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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?

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what does 𝑛 exactly represent? Is a threshold?

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 ($N$).

Why would rational players prefer everybody to cooperate?

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 $n<N-1$. 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 $\mathbf{D}$.

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