2.1 Actions, Payoffs, Statistical Model, and Decisions

Our setup follows Montiel Olea et al. (Reference Montiel Olea, Qiu and Stoye2026), who in turn follow Ferguson (Reference Ferguson1967) and others. Consider a policy maker who needs to choose an action $a\in [0,1]$ interpreted as probability of assigning treatment in the target population.Footnote ⁵ Her payoff when taking action $a\in [0,1]$ is captured by the welfare function

(2.1) $$ \begin{align} W(a,\theta):=aW(1,\theta)+(1-a)W(0,\theta), \end{align} $$

where $\theta \in \Theta $ is an unknown state of the world or parameter and the functions $W(1,\cdotp ):\Theta \rightarrow \mathbb {R}$ and $W(0,\cdotp ):\Theta \rightarrow \mathbb {R}$ are known. Here, we may interpret $W(1,\theta )$ and $W(0,\theta )$ as the welfare of actions $a=1$ (treating everyone in the population) and action $a=0$ (treating no one in the population). Therefore, (2.1) implies that welfare is linear in actions, a standard assumption in the literature. Denote by $U(\theta ):=W(1,\theta )-W(0,\theta )$ the welfare contrast at $\theta $ . If $U(\theta )$ were known to the policy maker, her optimal action would simply be

(2.2) $$ \begin{align} \mathbf{1}\left\{ U(\theta)\geq0\right\}. \end{align} $$

The policy maker does not know $U(\theta )$ but can learn about $\theta $ . Specifically, we assume that she observes a random vector $Y\in \mathbb {R}^{n}$ with multivariate normal distribution

(2.3) $$ \begin{align} Y\sim N(m(\theta),\Sigma), \end{align} $$

where the function $m(\cdotp ):\Theta \rightarrow \mathbb {R}^{n}$ and the positive-definite matrix $\Sigma $ are known.

Our focus is on the case when the data are not entirely informative about the sign of $U(\theta )$ : Even if the policy maker perfectly learned $m(\theta )$ , she could not (necessarily) pin down the sign of $U(\theta )$ . To formally model such treatment choice problems with (decision-relevant) partial identification, let

(2.4) $$ \begin{align} M:=\left\{ \mu\in\mathbb{R}^{n}:m(\theta)=\mu,\theta\in\Theta\right\} \end{align} $$

collect all the means of Y that can be generated as $\theta $ ranges over $\Theta $ . We refer to elements $\mu \in M$ as reduced-form parameters because they are identified in the statistical model (2.3) without further assumptions. Define the identified set for the welfare contrast given $\mu $ as

(2.5) $$ \begin{align} I(\mu):=\left\{ u\in\mathbb{R}:U(\theta)=u,m(\theta)=\mu,\theta\in\Theta\right\} \end{align} $$

and the corresponding upper and lower bounds as

(2.6) $$ \begin{align} \overline{I}(\mu):=\sup I(\mu),\quad\underline{I}(\mu):=\inf I(\mu). \end{align} $$

Henceforth, when we refer to a treatment choice problem with partial identification, we mean that there exists some nonempty open set in M such that for all $\mu $ in that open set, $\underline {I}(\mu ) <0<\overline {I}(\mu )$ . For simplicity, we also assume that the infimum and supremum in (2.5) are attained.

A decision rule $d:\mathbb {R}^{n}\rightarrow [0,1]$ is a (measurable) mapping from data Y to the unit interval $[0,1]$ . We call d non-randomized if it (almost surely, a.s.) maps into $\{0,1\}$ ; otherwise, we call d randomized, including if it randomizes for some but not all data realizations. We use $\mathcal {D}_n$ to denote the set of all decision rules, and we consider decision rules the same if they a.s. agree. As a result, a rule is unique only up to a.s. agreement. The oracle policy $\textbf {1}\{U(\theta ) \geq 0\}$ is of special interest and for any given $\theta $ is contained in $\mathcal {D}_n$ , but is not feasible in the statistical sense because $U(\theta )$ is not known. In general, there will not be an unambiguously best feasible decision rule, a problem that gave rise to a large literature on different optimality criteria and their implementation. Before introducing the robust Bayes approach, we give two examples that fit into our general framework.

2.2 Examples

Example 1 (Stoye, Reference Stoye2012a)

The one-dimensional version of the general setup has been frequently analyzed before (Manski, Reference Manski2000; Brock, Reference Brock2006; Stoye, Reference Stoye2012a; Tetenov, Reference Tetenov2012a; Kitagawa, Lee, and Qiu, Reference Kitagawa, Lee and Qiu2023). One motivation for it is to think of a policy maker who uses experimental data to choose whether to implement a new policy in a population of interest, but is concerned about the external validity of the experiment at hand. The treatment effect of action $a=1$ is $\mu ^*\in \mathbb {R}$ , while the effect of action $a=0$ is normalized to $0$ ; thus, the policy maker’s expected payoff equals $W(a,\mu ^*):= a \cdot \mu ^*$ . The policy maker observes a realization of the one-dimensional statistic

(2.7) $$ \begin{align} \hat{\mu} \sim N(\mu,\sigma^2), \end{align} $$

where $\sigma>0$ is known and $\mu \in \mathbb {R}$ is an identifiable treatment effect, that is, it could be perfectly learned from infinite data. In this example, $\theta =(\mu ,\mu ^*)^{\top }$ , $\Theta \subseteq \mathbb {R}^2$ , $m(\theta )=\mu $ , and $U(\theta )=\mu ^*$ .

Since the target population and the population from which data (2.7) are collected can be different, partial identification naturally arises. We assume that the identifiable and true treatment effects are constrained by $\left \vert \mu ^*-\mu \right \vert \leq k$ for some known $k \geq 0$ , implying

$$\begin{align*}I(\mu)=[\mu-k,\mu+k],\quad \overline{I}(\mu)=\mu+k,\quad \underline{I}(\mu)=\mu-k,\quad\forall \mu \in \mathbb{R}. \end{align*}$$

The planner must choose a statistical decision rule $d\in \mathcal {D}_1:\mathbb {R}\rightarrow [0,1]$ that maps observed data $\hat {\mu }$ to an action $a\in [0,1]$ . Taking (2.7) as an approximation, this stylized example could reflect model uncertainty (e.g., a treatment effect is estimated in a possibly somewhat misspecified model), external validity concerns (e.g., a randomized clinical trial was performed on volunteers), or a shift in the environment (e.g., we transfer estimates from study populations to treatment populations with slightly different covariates or are concerned about distributional drift over time).

Example 2 (Ishihara and Kitagawa, Reference Ishihara and Kitagawa2021)

This example is taken from Ishihara and Kitagawa’s (Reference Ishihara and Kitagawa2021; see also Manski, Reference Manski2020) “evidence aggregation” framework. A policy maker is interested in implementing a new policy in country $i=0$ and observes estimates of the policy’s effect for countries $i=1,\ldots ,n$ . Let $Y=(Y_1,\ldots ,Y_n)^\top \in \mathbb {R}^n$ denote these estimates and let $(x_0,\ldots ,x_n)$ be non-random, d-dimensional baseline covariates. The policy maker is willing to extrapolate from her data by assuming that the welfare contrast of interest equals $U(\theta )=\theta (x_0)$ and that

$$ \begin{align*} Y = \begin{pmatrix} Y_1 \\ \vdots \\ Y_n \end{pmatrix} \sim N(m(\theta), \Sigma), \quad m(\theta) = \begin{pmatrix} \theta(x_1) \\ \vdots \\ \theta(x_n) \end{pmatrix}, \quad \Sigma = \mathrm{diag}(\sigma_1^2,\ldots,\sigma_n^2), \end{align*} $$

where $\theta : \mathbb {R}^d \rightarrow \mathbb {R}$ is an unknown Lipschitz function with known constant C. For notational simplicity, we can further write $\mu _i$ for $\theta (x_i)$ . Thus, $Y_i \sim N(\mu _i,\sigma _i^2)$ , $Y\sim N(\mu ,\Sigma )$ , and $U(\theta )=\mu _0$ . The policy question is: Given data Y, what proportion of the population in country $i=0$ should be assigned the new policy?

Let $\left \Vert \beta \right \Vert :=\sqrt {\beta ^{\top }\beta }$ be the Euclidean norm of a vector $\beta $ . In this example, the identified set for the welfare contrast $\mu _0$ is

$$ \begin{align*} I(\mu) = \{ u \in \mathbb{R} : \left\vert \mu_i-u \right\vert \leq C\left\Vert x_i-x_0\right\Vert, ~~ i=1,\ldots,n \}. \end{align*} $$

The lower and upper bounds on $I(\mu )$ are simply intersection bounds:

$$\begin{align*}\underline{I}(\mu) = \max_{i=1,\ldots,n} \{ \mu_i - C\left\Vert x_i-x_0 \right\Vert \}, \quad \overline{I}(\mu) = \min_{i=1,\ldots,n} \left \{ \mu_i + C \left\Vert x_i-x_0 \right\Vert \right \}. \end{align*}$$

2.3 Robust Bayes Optimality

Our setting up to here is as in Montiel Olea et al. (Reference Montiel Olea, Qiu and Stoye2026). We now connect it to the robust Bayes literature by imposing a set of priors $\Gamma $ on $\theta $ . Following Giacomini and Kitagawa (Reference Giacomini and Kitagawa2021), we choose a particular single proper prior $\pi _{\mu }$ for $\mu \in M$ but leave the conditional prior of $\theta $ given $\mu $ , denoted as $\pi _{\theta \mid \mu }$ , unrestricted except for

(2.8) $$ \begin{align} \pi_{\theta \mid \mu}\{U(\theta)\in I(\mu)\}=1,\quad \pi_{\mu}\text{-a.s.} \end{align} $$

Then, the class of priors $\Gamma $ consists of all priors on $\theta $ induced by the single prior $\pi _{\mu }$ and any conditional prior $\pi _{\theta \mid \mu }$ that meets (2.8). Intuitively, we choose a single prior on the point-identified parameter and place no new restriction on the partially identified parameter $U(\theta )$ . One can pick any proper prior $\pi _{\mu }$ ; for tractability, we let $\pi _{\mu }$ be supported on two symmetric points $\{\bar {\mu },-\bar {\mu }\}$ with equal probability, where $\bar {\mu }\in \mathbb {R}^n$ is chosen by the decision maker. Henceforth, $\Gamma $ is understood to refer to the implied set of priors:

(2.9) $$ \begin{align} \Gamma:=\left\{\pi_\theta=\int \pi_{\theta\mid \mu}d\pi_{\mu}: \pi_{\mu} \sim \text{unif}(\{-\bar{\mu},\bar{\mu}\}),\pi_{\theta \mid \mu}\text{ satisfies } (2.8)\right\}. \end{align} $$

While the set of priors $\Gamma $ broadly puts us into “robust Bayes” territory, it still does not pin down a uniquely best decision rule because the sign of $U(\theta )$ can remain ambiguous. Let

$$\begin{align*}L(a,\theta):=\sup_{a^{\prime}\in[0,1]} W({a^{\prime},\theta}) - W(a,\theta)=U(\theta)\left\{ \mathbf{1}\{U(\theta)\geq0\}-a\right\} \end{align*}$$

be the regret of action $a\in [0,1]$ . We evaluate decision rules d by their expected regret, defined as

(2.10) $$ \begin{align} R(d,\theta) &:= \mathbb{E}_{m(\theta)}[L(d(Y),\theta)] \notag \\ &= U(\theta)\left\{ \mathbf{1}\{U(\theta)\geq0\}-\mathbb{E}_{m(\theta)}[d(Y)]\right\}, \end{align} $$

where for any $x\in \mathbb {R}^n$ , $\mathbb {E}_{x}[\cdot ]$ denotes expectation taken over $Y\sim N(x,\Sigma )$ .

Even for given set of priors $\Gamma $ and commitment to expected regret, the Robust Bayes literature contains multiple decision criteria that do not in general agree. The difference lies in when (and how) the expectations regarding the unknown parameter $\theta \in \Theta $ are taken. For decision rule $d\in \mathcal {D}_n$ , let

$$\begin{align*}r(d,\pi):=\int_{\theta\in\Theta} R(d,\theta)d\pi(\theta) \end{align*}$$

be its Bayes expected regret under a prior $\pi \in \Gamma $ . Following Berger (Reference Berger1985, Def. 12, p. 216), we introduce the first robust Bayes optimality notion.

Definition 1 ( Ex-Ante $\ \Gamma $ -Minimax Regret)

A decision rule $d^* \in \mathcal {D}_n$ is $\Gamma $ -minimax regret (henceforth $\Gamma $ -MMR) optimal if

(2.11) $$ \begin{align} \sup_{\pi \in \Gamma}r(d^*,\pi)= \inf_{d \in \mathcal{D}_n} \sup_{\pi \in \Gamma} r(d,\pi). \end{align} $$

An alternative “posterior” robustness notion is also common in Bayesian analysis. For each action $a\in [0,1]$ , define posterior expected regret under prior $\pi $ (Berger, Reference Berger1985, Def. 8, p. 159)

$$\begin{align*}\rho(a,\pi_{\theta\mid Y}):=\int_{\tilde{\theta}\in\Theta}L(a,\tilde{\theta})d\pi_{\theta \mid Y}(\tilde{\theta}), \end{align*}$$

where $\pi _{\theta \mid Y}$ is the posterior distribution of $\theta $ given prior $\pi $ and data Y.Footnote ⁶ Then we have the following alternative optimality criterion (Berger, Reference Berger1985, Def. 10, p. 205).

Definition 2 (Ex-Post $\ \Gamma $ -Posterior Expected Regret)

A decision rule $d^* \in \mathcal {D}_n$ is $\Gamma $ -posterior expected regret (henceforth $\Gamma $ -PER) optimal if

(2.12) $$ \begin{align} \sup_{\pi \in \Gamma}\rho(d^*(Y),\pi_{\theta\mid Y}) =\inf_{a \in [0,1]} \sup_{\pi \in \Gamma} \rho(a,\pi_{\theta\mid Y}),\quad \forall Y \in \mathbb{R}^n. \end{align} $$

If the decision maker’s action space is restricted to $\{0,1\}$ , that is, randomization is not allowed, the above definitions are revised by replacing $[0,1]$ with $\{0,1\}$ .

The labeling of $\Gamma $ -MMR as “ex-ante” versus $\Gamma $ -PER as “ex-post” can be related to the timing of a fictitious game against an adversarial Nature (see Section 3.3 for additional discussion). If $\Gamma $ were a singleton, the criteria would coincide and would also agree with (single-prior) Bayes optimality. They do not in general agree otherwise. The term “Gamma minimax” usually (and even “robust Bayes” more often than not) refers to $\Gamma $ -MMR (e.g., see Berger, Reference Berger1985, Sect. 4.7.6).Footnote ⁷

While it is not our agenda to advocate for either criterion, some possible considerations are as follows. The ex-ante approach may be perceived as more suitable if the planner has commitment power and has also been justified axiomatically (Hayashi, Reference Hayashi2008; Stoye, Reference Stoye2011). We also find some numerical and theoretical evidence that the $\Gamma $ -MMR rule may have desirable frequentist properties, for example, in states of the world off the prior’s support (see discussions at the end of Section 3.2). Regarding computational feasibility, the ex-post approach is often easier due to the applicability of backward induction and is routinely employed to quantify posterior robustness of statistical decisions. On the other hand, recent work on numerical discovery of minimax rules (Aradillas Fernández et al., Reference Aradillas Fernández, Blanchet, Olea, Qiu, Stoye and Tan2025; Guggenberger and Huang, Reference Guggenberger and Huang2025) may render the ex-ante approach more scalable. See additional discussions in Appendix E.

The two-point structure of $\pi _\mu $ in (2.9) appears in several related treatment choice problems with minimax regret optimality criteria. For example, the least favorable prior (LFP) takes such a form in a completely unconstrained minimax regret problem with point identified (Stoye, Reference Stoye2009) or partially identified (Stoye, Reference Stoye2012a; Yata, Reference Yata2021; Montiel Olea et al., Reference Montiel Olea, Qiu and Stoye2026) welfare contrast. In analogously constrained minimax regret problems in which we restrict $\mu \in [-\left |\bar {\mu }\right |,\left |\bar {\mu }\right |]$ , extending the analyses in the preceding literature would also imply a similar two-point symmetric structure for the LFP. Given these precedents, we think our choice of $\pi _\mu $ represents a general feature of this class of minimax regret decisions, in addition to offering computational tractability.

3.1 Ex-Ante Robust Bayes Optimality

Theorem 1 reveals that the $\Gamma $ -MMR rules qualitatively change depending on whether condition (3.2) 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 (3.2) equals the unique minimax regret optimal rule for known $\mu $ (Manski, Reference Manski2007b); therefore, it arguably measures the model’s identification strength. The right-hand side of (3.2) 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 (3.4) and (3.5). 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 et al., Reference Montiel Olea, Qiu and Stoye2026) 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 an MMR decision rule and a distribution over parameter values (the LFP) form a Nash equilibrium.Footnote ⁹ If identification is strong in the sense of (3.2), this game has an informative equilibrium: the LFP 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 LFP 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 LFP must also best respond to the decision rule. In this very structured setting, the latter is guaranteed by conditions (3.4) and (3.5), establishing claim (ii) and nonuniqueness of optimal decision rules. Indeed, even the non-randomized 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 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 et al. (Reference Montiel Olea, Qiu and Stoye2026), 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’s (Reference Manski2025) 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, Reference Dvoretzky, Wald and Wolfowitz1951; Khan, Rath, and Sun, Reference Khan, Rath and Sun2006).Footnote 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.

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 $\bar {\mu }$ .

Figure 1

$\Gamma $ -MMR optimal rules in Example 1.

Note: 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 (3.10). Therefore, the unique $\Gamma $ -MMR optimal rule is $d^*_0$ . In the bottom two panels, (3.10) 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.

Figure 1 Long description

A four-panel grid of line graphs. Each graph has an x-axis labeled mu-hat ranging from minus 3 to 3 and a y-axis labeled d dot ranging from 0.0 to 1.0. A legend at the bottom identifies four rules: d super star sub 0 (dashed teal), d super star sub R T (solid dark red), d super star sub linear (dashed green), and d super star sub step (dash-dot grey).

* Top-left panel (k equals 1, sigma equals 1, mu bar equals 1.5): Shows only the d super star sub 0 rule, which is a step function that is 0 for mu-hat less than 0 and 1 for mu-hat greater than 0.

* Top-right panel (k equals 1, sigma equals 1, mu bar equals 0.5): Identical to the top-left, showing only the d super star sub 0 step function.

* Bottom-left panel (k equals 1, sigma equals 0.5, mu bar equals 0.5): Displays multiple overlapping rules. d super star sub R T follows a steep S-shaped logistic curve. d super star sub linear follows a steep linear diagonal through the origin before leveling off. d super star sub step is a three-level step function starting at approximately 0.13, jumping to 0.5 at mu-hat equals 0, and jumping to 0.87 for mu-hat greater than 0.

* Bottom-right panel (k equals 2, sigma equals 1, mu bar equals 0.5): Similar to the bottom-left but with shallower slopes. d super star sub R T is a gradual S-curve. d super star sub linear is a gradual diagonal. d super star sub step has three levels at approximately 0.17, 0.5, and 0.83.

3.2 Ex-Post Robust Bayes Optimality

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 problems 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 the following corollary.

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 (3.10) 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 that (3.10) 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 $d^*_{\text {PER}}$ .

Figure 2

$\Gamma $ -PER optimal rules in Example 1.

Note: 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.

Figure 2 Long description

Each of the four panels shares a common X-axis labeled mu-hat ranging from minus 3 to 3 and a Y-axis labeled d dot open parenthesis dot close parenthesis ranging from 0.0 to 1.0. A legend at the bottom identifies three rules: d sub 0 super star (dashed teal line), d sub P E R super star (solid maroon line), and d sub step super star (dash-dotted grey line).

* Top-left panel (k = 1, sigma = 1, mu bar = 1.5): The dashed teal line d sub 0 super star is at 0 for mu-hat less than 0 and jumps to 1 for mu-hat greater than 0. The maroon line is identical and hidden behind it.

* Top-right panel (k = 1, sigma = 1, mu bar = 0.5): The dashed teal line remains a step from 0 to 1 at mu-hat equals 0. The solid maroon line d sub P E R super star shows a smaller step, starting at 0.25 and jumping to 0.75 at mu-hat equals 0.

* Bottom-left panel (k = 1, sigma = 0.5, mu bar = 0.5): The solid maroon line is at 0.25 for mu-hat less than 0 and 0.75 for mu-hat greater than 0. The dash-dotted grey line d sub step super star shows a wider step, from approximately 0.15 to 0.85.

* Bottom-right panel (k = 2, sigma = 1, mu bar = 0.5): The solid maroon line is at approximately 0.38 for mu-hat less than 0 and 0.62 for mu-hat greater than 0. The dash-dotted grey line shows a step from approximately 0.18 to 0.82.

Applying Montiel Olea et al. (Reference Montiel Olea, Qiu and Stoye2026, Thm. 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 et al., Reference Montiel Olea, Qiu and Stoye2026) 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, that is, there exists a rule $d\neq d^*_{\text {PER}}$ such that

(3.16) $$ \begin{align} \overline{R}(d,\mu)\leq\overline{R}(d^*_{\text{PER}},\mu) \end{align} $$

for all $\mu \in \mathbb {R}$ with the inequality strict for some $\mu $ (see Lemma D.2 in Appendix D 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 et al. (Reference Montiel Olea, Qiu and Stoye2026) 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 et al., Reference Montiel Olea, Qiu and Stoye2026). This may lend a Robust Bayes interpretation to the latter.

Figure 3

Profiled regret of $\Gamma $ -PER and other rules in Example 1.

Note: 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 et al., Reference Montiel Olea, Qiu and Stoye2026) for the parameter values considered in Montiel Olea et al. (Reference Montiel Olea, Qiu and Stoye2026, Fig. 2). In the left plot, we see that $\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.

Figure 3 Long description

A two-panel line graph. Both panels share a horizontal axis labeled mu ranging from minus 30 to 30 and a vertical axis labeled Profiled Regret ranging from 0 to 15.

* Left Plot (k = 18.6, sigma = 3.9, mu bar = 4.7): A solid dark red line representing d super star P E R forms a U-shaped curve that stays above the other rules, reaching a minimum of approximately 9 at mu equals 0. Two other lines, a grey dashed line for d super star linear global M M R and a teal dash-dot line for d super star linear ex-ante Gamma M M R, are perfectly overlapped. They form a bell-shaped curve peaking at approximately 9 when mu is 0 and tapering to 0 as mu moves toward minus 30 or 30.

* Right Plot (k = 18.6, sigma = 3.9, mu bar = 14): The solid dark red line d super star P E R shows a more complex M-shaped profile in the center with two peaks near 11 and a local minimum at mu equals 0, then transitions into linear segments at the far left and right. The grey dashed and teal dash-dot lines remain overlapped, maintaining the same bell-shaped curve seen in the first plot, peaking at 9 when mu is 0.

A legend at the bottom identifies the three rules: d super star P E R (solid red), d super star linear (global M M R) (grey dashed), and d super star linear (ex-ante Gamma M M R) (teal dash-dot).

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 $\{0,1\}$ .

Theorems 1 and 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.

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 $\underline {I}(\bar {\mu })\geq 0$ to the strictly weaker (3.17). 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 (3.17) will hold if, and only if, $\Sigma $ is large enough; hence, as long as $\underline {I}(\bar {\mu })\geq 0$ , 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.