On Auxiliary Games and the Modes of a Game
| Author | Knut Midgaard |
| DOI | 10.1177/001083676600100305 |
| Published date | 01 March 1966 |
| Date | 01 March 1966 |
| Subject Matter | Article |
Knut
Midgaard
On
AuxiliaryGames and
the
Modes
of
aGame
I.
IntroductionI
1.1.
The
background
One
of
the
lines pursued by Thomas C.
Schelling in his 'reorientation of game
theory' consisted in analysing a family
of
'basic' or 'typical' or 'strategic' moves:
unconditional commitment, threat, dele-
gation of decision, etc. He inquired into
the
characteristic relation which each of
them bears to
the
game it is related to,
and
the
conditions
for
each kind of
move to be successful."
The
topics to be studied in
the
present
article are related to those discussed by
Schelling; in fact, to some extent they
overlap.
The
major
part
of
the
article
will be devoted to the determination and
analysis of some basic relations which
can exist between one game and
another; and we shall, inter alia, discuss
the relationship between games
that
are
characterized by such relations to
other
games and Schelling's 'strategic moves'.
The
present author finds it natural to
emphasize this close connection.
The
article can be looked upon as an attempt
at supplementing Schelling's contribution
to
the
task of bridging
the
gap between
game theory and
the
empirical study of
conflict and cooperation.
1.2.
Terminological remarks
a) Game.
The
term game will be used
in a somewhat loose sense; it will denote
situations where strategic considerations
play a significant role, or - in
other
words -situations significant aspects
of
which can be described in game-theoretic
terms. Thus, the use
of
the
term negoti-
ation game implies the assertion
that
at
least some,
but
not
necessarily all sig-
nificant aspects of negotiation are, in
the
cases considered, strategic or game-
theoretic.
b) Embracing structure. By this expres-
sion I refer to
the
union of
(r)
the
game structure, i. e.,
the
'rules'
of
the
game and
the
players' utility functions
(preference structures); and (2) the
players' structural knowledge state, i. e.,
their assumptions about the game struc-
ture, their assumptions about each other's
assumptions in this regard, etc.
An
example may clarify
the
concept.
B
bl b2
A
3'ttj4
10
1 3
32 o3
Fig. I
Fig. 2
A game has
the
following embracing
structure:
(r)
The structure
of
the
game
(in
normal
form)
is such as represented in
Figv
r.
ON
AUXILIARY GAMES
AND
THE MODES OF A GAME 6S
(2)
Player A is known to consider the
structure of Fig. 1and
that
of Fig. 2
as equally probable. Player B, on
the
other hand, is
known
to
know
the
game
structure."
The
emphasis will in many games be
on
the structural knowledge state, be-
cause it may be difficult to tell
what
the
game structure 'really' is.
It
may,
in
particular, be a problematic thing to
tell which strategies are 'really' possible
to each player.
c)
The
analysis
of
the game. Aplayer
enters a game
with
(I)
certain 'method-
ological' or 'analytic' preconditions; there
are certain kinds of analysis he is able
to carry through, and there may be
some which he is particularly liable to
apply. He can be said to be in a certain
analytic (methodological) state.
He
fur-
ther enters
the
game
with
(2)
certain
assumptions
with
regard to the other
players' analytic state,
with
regard to
the
latter's assumptions about his own,
etc.; he can be said to be in a certain
analytic (methodological) knowledge
state.
This totality should always be
kept
in mind
when
reference is made to in-
fluences
upon
aplayer's analysis
of
a
game. In some cases a player's analysis
is influenced by a change in his ability
to analyse
the
game or in his immediate
inclination as to
the
choice of analysis.
In other cases
the
influence is due to a
change in his assumptions about the
other
players' analytic state. In a number
of games the situation is less transparent.
If an event influences aplayer's analysis
of
a given game, it may be difficult to
tell to
what
extent this is due to an
immediate influence on his inclinations
with
regard to the choice of method,
and to
what
extent it is due to an in-
fluence on his assumptions about a
similar influence on
the
other players'
inclinations, etc.
5-'"017.
COOPERATION AND CONFLICT
I"':
I
d) Prejerence structure.
The
idea of a
strictly game-theoretic utility function
defined over
the
set of end points in a
game tree raises intriguing problems if
the values which the player in question
attributes to the various end points re-
flect
not
only his evaluation
of
the
outcomes assigned to them, respectively,
but
also his evaluation of the cor-
responding plays of the game.
(A
similar
proposition holds for games in normal
form.)
We
shall here assume
that
the
players make an ordering of
the
various
end points with respect to the preference
and indifference relations and
then
in a
less specific way try to determine 'how
much' better or worse
the
various end
points are
than
each of
the
others. They
thereby clarify or determine
what
I shall
call, without any further attempt at
defining the term precisely, their pref-
erence structure.'
e) Dialectically conditioned transcend-
ence
(or
for short: dialectical tran-
scendence). Consider atwo-person game
G. By analyzing it - or trying to anlyze
it - a player gains knowledge which he
did
not
have at
the
outset,
when
the
data and methods on the basis of which
his analysis is undertaken, were de-
termined. Suppose the analysis does
not
lead to an unambiguous solution which
is
known
to be fully satisfactory to both.
Each player may in
that
case react to
his situation by trying to transcend the
game
(or
its mode)" and/or
the
method
used
for
analysing it, or he may suddenly
find
that
such a transcendence has already
taken place, due to
the
new
knowledge
acquired through his
(and
the other
player's) analysis. In both cases we shall
talk about a dialectically conditioned
transcendence
of
the given game (or its
mode)
andlor the original analysis.
Several kinds of dialectical tran-
scendence are possible. As we have al-
ready pointed
out
aplayer may tran-
scend the methodological basis of his first
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