Payoff tables

A profit table (pay­off table) can be a useful way to represent and analyse a scenario where there is a range of possible outcomes and a variety of possible responses. A pay­off table simply illustrates all possible profits/losses and as such is often used in decison making under uncertainty.

Illustration

Geoffrey Ramsbottom runs a kitchen that provides food for various canteens throughout a large organisation. A particular salad is sold to the canteen for $10 and costs $8 to prepare. Therefore, the contribution per salad is $2.

Based upon past demands, it is expected that, during the 250 ­day working year, the canteens will require the following daily quantities:

• On 25 days of the year, 40 salads.
• On 50 days of the year, 50 salads.
• On 100 days of the year, 60 salads.
• On 75 days 70 salads.

The kitchen must prepare the salads in batches of 10 meals in advance. The manager has asked you to help decide how many salads the kitchen should supply for each day of the forthcoming year.

Constructing a pay­off table:

If 40 salads will be required on 25 days of a 250 day year, then the probability that demand = 40 salads is

P(Demand of 40) = 25 days ÷ 250 days = 0.1

Likewise,

• P(Demand of 50) = 0 .20;
• P(Demand of 60) = 0.4 and
• P(Demand of 70) = 0.30

Now let's look at the different values of profit or losses depending on how many salads are supplied and sold.

For example, if we supply 40 salads and all are sold, our profits amount to 40 x $2 = 80.

If however we supply 50 salads but only 40 are sold, our profits will amount to 40 × $2 -­ (10 unsold salads × $8 unit cost) = 80 - 80 = 0.

Similarly, we can now construct a pay­off table as follows:

Decision Rules

To decide how many salads should be made very day, Geoffrey will first have to define his attitude to risk, and use one of the following rules to make up his mind :

• The maximax rule for an optimist - i.e. someone who wants the best possible upside potential without being very concerned about possible losses or downside.
• The maximin rule for a pessimist looking to minimise his losses - i.e. someone who wants to minimise the potential downside exposure.
• The minimax regret rule is for someone who doesn't like making the wrong decision. This approach seeks to minimise such "regret". 
• Alternatively, expected values of profits could be used to make a decision. These are averages and essentially ignore the spread or risk of outcomes. Risk must thus be brought back into the decision making process another way.

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