Consider a game in which, simultaneously, player $1$ selects any real number $x$ and player $2$ selects any real number $y$. The payoffs are given by:
$u_1 (x, y) = 2x − x^2 + 2xy$
$u_2 (x, y) = 10y − 2xy − y^2.$
(a) Calculate each player’s best-response function as a function of the opposing player’s pure strategy.
(b) Find and report the Nash equilibria of the game.
(c) Determine the rationalizable strategy profiles for this game.
I got the first two part
$\partial u_1 / \partial X = 0 \implies 2 – 2X + 2Y = 0$ $\implies$ Best Response of Player $1 = 1 + Y$
$\partial u_2/ \partial Y = 0 \implies 10 – 2Y – 2X = 0$ $\implies$ Best Response of Player $2 = 5 – X$
$BR_2 = 5-(1+Y)$
$Y^* = 2$ and $X^* = 3$.
The Nash equilibrium is $(3,2)$.
I am unsure abut rationalizable strategy. I think it is rationalizable strategy for both players are all real numbers because there is no single number that is strictly dominated by another number. For example, if player 2 chooses 10, 11 is the best strategy for player 1 and dominated other number. But, if player 2 chooses 11, 12 is the best strategy for player 1 and 12 dominated 11 in this case. This logic then applies to all real numbers. I am unsure if this logic is correct. Any help is greatly appreciated.