Script

DECISION MODELS – Making Decisions Under Uncertainty

Slide 1

  • Welcome back.
  • In this module we introduce a class of models known as decision models.

Slide 2

  • Decision analysis
  • Is about making optimal decisions when the future is unknown – which is just about every decision you ever make.
  • A decision model consists of three things.
  • First is a set of possible decision alternatives
  • Which can be discrete or continuous
  • Finite or infinite. If the decision is to choose any number, including pi, the square root of 1543, or any number – the alternatives are continuous and infinite. If the decision is to choose any integer – the alternatives are discrete and infinite. If the decision is to choose any number between 1 and 4, the alternatives are continuous butfinite. Finally, if the decision is to choose any integer between 1 and 4, the alternatives are discrete and finite. In this case we can simply list them --- 1, 2, 3, or 4. We will concentrate on only this case – where we can simply make a finite list of all decision alternatives.
  • The second part of a decision model is a set of states of nature that define the future.
  • And the third part is a payoff,
  • giving the return for making a particular decision and then having a particular state of nature occur.
  • For discrete, finite models, these can be summarized in what is called a payoff table.

Slide 3

  • Let’s get right to an example.
  • Suppose Tom Brown has $1000 to invest and he is considering investing it in
  • Gold, which generally moves in the opposite direction of the stock market
  • Or a bond that typically moves with the market
  • Or a stock that typically moves with the market but has both more upside and downside risk
  • Or a CD that pays the same no matter what the market conditions are
  • Or a stock option hedge which again generally moves with the market.
  • These are the decision alternatives. The states of nature have to do with the future state the stock market – but how do we measure this?
  • And how are we going to determine payoffs?

Slide 4

  • The market can be reflected using many indices such as the Dow Jones Industrial Average, the Sand P 500, or many other indices
  • Here we’ll use the Dow
  • Now we could list all possible values for the Dow from say 500 to 50,000, but this would be overkill and we would not easily be able to differentiate between the payoffs for say 10,003.45 and 10,003.46.
  • So here we will define the states of nature less rigorously as
  • State of nature 1 – a large rise a year from now from today’s current Dow value
  • Say of 1500 points or more
  • State of nature 2 will be small rise
  • Of say between 500 and 1500 points
  • State of nature 3 will be a no change condition of
  • Less than 500 points in either direction from today’s value
  • State of nature 4 will defined to be a small fall in the market
  • Of say between 500 and 1200 points
  • Whereas state of nature 5 will be a large fall
  • Of more than 1200 points from today’s current value

Slide 5

  • Now to develop a payoff table, we may seek the help of financial experts
  • Using the rules of thumb we presented earlier and the financial advisor’s advice, suppose we get the payoff table below.
  • These numbers are in terms of the gains on a $1000 investment, so a negative number represents a loss of money.
  • Now the question is, given all this information, “which decision should be made?”

Slide 6

  • Before we proceed, let us introduce the concept of a dominated alternative
  • These are alternatives that would never be chosen, because another alternative is always better, or at least as good.
  • Now we see from the payoff table, that sometimes gold is better than the bond and vice versa; and sometimes the stock is better than the CD and vice versa.
  • But if we look at the stock option hedge and the bond
  • We see that if S1 occurs, the bond is better; if S2 occurs, the bond is better; if S3 occurs they give the same payoffs; if S4 occurs, the bond is better; and if S5 occurs, again there is no difference.
  • So the stock option hedge is never better than the bond – so we would always choose the bond over the option – the option is dominated by the bond – so we will eliminate the stock option hedge from our list of decision alternatives.

Slide 7

  • The resulting payoff table looks like this
  • With the stock option hedge eliminated
  • In this table, no decision alternative dominates any other alternative.

Slide 8

  • We begin our discussion of which alternative to select by considering the unrealistic situation known as decision making under certainty.
  • This assumes we know for sure which state of nature will occur.
  • In this case the decision strategy is simple – for that certain state of nature, select the decision alternative that has the highest payoff.
  • So for our payoff matrix
  • If we know for sure that state of nature 1 will occur,
  • We would select D3 – the stock having the highest payoff of $500. And if we knew for sure that state of nature 2 would occur
  • Again we would select D3 – the stock having the highest payoff of $250. Now if we knew for sure that state of nature 3 would occur
  • We would select D1 – gold having the highest payoff of $200. And if we knew for sure that state of nature 4 would occur
  • Again would select D1 – gold having the highest payoff of $300. Finally if we knew for sure that state of nature 5 would occur
  • We would select D4 – the CD having the highest payoff of $60.

Slide 9

  • While decision making under certainty is unrealistic, decision making under uncertainty is not much more realistic
  • Here we assume we have no idea, no gut feel, not an inkling of which state of nature will occur.
  • In this case there are several approaches we can employ depending on the attitude of the decision maker.
  • One is an optimistic approach
  • Another is a pessimistic or conservative approach
  • A third would be one that minimizes our overall expected disappointment
  • Still another is one in which we assume that each state of nature is equally likely to occur

Slide 10

  • We begin by describing the optimistic approach, known as the MAXIMAX strategy
  • The optimist says, “no matter what decision I make, the best state of nature for that decision will occur”.
  • So let’s again refer to our payoff table.
  • Now for each alternative we will list the best thing that can happen.
  • For gold, it is state of nature 4 giving a $300 payoff
  • For the bond, it is state of nature 1 giving a $250 payoff
  • For the stock it is also state of nature 1 a $500 payoff
  • And for the CD, it doesn’t matter – we would get a $60 payoff
  • So if we think we would make $300, $250, $500, or $60, we should choose the decision that goes with the $500 – the stock. Note that what we did was list the maximum return for each payoff and then chose the decision corresponding to the maximum of these maximums – hence the term MAXIMAX decision strategy

Slide 11

  • Now we turn to the pessimistic or conservative approach, known as the MAXIMIN decision strategy
  • The pessimist says, “no matter what decision I make, the worst state of nature for that decision will occur”.
  • So let’s again refer to our payoff table.
  • Now for each alternative we will list the worst thing that can happen.
  • For gold, it is state of nature 1 giving a payoff of -$100
  • For the bond, it is state of nature 5 giving a payoff of -$150
  • For the stock it is also state of nature 5 a payoff of -$600
  • And for the CD, it doesn’t matter – we would get a $60 payoff
  • So if we think we would lose $100, lose $150, lose $600, or gain $60, we should choose the decision that goes with the $60 gain– the CD. Note that what we did was list the minimum return for each payoff and then chose the decision corresponding to the maximum of these minimums – hence the term MAXIMIN decision strategy

Slide 12

  • The next approach is what I like to call the “I shoulda approach”. It is technically referred to as the minimax regret strategy.
  • The regret a decision maker suffers for making a particular decision is the difference between the payoff he did get and the maximum payoff for the corresponding state of nature.

Slide 13

  • Here is out payoff table with the best payoffs for each state of nature given in red.
  • We now construct a regret table as follows
  • For state of nature 1 the best thing that can happen is to have invested in the stock – so if we made that decision we would have no regret.
  • If we invested in the CD we would have made $60,
  • but we could have made 500 so we would have a regret of $440.
  • Similarly for the bond we have a regret of $500 minus $250 or $250
  • And for gold, we would have a regret of 500 minus negative 100 or $600.
  • Now for state of nature 2 the best thing that can happen is again to have invested in the stock – so if we made that decision, again we would have no regret.
  • If we invested in the CD we would have made $60,
  • but we could have made 250 so we would have a regret of $190.
  • Similarly for the bond we have a regret of $250 minus $200 or $50
  • And for gold, we would have a regret of 250 minus 100 or $150.
  • For state of nature 3 the best thing that can happen is to have invested in gold – so if we made that decision we would have no regret.
  • If we invested in the CD we would have made $60,
  • but we could have made 200 so we would have a regret of $140.
  • Similarly for the stock we have a regret of 200 minus 100 or $100
  • And for the bond, we would have a regret of 200 minus 150 or $50.
  • For state of nature 4 the best thing that can happen again is to have invested gold – so if we made that decision we would have no regret.
  • If we invested in the CD we would have made $60,
  • but we could have made 300 so we would have a regret of $240.
  • Similarly for the stock we have a regret of 300 minus negative 200 or $500
  • And for the bond, we would have a regret of 300 minus negative 100 or $400.
  • Finally, for state of nature 5 the best thing that can happen is to have invested in the CD – so if we made that decision we would have no regret.
  • If we invested in the gold, we would have a payoff of 0
  • so we would have a regret of 60 minus 0 or $60
  • For the bond we have a regret of 60 minus negative 150 or $210
  • And for the stock, we would have a regret of 60 minus negative 600 or $660. We now have a complete regret table.

Slide 14

  • We now employ a pessimistic strategy
  • Which says “no matter what decision I make, the worst state of nature for that decision will occur”.
  • Looking at the regret table
  • Which is comprised of regrets not payoffs
  • We list the worst thing that can happen, that is the maximum regret for each decision alternative.
  • For gold, it is state of nature 1 with a regret of $600
  • For the bond, it is state of nature 4 with a regret of $400
  • For the stock it is state of nature 5 with a regret of $660
  • And for the CD, it is state of nature 1 with a regret of $440
  • So if we think we our regrets would be $600, $400, $660, or $440, we should choose the decision that goes with the minimum of these, the one that goes with the $400 regret, that is the bond. Note that what we did was list the maximum regret for each payoff and then chose the decision corresponding to the minimum of these maximums – hence the term MINIMAX REGRET decision strategy

Slide 15

  • Another strategy is called the principle of insufficient reason
  • It treats each state of nature as equally likely to occur
  • So we can look at the average value for each decision alternative which is the corresponding sum divided by 5. Since they are all divided by 5, the best alternative is the one with the minimum sum of the payoffs.
  • So for our payoff table
  • We tabulate the sum of all the payoffs for each alternative which are
  • 500 for gold
  • 350 for the bond
  • 50 for the stock
  • and 300for the CD
  • So under this criterion, gold would be chosen.

Slide 16

  • A template for decision models can make this process simple. On the payoff table worksheet…..
  • Enter the decision names
  • in column A,
  • the state of nature names in row 3,
  • and the payoffs in the resulting matrix
  • In the Results section the decision and corresponding payoffs or regrets are listed

Slide 17

  • So let’s summarize the recommendations
  • If the decision maker is optimistic, the recommendation is the stock
  • If the decision maker is pessimistic, the recommendation is the CD
  • If the decision maker is pessimistic but is concerned more about regret than payoffs, the recommendation is the bond
  • If the decision maker employs the principle of insufficient reason, the recommendation is gold
  • Four different strategies, four different recommendations – so which recommendation should be made?
  • As management scientists we do not have to make the decision, all we do is present the alternatives – so the alternative that should be chosen depends on the attitude of the decision maker.

Slide 18

  • Let’s review what we’ve discussed in this module.
  • We said that a decision model consists of
  • Decision alternatives, states of natures, and payoffs
  • We discussed dominated alternatives as those that are universally worse than one other decision alternative
  • We discussed decision making under certainty
  • And decision making under uncertainty
  • And we illustrated the maximax decision strategy for optimistic decision makers
  • The maximin decision strategy for pessimistic decision makers
  • The minimax regret decision strategy for pessimistic decision makers concerned more with regret than payoffs
  • And the principle of insufficient reason for indifferent decision makers
  • And we illustrated how to use the decision template to make recommendations for making decisions under uncertainty

That’s it for this module. Do any assigned homework and I’ll be back to talk to you again next time.