# Understanding a multi-player Tragedy of the Commons game

In the formulation of the Tragedy of the Commons below, what does $$n$$ exactly represent? Is a threshold? The text is from plato.stanford.edu website on Kuhn's Prisoner's dilemma.

Also, it says that $$\bf D$$ strongly dominates $$\bf C$$ for all players, and so rational players would choose $$\bf D$$ and achieve $$0$$, while preferring that everyone would choose $$\bf C$$ and obtain $$C+B$$.

I don't understand the last part. The reason $$\bf D$$ dominates $$\bf C$$ is because either the player gets $$B$$ or $$0$$ which are better from $$C+B$$ or $$C$$. Why would rational players prefer everybody to cooperate?

Yes, $$n$$ is the threshold above which the benefit $$B$$ realizes. Think of this as a situation where a public good with benefit $$B$$ is provided only if more than half of the population votes for it. $$C$$ is the cost of voting, and $$n=\lceil\frac12N\rceil$$ is the smallest integer greater than half of the population size ($$N$$).
While a rational player would choose $$\mathbf{D}$$ regardless of what others would choose, he/she would still prefer the outcome with benefit $$B$$ to the outcome with $$0$$ benefit. Hence, he/she would prefer everyone else chooses $$\mathbf{C}$$, i.e. bearing the cost of providing benefit $$B$$, while he/she enjoys $$B$$ without incurring any cost by choosing $$\mathbf{D}$$.
Additionally, suppose there are $$N$$ players, and that $$n. Then with all the other $$N-1$$ players playing $$\mathbf{C}$$, it doesn't really matter if the remaining player chooses $$\mathbf{C}$$ or $$\mathbf{D}$$. So arguably it's less morally objectionable for the last player to choose $$\mathbf{D}$$.