A proof of Calibration via Blackwell's. Dean P. Foster æ. Abstract. Over the past few years many proofs of calibration have been presented

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1 A proof of Calibration via Blackwell's Approachability heore Dean P. Foster æ February 27, 1997 Abstract Over the past few years any proofs of calibration have been presented èfoster and Vohra è1991, 1997è, Hart è1995è, Fudenberg and Levine è1995è, Hart and Mas-Colell è1996èè. Does the literature really need one ore? Probably not, but this algorith for being calibrated is particularly siple and doesn't require a atrix inversion. Further the proof follows directly fro Blackwell's approachability theore. For these reasons it ight be useful in the class roo. æ his work was done while I was visiting the Center for for Matheatical Studies in Econoics and Manageent Science, Northwestern University. Peranent aæliation: Dept. of Statistics, he Wharton School, University ofpennsylvania, Philadelphia, PA Eail:foster@hellspark.wharton.upenn.edu. hanks to Sergiu Hart who provided the proof of the only result in the paper. he algorith is a odiæcation of the original algorith in Foster and Vohra è1991è. 1

2 Suppose at tie t a forecast, f t, is ade which takes on the value of the idpoint of each of the intervals ë0; 1=ë, ë1=; 2=ë, :::,ë,1 ;1ë, naely, for i equals 1 to. Let Ai t the vector of indicators as to which forecast 2i,1 2 is actually ade: 8 A é: i = 1 if f t = 2i,1 2 t 0 otherwise Let X t be the outcoe at tie t. We can now deæne the epirical frequency i t as: i = X t A i t Hopefully, i lies in the interval ë i,1 ; i ë. If so, the forecast is approxiately calibrated. If not, I will easure how far outside the interval it is by two distances: d i t and e i t èfor deæcit and excessè which are deæned as: d i = 1 e i = 1 è i,1 A i t, X tèa i t =ë i,1,i ëa i èx t, i èai t =ëi, i ëai P where A i = A i t =. I will show that the following forecasting rule will drive both of these distances to zero: 1. If there exist an i æ such that e iæ 0 and d iæ 0, then forecast 2iæ, Otherwise, ænd an i æ such that d iæ é 0 and eiæ,1 forecast either 2iæ,1 or 2iæ +1 with probabilities: 2 2 P f +1 = 2iæ, 1 2 =1,P f +1 = 2iæ +1 2 é 0 then randoly = d iæ d iæ +e iæ,1 2

3 It is clear that an i æ can be found in step 2, since i = 1 always under forecasts and i = always over-forecasts. he L-1 calibration score: C 1; X i=0 j i t, 2i+1 jai = X axèd i ; ei è so showing that all the e i and di converge to zero, iplies that C 1; converges to 1 2. heore 1 èfoster and Vohraè For all æé0, there exists a forecasting ethod which is calibrated in the sense that C 1; éæif is suæciently large. In particular the above algorith will achieve this goal if 1 æ. Consider this as a gae between a statistician and nature. he statistician picks the forecast f t and nature picks the data sequence X t. he statisticians goal is to force all of the e i and d i to be negative èor at least approach this in the liitè. Nature's goal is to keep the statistician fro doing this. his set up is a gae of ëapproachability" which was studied by Blackwell. He found a necessary and suæcient condition for a set to be approachable. heore 2 èblackwell 1956è Let L ij beavector valued payoæ taking values in R n, where the statistician picks an i fro I at round i and nature picks a strategy j fro J at tie t. Let G be aconvex subset of R n. Let a 2 R n and let c 2 G be the closest point in G to the point a. hen G is approachable by the statistician if for all such a, there exist a weight vector w i such that for all j 2J, X è w i L ij, cè 0 èa, cè 0: è1è i2i o prove heore 1, we need to translate the calibration gae into a Blackwell approachability gae. he set of strategies for the statistician, I, 3

4 is the set of the diæerent forecasts. he set of strategies for nature, J,is the set f0; 1g. Deæne e i X = èx, i èai d i X = è i,1, XèAi he vector loss is the vector of all the èd i ;e i è's, in otherwords, it is a point in R 2. he goal set G R 2 is G = fx 2 R 2 jè8kèx k 0g. Let e i = 1 P ei X t and d i = 1 P di X t. he èd i X ;ei Xè i will be our L ij in the Blackwell gae, and èd i ; e i è i will be the point c. he closest point ing to the current average a =èd i ;e i è i2i is c = èd i è, ; èe i è i2i, : where we have deæned the positive and negative parts as x + = axè0;xè and x, = inè0;xè. he weight vector w is the vector of probability of forecasting i=k. Proof: èhart 1996è Now tocheck equation è1è. Writing it in ters of d i 's and e i 's equation è1è is: X èw i d i, èd i è, èèd i, èd i è, è+èw i e i,èe i è, èèe i, èe i è, è 0 fro x, x, = x + equation è1è is equivalent to X èw i d i, èd i è, èèd i è + +èw i e i,èe i è, èèe i è + Since, èx, èèx + è = 0, it is suæcient to show: X w i èe i èe i è + + d i èd i è + è 0: 0 4

5 If step 1. of the algorith is used the weight vector is just w iæ =1ifi æ is the forecast chosen and zero otherwise. So w i 6= 0 only when both èd i è + and èe i è + are zero, so the entire su is zero. If step 2. is used, the non-zero ters are w iæ and w iæ,1. But, èe iæ è + is zero and èd iæ,1è + is zero. So, it is suæcient to show: w iæ d iæ èd iæ è + + w iæ,1 e iæ,1 èe iæ,1 è + 0 But, d iæ =,e iæ,1, so it is suæcient to show: w iæ èd iæ è +, w iæ,1 èe iæ,1 è + 0 But, this follows èwith equalityè fro the deænition of our probabilities. 2 References Blackwell, D. è1956è. ëan analog of the iniax theore for vector payoæs." Paciæc Journal of Matheatics, 6, 1-8. Foster, D. and R. Vohra, è1991è ëasyptotic Calibration," technical report, Graduate School of Business, University of Chicago, Chicago IL. Foster, D. and R. Vohra, è1997è ëasyptotic Calibration," anuscript. Fudenberg, D. and D. Levine, è1997è ëan easier way to calibrate," anuscript, Hart, S., personal counication, Hart, S. and A. Mas-Colell, è1997è ëa siple adaptive procedure leading to correlated equilibriu." 5