that arise in stochastic games Mike Harrison Thera Stochastics May - PowerPoint PPT Presentation

Interesting one-dimensional diffusions that arise in stochastic games Mike Harrison Thera Stochastics May 31, 2017 Aaron Kolb (2016), Strategic real options, working paper, Kelly School of Business, Indiana Univ. (submitted for publication) Brendan Daley and Brett Green (2012), Waiting for news in the market for lemons, Econometrica , 80, 1433-1504. George Akerlof (1970 ), The market for “lemons”: Quality uncertainty and the market mechanism, Quart. J. Econ , 84, 488-500 AaronKolb August 10,2016 17 /33

Five Model Components Added Sequentially 1. Adoption decision 2. Learning 3. Strategic seller (exit) 4. Private information 5. Endogenous quality (upgrades) AaronKolb August 10,2016 6 /33

1. Adoption Decision ∈ { H , L } , buyer has prior p 0 = P ( θ = H ) . Asset has type θ Buyer chooses whether to adopt or not. If she adopts, she gets 1 { θ = H } − k , where k ∈ ( 0 , 1 ) ; otherwise, 0. V B 1 V B ( p ) 0 p 1 k − k Figure: Buyer Value Function AaronKolb ugust 10,2016 7 /33

2. Learning Continuous time, t ∈ [ 0 , ∞) , discount rate r > 0. Players observe news process: dX t = µ  dt + σ dW t , where µ H > µ L .   𝐼 −  𝑀 = 1 (signal-to-noise ratio). Assume for simplicity that φ   Posterior belief process  𝑢  𝑄  = 𝐼  𝑌 𝑡 ,0  𝑡  𝑢 satisfies 𝑒  𝑢 =  𝑢 (1-  𝑢 ) 𝑒𝐶 𝑢 where 𝐶 is another standard BM  𝑢  State process 𝑎 𝑢  log 1−  𝑢 , 𝑢 ≥ 0 Buyer chooses stopping time ρ to adopt   = inf {𝑢  0: 𝑎 𝑢   ∗ } AaronKolb August 10,2016 8 /33

Optimal Adoption Policy in Model with Learning Z t α * t Sample Path of State Process Z AaronKolb August 10,2016 14 /33

3. Strategic Seller ∈ { H , L } , common prior p 0 = P ( θ = H ) . Seller has type θ Seller has flow cost c > 0. Seller chooses stopping time  to exit. Payoffs (excluding flow cost and discounting):  ( 0 , 0 ) if  ≤ ρ  ( 1 { θ = H } − k , k ) if ρ <  Seller exits when 𝑎 𝑢 ≤ β Buyer adopts when 𝑎 𝑢 ≤  β <  <  ∗ Buyer is made worse off AaronKolb 9 /33

Equilibrium pair (  ,  ) with learning, no private info Z t α β t Sample Path of State Process Z 17 /33

4. Asymmetric Information Seller knows his type, buyer has known prior p 0 = P ( θ = H ) . Seller types choose (or randomize over) stopping times  H ,  L . It suffices to consider selling strategies of the following form:  H = ∞,  L = inf {t  0: L t   }, where {L t , t  0} is  and adapted to X,   exp(1), and  is independent of X.  𝑢 Using the log-likelihood transformation, 𝑎 𝑢  log ,𝑢 ≥ 0, 1−  𝑢 𝑎 𝑢 = ሚ + 𝑀 𝑢 𝑎 𝑢 State process = State based on news alone + Conditioning on no exit 10 /33

Equilibrium strategy pair with asymmetric information Buyer:  = inf {𝑢  0:𝑎 𝑢   } 𝑎  = local time of Z at level  Seller: 𝑀 𝑢 = 𝑀 𝑢 α 𝑎 𝑢 There is killing in local time at the reflecting  t boundary (killing rate 1 ) ෨ 𝑎 𝑢 Reflecting Equilibrium AaronKolb 11 /33

Equilibrium necessarily involves randomization  Consider a putative equilibrium of the following form (  <  ): buyer adopts when Z   and low-type seller exits when Z   .  Then buyer will adopt whenever Z   , because seller’s non -exit in that region guarantees  = H.  Thus an equilibrium in pure (non-randomized) strategies cannot have the hypothesized form, and continued reasoning shows that it cannot have any other form either. 11 /33

5. Endogenous Quality Suppose that L can privately upgrade to H for lump-sum cost K ∈ ( 0 , 1 ) . The seller now chooses an exit time  𝑀 for use if low type, and an upgrade time  𝑀 for use if low type (  𝐼 =  𝐼 =  ).  𝑀 = inf {t  0: Q t   }, where {Q t , t  0} is  and adapted to X,   exp(1), independent of X and  . Seller type is now a process {  t , t  0} , and news arrives as 𝑒𝑌 𝑢 =  𝑢 𝑒𝑢 + σ 𝑒𝑋 𝑢 . Buyer’s beliefs incorporate hidden upgrade possibility: 𝑎 𝑢 = ሚ 𝑎 𝑢 + 𝑀 𝑢 + 𝑅 𝑢 . AaronKolb August 10,2016 12 /33

Three possible forms of equilibrium in the model with endogenous quality Critical values K* and K** satisfy 0 < K** < K* <  . 0  K  K**  resetting equilibrium with  parameters  ,  and 𝑨 ∗ (0 <  < 𝑨 ∗ <  <  ) K  K*  reflecting equilibrium with  parameters  and  (0 <  <  <  ) K** < K < K*  skew-resetting equilibrium with  𝑨 , 𝑨 ∗ and  parameters  ,  , Ƹ 𝑨 < 𝑨 ∗ <  <  and  > 0) (0 <  < Ƹ AaronKolb August 10,2016 17 /33

Resetting Equilibrium Z t α z ∗ β t Q t = sum of jumps, each of size (z*-  ), initiated at successive times when Z =  . AarKolb August 10,2016 17 /33

Local Time and Skew-Brownian Motion  Define local time of process Z at level z as follows: 𝑎 𝑨 = 𝑚𝑗𝑛 1 2  𝑛𝑓𝑏𝑡{𝑡  0,𝑢 : 𝑎 𝑡 − 𝑨   } 𝑀 𝑢  0  An SDE involving own local time at z : 𝑎 𝑨 Z t = W t + δ 𝑀 𝑢 (1)  Harrison and Shepp (1981, Annals of Probability ): (1) has a solution iff | δ | ≤ 1, in which case solution is unique  Limit of a rescaled binary random walk that is symmetric except for one distinguished point: 1+  P{up} = 1- P{down} = at the distinguished point 2 August 10,2016 5 /5

Skew-Resetting Equilibrium 𝑎 𝑢 𝑎 𝑢 = ሚ 𝑎 𝑢 + 𝑀 𝑢 + 𝑅 𝑢  𝑀 𝑢 𝑎 , 𝑀 𝑢 =  𝑀 𝑢  < 1 There is killing in local time at level ෥ 𝑨 𝑨 ෥ (killing rate  ) ෨ 𝑎 𝑢 𝑅 𝑢 = sum of jumps, each of size (z*-  ), β t initiated at successive Sample Path of State Process Z times when Z =  . AaronKolb August 10,2016 22 /33

Highlights  Unique equilibrium involves randomization  Surprising appearance of a “punched” or “partially reflected” diffusion process  Novel phenomenon: (partial) reflection with killing in local time  Novel interpretation of (partial) reflection: informational displacement August 10,2016 5 /5