# Equilibrium in strategic game with complete information

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

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 $$c_i$$.