STATIC GAMES OF INCOMPLETE INFORMATION

Transcription

1 ECON 10/410 Decsos, Markets ad Icetves Lecture otes Nls-Herk vo der Fehr SAIC GAMES OF INCOMPLEE INFORMAION Itroducto Complete formato: payoff fuctos are commo kowledge Icomplete formato: at least oe player s ucerta about aother player s payoff fucto Bayesa games Examples (sealed-bd) auctos bargag (dyamc) Represetato ormal form or extesve form belefs Solutos: Refemet of Nash Equlbrum Bayesa Nash Equlbrum belefs at equlbrum Normal-Form Represetato he ormal-form represetato of a -player statc Bayesa game specfes the players strategy (acto) spaces S 1,..., S ( A,..., 1 A ), ther type spaces,..., 1, ther belefs p,..., 1 p ad ther payoff fuctos u 1,..., u. he game s G S,..., S ;,..., ; p,..., p ; u,..., u. deoted { } =

2 Player s type, t, s prvately kow by player, determes player s payoff fucto u (,..., ; s1 s t ) ad s a member of the set of possble types,. Example hgh-cost versus low-cost frms successful versus usuccessful developers of ew techology Player s belef ( ) p t t descrbes s ucertaty about the -1 other players possble types, t, gve s ow type, p t t gves the probablty (as see by player ) that oppoets types are t, codtoal o player s type beg t. Example probablty dstrbuto over compettors costs t. he fucto ( ) probablty dstrbuto over whether compettors have succeeded Note: both cases belefs may deped o ow type (eg. more lkely that compettor has low costs (has bee successful), f the frm tself has low costs (has bee successful)). Problem: How are belefs determed? How do we esure that belefs are compatble? mg (Harsay, 1967) 1) Nature draws a type vector t = ( t 1,..., t ), where t s draw from the set. ) Nature reveals t to player, but ot to ay other player. ) Players smultaeously choose strateges (actos). 4) Payoffs are receved. Note that ths set up trasforms a game of complete formato to a game of mperfect formato the move of Nature s ot (perfectly) observed ad hece players do ot kow the whole hstory of the game whe they have to make ther move. It s assumed that the probablty dstrbuto accordg to whch Nature moves s commo kowledge; that s, all players kow that Nature draws t = ( t 1,..., t ) accordg to the probablty dstrbuto p( t ).

3 After dvdual types have bee revealed, dvdual belefs ca be calculated usg Bayes rule: ( ) p t t ( ) pt ( ) t ( ) pt ( ) pt, t pt, t = = pt t s the probablty that ature draws type t for player ad where ( ) t for s oppoets, ad ( ) types draws type t for player. pt s the (margal) probablty that ature It follows that other players ca compute the varous belefs that player may hold, depedg upo ther types. Example: wo players, 1 ad, play a game. Player may be of oe of two types, t = 1 or t = 1. he probablty that Player s of type t = 1 s p. Gve Player s type, the game may be descrbed by the followg table: Player L R U 1,t -,0 Player 1 M 0, 0, D -,0 1,-t he ormal-form represetato may be descrbed as follows: Players strategy spaces are S1 = { U, M, D } ad S = { L, R }. Player 1 s type set s a sgleto (the type s commo kowledge), whle Player s type-space s = { 1,1}. he belefs of Player 1 about Player s type s gve by ( t = ) = p ad ( t ) Pr 1 table above. Pr = 1 = 1 p. he payoffs are as descrbed the Exteso: games whch players have prvate formato ot oly about ther ow payoffs, but about other players payoffs also (example: asymmetrc formato about demad codtos olgopoly). Cosequetly, payoffs are u s,... s ; t,..., t. fuctos of other player s types also, ( ) 1 1

4 Bayesa Nash Equlbrum I a statc Bayesa game = { 1,..., ; 1,..., ; 1,..., ; 1,..., } for player s a fucto ( ), ( ) G S S p p u u, a strategy s t, where for each type t s t specfes t would choose f draw by the acto from the feasble set S that type ature. I a separatg strategy, each type chooses a dfferet strategy,.e. s tˆ s t f tˆ t. I a poolg strategy, all types choose the same ( ) ( ) j strategy,.e. ( ) = s t s,allt. I the statc Bayesa game = { 1,..., ; 1,..., ; 1,..., ; 1,..., } strateges * = ( 1 *,... *) t, s * ( t ) solves G S S p p u u the s s s are a (pure-strategy) Bayesa Nash equlbrum f for each player ad for each of s types ( ) ( ( ) ( ) ) s p t t u s1 t1 s s t t t max *,...,,..., * ; I other words, gve a player s belef about the other players types, hs or her strategy s a best respose to the other players strateges. hs s true whatever type player may be. Example cotued: I the above example, Player has a (weakly) domat strategy: f hs type s t = 1 the best respose s R whatever the choce of Player 1; f hs type s t = 1 the best respose s L. Suppose Player 1 beleves that Player wll play L whe he s of type t = 1 whch occurs wth probablty p ad play R whe he s of type t = 1 whch occurs wth probablty 1-p. he Player 1 s payoff from playg hs varous strateges are [ ] [ ] [ p] + p = [ ] [ ] U: 1 p 1+ p = 1 p M: D: 1 p + p 1= p It follows that U s a best respose f p 1, D s a best respose f p, whle M s a best respose f 1 p. Cosequetly, we have the followg equlbra 4

5 1 p < : s1 * = U, s * ( 1 ) = R, s * ( 1 ) = L 1 < p < : s1 * = M, s * ( 1 ) = R, s * ( 1 ) = L p > : s1 * = D, s * ( 1 ) = R, s * ( 1 ) = L Whe p = 1 ad p = we have multple (.e. two) equlbra. (Note that, ths game, Player may actually beeft from the fact that Player 1 does ot kow hs type; hece he has a cetve to hde hs true type,.e. to make p close to 0.5.) Frst-Prce, Sealed-Bd Aucto Suppose two bdders partcpate a aucto for oe object. Bdder has valuato (.e. maxmum wllgess to pay) v for the object, = 1,. Bdders valuatos are depedetly ad uformly dstrbuted o [ 0,1 ]. Bdders are rsk eutral. Bdders smultaeously submt ther bds, whch must be oegatve. he bd of player s deoted b, = 1,. he bdder wth the hghest bd ws the object ad pays a prce equal to hs bd (f bdders submt the same bd, the object s allocated o a bass). he loser gets ad pays othg. I ths game, player s strategy space s S = [ 0, ) = [ 0,1] v s uformly dstrbuted o [ ] j payoff s, whle the type space s. Because valuatos are depedetly draw, player s belef s that 0,1, o matter what the value of v s. Player s v b f b > bj 1 u ( b1, b; v1, v) = [ v ] f = b b bj 0 f b < bj Suppose bdder beleves bdder j s strategy the fucto from type to acto s gve by bj ( v j ). Let Fj ( b ) deote the probablty that bdder j bds below b. he bdder chooses hs bd so as to solve the problem max [ ] ( ) v b F b. b j he frst-order codto for ths problem may be wrtte 5

6 ( ) [ ] ( ) F b + v b F b =0. j j Note the probablty that bdder j bds below b s equal to the probablty that bdder j s valuato s below v, where v s such that bj ( v) = b. Sce F b v v. By ( ) = ( ) ( ) valuatos are uformly dstrbuted o [ 0,1 ], t follows that j j ( ) mplct dervato, we fd F b ( v) b v = 1 or ( ) = 1 ( ) ( ) j j j F b v b v. j j j We are lookg for a symmetrc equlbrum, whch both players follow the same strategy; that s, b1( v) = b( v) = b( v ) ad F1( b) = F( b) = F( b ). Substtutg to the above equatos ad solvg, we fd 1 v + v b( v) = 0, bv whch may alteratvely be wrtte ( ) b ( v) v + b( v) = v. d Note that the left-had sde s equal to dv ( ( ) ) sdes wth respect to v, we fd 1 bvv ( ) = v + A bvv. he, tegratg both Note that the bdder type wth the lowest valuato must bd zero; that s, b ( 0)= 0. It follows that bv ( )= 1 v So, at equlbrum each bdder submts a bd equal to half hs valuato. Wth bdders, oe ca show that strateges are gve by ( ) bv 1 = v, whch mples that bdders bd more aggressvely (.e. closer to ther valuatos) the more bdders there are. he Revelato Prcple See Gbbos, ch... 6