Nash equilibrium under knightian uncertainty: a generalization of the existence theorem

Nash Equilibrium under Knightian Uncertainty:

A Generalization of the Existence Theorem

1

Paulo César Coimbra-Lisboa

2

EPGE - FGV

3

First Version: March 30, 2002 This Version: March 30, 2003

Abstract

The most well successful definition of Nash equilibrium for two-person normal form games in the presence of Knightian uncertainty is due to Dow-Werlang [1994]. With the formalization of Gilboa and Schmeidler they proved the existence of equilibrium for the case of an uniform squeeze (let I be the power set of a finite state space -.Q -, v be a convex capacily,

q

be a(n) (additive)

probabilily measure and c a number between O and 1, and Ihen, for any Ae-I -excepl the whole sei -: v(A)=(1-c)q(A) and in lhe case of A being the whole sei both vand q are 1). Taking a differenl definition of support from that of Dow-Werlang's Nash Equilibrium under Uncertainty paper, Marinacci [2000] extended lhe proof of the existence for any given uncertainly aversion function.

The purpose of Ihis paper is to extend the proof of lhe Dow-Werlang's exislence theorem of Nash equilibrium under uncertainty, using the same definition of support in Iheir paper (the most useful, general notion and has some advantages over lhe others definilions). Let I,

v

and

q

as above defined and lj/:I-f{O,1} be an uncertainty aversion function such that, for any Ae-E. !f!(A)=c(v,A) - lhe right hand side of this equalily is the uncertainty aversion measure of vat event A (see Dow-Werlang [1992]). I will present a reslriction over the set of convex capacities, more specifically, I will work wilh the class of convex capacilies thal are squeeze of (additive) probabilily measure (that I will refer as the set e{'P,LI)). Then any capacily ve-e{'P,LI) can be represenled as: v(A)=(1-!f!(A))q(A), excepl for lhe case when A is the whole set (implying v and q are 1). This enables us lo give a parametric approach of the exislence resull thal generalize Dow-Werlang's exislence Iheorem and will be very useful for comparalive static exercises over lhe uncertainly aversion function.

Journal of Economic Literature Classification Numbers: C72, 081.

Key Words: EI/sberg paradox; Knightian uncertainty; capacities (non-additive probabilities); uncertainty aversion; Choquet integral; equilibrium concepts.

I Preliminary version.

2 PhD student in Economics -EPGE - FGV.

E-mail: coimbra@['"vmail.hr url: http://www.fgv.br/cogeiusen;/coimbra. 3 Graduate School of Economics at Getulio Vargas Foundation.

1. INTRODUCTION

Dow·Welang [1994] extended the notion of Nash equilibrium for two-player finite normal games when players are uncertainty on the behavior of his opponents. They showed the existence of equilibrium for any given degree of uncertainty (however constant over ali possible events, except the null and the whole event). Using a different definition of support, Marinacci [2000] proved the existence of Nash equilibrium for any given uncertainty aversion function.

In this paper I will extend Dow-Werlang [1994]'s Nash equilibrium under uncertainty using the same definition of support that they used and a parametrical approach, based on the uncertainty aversion function, which enables us to do comparative exercises in a easy way. I will work with convex capacities that are "squeeze" of (additive) probability measures.

1.1. EIIsberg Paradox

In ElIsberg [1961]'s paper there exists a mind experiment that consist of making a choice over lotteries to show that decisions under uncertainty (in the Knight sense) is not consistent with Savage's paradigm. Suppose that there exist an urn tha! contains 90 balls on 3 different colors: red, black and yellow. Besides this is also known that there exists 30 red balls and that ali that is known about the other 60 balls is that they are black or yellow. ElIsberg purpose two mind experiments.

Imagine, first, the two following lotteries: "win $100 if a red ball ball is drawn from the urn and win nothing otherwise" and "win $100 if a black ball is drawn from the urn and win nothing otherwise". Of course it is expected that most of participants of this experiment will preffer to bet on "red ball" lottery, because it is a known fact that 1/3 of the balls are with certainty reds. However, if in each of these lotteries the option "yellow ball" is included in the following way: "win $ 1 00 if a red ball or a yellow ball is drawn from the urn and win nothing otherwise" and "win $ 1 00 if a black ball or a yellow ball is drawn from the urn and win nothing otherwise" then it is expect~d that most of the participants will now prefer to bet on "black ball - yellow ball" lottery. This occurs because it is known with certainty that there exist exactly 60 balls that are blue or yellow. So decisions under uncertainty are different from that ones when the probability distributions are objectively known.

The paper is organized as follows. The next section introduces the required definitions and basic statements on Knightian uncertainty's decision theory. In Section 3 I introduce formally the class of convex capacities that are squeeze of (additive) probability measures, a key concept to the extension that I am purposing in this paper. Section 4 gives the definition of Nash equilibrium under uncertainty and present the theorem on the existence of Nash equilibrium that extend the Dow-Werlang[1994]'s existence theorem [1994]. Section 5 present some related results and concludes.

2. SET-UP ANO PRELlMINARIES ON CAPACITY INTEGRATION

The decision setting that I will use in the paper are developed in a Savage-style (se e Savage [1954]). Let Q be a non-empty finite set that contains ali the states of the world4, 1: be an

algebra of subsets of Q called events5 and a set X of consequences. I denote by fI the class of ali

simple acts: finite-valued functions j":Q---fX which are measurable with respect to

I!.

For XéX I

4LetQ={OlJ. CtJ.z, ••• , f4.}.

5 In this paper 1: = 2Q satisfying the foIlowing properties:

i) QéI;

ii) For ali A. B é I: A uB é 1:;

iii) For ali A é I: (Q\A) é L.

Then. it s easy to prove the following:

Lemma 2.1: lf 1:= 2Q satisfy (i) to (iii) above then: iv) 0 E 1:;

v) For ali A, B E L.- AnB é 1:.

Proof: Omitted.

NASH EQUILIBRIUM UNDER UNCERTAINTY

3

define xe-3 to be the constant act such that x( úJ)=x for ali úJE Q. So, with abuse of notation, the set X of consequences is identified with the subclass of the constant acts in 3.

A set-funetion v: I-f9{ with v(0)=O is called a capacity (also called a non-additive probability) on (Q,I) if it is normalized and monotone, that is: i) normalized: v(Q)=1; ii) monotone: For ali A, B E Isueh that AçB: v(A)5v(B). I denote by V(Q,I) the elass of ali capacities on (Q,L/. A capacity is convex if, besides (i) and (ii) it satisfies the following property: (iii) For ali A,BEI: v(AuB)+v(AnB);? v(A)+v(Bf. I denote by A (cV(Q,I)) the elass of ali convex capacities on (Q,I). A capacity is (finite/y) additive (also ealled a(n) (additive) probability measure) if, besides (i) and (ii) it satisfies the foJlowing property: (iii') For ali A, B E I sueh that AnB=@. v(AuB)=v(A)+v(B). We denote by LI (c A) the elass of ali (additive) probabi/ity measures on (Q,I).

2.1. Uncertainty A version Measure

If VEA is not a(n) (additive) probability measure then there exists at least a pair A, BEIsuch that: v(AuB)+v(AnB»v(A)+v(B). In particular, if B=(Q\A) then v(A)+v(Q\A) may be less than 1, implying that not ali probability mass is allocated to a event and its complement. Dow-Werlang [1992] proposed an uncertainty aversion measure of a capacity vat event A:

Definition 2.1 (Dow-Werlang [1992]) Let v E V(Q,I) and A E I .

The uncertainty aversion measure Df v at event A, is defined by: c(v,A) = 1-v(A)-v(Q\Al.

It is easy to prove that if a capacity is convex then for ali A E I: C(V,A)E[O,1]. Convex capacities are also know as non-additive probabilities reflecting uncertainty aversion. Throughout this paper I will restriet attention to convex capacities.

The foJlowing proposition present the properties that are satisfied by the uncertainty aversion measure of v at event A when v is a convex capacity.

Proposition 2.2: Let VE A and A, B E .E.

The uncertainty aversion measure of v at event A satisfy the following properties: i) c(v,0) = c(v,Q) = O;

ii) For ali A E .E: c(v,A) = c(v,(Q\A)); iii) For ali A, B E I: c(v,(AuB))+c(v,(AnB))~c(v,A)+c(v,B).

a·I(O)E I, where a·I(O)=( úJEQ: a( úJ)EO;' I denote by B(.Q,I) the c1ass of ali real-valued function, bounded on Q, which are L-measurable. Note that 3 (c B( Q,I)).

7 It is easy to prove the following:

Lemma 2.2: Ifv E V( Q,L) then,for all A E.E.- V(A)E[O,l]. Proof: Omitted.

8 Proposition 2.1:

lf

a set-function v: I-f9i with v( 0)=0 is such that, for ali A, B E I:

v(AuB) + v(AnB) ;?v(A) + v(B) Then v is monotone.

Proof: Let A, B E L be such that A c B. Define C=B \ A. Then A u C = B and A n C

=

0. From (iii) we now that: v(AuC) + v(AnC) = v(B) + v(0) ~ v(A) +v(C).

Using (i): v(B) ~ v(A) + v(C). We now that v(C)~O so v(B)~v(A).

9 It is easy to prove the following:

Lemma 2.3: Let VE V(.Q,L)

Iffor ali AEE. c(v, A)=O then VELI. Proof: Omitted.

Proof: For ali A E 1:. c(v,A)=1-v(A)-v(Q\A)

We now that for ali VE A: v(0)=O and v(Q)=1 so: c(v,0)=c(v,Q)=O.

For ali A E 1:. c(v,A)=1-v(A)-v(Q\A)= c(v,(Q\A))

To check (iii) note first that:

c(v,A)=1-v(A)-v(Q\A) c(v,B)=1-v(B)-v(Q\B)

c(v,(AuB))=1-v(AuB)-v(Q\(AuB)) Morgan's law imply that:

=

1-v(AuB)-v((Q\A)n(Q\B)) c(v,(AnB))=1-v(AnB)-v(Q\(AnB))

Morgan's law imply that: = 1-v(AnB)-v((Q\A)u(Q\B))

From the definition of convex capacity we know that, for ali A,B E 1:.

v(AuB)+v(AnB);?v(A)+ v(B)

and

v((Q\A)u(Q\B))+v((f2\A)n(Q\B));?v(f2\A)+v(f2\B)

Re-writing these expressions:

and

Sum:

1-v(AuB)-v(AnB) s1-v(A)-v(B)

1-v((Q\A)u(Q\B))-v((Q\A)n (f2\B))s1-v(Q\A)-v(Q\B)

(1-v(AuB)-v(AnB))+(1-v((Q\A)u(f2\B))-v((Q\A)n(Q\B)))s s(1-v(A)-v(B))+(1-v(Q\A)-v(f2\B))

Re-writing and using Morgan's law.

(1-v(AuB)-v(Q\(AuB)))+(1-v(AnB)-v(Q\(AnB)))s s(1-v(A)-v( f2\A) )+( 1-v(B)-v(Q\B))

Finally, using the definition of uncertainty aversion measure:

c(v, (AuB))+c(v, (AnB))sc(v,A)+c(v, B).

2.2. Choquet Integral

Q. E.D.

Since that capacities can be a non-additive measure I can't use an integral in the sense of Lebesgue. The appropriate notion of integral is due to Choquet [1953] . For any given real-valued

function, bounded on Q, a E B(Q,.E), the Choquet integral of a with respect to a capacity VE V(Q,.E)

is defined as follows:

o ~

fadv", f[V({lOE

Q:a(lO)~a})-l.kia+

fV({lOE

Q:a(w)~a})da10

o

NASH EQUILIBRIUM UNDER UNCERTAINTY

5

where the right hand side is a well defined integral in the sense of Riemman (because a is bounded and vis monotone)".

Ali the real-valued function considered in this pape r will be simple acts. Besides this, every

simple act j"eS will be defined such that j"(r»t)5 j"(~)5 ... 5/(w,,). I am particularly interested in a special subclass of the class of simple acts, which are constituted by the co-monotonic acts. Two

acts, f, 9 e Sare said to be co-monotonic if, for ali ro,ro'e.Q: (f(úJ)-f(úJ'))(g(úJ)-g(úl)) cO. I denote by F

(c S) the class of monotonic acts which is constituted by simple acts that are pairwise

co-monotonics.

For ali f, 9 e S and v, V'e A, the list below present some properties of Choquet integral

(proved in Simonsen-Werlang [1991]):

a) Positive homogeneous:

fcfdv

=

c ffdv, for c e 91+;

b) Monotonic:

f

~

g

~

ffdv

~

fgdv;

c)

f(f+c)dv

=

ffdv+c, for

Ce9l+;

d) Co-monotonic additive: If f, 9 e F are non-negative then:

f(f+g)dv

=

ffdv

+

fg dv ;

e) Monotonic in the capacity: If v c v'then

ffdv

~

ffdv'.

f)

ffdv

+

fc-

f)dv

~

O;

g) Jensen Inequality: For ali concave functions,

u:

91-791:

fuCf)dv

~

u(ffdv ).

In this pape r I will use the Gilboa [1987]'s axiomatization of the use of the Choquet integral as a choice criterion, which in turn is based in a Savage-style. Wakker [1989] and Sarin and Wakker [1992] have an axiomatization which are very similar to Gilboa [1987]. Schmeidler [1989] present an axiomatization in an Anscombe-Aumann-style (see Anscombe-Aumann [1963]).

3. CONVEX CAPACITIES THAT ARE SQUEEZE OF (ADDITIVE) PROBABILlTY MEASURES

In this section I will present a special class of convex capacities that will enables us to do comparative static exercises in a easy way. First I will discuss the support of a capacity (subsection 3.1), then I will present a function that has the same properties of an uncertainty aversion measure (subsection 3.2). The next step is present the condition on which convex capacities are squeeze of

(additive) probability measures (subsection 3.3). Finally I can present the Choquet integral with

convex capacities that are squeeze of (additive) probability measures.

3.1. Support of

a

capacity

The notion of support of a capacity is the first step necessary to study the conditions in which a convex capacity can be understood as a squeeze of a(n) (additive) probability measure.

Definition 3.1 Support of a Capacity

Let v e V(f2,I) and A, B e I.

The support of the capacity v is an event B such that: i) v(f2\B) = O;

ií) For ali A, B e I, A c B: v(f2\A) > O.

o ~

fadv

==

fCvCa

~

a) -l)da+ fvCa

~

a)da

o

The EIIsberg's um, presented at the introduction of this pape r, give us an example of a

capacitythat does not have an unique support (if k = O). Example 3.1 ElIsberg's Um

Let Er be the event "a red ball is drawn from the urn". The events Eb and Ey are similarly defined.

Let (fl,r, v) be a capacity space that reflects the EIIsberg's um, Le.:

fl = {Er, Eb, Ey};

L=~;

v(0) = O; v(fl) = 1; v({EJ) =1/3; v({Eb}) = v({Ey}) = k; v({Er,Eb})=v({Er, Ey})=(1/3 + k); v({Eb, Ey}) = 2/3. If k E {0,1/3jthen v is a convex capacity.

If k E (O, 1/3jthen the (unique) supportis the event {Er, Eb, Ey}.

If k = O then the supports are the events {E" Eb} and {Er, Ey}.

The following example, due to Eichberger-Kelsey [2000], shows that the supports may ali

get a zero mass 12:

Example 3.2 (Eichberger-Keysey [2000]) Let (fl,r, v) be a capacity space defined by:

fl = {COt, ~, W:3};

L =2n;

v(0) = O; v(fl) = 1; v({COt}) = v({wi})

=

v({W:3}) = O;

v({COt, ~}) = 1; v({COt, W:3}) = v({~, W:3}) = O. The supports are the events {COt} and {wi}, both with zero mass. 13

The following proposition, due to Marinacci [2000], shows when there is an unique support.

Proposition 3.1 : (Marinacci [2000J) Let v E V(fl,L).

The following two conditions are equivalent: i) supp v is unique;

ii) For ali A E L and ali OJ E fl such that v(A)

=

v({ OJ})

=

O: v(A u (OJj) = v(A) + v{{OJ}).

Proof: Marinacci, M. (2000j, Games and Economic Behavior.

This suggest the following definition:

Definition 3.2 Convex Capacity with an Unique Support

A convex capacity has an unique support ir, besides (i) to (iii) it satisfies the following property:

(iv) For ali A E L and ali OJ E fl such that v(A)=v({OJj) = O: v(A u (OJj) = v(A) + v({OJj)

We denote by A(S) (c A) the class of ali convex capacities which has an unique support.

12 Marinacci [2000] presents a very similar example.

NASH EQUILIBRIUM UNDER uNCERTAINTY

7

3.2. Uncertainty Aversion Function

In the section 2.1 I discuss about the uncertainty aversion measure and its properties when

v is a convex capacity. Now I will define a set-function that has the same properties.

Definition 3.3 Uncertainty Aversion Function

A set-function If/: I-7[O, t}, is an uncertainty aversion function if it satisfies the following properties:

i} If/{e) = 'I1Q} = O;

ii} For ali A E I: 'I1A} = 'I1Q \ A};

iii} For ali A, B E I: 'I1AuB} + If/ (AnB) 5 'I1A} + 'I1B}.

I denote by 'P(Q,I} the c\ass of ali uncertainty aversion function on (Q,I). Associate with

each uncertainty aversion function If/ E 'P(Q,I} we have a subclass of convex capacities v E A with

the property that, for ali A E I, the uncertain aversion measure is such that: c(v, A}

==

'I1A}. 80, for

each If/ E 'P(Q,I}, I denote by A(VIJ (c A) the class of convex capacities which are such that for ali A

E 1:: c(v,A} = If/(A}.

Example 3.3 ElIsberg's Um (Continuation)

For each k E [O, tl3}we now that the uncertainty aversion measure associate to the ElIsberg's urn has the following characteristic:

c(v, (E,)} = c(v, (Eb,Ey)} = O; c(v, (Eb)} = c(v, (E"Ey)} = 213 -2k; c(v, (Ey)}

=

c(v, (E"Etl)

=

2/3 -2k.

80, for each kE[O, t13}, we can define an uncertainty aversion function If/E 'P(Q,I} in the following way: For ali AEI: 'I1A}=c(v,A}, where vis the convex capacity associate with the Ellsberg urn. That is:

'I1(E,)} = 'I1{Eb,Ey)} = O; 'I1(Eb)} = 'I1(E"Ey)} = 213 -2k; 'I1(Ey)} = 'I1(E"Eb)} = 213 -2k.

Of course if vis the convex capacity associate to the ElIsberg's urn then, for each

kE[O, t13}, VE A (1jI, k), where If/E 'P(Q,I} is the above uncertainty aversion function.

The following example shows that there exit at least a kE[O, tl3} such that A(ljI,k} contains more than one element:

Example 3.4 /f k=113 Then A(ljI,k=113} is a Non-Empty, Non-Unitary Class

Let (Q,I) be a state space defined by:

Q = (E" Eb, Ey); L = 2n;

Let If/E 'P(Q,I} be defined as in example 3.3.

If k=1/3 then, for ali A E I : 'I1A) =0.

Let pE,,1 be a(n) (additive) probability measure on (Q,I). Then, for any p the

uncertainty aversion measure is such that, for ali A E I: c(v,A)=O.

80,

A(ljI,k=113} defined in example 3.3 is a non-empty, non-unitary class:

A(ljI,k=tI3)={pE,,1;,,1 is the set of probability measures on (Q,I}).

3.3. Squeeze of (Additive) Probability Measures

In this subsection I will summarize the results that enables us to characterize convex

capacity that are squeeze of (additive) probability measures associate to uncertainty aversion

Definition 3.4 Squeeze of a(n) (additive) probability measure (Squeeze). Let '1'6 'P(.a,E).

The capacity V6 V(.a,E) is

a

squeeze of a(n) (additive) probability measure p6,,1 associate to an uncertainty aversion function if:

{

(l-

V(A» p(A) se A

"*

Q

v(A)

=

1

seA=Q

Example 3.5 Uniform Squeeze

Let 1jI'6 'P(.a,E) be such that, for ali A6.E, 0ot:At:·n: IjI'(A)=c (and, as usual,

1jI(0)=IjI(.a) where C6{O, 1). Then, for ali p6,,1 defined on (.a,E) there exists a convex

capacity V6A(IjI') that is an uniform squeeze of p 6 ,,1 defined by:

{

O-C)p(A)Se A"*

Q

v(A)

=

1

seA

=Q

To check it, define a set-function q:E-f{O, 1J by:

{

v(A) se A

"*

.Q

q(A)

=

(l-c)

1

seA

=Q

where vis defined as above.

l1's easy to verify that, for ali C6{O, 1), q 6,,1.

The following definition extend the properties of a convex capacity that are necessary to be satisfied to be represented as a squeeze of some p6,,1 defined on (.a,E).

Definition 3.5 Properties of a Convex Capacity that are Squeeze. Let v 6 A(S), with fã ~ O

=

supp V ç.a, and C, C', O, E 6 E.

1 can say that v is

a

squeeze of a(n) (additive) probability measure p E ,,1 associate to some '1'6 'P(.a,E) if it also satisfy the following properties:

v) ForaIl0~C, C' cO, with C uC' cO and C n C' = fã:

v(Cu C') = v(C) + v(C')

vi) For ali fã~C cO and ali E ç(.a \ O): v(CuE) = v(C)

Proposition 3.2 Let 'I' 6 'P(.a,E).

If v E A(~ satisfy the properties (i) to (vi) then v is a squeeze of some p E ,,1 defined on (.a,E) associate to some If/.

Proof: Coimbra-Lisboa, P. (2003]. 14

I denote by e(,,1,''P) (c A(S)) the class of ali convex capacities that are squeeze of

(additive) probability measures p 6 L1 defined on (.a,E) associate to some 'I' E 'P(.a,E).

Proposition 3.3 Let v E e(,,1, 'P), with fã ~ O=supp V

ç.a

and C,C',O,EEE. The uncertainty aversion measure of valso satisfy the following properties:

iv) For ali fã ~ C, C'e O, with C u C'e O and C

n

C' = fã: c(v, C)

=

c(v, C') ~ c(v, O)

NASH EQUILIBRIUM UNDER UNCERTAINTY

v) For ali 0;ró C c O and ali E ç (Q \ O): c(v,(CuE))

=

c(v, C)

Proof: Coimbra-Lisboa, P. (2003].

Proposition 3.4 Let If/é 'P(Q,E), péJ1, with 0;ró 0= supp p ç Q and C,C',O,Eé I. If If/ a/so satisfy the following properties:

iv) For ali 0;róC, C' cO, with C uC' cOand C

n

C'= 0: 'I1C) = 'I1C')

c

If/(O)

v) For ali 0;róC cO and ali E ç(Q \ O): 'I1CuE)

=

'I1C)

And if v é V(Q,E) is defined by:

)

{

O-V/(A))p(A)SeA*"Q

v(A

=

1

seA

=Q

Then Vé A(\fIJ.

Proof: Coimbra-Lisboa, P. (2003].

Proposition 3.5 Letlf/é 'P(Q,E), péJ1, with 0;ró O = supp pçQand C,C',O,Eé I. If If/ a/so satisfy the following properties:

iv) For ali 0;róC, C' cO, with C uC' cOand C

n

C' = 0: 'I1C) = 'I1C')

c

If/(O)

v) For ali 0;róC cO and ali E ç(Q \ O): 'I1CuE) = 'I1C)

And if v é A(If/) is defined by:

v(A) _

{O-

V/(A))p(A)if A*"

Q

1

ifA=Q

Then v a/so satisfy the following properties:

v) For ali 0;róC, C' cO, with C uC' cO andC

n

C' = 0: v(Cu C') = v(C) + v(CJ

vi) For ali 0;ró C c O and ali E ç (Q \ O): v(CuE)

=

v(C)

Proof: Coimbra-Lisboa, P. {2003].

9

Remark 3.1 If supp v = {w}, for any W é Q then properties (v) and (vi) are innocuous.

Remark 3.2 If supp v = Q then for ali A é E, 0;róA;rón: If/(A)=Cé{O,1]. So, (v) and (vi) are

trivially satisfied.

Example 3.6 Convex Capacity that is Squeeze of a Probability Measure

Let (Q,E, v) be a capacity space defined by:

Q={(0, @.?, OJ.3}; L=2n;

v(0)

=

O; v(Q)

=

1; v({(0})

=

O; v({ruJ) = a; v({0J.3})

=

b; v({(0, ruJ)=a; v({(0, OJ.3})=b; v({@.?, OJ.3})=a+b+c.

The (unique) support is the event {@.?, OJ.3}.

If a + b + c S 1 then v é .A and satisfies (v) and (vi), so v é e(LI, 'P), Le., v is a

squeeze for some p é LI defined on (Q,E) and some If/ é 'P(Q,E) satisfying (iv) and (v) such

In fact, the following uncertainty aversion function on (Q,.E) satisfies these requirements:

1f/({úJtl) = 1f/({úJ2,úJ3}) = 1 -

a -

b -c; l{I({úJ21)

=

1f/({úJ/,úJ 3})

=

1 -

a -

b; l{I({úJ3})

=

1f/({úJ1,úJ

21) =

1 -

a -

b.

80, v é A(Vf}. Note also that If/satisfies (iv) and (v), which means that there exists

some p é LI such that v can be defined as:

v(A) _

{o-

vr(A))p(A)if A:;t Q

I

ifA=Q

The set-function q:.E~9f is defined on (Q,.E) by:

{

v(A) if A:;t.Q q(A)

=

(1- vr(A))

I

-

ifA=.Q

It is easy to check that qé LI. 80, v is a squeeze of qéLl associate to If/é IJ'(Q,.E)

Example 3.7 Convex Capacity that is Squeeze of a Probabi/ity Measure

Let (Q,I, v) be a capacity space defined by:

Q= {rur, .~, ~, ~};

L = 2fl; v(0) = O; v(Q) = 1;

v({rur}) = O; v({COz)) = a; v({W:3}) = b; V({úJ4}) =c; v({rur,

úld)

= a; v({rur, W:3}) = b; v({rur, úJ4}) = c;

v({~, ~})=a+b; v({~, úJ4})=a+c; v({~, úJ4})=b+ c;

v({rur,~, ~}) = a+b; v({rur,~, úJ4}) =a+c;

v({rur,~, úJ4}) = b+c; v({~,W:3, úJ4}) =a+b+c+d;

The (unique) supportis the event {~, W:3, úJ4}

If a+b+c+ds1 then v é A and satisties (v) and (vi), so v €i' (9(LI, 'P), Le., v is a

squeeze of some p €i' LI defined on (Q,.E), and some If/é IJ'(Q,.E) satisfying (iv) and (v) such

that, for ali Aé.E. I{I(A)=c(v,A), where c(v,A) is the uncertainty aversion measure of v.

The following uncertainty aversion function on (n,L) satisfies these requirements:

1jI({ rur})=I{I({~,~, úJ4})= 1-a-b-c-d; 1jI({~})= I{I({ úJl, W:3, úJ4})= 1-a-b-c;

1jI({~})=I{I({rur, ~, úJ4})=1-a-b-c; l{I({úJ4})=If/({rur, ~, 0>.3})=1-a-b-c; 1jI({ rur, ~ })=If/({~, úJ,j})=1-a-b-c; 1jI({ rur,W:3})=I{I({~, úJ4})= 1-a-b-c;

ljI({rur,úJ4})=If/({~, ~})=1-a-b-c;

80 V€i'A(If/). Note also that If/ satisfies (iv) and (v), which means that there exist

some

p

€i' LI such that v can be definided as:

v(A) _

{o-

vr(A)) p(A) if A :;t Q

I

ifA=Q

The set-function q:.E~9f is defined on (Q,.E) by:

{

v(A) if A:;t Q

q(A)

=

0-

vr(A)) .

I

ifA=.Q

It is easy to check that qé LI. 80, v is a squeeze of qéLl associate to If/é IJ'(Q,.E).

"IfIIUOTECA MARIO

HENRIQUE SIMONSt'·

NASH EQUILIBRIUM UNDER UNCERTAINTY

3.4. Choquet Integral with Convex Capacities that are Squeeze Theorem 3.1 Let p € LI.

If v € e(LI, 'P) then the following conditions are equivalents:

o ~

i)

ffdv ==

f(v(f

~a)-l)da

+

fv(f

~a)da

o

ii)

ffdv ==

Y'J({lü

J })

+

(1-

Y'J)

ffdp

+

11

+

~(Y'J

-Y'j ....

n{tP({lü;}))(f({lü))- f({lü

j _J })

where: VI = If/{{~J} and

'I1, ...

,n= 1f/{{f4' ... ,CtJ"j);

f fdp is the integral of

fwith respect to the probability measure p€LI.

o ~

Proof:

ffdv ==

f(v(f

~

a)-l)da

+

fv(f

~

a)da

If a < f({~J) then v(fêa) = v({~, /í),?, ... , CtJ"j) = v(fl) = 1;

If a € [f({ ~J), f({ o;.J)[ then v(fêa) = (1-Vz .... , n)P({ /í),?, CtG, .. ·, w,,});

If a € [f({o;.J}, f({CtG})[ then v(fêa) = (1-V3 .... ,n)P({CtG' úJ4,"" w,,});

And so on ...

If a € [f({OJ".d}, f({w,,})[ then v(fêa) = (1-Vn)p({w"J); If a ê f( ( w,,}) then v(fêa) = O;

o ~

ffdv -

f(v(f~a)-l)da

+

fv(f~a)da

f(~)

f(v(f

~

a) -l)da

+

o

o

f(O},)

Jcv(f~a)-l)da

+ ... +

f(~)

f(O},+,)

+

f(v(f

~

a) -l)da

+

fv(f

~

a)da

+

o

f(W_{j )}

f(O},) ~

+

fv(f

~a)da

+

fv(f

~a)da

-f(O},_,) f(w,)

f(O},)

f(v(f

~

a) -l)da

+

f(w_j_,)

f(Wj_2)

fv(f

~

a)da

+ ... +

f(wj+,)

f(úJ. )

f((l-Y'2 ....

n)P({lü

_{2, ...}

,lün})-1)da

+ ... +

f(W,)

+

f«(1-~j

.... n)P({O}j""'O}n})-l)da

+

f (W_{j_1 )}

o

+

f((l-~j+I.

.... n)P({O}j+p""O}n})-l)da

+

f(wj +l )

+

f(1-~j+I

... Jp({O}j+i' ... ,O}J)da

+

f(W,+,)

+

fo-

~j+2

... n )p( {O}j+2 ""'O}n })da

+ ... +

f(w,) ~

+

fO-~n)P({O}n})da

+

fo-vr{})p({})da=

== O[f( rõt)-(-oo)j + ((1-1f/2, ... ,n)P({ úJ;!, ... , m,,})[f( úJ;!)-f( rõt)} + ... +

+((1-'l/j,.,n)P({ cq .... , m,,})-1 )[f( cq)-f( cq·d)+

+((1-'l/j+l .... "JP({ cq+l .... ,l4,})-1 )[O-f( cq)}+

+((1-'l/j+l, ... ,n)P({ cq+l, ... , l4,})-1 )[f( cq+l)-O)+

+(1-'l/j+2 .... ,n)P({ cq+2, ... , l4,})[f( cq+2)-f( cq+l)j+ ... +

+(1-lf/n)P({ l4,})[f( m,,)-f( l4,'I)}+ O[oo-f( l4,)j=

After a little algebra:

vrJ( {O}I})

+

(1-

vrl) f!dp

+

+

~(vrl

-vrj, ..

n{~P({O}J»)(f({O}j})-

!({O}j_l})

Q.E.D.

These following examples, which extends, respectively examples 3.6 and 3.7, intends to

show the equivalence between these two formulas of Choquet integral if vEe(Ll, 'P).

Example

3.8

(Continuation of Example 3.6)

Let (a,E) be a state space defined as:

a=

{rõt, 10, lí>.J};

L =2!l.

Let If/ E 'P(a,I) be an uncertainty aversion function defined by:

lf/{{úJdJ = lf/{{úJ2,úJ 3}) = 1 -

a -

b -

c;

lf/{{úJ21) = lf/{{úJ1,úJ 3}) = 1 - a - b;

NASH EQUILIBRIUM UNDER UNCERTAINTY

LeI p E LI be defined by:

a

b

P({úJl}) = O; p({úJ"j) = - - ; p ({úJ 3)) = - - .

a+b

a+b

So v E e(LI, 'P) can be defined as:

{

(1-

VI{A»p(A)se A '1=

o.

v(A)

=

1 seA

=0.

LeI f:12-f9fbe a real-valued funclion defined by:

f(f4) =i, for i E {1, 2, 3}.

II is easy lo check Ihal lhe" usual' Choquet integralis 1+ a+ 2b+ c.

Using (ii):

(

2a+3b)

( b )

=(1-a-b-c).1+(1-(1-a-b-c)) +((1-a-b-c)-(1-a-b)) - - (3-2)=

.

a+b

a+b

(

2a+3b)

( b )

=(1-a-b-c) + (a+b+c) + (-c) - - =

a+b

a+b

Example 3.9 (Continuation of Example 3.7)

LeI (12,I) be a state space defined as:

12={aJt,~,~, (O4);

I: = 2!l.

LeI If/ E P(12,I) be an uncertainly aversion funclion defined by:

ljI{{aJt})=If/({~,~, (O4})= 1-a-b-c-d; 1jI{{(lJz})=IjI{{ (01, li.!J, ú4})=1-a-b-c;

1jI{{ li.!J})=If/({ aJt,~, (J)4})= 1-a-b-c; 1f/({(04})=IjI{{ aJt,úJ;!,li.!J})= 1-a-b-c;

1jI{{ aJt, ~ ))=IjI{{~, (O4})= 1-a-b-c; 1jI{{ aJt, li.!J))=IjI{{ úJ;!, ú4})= 1-a-b-c; 1jI{{ aJt,(04})=If/({~, li.!J})=1-a-b-c.

LeI p E LI be defined by:

a

b

c

p({úJtJ)=O; p({úJ"j)

b

;

P({úJ3}) ; p({úJ4})= ;

a+ +c

a+b+c

a+b+c

So V E e(LI, 'P) can be defined as:

{

(1-

'I'"(A»p(A) se A '1=

o.

v(A)

=

1

se A =.Q

LeI f:12-f9fbe a real-valued funclion defined by:

f(f4)

=

i, for i E {1, 2, 3}.

II is easy lo check Ihal lhe" usual' Choquet integral is 1 + a+ 2b+ 3c+d.

13

Using (ii):

f

fdv

==

'l/J(áJ

1 )

+

(1-'1/1)

f

fdp

+

('1/1

-'1/3.4)

p(áJ

3

+

áJ4 )(f(áJ

3 ) -

f(áJ

2

» +

+

('1/1

_{-'l/4)P(áJ4)(f(áJ4)- f(áJJ)}

+

(

2a+3b+4c)

= (1-a-b-c-d) .1 + (1-(1-a-b-c-d}) +

a+b+c

(

c+d )

+((1-a-b-c-d}-(1-a-b-c}) (3-2)+

a+b+c

+((1-a-b-c-d}-(1-a-b-c}) (

d

)

(4-3)=

a+b+c

=1+a+2b+3c+d

4. NASH EQUILlBRIUM UNDER UNCERTAINTV I can describe a two-person finite normal form game as:

T= (SI, S2; UdS/>S2), U2(SI,S2}}

a.E.D.

where for each player i é {1,2} the Si is a finite set of pure strategies and the U;{SI,S2} are payoffs

(utilities) functions depending on the pure strategy combination (S/>S2) é (S/> S;J played.

To unsure the existence of Nash equifibrium it is necessary to allow the possibility of playing

mixed strategies. A mixed strategy can be understood as a(n) (additive) probability measure over the pure strategies. To complete the description, payoffs (utilities) are constituted by von

Neumann-Morgenstern expected utility. The modify game can be described as:

r

= (.1(SI), .1(S2}; Udpl,P2}, U2(Pl,P2}}

where, for each player i é {1,2}, the .1(S;} is a set of mixed strategies over the finite pure

strategies15and the U;{Pl,p;J are von Neumann Morgenstern expected utilities functions depending

on the mixed strategy combination (P/>P2) é (.1(SI), .1(S;J} played.

Nash [1950] proved the existence of equilibrium in n-person finite normal form games on which mixed strategies are possible. A mixed strategy Nash equilibrium is defined as follows:

Definition 4.1 Mixed Strategy Nash Equilibrium

Let (Pl,P2}) é (.1(SI), .1(S2}} be

a

mixed strategy combination in

a

game

r.

The mixed strategy combination (Pl " p/) represents

a

Nash equilibrium in mixed strategies ir, for each player i é (1,2):

U;{p;', P.i'J ~ Ui(Pi, p.;'}, for a/l Pi é .1(S;}.

For each player i é {1,2} let supp Pi denote the support of Pio

15 50, if player i has kj pure strategies then the mixed strategies sets, Pi, will be a list:

( 1 2 k;)

NASH EQUILIBRIUM UNDER UNCERTAINTY

15

In a Nash equifibrium, each Si E sUPP Pi is a best response to P-i, i.e., for each player

iE{1,2}, Si maximizes the von Neumann Morgenstern expected utifity of player i given that player

_/6

is playing the mixed strategy P-i.

I will adopt the same subjective interpretation as Dow-Werlang [1994]:

"A subjective interpretation (to mixed strategies Nash equilibrium)

can be given ( ... ): the mixed strategy of player i, Pi, may be viewed

as

the belief that player-. i has about the pure strategy play of player i. ,,17

4.1. The Main Theorem

If uncertainty are considered then, for each player i E {1,2}, beliefs of player i about player-i behavior are represented by convex capacities. To generalize Dow-Werlang [1994]'s Nash

equifibrium under uncertainty I will consider convex capacities that are squeeze of (additive)

probability measures, i.e. for each player i E {1,2}: Vi E e(Ll,

'P).

Definition 4.2 Nash Equilibrium under Uncertainty

For each player iE{1,2}: let vJ E 'F i(Sj,?-), where vJ:~-i-7[O, 1} is the uncertainty aversion function of V-i E e(Ll, 'F-i) 18.

A pair (VI, vz) of convex capacities that are squeeze of (additive) probability measures (Vi E e(Ll, 'Fi)), Vi over Si (for each player i E (1,2}) is

a

Nash equilibrium under uncertainty if there exists

a

support of VI and

a

support of V2 such that, for each player i E (1,2):

For ali Si in the support of Vi, Si maximizes the Choquet integral of player i, given that V_i represents player i's beliefs about the strategies of player -i. 19

Now I will present the main result that generalize Dow-Werlang [1994]'s result, with the use of a uncertainty aversion function of each player i E (1,2) as a parameter in the game.

Theorem 4.1 Existence of Nash Equilibrium Uncertainty

Let F= (SI, S2; UI(Sr,Sz), UiSI,S2~ be

a

two-person finite normal form game. For each i E {1,2}:

r;J

E 'F '(S,,2 O'), where

r;J:

~-I-7[O, 1} is the uncertainty aversion

function of V_i E e(Ll, 'F-i).

For ali

(yJ,

,j)

there exists

a

Nash equilibrium under uncertainty.

Proof: My proof is the same, in spirit, to Dow-Werlang [1994]'s existence proof.

We now that if Vi E e(Ll, 'F') then for some pE Do it's true tha!:

{

(1_lj/i (A)) p(A)

se

A

'*

Q

v(A) =

1

se

A

= Q

So, Choquet integral has the form:

16Notation 4.1 If player i

=

1 thenplayer -i

=

2 and, conversely. if player i = 2 then -i = 1.

17 See Dow-Werlang [1994], p. 312.

18 Note carefuIly that V_i represents player i's beliefs about what player -i wiIl do, so that the uncenainty

aversion function of V.i is a characteristic of player i.

19 Note that if there is no uncertainty then lhe pai r (Vh 1'2) reduces to a pair of (additive) probability measures

and the Choquet integral reduces to a von Neumann Morgenstem expected utility. This implies that there exists at least one Nash equilibrium under uncenainty (if there is no uncertainty!), as presented in the foIlowing:

f fdv -

vrJ (

{lO

_{I })}

+

(1-

_{vrl) f fdp +}

+

~(vrl

- v r j ...

Il{~P({lO;}))(f({lO))-

f({lO

j _ l })

Suppose, without loss of generality, that player -i has n pure strategies. So we can order the payoffs of player i E {1 ,2} to each pure strategy, SiE Si, as:

ui(s;)

$

u}(s;)

$ ... $

u; (S;)

where

u/ (s;)

=

U; (si'

s)

is the j-position U=1 , ... ,n)

We modify the original game

r

to

r(V

.vr')

= ($1> $2: WdSI,Sz), W2(SI,Sz)), where:

w;(si's)

=

vrJu;(s)

+

(1-vr;)u!(s)

if

j=I,2.

And, if j =

3, ...

,n.

j

w;(sps)

=

vr:u;(s)

+

(1-vrJ)u!(s)

+

I(vrf

-vr~

... n)(Uik(s;)-Uik-l(s)

k:3

Note that

vr{

is the uncertainty aversion function associate to the strategy of player

-iwho gives the worst payoftto player i (6 (1,2)) when his choice is the pure strategy Si.

Let (Pt.P2) be a standard mixed strategy Nash equi/ibrium of the flJodified game. We will show that the pair (VI, V2), where

{

(1-

vr2 (A»

PI

(A) se

A

=F SI

{(1-

vr

l

(A»

Pz

(A) se A

=F S

2

VI (A)

=

and vz(A)

=

1

se A

= SI

1

se A

= S 2

is a Nash equilibrium under uncertainty for the original game, with the specified uncertaínty

aversíon functions associated.

To check that this is a Nash equilibrium under uncertainty, note that (except in case

where, for each iE {1 ,2} and for ali A6zSi, .0;rA;rSj: yJ(A)=1) the support of Vj is unique (by

definition!), and coincides with the support of pj for each player i6{1,2}. Since (Pt.PZ) is a

standard mixed strategy Nah equilíbrium for the modífied game, it follows that any Sj6 supp

pj is a best response to P.i (for the modified utility Wi). In other words, Si maximizes the

following expression over s 6 Si:

fw;(s;)dp_;

=

p~i(vr;u:(s)+(l-vr:)ui(s)

+ ... +

k

+

P~J(vr:u;(sJ+(1-vr:)u;(s)+

I(vr: -vr: ..

n)(u~(sJ-u~-I(s)]

+ ... +

1=3

n

+

p~;f(vr;u;

(s)

+

(1-

vr; )u;n (s)

+

I

(vr; - vr: ..

n

)(U: (Si) -

U:-I

(S)]

=

1=3

=

vr;U;(S)

+

(l-vr:)fU;(Sp.)dp_;

NASH EQUILIBRIUM UNDER UNCERTAINTY

17

Thus, Si is ais o a best response in the original game. In case where, for each player iE{1,2} and for ali Ae2si, 0;éA;éSj: ';(A)=1 any singleton {sJ is a support of Vj . Therefore any best response for player i is in a support.

Thus (VI, V2) is a Nash equilibrium under uncertainty for the original game.

a.E.D. The following proposition show that this theorem generalize Dow-Werlang [1994]'s existence theorem:

Proposition 4.1 Generalize Dow-Werlang [1994)'s Existence Theorem.

Every Dow-Werlang [1994]'s Nash equilibrium under uncertainty is a Nash equilibrium under uncertainty as in our Definition 4.2.

Proof: Consider, for each player i E {1,2}: ,; E 'P j(Sj,2

S-),

where

vI:

~-j-?[O, 1] is the uncertainty aversion function defined such that, for ali Ae 2Si, 0;éA;éSj:

';(A)=C;, cjE[O,1].

Thus defined, ,; exhibits constant uncertaintyaversion. 80, for each player iE{1,2), Vj is an uniform squeeze of pj. These beliefs (VI, v~ form a Nash equilibrium under uncertaintyas in our Definition 4.2.

a.E.D.

5. RELATEO RESUL TS ANO CONCLUSION

The definition of Nash equilibrium under uncertainty provided here, that extends the one presented in Dow-Werlang [1994]'s paper is related with the extension presented in Marinacci [2000]'s paper which proved the existence of Nash equilibrium under uncertainty for any given uncertainty aversion function.

The most important result of this paper is to present the uncertainty aversion function as an explicit parameter in the description of the game, which enables us to do static comparative static exercises in an easy way.

It remains an extension to n players.

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