- Email: [email protected]
Nash equilibrium vs. maximin
L
~cxpid
2;
publication November 1992
This paper derives comparative game statics results for both makimin and Nash strategies, thereby allowing a more systematic comparison between these two approaches.
I* ?ntrrutwgi~g mm-_-
w-w_
Nash (i950) demonstrated
that every n-person game has an equilibrium * Thus in every game, there exists a vector of player strategies (sf. sr,. . . SF.. . s,*) such that expectP’utility maximization bv pkyer I glen {ST,.. .si?;_ ,,sT+ I. ..sff) will lead to i .hoosing SF. The Nash equilibrium approach to analyzing non-constant-sum games has been a much more popular analytic device than the maximin approach.’ ecent work by Holler ( 1990) and Hofler and Host ( 1SW) has provided a pcrstiasitc argument in favor of maximin over Nash equilibria when mixed strategies are involved. In this paper, we first consider the arguments made and suggest that the choice between the two approaches must be based on their reiative success in predicting outcomes. We then derive comparative statics results in order to provide tesiable propositions. _^<“C p;,‘r
ii,
. ,riiii&
. sZrai;;g1G3*
~~rn~~~ -
2
oiler demonstrates that when mixed ar?d unique, they yield the same
ex
argues that the p!ayer should not choose t rathec the maximin strategy.
01762680/93/!§06.00
0
1993--Elsevier
Science Publishers B.V. AI rights reserved
D. Wittman, Nash equiiibrium us. maximin
Table 2” __~_____~
c*
c2
Rl
Nll.ii21
UlSIU22
R2
u;39u8;3 __--____-
u14.9124
aU*, >u,,>o
and
zi4>U12>O*
u21>
and u2.+> IA,~~0. The rankings of the uil insure that a mixed strategy equiiibrium iS psibie. u,, < LcLz iMd U24< Urg is possible.
u22 >O
than under the payoff from a Nash equilibrium. Thus the Nash equilibrium is unlikely to exist. It can also be shown that a Nash equiiibrium in mixed strategies may yield an outcome that is Pareto to a Nash equilibrium in pure strategies. owever, maximin is not without its faults. Most important, when the in strategies do not coincide with the Nash equilibrium strategies, then in is not a Nash equilibrium. Thus when both sides arc pursuing maximin strategies. one (or both sides) could improve his/her -welfare by ere is a second choice). so may yield odd results, especially when the paranoia of maximin is inzpproprfate. Consider for example, the game depicted in tabfe i ximin requires tha! Row time and R2 (C2) eighty e. in contrast, there are three Nash equilibria with the most ing the pure strategy equilibrium pair (K ,, Ct ). But sh equilibrium (ii/ 5,4/S) ( 1/5,4/S)) implies the strategy maximin, i.e., I. d a considerable amount of time debating the logical preach and the relative merits of the assumptions, the tiona, i;rnd the main vehicle for never
to be worse
off
redicriw
ifSiii~dS
is
a
*
comparative s:atics analysrs. In
a!%! blluw-
in the endogenous variables due to a c r both appraaches. Those derived teiatio for hypothesis testing either in a controh
Correspondingly,
th
Differentiatmg denoting partials interior maximu equilibrium):
Equivalently,
That is, Q is choosen so that 32.
. . . .__G:-E~UiViii~~~uy,
ow makes CoEir
A Nash strategy by and C2. Equivalently. P’
u24 f.421 -1422
-
u23 u23 + u2&$
.
in
rent
c
g-t
D. Witfmcrn, Nash equilibrium
5&l
w. maximin
1 by p’ and the probability obabi~ity that Row chooses lies Cl by q? maximin strategies imply the fQ~~ow~~~
at
ie
nw
WlllP
crf .““’
c3w ic indifferent
whether
C. .
quivalently, 4”
u24-"22
z
U?1
-u22
.
(12)
-'t23+U24
us the d~ff~r~n~e between Nash and maximin is that under Nash, Row ~~o~urnn) chooses a strate y that makes Column (Row) indifferent to which -‘I - ---- ii 4zolumn (cow) IS chosen; whiie under tilaxlmiii, Ro-w jC&iiiiil) c;lIUU~Gc3 akes Row (Column) indif%erent to which column (row) is Nash, the concern with the other player’s payoff is over-
maximin, thab cancer:: with the other player’s payoff is
y to undertake a comparative game statics analysis. so, it is u~~~~ to ask why one undertakes such an activity. In are tested. f.ot by looking at a the equilibrium changes Thus, for example, economists do not P~WWM~ rrwahnonc hnt ahey _-II_--_ f~+ --a _--_-F_, -__ pahy as will fait when the price rises. The bserving O-P level is that know the utility function
13. Wi!!msn, fV*si: equilibrium vs. maximin
563
methodology is raregly used for bimatrix games with mixed strategies, even though the rationale for using ccmparative statics still holds. We first consider the effect of an increase in ia,j on p and q (in practice, we are un:ikeIy to observe the payotT directly, rather we may observe a _;* in 2 particu!zr direction). technological change that is likely to effect IP; U,j
Proposilion I.
Assuvne that the participants are Nash players.
:z: 1:;; iiir(cI4ac no effect on (b) An increase (c) An increase (d) An increase (a) An increase
on p. 1 ifar ls, a change in Row’s ihlity has Row’s equilibrium behavior. in u, 1 will result in a decrease in q. in u I z wiii result in Q decrease in q. in i?i j wi!! resu!t ir! a~ incremz it; q. in u 14 will result in an increase in q.4 iit
l4 1 j
k4s
no
fg$x~
ProoJ. (a) From eq. (7) it is readily apparent that an increase in Uli will have no erTect on p. (b) Looking at eq. (5) it is readily established that both the numerator and the denominator are greater than zero: by assumption u,,>u,,>O and u,,>u13>0. Therefore as ull increases, the denominator increases and q decreases. (c) Again looking at (5), we derive the partial derivative of q with respect to u12:
1 4u,*=
-
Ull
-U13
Ull
U14-U12 ~.___~ ~____. -U12
-U,3+U,4~2
----CO.
!U;;-!“;t-lri;+l!i4?2
The !zs! “--1inmualitv -~ d
__~_ 5
-u13+1414
-U12
UII =
_
holds since g?ri >
(13)
assumption. (d) Again looking St (S), it is immediately apparent that an increase in u13 will lead to an inc! ease in q. (e) Again looking at (5), we derive the partfa’, dcs,vative of y with respect to ulj
by
Ul4: 4-“I. “. .
=
_
_..
._
U11 -U12
Ull b,*
-
! _-- .
-U13+-U,4
-U13 ___
Ul4-Ul2
-
- ~_. ~ _.__
u~2-u,~+Ur* )2
-
@I,
-U,2-U13+h4)2
>O.
*rhis ccw!f~~!ntl~~~~ve &4mkK $v Nash pOaycrs was discusseri in .~ittrn~n characteristics of mixed strategy Wa& equilibria are found in Wittman ( 195%.
(1985). Other
D. Wittman, Nash equilibrium vs. maximin
564
We next consider maximin. Asstciiie that the PlUyWseiiljdOv
Froposition 2. (a)
(6) (c) (d) (e)
An An An An An
increase increase increase increase imrease
in U2i has no effect on p”. in u1 1 will lead to a decrease in p”. in u, 2 wiC1lead to an irtcrecse iz p”. in u1 3 will lead to an increase in p” if and only if u12 > u I 1. in u14 will lead to an increase in p” if and only if u 12 c u 11.’
(a) Looking at eq. (10) it is immediately will have no efbect on pve (b) Again looking at (IQ), it is immediately clear result in a d--rnncp ;b.W,LbcK+I* 111pc. (c) Again looking at (IO), it is immediately clear ..A_.. Okir _I :_.__*_$.$in p”_ :csxlc . 1 &ii *drivb (d) Taking the partial derivative of (10) with foiiowing equation:
clear that a change in
proof.
Pk=
strategies.
iiiiaXiB?iil
- --Ull
i -!?ii
-.
-
that an increase in u1 1 will that an increase in uz2 will respect to ut3 we get the
u14-u13
-i-
--u13+u14
U2t
b11-u12
-f413+Ul*)*
84!112-cJ11
@It -u12
-h3
+&4b2’
his last term is (strictly) positive if and only if u12 > u, t. (e) Taking the partiai derivative of (10) with respect to ur4 we get the following equation: #, -.,,u;,.,J;;..
__ 14
ull-“Ii =--------_-. 6,. .-II Irr 12’ 1-11 -“la-” =13 1 WI41
1614-1613 f 1111 -u12
-u13
+u14)2
WI
5.
ca
s
Hoiier (S!Xl~ argued on a priori grounds that when both maximin and Nash are mixed and unique, maximin is superior to Kash. Wdlei and (1990) then undertook an experimental study. Using a sample of stu strategies to a particular game, they obtained frequencies for each row and column. The observed quencies were closer to the frequencies predicted by maximin than Nash. re we have suggested that the empirical validity of i&~- approaches shouid be based on then predictability and that the best method for deriving behavioral reiations is by means of a comparative game statrcs analysis rather than levels for a particular game. To that end we have derived the comparative static relations and look forwards to future experimenta! and non-experimental tests that determine i~j*hi& i. _--iii t‘neorv =---=-= J most ace;+ rately predicts human behavior.
Nash, J, t9SO. Equilibrium points in N-person games, Proceedings of the National Academy of Sciences, U.S.A. 36. 48-49. UI,BUII~~in iwo-person games, Holler, Manfred, l990? The unprofitability of mixed strategy b mq.*z’zL-‘Economics Letters 32, 3 19-323. Holler, Manfred, n.d., Fighting when decisions are strategic, Public Choice, forthcoming. Holler, Manfred and Viggo Host, 1990, Maximin vs. Nash equilibrium: Theoretical results and empirical evidence, in: Richard Quandt and Dusan Triska, eds., Optimal ciecisions in markets and planned economies (Westview P:ess, Boulder, CO) 245-255. Wittman, Donald, 1985, Counter-intuitive results in game theory, Esropean Journal of Political Economy I, 77-85. Wittman. Donald, 1989, Arms control ver:%ation and other games involving imperfect detection, American Political Science Review 33, 923-945.
~cxpid
2;
publication November 1992
This paper derives comparative game statics results for both makimin and Nash strategies, thereby allowing a more systematic comparison between these two approaches.
I* ?ntrrutwgi~g mm-_-
w-w_
Nash (i950) demonstrated
that every n-person game has an equilibrium * Thus in every game, there exists a vector of player strategies (sf. sr,. . . SF.. . s,*) such that expectP’utility maximization bv pkyer I glen {ST,.. .si?;_ ,,sT+ I. ..sff) will lead to i .hoosing SF. The Nash equilibrium approach to analyzing non-constant-sum games has been a much more popular analytic device than the maximin approach.’ ecent work by Holler ( 1990) and Hofler and Host ( 1SW) has provided a pcrstiasitc argument in favor of maximin over Nash equilibria when mixed strategies are involved. In this paper, we first consider the arguments made and suggest that the choice between the two approaches must be based on their reiative success in predicting outcomes. We then derive comparative statics results in order to provide tesiable propositions. _^<“C p;,‘r
ii,
. ,riiii&
. sZrai;;g1G3*
~~rn~~~ -
2
oiler demonstrates that when mixed ar?d unique, they yield the same
ex
argues that the p!ayer should not choose t rathec the maximin strategy.
01762680/93/!§06.00
0
1993--Elsevier
Science Publishers B.V. AI rights reserved
D. Wittman, Nash equiiibrium us. maximin
Table 2” __~_____~
c*
c2
Rl
Nll.ii21
UlSIU22
R2
u;39u8;3 __--____-
u14.9124
aU*, >u,,>o
and
zi4>U12>O*
u21>
and u2.+> IA,~~0. The rankings of the uil insure that a mixed strategy equiiibrium iS psibie. u,, < LcLz iMd U24< Urg is possible.
u22 >O
than under the payoff from a Nash equilibrium. Thus the Nash equilibrium is unlikely to exist. It can also be shown that a Nash equiiibrium in mixed strategies may yield an outcome that is Pareto to a Nash equilibrium in pure strategies. owever, maximin is not without its faults. Most important, when the in strategies do not coincide with the Nash equilibrium strategies, then in is not a Nash equilibrium. Thus when both sides arc pursuing maximin strategies. one (or both sides) could improve his/her -welfare by ere is a second choice). so may yield odd results, especially when the paranoia of maximin is inzpproprfate. Consider for example, the game depicted in tabfe i ximin requires tha! Row time and R2 (C2) eighty e. in contrast, there are three Nash equilibria with the most ing the pure strategy equilibrium pair (K ,, Ct ). But sh equilibrium (ii/ 5,4/S) ( 1/5,4/S)) implies the strategy maximin, i.e., I. d a considerable amount of time debating the logical preach and the relative merits of the assumptions, the tiona, i;rnd the main vehicle for never
to be worse
off
redicriw
ifSiii~dS
is
a
*
comparative s:atics analysrs. In
a!%! blluw-
in the endogenous variables due to a c r both appraaches. Those derived teiatio for hypothesis testing either in a controh
Correspondingly,
th
Differentiatmg denoting partials interior maximu equilibrium):
Equivalently,
That is, Q is choosen so that 32.
. . . .__G:-E~UiViii~~~uy,
ow makes CoEir
A Nash strategy by and C2. Equivalently. P’
u24 f.421 -1422
-
u23 u23 + u2&$
.
in
rent
c
g-t
D. Witfmcrn, Nash equilibrium
5&l
w. maximin
1 by p’ and the probability obabi~ity that Row chooses lies Cl by q? maximin strategies imply the fQ~~ow~~~
at
ie
nw
WlllP
crf .““’
c3w ic indifferent
whether
C. .
quivalently, 4”
u24-"22
z
U?1
-u22
.
(12)
-'t23+U24
us the d~ff~r~n~e between Nash and maximin is that under Nash, Row ~~o~urnn) chooses a strate y that makes Column (Row) indifferent to which -‘I - ---- ii 4zolumn (cow) IS chosen; whiie under tilaxlmiii, Ro-w jC&iiiiil) c;lIUU~Gc3 akes Row (Column) indif%erent to which column (row) is Nash, the concern with the other player’s payoff is over-
maximin, thab cancer:: with the other player’s payoff is
y to undertake a comparative game statics analysis. so, it is u~~~~ to ask why one undertakes such an activity. In are tested. f.ot by looking at a the equilibrium changes Thus, for example, economists do not P~WWM~ rrwahnonc hnt ahey _-II_--_ f~+ --a _--_-F_, -__ pahy as will fait when the price rises. The bserving O-P level is that know the utility function
13. Wi!!msn, fV*si: equilibrium vs. maximin
563
methodology is raregly used for bimatrix games with mixed strategies, even though the rationale for using ccmparative statics still holds. We first consider the effect of an increase in ia,j on p and q (in practice, we are un:ikeIy to observe the payotT directly, rather we may observe a _;* in 2 particu!zr direction). technological change that is likely to effect IP; U,j
Proposilion I.
Assuvne that the participants are Nash players.
:z: 1:;; iiir(cI4ac no effect on (b) An increase (c) An increase (d) An increase (a) An increase
on p. 1 ifar ls, a change in Row’s ihlity has Row’s equilibrium behavior. in u, 1 will result in a decrease in q. in u I z wiii result in Q decrease in q. in i?i j wi!! resu!t ir! a~ incremz it; q. in u 14 will result in an increase in q.4 iit
l4 1 j
k4s
no
fg$x~
ProoJ. (a) From eq. (7) it is readily apparent that an increase in Uli will have no erTect on p. (b) Looking at eq. (5) it is readily established that both the numerator and the denominator are greater than zero: by assumption u,,>u,,>O and u,,>u13>0. Therefore as ull increases, the denominator increases and q decreases. (c) Again looking at (5), we derive the partial derivative of q with respect to u12:
1 4u,*=
-
Ull
-U13
Ull
U14-U12 ~.___~ ~____. -U12
-U,3+U,4~2
----CO.
!U;;-!“;t-lri;+l!i4?2
The !zs! “--1inmualitv -~ d
__~_ 5
-u13+1414
-U12
UII =
_
holds since g?ri >
(13)
assumption. (d) Again looking St (S), it is immediately apparent that an increase in u13 will lead to an inc! ease in q. (e) Again looking at (5), we derive the partfa’, dcs,vative of y with respect to ulj
by
Ul4: 4-“I. “. .
=
_
_..
._
U11 -U12
Ull b,*
-
! _-- .
-U13+-U,4
-U13 ___
Ul4-Ul2
-
- ~_. ~ _.__
u~2-u,~+Ur* )2
-
@I,
-U,2-U13+h4)2
>O.
*rhis ccw!f~~!ntl~~~~ve &4mkK $v Nash pOaycrs was discusseri in .~ittrn~n characteristics of mixed strategy Wa& equilibria are found in Wittman ( 195%.
(1985). Other
D. Wittman, Nash equilibrium vs. maximin
564
We next consider maximin. Asstciiie that the PlUyWseiiljdOv
Froposition 2. (a)
(6) (c) (d) (e)
An An An An An
increase increase increase increase imrease
in U2i has no effect on p”. in u1 1 will lead to a decrease in p”. in u, 2 wiC1lead to an irtcrecse iz p”. in u1 3 will lead to an increase in p” if and only if u12 > u I 1. in u14 will lead to an increase in p” if and only if u 12 c u 11.’
(a) Looking at eq. (10) it is immediately will have no efbect on pve (b) Again looking at (IQ), it is immediately clear result in a d--rnncp ;b.W,LbcK+I* 111pc. (c) Again looking at (IO), it is immediately clear ..A_.. Okir _I :_.__*_$.$in p”_ :csxlc . 1 &ii *drivb (d) Taking the partial derivative of (10) with foiiowing equation:
clear that a change in
proof.
Pk=
strategies.
iiiiaXiB?iil
- --Ull
i -!?ii
-.
-
that an increase in u1 1 will that an increase in uz2 will respect to ut3 we get the
u14-u13
-i-
--u13+u14
U2t
b11-u12
-f413+Ul*)*
84!112-cJ11
@It -u12
-h3
+&4b2’
his last term is (strictly) positive if and only if u12 > u, t. (e) Taking the partiai derivative of (10) with respect to ur4 we get the following equation: #, -.,,u;,.,J;;..
__ 14
ull-“Ii =--------_-. 6,. .-II Irr 12’ 1-11 -“la-” =13 1 WI41
1614-1613 f 1111 -u12
-u13
+u14)2
WI
5.
ca
s
Hoiier (S!Xl~ argued on a priori grounds that when both maximin and Nash are mixed and unique, maximin is superior to Kash. Wdlei and (1990) then undertook an experimental study. Using a sample of stu strategies to a particular game, they obtained frequencies for each row and column. The observed quencies were closer to the frequencies predicted by maximin than Nash. re we have suggested that the empirical validity of i&~- approaches shouid be based on then predictability and that the best method for deriving behavioral reiations is by means of a comparative game statrcs analysis rather than levels for a particular game. To that end we have derived the comparative static relations and look forwards to future experimenta! and non-experimental tests that determine i~j*hi& i. _--iii t‘neorv =---=-= J most ace;+ rately predicts human behavior.
Nash, J, t9SO. Equilibrium points in N-person games, Proceedings of the National Academy of Sciences, U.S.A. 36. 48-49. UI,BUII~~in iwo-person games, Holler, Manfred, l990? The unprofitability of mixed strategy b mq.*z’zL-‘Economics Letters 32, 3 19-323. Holler, Manfred, n.d., Fighting when decisions are strategic, Public Choice, forthcoming. Holler, Manfred and Viggo Host, 1990, Maximin vs. Nash equilibrium: Theoretical results and empirical evidence, in: Richard Quandt and Dusan Triska, eds., Optimal ciecisions in markets and planned economies (Westview P:ess, Boulder, CO) 245-255. Wittman, Donald, 1985, Counter-intuitive results in game theory, Esropean Journal of Political Economy I, 77-85. Wittman. Donald, 1989, Arms control ver:%ation and other games involving imperfect detection, American Political Science Review 33, 923-945.