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Topologies on TypesEddie Dekel, Drew Fudenberg and Stephen MorrisTalk for Southwestern Economic Theory ConferenceArizona State UniversityMarch 2006Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesMotivation Harsanyi (1967/68), Mertens and Zamir (1985): single "universal typespace" incorporates all incomplete information Economic researchers work with small type spaces: leap of faith thatthese capture avor of universal type space1Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types"Finite types" dense in universal type space under product topology, but But strategic outcomes not continuous with respect to product topology Finite common prior types are dense in product topology (Lipman 2003) Types with unique rationalizable outcomes are open and dense in theproduct topology (Yildiz 2006)2Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesDening a Strategic TopologyTWO TYPES ARE CLOSE IF THEIR STRATEGIC BEHAVIOR IS SIMILARIN ALL STRATEGIC SITUATIONS1. "Strategic situations": Fix space of uncertainty and all possible beliefsand higher order beliefs about . Vary nite action sets and payofunctions depending on actions and .2. "Strategic behavior": interim (correlated) rationalizable actions, i.e.,those actions consistent with common knowledge of rationality.3. "Similar" strategic behavior: Allow "-rationality3Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesResults MAIN RESULT: Finite types are dense in the strategic topology{ nite common prior types are NOT dense in the strategic topology{ open sets of types have multiple rationalizable outcomes Product topology implies upper hemicontinuity of rationalizable outcomes Lower strategic convergence implies upper strategic convergence and isstrictly stronger than the product topology Finite types are nonetheless "non-generic" (category 1) in both productand strategic topology4Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesElectronic Mail Game two players 1 and 2 two actions N and I two payo states 0 and 1 payos: = 0 N IN 0; 0 0; 2I 2; 0 2; 2 = 1 N IN 0; 0 0; 2I 2; 0 1; 15Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types with probability , it is common knowledge that = 1 otherwise{ = 1 with probability { player 1 only informed if is 0 or 1{ if = 1, player 1 sends a message to player 2, message lost withprobability { if player 2 receives a conrmation, he sends a reply - lost withprobability .... write t11for the type of player i for whom is common knowledge write tikfor the type of player i for whom is not common knowledgebut he has sent k 1 messages{ for type t1k, 1 knows that 2 knows that (k 2 times)::: = 16Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types{ for type t2k, 2 knows that [1 knows that 2 knows that ( k 2times)::: = 1] Thus Ti= fti1; ti2; ::::g [ fti1g. Common Prior = 0 t21t22t23t24 t21t11(1 ) 0 0 0 0t120 0 0 0 0t130 0 0 0 0t140 0 0 0 0.....................t110 0 0 0 07Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types = 1 t21t22 t21t110 0 0t12(1 ) (1 ) (1 ) (1 )2 0t130 (1 ) (1 )3 0............... 0.....................t110 0 0 0 t1k! t11in product topology8Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types Iterative argument establishes that N is the only "-rationalizable actionfor type tikfor any " <1+2and k < 1{ N is only rationalizable action for type t11{ type t21assigns probability1+2to type t11; thus payo to invest is atmost1 2 (1) +12 (2) = 1 + 2 < 12{ type t12asigns probability1+2to type t21.... Both N and I are rationalizable for type ti1. Thus example shows failure of lower hemicontinuity w.r.t. producttopology No failure of upper hemicontinuity w.r.t. product topology9Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types However, ti1is itself a nite type, so this example does not show afailure of denseness10Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesTypes two players, 1 and 2 nite payo states Type t = (1; 2; :::) 2 1k=0 (Xk) whereX0= X1= X0 (X0)Xk= Xk1 (Xk1) T is set of "coherent" types11Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types There exists homeomorphism : T ! (T ) we ignore "redundant types" - do not matter for our soln concept "nite types": belong to a nite belief closed subset of the universal typespace "product topology": tni! tiif ki(tni) ! ki(ti) as n ! 1 for all k{ where ki(ti) 2 (Xk1) is type ti's kth level beliefs{ often studied, but we do not expect continuous behavior12Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesGames A (two player) game G 2 G consists of{ nite action sets Ai{ payo functions gi: A ! [M; M]13Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesInterim Rationalizability R0i(ti; G; ") = AiRk+1i(ti; G; ")=8>>>>>>><>>>>>>>:ai2 Ai9 2 T j Ajsuch that(i) (tj; ; aj) : aj2 Rkj(tj; G; ") = 1(ii) margT = i(ti)(iii)R(tj;;aj)gi(ai; aj; )gi(a0i; aj; )d "for all a0i2 Ai9>>>>>>>=>>>>>>>;14Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on Types Ri(ti; G; ") = \k1Rki(ti; G; ") Properties (see DFM "Interim Correlated Rationalizability"http://www.princeton.edu/~smorris/pdfs/interimrationalizability.pdf){ Doesn't depend on redundant types{ Ri(ti; G; ") is the set of actions that might be played in any "-equilibrium on any type space by a type whose higher order beliefs aregiven by ti{ Thus Ri(ti; G; ") is the set of actions consistent with commonknowledge of "-rationality for a type whose higher order beliefs aregiven by ti15Eddie Dekel, Drew Fudenberg and Stephen Morris Topologies on TypesConvergence Properties Let hi(tijai; G) be the "rationalizability"