Applying Token-Level Averaging for Text Generation

In a text generation process that combines outputs from K different prompts, the next token ŷ_j is chosen according to the following decision rule:

ŷ_j = argmax(y_j) Σ(k=1 to K) log Pr(y_j | x_k, ŷ_1, ..., ŷ_{j-1})

What is the primary analytical reason for summing the log-probabilities (log Pr) of a candidate token across all prompts, rather than multiplying the raw probabilities (Pr)?

A language model uses an ensemble of 3 prompts (K=3) to generate the next token in a sequence. The model selects the token ŷ_j that maximizes the sum of log-probabilities across all prompts, according to the formula: ŷ_j = argmax(y_j) Σ(k=1 to K) log Pr(y_j | ...).

Given the following log-probabilities for two candidate tokens, 'cat' and 'dog', which token will the model select?