Handling Uncertainty Ender Ozcan given by Peter Blanchfield
Objectives Be able to construct a payoff table to represent a decision problem. Be able to apply the maximin and maximax criteria to the table. Be able to apply the expected monetary value (EMV) criterion to the table. Be able to discuss the rationale, the strengths and the limitations of these criteria. 2
Introduction Many decisions are made under conditions of uncertainty, i.e. having made a decision you cannot be sure what the outcome will be. In this session, we will look at three simple criteria that can be used to help people facing decisions of this sort. We will also show how some problems can be represented as decision tables 3
Expected values These will underpin many of the methods that we will be applying to decision problems. An expected value (of X) is a long run average result. E X = Where p i is the probability of observation x i occurring For example for throwing a fair six sided dice a number of times - where the probability of any given number is 1/6 and assuming the numbers on the dice are 1, 2, 3 6 then E(x) is 21/6 or 3.5 the average of the six numbers i p i x i 4
Roulette A US roulette wheel has 38 equally likely outcomes. (p i = 1/38 for each number) A winning bet placed on a single number pays 35-to-1 (this means that you are paid 35 times your bet and your bet is returned, so you get 36 times your bet). What is the expected outcome of placing a $1 bet on any number? The outcome you loose will have a probability of 37/38 The outcome for these possibilities is -$1 you lose your money The outcome of winning is $36 5
Outcome of Roulette E(x) = (p lose. x lose ) + (p win. x win ) E(x) = (37/38. -$1) + (1/38 $36) = -$0.03 The bank will win! don t gamble on roulette 6
Business Example A sales person is paid commission on sales If the following sales commission has the given probability $80 probability 0.3 $90 probability 0.5 $100 probability 0.2 What is the sales person s expected outcome? 7
Decisions under uncertainty A company supplying paint to the car industry It has to buy the paint in on the probability it will be sold Otherwise they have a surplus to store, which costs them money Or they will not meet demand, which means the companies go elsewhere in future Daily demand for a given colour is for 1, 2 or 3 batches Any left over at the end of the day will cost $10 to store However each batch sells for $1000 and costs $500 Not having a batch will make future loss of $1000 per batch 8
The company s problem How many batches should they buy every day? We can work out a pay off table Demand Decision One Batch Two Batches Three Batches One batch $500 $0 -$500 Two batches $490 $1000 $500 Three batches $480 $500 $1500 9
The decision This will depend on the company strategy Pessimistic called maximin The worst outcome will always happen So choose the strategy with the best worst outcome In this case buying three batches has a worst outcome of making $480 profit 10
The Optimist The optimist always assumes the best possible outcome - MaxiMax In this case still buy three batches as they will make $1500 11
The realist - Expected monetary value (EMV) criterion This strategy takes into account the probabilities of the different outcomes of the decisions. It involves calculating the expected payoff of each decision and then selecting the decision with the best expected payoff. Let us randomly select thr4ee probabilities of the events Demand for 1 batch probability 0.3 Demand for 2 batches probability 0.6 Demand for 3 batches probability 0.1 12
Realistically Probabilities applied One p = 0.3 Two p = 0.6 Three p = 0.1 Risk One $150 $0 -$50 $100 Two $147 $600 $50 $797 The winner Three $144 $300 $150 $594 13
Limitations of EMV Since an expected value represents the average payoff that will result if the decision was repeated a large number of times, would it be reasonable to apply it to a one-off decision? It does not take into account the attitude to risk of the decision-maker. Only one objective, maximising monetary returns, is assumed. Many decisions involve other less quantifiable factors such as reputation. A linear value function for money is assumed i.e. each extra $1 received brings the same increase in satisfaction to the decision-maker. Generally the probabilities and payoffs are rough estimates. 14