Value of guessing a number that pays itself

I secretly choose an integer $n$ between $1$ and $100$. You get one guess, and if you guess correctly I pay you $n$; otherwise you get nothing. Both of us play optimally. How much is this game worth to you?

$1/H_{100}\approx0.193$, where $H_{100}=\sum_{k=1}^{100}\tfrac1k$. I choose $n$ with probability proportional to $\tfrac1n$, so every guess $k$ earns you $k\cdot\tfrac{1/k}{H_{100}}=\tfrac1{H_{100}}$. You guess with the same $1/k$ weights, making each of my choices cost me the same amount, so neither can do better.