# Week 5 lab: managers, matching and authority ## Question and learning objectives What do managers contribute, and how should organizations assign authority and discretion? The lab distinguishes assignment to a manager from managerial skill, match quality and decision rights. You will reconstruct a paired assignment estimator, distinguish two different assignment enumerations, diagnose matching and measurement failures, and propose one field design with a credible comparison and an explicit remaining threat. Weidmann et al. (2026) is the primary design anchor. Its randomized laboratory assignment, role selection, predictive skill evidence and separate field validation are distinct evidence layers. The exercise below is an original synthetic matched-pair construction, not that paper's task, participants, protocol, microdata or numerical results. Bandiera–Best–Khan–Prat (2021), Minni (2026) and Fenizia (2022) provide different transfer routes; they do not share one common design. ## Setup and source mode From the lab root: ```bash ENV_NAME=research bash smoke.sh ``` **Mode A: synthetic design reconstruction** runs entirely offline with Python's standard library. The required marker is **`SYNTHETIC_TEACHING_CONSTRUCTION_NOT_PUBLISHED_REPLICATION`**, and `empirical_replication` is false. **Mode B**, optional original replication, was omitted. The supplied pack records article/archive pointers; no original file contents or licenses were inspected here, and no original data or code were downloaded or executed. The numerical generator accepts `--output` only for an empty or nonexistent directory. The wrapper uses a fresh staging root and an explicit inventory to update the documented lab without replacing its README. It refuses to overwrite modified generated inputs. Read `README.md` before running a separate seed. Smoke tests real artifacts and fresh executions; no statistically significant estimate is required. ## Reproduce: a paired incoming-manager assignment ### Population, chronology and data ledger Open `data/teaching/worker-manager-schema.csv` and `worker_manager_panel.csv`. There are 12 pairs, 24 teams, eight continuing workers per team and nine periods, −3 through 5: 192 workers, 216 team-periods and 1,728 worker-periods. All teams receive a new manager at period zero. Within each pair, one team receives a higher-scored incoming type and the other a lower-scored type. Both arms therefore experience manager replacement. The incoming score is fixed independently of the focal output before exposure; the simulation uses labels zero and one, not an estimated manager-quality regression. Assignment is constant within team. `manager_id` changes from a prior manager to an incoming manager at zero, while the worker roster stays fixed. Potential outcomes, abilities and shocks are generated before assignment using a separate random-number stream. Both arms have a common period-zero disruption, and the high-type effect is zero in the three baseline periods. Write a reporting ledger identifying worker, team, pair, manager and work period. Explain why the assignment label can be present on pre-period rows without implying pre-period treatment. Actual appointment decisions, announcements, awareness and arrival dates may differ in a field study; this toy uses one common arrival date and does not observe those separate events. ### Point estimate and event contrasts The target is the average team-level effect of assignment to the higher rather than lower incoming type across these teams and six post periods. It is not the effect of training a manager, creating a skill, adding a manager or changing authority. Average worker output within each team-period. For each team, subtract mean output in periods −3, −2 and −1 from mean output in periods 0 through 5. Within each pair, subtract the low-assignment team's change from the high-assignment team's change. Average the twelve pair differences with equal weights: ```{math} \widehat\tau=\frac{1}{12}\sum_{p=1}^{12}d_p, \qquad d_p=\Delta Y_{p,H}-\Delta Y_{p,L}. ``` Reconstruct `output/reproduced/assignment_pair_contrasts.csv` and compare with `assignment_summary.json`. The reference standard error is the sample SD of the twelve differences divided by $\sqrt{12}$. The lab does not claim an exact small-sample confidence interval or justify treating repeated worker rows as independent assignments. For `event_time_contrasts.csv`, compare each period with the **mean of the three pre-periods** within team, then take the high-minus-low contrast within pair and average. This differs from a regression omitting period −1. The pre-period contrasts average to zero by construction; that normalization is not evidence against selection or anticipation. The post-period contrasts average to the main ITT. Use the chapter's synthetic event-time figure, and state its baseline, weights and twelve assignment units explicitly. Do not turn its pair SEs into fabricated empirical confidence intervals. ### Two enumerations with different meanings For the **sharp null of no assignment effect on any team**, hold the realized team changes fixed and enumerate all $2^{12}=4096$ within-pair assignments. The two-sided p-value counts absolute statistics at least as large as observed, including ties. This is exhaustive enumeration, so no Monte Carlo correction is added. Inspect `exact_sharp_null_distribution.csv`: its center is zero because the permitted assignments include every sign reversal. This test assumes the declared assignment support and no interference across teams. It is not a general test of a zero average effect under unrestricted heterogeneity. Do not shuffle worker rows or send managers to teams that were not eligible matches. A large p-value does not establish equivalence, and rejection alone does not identify a mechanism. The **instructor truth check** instead uses both potential outcomes, changes treatment assignment and reconstructs the corresponding outcome realization. Averaging its estimates over all assignments equals the known synthetic team-average effect. See `data/instructor_only/design_truth.json`. This checks the estimator's expectation under the constructed DGP; it is **not** the sharp-null reference distribution. The main analyst panel excludes potential outcomes and true effects. Explain the distinction without joining truth into a learner feature set. ## Diagnose ### 1. Rescue appointments and mean reversion `data/teaching/rescue_appointment_example.csv` follows eight sites. Four receive a temporary −10 shock just before a replacement and return to output 50 afterward; the replacement's causal effect is zero. The last-pre to first-post DiD is $(50-40)-(50-50)=10$. Compute it and use earlier observations to explain the misleading comparison. A nominal control group and site fixed effects do not remove appointment responses to temporary shocks. The known zero-effect field is kept in a separate instructor file. ### 2. Connectedness supports comparisons, not exogeneity `connectedness_and_rank.csv` compares four managers permanently tied to four sites with connected rotations. The matrix has an intercept, three site indicators and three manager indicators: seven columns after omitted-category normalizations. Rank is four with four disconnected components and seven with one connected component. Reconstruct the edges and ranks, then explain why connected rotations permit additive comparisons without proving exogenous appointments, a portable manager ranking or separately identified match effects. ### 3. A noisy score is not pure managerial heterogeneity `balanced_noisy_score_example.csv` contains two noisy scores and a grid identifier. The instructor companion holds the balanced latent components $(-1.5,-0.5,0.5,1.5)$ and two independent errors taking values ±0.6. Single-score **population** variance is 1.61; cross-half covariance is 1.25, equal to the latent variance in this balanced construction. Recompute the moments using divisor $N$, not a sample-variance adjustment. This is an algebraic measurement-error illustration, not the Kline–Saggio–Sølvsten estimator or a generally unbiased correction for arbitrary personnel panels. Independent score construction, cross-fitting and leave-out variance estimation solve different problems. Describe how a future-output score can leak the focal outcome, which dependence unit a split must respect, and which assignment threats remain after avoiding leakage. ### 4. Information versus conflict in delegation `delegation_information_conflict.csv` uses fixed signals, quadratic loss, posterior-error orthogonality and no cross-team externality. With zero decision costs, central loss is 0.64 and delegated loss is $0.16+b^2$. Verify 0.25 at $b=0.3$ and 0.80 at $b=0.8$. Delegation can lower the principal's expected loss with sufficiently aligned objectives; this does not solve the full authority game, endogenous information acquisition or a social-welfare optimum. State what a field decision-rights ledger would need to observe. ### 5. Attention queues and abandoned decisions `queue_illustration.csv` uses an invented M/M/1 queue: independent Poisson arrivals, exponential service and one server. Escalation arrivals are $\lambda=n\nu p_e$, and mean **sojourn time**, including waiting and service, is $1/(\mu-\lambda)$ only when $\lambda<\mu$. With $\nu=0.5$ problems per worker-hour, $\mu=4$ resolutions per hour and $p_e=0.2$, spans 10 and 30 imply 20 and 60 minutes. At span 40 the queue is nonstationary: the CSV leaves the mean blank and flags `stationary=0`. Never code that row as zero or a capped finite estimate. Reducing escalation to 0.1 at span 30 gives 24 minutes. Recompute all four cases. These are not estimates of an optimal span. Discuss priorities, endogenous requests, managerial multitasking and requests abandoned before approval. ### 6. Composition and comparative advantage `composition_accounting.csv` moves from total output 80 across ten workers to 60 across six: mean output rises from 8 to 10 while total service capacity falls. The main randomized module has no turnover; this separate accounting exercise shows why field estimates should retain exits, headcount and totals instead of silently changing denominators. It does not identify the effect of an actual manager-induced exit policy. In `comparative_advantage.csv`, manager A produces 14 in a technical team and 9 in a customer team; B produces 12 and 13. One manager per team gives total output 27 or 21 depending on assignment. A universal rank misses the match-specific advantage. Neither arithmetic example establishes field causality or welfare-optimal allocation. ## Transfer: one manager-effect or authority-reform design Complete `transfer-memo-template.md`: two pages plus a compact dictionary. Specify the transition, comparison group, event chronology and largest remaining matching threat. Choose whether the target concerns a person, match, practice or decision right. Record reporting and decision-rights ledgers, operational output, independently recorded quality, worker welfare, resources and exits. For an observational transition design, distinguish L→H versus L→L from H→L versus H→H, with independently constructed prior scores and explicit treatment histories. Those comparisons are design proposals, not a generic repeated-switch TWFE recipe. For randomized authority, preserve actual eligible assignments and strata; this lab's twelve pairs cannot stand in for an original field protocol. The proposal must state two competing mechanisms and a prediction that would discriminate between them. Separate total organizational effects from direct within-task or continuing-worker effects. Persistent gains could reflect learning, allocation, composition, resources or reporting; they do not uniquely identify teaching. State what is learned and what remains unidentified about welfare and policy portability. No restricted-data application or contact with an employer is required for this assignment. ## Submission Submit the ledger/dictionary, twelve pair contrasts and primary inference explanation, the event-time baseline explanation, short diagnostic calculations, the two-page proposal and a reproducibility log. Label synthetic results, record command and seed, and keep the two assignment enumerations separate. Assessment weights are 25% reconstruction, 30% diagnosis, 30% feasible transfer design and 15% transparency. A polished coefficient table without a credible comparison or with future-outcome leakage does not satisfy the design component.