A Smaller Step

Put three plans on the workshop’s desk. One is to buy the full machine now. Another is to pay for information and decide later. The third is to rent a modest setup that can switch to other work if the rule changes.

We can describe all three as sensible. That is not yet enough to choose between them. We need to state what each plan costs, what it might earn and which outcomes we are willing to bear.

The following calculation is invented. Its numbers describe no observed firm, and its probability is not a prediction of any government’s behaviour. The calculation has been executed in a small program, with its assumptions and outputs available in the accompanying method. Its value is that a reader can change an assumption and see why the recommendation changes.¹

We will judge the plans first by their probability-weighted net dollar payoff. In ordinary language, that is the average result implied by the chances we assign to the possible outcomes. It is one decision rule. It does not protect against the worst loss, guarantee survival or tell us how to value someone else’s working life.

Keep that limitation in view as we do the arithmetic.

One year and two possible rules

All three plans cover the same twelve months. The owner must choose a plan at the beginning. The final rule will become known, and take effect, at month six. She assigns a 35 percent chance to a change that makes the specialized machine much less useful.

For this calculation, the full machine costs $80,000. If the rule does not change, the year’s net operating receipts will be $140,000. Those are receipts after the direct cost of doing the work, before paying for the machine. Subtract the machine cost and the owner ends with a net gain of $60,000.

If the rule changes, net operating receipts fall to $20,000. The machine cannot be returned, sold or switched to another task in this example. After its cost, the owner loses $60,000.

These are deliberately simple assumptions. A real machine may have resale value. Real receipts arrive over time. Credit, taxes and discounting can matter. Leaving them out makes the comparison easy to inspect, but it also makes the example unsuitable as a ready-made investment recommendation.

With a 65 percent chance of gaining $60,000 and a 35 percent chance of losing $60,000, buying now has an expected net payoff of $18,000. That is an average across the two possibilities. Neither actual outcome is $18,000. The owner will live through one outcome, not a smooth blend of both.

This is why the word expected can mislead. It is a mathematical average, not the amount we promise will arrive.

Paying for an imperfect answer

The second plan pays $2,000 for a report. It arrives at month three, before the rule is settled at month six. The owner then buys the full machine or walks away.

Waiting cuts the net operating receipts by $10,000 in either rule outcome. The delay is the same three-month pause throughout the base calculation. If she buys after it, the project yields $50,000 without the change or loses $70,000 with it, before the report fee. If she walks away, there is no project payoff, but the $2,000 fee is gone.

The report gives a green or red signal. If no rule change will occur, it says green 90 percent of the time. If a change will occur, it says red 90 percent of the time. It can therefore be wrong in either direction.

A green signal does not mean that the chance of no change is exactly 90 percent. The answer also depends on the starting chance. At our assumed 35 percent chance of change, a green report leaves about a 5.65 percent chance that the rule will change. A red report raises that chance to about 82.89 percent. The program calculates these values from the possible combinations; the appendix explains the method.

At those updated chances, the owner buys after green and walks away after red.

Four outcomes now matter. No change with a green report happens with probability 58.5 percent and produces a $48,000 gain, including the fee. No change with a red report happens with probability 6.5 percent; the owner walks away and loses only the $2,000 fee. A changed rule with a false green report happens with probability 3.5 percent and produces a $72,000 loss. A changed rule with a red report happens with probability 31.5 percent and again costs only the fee.

Weight those outcomes by their probabilities and add them. The expected net payoff is $24,800.

Waiting has improved the average result compared with buying now. It has not abolished the possibility of a serious mistake. The worst loss is actually larger: the wrong green signal can lead to a late purchase, lost operating receipts and a wasted report fee.

The report helps by allowing the owner to avoid some bad commitments. It cannot be treated as a guarantee simply because its average contribution is positive.

A less ambitious machine

The third plan costs $20,000 for a modest rental setup. That price includes an allowance to return it and switch work if the rule changes. In the invented unchanged-rule outcome, its net operating receipts are $50,000. In the changed-rule outcome, after switching to permitted work, they are $30,000.

The resulting net payoffs are $30,000 or $10,000. At the same 35 percent chance of change, their weighted average is $23,000.

The small step gives up some upside. If the favourable outcome occurs, the owner would have earned more with the full machine. In return, the small setup keeps a positive payoff in both states of this particular example.

Do not give the word small credit for that protection. We wrote it into the assumptions: the rental is cheaper, the switch is feasible, and its cost is included. A badly designed pilot could lose money in both states. A lease with severe exit charges could be harder to leave than a purchase. Reversibility must be found in the actual terms and practical alternatives.

Under our starting assumptions, waiting leads the average-money comparison at $24,800. The smaller step follows at $23,000, and buying now returns $18,000 on average. The lead over the smaller step is only $1,800. A recommendation that ignores how easily that gap could change would be more confident than its own calculation.

Move an assumption

First change only the probability of the rule change. At a low enough probability, the full machine’s larger favourable payoff makes buying now best under the average-money rule. In the middle, the report’s ability to prevent some bad purchases earns its fee and the cost of delay. At higher probabilities, the modest setup’s ability to keep working is worth more.

The exact boundaries are 25 percent and 40.625 percent under these invented amounts. They are not magic thresholds that can be taken into another business. They show where these three specified plans exchange places.

The calculation also updates the choice after each report as the starting probability changes. It does not assume that green must always mean buy. If a rule change is almost certain, even a usually reliable green signal may leave too much risk. If a change is very unlikely, a red report may not justify walking away. Information must be read in relation to the belief it updates.

Now hold the probability at 35 percent and increase the cost of the same delay. Waiting and the smaller step tie at a delay cost of about $12,903, rather than the starting $10,000. Beyond that point, the lost opportunity makes the smaller step better. A business with a fast-disappearing customer order could therefore choose differently from one whose work can wait.

Alternatively, leave the delay cost alone and reduce the report’s accuracy. The two plans tie when accuracy falls to about 86.84 percent. Below that, buying the report is less attractive than the small step at the base probability. This is a useful question for anyone selling advice: how much better must the information be for its price and delay to be worthwhile?

The decimal places serve reproducibility. In a real discussion, our knowledge of the inputs would rarely justify treating the boundary as precise to the last dollar. We should not let accurate arithmetic disguise approximate assumptions.

What the information itself is worth

There is another tempting shortcut: waiting produces $24,800, buying now produces $18,000, so the report is worth $6,800.

That difference is the advantage of the whole wait-and-report strategy over buying immediately. It includes the effect of delaying the project. To isolate the value of the report within the delayed decision, compare two choices at the same date.

Without a report, the best buy-or-walk-away action after the same pause has an expected payoff of $8,000. With the report, before its fee, the owner can choose actions worth $26,800 on average. The information adds $18,800 at that common decision point, or $16,800 after the fee.

A perfect report would do better, but still would not restore the three months. With the same delay and fee, perfect information yields $30,500 in this example. It is useful to keep this ceiling visible. A seller cannot justify an unlimited fee merely by promising a clearer view.

In many real decisions, we will not know the quality of a report well enough to calculate its value this neatly. We can still ask the questions the comparison exposes. What action could it change? What errors can it make? How soon will it arrive? Is the same information available more cheaply? Will its conclusion still be useful when it arrives?

The plan you can survive

Suppose the owner cannot withstand a $60,000 loss. The average-money winner does not solve that problem. Waiting can still lose $72,000 in the false-green outcome. A lender or a family depending on the business may care deeply about that tail of the distribution.

At the base assumptions, the probability of any negative net payoff is 35 percent for buying now and 41.5 percent for waiting. Waiting’s larger figure includes the many cases in which only the $2,000 report fee is lost. Counting losses without their size would therefore give an incomplete picture too. The small step has no negative payoff in our two-state model, because that protection was explicitly assumed.

Cash timing adds another constraint. Under an intentionally simple schedule in which equipment costs are paid before all year-end receipts, the largest amount tied up is $80,000 for buying now, $82,000 for waiting and then buying, and $20,000 for the rental step. Real working-capital needs could change these figures substantially. Even this crude schedule makes one point clear: postponing a purchase does not necessarily reduce the maximum cash it will require.

A person can therefore accept the calculation and reject its average-money choice. They may prefer a lower expected gain with a loss they can bear. They may lack the cash for the winning plan. They may value keeping a reliable promise to workers or customers. Those reasons should be stated, rather than smuggled into the arithmetic or dismissed as a failure to understand it.

The calculation has done its job when the disagreement becomes clearer.

The owner can now say that waiting looks best only within a particular range of beliefs and costs. She can ask whether the report deserves its assumed accuracy. She can check whether the rental really allows the promised switch. She can ask what her plan would require of the people who are waiting with her.

She still has to make a judgment. At least the numbers have become a way to examine that judgment, rather than a way to hide it.

Source notes