If you happened to read the book, please help me understand the statement in the example 37.1.
- How to understand the payoff function written as $a_i(c+a_j-a_i)$? How to interpret such a notation?
- Where does the quadratic function come from? How does the authour know where the function is equal to zero? And how is this quadratic function related to the aforementioned payoff function? It says,
given $a_j$ individual i’s payoff is a quadratic function of $a_i$ that is zero when $a_i = 0$ and when $a_i = c + a_j$, and reaches a maximum in between.
The payoff function is the quadratic function. I don't think there's any deeper interpretation other than each player's payoff is modelled as a quadratic function of this specific form to represent the idea that
(1) When they contribute zero effort their payoff is zero
(2) When they contribute $c + a_j$ effort their payoff is zero
(3) If they choose $a_i \in (0,a_j+c)$, they will have a positive payoff, and their aim is to find some $a_i$ in this region that maximises their payoff given $a_j$
The author goes on to show that when both players play optimally (maximise their payoffs by choosing $a_i$ given $a_j$) the game will have have a unique equilibrium at $(a_1,a_2) = (c,c)$.