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Methods appendix

Methods: A workshop under a possible rule change

Assumptions, exact arithmetic and limits of the invented workshop model.

By Kevin L. Michel4 minute read

A workshop under a possible rule change

Purpose and limits

This is an original, invented teaching example. No number comes from observed firms or a policy forecast. It asks which of three plans has the highest probability-weighted net dollar payoff under stated assumptions. That is an average-money rule, not a rule for protecting the worst outcome or minimizing regret. A person with limited cash or a strong need to avoid losses may choose differently.

All money in the decision model is in thousands of invented dollars. Each plan covers the same 12 months, from month 0 through month 12. There is no later value, discounting, tax, borrowing, or salvage value. Net operating receipts mean money left after direct operating costs, before the equipment cost specified below. These omissions keep the arithmetic visible; they limit its use in real decisions.

The clock and three plans

At month 0, the workshop assigns a 35% chance to a rule change. The final rule becomes known at month 6 and takes effect then. A report arrives at month 3, before the final answer. The workshop chooses among:

  1. Commit now. Pay 80 for a specialized machine. Its net operating receipts over the year are 140 without the change and 20 with it. Subtract the machine cost: the final net payoffs are +60 and −60. The machine cannot be returned, sold, or switched to another job within this model.
  2. Wait for a report. Pay a nonrefundable report fee of 2 at month 0. At month 3, either buy the same machine for 80 or walk away. Starting late cuts net operating receipts by 10 in either state. Thus the delayed purchase pays +50 or −70 before the report fee. Walking away pays zero before that fee. Buying at month 3 remains exposed to the month-6 rule.
  3. Take a reversible small step. Pay 20 for a modest rental setup, including a prepaid return-and-switch allowance. Its net operating receipts are 50 without the change and 30 with it. The changed-state figure includes switching to permitted work. Net payoffs are +30 and +10. These advantages are assumptions about this particular small step, not a claim that all pilots are cheap or safe.

Imperfect information, paid for

The report says green or red. Without a rule change, it says green 90% of the time. With a change, it says red 90% of the time. This does not mean a green report gives a 90% chance of no change: that depends on the starting probability.

At the base assumptions, green occurs with probability 62%. After green, the chance of change is about 5.65%, so buying has positive average value. After red, that chance is about 82.89%, so walking away is better. The fee is already spent at this decision point.

The four possible waiting outcomes are: no change/green, probability 58.5%, payoff +48; no change/red, 6.5%, payoff −2; change/green, 3.5%, payoff −72; change/red, 31.5%, payoff −2. Multiplying each payoff by its probability and adding gives 24.8. Commit now averages 18; the small step averages 23.

Without a report, the best delayed buy-or-walk-away action averages 8. The report-guided action averages 26.8 before its fee, so the information is worth 18.8 gross or 16.8 net at the same delay. A perfect report, still paying the same delay cost and fee, would yield 30.5. Information is neither free nor perfect in the base case.

Sensitivity and financial exposure

The script recalculates the best action after each signal throughout the probability grid. It does not mechanically retain the base green/red policy. Holding other assumptions fixed, commit wins below 25% change probability; wait wins between 25% and 40.625%; the small step wins above 40.625%. Adjacent strategies tie at those boundaries. No plan always wins.

At 35% probability, waiting and the small step tie when the delay cost reaches about 12.9032, or when report accuracy falls to about 86.8421%. The delay sweep changes the cost of the same three-month pause, not the calendar horizon.

The separate exposure table matters. Worst realized payoffs are −60, −72, and +10. Probabilities of a negative payoff are 35%, 41.5%, and zero. Waiting’s average payoff conditional on a changed rule is −9; that is not its worst outcome. For a deliberately simple cash schedule, all net operating receipts arrive at year-end. Maximum cash tied up is then 80, 82, and 20. This schedule omits real working-capital demands and is not a financing estimate.

Who bears waiting, and reproduction

A separate illustration lists the owner’s 10,000-dollar missed net contribution; 120 unpaid worker hours at 25 dollars per hour; 30 buyer workaround hours; and a supplier’s 5,000-dollar receipt postponed 90 days. The owner temporarily retains that same payment. Do not sum these rows: units differ, hours and wages describe the same burden, and the transfer appears on both sides. A delayed receipt is not a lost sale. No social-welfare total is estimated.

Run python3 analysis/run_analysis.py from the project directory. Python 3.9+ and its standard library suffice. inputs.json holds the assumptions. Exact rational arithmetic produces JSON, CSV tables, and desktop/mobile SVGs; there is no random sampling. CSV values are displayed to nine decimals. hashes.json records SHA-256 hashes of the input, script, method, results, and figures, excluding itself.