Admissibility and Minimaxity Are Different

Two Different Questions About Risk

A decision rule can be judged in at least two ways. Admissibility asks whether another rule has risk no greater at every parameter value and strictly smaller at at least one. Minimaxity asks whether the rule attains the smallest possible worst-case risk. These criteria use the same risk function, but they compare rules differently: one is a pointwise comparison, while the other reduces each rule's risks to a single maximum.

Neither criterion, by itself, answers the other question. A rule can minimize the maximum risk and still be dominated at some parameter value. Conversely, a rule can be undominated yet have a worst-case risk greater than the minimax value. The following finite problem demonstrates both possibilities, even when randomized rules are included.

A Finite Decision Problem

Suppose the parameter space is \(\Theta=\{0,1\}\), and the observation carries no information: the sample space consists of a single possible observation. The action set is \(\{a,b,c\}\). Since the observation is fixed, a deterministic rule selects one action, and a randomized rule selects a probability distribution over the three actions. The loss table gives the loss at each parameter value.

Action	Loss at \(\theta=0\)	Loss at \(\theta=1\)
\(a\)	1	\(1/2\)
\(b\)	1	1
\(c\)	\(3/2\)	0

Definition: The risk of a rule \(\delta\) at parameter value \(\theta\) is \(R(\theta,\delta)=E_\theta[L(\theta,\delta(X))]\). A rule \(\delta'\) dominates \(\delta\) if \(R(\theta,\delta')\leq R(\theta,\delta)\) for every \(\theta\), with strict inequality for at least one \(\theta\). A rule is admissible in a specified class if no rule in that class dominates it. The maximum risk of \(\delta\) is \(\sup_{\theta\in\Theta}R(\theta,\delta)\); a rule is minimax if it attains the infimum of maximum risk over the class.

Write \(p_a,p_b,p_c\) for the probabilities with which a randomized rule chooses the respective actions, where each probability is nonnegative and \(p_a+p_b+p_c=1\). Its two risks are expected losses under this randomization:

R(0,\delta)=p_a+p_b+\frac{3}{2}p_c =1+\frac{1}{2}p_c, \qquad R(1,\delta)=\frac{1}{2}p_a+p_b =\frac{1}{2}+\frac{1}{2}p_b-\frac{1}{2}p_c.

The second expressions use \(p_a=1-p_b-p_c\). The formulas let us assess deterministic and randomized rules within one calculation. In particular, allowing randomization does not mean that every pointwise comparison can be reduced to comparing only the three rows of the table.

Worked Example: Finding the Minimax Value

For every randomized rule, \(p_c\geq 0\), so the risk formula gives \(R(0,\delta)=1+\frac{1}{2}p_c\geq 1\). Therefore its maximum risk is at least 1. The deterministic rule that always chooses \(a\) has risks \((1,\frac{1}{2})\), so its maximum risk is 1. These two calculations show that the minimax value is exactly 1.

The rule that always chooses \(b\) has risks \((1,1)\), and hence also has maximum risk 1. It too is minimax. In this problem, minimaxity does not single out one risk function: both \((1,\frac{1}{2})\) and \((1,1)\) attain the same smallest maximum risk.

Minimaxity Does Not Guarantee Admissibility

Compare the deterministic rules \(a\) and \(b\). At \(\theta=0\), their risks are equal. At \(\theta=1\), the risk of \(a\) is \(\frac{1}{2}\), strictly less than the risk of \(b\), which is 1. Thus \(a\) dominates \(b\). Nevertheless, both are minimax: each has maximum risk 1, the value established above.

Worked Example: A Minimax Rule That Is Inadmissible

The rule \(b\) has risk \(R(0,b)=1\) and risk \(R(1,b)=1\). Its maximum risk is \(\max\{1,1\}=1\), so it is minimax. Rule \(a\) has risks \(R(0,a)=1\) and \(R(1,a)=\frac{1}{2}\). At both parameter values its risk is no greater than \(b\)'s, and at \(\theta=1\) it is strictly smaller. Therefore \(a\) dominates \(b\), which makes \(b\) inadmissible even though it is minimax.

The strict improvement at \(\theta=1\) does not lower the maximum risk: both rules still have risk 1 at \(\theta=0\). That is why the domination of \(b\) does not contradict its minimaxity.

Result: If \(\delta'\) dominates \(\delta\), then \(\sup_{\theta}R(\theta,\delta')\leq\sup_{\theta}R(\theta,\delta)\). This follows because \(R(\theta,\delta')\leq R(\theta,\delta)\) holds at every parameter value, so taking the supremum preserves the inequality. The inequality need not be strict: a strict improvement at a parameter value that does not determine the larger maximum may leave the maximum unchanged. In the example, the strict improvement from \(b\) to \(a\) occurs at \(\theta=1\), while both rules have their maximum at \(\theta=0\). \(\square\)

An Admissible Rule That Is Not Minimax

Now consider rule \(c\), whose risk function is \((\frac{3}{2},0)\). Its risk at \(\theta=0\) is above the minimax value 1, while its risk at \(\theta=1\) is below that value. Its maximum risk is \(\frac{3}{2}\), so it is not minimax. To establish admissibility, however, the relevant question is whether some available rule has risk no greater at both parameter values and strictly less at one.

Worked Example: Checking Whether Rule \(c\) Can Be Dominated

Let \(\delta\) be any randomized rule that could dominate \(c\). In particular, it must satisfy \(R(1,\delta)\leq R(1,c)=0\). Since \(R(1,\delta)=\frac{1}{2}p_a+p_b\) and both terms are nonnegative, this inequality implies \(p_a=0\) and \(p_b=0\). The probabilities sum to 1, so \(p_c=1\), meaning \(\delta\) is exactly rule \(c\). It cannot have a strictly smaller risk at either parameter value. No randomized rule dominates \(c\); thus \(c\) is admissible even in the class of randomized rules.

Its maximum risk is \(\max\{\frac{3}{2},0\}=\frac{3}{2}>1\), so this admissible rule is not minimax. Its risk at \(\theta=0\) is above the minimax value, but its risk at \(\theta=1\) is below it; admissibility does not require each risk to be above the minimax value.

Worked Example: A Randomized Rule and Its Worst-Case Risk

Consider the rule that chooses \(a\) with probability \(1/2\) and \(c\) with probability \(1/2\). Here \(p_a=1/2\), \(p_b=0\), and \(p_c=1/2\). Substitution gives

R(0,\delta)=1+\frac{1}{2}\left(\frac{1}{2}\right)=\frac{5}{4}, \qquad R(1,\delta)=\frac{1}{2}+\frac{1}{2}(0)-\frac{1}{2}\left(\frac{1}{2}\right)=\frac{1}{4}.

The maximum risk is \(\frac{5}{4}\), which is greater than the minimax value 1. This calculation illustrates why a rule's risk at one parameter value should not be confused with its maximum risk: here \(R(1,\delta)=\frac{1}{4}\), but \(R(0,\delta)=\frac{5}{4}\) determines the maximum. This example calculates the rule's minimax status; admissibility would require a separate dominance check.

What the Two Criteria Do—and Do Not—Say

The examples give both directions of the distinction. Rule \(b\) is minimax but inadmissible, whereas rule \(c\) is admissible but not minimax. These conclusions are not in tension. Minimaxity compares maximum risks; admissibility compares the full risk functions coordinate by coordinate. A small maximum does not rule out a pointwise improvement that leaves the maximum unchanged, and lack of a pointwise improvement does not require a small maximum.

When analyzing a finite risk table, keep the questions separate:

For minimaxity, compute or bound each rule's maximum risk, then compare it with the smallest attainable maximum.
For admissibility, look for a rule whose risks are no larger at every parameter value and strictly smaller at at least one.

A common mistake is to treat “minimax” as synonymous with “best at every parameter value.” It is not: the criterion only minimizes the worst risk, and ties in that maximum can conceal a pointwise improvement. The reverse mistake is to assume an admissible rule must be minimax. Rule \(c\) disproves that claim: its zero risk at one parameter value helps make it undominated, even though its risk at the other parameter value makes its maximum larger than necessary.

Key takeaway: Admissibility is about the absence of a uniform pointwise risk improvement; minimaxity is about attaining the smallest worst-case risk. In a single finite problem, one rule can be minimax and inadmissible while another is admissible and nonminimax.

Check Your Understanding

Use the loss table and risk formulas above to answer the following questions.

Why does \(R(0,\delta)\geq 1\) for every randomized rule establish a lower bound on the minimax value?
Which parameter value gives the strict risk improvement when \(a\) dominates \(b\), and why does the maximum risk remain unchanged?
What condition on the probabilities of a randomized rule follows from requiring its risk at \(\theta=1\) to be no greater than the risk of \(c\)?
For the rule that chooses \(a\) and \(c\) with equal probabilities, which parameter value determines its maximum risk?
In one sentence, distinguish the comparison used to assess admissibility from the comparison used to assess minimaxity.