We present a brief proof about the local optimality of one-hot encodings in the decision-theoretic framework presented in Section 3.2. We seek to prove that, under assumptions of an identity reward matrix, tokens constrained to a unit hypercube, and gaussian additive noise, one-hot tokens are an optimally robust communication strategy. We only seek to prove local optimality, as one many trivially generate multiple, equally optimal tokens by, for example, flipping all bits. The following derivation uses Karush-Kuhn-Tucker (KKT) conditions, a generalization of Lagrange multipliers [17]. We maximize the function, subject to constraints. T>j Ti Ti + ||Tj||2 Ti # ~µi + ~λi = ~0 (13) (14) We seek to show that one-hot vectors are an optimum, so we now show that one-hot vectors indeed respect the constraints and set the derivatives to zero.

Recall that ˆmc(u) is exactly the listener's decoder in the IB framework (see Section 3.1.1). Therefore, anyother decoder would lend an upper bound on the informativeness loss term. Notice that under our assumptions,ˆmc is a Gaussian mixture, whereas the speaker's beliefs are simply Gaussian. All the systems with the samek form an equivalence class and the canonical system within each class is the one with minimalk. These canonical systems are the natural one to prefer, because they can attain the optimum for a given complexity with aminimal codebook.

We argue that the reference games typically used in these studies are ill-suited to drive linguistic systematicity for two reasons.

Simulation experiments confirmed these results: ouragents learned anarrangement oftokensthatclustered inhuman-understandable patterns.

Iterated reference games - in which players repeatedly pick out novel referents using language - present a test case for agents' ability to perform context-sensitive pragmatic reasoning in multi-turn linguistic environments. We tested humans and vision-language models on trials from iterated reference games, varying the given context in terms of amount, order, and relevance. Without relevant context, models were above chance but substantially worse than humans. However, with relevant context, model performance increased dramatically over trials. Few-shot reference games with abstract referents remain a difficult task for machine learning models.