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S  0 ^ `  X*  0 ^   Z*2  0T  ^ `` H@___PPT9"@ j*" h  Bd޽h ?"` 3380___PPT10.1%B. Default Designz 0 0(  0 0 0\p P    X*  0 0t     Z* d 0 c $ ?   0 0Xy  0  RClick to edit Master text styles Second level Third level Fourth level Fifth level!     S 0 6h| _P   X* 8 0 6 _ ` H@___PPT9"@ j*"   H 0 0޽h ? 3380___PPT10.O>`FK $@(  `   BC8 DEdFli @?:yK_wF_lPXM6S6 b q    @|T6   8 {FK34@C"x    ?Th ,$@ 0z   8,$D  0`  0C`2  0CjR`2  0vJ}Z  s *"2  0s'h,$D 02   H @?S" ? l8,$@ 0   3 : WG,{xIp?Two puzzles One observationArial Black"`     <Ԕ8c?C"? .   3 : WG,xID[@8c?Two puzzles One observationArial Black"`  ,$D 02   N @vvv?S" ?G,$@ 0"z  q   q,$D  0`  0+(! -`2  05} ^ `2  0 qG`  0J eh   0e0e    BC DEF  @ 5% 8c8c     ?A)BCD|E||((i  W &m b<   5  : RT m  * Sq P  } GxTm* 5 i RT@`# "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abv ,$@ 02  0Z e ,$D 0]   `dN~  o,kȜ?"6@ NNN?N ,$D  0 [!D. Samet I. Samet D. 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33z-r-___PPT10R-.1% 5+D+' = @B DY+' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(DZ' =4@BB#BB%(D' =-g6B fade*<3<*D' =+4 8?HCBBCB4DB*k3>'Bstyle.rotation<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D ' =%(D' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(DV' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*DF' =%(D' =%(DN' =A@BB@BB0B%()?)?)D' =.M7 BBBBB_M 3.33333E-6 1.54487E-6 L 3.33333E-6 -0.15773 *3>*B ppt_xB ppt_y=@0BBAApBBB}<*D@' =A@BB@BB0B%()?)?D' =.O7 BBBBBaM 3.33333E-6 -2.89017E-7 L 3.33333E-6 -0.14428 *3>*B ppt_xB ppt_y=@0BBAApBBBx<*++0+~  ++0+~  ++0+~  ++0+~  ++0+ ~  ++0+~  ++0+~  +WK0   0  (     `3Ԕ___ s|9)?"6@ NNN?N7    < ~  ?"6@ NNN?N9 @,$  0 ?S" 2g$f   <Љ~  ?"6@ NNN?N9,$ 0 @2S" 2g$f  <~  ?"6@ NNN?NO1  2 Probability 1/2: the double sum is in the red.3 2cgfg3gfcgc  c $~  `}  ~  QTo switch or not to switch?ap  HAX)?envelope @ r  JAX)?envelope2 @ -  0~ ` 5Two sums of money, S and 2S are put in two envelopes.|6(2g___gfg___gfg___  < ~  ?"6@ NNN?Nc ^Probability 1/2: the double sum is in the blue& 0(2 cgfggfcg3cH  0޽h ? 33___PPT10.1% 5+ mJbD' = @B Dg' = @BA?%,( < +O%,( < +DN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* DH' =%(D' =%(DN' =A@BB@BB0B%()?)?)D' =.M7 BBBBB_M 3.33333E-6 1.54487E-6 L 3.33333E-6 -0.15773 *3>*B ppt_xB ppt_y=@0BBAApBBB}<* DB' =A@BB@BB0B%()?)?D' =.Q7 BBBBBcM -3.33333E-6 2.19653E-6 L -3.33333E-6 -0.15307 *3>*B ppt_xB ppt_y=@0BBAApBBB<* ++0+   ++0+   ++0+   ++0+   +uK0 O G P $(  $ $  `3Ԕ___ s|9)?"6@ NNN?N %X,$D  0 $ c $  `}   QTo switch or not to switch?a  $ T 3Ԕ___X)?"6@ NNN?N7,$D 0 4An envelope is selected at random and handed to you.F5 2c gfc  $ <(  ?"6@ NNN?N@  ,$ 0 .You can take the money in the envelope . . . n/ 2cgcgcn  $ <  ?"6@ NNN?N ,$ 0 .. . . or take the money in the other envelope.F/ 2cg3cF  $ <!  ?"6@ NNN?NO,$ 0 h(The situation is symmetric. Why switch? $)(2)g3p $ HAX)?envelope @ r $ JAX)?envelope2 @ H $ 0޽h ? 33___PPT10.1% 5+OD' = @B D=' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-o6Bdissolve*<3<* $D' =%(D' =%(D-' =4@BB8BB%()?)?D' =.I7 BBBBB[M 3.33333E-6 -2.94798E-6 L 0.13264 -0.29711 *3>*B ppt_xB ppt_y=@0BBAApBB}҇=B=<*$D)' =4@BB BB%(D' =-g6B fade*<3<*$D' =1:Bhidden*o3>+B#style.visibility<*$%(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bwipe(up)*<3<*$D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(++0+ $  ++0+ $  ++0+ $  ++0+ $  +)K0 og` ((  ( (  `3Ԕ___ s|9)?"6@ NNN?N= b,$D  0r ( JAX)?envelope27  ( c $|H  `}   QTo switch or not to switch?a8  ( <K  ?"6@ NNN?NS ,$ 0 ZAn argument for switching. " 2g  ( <pP  ?"6@ NNN?N ,$ 0 A'Suppose there is X in the red envelope.((2g3cgcg3cg3cgcgcp  ( <_  ?"6@ NNN?NV SK\ ,$ 0 PIn the blue: 2X or X, each with probability 1/2. The expected sum in the blue:&Q(2cg3cg3cg3cg3cgfg3cg3c  ( Bj  o?"6@ NNN?N 0 ,$D  0 2 (2X) + ( X) = 1 X(2gfcg3cgfcg3cgfcgc ( <v  ?"6@ NNN?NNSnT,$ 0 NThis is true for any X. Switch before you see the amount in the red envelope.O(2 cgg cgc gc+ ( N  ?"6@ NNN?N N q,$  0 ;X 2c$X ( T 3Ԕ___X)?"6@ NNN?N7 4An envelope is selected at random and handed to you.F5 2c gfcH ( 0޽h ? 33___PPT10.1% 5+QD' bw= @B D' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* (D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-o6Bwipe(up)*<3<*(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* (D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* (DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+P+0+ (  ++0+ (  ++0+ (  ++0+ (  ++0+(  ++0+(  +"K0 @ 8  8 (  8X 8 Tt 3Ԕ___X)?"6@ NNN?N7 4An envelope is selected at random and handed to you.F5 2c gfc 8  `3Ԕ___ s|9)?"6@ NNN?N= b~ 8 VAX)?envelope27  8 c $Ĥ  `}   QTo switch or not to switch?a 8 <X  ?"6@ NNN?NS YAn argument for switching. " 2g 8 <1  ?"6@ NNN?N  A'Suppose there is X in the red envelope.((2g3cgcg3cg3cgcgc  8 N1  ?"6@ NNN?N N q ;X 2c$z  8 < 1  ?"6@ NNN?N J ,$ 0 :The same argument holds if you receive the blue envelope. F; 2+cg3 c  8 B1  o?"6@ NNN?N .U,$D  0 ;No matter what envelope you get switch without opening it!H<(2!cg3c 8 HAX)?envelope7 ,$D 0ir 8 B1  +?"6@ NNN?NyO,$D  0  Mind blowing! Z(2c g,c$c H 8 0޽h ? 333+___PPT10 .1% 5+aaQD' = @B D' = @BA?%,( < +O%,( < +Ds' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =-g6B fade*<3<* 8D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-g6B fade*<3<*8D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 8%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* 8D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* 8D ' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*8D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*8D' =%(DP' =A@BB5BB0B%(D'  =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*8D'  =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*8D'  =-g6B fade*<3<*8D' =1:Bhidden*o3>+B#style.visibility<*8%( ++0+ 81  ++0+ 81  ++0+81  ++0+81  + nf(    0D^1  #,$ 0 2Z John E. Littlewood (Ervin Schrdinger) 1953 Kraitchik, M. 1988 Zabell, S.L. 1989 Nalebuff, B. 1992 Christensen, R., Utts J 1994 Jackson, F., P. Menzies, G. Oppy Castel, P., B. Diderik Sobel, J. H. 1995 Brams, S. J., D. M. Kilgour Chae, K. C. Broome, J. 1996 Bruss, F. T. 1997 Arntzenius, F., D. McCarthy Merryfield, K.G., N. Viet, S. Watson Scott, A. D., M. Scott McGrew, T., D. Shier, H. S. Silverstein 1998 Norton, J.D. 2000 Clark, M., N. Shackel, Horgan, T. 2001 Blachman, N. M., D. M. Kilgour 2002 Chase, J. Chalmers, D. . ,  2 )cgcccg cccgcgcg%Acg% cg cgcg cgDcgcg,cg%cg+cg$cgcgcp      (8XH  0޽h ? 335-___PPT10 .'C+ۥD' = @B Dd' = @BA?%,( < +O%,( < +D' =%(%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' `=+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' `=+4 8?dCB#ppt_y+1BCB#ppt_y-1B*Y3>B ppt_y<*+8+0+1  +PK0 :2H(  HH" H HI1 G 3?"6@ NNN?N{  o,$@ 0 ^kْ$(2g(H" H H0N1 G ?"6@ NNN?NQ ) ,$D  0 ^7j$(2g(  H HAX)?envelope v V+,$D   0 H  `3Ԕ___ s|9)?"6@ NNN?N ,$D  0 H 0x/1 ,$ 0 !]I write two real numbers x < y on two slips of paper, and put each in one of the envelopes. ^ 2ccgfcgfcgf@c H c $K  `}<$ 0  1  NWhich is the larger one?a  H <4O  ?"6@ NNN?N&t,,$  0 DAn envelope is selected at random and you observe the number in it. zE 2&gggg +g  H <W  ?"6@ NNN?NDJ ,$ 0 EYou can bet that the number is the larger of the two, or the smaller.nF 2#cg3cg3c  H B]  ?"6@ NNN?N $,$ 0 If you are right I pay you 1f(2 g3g g3gf H BH  ?"6@ NNN?N v,$  0 if you are wrong you pay me 1.x (2 g3c g3gfg3  H JAX)?envelope2 _ 4,$@  0Lz   H  ,$D 0 H tW?PuzzleArial Black"6@`NNN?Nc  H jW?2Arial Black"6@`NNN?Np  H H 0޽h ? 33d>\>___PPT10<>.1% 5+ߟcD;' = @B D;' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-g6B fade*<3<*HD' =%(D' =%(DZ' =4@BB#BB%(D' =-g6B fade*<3<*HD' =+4 8?HCBBCB4DB*k3>'Bstyle.rotation<*HD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*HD' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*HD' =1:Bhidden*o3>+B#style.visibility<*H%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-g6B fade*<3<*HD' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-o6Bwipe(up)*<3<*HD' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-g6B fade*<3<*HDN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*HD ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HD' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*HD' =1:Bhidden*o3>+B#style.visibility<*H%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*HD' =1:Bhidden*o3>+B#style.visibility<*H%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bdissolve*<3<* HD' =%(D' =%(D5' =4@BB?BB%()?)?D' =.Q7 BBBBBcM -1.66667E-6 -4.39306E-6 L 0.13889 -4.39306E-6 *3>*B ppt_xB ppt_y=@0BBAApBB8=B<* HD)' =4@BB BB%(D' =-g6B fade*<3<* HD' =1:Bhidden*o3>+B#style.visibility<* H%(D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =-o6Bdissolve*<3<* HD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* H%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<* HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<* HD{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*HD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*H+0+0+H  ++0+H  ++0+H  ++0+H  ++0+H  ++0+H  ++0+ H  ++0+ H  ++0+ H  ++0+H  +N  LF(  L L S #  `}   MWhat can you guarantee?a| L B&  o?"6@ NNN?N/.,$D  0 0Always bet the number you observe is the larger L1(2)gc,g$g L <$-  ?"6@ NNN?Nb ,$ 0 %The probability you are right is 1/2.H&(2!cgfc L 3 r2 ?3Ԕ___ s|9)?"<?@@`NNN?N Y,$D  0 8Can you guarantee that your expected payoff is positive?P9(2/g g fg  L Td_d9 GJH ?"6@ NNN?Ny  F (2c   L T8;  o?"6@ NNN?N_ Y ,$D 0 your expected payoff is 0.^(2g +c g fc H L 0޽h ?L 33( ___PPT10..?;+.XD ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-o6Bwipe(up)*<3<*LD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-g6B fade*<3<*LD' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* L%(D' =-6B%slide(fromBottom)*<3<* LD' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-s6Bwipe(left)*<3<*L++0+L  ++0+L  ++0+L  ++0+ L  +"B B:P(  P P  `3Ԕ___ s|9)?"6@ NNN?NE ,$D  0 P c $  `}   MWhat can you guarantee?aW P < a  ?"6@ NNN?N?,$ 0 yA threshold strategy:H(2c g c   P <d ?"0@NNN?N %,$  0 Ab$(2gf P <h ?"0@NNN?N ,$ 0 Fix a number b (for big).(2 g c g c g g g c l ePe* PPee*,$D  0B  P <DjJ?"0@NNN?N7 P 0Tq  ?"6@ NNN?N6Pe$ D . . . . .  2 c P 0v  ?"6@ NNN?NeV$* D . . . . .  2 cB P@ 6D?"0@NNN?NK L ,$@  0  P 0@z  ?"6@ NNN?No "v,$   0 ;. 2c0v P B}  ?"6@ NNN?N B p ,$  0 .If observed number e" b H(2cgfc_ P B }  ?"6@ NNN?N 1 ,$  0 {If observed number < b H(2cgfc  P 0}  ?"6@ NNN?Nm ct,$   0 ;. 2c0I P B }  ?"6@ NNN?N & s,$  0 ebet it is the larger.4 2g +cJ P B}  ?"6@ NNN?N  ,$  0 fbet it is the smaller.4 2gcH P 0޽h ? 33//___PPT10`/..?;+ĀD<-' = @B D,' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bwipe(up)*<3<*PD' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bdissolve*<3<*PD' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-o6Bdissolve*<3<*PD_ ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* P%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* PD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* PD' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PD+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*PD' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*PD+' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*PD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*PD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-g6B fade*<3<*P++0+P}  ++0+ P}  ++0+P}  ++0+P}  ++0+P}  ++0+P}  ++0+P}  ++0+P}  ++0+P}  + - H@0X(  XRl  o | Xo  |,$@  0 X 6U  ?"6@ NNN?No v ;. 2c0 X 6Y  ?"6@ NNN?N u | ;. 2c0  X # l?3Ԕ___ s|9)?"<?@@`NNN?Ng] ,$D  0 X c $V  `}   MWhat can you guarantee?a X Td_8a GJH ?"6@ NNN?Ny  F (2c  X <c ?"0@NNN?N % Ab$(2gfF ePe* X Pee*B  X <DjJ?"0@NNN?N7  X 0,h  ?"6@ NNN?N6Pe$ D . . . . .  2 c  X 0l  ?"6@ NNN?NeV$* D . . . . .  2 cB  X@ 6D?"0@NNN?NK L ( X 0

+B#style.visibility<*X%(D' =-o6Bwipe(up)*<3<*XD' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-o6Bdissolve*<3<*XD' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B%slide(fromBottom)*<3<*XD<' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*XD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-g6B fade*<3<*XD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*X++0+X  ++0+X  ++0+X  ++0+X  +*  @\(  \Tz  o | \ p ~B },$D  0 \ 0*  ?"6@ NNN?No v ;. 2c0 \ 0D.  ?"6@ NNN?N u | ;. 2c0 \ # l?3Ԕ___ s|9)?"<?@@`NNN?Ng]  \ c $p+  `}   MWhat can you guarantee?a \ Td_A GJH ?"6@ NNN?NyK  F (2c  \ <( ?"0@NNN?N % Ab$(2gfF ePe*  \ Pee*B  \ <DjJ?"0@NNN?N7  \ 0  ?"6@ NNN?N6Pe$ D . . . . .  2 c  \ 0`  ?"6@ NNN?NeV$* D . . . . .  2 cB  \@ 6D?"0@NNN?NK L ( \ 0X  ?"6@ NNN?N)R ,$ 0 VCase 24 2g( +c(= \ 0  ?"6@ NNN?N : ,$ 0 k x < y < bF 2c g fc Tz    \ 8 ,$D  0 \ 0\  ?"6@ NNN?N   ;x 2c \ 0"  ?"6@ NNN?N   ;y 2c \ 0  ?"6@ NNN?N ,$ 0 3Observing either x or y you bet it is the smaller. 4 2gcgcgg3cN \ 0  ?"6@ NNN?N)  ,$  0 |Your expected payoff is 0.F 2c(g(fc(H \ 0޽h ?\ 33/'___PPT10..?;+7XD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-o6Bdissolve*<3<*\D' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-6B%slide(fromBottom)*<3<*\D' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*\D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*\D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*\D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-g6B fade*<3<*\D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*\D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*\++0+\  ++0+\  ++0+\  ++0+\  +N/ `d(  dTz  o | d q ~,$D  0 d 0#  ?"6@ NNN?No v ;. 2c0 d 0d$  ?"6@ NNN?N u | ;. 2c0 d # l?3Ԕ___ s|9)?"<?@@`NNN?Ng]  d c $*  `}   MWhat can you guarantee?a d Td_L. GJH ?"6@ NNN?NyK  F (2c  d <p0 ?"0@NNN?N % Ab$(2gfF ePe*  d Pee*B  d <DjJ?"0@NNN?N7  d 04  ?"6@ NNN?N6Pe$ D . . . . .  2 c  d 08  ?"6@ NNN?NeV$* D . . . . .  2 cB  d@ 6D?"0@NNN?NK L ( d 0<  ?"6@ NNN?N)R ,$ 0 VCase 34 2g( +c(j d 0@  ?"6@ NNN?N : ,$ 0 x < b d" yj 2c g fc cc Tz    d  ,$D  0 d 0G  ?"6@ NNN?N   ;x 2c d 0dK  ?"6@ NNN?N   ;y 2c d 0N  ?"6@ NNN?N<L B ,$ 0 (Observing x you bet it is the smaller. t)(2 gcgg3cH d 0\U  ?"6@ NNN?N2 w ,$  0 vYou gain 1 for sure!F 2 c(g(f c( d 0[  ?"6@ NNN?NJ QP ,$ 0 'Observing y you bet it is the larger. t((2 gcgg3cH d 0޽h ?d 33OG___PPT10'..?;+`M/D' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-o6Bdissolve*<3<*dD' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-6B%slide(fromBottom)*<3<*dD' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*dD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*dD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*dD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-g6B fade*<3<*dD' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-g6B fade*<3<*dD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*dD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*d++0+d  ++0+d  ++0+d  ++0+d  ++0+d  +ID d\p3h (  h h c $t  `}   MWhat can you guarantee?aF ePe*  h Pee*B  h <DjJ?"0@NNN?N7  h 0x  ?"6@ NNN?N6Pe$ D . . . . .  2 c  h 0T}  ?"6@ NNN?NeV$* D . . . . .  2 cl G hG,$D  0 h  0e0e    B CDEXF( A@  5% 8c8c     ?A)BCD|E||+7;CIX58dY; , R x   @       # "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abv  h  0e0e    B CDEXF( @  5% 8c8c     ?A)BCD|E||  ;JXg;Y;y S@       # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab| GTz  o | h q u 9~,$D   0 h 0  ?"6@ NNN?No v ;. 2c0 h 0l  ?"6@ NNN?N u | ;. 2c0Tz    h } ,,$D   0  h 0艓  ?"6@ NNN?N   ;x 2c !h 0@  ?"6@ NNN?N   ;y 2c2 1h N >?"6@ NNN?NRw ,$D  0 0h TD GAXHhs?"6@ NNN?NW Q,$D  0 POr any distribution that assigns positive probability to non-trivial intervals. Q(2Qc h 6  ?"6@ NNN?N@},$ 0 9A mixed strategy: choose b from a normal distribution. p:(2cg3 cgfcH h 0޽h ?0h 33&&___PPT10e&..?;+tADY%' = @B D%' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-o6Bdissolve*<3<*hD\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*hD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*hDa' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1h%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*1hD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*1hD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0h%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*0hD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*0hDK ' =%(D' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*0hD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*0hD' =-g6B fade*<3<*0hD' =1:Bhidden*o3>+B#style.visibility<*0h%(DC' =4@BB5BB%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*1hD' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*1hD' =-g6B fade*<3<*1hD' =1:Bhidden*o3>+B#style.visibility<*1h%(D?' =%(D' =%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*h%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*hD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*hDF' =%(D' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*hD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*h++0+0h  ++0+0h  ++0+h  ++0+h  +y RKJK pJ(  ph   p  0e0e    B[CDE Ff  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||CCgrF {*Oq!9Yyb*_$M$9%&i"""N2([qjojeQLGLLLbNOaPR<RRSRQ\IFL#O\g04@                      `s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D   0,  p C 0e0e    B\ CDEFH  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.."G2 "8kt55oOk;f #3-8=BA.CnE~CIO\ _is~fX}|{||!$@               s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abi ,$D 0 p c $ȫ  `}   MWhat can you guarantee?aF ePe* p Pee*B p <DjJ?"0@NNN?N7 p 0  ?"6@ NNN?N6Pe$ D . . . . .  2 c p 0L  ?"6@ NNN?NeV$* D . . . . .  2 cF G p G p  0e0e    B CDEXF( @  5% 8c8c     ?A)BCD|E||+7;CIX58dY; , R x   @       # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abv   p  0e0e    B CDEXF( @  5% 8c8c     ?A)BCD|E||  ;JXg;Y;y S@       # "0e@       @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab| G F  o |  p q u 9~  p 0  ?"6@ NNN?No v ;. 2c0 p 0  ?"6@ NNN?N u | ;. 2c0 F    p } , p 0\  ?"6@ NNN?N   ;x 2c p 0,ē  ?"6@ NNN?N   ;y 2c p@  t0e0e    BCDEF @  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||B@ s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D 0  p C N0e0e    B CDE$Fl  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||II   2 OZ :   J ~  r K g: ) 3 (     D  o 9E  ,`9,;"$> Z{jD+<KMXeoMwz 6Yq!IE  34@                        s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D  0T p # lǓ ?ԔA|X)?"<?@@`NNN?N^; ,$D  0 @vWith probability p b d" x < y. Your expected payoff is 0. <(2cgfcgf cggfc p <ѓ  ?"6@ NNN?N w% ;,$  0 =p (2c( p <Փ  ?"6@ NNN?Nr ,,$  0 =q (2c( p <ʓ  ?"6@ NNN?N n  L,$   0 Ar$(2g(3E p # l ݓ ?ԔA|X)?"<?@@`NNN?NkR ,$D   0 1;With probability q x < y < b. Your expected payoff is 0. <(2cgcgcgfcggfg p # lC ?ԔA|X)?"<?@@`NNN?NP,$D   0 tWith probability r, x < b d" y. Your gain is 1 for sure. ;(2cgfg3cgfcgfcggfcgcB p BD?"0@NNN?NI ,$D 0 p TyC 3)?"6@ NNN?NJ,$D   0 EThis mixed strategy guarantees that your expected payoff is positive.LF(2<g0g0fc0H p 0޽h ? 33--___PPT10-..?;+56D,' = @B D+' = @BA?%,( < +O%,( < +D ' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-s6Bwipe(left)*<3<*pD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*pD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*pD' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bwipe(up)*<3<*pD ' =%(D' =%(Di' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-u6Bwipe(right)*<3<*pD' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*pD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*pD' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-o6Bwipe(up)*<3<*pD' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* p%(D' =-s6Bwipe(down)*<3<* pD#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*pD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*pD_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*pD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*pDu' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*pD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*pD' =+K4 8?CBB#ppt_x+(cos(-2*pi*(1-$))*-#ppt_x-sin(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_x<*pD' =+K4 8?CBB#ppt_y+(sin(-2*pi*(1-$))*-#ppt_x+cos(-2*pi*(1-$))*(1-#ppt_y))*(1-$)CB?B*Y3>B ppt_y<*p++0+pC  ++0+pC  ++0+pC  ++0+pC  ++0+pC  ++0+pC  ++0+pC  +]  M(     # l?3Ԕ___ s|9)?"<?@@`NNN?NFp ,$D  0  c $#C  `}  C  MWhat can you guarantee?a  Td_(C GJH ?"6@ NNN?Ny  F (2c   <*C  ?"6@ NNN?N.w7,$ 0 ;You can guarantee that your expected payoff is positive. <(2gc gcgcgfc9  <4C  ?"6@ NNN?N  ,$ 0 [ Really???6 (2g$3c$  B49C  ?"6@ NNN?N ,$  0 @Therefore, I cannot guarantee for myself zero expected payoffs. A(2 cg3ggcg3cgfcH  0޽h ? 330(___PPT10..?;+D4' = @B D ' = @BA?%,( < +O%,( < +D{' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*Di' =%(D' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D2' =%(D' =%(D' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*++0+C  ++0+C  ++0+C  +& K0    ;tr (  tr t S LC  `}  C  B t BD?"0@NNN?Nn n %,$D  0B t BD3?"0@NNN?N - ,$@  0B  t@ NDjJ?"0@NNN?N.J,$D  0c  t 3 r4C ?3Ԕ___ s9p?"<?@@`NNN?N,$D 0 ISP is a probability over pairs of numbers x1 and x2 in the red and blue envelopes.T(2g*co3co cg3cg cH 6t <C  ?"6@ NNN?N # ,$ 0 jx1L 2co3g[ 7t <C  ?"6@ NNN?N~} ,$ 0 }x2 ^ 2cocgz 8t <ȨC  ?"6@ NNN?Nj ,$  0  x2 > x1v 2coco3gz :t <dC  ?"6@ NNN?NGXg,$   0  x1 > x2v 2co3cogH t 0޽h ? 33___PPT10e.?+sD!' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-o6Bdissolve*<3<* tD ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-s6Bwipe(left)*<3<*tD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-s6Bwipe(down)*<3<*tD' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6t%(D' =-o6Bdissolve*<3<*6tD@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*7t%(D' =-o6Bdissolve*<3<*7tD ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-s6Bwipe(down)*<3<* tD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8t%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*8tD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*8tD' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:t%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*:tD' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*:t++0+ tC  ++0+6tC  ++0+7tC  ++0+8tC  ++0+:tC  +Q K0 ))<G)(  L 5  0e0e    B>C DEF  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| > . @`s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$@  0B -@ BD?"0@NNN?N ,$@ 0x  c $ C  `}  C  B  BD?"0@NNN?Nn n %B  BD3?"0@NNN?N - B @ NDjJ?"0@NNN?N.J " <  ?"6@ NNN?N #  jx1L 2co3g' # <  ?"6@ NNN?N~}  }x2 ^ 2cocg3 ' C x ?3Ԕ___ s9p?"<?@@`NNN?N ,$D 0 aProperty 1: Given any set of values A of x1, the probability of x2 > x1 and x1 > x2 is 1/2.b(2 g3cg3cg3co3coco3g3co3cocgcF ) <  ?"6@ NNN?NGXg  x1 > x2v 2co3cogF + <  ?"6@ NNN?Nj   x2 > x1v 2coco3gB , BD?"0@NNN?N ,$D 0B . HD3)?"0@NNN?N  ,$D  0 / <%  ?"6@ NNN?N@ $ w `,$ 0 AA$(2g3% 7 <)  ?"6@ NNN?NAS % ,$  0 G . . . . . . 82 aL 6  0e0e    BDCDEF  A5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||D7p @`s " 0e@        @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab I,$D   0O 8 Bt.  ?"6@ NNN?NM : m ,$  0 k1/2L(2gcg[ 9 N2  ?"6@ NNN?N' ,$  0 k1/2L(2gcg : <9  ?"6@ NNN?N$ D ,$   0 P(x1 > x2) | A) = 1/2 2co3cocg3cgcg ; <PB  ?"6@ NNN?N  ,$   0 P(x2 > x1) | A) = 1/2 2coco3cg3cgcg/ < 3 rM ?3Ԕ___ s9p?"<?@@`NNN?N ISP is a probability over pairs of numbers x1 and x2 in the red and blue envelopes.T(2g*co3co cg3cg cH  0޽h ? 33t'l'___PPT10L'.?+)D%' = @B DS%' = @BA?%,( < +O%,( < +D' =%(%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-g6B fade*<3<*'D,' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*.D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*.D' =%(Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-o6Bdissolve*<3<*/D' =%(D' =%(DE' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B%slide(fromBottom)*<3<*-DE' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-6B%slide(fromBottom)*<3<*,D' =%(Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-s6Bwipe(down)*<3<*7D ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-s6Bwipe(down)*<3<*5D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-o6Bwipe(up)*<3<*6D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-o6Bdissolve*<3<*8D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-o6Bdissolve*<3<*9D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*:%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*:D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*:D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*;D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*;++0+'  ++0+/  ++0+7  ++0+8  ++0+9  ++0+:  ++0+;  +O K0 +*! *(  %  Bp  ?"6@ NNN?N 4 ( ,$ 0 AB$(2gx  c $  `}   B  BD?"0@NNN?Nn n %B  BD3?"0@NNN?N - B @ NDjJ?"0@NNN?N+S  <  ?"6@ NNN?N #  jx1L 2co3g'  <  ?"6@ NNN?N~}  }x2 ^ 2cocg3  C x ?3Ԕ___ s9p?"<?@@`NNN?N ,$D 0 aProperty 2: Given any set of values B of x2, the probability of x2 > x1 and x1 > x2 is 1/2.b(2 gcg3cgcococo3g3co3cocgcF  <  ?"6@ NNN?NGXg  x1 > x2v 2co3cogF  <  ?"6@ NNN?Nj   x2 > x1v 2coco3g  <  ?"6@ NNN?N6 V,$   0 P(x1 > x2) | B) = 1/2 2co3cocgcgcg  <  ?"6@ NNN?N,$   0 P(x2 > x1) | B) = 1/2 2coco3cgcgcgR  # 0e0e    B CDEF  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||p z p  @`s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab I ,$@  0 l  l   l ,$D 0B B ND?"0@NNN?N[  B  NZD?"0@NNN?N     H  ?"6@ NNN?N `l  G . . . . . . 82 a  H|  ?"6@ NNN?N   G . . . . . . 82 aB  HD)?"0@NNN?N n n ,$D  0[  N  ?"6@ NNN?N B  ,$ 0 k1/2L(2gcgR  # 0e0e    B& CDEF  5% 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| &   @`s " 0e@        @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abt & ,$D 0O  BI!  ?"6@ NNN?N = ,$  0 k1/2L(2gcg/ ! 3 r$O! ?3Ԕ___ s9p?"<?@@`NNN?N ISP is a probability over pairs of numbers x1 and x2 in the red and blue envelopes.T(2g*co3co cg3cg cH  0޽h ? 339$1$___PPT10$.?+$FD"' = @B DP"' = @BA?%,( < +O%,( < +D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D@ ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D@ ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-u6Bwipe(right)*<3<*D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*Dp' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*+P+0+!  ++0+ !  ++0+!  ++0+!  ++0+!  ++0+!  +$   0 7 (     # l?3Ԕ___ s|9)?"<?@@`NNN?N H,$@  0   # l?3Ԕ___ s|9)?"<?@@`NNN?NDL) ,$D  0  c $x!  `}  !  Why cannot I guarantee 0?b ae3 aea  <!!  ?"6@ NNN?NJu ,$ 0 When I use the mixed strategy P: Given any information you get about one of the numbers, the probability it is the larger is 1/2. Your bet must result in zero expected payoff. 8(2cg3cgfcg3Ccgfcgcgfcgc  B(-!  ?"6@ NNN?N, f,$  0 5There is no probability P with these two properties. b6(2g(3g(fg(3c(@  <l2!  ?"6@ NNN?Nh %I ,$  0 bA contradiction!6(2g$c$H  0޽h ? 33___PPT10..?;+9}^D' = @B D' = @BA?%,( < +O%,( < +D{' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-g6B fade*<3<*"D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(DA' =%(D' =A@BBB B0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* D ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*++0+!  ++0+!  ++0+ !  +  P (     # l?3Ԕ___ s|9)?"<?@@`NNN?NV,$D  0  c $F!  `}  !  ]Explaining puzzle 1.aef  Td_Pl! GJH ?"6@ NNN?Ny  F (2c   <|\!  ?"6@ NNN?N ,$  0 0An assumption is made: No matter what envelope you hold and which number you observe, The probability you have the bigger number is 1/2(2g`gc gfcgI  <De ?"6@ NNN?Ne k,$  0 kyThis assumption presupposes a probability distribution P over pairs of sums which was shown, by puzzle 2, not to exist. z(26g3cgfc.g3cgfc g3cH  0޽h ? 33  ___PPT10 ..?;+!DT ' = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D_' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* +p+0+ ++0+  + HHp&907H(  0 90  0e0e    BCDEF  5% 8c8c     ?A)BCD|E|| @`# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab = ,$D 0 80  0e0e    B C DEF  5% 8c8c     ?A)BCD|E||  @`# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  ,$@ 0 0  0e0e    BD CDEF  5% 8c8c     ?A)BCD|E|| = D   = @`# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abW 7,$@  0 30 <?"6@ NNN?Nu ,$D  0 40 <?"6@ NNN?N W,$@  0B 0@ NDjJ?"0@NNN?Nxl,$@ 0l - .0-,$@) 0 -0 <?"6@ NNN?N-  ,0 <?"6@ NNN?N @ 1 (0mVB %0 BD?"0@NNN?N  B &0 BD?"0@NNN?N& . 1 '0 <_ ?"6@ NNN?N zq  ].$(2g0 $ 0 c 0e0e    B C DEF  A5% 8c8c     ?A)BCD|E|| @`# "0e@       @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  ,$D  0B  0 BD?"0@NNN?N #Z ,$@  0B  0 BD?"0@NNN?Nq ,$D   0 0 6  ` XA direct proof.aefz 0 < ?"6@ NNN?Ne3 ,$  0  x1 > x2v 2co3cogz 0 <  ?"6@ NNN?N6 )DV ,$  0  x2 > x1v 2coco3gF 0 B ?"6@ NNN?NL^ l,$ 0 b. . . . " ( 2 c F  0 Bh ?"6@ NNN?Nt",$ 0 b. . . . " ( 2 c L  0 < ?"6@ NNN?N9 Y ,$   0 nQ14(2ck L 0 <! ?"6@ NNN?NlG ,$   0 nQ24(2ck L 0 <% ?"6@ NNN?N,$ 0 nA14(2ck L 0 <0* ?"6@ NNN?N] },$  0 nA24(2ck  0 0. ?"6@ NNN?N Z  (2c ;  0 <2 ?"6@ NNN?N^  ! ,$  0 ].$(2g0 l 0 <|7 ?"6@ NNN?N w:,$ 0 P(Q1) P 2g33o33g33 n 0 << ?"6@ NNN?N (:,$ 0 + P(A1) P 2g33o33g33 o 0 <lB ?"6@ NNN?N :,$ 0  = P(A2) P 2g33o33g33  " 0 <0H ?"6@ NNN?N ,$  0 D2P(Q2) = P(A1) - P(A2)  2g33o33 g33o33g3333g33o33g33 > 0 <R ?"6@`NNN?N D(,$D 0 `-&(233 Y /0 0+B#style.visibility<*0%(D' =-g6B fade*<3<*0Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-g6B fade*<3<* 0D^' =%(D' =%(D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*0D' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*0D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* 0D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* 0D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-s6Bwipe(left)*<3<* 0D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =-s6Bwipe(down)*<3<* 0D| ' =%(D$ ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* 0D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* 0D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*0D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*40%(D' =-g6B fade*<3<*40D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*30%(D' =-g6B fade*<3<*30D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*30D' =1:Bhidden*o3>+B#style.visibility<*30%(D)' =4@BB BB%(D' =-g6B fade*<3<*40D' =1:Bhidden*o3>+B#style.visibility<*40%(D5 ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*0Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*80%(D' =-g6B fade*<3<*80D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*90%(D' =-g6B fade*<3<*90D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*90D' =1:Bhidden*o3>+B#style.visibility<*90%(D)' =4@BB BB%(D' =-g6B fade*<3<*80D' =1:Bhidden*o3>+B#style.visibility<*80%(DR' =%(D' =%(Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-s6Bwipe(down)*<3<*0D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-o6Bwipe(up)*<3<*0D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(D' =%(D<' =A@BB#BB0B%()?)?D' =.K7 BBBBB]M 0.00538 -2.22222E-6 L -0.11372 -2.22222E-6 *3>*B ppt_xB ppt_y=@0BBAApBB?sB<*0Db' =A@BBBB0B%()?)?D' =.q7 BBBBBM -3.61111E-6 -2.22222E-6 L 0.05955 0.11435 L 0.14063 -0.00046 *3>*B ppt_xB ppt_y=B0BB AAApBB=B>i=<*0D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-g6B fade*<3<*0D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*00%(D' =-g6B fade*<3<*00D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*/0%(D' =-g6B fade*<3<*/0D' =%(D ' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.0%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* 0%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<* 0%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* 0%(D' =%(Ds' =%(D' =4@BB8BB%())Do'  =.77 BBBBBIM 0.0 1.85185E-6 L 0.0967 -0.11945 *3>*B ppt_xB ppt_y=@0BBAApBB[E=BJt<*.0D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*10%(D' =-g6B fade*<3<*10D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*20%(D' =-g6B fade*<3<*20++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+/0 ++0+00 ++0+10 ++0+20 +#   p  (  I  C x?3Ԕ___ s9p?"<?@@`NNN?N ^,$D 0 )aProperty 2: Given any set of values B of x2, the probability of x2 > x1 and x2 < x1 is 1/2.b(2 gcgg3cgcococo3g3coco3cgc&  3 r?3Ԕ___ s9p?"<?@@`NNN?N,$@ 0 TThere is no probability distribution P over pairs of numbers x1 and x2 such that, U(2%cgco3co c3  C x|?3Ԕ___ s9p?"<?@@`NNN?N Y,$@ 0 aProperty 1: Given any set of values A of x1, the probability of x2 > x1 and x2 < x1 is 1/2.b(2 g3cg3cg3co3coco3g3coco3cgcB  HԔ?"6@ NNN?N3 ,$D 0 XWe have shown the following... 2c(Z  HhԔ?"6@ NNN?NIs ,$D  0 p6We now show that the following generalization holds...7 27c(H  0޽h ? 33ia___PPT10A.C0=+ՂD' = @B D' = @BA?%,( < +O%,( < +D<' =%(D' =%(D' =A@BB BB0B%()))E' =1B BPB`B1:Bhidden*3>+B#style.visibility= `B<*D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D ' =%(Dh ' =%(DP' =A@BB5BB0B%(D' =+4 8?PCB ppt_wBCBB*Y3>B ppt_w<*D' =+4 8?PCB ppt_hBCBB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(DS' =%(D' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* +P+0+ ++0+ ++0+ ++0+ ++0+ ++0+  +    < (    3 r ?3Ԕ___ s9p?"<?@@`NNN?N TThere is no probability distribution P over pairs of numbers x1 and x2 such that, U(2%cgco3co c  C x?3Ԕ___ s9p?"<?@@`NNN?N Y aProperty 1: Given any set of values A of x1, the probability of x2 > x1 and x2 < x1 is 1/2.b(2 g3cg3cg3co3coco3g3coco3cgc  TH0GHجfԔ?"6@ NNN?NR9 ,$D 0 RWe allow the events x2 > x1, x2 < x1 and also x2 = x1 to have any probability.S(2coco3coco3cgcoco3 cgg c  C x@?3Ԕ___ s9p?"<?@@`NNN?N ^ )aProperty 2: Given any set of values B of x2, the probability of x2 > x1 and x2 < x1 is 1/2.b(2 gcgg3cgcococo3g3coco3cgcH  0޽h ? 33___PPT10.C0=+EgD&' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+8+0+ +_ K0 ^ V (    3 r`T?3Ԕ___ s9p?"<?@@`NNN?N TThere is no probability distribution P over pairs of numbers x1 and x2 such that, U(2%cgco3co c  C x`?3Ԕ___ s9p?"<?@@`NNN?N Y sProperty 1: Given any set of values A of x1, the probabilities of x2 > x1, x2 = x1, and x2 < x1 are p, q, r. $t(2 g3cg3cg3co3coco3coco3coco3cgfcgfcgfc  C x w?3Ԕ___ s9p?"<?@@`NNN?N ^ )aProperty 2: Given any set of values B of x2, the probability of x2 > x1 and x2 < x1 is 1/2.b(2 gcgg3cgcococo3g3coco3cgcH  0޽h ? 33___PPT10i.C0=+D=' = @B +9K0    (    3 r?3Ԕ___ s9p?"<?@@`NNN?N TThere is no probability distribution P over pairs of numbers x1 and x2 such that, U(2%cgco3co c  C x?3Ԕ___ s9p?"<?@@`NNN?N Y sProperty 1: Given any set of values A of x1, the probabilities of x2 > x1, x2 = x1, and x2 < x1 are p, q, r. $t(2 g3cg3cg3co3coco3coco3coco3cgfcgfcgfc  C x?3Ԕ___ s9p?"<?@@`NNN?N ^ sProperty 2: Given any set of values B of x1, the probabilities of x2 > x1, x2 = x1, and x2 < x1 are p, q, r. :t(2 gcgg3cgco3coco3coco3coco3cgfcgfcgfc  TG5cHfԔ?"6@ NNN?N6$I,$D 0 'This is wrong if any of p, q, r is 1. ((2cgfcgfcgfcgfcH  0޽h ? 33___PPT10.C0=+|3D&' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*+8+0+ + K0   @ (    3 rX?3Ԕ___ s9p?"<?@@`NNN?N TThere is no probability distribution P over pairs of numbers x1 and x2 such that, U(2%cgco3co cb  C x o?3Ԕ___ s9p?"<?@@`NNN?N Y vProperty 1: Given any set of values A of x1, the probabilities of x2 > x1, x2 = x1, and x2 < x1 are p, q, r, all 1. b}(2 g3cg3cg3co3coco3coco3coco3cgfcgfcgfcfcz  C xc?3Ԕ___ s9p?"<?@@`NNN?N ^ Property 2: Given any set of values B of x1, the probabilities of x2 > x1, x2 = x1, and x2 < x1 are p, q, r, all 1. x~(2 gcgg3cgco3coco3coco3coco3cgfcgfcgfcfcH  0޽h ? 33___PPT10i.C0=+D=' = @B +8   P 7 (    S  `}   qThe one-observation theoremef3 6  <  ?"6@ NNN?N.w,$ 0 XFX1, & , Xn are n random variables.$ 2g$o$c$g$o$kc$g$c$& ^  < ?"6@ NNN?NP},$ 0 DY = f (X1, & , Xn) is their order.# 2g$g$c$g$c$g$o$c$g$o$c$&   <  ?"6@ NNN?Nh  ,$ 0 If Y depends on (X1, & , Xn) , then it depends on at least one of the variables.R 2c$g$ c$g$o$c$g$o$c$k5c$&6 ~  Tl0GTHo?"6@ NNN?N c ,$D  0 2e.g. X7 < X2 = X4 < X2 &  2c$g$o$c$g$o$c$g$o$c$g$o$c$ 8  <9 ?"6@ NNN?N [B,$ 0 Z  (2c$ H  0޽h ?  33!___PPT10.@QF+aD' = @B Dp' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D+' =%(D' =A@BBBB0B%(E' =1B B`BPB1:Bhidden*3>+B#style.visibility= `B<* D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(++0+ ++0+ ++0+  ++0+  ++0+  +S #"`&("(  (& ( <<@  ?"6@ NNN?N*c |$Interactive epistemology: a reminder %(2%c$$ > ( <O ?"6@ NNN?N,$  0 `states" 2g 7 ( < ?"6@ NNN?N ,$ 0 Y. (2cX 7 ( <4 ?"6@ NNN?NH > ,$ 0 Y. (2cX 7 ( < ?"6@ NNN?N/% ,$ 0 Y. (2cX 7  ( <` ?"6@ NNN?N1'_ ,$ 0 Y. (2cX 7  ( <ܞ ?"6@ NNN?N|J @  ,$ 0 Y. (2cX 7  ( < ?"6@ NNN?N b ,$ 0 Y. (2cX 7  ( <t ?"6@ NNN?N ,$  0 Y. (2cX B  ( < ?"6@ NNN?N6/,$   0 d partitions" 2 g   ( H o?"6@ NNN?N |p,$@   0 ( B o?"6@ NNN?NP  ,$@   0 ( B o?"6@ NNN?N  A ,$D   02 ( B o?"6@ NNN?N V ,$@  02 ( B o?"6@ NNN?N ,$D  02 ( B o?"6@ NNN?Nv 5 ,$@  0A ( < ?"6@ NNN?N6,$  0 c knowledge" 2 g  H ( < ?"6@ NNN?N ,$  0 jcommon knowledge" 2g D ( < ?"6@ NNN?Nq  ,$  0 f common prior" 2 g  B ( < ?"6@ NNN?N  ,$  0 d posteriors" 2 g  < ( < ?"6@ NNN?N ,$ 0 ^1/10"(2e < ( <  ?"6@ NNN?N  ,$ 0 ^2/10"(2e <  ( < ?"6@ NNN?N  ,$ 0 ^3/10"(2e < !( < ?"6@ NNN?Np %W ,$ 0 ^1/10"(2e < "( <\ ?"6@ NNN?N!  ,$ 0 ^2/10"(2e < #( <D ?"6@ NNN?Nw ^ ,$ 0 ^1/10"(2e 9 $( <! ?"6@ NNN?N ,$ 0 [0"(2e ; %( <% ?"6@ NNN?N+v ,$ 0 ]2/3"(2e ; &( <) ?"6@ NNN?N} ud ,$ 0 ]1/3"(2e H ( 0޽h ? 33_____PPT10_.@zP:N+}DZ' = @B DZ' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(Dh' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* (D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* (D ' =%(DN ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D ' =%(DN ' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =%(D`' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*$(%(D' =-g6B fade*<3<*$(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-g6B fade*<3<*(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<* (%(D' =-g6B fade*<3<* (Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*"(%(D' =-g6B fade*<3<*"(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*!(%(D' =-g6B fade*<3<*!(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*#(%(D' =-g6B fade*<3<*#(D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&(%(D' =-g6B fade*<3<*&(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(%(D' =-g6B fade*<3<*%(++0+( ++0+( ++0+( ++0+( ++0+ ( ++0+ ( ++0+ ( ++0+ ( ++0+ ( ++0+( ++0+( ++0+( ++0+( ++0+( ++0+( ++0+ ( ++0+!( ++0+"( ++0+#( ++0+$( ++0+%( ++0+&( +>% W O   (    <o ?"6@ NNN?N) 2Agreeing to disagree is impossible. (Aumann, 1976)|$ 2(2g$c$g$c$c2 2  B Ԕ?"6@ NNN?Nb,$D  0b   fZG(HIko?"0@NNN?Ni=,$D  0c  BXx o?"6@ NNN?N - ,$D 0 Having common knowledge0 2cc 2  B Ԕ?"6@ NNN?N[ ,$D  0b  @  fZG!*HţIo?"0@NNN?Nt ,$D  0  B`~ o?"6@ NNN?N  ,$D 0 )Having different posteriors for an event.B* 2c cc)    3 rT?3Ԕ___ s9p?"<?@@`NNN?Nu W,$D  0 5Is agreeing to agree possible? E. Lehrer and D. Samet6(2c$g$c$g$ c$c5 H  0޽h ?/@  33___PPT10g.SR#K+^?D' = @B DN' = @BA?%,( < +O%,( < +D} ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D} ' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<* D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* ++0+ ++0+  ++0+  +"   ` & (  hl 9  9 ,$D  0B  HDo?"0@NNN?N9 K4  <ě ?"6@ NNN?N 1 s profits4 2g c  hl ( w@  w( @,$D  0B  HDo?"0@NNN?Nw@4  < ?"6@ NNN?N( {iu 2 s profits4 2g c   # < ?"6@ NNN?N  ,$ 0 >Each point is a state that specifies the profits of the firms.N? 2gg*g>  $ <  ?"6@ NNN?N7 = ,$ 0 $Each firm knows only its own profit.F% 2 cgc$  % < ?"6@ NNN?N |,$ 0 >1 s partition  vertical lines.X 2gccc  & <t͘ ?"6@ NNN?N ,,$ 0 B2 s partition  horizontal lines.X" 2gccc! H  0޽h ? 33^V___PPT106.Q+PMD*' = @B D' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D' =%(DL' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-{6Bslide(fromTop)*<3<*#D' =%(D' =%(DL' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-{6Bslide(fromTop)*<3<*$D(' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-g6B fade*<3<*%D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&++0+# ++0+$ ++0+% ++0+& +7T   p(  BF 9   9 B  HDo?"0@NNN?N9 K4  <| ?"6@ NNN?N 1 s profits4 2g c  BF ( w@  w( @B  HDo?"0@NNN?Nw@4  <4 ?"6@ NNN?N( {iu 2 s profits4 2g c  jz `| E  |` E,$D  0B   HDo?"0@NNN?N|EB   HDo?"0@NNN?N`; 4   < ?"6@ NNN?ND( " 1 s profits4 2g c  4   < ?"6@ NNN?Nq 2 s profits4 2g c  tz 5     5  ,$D 0   f3fu @?"6@ NNN?N5    3 r3fud @?"6@ NNN?N  <  ?"6@ NNN?N ]E$(2g   < ?"6@ NNN?N 3S ]E$(2g {z M   M ,$D 0  # lr.3fuy @?"6@ NNN?NM   < ?"6@ NNN?N ]E$(2g   <$ ?"6@ NNN?N F ,$ 0 1P - a common prior, symmetric w.r.t. the axes. 42 2g1c&   < ?"6@ NNN?N= SC,$ 0 *The posteriors of E in each state are 1/2.j+ 2cgc c c*   B o?"6@ NNN?NV Pn,$D   0 4It is common knowledge that the posteriors coincide.`5 2ggffgc4 B @ NDff)?"0@NNN?N$ 6 >,$D   0.  3 r?3Ԕ___ s9p?"<?@@`NNN?NV ,$D 0 `There is no common prior for which the agents agree to agree on nontrivial posteriors for E.a(2 c g %c g ffc g c 2  H?"6@ NNN?N.,$D  0B  NDjJ?"0@NNN?N ,$D  0;  <( ?"6@ NNN?N? ^,$ 0 ]. . . (2c B  NDjJ?"0@NNN?N,$D  0A  BZ ?"6@ NNN?N` ,$  0 ]. . . (2c   3 rp?3Ԕ___ s9p?"<?@@`NNN?N- ,$D   0 `There is a common prior for which the agents agree to agree on nontrivial posteriors for E.pa(20c g ff c g c H  0޽h ? 3333___PPT103.Q+ aD1' = @B D1' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D@' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DR' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D$' =%(D4' =%(D' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D8 ' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D$' =%(D' =%(Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bwipe(left)*<3<*D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bwipe(up)*<3<*Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+ ++0+ ++0+ ++0+ ++0+ ++0+ ++0+ +<+ p(  B  HDo?"0@NNN?NRR,  < ?"6@ NNN?N P 1 s profits4 2g c  B  HDo?"0@NNN?N6 6 ,  < ?"6@ NNN?N  2 s profits4 2g c     f3fu @?"6@ NNN?NY2 =   3 r3fud @?"6@ NNN?Nf@ vJ  <o ?"6@ NNN?N[O Y{ ]E$(2g   <dp ?"6@ NNN?N  ]E$(2g ll (    (  ,$D  02   H?"6@ NNN?N( ` 2   H?"6@ NNN?N0 \ h 2   H?"6@ NNN?N  2   H?"6@ NNN?N _  &  B x o?"6@ NNN?N_,$D 0 BPAt each state the agents are ignorant of E: They cannot tell whether or not E.Q 2c g3cgcccgcP r  <Ђ ?"6@ NNN?N K  ,$ 0 (F  a 4-state event.4 2g$c$ fl      ,$D  0B  HD?"0@NNN?N    BZ` ?"6@ NNN?N   ]. . . (2c   BZ ?"6@ NNN?N  ]. . . (2c   < ?"6@ NNN?NR ^@,$ 0 6When F is added to the agents information they are still ignorant of E.nI 2c$g$@c$g$c$H H  0޽h ? 33___PPT10.Q+yD' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(Dh' =A@BB BB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*DR' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D#' =+4 8?nCB!#ppt_y-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*++0+ ++0+ ++0+ + 6.(  B  HDo?"0@NNN?NRR,  < ?"6@ NNN?N P 1 s profits4 2g c  B  HDo?"0@NNN?N6 6 ,  <8 ?"6@ NNN?N  2 s profits4 2g c     f3fu @?"6@ NNN?NY2 =   3 r3fud @?"6@ NNN?Nf@ vJ  <Ȯ ?"6@ NNN?N[O Y{ ]E$(2g   <$ ?"6@ NNN?N  ]E$(2g FF (      (  2   H?"6@ NNN?N( ` 2   H?"6@ NNN?N0 \ h 2   H?"6@ NNN?N  2  H?"6@ NNN?N _    B o?"6@ NNN?N_ BPAt each state the agents are ignorant of E: They cannot tell whether or not E.Q 2c g3cgcccgcP @F       B  HD?"0@NNN?N    BZ` ?"6@ NNN?N   ]. . . (2c   BZ  ?"6@ NNN?N  ]. . . (2c d  B  o?"6@ NNN?NV 8Vb There is a finite event F which after being added to the agents information, they are still ignorant of E. t 2c g 3c g 3Rc g$c s H  0޽h ? 33___PPT10i.Q+D=' = @B +E zr)(  d   B o?"6@ NNN?NV 8Vb There is a finite event F which after being added to the agents information, they are still ignorant of E. t 2c g 3c g 3Rc g$c s B  HDo?"0@NNN?NRR,  <$ ?"6@ NNN?N P 1 s profits4 2g c  B  HDo?"0@NNN?N6 6 ,  <) ?"6@ NNN?N  2 s profits4 2g c    B\. o?"6@ NNN?N_ PAt each state the agents are ignorant of E: They cannot tell whether or not E.Q 2c g3cg#cgcP {z M   <,$D 0  # lr.3fuy @?"6@ NNN?NM   <7 ?"6@ NNN?N ]E$(2g x ! <; ?"6@ NNN?N  ,$D  0 .1 s maximal profit in F4 2cg b "@ ZG0*HIB?"0@NNN?Nej q,$D  0^ # <XA ?"6@ NNN?N,$D  0 1 cannot tell not- E4 2cg 2 $ H?"6@ NNN?NV .,$D  0Y % <8G ?"6@ NNN?N<$b,$D  0 {2 cannot tell E4 2cg 2 & H?"6@ NNN?N4,$D  0B ' HD?"0@NNN?N4v v -,$D  0B ( HD?"0@NNN?N m,$D   0 ) N ?"6@`NNN?N I ,$D  0H  H,N ?"6@`NNN?N4  ,$   0 ^no$(2g$ H  0޽h ? " 33M.E.___PPT10%..Q+صD-' = @B D,' = @BA?%,( < +O%,( < +D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-g6B fade*<3<*D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*!D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*!D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bwipe(down)*<3<*"D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*#D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*#D' =%(DH' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*'D#' =+4 8?nCB!#ppt_y+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*'D' =+4 8?\CB#ppt_wBCB#ppt_wB*Y3>B ppt_w<*'D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*'DR' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*$D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*$D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*%D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*%D\' =%(D' =%(D' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*(D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*(D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-g6B fade*<3<*&Dg' =%(D+' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?dCB1+#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(++0+! ++0+# ++0+% ++0+ +  xp(    3 r ?3Ԕ___ s9p?"<?@@`NNN?N]$D  0TL___PPT9.&@ Z For countable partitions, it is possible to agree to agree on nontrivial posteriors for E iff there exists a finite F which after being added to the agents information, they are ignorant of E. n(2cc$g$c$g$c$g$c$g$c$g$c$g$c$Lc$g$c$c:[ O H  0޽h ? 33 ___PPT10.R1+hD' = @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+ + s%k% $2 $(      <$ ?"6@ NNN?N5 j2 s profit 2 c     B ?"6@ NNN?N_F  2  NGEH)#?"6@ NNN?Nt ?,$D  0 2Shift-rotate left this line an irrational distance^3(2gc g cz J  #  J. ,$D  0B   NDԔ?"0@NNN?NJ B   NDԔ?"0@NNN?N  B   BD?"0@NNN?N B   BD?"0@NNN?N]BB   HDԔ?"0@NNN?Ng $,$D  0B   HDԔ?"0@NNN?N 6,$D  02   H?"6@ NNN?N A ,$D 0;   <0 ?"6@ NNN?NBF ,$   0 ]E$(2g$33 B   ND?"0@NNN?N{ g,$D  02   H?"6@ NNN?N,$D 0   <$ ?"6@ NNN?N1 ,$ 0 qThe state space: the four thick lines in the four unit-squaresJr 2gVc gf3q    < ?"6@ NNN?N2y,$  0 8The prior: the uniform distribution49 2 g0c8    <D  ?"6@ NNN?N   j1 s profit 2 c  B   ND?"0@NNN?NPV- ,$D  0   H ?"6@`NNN?NS  '  H ?"6@`NNN?N# < B   HDԔ?"0@NNN?N+ - ,$D  0B   ND?"0@NNN?NI ,$D  02   H?"6@ NNN?Nf  ,$D 0B   ND?"0@NNN?N| 5 ,$D 02   H?"6@ NNN?Nd 9q ,$D 0 (  <' ?"6@ NNN?N?,$ 0 *The posteriors of E: 1/2 in each state+ 2ggg33cgfc* 2 )  H?"6@ NNN?N';s_,$D 0  *  3 r0?3Ԕ___ s9p?"<?@@`NNN?N F,$D  0 A It is possible to agree to agree on nontrivial posteriors of E. B 2ccgcgfccx +  3 rl3?3Ԕ___ s9p?"<?@@`NNN?Ne $D  0TL___PPT9.&@ "there is no finite F as required. #(2 c$g$c$g$c$ c$c(   B ,  ND?"0@NNN?NA{ B,$D  02 -  H?"6@ NNN?N" Z,$D 0B .  ND?"0@NNN?N ,$D  02 /  H?"6@ NNN?N  ,$D 0B 0 @ ND?"0@NNN?No ,$D 02 1  H?"6@ NNN?N ,$D 0H   0޽h ?2  33ZZ___PPT10Z.R+DX' = @B DfX' = @BA?%,( < +O%,( < +D' =%(D' =%(Dc' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*# %(D' =-s6Bwipe(down)*<3<*# D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D@' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2 %(D' =-o6Bwipe(up)*<3<*2 Dq' =%(D' =%(D+' =4@BB#BB%()?)?D' =.G7 BBBBBYM 3.33333E-6 5.20231E-7 L -0.08021 0.00208 *3>*B ppt_xB ppt_y=@0BBAApBBDD$Brr:<*# D' =%(D6' =A@BB BB0B%(D' =-g6B fade*<3<*2 D' =1:Bhidden*o3>+B#style.visibility<*2 %(D ' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB0-#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =%(D]' =4@BBBB%(D' =,5433*3>#Bstroke.color=@BPB<* D' =1:B true*a3>Bstroke.on<* D]' =4@BBBB%(D' =,5433*3>#Bstroke.color=@BPB<* D' =1:B true*a3>Bstroke.on<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D8' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*( %(D' =-g6B fade*<3<*( D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<** %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<** D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<** D' =%(D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dr' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(left)*<3<* D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* Dn' =%(D' =%(D3' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-o6Bwipe(up)*<3<* D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bwipe(right)*<3<* D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-g6B fade*<3<* D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bwipe(down)*<3<* D' =%(D+' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*) %(D' =-g6B fade*<3<*) D' =%( D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibil