Game Theory ' ' DT Basics : → Representation Information set ' DT interactive → extensive form decision nodes between which : ' < N strategic) form → → normal ( is :) i. c- N , tuition > cannot player a , distinguish • ' ' • ' strategy Ii strategy = S - ( 51 = I of the all Mixed over set the her lol • . . , S L I - S , it strategy player i . . - i. Sax ✗ one , . - . ✗ AS C- the - is i. a i : prob r : c- DI , A, is strategy ;) = distribution . a where , prob set ☐ $ - ie strategies the over space , where D8 , is the distributions over her payoff player Sn other players of probability distributions players except player i player for each : strategies of of game Sn ) s - EU , O : , , the in strategy strategy Expected General - - strategy probability Ui , Mixed , Belief of 52 , player a strategies 5=5 , , for single of = is the • % sie , = a over ' space vector Strategy profile s -= 5 set of strategy profiles Beliefs - complete contingent plan a : . distribution of all space : £ £ sits ; S Assumptions - Tilsit A- its - it F- i ;) uilsi,s-D_ / : Rationality maximising one's subjective expected payoff fact F is CK between the common knowledge : a : players the if each player others know player knows F , knows F each that each , player player each player knows knows that that every knows F , and so other on and BR : Dominance - player strategy dominated it mixed D8 ) strategy Dominance : there uilri , 5- it D - First s & - ) pure is a at , the S , - all players i is or pure it for other tension : (e. g. of si ( ufs.ie > strategic interests • i - a at . . us individual . Prisoner's Dilemma) , Pareto dominance strategy profile strategy profiles : a ' if s u ; ( s ) hits ) for all ' 7 dominates Pareto and the • s .t strategy profiles interests group ; i player inequality is strict for at least one player is Pareto efficient efficiency a strategy profile Pareto s : if there is profiles strategy other no that ' Pareto dominates s ' Best response u :( Si , O > BR :( of • • up ) 0 - his 7 i ) player set : i. = , O , - = strategies all of the - $ - Rational iz ability Ratio realizable BR if a belief any 0 i - for strategic to player i s - - t . S that is is that t are of player beliefs ; are not best i BR , to it } - player game , Bi = UD , : strategies iterated dominance always lead uncertainty ) for player possible finite two a for UD2 second is sit $ , ;) for every of best responses 1 I 0 it ☐ in $i sit i over { sir Proposition : Bz • Isi : B. ' strategy i 's strictly dominated the set of Bi strategies responses • i the set of : , - player : : strategies that tension : rational izability a survive does not unique strategy profile 1strategic - Big picture > Rationality : > > Rationality Rationality strategies Nash is NE hits ; , Strict Nash hi Isi , player . i) - = s, player , or E Duopoly , is ;) NE > . . . , a - That ;) for player i. every s.ie $ ; and ( s ;) ; - s , strategy profile A - for each player se i. $ is That a is , and each every sit % for ;) quantity : price : NE in ) competition competition consider : , , r - l M ) ie , strategy profile a ' where NE MS nil si at D8 ; ✗ each player i. Iff NE ) for for each si c- $; and each i. : in ( pure a : proposition F- very of player 's and a NE is : Duopoly rz $ BR ! only strategy , BR = hits , * (9 uilri s , * ' > strategy Profile ' wits , 7 se ;) for each - i. Bertrand Mixed strategy profile Equilibrium : Iff Gourmet T . S that so mutual BRS A : ;) - Applications Mixed • Equilibrium i. player strict NE • Nash izability single strategy profile a iff sit BR ; ( s s , on are Equilibrium each NE rational : strategy ) • consistency + Equilibrium Players coordinate their ' of rationality CK + Nash ' and BR dominance pure or finite game ( having finite strategy space) mixed strategies a has finite number at least one Extensive Form : ' Perfect recall : well as - as any is a single : mode from Proposition Sub sub an of Sub - sub every info game Perfection : Every : game : the finite BI it neither . that a w/ game identifies contains its info are not a . initiate to successors nodes that a has . equilibrium is said any of war perfect an player analysing beginning , a in are successors . game Perfect if it game NE specifies ( SPE ) a NE : in A strategy profile is called of the original sub every game a . continuation a . process of A woolen in the tree game info set a NE set at which this . The end to the strategy SPE - otherwise : a Backward induction : pure • on the move game ' extensive the : has ' of set . sequential rationality An optimal strategy for player should maximise his or her expected payoff conditional • past actions dashed lines) two sequential Rationality and - info Every Imperfect information > own events that they have observed other Perfect information form remember their Players player's CP payoff is : the For a given mode a in payoff that this on the path of a game , player eventually get contingent play passing through profile strategy mode a 1 based on the will ) Incomplete information Incomplete information refers : • of nature IW1 info fixed between the . to having moves generate asymmetric prob ) that . games players ( exogenous asymmetry ) Perfect info ( chess) e.g → complete information → Imperfect info le g. ( endogenous asymmetry) Type The different moves of nature that . . > ' privately Bayesian w/ game > form : Normal - of form version a nature NE and LBNE ) Bayesian rationalize ability : Second First price - sealed bid auction - dominant weakly - single observes normal Bayesian Auction • a : player ' RP5 ) . . strategy for each price sealed bid auction - - bzlvz) v42 = constitute a : bi.LY) : is a player b , th ) BNE vi = = " I b.dk ) and 12 " = YN WI N people) ' Revenue expected Perfect - equivalence revenue : the two Bayesian Equilibrium Assessment auctions for the seller yield the same : where ✗ is a strategy A pair I ✗ m) and set profile µ is a belief system ; i. e. , for each into I MI II) is a probability measure over nodes in I : , , . , • . Perfect la , M) the Bayesian Equilibrium is a following weak PBE (WPBE ) are satisfied : IPBE ) : An assessment if at each into . set I > > sequential rationality Given belief system µ ✗ is sequentially rational 2) Bayesian updating Given strategy profile ✗ 1) : : M is consistent v1 Bayesian rule whenever , , possible