1
The matrix is the same
The switch and the footbridge each offer one life against five. Written as acts against outcomes, the two cases are identical, and so is every rule that looks only at the payoffs.
Lesson · Ethics and the Arithmetic of Welfare · Consequentialism
The Trolley Problem as a Decision Matrix
Lay the switch and footbridge cases out as acts against outcomes. Total welfare gives both the same recommendation because their numbers are the same. A side constraint that forbids an act removes it before the maximising, and that is the only thing that makes the recommendations differ.
acts × outcomes: divert −1, do nothing −5
total welfare: take the largest payoff
side constraint: delete the forbidden act
then maximise over what is left
A decision matrix with a constraint
Core ideas
What this lesson establishes
Decision and Rationality supplied the matrix. The new ingredient is a rule that acts before the maximising does.
2
A constraint removes an act
A side constraint says some acts are not to be performed whatever the outcomes. The constrained rule strikes them from the table first, then maximises over the acts that remain.
3
The lab applies a list, not a theory
Which acts are forbidden is typed by you. The lab does not know that pushing a man uses him, or that diverting does not, and it takes any list you give it.
The material
Acts, outcomes and a forbidden list
Three trolley cases as one table each, and one rule that changes the answer.
In the switch case a runaway trolley will kill five people unless you pull a lever that sends it onto a side track, where it will kill one. In the footbridge case it will kill five unless you push a large man off a bridge into its path, killing him. Most people say pulling the lever is permissible and pushing the man is not.
The first thing to see is that the numbers cannot account for that. Let the payoff be minus the number of deaths and let the outcome be certain.
switch certain
divert −1
do nothing −5
footbridge certain
push −1
do nothing −5
The two tables are the same table with the first act renamed. Total welfare, maximising the payoff, says divert in the first and push in the second, and no rule that reads only the table can say otherwise. If you judge the cases differently, something that is not a cell in the table is carrying the difference.
Definition · Side constraint
A side constraint is a restriction on acts that holds whatever the outcomes. The constrained rule deletes every forbidden act from the table and then picks the act with the best payoff among those left.
With “do not push a man to his death as a means” on the forbidden list, the footbridge table loses its first row and the only act left is to do nothing, with payoff −5. The switch table has no forbidden act and its answer does not change. The numbers were never the difference; the list was.
A constraint is not a penalty. Subtract a large number from the pushing payoff and the act sinks in the table; but a large enough gain elsewhere brings it back, and the number can be outweighed. A constraint is applied before any weighing, so no gain can reach it.
Example · The loop
Suppose the side track loops back to the main one, and the trolley will kill the five unless the one man’s body stops it. The numbers are the same again. Whether diverting onto the loop belongs on the forbidden list depends on whether the man is being used to stop the trolley, and people disagree. The loop preset lists it as forbidden. “Double Effect, Means and Side Effects” takes that question up with a causal model.
The lab applies the list you give it, and whether an act belongs on the list is the ethical question. It also cannot say why a constraint should exist, or how it fares when the stakes grow large. It shows what a constraint does to a recommendation, and no more.
Interactive
A decision table under every rule
the same payoffs, eight ways of choosing, each computed exactly
Method
Running a case through the matrix
Five steps. Run the third before the fourth, so the constraint’s effect is something you can see.
1
List the acts and the outcomes
Write each act you can perform as a row and each way things might go as a column. A certain outcome is one column.
2
Fill in the payoffs
Put in the welfare of each act in each outcome. Here it is minus the number of deaths.
3
Apply total welfare
Choose the act with the largest payoff. Record it, with its score.
4
Write the forbidden list
Name the acts a side constraint rules out. This is a judgment the table cannot make for you.
5
Run the constrained rule
Strike the forbidden acts, maximise over the rest, and compare with step three. A case where the answer differs is one where the constraint bites.
Numbers and constraints
Score 0 / 0Choose the answer you can justify.
Three cases, one table
The payoffs are the same in every case. Only the forbidden list changes the recommendation.
switch divert -1 do nothing -5 divert
footbridge push -1 do nothing -5 push (total)
footbridge, push forbidden do nothing
loop, divert to loop forbidden do nothing
Each recommendation is the best payoff among the acts allowed. In the second row nothing is forbidden and in the third the allowed set has shrunk to one act. Switch the lab’s rule to expected utility and the footbridge and loop cases move to the act they forbid, while the switch case stays where it was.
Common mistakes
1
Thinking the two cases differ in their numbersThe switch and footbridge tables are identical: one dies against five. Everything that differs between the lab’s two answers comes from the forbidden list. If you judge the cases differently, the difference lies in a rule about acts, not in a cell.
2
Thinking a side constraint is a very large penaltyA penalty is a number and can be outweighed; a constraint is applied before the numbers are weighed. The distinction matters when the stakes rise, and it is the distinction deontologists draw.
3
Expecting the lab to know which acts are forbiddenThe lab applies the list you type. That pushing belongs on it and diverting does not is the thing in dispute, and the lab has no opinion about it.
Check your understanding
✓
Finish when you can name the case in which a constraint changes the answer.
Given a trolley case, you should be able to write it as a decision matrix, apply total welfare, write the forbidden list and apply the constrained rule, and say which cases change and why the numbers could not have explained it.
Note: Real versions of these cases rarely come with certain outcomes. A state column with probabilities makes the same structure a decision under risk, and the constrained rule treats it the same way: strike, then maximise.