Payoff or value function notation in game theory I am not sure what exactly I don't understand: the concept or notation.
Can you explain what $x_i\left(1 - \sum_j{x_j}\right)$ notation for the value means? What does $x_i$ as a function equal to or how do I interpret it? I see it as $x_i$ is a function and $1-\sum_j{x_j}$ as an argument but have no idea of the body of the function, the value it maps to. How about $x\left(1-t-x\right)$ at the end? Is it recursion?
Here is what the textbook says:

Example 1.4 (Tragedy of the commons) We will describe this game in the context of sharing bandwidth. Suppose that $n$ players each would like to have part of a shared resource. For example, each player wants to send information along a shared channel of known maximum capacity, say $1$. In this game each player will have an infinite set of strategies, player $i$’s strategy is to send $x_i$ units of flow along the channel for some value $x_i \in [0, 1]$.
 
Assume that each player would like to have a large fraction of the bandwidth, but assume also that the quality of the channel deteriorates with the total bandwidth used. We will describe this game by a simple model, using a benefit or payoff function for each set of strategies. If the total bandwidth $\sum_j{x_j}$ exceeds the channel capacity, no player gets any benefit. If $\sum_j{x_j} < 1$ then the value for player $i$ is $\bf{x_i\left(1 - \sum_j{x_j}\right)}$. This models exactly the kind of trade-off we had in mind: the benefit for a player deteriorates as the total assigned bandwidth increases, but it increases with his own share (up to a point).


To understand what stable strategies are for a player, let us concentrate on player $i$, and assume that $t = \sum_{j \neq i} x_j < 1$ flow is sent by all other players. Now player $i$ faces a simple optimization problem for selecting his flow amount: sending $x$ flow results in a benefit of $\bf{x\left(1 − t − x\right)}$.

 A: In this game, the tragedy of the commons is that when all the players are greedy they each get less than what they could get if they found a way to cooperate.
The variable $x_j$ is the amount (of bandwidth) that player $j$ asks for.
Then
$$
 T = \sum_j{x_j}
$$
is the total bandwidth requested by all the players.
When $T < 1$ the model says that there is some benefit to distribute among the players. In that case player $j$ gets benefit $x_j(1-T)$.
The model does not specify the nature of that benefit. It's just some number the player would like to maximize.
If   $T \ge 1$ (which will happen when players are collectively greedy and ask for more than is available) then no one gets any benefit.
If player $j$ knew the total $t$ of all the other players' requests and they asked for bandwidth $x$ then that would make the total requested $t+x$ and they would get benefit
$$
b(x) = x(1- t -x) .
$$
So knowing $t$, player $j$ should choose their $x_j$ to maximize that function. It's easy to see that the maximum benefit occurs when
$$
x_j = 1 - \frac{t}{2} .
$$
Note that the total benefit distributed is  $T(1-T)$. That total is largest when $T=1/2$, so collectively there is the most benefit when only half the bandwidth is requested.
