3 Main Results
In this section, we solve for the two versions of robust Bayes optimality under the set of priors Equation 9. Following Yata (2021), we assume:
- \(\Theta\) is convex, centrosymmetric (i.e., \(\theta\in\Theta\) implies \(-\theta\in\Theta\)) and nonempty.
- \(m(\cdot)\) and \(U(\cdot)\) are linear.
These conditions are restrictive but encompass many examples of empirical relevance; see this paper’s introduction, Montiel Olea, Qiu, and Stoye (2025), and Yata (2021). Exploiting symmetry of the setting, we also set
\[ \overline{I}(\bar{\mu})+\underline{I}(\bar{\mu})>0. \]
This is a normalization because, by Lemma Lemma 9, Assumption 1 implies \(\overline{I}(-\bar{\mu})+\underline{I}(-\bar{\mu})=-(\underline{I}(\bar{\mu})+\overline{I}(\bar{\mu}))\) and we could always replace \(m(\cdot)\mapsto-m(\cdot)\); furthermore, the case of \(\overline{I}(\bar{\mu})+\underline{I}(\bar{\mu})=0\) gives rise to trivial solutions.8
We say a rule is a linear-index threshold rule if it has the form \(\mathbf{1}\left\{ \beta^{\top}Y\geq c\right\}\) for some \(\beta\in\mathbb{R}^{n}\) and \(c\in\mathbb{R}\). Linear-index threshold rules are nonrandomized. They are of particular interest because they form a complete class when \(U(\theta)\) is point-identified Karlin and Rubin (1956) and have received particular attention in the recent literature Ishihara and Kitagawa (2021); Montiel Olea, Qiu, and Stoye (2025). For both \(\Gamma\)-MMR and \(\Gamma\)-PER, we will clarify when linear-index threshold rules are optimal and when they are not. For reasons that will become obvious, the following linear-index threshold rule is of particular interest:
\[ d_{w,0}^{*}:=d_{w,0}^{*}(Y):=\mathbf{1}\left\{ w^{\top}Y\geq0\right\},\text{ }w:=\Sigma^{-1}\bar{\mu}. \]
For each vector \(\beta\in\mathbb{R}^n\), let \(\left\Vert \beta\right\Vert_{\Sigma}:=\sqrt{\beta^{\top}\Sigma\beta}\). Thus, \(\lVert w \rVert_{\Sigma} =\sqrt{w^{\top}\Sigma w}=\sqrt{\bar{\mu}^{\top}\Sigma^{-1}\bar{\mu}}\). Denote by \(\Phi(\cdot)\) the standard normal c.d.f. and by \(\Phi^{-1}(\cdot)\) its inverse, i.e., the corresponding quantile function.
3.1 Ex-ante Robust Bayes Optimality
Theorem 1 (\(\Gamma\)-MMR optimal decisions) Consider a treatment choice problem with welfare function Equation 1, statistical model Equation 3, and set of priors Equation 9, that satisfy Assumption 1.
- If
\[ \frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}\geq\Phi(\lVert w \rVert_{\Sigma} ), \tag{11}\]
then \(d_{w,0}^{*}\) is uniquely \(\Gamma\)-MMR optimal.
- If
\[ \frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}<\Phi(\lVert w \rVert_{\Sigma} ), \tag{12}\]
then a rule \(d\in\mathcal{D}_{n}\) attains \(\Gamma\)-MMR if, and only if, it implies:
\[ \mathbb{E}_{-\bar{\mu}}[d^{*}(Y)] = \frac{-\underline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}, \tag{13}\]
\[ \mathbb{E}_{\bar{\mu}}[d^{*}(Y)] = \frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}. \tag{14}\]
In particular, any rule of the form \(\mathbf{1}\left\{ w^{\top}Y\geq c\right\}\) for some \(c\in\mathbb{R}\) is not \(\Gamma\)-MMR optimal. The following rules, and any convex combination of them, are \(\Gamma\)-MMR optimal:
\[ d^*_\text{RT}:=\Phi\left(\frac{w^{\top}Y}{\tilde{\sigma}}\right), \]
\[ d^*_\text{linear}:= \begin{cases} 0, & w^{\top}Y<-\rho^{*},\\ \frac{w^{\top}Y+\rho^{*}}{2\rho^{*}}, & -\rho^{*}\leq w^{\top}Y\leq\rho^{*},\\ 1, & w^{\top}Y>\rho^{*}, \end{cases} \]
\[ d^*_\text{step}:= \begin{cases} \frac{1}{2} - \beta^*, & w^{\top}Y<0,\\ \frac{1}{2} + \beta^*, & w^{\top}Y\geq0, \end{cases} \]
where
\[ \tilde{\sigma}=\sqrt{\left[\frac{\lVert w \rVert_{\Sigma} ^{2}}{\Phi^{-1}\left(\frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}\right)}\right]^{2}-\lVert w \rVert_{\Sigma} ^{2}}, \]
\(\rho^{*}>0\) is unique and solves:
\[ \int_{0}^{1}\Phi\left(\frac{2\rho^{*}x-\rho^{*}-\lVert w \rVert_{\Sigma} ^{2}}{\lVert w \rVert_{\Sigma} }\right)dx=\frac{-\underline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}, \]
and
\[ \beta^{*}=\frac{\frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}-\frac{1}{2}}{2\Phi(\lVert w \rVert_{\Sigma} )-1}\in\left(0,\frac{1}{2}\right). \]
- In case Equation 12, among linear-index threshold rules of the form \(\mathbf{1}\left\{ w^{\top}Y\geq c\right\}\), the optimal thresholds are \(\pm c^{*}\), where
\[ c^{*}:=\lVert w \rVert_{\Sigma} ^2-\lVert w \rVert_{\Sigma} \Phi^{-1}\left(\frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}\right)>0. \]
- In case Equation 12 and if \(n>1\), the following linear-index threshold rule is also \(\Gamma\)-MMR optimal:
\[ d_{w_{t^*},0}^{*}(Y):=\mathbf{1}\left\{ w_{t^*}^{\top}Y\geq0\right\}, \]
where \(w_{t^*}=\Sigma^{-1}\left(t^*\bar{\mu}+(1-t^*)\dot{\mu}\right)\), \(\dot{\mu}\neq0\) is such that \(\dot{\mu}^{\top}\Sigma^{-1}\bar{\mu}=0\),
\[ t^*:=\frac{1}{1\pm\sqrt{\frac{(1-s^{*})}{s^{*}}\frac{\lVert w \rVert_{\Sigma} ^{2}}{\lVert \Sigma^{-1} \dot{\mu}\rVert_{\Sigma} ^{2}}}}, \tag{15}\]
and
\[ s^{*}:=\frac{\left[\Phi^{-1}\left(\frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}\right)\right]^{2}}{\lVert w \rVert_{\Sigma} ^{2}}\in(0,1). \]
Thus, \(d_{w_{t^*},0}^{*}\) is also optimal among all linear threshold rules.
Theorem 1 reveals that the \(\Gamma\)-MMR rules qualitatively change depending on whether condition Equation 11 is met or not. This condition admits an intuitive interpretation: Up to clamping to the unit interval (i.e., values outside this interval are mapped to its edges), the left-hand side of Equation 11 equals the unique minimax regret optimal rule for known \(\mu\) Manski (2007)b; therefore, it arguably measures the model’s identification strength. The right-hand side of Equation 11 can be interpreted as the informativeness of the data about which of \(\{-\bar{\mu},\bar{\mu}\}\) obtained. Therefore, Theorem 1 says that if the model’s identification power is sufficiently large compared to the informativeness of the data, then, the unique \(\Gamma\)-MMR optimal rule is \(d_{w,0}^{*}\), a non-randomized linear index rule with threshold \(0\) that effectively ignores the partial identification issue. In contrast, if the model’s identification power is small compared to the informativeness of data, there are infinitely many \(\Gamma\)-MMR optimal rules, all of which satisfy Equation 13 and Equation 14. Examples include suitably smoothed versions of \(d_{w,0}^{*}\) like \(d_{\text{RT}}^{*}\) and \(d_{\text{linear}}^{*}\) (the functional forms of which showed up in Montiel Olea, Qiu, and Stoye (2025)) as well as \(d_{\text{step}}^{*}\) (which is a new result).
To understand how these two “regimes” arise, it is helpful to think of a zero-sum game against an adversarial Nature in which a MMR decision rule and a distribution over parameter values (the least favorable prior) form a Nash equilibrium.9 If identification is strong in the sense of Equation 11, this game has an informative equilibrium: the least favorable prior evenly randomizes over two points \((\mu,U(\theta)) \in \{(\bar{\mu},\overline{I}(\bar{\mu})),(-\bar{\mu},\underline{I}(-\bar{\mu}))\}\), to which the unique Bayes response (and, therefore, uniquely optimal MMR rule) is \(d_{w,0}^*\). In all other cases, the equilibrium is uninformative: the least favorable prior is supported on four points \((\mu, U(\theta))\in\{(\bar{\mu},\overline{I}(\bar{\mu})),(\bar{\mu},\underline{I}(\bar{\mu})),(-\bar{\mu},\overline{I}(-\bar{\mu})),(-\bar{\mu},\underline{I}(-\bar{\mu}))\}\) with a probability profile such that the posterior expectation of \(U(\theta)\) always equals \(0\). While any decision rule best responds to this prior, not every decision rule is MMR because the least favorable prior must also best respond to the decision rule. In this very structured setting, the latter is guaranteed by conditions Equation 13 and Equation 14, establishing claim (ii) and nonuniqueness of optimal decision rules. Indeed, even the nonrandomized threshold rules from Theorem 1(iv) do not reflect any updating along the game’s equilibrium path; they rather use uninformative features of the data as a randomization device.
In the uninformative equilibrium, we encounter several additional findings. First, despite the problem’s apparent symmetry, \(d_{w,0}^*\) is not optimal even among linear-index threshold rules that use the index \(w^{\top}Y\). Instead, Theorem 1(iii) characterizes exactly two optimal thresholds, one positive and one negative. Second, while the particular linear-index threshold rule \(d_{w,0}^*\) is not \(\Gamma\)-MMR optimal, in higher dimensional problems (\(n>1\)), there do exist linear-index threshold rules that are. This finding is in stark contrast to Montiel Olea, Qiu, and Stoye (2025), who find that, in a large class of special cases, no linear index threshold rule is MMR optimal. The crucial difference in settings is that the set of priors much constrains the decision theoretic problem’s state space; as a result, the signal space is much richer than the state space, and this can be exploited to mimic randomization without nominally randomizing. Compare Manski (2024)’s abstract observation that, if a policy maker is not allowed to explicitly randomize, sampling uncertainty can be beneficial by providing an implicit randomization device. We note that this phenomenon is reminiscent of classic “purification” results in game theory Dvoretzky, Wald, and Wolfowitz (1951); Khan, Rath, and Sun (2006).10 In contrast, it is not deeply related to the general intuition that “Bayesians don’t randomize.”
We next apply Theorem 1 to Example 1 and immediately get the following results.
Corollary 1 (\(\Gamma\)-MMR decisions in Example 1) In Example 1, the following statements are true:
- If
\[ \frac{\bar{\mu}+k}{2k}\geq\Phi\left(\bar{\mu}/\sigma\right), \tag{16}\]
then
\[ d_0^*(\cdot):=\mathbf{1}\{\hat{\mu}\geq0\} \]
is the unique \(\Gamma\)-MMR optimal decision rule.
- If Equation 16 fails, then a rule \(d \in \mathcal{D}_1\) attains \(\Gamma\)-MMR if, and only if, it implies
\[ \begin{aligned} \mathbb{E}[d(\hat{\mu}) \mid \mu=-\bar{\mu}] &= \frac{-\bar{\mu}+k}{2k}, \\ \mathbb{E}[d(\hat{\mu}) \mid \mu=\bar{\mu}] &= \frac{\bar{\mu}+k}{2k}. \end{aligned} \]
In particular, no linear threshold rule is \(\Gamma\)-MMR optimal. The following rules, and any convex combination of them, are all \(\Gamma\)-MMR optimal:
\[ \begin{aligned} d^*_\text{RT}&:=\Phi\left(\frac{\hat{\mu}}{\tilde{\sigma}}\right),\\ d^*_\text{linear}&:=\begin{cases} 0, &\hat{\mu}<-\frac{\sigma^{2}\rho^{*}}{\bar{\mu}},\\ \frac{\bar{\mu}\hat{\mu}+\sigma^{2}\rho^{*}}{2\sigma^{2}\rho^{*}}, & -\frac{\sigma^{2}\rho^{*}}{\bar{\mu}}\leq \hat{\mu}\leq\frac{\sigma^{2}\rho^{*}}{\bar{\mu}},\\ 1, & \hat{\mu}>\frac{\sigma^{2}\rho^{*}}{\bar{\mu}}, \end{cases}\\ d^*_\text{step}&:=\begin{cases} \frac{1}{2} - \frac{\bar{\mu}}{2k\left(2\Phi\left(\frac{\bar{\mu}}{\sigma}\right)-1\right)}, & \hat{\mu}<0,\\ \frac{1}{2} + \frac{\bar{\mu}}{2k\left(2\Phi\left(\frac{\bar{\mu}}{\sigma}\right)-1\right)}, & \hat{\mu}\geq0,\\ \end{cases} \end{aligned} \]
where
\[ \tilde{\sigma}=\sigma\sqrt{\left[\frac{\bar{\mu}}{\sigma \Phi^{-1}\left(\frac{\bar{\mu}+k}{2k}\right)}\right]^{2}-1 }, \]
and \(\rho^{*}>0\) is unique and solves \(\int_{0}^{1}\Phi\left(\frac{2\rho^{*}x-\rho^{*}-(\frac{\bar{\mu}}{\sigma}) ^{2}}{\frac{\bar{\mu}}{\sigma} }\right)dx=\frac{-\bar{\mu}+k}{2k}\).
- In case (ii), the best linear threshold rules in terms of \(\Gamma\)-minimax regret are \(\mathbf{1}\{\hat{\mu}\geq \pm c^*\}\), where
\[ c^* = \bar{\mu} - \sigma \Phi^{-1}\left(\frac{\bar{\mu}+k}{2k}\right). \]
Code
# Load necessary library
library(ggplot2)
library(latex2exp)
# Run from research/two-point-prior/
# Global parameters
alpha <- 0.7 # Transparency value for plotting
A_col <- "#0A567D"
B_col <- "#963C3C"
C_col <- "#2D6D66"
D_col <- "#6B5957"
E_col <- "#1C84D1"
F_col <- "#A2B1B9"
G_col <- "#FDFBF7"
# Parameters
params <- list(
list(k = 1, sigma = 1, overline.mu = 1.5),
list(k = 1, sigma = 1, overline.mu = 0.5),
list(k = 1, sigma = 0.5, overline.mu = 0.5),
list(k = 2, sigma = 1, overline.mu = 0.5)
)
# Function to check whether we are in the "small" or large regime
check_condition <- function(overline.mu, sigma, k) {
return((overline.mu + k) / (2 * k) >= pnorm(overline.mu / sigma))
}
# Decision rules
d_0 <- function(u) {
as.numeric(u >= 0)
}
d_RT <- function(u, overline.mu, sigma, k) {
sigma.tilda <- sigma* sqrt((overline.mu / (sigma * qnorm((overline.mu + k) / (2 * k))))^2 - 1)
return(pnorm(u / sigma.tilda))
}
rho.finder <- function(rho, overline.mu, sigma, k) {
w_norm <- overline.mu / sigma
A <- (rho - w_norm^2) / w_norm
B <- (-rho - w_norm^2) / w_norm
integration.output <- (w_norm / (2 * rho)) * (pnorm(A) * A - pnorm(B) * B + (1 / sqrt(2 * pi)) * (exp(-A^2 / 2) - exp(-B^2 / 2)))
return(integration.output - ((-overline.mu + k) / (2 * k)))
}
d_linear <- function(u, rho.star,overline.mu,sigma) {
pmax(0, pmin(1, (u*overline.mu + rho.star*sigma*sigma) / (2 * rho.star*sigma*sigma)))
}
d_step <- function(u, overline.mu, sigma, k) {
ifelse(u < 0,
0.5 - overline.mu / (2 * k * (2 * pnorm(overline.mu / sigma) - 1)),
0.5 + overline.mu / (2 * k * (2 * pnorm(overline.mu / sigma) - 1)))
}
# Generate grid for plotting
u_grid <- seq(-3, 3, 0.01)
# Function to create plots
create_plot <- function(k, sigma, overline.mu) {
if (check_condition(overline.mu, sigma, k)) {
d_0_grid <- sapply(u_grid, d_0)
plot(u_grid, d_0_grid, type = "l", lty = 2, lwd = 5.5, col = A_col,
xlab = "", ylab = "d(.)", ylim = c(-0.002, 1.002), yaxt = "n", cex.axis = 1.25, yaxs="i")
} else {
rho.star <- uniroot(rho.finder, c(0.01, 10), overline.mu = overline.mu, sigma = sigma, k = k)$root
d_RT_grid <- sapply(u_grid, d_RT, overline.mu = overline.mu, sigma = sigma, k = k)
d_linear_grid <- sapply(u_grid, d_linear, rho.star = rho.star,overline.mu = overline.mu, sigma = sigma)
d_step_grid <- sapply(u_grid, d_step, overline.mu = overline.mu, sigma = sigma, k = k)
plot(u_grid, d_RT_grid, type = "l", lty = 1, lwd = 5.5, col = B_col,
xlab = "", ylab = "d(.)", ylim = c(-0.002, 1.002), yaxt = "n", cex.axis = 1.25, yaxs="i")
lines(u_grid, d_linear_grid, lty = 2, lwd = 5.5, col = C_col, yaxs="i")
lines(u_grid, d_step_grid, lty = 4, lwd = 5.5, col = D_col, yaxs="i")
}
axis(1, cex.axis = 1.25) # Customize the x-axis ticks
axis(2, at = seq(0, 1, 0.1), las = 2, cex.axis = 1.25) # Customize the y-axis ticks
title(main = "")
mtext(side = 1, line = 3.5, at = mean(par("usr")[1:2]), text = expression(hat(mu)), cex = 1.5)
mtext(side = 3, line = 0.5, at = mean(par("usr")[1:2]),
text = bquote(paste("k = ", .(k), ", ", sigma, " = ", .(sigma), ", ", bar(mu), " = ", .(overline.mu))),
cex = 1.5)
# Add light grid lines
abline(v = seq(-3, 3, by = 1), col = E_col, lty = 3)
abline(h = seq(0, 1, by = 0.1), col = E_col, lty = 3)
}
# Set up plotting parameters
svg("figures/fig1.svg", width = 12, height = 13.5)
par(mfrow = c(2, 2), oma = c(8, 2, 2, 2), mar = c(5, 4, 4, 2) + 0.1, bg=G_col)
# Create plots
mapply(function(p) create_plot(p$k, p$sigma, p$overline.mu),
params)
# Add legend with increased text size and adjusted inset
par(fig = c(0, 1, 0.02, 0.12), oma = c(0, 0, 0, 0), mar = c(0, 0, 0, 0), new = TRUE)
plot(0, 0, type = "n", bty = "n", xaxt = "n", yaxt = "n")
legend("bottom",
legend = TeX(c(
'$d_{0}^{*}$',
'$d^*_{RT}$',
'$d^*_{linear}$',
'$d^*_{step}$'
)),
col = c(A_col, B_col, C_col, D_col),
lty = c(2, 1, 2, 4), lwd = 6, ncol = 4, bty = "n", xpd = TRUE,
title = "", cex = 2.25, title.cex = 0.5) # Adjust text.width to position lines above text
dev.off()
Notes: This figure reports the \(\Gamma\)-MMR optimal rules in Example 1 for various combinations of parameter values. In the top two panels, the combinations of parameter values satisfy Equation 16. Therefore, the unique \(\Gamma\)-MMR optimal rule is \(d^*_0\). In the bottom two panels, Equation 16 fails. As a result, \(d^*_0\) is no longer \(\Gamma\)-MMR optimal. Instead, \(d^*_{\text{RT}}\), \(d^*_{\text{linear}}\) and \(d^*_{\text{step}}\) are all \(\Gamma\)-MMR optimal.
Note that there is no analog to Theorem 1’s case (iv); indeed, no linear threshold rule is optimal in case (ii). This is because the scalar nature of the signal \(Y\) shuts down the aforementioned purification mechanism. See Figure 1 for an illustration of different \(\Gamma\)-MMR optimal rules in Example 1 for selected parameter values of \(k,\sigma\) and \(\bar{\mu}\).
3.2 Ex-post Robust Bayes Optimality
Theorem 2 (\(\Gamma\)-PER optimal decisions) Suppose all conditions in Theorem 1 hold true. Then:
- If \(\underline{I}(\bar{\mu})<0<\overline{I}(\bar{\mu})\), the unique \(\Gamma\)-PER optimal rule is
\[ d_{\operatorname{PER}}^{*}(Y)=\begin{cases} \frac{-\underline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}, & \text{if }w^{\top}Y<0, \\ \frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}, & \text{if }w^{\top}Y\geq0. \end{cases} \]
Otherwise, \(d_{w,0}^{*}\) is \(\Gamma\)-PER optimal.
- \(d_{w,0}^{*}\) is always the \(\Gamma\)-PER optimal non-randomized rule.
The results of Theorem 2 offer some important clarifications regarding \(\Gamma\)-PER in treatment choice problems with partial identification. First, even if regret is evaluated according to the posterior distribution, it is not necessarily true that optimal rules are non-randomized. In fact, whenever there is model ambiguity regarding the sign of \(U(\theta)\) (i.e., \(\underline{I}(\bar{\mu})<0<\overline{I}(\bar{\mu})\)), the unique \(\Gamma\)-PER optimal rule is randomized. Therefore, restricting the action space to \(\{0,1\}\) in such a problem is not without loss of generality even under the \(\Gamma\)-PER criterion. Comparing results with Theorem 1 also allows for instructive observations on when \(\Gamma\)-PER and \(\Gamma\)-MMR optimal rules agree or disagree; we elaborate these in Corollary 3. Applying Theorem 2 to Example 1, we finally obtain:
Corollary 2 (\(\Gamma\)-PER rules in Example 1) In Example 1, the following statements are true:
- If \(\bar{\mu}<k\), then \(\Gamma\)-PER is uniquely minimized by
\[ d^*_{\operatorname{PER}}(\hat{\mu}) = \begin{cases} \frac{k + \bar{\mu}}{2k} & \text{if } \hat{\mu} \geq 0\\ \frac{k-\bar{\mu}}{2k} & \text{if } \hat{\mu} < 0 \end{cases} \]
- If \(\bar{\mu}\geq k\), then \(d_0^*\) is \(\Gamma\)-PER optimal. Furthermore, it is always the \(\Gamma\)-PER optimal threshold rule.
Code
# Load necessary library
library(ggplot2)
library(latex2exp)
# Run from research/two-point-prior/
# Global parameters
alpha <- 0.7 # Transparency value for plotting
A_col <- "#0A567D"
B_col <- "#963C3C"
C_col <- "#2D6D66"
D_col <- "#6B5957"
E_col <- "#1C84D1"
F_col <- "#A2B1B9"
G_col <- "#FDFBF7"
# Parameters
params <- list(
list(k = 1, sigma = 1, overline.mu = 1.5),
list(k = 1, sigma = 1, overline.mu = 0.5),
list(k = 1, sigma = 0.5, overline.mu = 0.5),
list(k = 2, sigma = 1, overline.mu = 0.5)
)
# Function to check whether we are in the "small" or large regime
check_condition <- function(overline.mu, sigma, k) {
return(overline.mu >= k)
}
# Decision rules
d_0 <- function(u) {
as.numeric(u >= 0)
}
d_PER <- function(u, overline.mu, sigma, k){
ifelse(u>=0,
(k+overline.mu)/(2*k),
(k-overline.mu)/(2*k))
}
d_step <- function(u, overline.mu, sigma, k) {
ifelse(u < 0,
0.5 - overline.mu / (2 * k * (2 * pnorm(overline.mu / sigma) - 1)),
0.5 + overline.mu / (2 * k * (2 * pnorm(overline.mu / sigma) - 1)))
}
# Generate grid for plotting
u_grid <- seq(-3, 3, 0.01)
# Function to create plots
create_plot <- function(k, sigma, overline.mu) {
if (check_condition(overline.mu, sigma, k)) {
d_0_grid <- sapply(u_grid, d_0)
plot(u_grid, d_0_grid, type = "l", lty = 2, lwd = 5.5, col = A_col,
xlab = "", ylab = "d(.)", ylim = c(-0.002, 1.002), yaxt = "n", cex.axis = 1.25, yaxs="i")
}
else {
d_PER_grid <- sapply(u_grid, d_PER, overline.mu = overline.mu, sigma = sigma, k = k)
d_step_grid <- sapply(u_grid, d_step, overline.mu = overline.mu, sigma = sigma, k = k)
d_0_grid <- sapply(u_grid, d_0)
plot(u_grid, d_PER_grid, type = "l", lty = 1, lwd = 5.5, col = B_col,
xlab = "", ylab = "d(.)", ylim = c(-0.002, 1.002), yaxt = "n", cex.axis = 1.25, yaxs="i")
if (k>sigma) {
lines(u_grid, d_step_grid, lty = 4, lwd = 5.5, col = D_col, yaxs="i")
}
else {
lines(u_grid, d_0_grid, lty = 2, lwd = 5.5, col = A_col, yaxs="i")
}
}
axis(1, cex.axis = 1.25) # Customize the x-axis ticks
axis(2, at = seq(0, 1, 0.1), las = 2, cex.axis = 1.25) # Customize the y-axis ticks
title(main = "")
mtext(side = 1, line = 3.5, at = mean(par("usr")[1:2]), text = expression(hat(mu)), cex = 1.5)
mtext(side = 3, line = 0.5, at = mean(par("usr")[1:2]),
text = bquote(paste("k = ", .(k), ", ", sigma, " = ", .(sigma), ", ", bar(mu), " = ", .(overline.mu))),
cex = 1.5)
# Add light grid lines
abline(v = seq(-3, 3, by = 1), col = E_col, lty = 3)
abline(h = seq(0, 1, by = 0.1), col = E_col, lty = 3)
}
# Set up plotting parameters
svg("figures/fig2.svg", width = 12, height = 13.5)
par(mfrow = c(2, 2), oma = c(8, 2, 2, 2), mar = c(5, 4, 4, 2) + 0.1, bg=G_col)
# Create plots
mapply(function(p) create_plot(p$k, p$sigma, p$overline.mu),
params)
# Add legend with increased text size and adjusted inset
par(fig = c(0, 1, 0.02, 0.12), oma = c(0, 0, 0, 0), mar = c(0, 0, 0, 0), new = TRUE)
plot(0, 0, type = "n", bty = "n", xaxt = "n", yaxt = "n")
legend("bottom",
legend = TeX(c(
'$d_{0}^{*}$',
'$d^*_{PER}$',
'$d^*_{step}$'
)),
col = c(A_col, B_col, D_col),
lty = c(2, 1, 4), lwd = 6, ncol = 3, bty = "n", xpd = TRUE,
title = "", cex = 2.25, title.cex = 0.5) # Adjust text.width to position lines above text
dev.off()
Notes: This figure reports the \(\Gamma\)-PER optimal rules for the same parameter values considered in Figure 1. In the top left panel, \(\Gamma\)-PER and \(\Gamma\)-MMR optimal rules coincide and are both \(d^*_{0}\). For the rest of the panels, the unique \(\Gamma\)-PER optimal rules are all \(d^*_{\text{PER}}\) and are different from any \(\Gamma\)-MMR optimal rules.
In Figure 2, we depict \(\Gamma\)-PER optimal rules for Example 1 with the same parameter values considered in Figure 1. We see clearly that \(\Gamma\)-MMR and -PER optimal rules coincide (if randomization is allowed) only in the special case when \(k\leq\bar{\mu}\), an observation we generalize in Corollary 3. More specifically, in the top left panel of Figure 2, as \(k\leq\bar{\mu}\), \(\Gamma\)-MMR and -PER coincide and are the non-randomized threshold rule \(d^*_{0}\). In the top right panel, \(k>\bar{\mu}\) and the \(\Gamma\)-PER optimal rule becomes \(d^*_{\text{PER}}\). However, since Equation 16 still holds, \(d^*_0\) is still \(\Gamma\)-MMR optimal. For the bottom two panels, as it still holds \(k>\bar{\mu}\), \(d^*_{\text{PER}}\) is still \(\Gamma\)-PER optimal. However, the associated parameter values imply Equation 16 fails. As a result, \(d^*_{0}\) is no longer \(\Gamma\)-MMR optimal and many \(\Gamma\)-MMR rules exist. But even in these cases, \(\Gamma\)-MMR and -PER rules differ, as among the class of step function rules (which contain \(d^*_{\text{PER}}\)), only \(d^*_{\text{step}}\) is \(\Gamma\)-MMR optimal, still different from \(d^*_{\text{PER}}\).
Code
# Profiled Regret ---------------------------------------------------------
# Packages
rm(list = ls())
library(ggplot2)
library(MASS)
library(latex2exp)
library(foreach)
library(doParallel)
# Run from research/two-point-prior/
alpha <- 0.7 # Transparency value for plotting
A_col <- "#0A567D"
B_col <- "#963C3C"
C_col <- "#2D6D66"
D_col <- "#6B5957"
E_col <- "#1C84D1"
F_col <- "#A2B1B9"
G_col <- "#FDFBF7"
# Values taken from Pepe's code
sigma <- 3.9
LC <- 2.5
x1_x0 <- abs(-7.459)
caux <- (LC * x1_x0) / (sigma * sqrt(pi / 2)) # Controls the gap between "C" and "K"
C <- sigma * sqrt(pi / 2)
k <- round(caux * C, 1)
bar.mu_list <- c(round(0.25 * k, 1), round(0.75 * k, 1))
# Function for profiled regret, special cases
profiled_regret_special_case <- function(k, gamma, sigma, E_d) {
if (gamma < -k) {
profiled_regret <- (-gamma + k) * E_d
} else if (gamma >= -k & gamma <= k) {
profiled_regret <- max((-gamma + k) * E_d, (gamma + k) * (1 - E_d))
} else {
profiled_regret <- (gamma + k) * (1 - E_d)
}
return(profiled_regret)
}
# Functions to compute expectations
expectation_d_linear <- function(k, gamma, sigma) {
rho_star <- rho_star_fun(sigma, k)
E_d <- pnorm((gamma - rho_star) / sigma) +
(sigma / (2 * rho_star)) * (1 / sqrt(2 * pi)) * (
exp((-1 / 2) * (((rho_star + gamma) / sigma) ^ 2)) -
exp((-1 / 2) * (((rho_star - gamma) / sigma) ^ 2))
) +
(gamma + rho_star) * (1 / (2 * rho_star)) * (
pnorm((rho_star - gamma) / sigma) -
pnorm((-rho_star - gamma) / sigma)
)
return(E_d)
}
rho_star_fun <- function(sigma, k) {
rho_star <- uniroot(
function(x) 1 - (k / x) * (1 - 2 * pnorm(-x / sigma)),
lower = 0.001,
upper = k
)$root
return(rho_star)
}
expectation_d_PER <- function(k, gamma, sigma, bar.mu) {
E_d <- 0.5 + (bar.mu / (2 * k)) * (1 - 2 * pnorm(-gamma / sigma))
return(E_d)
}
expectation_d_Gamma_MMR_linear <- function(k, gamma, sigma, bar.mu) {
rho_star <- uniroot(rho.finder_Gamma_MMR, c(0.01, 2 * k), bar.mu = bar.mu, sigma = sigma, k = k)$root
gamma_1 <- bar.mu * gamma / (sigma ^ 2)
sigma_1 <- bar.mu / sigma
E_d <- pnorm((gamma_1 - rho_star) / sigma_1) +
(sigma_1 / (2 * rho_star)) * (1 / sqrt(2 * pi)) * (
exp((-1 / 2) * (((rho_star + gamma_1) / sigma_1) ^ 2)) -
exp((-1 / 2) * (((rho_star - gamma_1) / sigma_1) ^ 2))
) +
(gamma_1 + rho_star) * (1 / (2 * rho_star)) * (
pnorm((rho_star - gamma_1) / sigma_1) -
pnorm((-rho_star - gamma_1) / sigma_1)
)
return(E_d)
}
rho.finder_Gamma_MMR <- function(rho, bar.mu, sigma, k) {
w_norm <- bar.mu / sigma
A <- (rho - w_norm ^ 2) / w_norm
B <- (-rho - w_norm ^ 2) / w_norm
integration.output <- (w_norm / (2 * rho)) * (pnorm(A) * A - pnorm(B) * B + (1 / sqrt(2 * pi)) * (exp(-A ^ 2 / 2) - exp(-B ^ 2 / 2)))
return(integration.output - ((-bar.mu + k) / (2 * k)))
}
# Function to create overlaid plots for Profiled Regret
create_profiled_regret_plot <- function(k, sigma, bar.mu) {
u_grid <- seq(-30, 30, 0.1)
# Compute profile regret values for PER, Global MMR, and Gamma MMR
profile_regret_PER <- sapply(u_grid, function(gamma) profiled_regret_special_case(k, gamma, sigma, expectation_d_PER(k, gamma, sigma, bar.mu)))
profile_regret_Global_MMR <- sapply(u_grid, function(gamma) profiled_regret_special_case(k, gamma, sigma, expectation_d_linear(k, gamma, sigma)))
profile_regret_Gamma_MMR <- sapply(u_grid, function(gamma) profiled_regret_special_case(k, gamma, sigma, expectation_d_Gamma_MMR_linear(k, gamma, sigma, bar.mu)))
# Plot the first line (PER)
plot(u_grid, profile_regret_PER, type = "l", lty = 1, lwd = 5.5, col = B_col,
xlab = "", ylab = "Profiled Regret", ylim = c(-0.002, 15), yaxt = "n", cex.axis = 1.25, yaxs="i")
# Add the second line (Global MMR)
lines(u_grid, profile_regret_Global_MMR, lty = 4, lwd = 5.5, col = D_col)
# Add the third line (Gamma MMR)
lines(u_grid, profile_regret_Gamma_MMR, lty = 2, lwd = 5.5, col = C_col)
# Add axes and labels
axis(1, cex.axis = 1.25)
axis(2, at = seq(0, 15, by = 3), las = 2, cex.axis = 1.25)
title(main = "")
mtext(side = 1, line = 3.5, at = mean(par("usr")[1:2]), text = expression(mu), cex = 1.5)
mtext(side = 3, line = 0.5, at = mean(par("usr")[1:2]),
text = bquote(paste("k = ", .(k), ", ", sigma, " = ", .(sigma), ", ", bar(mu), " = ", .(bar.mu))),
cex = 1.5)
# Add grid lines
abline(v = seq(-30, 30, by = 5), col = E_col, lty = 3)
abline(h = seq(0, 15, by = 3), col = E_col, lty = 3)
}
# Set up plotting parameters to create a PDF with two figures and a shared legend
svg("figures/fig3.svg", width = 15, height = 9)
par(mfrow = c(1, 2), oma = c(8, 2, 2, 2), mar = c(5, 4, 4, 2) + 0.1, bg=G_col)
# Create overlaid plots for both bar.mu values
for (t in 1:2) {
bar.mu <- bar.mu_list[t]
create_profiled_regret_plot(k, sigma, bar.mu)
}
# Add common legend
par(fig = c(0.0, 1, 0.08, 0.18), oma = c(0, 0, 0, 0), mar = c(0, 0, 0, 0), new = TRUE)
plot(0, 0, type = "n", bty = "n", xaxt = "n", yaxt = "n")
legend("bottom",
legend = TeX(c(
'$d^*_{PER}$',
'$d^*_{linear}$ (global MMR)',
'$d^*_{linear}$ (ex-ante $Gamma$ MMR)'
)),
col = c(B_col, D_col, C_col),
lty = c(1, 4, 2), lwd = 5, ncol = 3, bty = "n", xpd = TRUE,
title = "", cex = 1.5, title.cex = 0.5,
inset = c(0, 0)) # Adjust the inset for better positioning
dev.off()
Notes: This figure reports the (frequentist) profiled regrets, as a function of the true mean \(\mu\) of data \(\hat{\mu}\), of \(\Gamma\)-PER rule, \(\Gamma\)-MMR linear rule, and the least randomizing global MMR optimal rule Montiel Olea, Qiu, and Stoye (2025) for the parameter values considered in Montiel Olea, Qiu, and Stoye (2025), fig. 2. In the left plot, we see \(\Gamma\)-PER rule is visually dominated. In both plots, the profiled regret of \(\Gamma\)-MMR linear rule and the least randomizing global MMR optimal rule look very similar and are essentially overlapping with each other.
Applying Montiel Olea, Qiu, and Stoye (2025), Theorem 1 to the current setting, we can conclude that all rules, including both \(\Gamma\)-MMR and \(\Gamma\)-PER optimal rules, are at least admissible. Moreover, by definition, the \(\Gamma\)-MMR rule will have the lower value of the (ex ante) game. Therefore, it might be more useful to compare the (frequentist) profiled regrets Montiel Olea, Qiu, and Stoye (2025) of ex-post \(\Gamma\)-PER and other rules as a function of the true but unknown mean \(\mu\) of data \(\hat{\mu}\). In the context of Example 1, we can report \(\bar{R}(d,\mu):=\sup_{\mu^*\in[\mu-k,\mu+k]}R(d,\mu,\mu^*)\) as \(\mu\) varies for each rule \(d\), where \(R(d,\mu,\mu^*)\) is the expected regret of rule \(d\) as a function of \(\mu\) and \(\mu^{*}\). We report several findings (we emphasize that these are not obvious: we took an ex-ante perspective but did not restrict ourselves to those values of \(\mu\) that the prior allows). First, in both panels of Figure 3, the profiled expected regret of the \(\Gamma\)-PER rule exceeds that of the \(\Gamma\)-MMR rule. Therefore, in this particular example, \(\Gamma\)-MMR arguably outperforms \(\Gamma\)-PER from a broader frequentist point of view. Second, we in fact prove that, whenever \(\bar{\mu}\) is sufficiently small and \(k\) is sufficiently large, the \(\Gamma\)-PER rule is profiled-regret dominated, i.e., there exists a rule \(d\neq d^*_{\text{PER}}\) such that
\[ \overline{R}(d,\mu)\leq\overline{R}(d^*_{\text{PER}},\mu) \tag{17}\]
for all \(\mu\in\mathbb{R}\) with the inequality strict for some \(\mu\); see Lemma Lemma 10 in Section 5.4 for an exact statement. Intuitively, the \(\Gamma\)-PER rule mixes between a coin flip rule and the naive threshold rule (\(d_0^{*}\)). When \(\bar{\mu}\) is small, \(\Gamma\)-PER rule is more analogous to the coin flip rule, which Montiel Olea, Qiu, and Stoye (2025) show to be dominated in terms of profiled regret; also, its profiled regret fails to vanish as \(\mu\to\pm\infty\) even though the optimal treatment is known ex ante in this case. Furthermore, Figure 3 reveals that the \(\Gamma\)-MMR rule \(d^{*}_{\text{linear}}\), though not globally MMR optimal, in some cases has essentially the same profiled risk function as the least randomizing global MMR optimal rule Montiel Olea, Qiu, and Stoye (2025). This may lend a Robust Bayes interpretation to the latter.
3.3 Ex-ante and Ex-post in the Game Against Nature
The different approaches analyzed here can all be expressed as different timing assumptions in the statistical game. The distinction is quite obvious for ex-ante versus ex-post: In the former (and original Waldian) perspective, this game is simultaneous move; in particular, Nature moves before data \(Y\) are realized. In the ex-post perspective, Nature sees \(Y\) before choosing a prior. It is immediately clear that this may be easier to solve because it allows for backward induction. There is also an immediate sense that solutions might not agree, as we indeed found.
But whether the decision maker is allowed to (or at least wants to) randomize or not can equally be thought of as changing the game’s timing, providing another way to think about the action space for the ex-ante and ex-post approaches. The more standard perspective is that the decision maker may randomize over decision rules and Nature must move before learning the outcome of this randomization. By the nature of zero-sum games, this setup will frequently yield randomized solutions. In consequence, it is essential to define the decision maker’s action space as \([0,1]\). In contrast, if Nature is allowed to move after the decision maker’s randomization is realized, then any incentive to randomize is gone and we may as well restrict the action space to \(\{0,1\}\).
Theorem 1 and Theorem 2 clarify that these distinctions actually matter in an interesting example. We spell this out in Corollary 3, which considers both cases when randomization is allowed and not allowed. The bottom line is that the assessment is quite sensitive to how the problem is set up. If underlying parameters lead to sufficiently small identification power, the criteria disagree.
Corollary 3 Consider a treatment choice problem with welfare function Equation 1 and statistical model Equation 3 that satisfies Assumption 1.
Suppose randomization is allowed. Then the \(\Gamma\)-MMR and \(\Gamma\)-PER optimal rules coincide if, and only if, \(\underline{I}(\bar{\mu})\geq0\).
Suppose randomization is not allowed. Then the \(\Gamma\)-MMR and \(\Gamma\)-PER optimal rules coincide if, and only if,
\[ \frac{\overline{I}(\bar{\mu})}{\overline{I}(\bar{\mu})-\underline{I}(\bar{\mu})}\geq \Phi(\lVert w \rVert_{\Sigma} ). \tag{18}\]
Thus, if randomization is allowed, \(\Gamma\)-MMR and \(\Gamma\)-PER optimal rules coincide only in the somewhat trivial case in which we a priori know that \(U(\theta)\) and \(\mu\) have the same sign, so that optimal treatment choice reduces to Bayesian inference on the point identified \(\mu\). They disagree in all other cases, including in settings where there are infinitely many \(\Gamma\)-MMR optimal rules.
One might expect more agreement once randomization is excluded; after all, this leads to a much simpler action space. Part (ii) shows that there is some truth to this: The condition for agreement changes from \(\underline{I}(\bar{\mu})\geq0\) to the strictly weaker Equation 18. However, the criteria continue to disagree in many cases. It may be instructive to think of these cases in terms of “comparative statics.” For example, consider holding all parameters of the problem fixed but scaling the signal variance \(\Sigma\) by a positive scalar, say to reflect a change in sample size. Then Equation 18 will hold if, and only if, \(\Sigma\) is large enough; hence, as long as \(\underline{I}(\bar{\mu})<0\), increasing sample size will eventually cause disagreement between \(\Gamma\)-MMR and -PER even if randomization is excluded. Similarly, for fixed \(\Sigma\), \(\Gamma\)-MMR and -PER rules will always disagree if the model’s identification power is small enough.