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6

Let's first determine the sets of actions of the players. An action of player 1 is simply a bid $x_1 \in \mathbb{R}_+$. An action of player 2 is a function: $f_2: \mathbb{R}_+ \to \mathbb{R}_+$ that determines for every action $x_1$ of Player $1$ an action $x_2 = f_2(x_1) \in \mathbb{R}_+$. Let us denote by $F_2$ the set of all actions of player 2. Let's now ...


6

You are only required to get Nash equilibria and not sequentially rational/subgame perfect equilibria. Hence Player 2's actions at information sets that do not occur (that do not reflect Player 1's actual strategy) do not need to be best responses. All you have to make sure is that no one is better off by deviating. In case 4., regardless of what Player 2 ...


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