# Stage 4 Derivation: Repeated Direction Choice with One Unmatched Increment This note replaces the one-shot direction game and the invalid infinite-intensity extrapolation. The model is a finite embedded-state semi-Markov game. Direction choice repeats after every lead is matched or imitated, and the state explicitly records the single unmatched increment. ## 1. States, actions, and one-increment discipline There are two symmetric laboratories and two directions `i in {H,F}`. The embedded states are: - `0`: neither laboratory owns an unmatched increment; - `L_i^j`: laboratory `j` owns one unmatched direction-`i` increment. At state `0`, each laboratory chooses a direction. A laboratory choosing `i` pays research flow cost `psi_i` while waiting and has an exponential discovery clock of rate `nu_i`. The first discovery ends the research phase. Its increment has scale `s_i` and flow boundary rent `b_i=s_i B_i`. The program has a one-unmatched-increment capacity constraint: no new frontier increment is launched until the current lead resolves. This is the smallest state space that makes repeated direction choice coherent. It is not a claim about an unbounded innovation queue. If the firms chose different directions, a winner's increment is absorbed after its deterministic imitation lag `tau_i`. If both chose `i`, it is absorbed at the earlier of imitation after `tau_i` and the rival's matching discovery, whose exponential hazard is `mu_i`. No action is taken in a lead state. On absorption the game returns to state `0`, where directions are chosen again. Deterministic holding times make the embedded process semi-Markov; adding the age of the lead makes it an ordinary Markov process without changing any value below. Direction-specific `(s_i,psi_i,nu_i,mu_i)` allow scales, research costs, discovery rates, and matching rates to differ. There is no infinite-intensity limit. ## 2. Rent windows and Laplace transforms For a split profile, define `a_i = (1-exp(-r tau_i))/r`, `L_i^S = exp(-r tau_i) = 1-r a_i`. For a same-direction profile, define `d_i = (1-exp(-(r+mu_i)tau_i))/(r+mu_i)`, `L_i^D = 1-r d_i`. Here `a_i` and `d_i` are expected discounted rent windows, and `L` is the discounted continuation factor at absorption. The identity `L=1-r A` follows from `A=E integral_0^D exp(-rt)dt = {1-E exp(-rD)}/r`. Both windows increase in the imitation lag. The head-to-head window `d_i` decreases in the finite matching hazard `mu_i`. ## 3. One-cycle Bellman operator Suppose a player chooses `i`, its opponent chooses `j`, and after the lead resolves the player's continuation value is `V`. Let `m=D` when `i=j` and `m=S` when `i!=j`. The player's one-cycle Bellman operator is `T_i(i,j;V)` ` = [-psi_i + nu_i b_i A_i^m` ` + {nu_i L_i^m + nu_j L_j^m}V] / (r+nu_i+nu_j)`. (SR1) The denominator integrates the exponential race to the first discovery. The cost term is a flow cost until that discovery. On the player's discovery, it receives the lead rent and continuation; on the rival's discovery, it receives only continuation after the rival lead resolves. The coefficient on `V` is strictly below one, so every stationary action profile has a unique value. ## 4. Closed-form renewal values If both players choose `i` every time state `0` recurs, the common value is `U_i^D = (nu_i b_i d_i-psi_i) / {r(1+2 nu_i d_i)}`. (SR2) If one player chooses `i` and the other `j`, with `i!=j`, the direction-`i` player's value is `U_i^S(j) = (nu_i b_i a_i-psi_i)` ` / {r(1+nu_i a_i+nu_j a_j)}`. (SR3) The split-profile denominator is common to the two players. These are renewal values, not one-shot annuities: they include repeated returns to the tied state. ## 5. One-stage deviations and equilibrium regions The Bellman operator is affine: `T(V)=alpha+beta V`, with `beta<1`. If its fixed point is `U`, then `V >= T(V) iff V >= U`. (SR4) This observation turns every one-stage deviation condition into a comparison of the closed-form renewal values. Define `X = U_F^D-U_H^S(F)`, (SR5) `Y = U_F^S(H)-U_H^D`. (SR6) Then: - `F` is a best response to `F` iff `X>=0`; - `F` is a best response to `H` iff `Y>=0`. The pure stationary Markov-perfect equilibria follow: 1. `X>0, Y>0`: unique stationary equilibrium `(F,F)`; reliability is strictly dominant whenever the game returns to the tied state. 2. `X<0, Y<0`: unique pure equilibrium `(H,H)`. 3. `X<00,Y>0` can be written as a threshold in the reliability increment rent `b_F=s_F B_F`: `b_F > max{bbar_F^F,bbar_F^H}`, (SR7) where `bbar_F^F = psi_F/(nu_F d_F)` ` + (1+2nu_F d_F)/(nu_F d_F)` ` * (nu_H b_H a_H-psi_H)/(1+nu_H a_H+nu_F a_F)`, (SR8) and `bbar_F^H = psi_F/(nu_F a_F)` ` + (1+nu_H a_H+nu_F a_F)/(nu_F a_F)` ` * (nu_H b_H d_H-psi_H)/(1+2nu_H d_H)`. (SR9) The first threshold makes `F` optimal against `F`; the second makes it optimal against `H`. Direction-specific breakthrough scale and cost enter without being normalized away. ## 7. Imitation asymmetry and lumpiness Useful derivatives are `dU_i^D/dd_i = nu_i(b_i+2psi_i)/{r(1+2nu_i d_i)^2}>0`, (SR10) `partial U_i^S/partial a_i` ` = nu_i{b_i(1+nu_j a_j)+psi_i}` ` / {r(1+nu_i a_i+nu_j a_j)^2}>0`, (SR11) and `partial U_i^S/partial a_j` ` = -nu_j(nu_i b_i a_i-psi_i)` ` / {r(1+nu_i a_i+nu_j a_j)^2}`. (SR12) Suppose active split research has nonnegative renewal numerator: `nu_i b_i a_i>=psi_i`. Since both `a_i` and `d_i` increase in `tau_i`, (SR10)--(SR12) imply: - both `X` and `Y` increase with `tau_F`; - both `X` and `Y` decrease with `tau_H`. Thus a wider imitation asymmetry expands the strict moat-herding region. Because `d_F` decreases in `mu_F`, (SR10) also implies that `X` falls as the rival matching hazard rises. Lumpy reliability progress (a low finite `mu_F`) makes the `(F,F)` profile more self-sustaining. The model does not send intensity to infinity with fixed step size; all results are finite-parameter comparisons. ## 8. Scope The one-increment cap is an explicit capacity constraint, not a hidden approximation. It makes the repeated state space finite and rules out an unbounded queue of unmatched innovations. Relaxing it requires queue states and is a separate extension. The core equilibrium message is already dynamic: after every imitation or match, firms return to the tied state and choose direction again, with continuation values internalized. The game is local in the task frontier: boundary values and lags are held fixed during one renewal cycle and update between cycles as the state of the outer model changes. This is the same episode logic as the optimal-rotation program, now with strategic direction choice at each renewal.