When uncertainty is adversarial, the robust planner plays a zero-sum game against Nature
The vaccination game was a game between people. Each individual weighed the cost of vaccination against the risk of infection, and the Nash equilibrium emerged from their simultaneous best responses. That framework explains a lot — why voluntary uptake stalls below the herd-immunity threshold, and how large the gap can be.
But at IVI, and in public-health agencies more generally, the operative decision-maker is usually not a pair of strategic individuals. It is a planner — a ministry of health, a WHO technical advisory group, a vaccine-allocation committee — who must commit to a policy before the epidemic unfolds, under substantial uncertainty about what the epidemic will do. The uncertainty can be parameter uncertainty (\(R_0\), vaccine efficacy, waning), structural uncertainty (which compartmental model?), or genuinely adversarial (which strain will emerge? where will adherence collapse?).
A useful frame for this is the game against Nature. The planner chooses an action; Nature then reveals the state of the world; the planner absorbs the resulting cost. If the planner is risk-neutral and holds a credible prior over Nature’s moves, she minimizes expected cost — the Bayes rule. If she is unwilling to commit to a prior, or if Nature’s move is genuinely adversarial, she minimizes the worst-case cost — the minimax rule. This post is about the minimax rule: where it comes from, how to compute it, when it is the right thing to do, and when it is not.
Minimax is the oldest result in modern game theory. Von Neumann proved the minimax theorem in 1928 (1), sixteen years before Theory of Games and Economic Behavior and twenty-two years before Nash. Its original setting was the two-player zero-sum game.
Let two players choose mixed strategies \(x \in \Delta_n\) and \(y \in \Delta_m\) over their respective pure strategies, and let \(A\) be an \(n \times m\) payoff matrix (Player 1 receives \(x^\top A y\); Player 2 receives \(-x^\top A y\)). Player 1’s maximin value is what she can guarantee against any opponent:
\[ \underline{v} \;=\; \max_x \min_y\; x^\top A y. \]
Player 2’s minimax value is what she can limit her opponent to:
\[ \overline{v} \;=\; \min_y \max_x\; x^\top A y. \]
It is always true that \(\underline{v} \le \overline{v}\). Von Neumann’s minimax theorem asserts that for finite zero-sum games played in mixed strategies, equality holds:
\[ \underline{v} \;=\; \overline{v} \;=\; v, \tag{1} \]
and the common value \(v\) is the value of the game. Pure-strategy equilibria can fail to exist — rock–paper–scissors has none — but mixed-strategy equilibria always do. Existence is proved via the separating-hyperplane theorem (equivalently, LP duality); it was the first equilibrium existence result in game theory and a precursor to Nash’s 1950 theorem for general \(n\)-player games (2).
The key link back to Part 1: in two-player zero-sum games, Nash equilibrium, minimax, and maximin all coincide. Outside this class — including the vaccination game, which is non-zero-sum and has externalities — they generally differ.
install.packages(c("dplyr", "tidyr", "ggplot2", "patchwork", "truncnorm"))Code tested with: R 4.4, ggplot2 3.5, dplyr 1.1.
Wald (3) generalised minimax from game theory to statistical decision theory. Consider a decision-maker choosing action \(a\) in a space \(\mathcal{A}\), facing an unknown state of nature \(\theta \in \Theta\), with loss \(L(a, \theta)\). Three canonical rules deserve their names:
| Rule | Formula | When it is the right thing |
|---|---|---|
| Bayes | \(a_B \;=\; \arg\min_a \int L(a, \theta)\, \pi(\theta)\, d\theta\) | We trust the prior \(\pi\) |
| Minimax | \(a_M \;=\; \arg\min_a \sup_{\theta \in \Theta} L(a, \theta)\) | We do not trust the prior; Nature is (or might as well be) adversarial |
| Minimax regret | \(a_R \;=\; \arg\min_a \sup_\theta \big\{ L(a, \theta) - \min_{a'} L(a', \theta) \big\}\) | We want to minimise opportunity loss rather than absolute loss |
Minimax is often criticised as pessimistic: it hedges against states of nature that may be vanishingly improbable. For an infectious-disease planner, this can mean hedging against \(R_0 = 15\) in a setting where credible estimates place \(R_0\) in \([2, 4]\). The remedy is not to abandon minimax but to restrict the uncertainty set \(\Theta\) — a device called \(\Gamma\)-minimax when \(\Gamma\) is a family of priors rather than a set of parameters (4). We return to this at the end.
Let us ground the abstraction in a problem close to our work at IVI: a cholera-prone setting where oral cholera vaccine (OCV) must be allocated before outbreak onset, under uncertainty about transmissibility. We will compute the minimax coverage \(\pi_{\text{mm}}\) and compare it against two references — a Bayes-optimal coverage \(\pi_{B}\) and the Nash coverage \(\pi^*\) from Part 1.
Fix an SIR model with a perfect pre-epidemic vaccine, coverage \(\pi\), and basic reproduction number \(R_0\). Among the unvaccinated, the attack fraction \(w\) satisfies the classical final-size relation
\[ w \;=\; 1 - \exp\!\big(-R_0\,(1 - \pi)\,w\big), \tag{2} \]
with \(w = 0\) whenever \(R_0(1 - \pi) \le 1\). The population-level attack rate is \(\text{AR}(\pi, R_0) = (1-\pi)\, w\). We take the planner’s per-capita loss as
\[ L(\pi, R_0) \;=\; c_v \cdot \pi \;+\; c_i \cdot \text{AR}(\pi, R_0), \tag{3} \]
where \(c_v\) is the per-dose vaccination cost and \(c_i\) is the per-case cost (DALYs, economic loss, …). The only uncertainty is over \(R_0\), which lies in \([R_{\min}, R_{\max}]\).
library(dplyr)
library(tidyr)
library(ggplot2)
library(patchwork)
library(truncnorm)
theme_set(
theme_minimal(base_size = 10) +
theme(
panel.grid.minor = element_blank(),
plot.title = element_text(face = "bold", size = 10),
legend.position = "bottom"
)
)
C_NASH <- "#F44336" # red
C_SOCIAL <- "#2196F3" # blue
C_CRIT <- "#4CAF50" # green
C_MINIMAX <- "#FF9800" # orange
C_GAP <- "#9C27B0" # purple# Attack rate from the SIR final-size relation (perfect pre-vaccination)
attack_rate <- function(pi, R0) {
Reff <- R0 * (1 - pi)
if (Reff <= 1) {
return(0)
}
w <- uniroot(
\(w) w - (1 - exp(-Reff * w)),
interval = c(1e-8, 1 - 1e-8)
)$root
(1 - pi) * w
}
# Planner's per-capita loss
loss <- function(pi, R0, cv = 0.10, ci = 1.0) {
cv * pi + ci * attack_rate(pi, R0)
}
# Worst-case loss over R0 in [R_lo, R_hi]
worst_case <- function(pi, R_lo, R_hi, cv, ci) {
R_grid <- seq(R_lo, R_hi, length.out = 60)
max(sapply(R_grid, \(R0) loss(pi, R0, cv, ci)))
}
# Minimax coverage: minimises the worst-case loss
minimax_pi <- function(R_lo, R_hi, cv = 0.10, ci = 1.0) {
optimize(\(pi) worst_case(pi, R_lo, R_hi, cv, ci),
interval = c(0, 1)
)$minimum
}
# Bayes coverage: minimises expected loss under a truncated-normal prior
bayes_pi <- function(R0_mean, R0_sd, R_lo = 1.1, R_hi = 20,
cv = 0.10, ci = 1.0, n = 500) {
R0_samples <- rtruncnorm(n,
a = R_lo, b = R_hi,
mean = R0_mean, sd = R0_sd
)
optimize(
\(pi) mean(sapply(R0_samples, \(R0) loss(pi, R0, cv, ci))),
interval = c(0, 1)
)$minimum
}
# Nash coverage from Part 1 (linear pi_inf)
nash_coverage <- function(R0, q, r) {
pi_c <- 1 - 1 / R0
if (r >= q) {
return(0)
}
uniroot(
\(pi) q * pmax(0, 1 - pi / pi_c) - r,
interval = c(0, pi_c)
)$root
}Take the uncertainty set \(R_0 \in [1.5, 6]\) — a range spanning published cholera estimates across endemic and outbreak settings. Set \(c_v = 0.10\) and \(c_i = 1\) (a single dose costs 10% of an averted case), and place a Bayesian prior centred at \(R_0 = 2.5\) with standard deviation \(0.8\).
R_lo <- 1.5
R_hi <- 6
cv <- 0.10
ci <- 1.0
pi_mm <- minimax_pi(R_lo, R_hi, cv, ci)
pi_B <- bayes_pi(
R0_mean = 2.5, R0_sd = 0.8,
R_lo = R_lo, R_hi = R_hi, cv = cv, ci = ci
)
pi_N <- nash_coverage(R0 = 2.5, q = 0.5, r = 0.10)
tibble(
strategy = c(
"Nash equilibrium (pi*, from Part 1)",
"Bayes-optimal (pi_B)",
"Minimax (pi_mm)",
"Herd-immunity threshold at R_max"
),
value = c(pi_N, pi_B, pi_mm, 1 - 1 / R_hi)
) |>
mutate(value = sprintf("%.3f", value)) |>
knitr::kable()| strategy | value |
|---|---|
| Nash equilibrium (pi*, from Part 1) | 0.480 |
| Bayes-optimal (pi_B) | 0.742 |
| Minimax (pi_mm) | 0.833 |
| Herd-immunity threshold at R_max | 0.833 |
The pattern is what the theory anticipates. Voluntary uptake (Nash) is lowest — individuals free-ride on the herd. The Bayes planner does better, pulling coverage upward to internalise the externality. The minimax planner does more, pushing coverage close to the herd-immunity threshold of the worst-case \(R_0\) in the uncertainty set.
Because \(L(\pi, R_0)\) is monotone increasing in \(R_0\) whenever \(\pi < 1 - 1/R_0\), the inner supremum in the minimax problem is attained at \(R_0 = R_{\max}\):
\[ \pi_{\text{mm}} \;=\; \arg\min_\pi\, L(\pi, R_{\max}). \tag{4} \]
The minimax planner acts as if \(R_0\) were pinned at its upper bound and ignores everything else. This is simultaneously minimax’s strength — it is robust to misspecification of the prior — and its weakness. It can demand expensive hedging against very unlikely scenarios.
grid <- expand_grid(
pi = seq(0, 0.95, length.out = 70),
R0 = seq(R_lo, R_hi, length.out = 70)
) |>
rowwise() |>
mutate(L = loss(pi, R0, cv, ci)) |>
ungroup()
ggplot(grid, aes(x = pi, y = R0, fill = L)) +
geom_raster() +
scale_fill_viridis_c(option = "magma", direction = -1, name = "Loss") +
geom_vline(xintercept = pi_N, colour = C_NASH, linewidth = 1.0, linetype = "dotted") +
geom_vline(xintercept = pi_B, colour = C_SOCIAL, linewidth = 1.0, linetype = "dashed") +
geom_vline(xintercept = pi_mm, colour = C_MINIMAX, linewidth = 1.1) +
annotate("text", x = pi_N, y = R_hi - 0.15, label = "Nash", hjust = -0.1, colour = C_NASH, size = 3.2) +
annotate("text", x = pi_B, y = R_hi - 0.45, label = "Bayes", hjust = -0.1, colour = C_SOCIAL, size = 3.2) +
annotate("text", x = pi_mm, y = R_hi - 0.75, label = "Minimax", hjust = -0.1, colour = C_MINIMAX, size = 3.2) +
labs(
x = expression("Vaccination coverage " * pi),
y = expression(R[0]),
title = "Three policies on one loss surface"
)
The most revealing comparison is over the width of the uncertainty set. A Bayesian planner tracks the mean of the prior; a minimax planner tracks its upper bound.
R0_centre <- 2.5
widths <- seq(0.2, 7, length.out = 30)
sweep <- tibble(
width = widths,
R_lo = pmax(1.1, R0_centre - widths / 2),
R_hi = R0_centre + widths / 2
) |>
rowwise() |>
mutate(
pi_minimax = minimax_pi(R_lo, R_hi, cv, ci),
pi_bayes = bayes_pi(
R0_centre, pmax(widths / 4, 0.05),
R_lo, R_hi, cv, ci
)
) |>
ungroup()
ggplot(sweep, aes(x = width)) +
geom_line(aes(y = pi_minimax, colour = "Minimax"), linewidth = 1.1) +
geom_line(aes(y = pi_bayes, colour = "Bayes"), linewidth = 1.1) +
scale_colour_manual(
values = c(Minimax = C_MINIMAX, Bayes = C_SOCIAL),
name = NULL
) +
labs(
x = expression(R[max] - R[min]),
y = "Optimal coverage",
title = "Minimax is increasingly conservative as uncertainty widens"
) +
coord_cartesian(ylim = c(0, 1))
When the uncertainty set is narrow the two policies agree. As it widens, the minimax policy pulls sharply upward, tracking the worst plausible \(R_0\), while the Bayes policy stays close to the prior mean. Which is appropriate depends on how much trust the prior deserves.
It is tempting to describe host-pathogen interactions as minimax games: the pathogen “wants” to maximise spread; the host “wants” to minimise it. Some of the formal machinery transfers usefully — ESS analysis uses fixed-point arguments that echo minimax — but the analogy needs two caveats.
(1) Pathogens do not optimise. Evolution by natural selection produces strategies that are evolutionarily stable given the current environment, but it does not anticipate or counter-select. A strict minimax frame in which the pathogen chooses the worst case for the host typically over-states what evolution can do.
(2) The game is rarely zero-sum. Virulence–transmissibility trade-offs (5,6) show that a more virulent strain can be bad for both pathogen persistence and host welfare — the pathogen’s payoff is not a simple negation of the host’s.
That said, two minimax-like frames are genuinely useful for pathogen-evolution problems:
A fuller treatment of evolutionary games awaits Post 4.
Minimax has made a second career outside game theory, and three instances are worth knowing if you build models at the interface of epidemiology and machine learning.
Distributionally robust optimisation (DRO). Rather than minimise expected loss under a fixed data distribution, one minimises expected loss under the worst-case distribution within a divergence ball around the empirical distribution (7,8):
\[ \hat\theta \;=\; \arg\min_\theta\; \sup_{Q\,:\, D(Q\,\|\,\hat P)\, \le\, \rho}\; \mathbb{E}_Q\big[L(\theta, X)\big]. \tag{5} \]
For fitting disease models to outbreak data, DRO is a principled guard against covariate shift between the training setting and the deployment setting.
Adversarial training. A neural-network classifier is trained against worst-case small perturbations of its inputs (9):
\[ \min_\theta\, \mathbb{E}_{(x,y)}\!\Big[\, \max_{\|\delta\| \le \epsilon}\, L\big(f_\theta(x + \delta),\, y\big)\,\Big]. \tag{6} \]
This is a standard ingredient when building diagnostic models that must be robust to slight image-acquisition differences across sites.
Minimax-weighted PINN training. A physics-informed neural network fits data loss and PDE-residual loss jointly. Several recent papers recast the balancing of the two loss terms as a minimax problem — the network minimising while a dual variable maximises across collocation points — and find that this stabilises training and improves out-of-distribution generalisation (10). The frame is of direct interest for PINN-based epidemic reconstruction.
The common thread is that minimax is what we reach for when we want guarantees in the presence of something we cannot control or cannot model — an unobserved parameter, a distributional shift, a crafted perturbation, or (in game theory proper) a strategic opponent.
Pure minimax has three well-known pathologies, each with a standard remedy.
| Pathology | Remedy |
|---|---|
| Hedges against vanishingly improbable states | \(\Gamma\)-minimax: optimise over the worst prior in a restricted family \(\Gamma\) |
| Ignores the margin by which other actions are worse in good states | Minimax regret (11) |
| Changes sharply when the uncertainty set is expanded | Smooth ambiguity (12) |
For OCV allocation in particular, minimax regret is often the more defensible frame: how much worse could I be, relative to the best policy in hindsight? It produces less cautious — and more communicable — recommendations than pure minimax.
| Question | Answer |
|---|---|
| Does minimax coincide with Nash? | Only in 2-player zero-sum games. |
| Does minimax depend on a prior? | No — that is its virtue, and its cost. |
| Is the minimax planner “rational”? | Rational in the Wald sense; often over-cautious in the Bayesian sense. |
| When should an epidemiologist reach for it? | When the uncertainty set is plausibly broad, the consequences of underestimating it are severe, and no credible prior is available. |
| When should an epidemiologist avoid it? | When expert priors are well-calibrated and decision-relevant. |
| What is the operational short-cut? | For monotone losses, the minimax planner optimises at the worst-case parameter. |
R 4.5.3ggplot2 4.0.2
dplyr 1.2.0
tidyr 1.3.2
patchwork 1.3.2
truncnorm 1.0.9