1
Mathematical Notes
Since the mathematics in the paper are very complex, we add the following notes for
the notice of those who are interested in the mathematics.
Derivation of equation 1
For example, the probability that the focal phenotype (r1s, r2s,r3s)/(r1d, r2d,r3d) is
dominant, the 3rd member which prefers an activity change and plays GIVE IN is:
1 1
  (1  r3d ) . In this case, the probability that the other group member which prefers
3 3
to change activity 1st plays INSIST is R1s (drawn randomly from the population) and,
similarly, that the group member which prefers to change 2nd plays GIVE IN is
(1  R2s ) . The total probability for such a situation is, thus:
1 1
  R1s  (1  R2 s )  (1  r3d ) .
3 3
In this situation, the group does not split and changes activity when the 1st member
prefers to change activity (because the 1st member is the only member which insists
on its own preference). The focal gains the grouping benefits of remaining in a group
with three members (i.e. B3) but pays the costs of changing activity much too early
(i.e., C2). It follows that the expected gains to the focal for this case are:
1
 R1s  (1  R2 s )  (1  r3d )  B3  C 2 . In the same manner one can calculate the
9
expected gains to the focal for all other possible situations. By summing the gains to
the focal for all the possible permutations of focal preferences, focal behaviour,
dominance relationships and preferences and behaviour of the other two group
members, weighted by their respective probabilities, one can calculate the overall
expected gains to the focal, as given in eqn 1 in the main text. (Note also that if the
group splits in such a way that the focal becomes solitary, the focal’s gains are 0, as
there are no grouping benefits but also no consensus costs).
2
ESS’s in group of three
gains increase with ra if dgains/dra > 0 and decrease with ra if dgains/dra < 0. Thus, if
dgains/dra > 0, gains are maximal if ra = 1, and if dgains/dra < 0, gains are maximal if
ra = 0.
Using eqn 2, it is in a population of equally-sharing decision makers with a majority
threshold:
dgains/dr1d = ½.dgains/dr1s = - (B3- C1)/9 < 0
dgains/dr2d = ½.dgains/dr2s = + 2.C1/27 > 0
if B3>C1;
since C1>0;
dgains/dr3d = ½.dgains/dr3s = - (B3- C1)/9 < 0
if B3>C1.
Therefore, if B3>C1, gains are maximal, if r1s=r1d=r3s=r3d=0 and r2s=r2d=1.
The other ESS’s can be dealt with in a similar manner.
Relaxing the assumption that consensus costs are symmetric
It follows from equation (8) that the expected gains to a potentially invading
phenotype (r1s, r2s, r3s)/(r1d, r2d, r3d) in a monomorphic population of equally-shared
decision makers with a majority as a threshold [phenotypes (0,1,0)/(0,1,0)], are:
B3  C1late / 3  C1early / 2
gains[(r1s, r2s, r3s)/(r1d, r2d, r3d),(0,1,0)/(0,1,0)] =
 (2r1s  r1d ) / 9  ( B3  C1late )
 (2r2 s  r2 d ) / 18  C1early
 (2r3s  r3d ) / 9  ( B3  C1early )
Thus, it is:
dgains/dr1d = ½.dgains/dr1s = - (B3- C1late)/9 < 0
dgains/dr2d = ½.dgains/dr2s = + C1early/18 > 0
if B3>C1late;
since C1early>0;
(N.1)
3
dgains/dr3d = ½.dgains/dr3s = - (B3- C1early)/9 < 0
if B3>C1early (which is always
given if B3>C1late, since C1late >C1early).
Therefore, if consensus costs of changing activity too late are not too high relative to
grouping benefits (i.e., if C1late<B3), r1s=r1d=0, r2s=r2d=1, r3s=r3d=0 yields the highest
expected gains and, thus, the best strategy for a phenotype is to make decisions
equally-shared with a majority as a threshold [phenotype (0,1,0)/(0,1,0)], which is,
therefore, an ESS. However, if consensus costs of changing activity too late are high
relative to grouping benefits (i.e., if C1late>B3), equally-shared decision making with a
majority as a threshold is no longer an ESS (note, while it could be that C1late,
C2late>B2, B3, it still has to be C1early, C2early<B2, B3, since otherwise animals would
not be social: Conradt 1998; Ruckstuhl 1998, 1999; Ruckstuhl & Neuhaus 2002;
Krause & Ruxton 2002).
Further, it follows from equation (8) that the expected gains to a potentially
invading phenotype (r1s, r2s, r3s)/(r1d, r2d, r3d) in a monomorphic population of equallyshared decision makers with a sub-majority as a threshold [phenotypes
(1,0,0)/(1,0,0)], are:
B3  C 2 early / 3  C1early / 3  C1late / 6
gains[(r1s, r2s, r3s)/(r1d, r2d, r3d),(1,0,0)/(1,0,0)] =
 (2r1s  r1d ) / 18  C1late
 (2r2 s  r2 d ) / 9  ( B3  B 2  C1early )
 (2r3s  r3d ) / 9  ( B3  C 2 early )
Thus, it is:
dgains/dr1d = ½.dgains/dr1s = + C1late/18 > 0
since C1late>0;
dgains/dr2d = ½.dgains/dr2s = - (B3- B2-C1early)/9 < 0
dgains/dr3d = ½.dgains/dr3s = - (B3- C2early)/9 < 0
if B3>B2+C1early;
if B3>C2early.
4
Therefore, if consensus costs of changing activity too early are not too high relative
to grouping benefits and group size is not above optimal group size (i.e., if C2early<B3
and B2+C1early<B3), r1s=r1d=1, r2s=r2d=0, r3s=r3d=0 yields the highest expected gains
and, thus, the best strategy for a phenotype is to make decisions equally-shared with
a sub-majority as a threshold [phenotype (1,0,0)/(1,0,0)], which is, therefore, an ESS.
Further, it follows from equation (8) that the expected gains to a potentially
invading phenotype (r1s, r2s, r3s)/(r1d, r2d, r3d) in a monomorphic population of equallyshared decision makers with a super-majority as a threshold [phenotypes
(1,0,0)/(1,0,0)], are:
B3  C 2 late / 3  C1late / 3  C1early / 6
gains[(r1s, r2s, r3s)/(r1d, r2d, r3d),(0,0,1)/(0,0,1)] =
 (2r1s  r1d ) / 9  ( B3  B 2  C 2 late )
 (2r2 s  r2 d ) / 9  ( B3  B 2  C1late )
 (2r3s  r3d ) / 18  C1early
Thus, it is:
dgains/dr1d = ½.dgains/dr1s = - (B3- B2-C2late)/9 < 0
if B3>B2+C2late;
dgains/dr2d = ½.dgains/dr2s = - (B3- B2-C1late)/9 < 0
if B3>B2+C1late (which is
always given if B3>B2+C2late, since C2late >C1late);
dgains/dr3d = ½.dgains/dr3s = + C1early/18 > 0
since C1early>0.
Therefore, if consensus costs of changing activity too late are not too high relative to
grouping benefits and group size is not above optimal group size (i.e., if
B2+C2late<B3), r1s=r1d=0, r2s=r2d=0, r3s=r3d=1 yields the highest expected gains and,
thus, the best strategy for a phenotype is to make decisions equally-shared with a
super-majority as a threshold [phenotype (0,0,1)/(0,0,1)], which is, therefore, an ESS.
5
Further, it follows from equation (8) that the expected gains to a potentially
invading phenotype (r1s, r2s, r3s)/(r1d, r2d, r3d) in a monomorphic population of
unshared decision makers [phenotypes (0,0,0)/(1,1,1)], are:
gains[(r1s, r2s, r3s)/(r1d, r2d, r3d),(0,0,0)/(1,1,1)] =
B3  C 2 late / 9  5  C1late / 18  C1early / 3  C 2 early / 9
 r1d / 18  C1late
 r1s / 9  (2  B3  B 2  C 2 late  C1late )
 r2 d / 18  C1early
 r2 s / 9  (2  B3  2  B 2  C1late  C1early )
 r3d / 18  C1early
 r3s / 9  (2  B3  C 2 early  C1early )
Thus, it is:
dgains/dr1d = + C1late/18 > 0
since C1late>0;
dgains/dr1s = - (2.B3- B2-C1late-C2late)/9 < 0
dgains/dr2d = C1early/18 > 0
if 2.B3>B2+C1late+C2late;
since C1early>0;
dgains/dr2s= - (2.B3- 2.B2-C1late-C1early)/9 < 0
dgains/dr3d = C1early/18 > 0
if 2.B3>2.B2+C1late+C1early;
since C1early>0;
dgains/dr3s = - (2.B3- C1early-C2early)/9 < 0
if 2.B3>C1late+C2late (which is always
given if 2.B3>B2+C1late+C2late, since C1/2late >C1/2early).
Therefore, if consensus costs of changing activity too late are not too high relative to
grouping benefits and group size is not above optimal group size (i.e., if
B2+C2late+C1late <2B3 and 2B2+C1early+C1late <2B3), r1d=r2d=r3d=1, r1s=r2s=r3s=0
yields the highest expected gains and, thus, the best strategy for a phenotype is to
make decisions unshared [phenotype (0,0,0)/(1,1,1)], which is, therefore, an ESS.
6
Decisions in larger groups
For equally shared decisions with x as a consensus threshold:
Using eqn 9, it is:
(a) for i<x : dgains/drid =
d (k ) .
dgains/dris =  d (k )  pi (k )  Bn  BX i  Clate | x  i | ;
1  d (k )
and it is  d (k )  pi (k )  Bn  BX i  Clate | x  i | < 0
(b) dgains/drxd =
if Bn  BX i  Clate | x  i | );
d (k ) .
dgains/drxs =
1  d (k )
n

1  x 1
 d (k )  p x (k )    Cearly (| x  j |   Clate (| x  j |)  , which is always >0, since
n  j 1

j  x 1
consensus costs cannot be negative;
(c) for i>x : dgains/drid =
d (k ) .
dgains/dris =  d (k )  pi (k )  Bn  BX i  C early | x  i | ;
1  d (k )
and it is  d (k )  pi (k )  Bn  BX i  C early | x  i | < 0
if Bn  BX i  C early | x  i | ).
For unshared decisions:
Using eqn 10, it is:
1  i 1
(a) dgains/drid = d (k )  pi (k )     C early | i  j | 
n  j 1
>0, since consensus costs cannot be negative;
n

 Clate | i  j | 
j i 1

, which is always
7
 i 1

 p( j , k )  Bn  BX ij  C early (| j  i |)) 
j 1
 ; and it is
(b) dgains/dris=  (1  d (k ))  pi (k )  

n
  p( j , k )  Bn  BX ij  Clate (| j  i |)) 
 j i 1





 i 1

 p( j , k )  Bn  BX ij  C early (| j  i |)) 
j 1
 < 0, if
 (1  d (k ))  pi (k )  

n
  p( j , k )  Bn  BX ij  Clate (| j  i |)) 
 j i 1





i 1
n
j 1
j i 1
 p( j, k )  Bn  BX ij  Cearly (| j  i |))   p( j, k )  Bn  BX ij  Clate (| j  i |))  >0, which
is the case, if generally Bn  BX ij  C early (| j  i |)) for all j<i with p(j,k)>0 and
Bn  BX ij  Clate (| j  i |)) for all j>i with p(j,k)>0.