text: Query-independent confounders. One downside of the per-query approach is that the adversary must repeat, for each 
  query, the search for a good confounder. In practice, the adversary might prefer a query-independent attack. Our con- 
  founder gadget approach extends to this setting readily: perform the search routine above for an empty query. In other 
  words, just ignore xi in the query-dependent attack above, replacing Sθ(c∥xi) in Eq. 1 with Sθ(c). This finds a sin- 
  gle query-independent confounder c that can be prefixed to all queries, i.e., ˆxi = c∥xi. We will show that this works 
  surprisingly well. 
  
  It is tempting to assume the reason a query-independent confounder works well is that a good scoring function should be 
  roughly monotonic in query extensions, i.e., one might expect that Sθ(c∥x) ≥ Sθ(c) for almost any suffix x. This intuition 
  is not correct. In our experiments, we found that Sθ(c∥x) < Sθ(c) for many x and some of the routers discussed below. 
  Nevertheless, by ensuring that Sθ(c) is pretty high (set the number of iterations T higher) the resulting query-independent 
  confounder works well. That is, we at least get that Sθ(c∥x) > Sθ(x). 