The most common mistake in thinking about fitness in a variable environment is to average the wrong way. A strategy that does brilliantly in good years and terribly in bad ones can have a higher average yearly output than a steady rival and still lose, decisively, over the long run. The reason is that a lineage multiplies its numbers year on year, and what governs a product is the geometric mean, not the arithmetic one. Everything about bet-hedging follows from that single fact (Philippi and Seger 1989 Trends in Ecology and Evolution 4(2):41-44).
Take a risky strategy that doubles in a good year and shrinks to 0.4 in a bad one, each with even odds, against a steady strategy that grows by a flat 1.1 every year regardless.
A_good <- 2.0; A_bad <- 0.4; p <- 0.5; B <- 1.1
c(risky_arithmetic = p*A_good + (1-p)*A_bad,
risky_geometric = A_good^p * A_bad^(1-p),
steady_both = B)risky_arithmetic risky_geometric steady_both
1.2000000 0.8944272 1.1000000 The risky strategy has the higher arithmetic mean, 1.20 against 1.10: on an average year it grows faster. But its geometric mean is 0.894, below one, which means that compounded over years it shrinks. The steady strategy’s geometric mean is its arithmetic mean, 1.10, and it grows. Judged by the average year the risky strategy looks better; judged by the long run it goes extinct while the steady one thrives.
Averages can feel abstract, so simulate many independent lineages of each strategy over 40 years and look at where they end up.
years <- 40
set.seed(403)
step <- function(N, good, bad, pg) N * ifelse(runif(length(N)) < pg, good, bad)
sim <- function(good, bad, pg, reps) { N <- rep(1, reps); for (t in 1:years) N <- step(N, good, bad, pg); N }
final_A <- sim(A_good, A_bad, p, 2000); final_B <- sim(B, B, p, 2000)
c(median_risky = median(final_A), median_steady = median(final_B),
risky_percent_extinct = 100 * mean(final_A < 0.01)) median_risky median_steady risky_percent_extinct
0.01152922 45.25925557 43.75000000 After forty years the median steady lineage has grown to about 45 times its start; the median risky lineage has collapsed to 0.012, a hundredth of where it began, and 44 per cent of risky lineages are effectively extinct. A few risky lineages get a lucky run of good years and shoot up, which is what props up the arithmetic mean, but they are rare, and betting a lineage on being one of them is a losing strategy.
idx <- 1:120
tr <- rbind(
do.call(rbind, lapply(idx, function(i) data.frame(t = 0:years, N = RA[i, ], grp = "risky", id = i))),
do.call(rbind, lapply(idx, function(i) data.frame(t = 0:years, N = RB[i, ], grp = "steady", id = i))))
med <- rbind(data.frame(t = 0:years, N = apply(RA, 2, median), grp = "risky"),
data.frame(t = 0:years, N = apply(RB, 2, median), grp = "steady"))
ggplot(tr, aes(t, N, group = interaction(grp, id), colour = grp)) +
geom_line(alpha = 0.08) +
geom_line(data = med, aes(t, N, group = grp, colour = grp), linewidth = 1.2) +
scale_y_log10() +
scale_colour_manual(values = c("risky" = col_risky, "steady" = col_steady)) +
labs(x = "Year", y = "Lineage size (log scale)", colour = NULL,
title = "The long run belongs to the geometric mean")
Bet-hedging is the strategy of trading away some arithmetic mean to cut variance, because cutting variance raises the geometric mean. How steep is that trade? Hold the arithmetic mean fixed and widen the spread of good and bad years around it: the geometric mean falls, and past a point it drops below one and the lineage is doomed on average.
gm_fixed_mean <- function(delta, m = 1.1) sqrt((m + delta) * (m - delta)) # same arithmetic mean m
sapply(c(0, 0.3, 0.6, 0.9), gm_fixed_mean)[1] 1.1000000 1.0583005 0.9219544 0.6324555At the same arithmetic mean of 1.1, a spread of zero leaves the geometric mean at 1.10; widen the good and bad years to plus or minus 0.9 and the geometric mean falls to 0.63. The approximation is worth remembering: the geometric mean is roughly the arithmetic mean minus the variance divided by twice the mean. Variance is subtracted from fitness directly, which is why a strategy that lowers variance, even at some cost to the average, can win.
dl <- data.frame(delta = seq(0, 1.05, 0.01)); dl$gm <- gm_fixed_mean(dl$delta)
ggplot(dl, aes(delta, gm)) +
geom_hline(yintercept = 1, colour = col_ref, linetype = "dashed") +
geom_line(colour = col_risky, linewidth = 1) +
labs(x = "Spread of good and bad years (arithmetic mean fixed at 1.1)", y = "Geometric mean fitness",
title = "More variance, less long-run growth")
Two clarifications keep this from being mis-applied. First, the variance that matters is the kind shared across the whole population in a given year, a good or bad year for everyone, because that is what erodes the lineage’s geometric mean. Variation that is independent among individuals mostly averages out and does not select for hedging. Second, there are two ways to hedge: be conservative, adopt a lower-variance phenotype outright, or diversify, spread the bet across many offspring or many years so that some always land in a good year. The classic example of diversifying in time is a plant that declines to germinate all its seeds at once, the subject of the next post. Detecting either kind in real data is genuinely hard, because it requires measuring variance in fitness across generations rather than the mean, which the checking post takes up (Simons 2011 Proceedings of the Royal Society B 278(1712):1601-1609).
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