Expected value and decision trees

Modules 2 and 3 gave you chances and amounts. A client does not need them for their own sake. A client needs to choose: settle or fight, sign or walk away, fix now or wait. A decision tree puts the choice and its possible results on one page.

1. State the decision and the options. For Alderwick: settle for $90,000 now, or go to trial.
2. List the results of each option. Settling has one result. Trial has three: win in full, win in part, lose.
3. Put a chance on each result. The chances after a decision must add up to 100 per cent.
4. Put a money value on each result, net of what you still have to pay.
5. Work backwards from the results: multiply each chance by its value and add them up.

Expected value is what you would get on average if you could make the same choice many times. For trial: 45% x $200,000 + 20% x $100,000 + 35% x $0 = $110,000, less the $40,000 cost of trial = $70,000. Settling gives $90,000 for certain.

On these numbers settling is better by $20,000. A single trial will not give $70,000: it gives $160,000, $60,000 or minus $40,000. The $70,000 is the average of those, not a result.

The most useful number is often not the average but the break-even chance: the chance of winning at which the two choices are equal. If your estimate is far above it, the choice is safe; if it is close, the choice is delicate and worth more work on the chances.

For Alderwick, with the other chances keeping their proportions, trial becomes better once the chance of a full win rises above about 57 per cent. The starting estimate is 45 per cent, so settling is better, but the gap is not enormous. The chart in the lab shows exactly where the lines cross.

This is also the cleanest way to talk to a client who disagrees with your estimate. Instead of arguing about 45 or 55 per cent, ask: “Do you believe the chance of a full win is above 57 per cent?” The disagreement becomes a clear question that can be tested against the evidence.

• Attitude to loss. A client who could not survive a loss may sensibly take the sure sum even when trial has the higher average.
• Time. Money received in two years is worth less than money now, and trial takes time. Include the cost of delay.
• Everything not in the tree. The relationship with the buyer, precedent for future disputes, management distraction. If they matter, add them as values or note them beside the tree.

• Branch chances that do not add to 100 per cent.
• Leaving out the middle outcomes, so the tree shows only total victory or total loss.
• Forgetting to subtract the cost of the route (trial costs money even if you win).
• Presenting the expected value as what will happen.
• Reporting the answer without the break-even, so the reader cannot see how fragile it is.