Peters, H.J.M.

Wakker, P.P.

English

Research Papers in Economics

Economics  Letters  22  (1987)  251-255 
North-Holland 
251 
CONVEX  FUNCTIONS  ON  NON-CONVEX  DOMAINS 
H.J.M.  PETERS 
Uniuersitv  of  Limhurg,  6200  MD  Muustricht,  The  Netherlands 
P.P.  WAKKER 
University  of  Ntpegen,  6500  WC  Nijmegen,  The  Netherlunds 
Received  28  May  1986 
It  is  shown  that  a  convex  function,  defined  on  an  arbitrary,  possibly  finite,  subset  of  a  linear  space,  can  be  extended  to  the 
whole  space.  An  application  to  decision  making  under  risk  is  given. 
1. Introduction 
The  definitions  of  convexity  of  a  function,  customary  in  literature,  assume  or  imply  convexity  of 
the  domain.  In  applications  however,  observations  about  the  values  of  a  function,  and  inequalities 
verified  by  these,  are  often  available  only  on  non-convex,  e.g.  finite,  domains. 
This  paper  considers  a  way  to  define  convexity  on  non-convex  domains.  The  main  theorem,  in 
section  2,  shows  that  a  function  is  convex,  if  and  only  if  it  is  the  restriction  of  a  convex  function 
defined  on  the  whole  space.  Section  3 gives  an  application  to  decision  making  under  risk,  and  section 
4  concludes. 
2.  The  extension  theorem 
Let  V  be  a linear  space  over  the  reals.  Let  T  be  some  arbitrary  subset  of  V, and  f  a function  from 
T  to  Iw u  { -  co, m}.  The  following  definition  adapts  the  definitions  of  convexity,  given  in  literature 
only  for  convex  sets  T  [see  for  instance  Rockafellar  (1970,  sect.  4)],  to  general,  possibly  finite,  sets  T. 
For  compact  sets  it  was  given  in  Peters  and  Tijs  (1981)  and  for  general  sets  in  Wakker,  Peters  and 
Van  Riel  (1985). 
Definition  1.  The  function  is  convex  if  for  all  convex  combinations  Xy=ip,x’  of  elements  xJ  of  T, 
for  which  not  both  -  cc  and  +  cc  are  contained  in  { f(~J)}/n=~,  we  have 
whenever  Cp,xJ  is  in  T. 
(1) 
0165.1765/86/$3.50  0  1986,  Elsevier  Science  Publishers  B.V.  (North-Holland) Asusualwetake:  Xco:=~,forh~lR++,  hm  :=  0 for  h =  0,  hcc  :=  -  co for  h E  R __;  h +  rx  :=  00 
for  X E  R  or  X =  CL),  and  A +  cc is undefined  for  h =  -  cro; X( -  m)  :=  -  M  for  h  E  [w  ++,  X( -  cc)  :=  0 
for  X =  0,  h(  -  m)  :=  cc  for  X E  R__;  X -  co :=  -  cz3 for  X E  lR  or  h =  -m,  and  X -  ry,  is  left 
undefined  for  X =  co. 
A  function  f  is  concave  if  -f  is  convex.  All  results,  derived  in  the  sequel  for  convex  functions  f, 
can  be  reformulated  for  concave  functions  g,  by  setting  g :=  -f. 
Theorem  I  (extension  theorem).  LetVbeaiinearspaceoverR;  1etTcV.  Letf:T+RU{-30.00) 
be  convex.  Then  there  exists  a convex  function  f”: V +  IF8  U  { -  cc,  cc}  which  extends  J: 
Proof.  We  define  f-on  V  as  follows: 
~:~+-+inf{~::=~CJn=~p,f(x’),  x  is  a  convex  combination  Cp,xJ  of  elements  XI  of  T  such  that  not 
f(x’)  =  cc  for  some  j,  and  f(x’)  =  -  cc  for  some  other  j}.  (2) 
Note  that  f(x)  sf(.x)  for  all  x  E  T;  by  convexity  of  f,  f(x)  >=  f(x)  for  all  x  E  T.  So  indeed  f 
extends  f.  Convexity  of  f  remains  to  be  demonstrated.  So  let  y  be  a  convex  combination  1::  lq,y’, 
with  all  q, > 0.  We  have  to  prove 
whenever  not  f”(y’)  =  m  for  some  i,  f(  y’)  =  -  00  for  some  other  i. 
The  case  where  f(  y’)  =  rxj for  some  i  is  immediate  anyhow,  so  we  suppose  fly’)  <  rx: for  all  i. 
This  implies  that  every  y’  is  in  conv(T).  Say  y’  is  the  convex  combination  YXk  P,~x’~  of  elements  xfh 
of  T.  Then  ,v =  C:f-lCAq,p,k~*‘.  By  definition  of  f: 
f(Y) s F  q,CP,kf(x’/o. 
r=l  k 
(4) 
Now  first  suppose  f((~“)  =  -  co,  for  some  1. Then,  for  every  ME  N,  we  can  take  the  x”  above  such 
that  1,  ~,~,f(x’~)  <  -M.  Since  q, > 0,  by  (4)  we  get  fl( y)  =  -  30,  and  (3)  follows. 
Next  suppose  f”(y’)  >  -  cx) for  all  i. Then,  for  any  6 >  0 and  every  i, we  can  take  the  xIh  such  that 
xkp,l,f(~‘k)  sfl_vl)  +  E. This  again  implies  (3).  Q.E.D. 
Note  that  the  [defined  in  (2)  is  the  maximal  convex  extension.  If  f  in  Theorem  1 is  real-valued,  then 
even  on  conv(  T),  it  may  be  impossible  to  have  f” real-valued,  as  the  following  example  shows. 
Example  1.  Let  V=lR’,  T=  ((-1,  -j)},,.U{(l,  j)),,..  fix,,  x2)=  -1x2(  forall(x,,  x,)ET. 
Then  considering  the  pair  ( -  1.  -j)  and  (1.  j).  one  sees  that  f(0,  0)  5  -j  must  hold,  for  all  ,j E  N. 
For  bounded  f,  f  can  be  taken  real-valued  on  conv(  T ): 
Coroilury  I.  For  the f:  defined  in  (2),  supflconv(  T))  = sup(  f(  T)),  and  inffz(conv(  T))  = inff(  T). 
Proof.  Obvious  from  (2). 
If  V = R,  then  a  real-valued  f  in  Theorem  1 can  be  extended  to  a  real-valued  f-on  conv(  T). H.J.M.  Peters,  P.P.  Wakker /  Convex  functions  on non-convex  domains  253 
Corollary  2.  Let  V=  R,  TC  V,  f:  T -+ 08 convex.  Then f  has  a  convex  extension  f-:  conv(  T)  +  08. 
Prooj  Obvious  if  T  contains  no  more  than  two  elements,  then  f-can  be  taken  affine  (i.e.,  both 
convex  and  concave).  So  let  x1 >  x2  >  x 3 in  T.  Let  1’  be  the  affine  function  through  (x’,  f(x’))  and 
(x2,  f(x*)),  I*  be  the  one  through  (x2,  f(x2))  and  (x3,  f(x3)).  Then,  with  f-as  in  (2),  for  all  x E  R 
we  have  f(x)  2  min{f’(x),  r*(x)}  >  -  00. 
Further,  for  all  x E  conv(T).  x  E  [x4,  x5]  for  some  x4,  x ’  in  T.  Hence  f-(x)smax{  f(x4), 
f(x’)}  <  co  for  such  x.  Q.E.D. 
The  following  example  shows  that,  even  if  V =  03, and  f  is  real  valued  and  bounded  on  T,  then 
still  no  convex  real-valued  extension  f-of  f  to  all  of  V  may  exist. 
Example  2.  Let  T =  [0,  11, f  : x  *  -  4x  on  T.  Then  any  convex  extension  f-of  f  on  V can  be  seen 
to  assign  00  to  all  of  [w  __. 
Lemma  I.  Let  V =  CR, and  f  non-decreasing  and  convex.  Then f:  as  defined  by  (2),  is  non-decreasing 
on  conv(  T ). 
Proof.  By  Corollary  1,  f” has  the  same  infimum  on  conv(  T),  as  f  on  T. Hence  the  infimum  of  f-can 
be  found  on  the  ‘left  side’  of  conv(T):  the  convex  f-must  be  non-decreasing.  Q.E.D. 
3.  An  application  to risk  aversion 
Let  V  be  a  non-empty  set  of  consequences.  -Ep”(%?) is  the  set  of  (simple)  lotteries  on  %‘; a  (simple) 
lottery  on  % is  a probability  measure  on  (%?, 2’@),  assigning  probability  one  to  a  finite  subset  of  %‘. By 
( p,;  XJ);=,  we  denote  the  simple  lottery,  assigning  probability  pj  to  every  xl.  Implicit  in  this  is  that 
x’  E  V  for  all  j,  p,  2  0  for  all  j,  and  X.p, =  1. For  any  a: E  V’, Lu  denotes  (1;  a),  i.e.,  the  lottery  which 
with  probability  one  results  in  consequence  a. 
We  assume  there  are  two  decision  makers  (persons,  players,  etc.)  Tk,  k =  1,  2,  with  preference 
relations  % k , i.e.,  binary  relations  on  S?(V).  The  interpretation  of  1 t  kl’  is  that  Tk,  when  having 
to  choose  one  element  from  {I,  I’},  is  willing  to  choose  1. 
We  assume  that  there  exist  von  Neumann-Morgenstern  (vNM)  utility  functions  U’,  U*,  i.e.,  for 
k  =  1,  2,  Uk:  C+  [w is  such  that 
J=l  i=l 
Here  XpJUk(xj)  is  the  expected  utility  of  (p,;  xJ)~=,. 
The  following  definition  gives  a way  to  compare  T’  and  T2  with  respect  to  their  ‘risk  aversion’.  It 
has  been  introduced  in  Yaari  (1969,  p.  316,  in  terms  of  so-called  ‘acceptance  sets’),  and  is  a variation 
on  earlier  definitions  of  Pratt  (1964)  and  Arrow  (1971). 
Definition  2.  T’  is  more  risk  averse  (MRA)  than  T*  if 
(p/;  SXJ)/nzl  t’cY*  (p/;  xJ)Yzl  z*ti.  (6) 
The  above  definition  applies  if  T*  is  willing  to  take  a  risky  lottery  (p,;  xJ),“=~  instead  of  a  certain 
consequence  (Y,  whenever  T’  is  willing  to  do  so. 254  H.J.  M.  Peters,  P. P.  Wukker  /  Convex  funcrions  on non-convex  domam 
Definition  2,  and  the  results  to be  derived  below,  have  first  been  studied  in  literature  for  the  case 
%?= R.  Later  Kihlstrom  and  Mirman  (1974)  extended  these  to  the  case  %‘= WT.  All  this  was  done 
under  differentiability  and  monotonicity  assumptions  about  the  vNM  utility  functions.  In  Wakker, 
Peters  and  Van  Riel  (1985,  Theorem  3.1.b)  the  results  were  extended  to  arbitrary  spaces  %?,  without 
any  restriction  on  the  vNM  utility  functions.  Motivation  for  this  was,  firstly,  that  differentiability  of 
the  utility  functions  does  not  always  have  a  clear  behavioural  meaning;  secondly,  that  it  seems 
desirable  for  applications  to be  able  to  handle  cases  where  one  has  information  about  the  choices  of 
decision  makers,  only  with  respect  to finitely  many  consequences;  and  thirdly,  that  it seems  desirable 
to be  able  to  handle  cases  where  one  does  not  (yet)  have  a quantification  of  the  consequences. 
The  following  theorem  characterizes  the  ‘more  risk  averse  than’  relation. 
Theorem  2.  The following  two  statements  are  equivalent: 
(i )  T’  is  more  risk  averse  than  T 2. 
(ii)  U2  = $0  U’  for  a  convex  non-decreasing  function  $J : U’(V)  +  U2(%‘). 
Proof.  First  suppose  (i).  Then  fl%  ‘Z =$ fl%  2Z,  so  U’(p)  2  U’(a)  *  U*(p)  >=  U2(a).  So  there 
must  exist  a non-decreasing  4  : U’(  9)  +  U*(V)  such  that  U* 14  0 U’.  Now  let  some  element  U’(p) 
of  U’(%?)  be  a convex  combination  E;2zlpjU1(~J)  of other  elements  of  U’(W).  Then  (p,;  P’);,~  + ‘CL, 
so  by  (i):  (p,;  pJ),“=i  % 2,ii, i.e.,  Ep,U  (PI)  2  U2(p).  Substituting  $  gives 
Cp,@J’($))  ‘=  [4@W)  =I  ~(~P,W’))~  (7) 
so  4  is convex  on  U’(U),  and  (ii)  is proved. 
Next  suppose  (ii).  Let  ( pJ;  PLJ)~=~  b  ‘ii,  so  C:=,p,U1(pJ)  2  U1(p).  Let  4  be  as  defined  in  (2);  by 
Lemma  1,  q  is  non-decreasing  on  conv(T).  We  get  Eyzlp,U2(pJ)  = Cp,$(U’(pLJ))  2  1c/(Cp,U’(pj)) 
>=  $(U’(~))  =  U2(p).  This  implies  (p,;  /.LJ),“=~  k  *,L., which  is what  (i)  requires.  Q.E.D. 
Theorem  2  shows  a  natural  occurrence  of  convex  functions  on  non-convex  domains:  if  observa- 
tions  of  choices  of  T’  and  T2  are  available  only  with  respect  to  consequences  from  %?, then 
(inequalities  concerning)  li, will  only  be  observed  on U’(V),  which  may  be  any  arbitrary  subset  of  R. 
In  the last  occurrence  of  4  in  the  proof  we essentially  use  the extension  theorem:  the  argument  of  +6 
there  does  not  have  to  be  in  the  domain  of  4.  Because  of  this  complication,  in  Wakker,  Peters  and 
Van  Riel  (1985)  a more  complicated  proof  (without  the  extension  theorem  available)  was given,  using 
first-order-difference  results  derived  in  the  appendix  there. 
4.  Conclusion 
Convexity  and  concavity  have  proved  to be  fruitful  concepts  in  economics.  Therefore,  it  seems  to 
be  worthwhile  to  define  convexity  and  concavity  of  a  function  also  on  non-convex  domains,  since 
these  may  occur  in  economics  in  a  natural  way.  The  main  theorem  presented  in  this  paper,  which 
extends  a  convex  function  to  at  least  the  convex  hull  of  its  domain,  enables  one  to  apply  results 
known  for  convex  functions  on  convex  domains  to  such  functions  on  non-convex  domains.  An 
example  of  an  application,  to  decision  making  under  risk,  has  been  given. H.J.M.  Peters,  P. P.  Wakker  /  Convex  functions  on non-conoex  domurns  255 
References 
Arrow,  K.J.,  1971,  Essays  in  the  theory  of  risk  bearing  (Markham,  Chicago,  IL). 
Kihlstrom,  R.E.  and  R.J.  Mirman,  1974,  Risk  aversion  with  many  commodities,  Journal  of  Economic  Theory  8.  361-388. 
Peters,  H.  and  S.H.  Tijs,  1981,  Risk  sensitivity  of  bargaining  solutions,  Methods  of  Operations  Research  44,  409-420. 
Pratt,  J.W.,  1964,  Risk  aversion  in  the  small  and  in  the  large,  Econometrica  32,  122-136. 
Rockafellar,  R.T.,  1970,  Convex  analysis  (Princeton  University  Press,  Princeton,  NJ). 
Wakker,  P.P.,  H.J.M.  Peters  and  T.  van  Riel,  1985,  Comparisons  of  risk  aversion,  with  an  application  to  bargaining,  Report 
(Department  of  Mathematics,  University  of  NiJmegen.  Nijmegen). 
Yaari,  M.E.,  1969,  Some  remarks  on  measures  of  risk  aversion  and  on  their  uses,  Journal  of  Economic  Theory  1,  315-329. 

Convex functions on non-convex domains.

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