text: We focus on the binary router setting in which the router applies a learned scoring function to input queries and routes 
  any query whose score exceeds some threshold τ to the strong LLM Ms. This setting has been the focus of several prior 
  works [27, 41, 47] and is used in the control planes that are deployed in practice (see Section 7). 
  More formally, we consider a router RM 
  ω for M = {Mw, Ms}, where ω consists of a scoring function S, scoring function’s 
  parameters θ, and a threshold τ ∈ R+. For notational brevity we just write Rω below, with M clear from context. Here 
  S and θ define a scoring function Sθ : X → R+. Since our focus is LLMs, we assume that queries X are strings of text 
  tokens. The routing algorithm then works as follows: 
  
  (cid:26)Mw(x)  if Sθ(x) < τ 
  Rω(x) = 
  Ms(x) otherwise 
  
  where ω = (S, θ, τ ). We will detail scoring functions in Section 5; prior work has suggested linear models, light-weight 
  LLMs, and more. Note that, consistent with this application, scoring functions are computationally efficient and cheap (as 
  compared to Ms, Mw). Deployments calibrate τ to limit the fraction of queries routed to the strong model Ms, giving rise 
  to the type of control plane integrity policy discussed in Section 3. 