Identification of situations where cooperation is preferable to median game theory - Themis projec

Walsh, J. E.

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THEMIS SIGNAL ANALYSIS STATISTICS RESEARCH PROGRAM
IDENTIFICATION OF SITUATIONS WHERE COOPERATION IS
PREFERABLE TO'USE OF MEDIAN GAME THEORY
by
John E.'Walsh
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Technical Report No. 54
Department . of Statistics THEMIS Contract
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Department of Statistics
Sou# , ern Methodist University
Dallas, Texas 75222
Reproduction in A&d^ , or in part is permitted
'for any purpose of`the United States Government.
This document has been approved for public release
and sale; its distribution is unlimited.
1	 '
THEMIS SIGNAL ANALYSIS STATISTICS RESEARCH PROGRAM
IDENTIFICATION OF SITUATIONS WHERE COOPERATION IS
PREFERABLE TO USE OF MEDIAN GAME THEORY
by
John E. Walsh
Technical Report No. 54
Department of Statistics THEMIS Contract
January 13, 1970
Research sponsored by the Office of Naval Research
	
iContract N00014-68-A-0515
Project NR 042-260
Reproduction in whole or in part is permitted
for any purpose of the United States Government.
This document has been approved for public release 	 ^.
and sale; its distribution is unlimited.
i=
DEPARTMENT OF STATISTICS
Southern Methodist. University	 t-
IDENTIFICA`1'ION OF SITUATIONS WHERE COOPERATION IS PREFEi^ABLE TO USE
OF MEDIAN GAME THEORY
John E. Walsh
Southern Methodist University
ABSTRACT
Discrete two-person game theory based on median (rather than ex-
pected-value) considerations is widely applicable when the players
choose their strategies separately and independently. When cooperation
can occur, however, its use in choice of strategies can have advantages
compared to the median approach. This holds even for some median compet-
itive games (where both players can be simultaneously protective and
vindictive in the median sense). The purpose of this paper is to identify
situations where cooperation is definitely preferable, for two types of
cooperation. One type is that where no side payments are made. This
type of cooperation can occur for any situation where median game theory
is applicable. Side payments can be made for the other type of coopera-
tion that is considered. This type can occur for situations where all
payoffs can be expressed in a common unit (and satisfy the arithmetical
operations). A rule is given for deciding when cooperation is advantageous,
for both types of cooperation. Some implications of this rule are discussed.
* Research partially supported by Mobil Research and Development Corpo-
ration. Also associated with ONR contract N00014-68-A-0515 and NASA
Grant NGR 44-007-028.
fA form of discrete two-person game theory based on median considera-
tions has been developed (see reference 1) for situations where the
players choose their strategies separately and independently. Median
game theory is found to have strong application advantages over expected-
value game theory for these situations. In particular, the class of 	 I
I'
median competitive games, (see reference 2). where both players can be
simultaneously protective and vindictive, is huge compared to (and
	 l
I-
includes) the corresponding class for expected-value game theory. More-
over, the median approach is applicable to games where the values in one
or both payoff matrices do not satisfy the arithmetical operations (but
can be ranked within each matrix). A generally applicable "optimum"
solution for two-person median game theory is given in reference 3.
The usefulness of the median approach changes somewhat when the play-
ers can cooperate in their strategy choice. That is, they may be able to
guarantee a game outcome (a pair of payoffs, one to each player) that is
preferable, for both, to use of solutions that are median optimum for the
case of no cooperation. The purpose of this paper is to identify cases
where cooperation is definitely preferable to use of the median approach.
The results are based on an extension of the "relative desirability"
function concept for ranking game outcomes that is introduced in refer-
ence 3. Each player chooses a procedure for deciding on his preference
between any two "outcomes." For the first type of preference procedure,
only the possible outcomes for the game can be considered. The payoffs
1 ,
are expressed in a common unit, and satisfy the arithmetical operations,
-2-
for the second type of procedure. Then, all possible "outcomes" can be
considered which are such that the sum of the two payoffs equals a sum
that occurs for an outcome of the game.
More specifically, let (p i ,pii ) denote a general "outcome," where
PI
 is the overall amount received by player I and pII is the amount
received by player II. The values of p  and p II can be influenced by
a payment made from one player to the other. Also, let (pI(g),pII(g))
denote an outcome that can occur for the game. Only the (pI(g),pII(g))
can be compared using the first type of procedure for establishing
relative desirability. All (pi,pii) such that p  + pII equals
PI (g)+ pII(g), for some game outcome, can be compared for the second
type of procedure. Of course, the (pI (g) ,pII (g) ) are included in the
totality of the (pI ,pII ) .
For player I (II), relative desirability is a nonincreasing function
of p  (pII) for a fixed value of pII (p I). This restriction seems re-
sonable and helps assure that the procedures result in preference rank-
ings of the (pI (g) ,p II (g) ) and of the (pI ,pII ) . A condition on both types
of preference procedures is that a preference ranking is obtained for the
outcomes that can be considered (no circularity in definite preferences).
Otherwise, the preference procedure used by a player can be of any nature.
In particular, the requirement that the outcomes of set I (set II) have
maximum preference to player I (II), a condition of the "relative de-
sirability" function of reference 3, need not hold. Actually, the
procedure for one player can be arbitrarily different from that used by
the other player. For deciding on the advantage of cooperation, however,
l-
11
-3-
the preference rankings for both players are assumed to be known.
Two types of cooperation are considered. No payments from one
player to the other are made for the first type, which is associated
with the first type of preference procedure. Since the amounts received
are not influenced by side payments, only the (pI(g),pII(g)) are
considered. Cooperation of the first type can occur for any situation
where the median approach is usable. That is, the only requirement on
the kind of payoffs is that, separately, they can be ranked in the
payoff matrix for each player.
The second type of cooperation can involve side payments but imposes
a condition on the kind of payoffs that can occur. That is, the payoffs
are expressible in a common unit and satisfy the arithmetical operations.
Occurrence of a side payment in combination with specified (p (g) ,p (g))
I	 II
results in amounts that still sum to p I (g) + pII (g) . Thus, all the
(pI'pII) with p  + pII = pI (g) + pII (g) , for some game outcome, can be
considered for the second type of cooperation.
Cooperation is beleived to have a definite advantage, compared to
the median solution, when both players can gain through agreement in the
selection of an outcome. A rule for deciding when cooperation is advan-
tageous, and some discussion, is given in the next section. Also, some
implications of this rule for competitive games (reference 1), for the
special case of generated median competitive games given in reference 1,
and for the general median competitive games of reference 2 are examined
in the final section.
THE RULE AND SOME DISCUSSION
Let the game outcomes (pI(g),pli(g)) be ranked according to
-4-
I
i
a
increasing preference separately by each player. There is a smallest
subset S  (SII ), consisting of most preferable game outcomes to player
I, (II), such that player I (II) can assure an outcome of S  (S II ) with
probability at least 1/2. These sets provide the basis for the rule
used to decide when cooperation is preferable to this median approach.
i
Incidentally, this always usable optimum solution for median games is
an extension of the solution given in reference 3, and is itself a sub-
ject for research. The extension allows a less restricted form of
"relative desirability" function.
General Rule: Cooperation is definitely advantageous to both
players when an achievable outcome exists that is at least as desirable
to player I as one or more outcomes in S  and also is at least as de-
I_
sirable to player II as one or more outcomes in SII.
This rule follows from the consideration that an outcome with at
least the minimum desirability for both ~I and S II can be guaranteed by
cooperation. The median approach only assures, with probability at least
1/2, that player I obtains an outcome with desirability at least equal
to the minimum for S  and player II gets an outcome with desirability at
least the minimum for SII.
Of course, only the (pI(g)'pII(g)) are achievable for the first type
of cooperation, and all the (pI,pII) that satisfy p  + pII = PI (g)+ pII(g),
for some game outcome, are achievable for the second type of cooperation.
The general rule can be simplified. For the first type, cooperation
is definitely advantageous when the same game outcome value (for example,
the same outcome) occurs in both S I and S II . This could happen, for 1
example, when an outcome occurs with p I (g) large compared to nearly all
IM
-5-
of the payoffs to player I, with pII (g) large compared to nearly all
the payoffs to player II, and where an increase in pciyoff to a player
is noticeably more desirable than a comparable decrease Li the payoff
to the other player.
For the second type of cooperation, a simplification occurs when
there is a yame outcome value that occurs in both S  and SII. A
simplification also occurs for the rather common situation where an
equal increase in the amount received by each player is preferable
(including equal preference) to both players. T):ien, cooperation is
always advantageous if there is an achievable (pI,pII), not necessarily
a game outcome, such that p  is at least as large as the smallest pI(g)
in S  and also pII is at least as large as the smallest pII (g) in SII.
To verify the above statement, consider an outcome (pI''pII') in
S  that has smallest pI (g) and an outcome (pI",pII") in SII with small-
est pII (g) . A game outcome occurs such that the sum of its payoffs is
at least equal to pI ' + pII ". This implies that, perhaps by a side pay-
ment, an achievable (pI'pII) occurs of the form
PI - PI + 2a ,	pII = PII + 2b,
with a and b nonnegative. In addition to the side payment to obtain
(pI'pII), suppose that player I makes a side payment of a-b to player II.
Then, choice of a game outcome whose payoffs sum to p  + p II , and use of
the stated side payments, gives
(pI - a+b,
which is preferable
at least as much as
(pI - a+b
which is preferable
pII + a-b)
to (p1',pII'
pII'. Simil
PII + a-b)
ofto (p,
I	 IJ
( pI ' + a+b, pII ' + a-b)
for player I, since pi ' was increased
►rly,
(p I II - a+b, pII 11 + a+b) ,
for player II.
-6-
COMPETI`T'IVE. MEDIAN COMPETITIVE CASES
For a competitive game, the totality of game outcomes can be
arranged so that the payoffs to player I are nondecreasing and also the
payoffs to player II are nonincreasing. Then, S  and SII are mutually
exclusive, since the arrangement is according to nondecreasing preference
for player I and nonincreasing preference for player II (due to the
restrictions on preference procedures). However, the same outcome value
could possibly occur in both S  and S II . This can happen only when the
least preferable outcome of S  has the same value as the least preferable
outcome of SII . Moreover, no combination of side payment and cooperative
choice would necessarily be beneficial to both players when compared to
use of the median approach. Thus, cooperation (of either type) is
seldom useful for competitive games.
r
The situation is different for median competitive games that are
not competitive. First, consider the kind of median competitive game
that is generated by a competitive game (reference 1). In S i , also in
SII , for the competitive game, consider the payoffs to player I and the
payoffs to player II. Within each of S  and S II , let the outcomes for
another game (the actual game) be obtained by combining payoffs for the
two players. In this way, the game outcomes are divided into the set
generated by S I , the set generated by S II , and the remainder (if any).
The situation is like that for competitive games if the preference
procedures for the players yield for the new S  the outcomes generated
by SI for the competitive game.and for the new SII the outcomes generated
by S 	 the competitive game. Then, the new S  and S 	 mutually
exclusive. Also, the payoffs to player I (II), in S  (S II ) are at least
10
-7-
Ias large as those to player I (II) in SII (S I). However, many preference
procedures do not result in this situation. Some procedures that do yield
this situation are those that, for player I (II), assign maximum prefer-
ence to any outcome with p  (g) 2 P I and pII (g) 
S P
II ' (pI (g) s P I ' and
pII 
(g) z P II). The payoffs PI , PII , P I ', P II ' are defined in reference
1 (also see reference 2). These are some of the properties that are
required for the "relative desirability" function of reference 3.
finally, consider general median competitive games (reference 2).
Cooperation is often advantageous for these games. An exception is
when some form of the "relative desirability" function of reference 3 is
used by each player.
-8-
REFERENCES
1. Walsh, John E., "Discrete two-person game theory with median
payoff criterion," Opsearch, Vol. 6, 1969, pp. 83-97.
2. Walsh, John E., "Median two-person game theory for median competi-
tive games," Journal of the Operations Research Society of Japan,
Vol. 12, No. 1, January 1970.
3. Walsh, John E., "Generally applicable solutions for two-person
median game theory." Submitted to Journal of the Operations
Research Society of Japan.
i
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3	 REPORT TITLE
Identification of Situations Where Cooperation is Preferable to Use of Median
Game Theory
d. OE5CRIP TIVE NOTES (T)•pe o! report ar:d,inchrsi ye dales)
Technical Report
5. AU THOR(S) (First name, middle initial, lest name)
John E. Walsh
6	 REPORT DATE 7a. TOTAL NO. OF PAGES 7b. NO. OF REFS
January 13, 1970 9 3
Pa. CONTRACT OR GRANT NO. 9a, ORIGINATOR'S REPORT NUMBER(S)
N00014-68-A-0515
b. PROJECT N..
54NR 042-260
.. 9b. OTHER REPORT NO'S) (Any other numbers that may be assi4ned
this report)
d.
10. DISTRIBUTION STATEMENT
This document has been approved for public release and sale; its distribution is
unlimited.	 Reproduction in whole or in part is permitted for any purpose of the
United States Governmgnt
11. SUPPLEMENTARY NOTES 12. SPONSORING MILITARY ACTIVITY
Office of Naval Research
13. ABSTRACT
Discret,J two-person game theory based on median (rather than expected-value)
considerations is widely applicable when the players choose their strategies sepa-
rately and independently. 	 When cooperation can occur, however, its use in choice
of strategies can have advantages comioared to the median approach. 	 This holds even
for some median competitive games (where both players can be simultaneously protect-
ive and vindictive in the median sense).	 The purpose of this paper is to identify
situations where cooperation is definitely prefereable, for two types of cooperation.
One type is that where no side payments are made. 	 This type of cooperation can
occur for an y situation where median game theory is applicable.	 Side payments can be
made for the other type of cooperationi that is considered.	 This type can occur
for situations where all payoffs can be expressed in a common unit (and satisfy the
arithmetical operations). 	 A rule is given for deciding when cooperation is advan-
tageous, for both types of cooperation.	 Some implications of this rule are discussed.
f
D D NovFORM 1 4 7 3 (PAGE 1)1	 65 '^F /	 UNCLASSIFIED
S/N 01 01 -807-681 1	 Securitv Classi(icah,n
1•:II1Ji


Identification of situations where cooperation is preferable to use of median game theory

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