Expected Values and the Maximax, Maximin and Minimax Regret Decision Rules

A caterer in Kano must decide how many crates of ingredients to buy before knowing how many guests will come. Buy too few and sales are lost; buy too many and the surplus is wasted. The choice is not a matter of calculation alone, because it depends on what the caterer knows about demand and what the caterer is willing to risk.

If the chances of low, medium and high demand can be estimated, the expected value gives a single figure to compare. If they cannot, the choice depends on the decision maker’s attitude to risk, and three rules apply: maximax for the optimist, maximin for the cautious, and minimax regret for the person who wants to limit the cost of being wrong.

This topic explains the pay-off table that organises the figures, the three rules and expected values, and when each is appropriate. It builds on the earlier topic on reducing uncertainty. The next topic covers decision trees for multi-stage decisions.

Pay-off tables and expected values

The pay-off table

When a choice must be made between mutually exclusive options and the result depends on a future state that is not yet known, a pay-off table (or profit table) records every combination. One side lists the options, the other the possible outcomes (for example weak, normal and strong demand), and each cell shows the profit, or cost, of that option in that state.

The table does not say which option is best; it organises the figures so that a decision rule can be applied. Where there is a limit on what can be sold, as when perishable goods are bought in advance, the cell must show the revenue only for the quantity actually sold, and the cost of all the quantity bought.

Expected value

Where probabilities can be assigned to the outcomes, the expected value is the weighted average of the pay-offs: multiply each pay-off by its probability and add. The decision rule is to choose the option with the highest EV of profit or the lowest EV of cost.

• Advantages. It uses all the outcomes and their probabilities; it expresses risk in a single figure, which makes comparison easy; and it is a logical rule for a risk-neutral decision maker.
• Disadvantages. The probabilities may be hard to estimate and, if they are unreliable, so are the EVs. The EV is an average and is often not an outcome that can actually occur (an EV of ₦320,000 may arise from outcomes of ₦100,000, ₦300,000 and ₦700,000 but is never itself achieved). It gives no indication of the range or spread of outcomes, so it is an inadequate measure of risk. For a decision that happens only once, the average will not be achieved.

EVs are most reliable when probabilities can be estimated with reasonable accuracy and the decision is repeated many times, so that good and bad results average out. A company deciding a policy on repairs for 10,000 units a month, using defect rates from records, can rely on the EV. A company deciding whether to launch a single new product cannot: it will experience one outcome, not the average.

The three decision rules and the choice between them

Maximax

For each option, find the best possible pay-off, then choose the option with the highest of these. It is based on the view that the decision maker should aim for the highest possible return, assuming events turn out in the best way. It is the rule of the risk seeker or optimist. It ignores the possibility of poor outcomes and the probabilities of any outcome.

Maximin

For each option, find the worst possible pay-off, then choose the option whose worst pay-off is the highest. It maximises the minimum return, and is the rule of the risk averse or pessimist. It guards against the worst case but may reject options with a small chance of a poor result and a large chance of a great one.

Minimax regret

Regret is the opportunity cost of having chosen the wrong option, given the state that actually happens: the best pay-off in that state minus the pay-off of the option chosen. To apply the rule:

1. In each state (each column), find the best pay-off.
2. For every cell, regret = best pay-off in that column minus the pay-off in the cell. The best option in each state has a regret of zero.
3. For each option, note the largest regret.
4. Choose the option with the smallest of these largest regrets.

It is the rule of the person who dislikes the thought of having chosen wrongly; it limits the maximum loss relative to what could have been earned.

Which rule?

The three rules are used where there are no reliable probabilities, and each assumes a different attitude to risk. Their common weakness is that they ignore probabilities: maximin may reject a project because of a very unlikely worst case. If probabilities are known, expected value is usually more suitable, although it ignores risk.

A manager can use all of them and compare: where they agree the choice is strong, and where they disagree the manager must decide how much risk is tolerable. For costs, the rules are reversed: minimin (the lowest of the lowest costs), minimax (the lowest of the highest costs) and regret measured as the excess over the lowest cost.

Layouts for pay-off tables and decision rules

1. Build the pay-off table. For perishable stock: revenue on units sold (the lesser of demand and quantity bought) less the cost of all units bought.
2. Expected value. For each option: pay-off × probability, summed.
3. Maximax / maximin. Best and worst pay-off in each row; choose the highest of the bests or the highest of the worsts.
4. Regret table. Best in each column minus each cell; maximum of each row; choose the smallest.

Money is in naira and each answer is rounded once, at the end.

Worked examples

1. Expected values.
   A: 0.3 × 100 + 0.5 × 300 + 0.2 × 700 = 30 + 150 + 140 = 320.
   B: 60 + 140 + 80 = 280.
   C: 0 + 125 + 170 = 295.
   Choose A.
2. Maximax. Best of each: A 700, B 400, C 850. Choose C.
3. Maximin. Worst of each: A 100, B 200, C 0. Choose B.
4. Minimax regret. Best by state: weak 200 (B), normal 300 (A), strong 850 (C). Regret: A 100, 0, 150, so maximum 150. B 0, 20, 450, so maximum 450. C 200, 50, 0, so maximum 200. Choose A, the smallest maximum regret.
5. Comment. Three rules give three answers. C suits a risk seeker, because it could earn most but may earn nothing; B suits the risk averse, since it never earns less than ₦200,000; A has the highest expected profit and the least regret, and is the best compromise if the probabilities are trusted.

1. Pay-offs.
   Buy 10: 10 crates sold in every state, 5,000 − 3,000 = ₦2,000 each.
   Buy 20: demand 10 gives 5,000 − 6,000 = ₦(1,000)
   demand 20 or 30 gives 10,000 − 6,000 = ₦4,000.
   Buy 30: demand 10 gives 5,000 − 9,000 = ₦(4,000)
   demand 20 gives 10,000 − 9,000 = ₦1,000
   demand 30 gives 15,000 − 9,000 = ₦6,000.
2. Expected values. Buy 10: 2,000.
   Buy 20: −200 + 2,000 + 1,200 = 3,000.
   Buy 30: −800 + 500 + 1,800 = 1,500.
   Choose 20 crates.
3. Maximax. Best: 2,000; 4,000; 6,000. Buy 30.
4. Maximin. Worst: 2,000; (1,000); (4,000). Buy 10.
5. Regret. Best by state: 2,000, 4,000, 6,000. Buy 10: 0, 2,000, 4,000, maximum 4,000. Buy 20: 3,000, 0, 2,000, maximum 3,000. Buy 30: 6,000, 3,000, 0, maximum 6,000. Buy 20.
6. Comment. Expected value and minimax regret agree on 20 crates, which is the choice most would make; the optimist buys 30 and the pessimist buys 10.

Expect variants: costs rather than profits (reverse the rules), a table with probabilities given for only some states, or a question that asks which rule suits a given manager.

Single-period stocking decisions and expected net present value

Single-period stocking. When stock must be ordered once for the whole season, with no repeat order. Demand is uncertain, build a payoff table: for each order quantity and each level of demand, calculate contribution = units sold × unit contribution, less the loss on any unsold units (cost less salvage value) and less any penalty for unmet demand (lost profit on extras).

Weight each row by the probabilities and choose the order quantity with the highest expected contribution; costs of ordering that are the same for every option are ignored. Example. Cost ₦100, selling price ₦160 (contribution ₦60), unsold units sold at ₦70 (loss ₦30), penalty for unmet demand ₦20 a unit. Demand 200 (0.3), 300 (0.5), 400 (0.2). Order 200: 12,000; 10,000; 8,000, expected ₦10,200. Order 300: 9,000; 18,000; 16,000, expected ₦14,900. Order 400: 6,000; 15,000; 24,000, expected ₦14,100.

Order 300. This differs from the economic order quantity model, which assumes known demand, repeat orders and no lost sales or surplus, and finds the order size that minimises ordering and holding costs; the single-period model accepts uncertain demand, one order and the cost of both over- and under-stocking.

Expected NPV with uncertain cash inflows. When several uncertain factors change the estimated inflows, find the expected change in inflows and apply it before discounting (outflows that are reliable are left alone). Example. Investment ₦12m; annual inflows ₦9m and outflows ₦4m for 5 years; cost of capital 10% (annuity factor 3.791). Factor 1: 10% chance inflows fall 20%; factor 2: 25% chance they fall 8%; factor 3: 30% chance they rise 6%.

Expected change = −0.10 × 20% − 0.25 × 8% + 0.30 × 6% = −2.2%. Expected inflow = 9 × 0.978 = 8.802m; net 8.802 − 4 = 4.802m; PV = 4.802 × 3.791 = 18.20m; expected NPV = ₦6.20m, against ₦6.96m on the unadjusted figures.

Measuring risk: standard deviation and the coefficient of variation

The expected value shows the average outcome but not the risk. The spread of possible outcomes is measured by the standard deviation (the square root of the probability-weighted squared differences from the expected value). A risk-averse person chooses the option with the lowest standard deviation for a given return.

The trade-off between risk and return is judged by the coefficient of variation = standard deviation ÷ expected value: the lower the figure, the less risk per naira of expected return. Example. Project A: expected value ₦200m, standard deviation ₦40m, coefficient 0.20. Project B: expected value ₦120m, standard deviation ₦18m, coefficient 0.15. B gives less return but less risk per naira.

Attitudes: a risk-averse decision maker avoids risk without adequate compensation; a risk-neutral one ignores risk and uses expected values; a risk-seeking one takes risk without requiring compensation.

A credit decision with a risk of non-payment

Whether to grant credit is an expected-value NPV problem. Example. An order for 3 machines at ₦4m each (₦12m): 10% deposit (₦1.2m) is paid now; the balance of ₦10.8m is due in 12 months.

Experience: 60% of customers pay on time; the other 40% go to a debt agency, which has a 50% chance of recovering the whole balance at once; its fee of ₦300,000 is payable whenever it is used and is not refundable; if it fails the debt is written off. Costs: ₦2.5m a machine, 80% for materials now (₦6m) and 20% at delivery in 12 months (₦1.5m). Cost of capital 15% (year-1 factor 0.8696).

Expected receipt of the balance = 0.60 × 10.8 + 0.40 × 0.50 × 10.8 = 6.48 + 2.16 = ₦8.64m. Expected agency fee = 0.40 × 0.3 = ₦0.12m. Net year-1 flow = 8.64 − 0.12 − 1.5 = ₦7.02m; PV = 7.02 × 0.8696 = ₦6.104m. NPV = 1.2 − 6.0 + 6.104 = +₦1.304m, so accept on financial grounds if there is spare capacity. Also consider the customer’s credit standing (a credit bureau report), the size of the order against capacity, effect on cash flow, and repeat business.

Common mistakes

• Computing revenue on units bought instead of units sold in a perishable-stock table.
• Regret from the wrong direction. Take the best pay-off in the column and subtract the cell.
• Choosing the largest regret instead of the smallest maximum regret.
• Using probabilities with maximax, maximin or regret. They do not use them.
• Applying profit rules to costs without reversing.
• Presenting the EV as the result that will happen. It is an average.

Quick revision

• EV = Σ probability × outcome; best for repeated decisions; ignores spread.
• Maximax: best of the best (risk seeker). Maximin: best of the worst (risk averse).
• Regret = best in the column minus the cell; choose the smallest maximum regret.
• The three rules ignore probabilities.

How the marks are earned

• Table: the pay-offs in full, with working.
• Rules: each applied step by step.
• Comment: attitude to risk and the limits of each rule.

Practice questions

These are original questions, written in the style of the Skills examination. Work each one before you open the answer.

Recall check: multiple choice

Section B style: written questions

Model answers

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