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of dense open sets of Z . We say that a property is satisfied by co-meager many elements in Z if the set of elements that satisfy this property is co-meager. 

Definition 5. Let Γ be a paradigm. A test T is strongly non-manipulable w.r.t Γ if, for every distribution ζ over Γ, 

<img src='content_image/320118.jpg'>

for co-meager many µ -s in Γ. 

Note that in Definition 5 we require that the set of µ on which the expert fail be co-meager relative to the topological space Γ. We do not assume that Γ itself is a co-meager subset of Δ(Ω). Indeed, most paradigms used in modeling, i.e., most interesting classes of stochastic processes (stationary processes, markov processes, mixing processes) are meager as subsets of Δ(Ω). 

## 3. Results 

3.1. Previous results. The following result extends an argument of Sandroni (2003) to convex and compact paradigms. The proof relies on Fan’s Minimax Theorem. See also Vovk, V. and G. Shafer (2005) for a related result. For completeness, we reproduce the proof in the appendix. 

Proposition 1. Fix ϵ > 0 and let Γ be a convex and compact paradigm. Let T be a finite test. If T accepts the data generating process in the paradigm Γ with probability 1 − ϵ then T is ( ϵ + δ ) -manipulable w.r.t Γ for every δ > 0 . 

The following result was proved by Shmaya (2008). The proof relies on Martin’s Theorem about determinacy of Blackwell games (Martin (1998)). 

Proposition 2. Fix ϵ > 0 and let Γ = Δ(Ω) be the paradigm of all theories. Let T be a prequential test. If T accepts the data generating process in the paradigm Γ with probability 1 − ϵ then T is ( ϵ + δ ) -manipulable w.r.t Γ for every δ > 0 . 

