For every µ ∈ Γ, if ζ is Dirac atomic measure over µ then F ( ζ, µ ) ≥ 1 − ϵ since T accepts data generating process in Γ with probability 1 − ϵ . Therefore it follows from (8) that for every δ > 0 there exists ζ ∈ Δ(Γ) such that min$_{µ}$ F ( ζ, µ ) > 1 − ( ϵ + δ ), i.e., that T is ( ϵ + δ )-manipulable. 

## References 

[1] N. Al-Najjar, R. Smorodinsky, A. Sandroni, and J. Weinstein, Testing Theories with Learnable and Predictive Representations Journal of Economic Theory , 145 (2010). 

[2] D. Blackwell and L. Dubins, Merging of opinions with increasing information, Annals of Mathematical Statistics , 33 (1962). 

[3] E. Dekel and Y. Feinberg Non-Bayesian Testing of a Stochastic Prediction, Review of Economic Studies , 73 (2006). 

[4] A. P. Dawid, Statistical Theory: The Prequential Approach, J.R.Statist. Soc. A , 147 (1984). 

[5] K. Fan, Minimax Theorems, Proceedings of the National Academy of Science U.S.A. , 39 , 42-47. 

[6] Kechris, A, Classical Descriptive Set Theory , Springer Verlag (1994). 

[7] E. Kalai and E. Lehrer, Weak and strong merging of opinions, J. Math. Econom. , 23 (1994). 

[8] E. Kalai, E. Lehrer and R. Smorodinsky, Calibrated forecasting and merging, Games Econom. Behav. , 29 (1999). 

[9] D. Martin, The Determinacy of Blackwell Games Journal of Symbolic Logic. , 63 (1998). 

[10] W. Olszewski Calibration and Expert Testing, Handbook of Game Theory, volume IV (2011) 

[11] W. Olszewski and A. Sandroni, Manipulability of Future-Independent Tests Econometrica , 76 (2008). 

[12] W. Olszewski, and A. Sandroni Strategic Manipulation of Empirical Tests, Mathematics of Operations Research , 34 (2009a). 

[13] W. Olszewski, and A. Sandroni A Nonmanipulable Test, Annals of Statistics, 37 (2009b). 

[14] A. Sandroni, The Reproducible Properties of Correct Forecasts, International Journal of Game Theory , 32 (2003). 

[15] E. Shmaya, Many inspections are manipulable, Theoretical Economics , 3 (2008). 

[16] S. Sorin, Merging, reputation, and repeated games with incomplete information, Games Econom. Behav. , 29 (1999). 

[17] V. Vovk and G. Shafer, Good Randomized Sequential Probability Forecasting is Always Possible, Journal of the Royal Statistical Society Series B , 67 (2005). 

