The following Supporting Information is available for this article:
Appendix S1 Simulation model details
Appendix S2 Fitness when no stochasticity
Appendix S3 Optimism when correlation is positive
Appendix S4 Density dependence
Appendix S5 Randomised strategies
S1: Simulation model details
A simulation model supports the theoretical predictions and confirms that particular
effects are still present in density-dependent cases.
The model is individual-based, with an integer number of individuals on each of n
patches; individuals on each patch are further segregated into b equi-spaced bins of uvalues. At each generation, every individual goes through an asexual reproductive
cycle before dying, with the expected number of recruits, rs(u), being dependent upon
the environment type, s, and trait, u, according to rA(u) = (1-u2)fA and rB(u) = (1-(1u)2)fB, where fA and fB are fixed real numbers. The actual number of recruits for a
given individual was sampled from a geometric distribution with the specified mean.
Each patch is initially assigned a type (A or B) with equal probability. At each
generation, each patch of type s remains of that patch-type with probability (1-1/ts),
giving mean durations for each type of patch as ts.
At each generation, following reproduction, offspring could disperse to a random
patch (which could be the same patch); this occurred with probability m for each
individual. By default, each offspring had the same u-value as its parent; in a small
proportion of cases, c, this value could mutate to an adjacent bin. Any attempt to
mutate to a value less than 0 or greater than 1 was prevented.
To prevent population explosions from slowing the run-time exponentially with each
generation, a rescaling was applied whenever the total population exceeded a defined
limit, z. This was done by dividing the number of individuals in each bin of each
patch by 10, resulting in some integer, X, and a number of 10ths, Y, i.e., the original
number was 10X + Y. A uniform random number, S  [0,1] was compared with Y, X
then being increased by 1 if and only if Y > S. The rescaled number in each u-bin per
patch was then set to the corresponding value of X.
Each u-bin on each patch was initially seeded with v individuals. Each simulation ran
for a fixed number of generations, g.
After g generations, the modal value of u was calculated by summing the number of
individuals per u-bin across patches and fitting a quadratic to the bin with the largest
number of individuals and its two neighbours to find where the quadratic peaked. The
subjective probability, ps, was then calculated from the modal u. The proportion of
individuals on a type B patch, p*, was also calculated.
The default values for each run were as follows: n = 1000, b = 201, fA = 1.4, fB = 1.0,
tA = 10, tB = 10, m = 0.2, c = 0, z = 200000000, v = 1000, g = 1000.
The simulation was modified slightly for a density-dependent case, for the results
shown in Appendix S5. In this variant, the number of individuals per patch was
limited to 1000, with excess individuals chosen at random from the patch for
elimination at each generation.
S2: Fitness when there is no environmental stochasticity
As in the main text we suppose that population members can be in one of two
circumstances, A or B. Let s A be the probability that an offspring from an individual
in circumstance A is also in circumstance A. Similarly, let s B be the probability that
the offspring of an individual in circumstance B is also in circumstance B.
Consider a genotype that has trait value u. Let d AA (u)  s A rA (u) ,
dAB (u)  (1 sA )rA (u) , dBA (u)  (1 sB )rB (u) and d BB (u)  s B rB (u) , and let Q(u) be
the matrix
Q(u ) 
d AA (u ) d AB (u )
d BA (u ) d BB (u )
.
(A2.1)
Let nA (t ) and nB (t ) be the number of genotype members in circumstances A and B
respectively in generation t. Let n(t) be the vector n(t)  (nA (t), nB (t)) . Then the
change in genotype number between generation t and generation t+1 is described by
the vector equation
n(t  1)  n(t )Q(u ) .
(A2.2)
Let N(t)  nA (t)  nB (t) be the total number of genotype members in generation t.
Then providing there is no environmental stochasticity
N (t  1)
  (u) as t   .
N (t )
(A2.3)
where  (u ) is the largest eigenvalue of the matrix Q(u). This eigenvalue is therefore
the asymptotic growth rate in genotype numbers, and is the standard measure of
fitness of the genotype (Metz et al. 1992). Although this measure is only strictly
applicable when there is no environmental stochasticity our computations confirm
(Figure 3) the measure is still a good approximation to fitness in an environment
composed of many stochastically varying patches, provided the dispersal rate between
patches is not too small.
Let
p(u)  limt
nB (t )
N (t )
(A2.4)
be the limiting proportion of genotype members in circumstance B. Then proportions
satisfy the eigenvector equation
(1  p(u), p(u))Q(u)  (u)(1  p(u), p(u))
(A2.5)
(Caswell, 2001). In non-vector notation, this equation becomes the pair of equations
(1 p(u))sA rA (u)  p(u)(1 sB )rB (u)  (u)(1 p(u))
(A2.6a)
(1 p(u))(1 sA )rA (u)  p(u)sB rB (u)  (u)p(u) .
(A2.6b)

We use these equations to establish:

Result A2a. Fitness as an average growth rate
Summing equations (A2.6a) and (A2.6b) gives
(u)  (1 p(u))rA (u)  p(u)rB (u) ,
which provides equation (3) of the main text.
(A2.7)
S3: Optimism when the correlation between parent and offspring is positive
In this appendix, we are concerned with scenarios in which the measure  (u ) given in
Appendix S2 is the appropriate fitness measure. We show that if the environment is
such that there is a positive correlation between the circumstance of an individual and
its parent, then it is optimal to be optimistic. To establish this we first relate the
covariance between an individual and its parent to environmental parameters. We then
investigate how the stable distribution of members of a genotype depends on their trait
value u. Finally, we investigate the properties of the optimal strategy.
Our technical assumptions on the functions rA (u ) and rB (u ) are as follows. We
assume that there is a trade-off, so as the trait value u increase one of rA (u ) and
rB (u ) is strictly increasing and the other is decreasing. For each p [0, 1] we assume
that
W ( p, u)  (1  p)rA (u)  prB (u)
is a unimodal function of u with a unique maximum at uˆ ( p) . Finally, we assume that
for each u there is a unique p such that uˆ( p)  u . With these assumptions it is easy to
show that if rA (u ) is a decreasing function and rB (u ) is an increasing function then
uˆ ( p) is a strictly increasing function of p. Similarly, if rA (u ) is increasing and rB (u )
decreasing then uˆ ( p) is strictly decreasing.
The covariance
Consider a population in which all population members have trait value u. We
suppose this population is at demographic stability, so that the proportion of
individuals in circumstance B is p(u ) . We analyse the correlation between the
circumstance experienced by an individual and it parent at this demographic
equilibrium. Notation follows that of Appendix S2.
Consider the experiment of choosing a population member at random. Define the
random variables X and Y as follows. Set Y = 0 if the individual is in circumstance A
and set Y = 1 if the individual is in circumstance B. Set X = 0 if the parent of the focal
individual was in circumstance A and set X = 1 if the parent of the focal individual
was in circumstance B. Then necessarily
E{Y}  P(Y  1)  p(u) .
(A3.1)
An individual in circumstance A will, on average, produce rA (u)(1  s A ) offspring in
circumstance B. An individual in circumstance B will, on average, produce rB (u)s B
offspring in circumstance B. Therefore,
P( X  1 | Y  1) 
p(u)rB (u)s B
 rB (u)s B /  (u) (A3.2)
(1  p(u))rA (u)(1  s A )  p(u)rB (u)s B
by equation (A2.6b). Similarly,
P( X  1 | Y  0)  p(u)rB (u)(1  s B ) /( (u)(1  p(u))) .
(A3.3)
It then follows that
E{X }  P( X  1)  p(u)rB (u) /  (u)
(A3.4)
and
E{XY}  P( X  1, Y  1)  p(u)rB (u)s B /  (u) .
(A3.5)
By equations (A3.1), (A3.4) and (A3.5)
Cov( X , Y )  E ( XY )  E ( X ) E (Y ) 
p(u )rB (u )
s B  p(u ).
 (u )
(A3.6)
We can then prove the following.
Result A3a. Conditions for the correlation to be positive
Cov( X ,Y )  0  sA  sB  1.
(A3.7)
Proof
Using equation (A2.7) to eliminate (u) from equation (A2.6b), and re-arranging
gives
rA (u)(1  p(u))(1  p(u))  s A   rB (u) p(u) p(u)  s B  .
Thus 1  p(u)  s A and p(u)  s B have the same sign, and
p(u)  s B  (1  p(u))  s A  1  p(u)  (1  p(u))  s A  s B .
Similarly
p(u)  s B  (1  p(u))  s A  1  p(u)  (1  p(u))  s A  s B .
Thus
p(u)  sB  s A  sB  1 ,
and the result follows from equation (A3.6)
Result A3b. Proportion in circumstance B
Suppose that Cov( X ,Y)  0 . Let R(u)  rB (u) / rA (u) . Then
(a) R(u)  0  p(u)  0
(b) R(u)  0  p(u)  0.
Proof of (a)
Combining equations (A2.6a) and (A2.6b) to eliminate  (u ) gives
(1  p(u))(1  s A )rA (u)  p(u)s B rB (u) ,
p(u)

(1  p(u )) (1  p(u )) s A rA (u )  p(u )(1  s B )rB (u )
which can be rewritten as
 (u) 
(1  s A )   (u)sB R(u) ,
s A   (u)(1  sB )R(u)
where (u)  p(u) /(1 p(u)) . Define the function f : (0, )  (0, ) by
f ( z) 
(1  s A )  s B z
.
s A  (1  s B ) z
(A3.8)
Then by the eigenvalue equation (A2.5),
 (u)  f ( (u) R(u)) .
(A3.9)
Differentiating the function f gives
f ( z ) 
( s A  s B  1)
.
( s A  (1  s B ) z ) 2
(A3.10)
Since Cov( X ,Y)  0 condition (A3.7) guarantees that s A  sB  1. We thus have
f ( (u ) R (u ))  0 .
(A3.11)
We also have
[ f (z)  zf (z)][sA  (1 sB )z]2  sA(1 sA)  2(1 sA)(1 sB )z  (1 sB )sB z2  0 .
Thus, zf ( z )  f ( z ) for all z > 0. In particular
 (u) R(u) f ( (u) R(u))  f ( (u) R(u)).
Using equation (A3.9) we thus have
(A3.12)
R(u) f ( (u)R(u))  1 .
(A3.13)
Differentiating equation (A3.9) with respect to u we have
(u)  f ((u)R(u))(u)R(u) (u)R(u) ,
and hence
 (u) 
 (u) R(u) f ( (u) R(u))
.
1  R(u) f ( (u)R(u))
(A3.14)
By assumption R(u)  0 . Thus by equations (A3.11) and (A3.13)
(u)  0.
(A3.15)
Thus since (u)  p(u) /(1 p(u)) , and hence p(u)   (u) /(1  (u)) we also have
p(u ) 
 (u )
0.
1   (u )2
(A3.16)
This holds for all u. In particular p(u )  0 .
The proof of part (b) is directly analogous.
 to prove our main result.
We are now in a position
Result A3c.
Suppose that Cov( X ,Y)  0 , then it is always optimal to be optimistic.
Proof
We prove the result for the case where rA (u ) is a strictly decreasing function and
rB (u ) is a strictly increasing function. (The proof for the other case is directly
analogous.) Note that in this case R(u ) is a strictly increasing function of u and uˆ ( p) is
a strictly increasing function of p.
The optimal trait value u* satisfies  (u*)  0 . In a population adopting this trait, the
probability that a randomly selected population member is in circumstance B is
p*  p(u*). From equations (1) and (4) of the main text,
W ( p, u )
u
 rA (u*)  rB (u*) p (u*) .
p  p *,u u*
(A3.17)
Since R(u ) is increasing, p(u*)  0 by Result A3b(a). Thus rB (u*)  rA (u*) implies
W ( p, u )
 0 and hence implies that W ( p*, u ) has a maximum with respect
u
p  p *,u u*
to u at uˆ ( p*)  u*  uˆ ( p s ) (since the function W ( p*, u ) is unimodal with respect to
u). Thus since uˆ ( p) is increasing, we have
rB (u* )  rA (u* )

ps  p * .
(A3.18)

ps  p * .
(A3.19)
Similarly,
rB (u* )  rA (u* )
Thus by conditions 2a and 2b of Box 1 in the main text, it is always optimal to be
optimistic.
We can apply the above analysis to a case considered in the main text. Suppose that
the environment is composed of infinitely many patches linked by dispersal. The
circumstance on each patch alternates between A and B; in each cycle of change
spending on average t A generations in circumstance A before switching to
circumstance B, and then spending on average t B generations in circumstance B
before switching back to A. Patches change independently of one another.
Result A3d.
If
1
1

1
t A tB
then it is optimal to be optimistic.
Proof
s A  (1  m)(1  (1/ t A ))  m A
and
sB  (1  m)(1  (1/ t B ))  m B .
where  A and  B denote the proportion of patches of type A and B, respectively.
Thus
  1 1 
  1 1 
s A  sB  1  (1  m)2      m  1  (1  m)1     .
  t A t B 
  t A t B 
Thus
1
1

 1  s A  sB  1 ,
t A tB
and the result follows from Results A3a and A3c.
S4: Density dependence does not qualitatively alter the results
Here, we show that the trends of Figure 3, i.e., of p* increasing with tB whilst ps
decreases, can still hold in density-dependent scenarios.
By limiting the number of individuals on any given patch to 1000 in the simulation,
with removal of individuals down to that maximum if exceeded, we obtained the
results shown in Figure 5A. There are several points to make about the figure. The
theoretical lines shown are not density-dependent; they are the same lines as shown in
Figure 3 (for the non-density-dependent case) and have been included here as a visual
aid. Due to the smaller number of individuals per patch, the number of patches was
increased from the baseline case of 1000 to 5000. Due to the potential for patches to
become deserted (especially for large tB), the simulation was run for 500 generations
rather than 1000, to ensure that results were obtained prior to extinction. To help
ensure that the number of individuals in each u-bin did not go to zero, a small
mutation rate was introduced, of 0.000001 (i.e., approximately 1 in a million
individuals mutated to another bin per generation). All other variables, such as the
number of u-bins, were set according to the baseline case.
Figure 5A: Density dependent simulation results. Note that the theoretical lines are
not density dependent. m = 0.2, tA = 10.
Although the results no longer agree in value with the theoretical lines (due in part to
the smaller number of generations) and there is a larger variance in the subjective
probabilities (due to the smaller numbers of individuals), the general trend of p*
increasing with tB whilst ps decreases, is still clearly present.
S5: Randomised strategies do not outperform the optimal pure strategy
Denoting the two environmental states by A and B , with respective probabilities p A
and p B , and letting rs (u) denote the expected number of offspring when taking pure
action u in state s, we let:
rA (u )  (1  u 2 ) f A ,
drA (u)
 2 f Au ,
du
rB (u)  (1  (1  u) 2 ) f B ,
drB (u)
 2 f B (1  u) .
du
Note that we deal with a slightly generalised form in this appendix; setting f A  1.4
and f B  1, we obtain the offspring functions of the main text.
With pure temporal variation, fitness is given by p A ln(rA (u))  p B ln(rB (u)) ,
(Houston & McNamara, 1999), so is maximised by a pure action u* when
p A drA (u )
p drB (u )
 B
0.
rA (u ) du
rB (u ) du u u*
Substituting for each r and rearranging, we obtain
p Au *
1  u *2

.
p B (1  u*) 2u * u * 2
(A5.1)
McNamara (1995) generalises specific cases by Haccou & Iwasa (1995) and Sasaki &
Ellner (1995) to show that in a temporally varying environment, a pure action u* is
optimal if and only if

rs (u)
 r (u*) f (s)ds  1
0
s
for all u, where f (s ) denotes the probability density of environmental state s.
Thus, in our case, u* is optimal if and only if
pA
rA (u)
r (u)
 pB B
1
rA (u*)
rB (u*)
u .
(A5.2)
Substituting for r (u) and differentiating with respect to u, the LHS of (A5.2) reaches
an extreme if
 f A pA
u
(1  u )
 f B pB
rA (u*)
rB (u*) u u
 0.
(A5.3)
max
If u max exists, it is an upper limit for (A5.2), as differentiating with respect to u again,
we obtain 
f A pA
f p
 B B , which is always negative as f A and f B are positive.
rA (u*) rB (u*)
Substituting for r(u*) in (A5.3) and rearranging, we obtain
p Aumax
1  u *2
.

pB (1  umax ) 2u * u *2
(A5.4)
Comparing equations (A5.1) and (A5.4), we find that u max  u * , so u max exists and
the LHS of (A5.2) reaches its maximum of p A  p B  1 . Thus, no randomised
strategy can outperform u* for this set of r(u) functions.