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Assume two functions: $U_A = -(a_1 - \hat{a}_2)^2$ and $U_R = -(a_0 - \hat{a}_2)^2$

Given $a)$ $a_0 \geq \hat{a}_2$ what are the possible conditions that satisfy $U_A > U_R$, $U_A < U_R$, and $U_A = U_R$ (if any); $b)$ $a_0 < \hat{a}_2$ what are the possible conditions that satisfy $U_A > U_R$, $U_A < U_R$, and $U_A = U_R$ (if any).

The question seems easy but I got stock at one point:

It is obvious that for part $a)$ when $a_1$ = $a_0$ we have $U_A = U_R$ and also for $a_1 > a_0$ we have $U_A > U_R$ but when $a_1$ < $a_0$ we have $U_A < U_R$ for $a_0 = \hat{a}_2$ but things get a bit ambiguous for the $a_1$ < $a_0$ part of $a_0 >\hat{a}_2$.

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Hint:

$\begin{array}\\ U_A-U_R &=-(a_1 - \hat{a}_2)^2-(-(a_0 - \hat{a}_2)^2)\\ &=(a_0 - \hat{a}_2)^2-(a_1 - \hat{a}_2)^2\\ &=((a_0 - \hat{a}_2)-(a_1 - \hat{a}_2))((a_0 - \hat{a}_2)+(a_1 - \hat{a}_2))\\ &=(a_0 - a_1)(a_0 - 2\hat{a}_2+a_1)\\ \end{array} $

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