Imperfect Information

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1 Imerfec Informaon Comlee Informaon - all layers know: Se of layers Se of sraeges for each layer Oucomes as a funcon of he sraeges Payoffs for each oucome (.e. uly funcon for each layer Incomlee Informaon - any of he four eces of nformaon above s unknown o a layer; also called asymmerc nformaon Prvae Informaon - ycal examle of ncomlee nformaon where layer knows somehng abou hs own acons ha oher layers don' observe; could be he oher way hough (e.g.ecals [docor mechanc] has more nformaon abou your ayoffs han you do Harsany - aer n Managemen Scence (967 on how o conver game wh ncomlee nformaon o game of merfec bu comlee nformaon by nserng random move by naure Incomlee n Payoffs - Harsany frs argued ha all yes of ncomlee nformaon can be convered o ncomlee nformaon on ayoffs Players - no sure f layer s n he game... equvalen o knowng he s n he game bu no sure abou hs sraegy sace (e.g. could have four oons or jus one effecvely no n he game because he has no decsons Player Sraeges Sraeges - no knowng se of sraeges could be ransformed by addng sraeges ha wll never be layed because of - ayoff; now all oons have same number of sraeges bu we don' know whch ayoffs are correc Sraeges Payoffs Oucomes - no knowng oucomes s equvalen o no knowng he uly funcons (ayoffs... echncally here s a dfference o knowng oucomes and no uly funcons f here are several sraeges (from dfferen nformaon ses ha have he same oucome (wll yeld he same ayoffs Infne Regress - n order o solve ncomlee nformaon roblem we have o worry abou belefs abou belefs abou belefs ec. Soluon - Harsany solved roblem by defnng yes of layers where layers dffer n ayoffs (uly funcon and/or belefs Smlfcaon - Harsany also argued ha all belefs could be caured by a sngle robably dsrbuon over he yes of he oher layers (could be correlaed by usng a jon dsrbuon Examle - 3 yes of "layer " and yes of "layer ": and of Each ye of "layer " has belefs abou he robably ha an oonen wll be of a secfc ye: = ( = ( = ( In or Ou? or or or

2 Fudge - n order o be racable hs mehod requres a fne number of yes Sandard - ycally we only allow layers yes o dffer on reference (uly funcon; heory allows references and/or belefs o vary bu he full mlcaons of hs haven' been suded Problem? - may have assumed away he roblem of ncomlee nformaon raher han solved Formally - Players - n layers Sraeges - each ye of layer has he same sraegy sace S (oal of n Se of Tyes - T s se of dfferen yes of layer (oal of n ses of yes Examle - from revous age T = { 3 } T = { 3 } Examle - yes of layer yes of layer and 3 yes of layer 3: T a b T c d T x y z = { } = { } 3 = { } Secfc Player - s he j h ye of layer (hs was denoed of 7 j on revous age Oher Player Tyes - T = T T T T + Tn... he Caresan roduc of he ses of all layer yes exce for layer T = ( a c( a d( b c( b Examle - { } Oher Players - 3 d T s a secfc ye of each layer (exce layer Examle - one of he elemens of T 3 above Belefs - = Pr[ ] = subjecve belef of j h ye of layer abou he robably ha hs secfc combnaon of oonens wll occur = se of all for layer Examle - 3x = ( y = ( z = ( So ye x of layer 3 hnks 's equally lkely o see any combnaon of yes of layers and ; ye y of layer 3 hnks he'll face ( a c (.e. ye a of layer and ye c of layer abou a ffh of he me; ec. = { } 3 3x 3 y 3z Preferences - u ( s ; uly for layer deends on he acual yes of each layer ( and he sraeges ha each layer uses (s Γ S S T T u u n n n n Sraeges Tyes Belefs Prefs Bayesan Game of Incomlee Informaon - ( (assume hs s common knowledge Examle of Belefs - n oker here are 5!/(7!5! ossble hands; ror o dealng all hands have equal robably; afer seeng your cards (and ohers avalable he robables assgned o dfferen yes of oonens (.e. hands hey have change... for examle f you have aces he robably ha an oonen has a hree or four of a knd wh aces or any oher hand ha requres more han aces wll be zero Professonal Poker - assume layer has queens and layer has jacks; on TV hey say n hs scenaro layer has a 90% chance of wnnng and layer only has a 0% chanceuggesng ha 's a bad dea for layer o say n he game... bu ha analyss s based on comlee nformaon; he layers have ncomlee nformaon so he relevan robables are each layer's (subjecve robably of wnnng gven hs

3 own cards (and hose he's seen; ha s each layer s basng hs decson on hs belef abou hs oonen's hand (usually a lo more robably on oor hands han good hands; f boh layers are sll n he game ha means hey boh hnk hey have greaer han 50% chance of wnnng Conssency - Pr[ ] = Pr[ ] = where Pr[ ] = Pr[ ] Pr[ ] Subjecve Objecve Examle - 3 layers each wh yes has oal of 8 combnaons (so each ye of each layer faces combnaons of ossble oonens: Pr[ 3] No Bg Deal - Almond showed nconssen belefs can be ransformed no conssen belefs; we usually jus assume belefs are conssen because of hs Max Execed Uly - max u ( s ( ; Pr[ ] s Suose belef = Pr[ ] s no conssen ( u ( s above s ncorrec Suose belef ˆ s conssen Always Conssen - le N ( T = # of vecors T = roduc of number of yes of all layers exce ; (examle above has N ( T 3 = Assume each s equally lkely so ˆ = Pr[ ] = / N( T Based on he defnon of conssency belef Defne new uly uˆ ( s N( T u ( s = Noe ha ˆ uˆ ( s = u ( s Pr[ 3] Pr[ 3] = Pr[ 3] + Pr[ 3 ] + Pr[ 3 ] + Pr[ 3 ] = 3 ˆ s conssen Tha means execed uly wh conssen and nconssen belefs s he same Cach - have o know he "werd" (nconssen robables o ncororae no references (new uly funcon 3 of 7

4 Equlbrum - wo deas on how o fnd Players Aroach - rea each ye of each layer as a searae layer (e.g. layers each wh yes becomes a layer game; "layers aroach" s Len's erm; Slusky called he ex os aroach (meanng layers choose her sraeges afer knowng her ye Game Srucure - no all layers wll face each oher (e.g. doesn' lay agans ; hs means he reacon funcon only deends on some of he oonens (n he examle would be oonens nsead of 3 Ineracon - (anoher Poker asde a layer of one ye may wan o affec ayoffs when he's anoher ye; Examles: Beng - be wh a bad hand so oonen knows you be wh a bad hand and hen doesn' know when you have a good hand Bluffng - f you bluff and he oonen folds and asks o see your cards here are wo oons: - Tell oonen ha o see he hand he needs o call (mach be... n oher words he has o ay for he nformaon - Show hm your hand so he knows you're bluffng; ha way f you be laer wh a good hand he mgh hnk you're bluffng and be agans you Problem - hese are use reeaed game reasonng (we're sll focusng on sngle-sho games σ ( s an equlbrum sraegy for (ye j of layer f and T T u (σ ( ; Pr[ ] u ( s ; Pr[ ] s S T Englsh - akng he oher layers' sraeges ( as gven layer 's bes rely s σ ( (.e. he execed ayoff s greaer han or equal o he execed ayoff of layng any oher sraegy s n hs sraegy sace S ; n order for hs o be an equlbrum all he layers mus be layng bes reles o her oonens' bes reles so hs condon holds for all yes of all layers ( and T Noe: alernave sraegy doesn' need o be ndexed by he layer ye ( j because we assumed ha all yes of he same layer have he same sraegy sace (o of. Sraeges Aroach - frs move n game s naure choosng layers yes; solve an merfec bu comlee nformaon game; "sraeges aroach" s Len's erm; Slusky called he ex ane aroach (layers choose sraeges before knowng her ye +: Fewer layers : More comlcaed sraeges (mus address sraeges for each layer ye Examle - 3 yes of layer ; yes of layer Naure 3 Dfferen? - many quesons on wheher hese wo mehods are dfferen n equlbra or mehod (e.g. easer o solve Subgame Perfecon - he layers aroach guaranees subgame erfec; some eole argue he sraeges aroach could have sraeges ha aren' subgame erfec Harsany - argued he wo were dfferen; dea of mmedae vs. delayed commmen (here "commmen" means ckng a sraegy Immedae Commmen - a sar of game before knowng ye (sraeges aroach of 7

5 Delayed Commmen - choose sraegy afer knowng ye (layers aroach Slusky - "The ermnology s no longer used n ar because 's no useful." Problem - Harsany argued hese are dfferen and sad delayed commmen s beer bu hs examle roblem combned cooerave and non-cooerave elemens Slusky - no sure wha equlbrum noon s n a mxed cooerave and noncooerave game Same - n fully non-cooerave game sraeges and layers aroach are THE SAME; some felds (e.g. regulaon use he layers aroach because "'s rgh" bu hey're ncorrecly aly Harsany concluson o fully non-cooerave games Hsory - Blackwell Nash Von-Neumann Kuhn all dd wo-layer oker examles usng he sraeges aroach 5 years before Harsany; hey usually used a smlfed verson lke layers each wh 3 yes (hgh medum and low hand; n ha caseraeges aroach s easer hen layers aroach ( layer 8 sraegy game vs. 6 layer sraegy Players Aroach Easer - for nfne yes wh connuous sraegy saces he "easy" way s o arameerze ye and solve usng he layers aroach (e.g. rncal agen roblem; consumer maxmzaon; frm rof maxmzaon Examle - smles case: layers yes of each sraeges: a a and b b Players Aroach - game shown here s Player for layer (only shows hs ayoffs b b even hough we need o know layer a x x 's o solve he game; a a mnmum we also need anoher ar of ayoff a x 3 x ables for layer ; ha wll conan all he nformaon alhough Slusky refers o have a ar for each layer ye (oal of four of hese Noaon: = Pr[ lays a ] (so = Pr[ lays a ] = Pr[ beleves s hs oonen] (so = Pr[ beleves s hs oonen] So he robables always defaul o he frs sraegy or he frs ye of he oonen Execed Uly - E[ u ] = x + ( x + ( x3 + ( ( x + ( y + ( ( y + ( ( y + 3 ( ( ( y Ths s lnear n o we can wre roblem as max f ( x - x y - y + c Prob n box (a b vs. Payoff n box (a b Player Prob n box (a b vs. Player Chosen by naure Payoff n box (a b Player b b a y y a y 3 y... only & are from oher layers Bes Rely - f f ( > 0 = 0 If f ( < 0 (0 f f ( = 0 There wll be a bes rely for he oher hree layer yes resulng n equaons wh unknowns ( 5 of 7

6 Sraeges Aroach - wll show hs s he same as he layers aroach; assume Eq s he ure sraegy equlbrum New Noaon - = Pr[ ] (robably layer s ye Same as Players Aroach - use execed ayoffs o show ex os ex ane Look a Eq frs: Player ye (rob s lays a ; hs oonen could be ye of layer (rob Pr[ ] and lays b or ye of layer (rob Pr[ ] and lays b Player ye (rob s lays a ; hs oonens are he same and lay he same sraeges (bu he condonal robables are dfferen E[Eq] = [ Pr[ ] u ( a b ; + Pr[ ] u ( a b ; + [ ] u ( a b ; + Pr[ ] u ( a b ; Pr[ Snce hs s an equlbrum we know: E[Eq] E[c] c has he followng sraeges ( a a bb so bascally only he sraegy for layer s changed (from a o a ; usng he same logc as above we have E[r] = [ Pr[ ] u ( a b ; + Pr[ ] u ( a b ; + [ ] u ( a b ; + Pr[ ] u ( a b ; Pr[ The corresondng crcled erms cancel ou so E[Eq] E[c] becomes [ Pr[ ] u ( a b ; + Pr[ ] u ( a b ; [ ] u ( a b ; + Pr[ ] u ( a b ; Pr[ Tha s he execed ayoff of a once layer knows he's ye s o he execed ayoff of a once he knows he's ye : E[ u ( a E[ u ( a We can use hs same logc o ge E[Eq] E[c] yelds same condon as layng a over a E[Eq] E[r] yelds same condon as layng b over b E[Eq] E[r] yelds same condon as layng b over b Combnng all four comarson yelds he same equlbrum sraegy as he ex os game: s = a = a = b = b equlbrum n ex ane game s he same equlbrum n he ex os game Now assume s = a = a = b = b s an equlbrum n he ex os game; we wan o show hs s also an equlbrum n he ex ane game; usng he work above we can work backwards o show Eq = ( aa bb has a hgher execed ayoff han r r c c; now consder E[Eq] vs. E[r];.e. comare ( s = a = a = b = b o ( s = a = a = b = b so boh yes of layer are changng her sraegy; changng he noaon a lle for convenence: E[Eq] = E u ( a + E[ u ( [ a E[ u ( a E[ u ( a E[r] = + Player b b b b b b b b a a c a a c a a r r Eq r a a Player 's sraegy f he's: Tye Tye Player c 6 of 7

7 From he equlbrum assumon we know E u ( a E[ u ( and [ a E[ u ( a E[ u ( a (.e. each erm n E[Eq] s 's resecve erm n E[r]... ha means E[Eq] E[r] We can reea hs logc o show E[Eq] E[c] ( s = a = a = b = b n he ex os game s he same as ( a a bb n he ex ane game Why? - hs works because he yes are ndeenden of each oher; we can' have and a he same me so here s no neracon (.e. can change boh her sraeges a he same me and 's equvalen o change one a a me Mxed Sraeges - frs look a ex os; suose lays (// and lays (/3/3 n he ex ane game ha's he same as: ex ane a a /3 /6 a a / /3 /6 a a / /3 /6 a a /3 /6 Problem? - gong he oher way 's ossble o come u wh a mxed sraegy n he ex ane game ha can' be relcaed by mxed sraeges n he ex os game (e.g. (0//0; hs s ar of he nuon why Harsany sad hey were dfferen No Problem - won' ge hs as a mxed sraegy because here's no gan n correlaon beween he ye yes; (0//0 mgh as well be (//// whch also has each ye layng a mxed sraegy; n a real layer game here could be correlaed sraeges so hs sn' he case 7 of 7