Equilibrium in strategic game with complete information

Question

Consider the following strategic game with complete information. Set of players is $N = \{1, . . . , n\}$, where $n ≥ 2$. Each player $i ∈ N$ chooses action $a$ $i ∈ \{0, 1\}$. Payoff of each $i ∈ N$ from action profile $ (a_i)_{i∈N}$ is $\sum_{ j∈N} a_j−c_ia_i,$ where $c_i > 0$ is a parameter. Assume $c_i$ is not an integer for each $i ∈ N$. The interpretation of the game is as follows. Each person in a society either takes an action that benefits everyone in the society but is costly for her, or not.

The solution it gives is as follows: Payoff of each $i ∈ N$ from action profile $ (a_i)_{i∈N}$ equals $\sum_{j∈N/\{i\}} a_j+(1-c_i)a_i,$ Hence there exists unique NE in which $a_i = 0$ for any player $i ∈ N$ with $1 − c_i < 0$ and $a_i = 1$ for any player $i∈ N$ with $1 − c_i > 0.$

But I'm not understanding where this payoff came from

Answer 1

Rewrite $i$'s payoff as \begin{align} u_i(a_i,a_{-i})&=\sum_{j\in N}a_j-c_ia_i\\ &=(a_1+\cdots+\color{red}{a_i}+\cdots+a_n)-c_ia_i\\ &=\underbrace{(a_1+\cdots+a_n)}_{\text{without $a_i$}}+\color{red}{a_i}-c_ia_i\\ &=\sum_{j\in N\setminus\{i\}}a_j+(1-c_i)a_i. \end{align} Thus, $i$'s best response only depends on the relative magnitude of $c_i$.