We propose a simple detection mechanism for the Gumbel watermarking scheme proposed by Aaronson (2022). The new mechanism is proven to be near-optimal in a problem-dependent sense among all model-agnostic watermarking schemes under the assumption that the next-token distribution is sampled i.i.d.

Hedonic Games (HGs) are a classical framework modeling coalition formation of strategic agents guided by their individual preferences.

A.1 Proof of Proposition 1 We note that 1 is restated and was proved in [25, Appendix A.1] Proof of 2: Non-negativity directly follows by non-negativity of mutual information. Proof of 5: The proof relies on the independence of functions of independent random variables. This concludes the proof. 1 A.2 Proof of Proposition 2 By translation invariance of mutual information, we may assume w.l.o.g. that the means are Next, we show that we may equivalently optimize with the added unit variance constraint. Example 3.4]), we have I (A B) null, where the last equality uses the unit variance property and Schur's determinant formula. Armed with Lemma 1, we are in place to prove Proposition 2. Since the CCA solutions Theorem 2.2], which is restated next for completeness.

We list detailed hyper-parameter settings here for reproducibility.

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