-4n)1

3

(3.2)

And of course for all € > 0, one can choose e

1, e2 and n in such a way that (3.1) is satisfied. However, such a choice induces irrational be-havior in the extensive form.

Namely, in the language of the extensive form, (3.1) says: if

b~u

is the local strategy at u₂

induce~

by

q~,

and

2

if b~ is the behavior strategy induced by q~, then

b~

_u (Q,)

2

b~

(r)

u 2

But this implies (since lim b~(L3) = 0) that player 1 must choose Q, with a

€-+O

probability, which is greater than O. However, this is certainly not the case: since player 1 expects player 3 to play R

3 he will play r (with pro-bability 1).

\~

\

From our main theorem it follows that the equilibrium point q=(R

1r,L2,R3) cannot be a proper equilibrium point of G. Let us show, that this is

indeed the case. Therefore, assume qE (as defined above) is an E-proper equilibrium point. Since L

12 is a better reply against (q~,q~) than L1r is, if E is sufficiently small, we must have E

1 :;; E.E2. Consequently, (3.1) cannot be satisfied if E is small and q is not a proper equilibrium point. One may show that, for the game of example 1, the perfect equilibrium points of

r

are precisely the proper equilibrium points of G. Hence, in

this example, rational behavior in the extensive form can be detected al-ready in the normal form. However, in the examples 2 and 3 we will show that things are not always as nice as in this example.

EXAMPLE 2

I~

I

extensive form 2 0 0 2 0 3 0 2 1 1 1 1 1 1 1 1

corresponding normal form

The only perfect equilibrium point in the extensive form is the equili-brium point (R

12, R2). This point is perfect in the normal form, too. However, since in the normal form R

1r is a duplicate of R12, the point (R₁r, R₂) is also a perfect equilibrium point in the normal form. This

equilibrium point is even proper and hence we have a proper equilibrium point that induces disequilibrium behavior at the information set u₂

(which is not relevant when (R

1r, R2) is played). However, equilibrium behavior at this information set can be obtained by considering limit behavior strategies, induced by (R

1r, R2).

Namely, for € > 0 let q€ be an E-proper equilibrium point, such that lim q€

=

q. E+O Write: qE = (q~, q~) E q1 El·Ll~ + €2· Ll r + €3· Rl t + ~4·Rlr € q2

=

O.L₂ + (1-o).R 2 € € Let b

i be the behavior strategy induced by qi (i € {1,2}, € > 0). Then: E E b lUl(Ll) = €l + €2 blul (R1) = €3 + E4

b~

(t) = - -€1 b€ €2 1 (r) = u 2 E1+E2 u2 €1+E2 = 1 - 0

If € is sufficiently small, then R

1r and Rlt are Furthermore, in this case, L

1t is a better reply Hence, we must have:

€

the only best replies against q2. against q~ than L₁r is.

(3.3)

From this it follows:

(3.4) _lim

b~u

_{= t}

€+O 2

lim b₂E

=

R₂ €+O

{ E E }

Hence, the sequence (b

l, b2) E>O has a unique limit, which is (Rl~' R2), the unique perfect equilibrium point of

r.

EXAMPLE 3 1 1

o

u 1 extensive form L.Q, 1 Lr 1 R.Q, 1 Rr 0 corresponding normal form.

The only perfect equilibrium point in the extensive form is L.Q,. The proper equilibrium points of the normal form are L.Q" Lr, R.Q,. The proper equilibrium point R.Q, induces a unique limit behavior strategy combination b, given by:

b

=

R,

lu 1

Hence, imperfect behavior is prescribed at the information set u

1 (and this information set is relevant when R.Q, is played).

However, if the player makes an imperfect choice at u

1' then this does not have serious consequences, since the mistake can be corrected at u • Because

2

of this possibility to correct we call the choice R an almost best reply.

For a general n-person game with perfect recall we define almost best replies as follows:

Let b = (b

1, ••. ,bn) be a completely randomized behavior strategy combination. Let i € {l, ••• ,n}, u € U

i, c € A . The choice c is an almost best replyu II

against b, if for any c' € A , b~ € B,t there exists a b. € B., such that:

u 1 1 1 1 (3.5) I , .. H. (bl < b. after u

>1

c ) ~ H. (bl < b. after u

>1

c) 1U 1 1U 1

"

(H. (b/,< b. after u >/ c) denotes the expected payoff of

1U 1

player i, if he chooses c at u, given that u is reached, given

II

that he will play b. at the information sets that come after u

1

In section 4 we will show, that a choice prescribed by a limit behavior strategy combination, induced by a proper equilibrium point, is always an almost best reply against a sequence of perturbations, converging to this limit behavior strategy combination.

4. THE THEOREM

We open this section by giving an alternative characterization of perfect equilibrium points in extensive games (cf. [3], theorem 7). In the sequel we will use the concept of local best replies.

Let b be a completely randomized behavior strategy combination. Let u € U.,

~

C € Au, b

iu e: Biu' c is a local best reply against b if

(4.1) for all c e: A

u

b

iu is a local best reply against b, if a similar inequality is satisfied. Lemma 1

Let

r

be an n-person game in extensive form. A behavior strategy combination b = (b

1, .••,bn) is a perfect equilibrium point of

r

if and only if there

. { k k} .

ex~sts a sequence ,b

1, •••,bn ke:JN of completely randomized behav~or strategy

combinations, that converges to b, such that for any player i, any infor-mation set u € U

i and all k € IN:

b

iu is a local best reply against

(b~,

••• ,

b~)

Proof

(b

1, •••,b ) is a perfect equilibrium point ofn

r.

For k e: IN let

(bi, ••.,b~) be such that (2.9)-(2.11) are satisfied.

a completely randomized behavior strategy (i € {1, •••,n}, ke:JN). k

A such that b. (c) > O. Since b. converges to b. and

u ~u ~ ~ to 0: Assume b

=

(r,11k) and k Then b. is ~ Let u e: U., c e: k ~ 11 (c) converges (4.2) b.k (c) > 11 (c)k if k is sufficiently large. ~u Hence

(4.3) c is a local best reply against

(b~,

.•• ,

b~),

if k is sufficiently large.

Since the game is finite, there exists a constant k

O such that (4.3) is satisfied for all i, c and u with b. (c) > 0, if k ~ k

O• Hence, the ~u

sequence

{(b~,

.••

,b:)}k~O

fulfils the conditions of the lemma.

k k

Next, let {(b

1, •.• ,b )}k_n €~ be a sequence as described in the lemma. Fork

k €~, define a perturbed game (f,n) by:

k k (c) (c u. ) n (c) = b iu(c) if b.~u = 0 € Au' U € ~ k 1 U i) n (c) = - if b (c) > 0 (c € A , U € k iu u

Then lim nk(c) = 0 for all c and hence the perturbed game (f,nk) is well k-+oo

defined if k is sufficiently large (say k ~ k

O). For k ~ kOwe have:

b~u(C)

> nk(c) => c is a local best reply against bk•

Hence, bk is an equilibrium point of (f,nk), if k

~

kO. So, b is a perfect

equilibrium point of f.

o

We have seen, that choices prescribed by a perfect equilibrium point of

f are best replies against possible mistakes of the other players and against possible mistakes of the player himself. An almost perfect equili-brium point of f is defined as a behavior strategy combination, which has the property, that a choice prescribed by it is a best reply against mistakes of the other players, but is not necessarily a best reply against future mistakes of the player himself.

Definition 1

Let

r

be an n-person game in extensive form with perfect recall. A behavior strategy

of f, if behavior

combination b

=

(b , ••. ,b ) is an almost perfect equilibrium point

1 n k k

there exists a sequence {(b

1, ••• ,b )}kn €~ of completely randomized

strategy combinations that converges to b, such that for any player i, any information set u € U., any choice c € A and all k € ~:

~ U k

if b. (c) > 0, then c is a local best reply against (b / < b~ after u » •.

~u ...

Lemma 2

Let

r

be an n-person extensive game with perfect recall. Let G be the normal form associated with f. Let q

=

(q1, •.• ,qn) be a proper equilibrium point of G. For € > 0, let q€ be an €-proper equilibrium point, such that

€ € €

lim q

=

q. Let b be the behavior strategy combination induced by q (€ > 0). €-+O

For all i E {li, ...,n}, u E U., C E A one of the following statements is

~ u

true:

i) there exists a sequence {€k}kE~ with lim E

k 0, such that c k-+oo

is an almost best reply against b Ek for all k E ~ ,

ii) lim €-+O E b. (c)

=

0 ~u Proof Assume c E A (u E U., i E {l, ...,n}) is u ~

EO > 0 such that c is not an almost best

(e) (€)

For E E(O, eO)' let c € Au and b i E

such that i) is not satisfied. Let reply against be, if e €(O, eo).

B. be such that, for all b. € B.:

~ ~ ~

(4.4)

It is easily seen, that there also exists a pure strategy (4.4). In the sequel we will assume, that

b~e)

is pure.

J. ,

Since the game has perfect recall, we have that, if b is

b~E)

that satisfies ~

a behavior stra-tegy combination such that u is reached with a positive probability, then

,

,

Hiu(b Ic) is independent of that part of b

i that comes before u (see [3J lemma4) Therefore, if ~ is a pure strategy, such that ~ E Ess(c), then i t follows

from (4.4)

(4.5)

- (E)

From (4.5) we see that there exists a pure strategy ~ which is a better re-I - . t bE: th . N 1 f ( 0 ) -- def~ne '"{E)E Ess (c (E» by'. p Y aga1ns an ~ 1S. ame y, or € € ,Eo' ~ "

~(€) (u')

₌

_c (€) _{i f u}

₌

U

~(E)(u')

₌

_{b~E~ if u comes after u}

1U

Then we have:

(4.6)

Now,

whe~

(bE/TI(E» is played, then u is reached with the same positive probability as when (bE/TI) is played. Moreover, TI and TI(E) yield the same

. E

payoff against b , when u is not reached. So we may conclude:

(4.10)

Hence

(4.11)

Since qE is an E-proper equilibrium point:

So, if E E (O,E O): U E (4.12) E b. (c) 1.U :s; ._{E.q. TI}«E» 1.

L

q~ (TI) TI E Ess(c) 1.

=--..--...;..--'----L

q~(TI.) U E Rel(TI.) 1. 1. 1. '. - (E) :s;. {TI iTI \' E (E) l.. E.qi (TI ) E Ess(c)}

L

Since the game is finite:

lim b: (c) = 0

E+ 0 1.U

We are now ready to prove our main theorem.

THEOREM

Let f be an n-person extensive game with perfect recall. Let G be the normal form game associated with f. Let q be a proper equilibrium point

of G and let b be a limit behavior strategy combination induced by q. Then b is an almost perfect equilibrium point of

r.

Proof

Using that the game is finite and applying lemma 2 repeatedly, we see that we can construct a sequence {(€k' qk,

bk)}kE~With

the following properties:

(4.13) (k EIN), lim €k

=

0

k+oo

(4.14) qk is an €k-proper equilibrium point of G (kE:N), lim qk

=

q

k+oo

(4.15)

(4.16)

bk is the behavior strategy combination induced by qk

(kE~)

lim bk

=

b

k+oo

for all i E {l, •.• ,n}, u E U. and c E A :

~. u

b. (c) > 0 implies c is an almost best reply against bk

(kE~).

~u

k

We will prove that the sequence {b }kEJN has the following property:

(4.17)

I

for all i E {I, ••• ,n}, u E U., c, C E A lb. E B. I k E ~:

k ~ t U I~ ~ k

if b. (c) > 0, then E. (b /< b. after u >/c ) S; ·E. (b /< b. after u >/c)

~u ~u ~ ~u ~

We first introduce some definitions to facilitate the proof of (4.17).

Fix i E {ll • • • ,

n}.

For u E U. we inductively define O(u) (the order of u)

J. by:

O(u)

=

a

if there does not exist an information set v € U. ,

J. that comes after u.

O(u)

=

m+l if there exists an information set v E u. with

~

O(v)=m that comes after u and if O(w) S; m for all

information sets w that come after u.

Let u € U. , C € A , let v E U. be an information set that comes directly

J. u ~

P(vlbk, c) := the probability that v is reached, given that u has been reached, that c has been chosen at u and that the players except i play in accordance with

bk•

j.I(xJbk, c) := the probability that x is reached and no information set w after uis reached, given that u has been reached, that c has been chosen at u and that the players except i play in accordance with bk•

We will prove (4.17) (for fixed i) with induction to O(u). Assume u E U.

~

O(u)

=

0 and c E A is such that b. (c) >

o.

u ~u k

It follows from (4.16) that c is a best reply against b (kE~).

,

So, indeed for all c E A , b. E B

i and k E ~: U ~ H. (bk/ < b. after u >/ C')

~

loU ~ H. (b k / < b. after u >/ c) ~u ~ E :N.

and that (4.17) is true for all v E U. with

, t ~ C E A , b. E B. and k u ~ ~ such that that b. (c) > O. Let J.U (k)

that there exists b

i E Bi Assume u E U., Q(u) = m+1

~

o

(v) ~. oJ}. Let c E A such

u It follows from (4.16) (4.18) We have: H. (bk/ <

b~k)

after u >/ c)

=

~u ~

I

k H. (bk/ < b~k) >/b.(k» = p(vlb ,c). after v + v directly J.V. ~ ~v after u. +

I

j.I(xlbk,c). h. (x) ~ endpoints ~ XEK after u

I

p(vlbk,c). k after v >/ b. ) ~ H. (b / < b. + v directly ~v ~ J.V after u

+ =

L

endpoints XEK after u k H. (b / < b i after c >/ c). ~u

Since the game is finite, the order of every information set is finite and the proof of (4.17) is complete. Recalling the definition of almost perfect equilibrium points, we see that we have proved the theorem. IJ

Remark

The converse of the theorem above is false in general: if b is a perfect equilibrium point of the extensive form, then there does not necessarily exist a proper equilibrium point q of the normal form, such that b is

in-duced by q. Namely, Myerson [2J has given an example of a normal form game G with a perfect equilibrium point q, that is not proper. The game G can also be viewed as an extensive game

r,

where each player has one informa-tion set. Then q is a perfect equilibrium point of

r

and q is not induced by a proper equilibrium point of G. The search for a concept for extensive

form games that corresponds precisely to the concept of proper equilibrium points for normal ~orm games, is a subject of current research.

REFERENCES

[1J Kuhn, H.W., Extensive games and the problem of information. Annals of Mathematics Study 28, Princeton University Press,

1953, 193-216.

[2J Myerson, R.B., Refinements of the Nash equilibrium concept. Int. J. Game Theory,

I

(1978) 73-80.

[3J Selten, R., Reexamination of the perfectness concept for equilibrium points in extensive games. Int. J. Game Theory! (1975)