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 TypesDening 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 payofunctions 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 payos: = 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 conrmation, 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+2and k < 1{ N is only rationalizable action for type t11{ type t21assigns probability1+2to type t11; thus payo to invest is atmost1  2  (1) +12  (2) = 1 + 2  < 12{ type t12asigns probability1+2to 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= Xk1  (Xk1) 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  (Xk1) 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; ") = AiRk+1i(ti; G; ")=8>>>>>>><>>>>>>>:ai2 Ai9 2 T j   Ajsuch 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; ") = \k1Rki(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"