Simple question on graphical representation of minmax decision rule In the picture below, I cannot understand why the minmax decision rule is on the line $R_1=R_2$.
$R_i=R(\theta_i,d)$, where $\theta_i$ is the parameter and $d$ is the decision rule. $S$ is the risk set. The thickened black line is the set of admissible decision rules; the points (black dots) on set $S$ are non-randomised decision rules.

Source: Essentials of Statistical Inference by Young, Smith.
 A: This warrants delving into an elaboration in a general setup:
$\delta_0$ is minimax if
$$\sup_{\theta\in\Theta} R(\theta, \delta_0) = \inf_{\delta\in D^\ast}\sup_{\theta\in\Theta}R(\theta, \delta).\tag 1\label 1 $$
Let $\Theta := \{\theta_1, \ldots, \theta_k\} ;$ let $S\subset \mathbb R^k.$ So,  each component of a point $\langle y_i\rangle_{i=1}^k\in S$ would correspond to $y_i = R(\theta_i, \delta) $ for some $\delta\in D^\ast.$
Assume $S$ does have its boundary points.
Define for $c\in \mathbb R,~Q_c:= \{\mathbf y: y_i\leq c\};$ all $\delta$'s corresponding to $S\cap Q_c$ are equivalent in the sense that $\max_i R(\theta, \delta) = c.$
The minimax risk is the infimum of such $c$'s, say, $c_0.$ Then the minimax rules would be associated with $S\cap Q_{c_0}.$ The diagonal $y_1 = \cdots=y_k$ is not special; minimax point might or might not lie on it:
$\bullet$ Minimax point on diagonal line:

Notice the minimax risk $c_0$ and the corresponding decision rule $\delta_0$'s minimax point is a boundary point of $S$ and lies on the diagonal line.
$\bullet$ Minimax points not on diagonal line:

Notice the minimax points are not lying on the diagonal line. There is nothing special about this line.

Reference/Figure Source:
$\rm [I]$ Mathematical Statistics: A Decision theoretic Approach, Thomas S. Ferguson, Academic Press, $1967,$ sec. $1.7,$ pp. $38-39.$
