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The $p$-th guest receives a share of $s_{p,N}=p N^{-p-1} (N-p+1) (N-p+2)_{p-1}$, where $(a)_b=\Gamma(a+b)/\Gamma(a)$ is the Pochhammer symbol. For large $N$ this approximates to $$s_{p,N}=\sqrt{\frac{1}{n+1}} (n+1)^{n+1} e^{1-p} p n^{-p-1} (n-p+1) \sqrt{\frac{1}{n-p+2}} (n-p+2)^{-n+p-1}$$ which equals the fair share $1/N$ for ...