Script

DECISION MODELS – Decision Trees

Slide 1

  • Welcome back.
  • In this module weshow how to use decision trees to aid in making a set of sequential decisions.

Slide 2

  • Thus far
  • We have used payoff tables to determine optimal strategies when only a single decision is to be made.
  • But sometimes we have to make a set of interrelated sequential decisions…
  • Where decisions made “along the way” will be influenced by decisions made up to that point in time.
  • Although this all can be handled by complicated mathematical relationships, decision trees provide an effective way to analyze such problems.

Slide 3

  • A decision tree
  • Consists of nodes and arcs.
  • Nodes can represent
  • The starting point in time
  • A point where a decision must be made
  • A point where a state of nature occurs
  • Or the end result of a set of sequential decisions where the cumulative effect of the decisions is recorded
  • Arcs are of two types
  • Decision arcs
  • Which give a possible decision and the resulting cost or profit of making that decision
  • And state of nature arcs
  • Which represent a possible state of nature and gives the Bayesian probability of that state of nature occurring.

Slide 4

  • Let’s look at a specific example.
  • Supposes BGD Development is interested in purchasing some land at a cost of $300,000
  • On which to build a shopping center which it can then sell for $450,000
  • A variance must be obtained before the shopping center is built and this will cost $30,000
  • Now if the variance is approved, the shopping center will be built
  • If it is denied, the shopping center will not be built.
  • It can purchase a 3-month option to buy the property before applying for the variance at a cost of $20,000
  • If it buys the land but does not build the shopping center on it, they can turn around and sell the land for $260,000
  • For $5000 they can hire a consultant to study the situation and render an opinion as to whether the variance will be approved or denied

Slide 5

  • BGD estimates
  • The probability that it will get the variance approved is .4
  • Which means its estimate that it will be denied is .6
  • Based on past history of the consultant,
  • The probability the consultant predicted approval for variances that wound up getting approved is .7
  • Which meansthe probability the consultant predicted denial for variances that wound up getting approved is .3
  • And the probability the consultant predicted denial for variances that wound up being denied is .8
  • Which meansthe probability the consultant predicted approval for variances that wound up being denied is .2

Slide 6

  • Using the Bayesian approach we show that
  • The probability of a variance being approved given that the consultant predicted it would be approved is .7
  • And that it is denied is .3
  • Note that the probability the consultant predicts approval is .7 times .4 plus .2 times .6 which is .4
  • The probability of a variance being denied given that the consultant predicted it would be denied is .8
  • And that it is approved is .2
  • Note that the probability the consultant predicts denial is .8 times .6 plus .3 times .4 which is .6

Slide 7

  • We now construct the decision tree. Because of its size it will take several slides to show. Decision trees represent a chronological sequence through the decision making process.
  • The process begins by deciding whether or not to hire the consultant.
  • If the consultant is not hired this costs nothing.
  • The next decision to be made then would be to decide to not buy the land, to buy the land and apply for the variance or to buy the land and purchase the option. If the land is not bought, this will cost $0
  • Meaning the net return for this branch is $0
  • If BGD buys the land and variance this will cost the $300,000 for the land and $30,000 for the variance or $330,000.
  • Now if the variance is approved, which happens with probability .4, it will build and sell the shopping center for a return of $450,000.
  • So the net profit on this branch is $450,000 minus the $330,000 or $120,000
  • But if the variance is denied, which happens with probability .6, BGD will sell the land for $260,000
  • This results in a loss of $260,000 minus $330,000 or a loss of $70,000
  • If it buys the 3-month option for $20,000 and applies for the variance for $30,000, this is a total cost of $50,000.
  • Again the variance could be approved with probability .4 in which case it will buy the land for $300,000 and build the shopping center and sell it for $450,000 giving a $150,000 profit.
  • This results in a net profit of $150,000 minus $50,000 or $100,000.
  • But if the variance is denied, which happens with probability .6, it will do nothing
  • And the net result of this sequence of decisions is a loss of $50,000.
  • Now the other option that could be done right at the start is hire the consultant

Slide 8

  • Continuing
  • From the start node
  • If the decision is made to hire the consultant this will cost $5000
  • Now the consultant could predict approval of the variance which will happen with probability .4
  • BGD could then elect to do nothing, costing $0
  • And the net result of this sequence of events is a loss of $5000
  • A second thing it could do if the consultant predicts approval of the variance is buy the land for $300,000 and apply for the variance for $30,000 for a total cost of $330,000
  • Again the variance could be approved, but his time this revised probability is .7 as shown earlier. If it is approved it builds and sells the shopping center for a $450,000 profit.
  • So the net result of this sequence of events is a profit of $450,000 minus $330,000 minus $5000 or $115,000.
  • But the variance could be denied, which now will happen with probability .3, in which case it will sell the land for $260,000
  • The net result of this sequence of events is $260,000 minus $330,000 minus $5000 or a loss of $75,000
  • The third thing it could do if the consultant predicts approval is buy the option for $20,000 and apply for the variance for $30,000 for a total cost of $50,000
  • Again the variance could be approved with probability .7 in which case it buys the land for $300,000 and builds and sells the shopping center for a $450,000 profit for a net profit of $150,000.
  • The net result of this sequence of events is $150,000 minus $50,000 minus $5000 or $95,000.
  • And if the variance is denied which happens with probability .3, it will not buy the land and this will cost nothing.
  • So the net result of this sequence of events are costs of $5000 and $50,000 for a loss of $55,000.
  • The consultant could also predict denial which happens with probability .6.
  • We have the same options with results calculated in a similar way. BGD could do nothing
  • Resulting in a net loss of $5000
  • It could buy the land and apply for the for the variance
  • Which could be approved (now with probability .2) and the shopping center built and sold
  • For a net profit of $115,000
  • Or it could be denied (now with probability .8), in which case it would sell the land
  • Resulting in a $75,000 loss.
  • Or it could buy the option and apply for the variance
  • And if the variance is approved it would buy the land and build and sell the shopping center
  • Resulting in a net profit of $95,000
  • Or it could be denied, in which case it would do nothing
  • Resulting in a net loss of $55,000

Slide 9

  • Now let’s calculate the expected profits for each sequence of decisions.
  • If BGD did not hire the consultant or buy the land the expected profit would be $0
  • If it did not hire the consultant but bought the land and applied for the variance, following the branches, the expected profit would be .4 times $120,000 plus .6 times negative $70,000
  • or $6000
  • And if it did not hire the consultant but bought the 3-moth option and applied for the variance the expected profit would be .4 times $100,000 plus .6 times negative $50,000
  • or $10,000
  • So if a consultant is not hired, it should purchase the 3-month option and apply for the variance for an expected profit of $10,000

Slide 10

  • Now if BGD did hire the consultant, …
  • And the consultant predicted approval of the variance application and BGD did nothing, this would result is a loss of $5000.
  • If it hired the consultant, the consultant predicted approval of the variance and BGD bought the land and applied for the variance, the expected profit would be .7 times $115,000 plus .3 times negative $75,000
  • or $58,000.
  • And if it hired the consultant, the consultant predicted approval of the variance and BGD purchased the option and applied for the variance, the expected profit would be .7 times $95,000 plus .3 times negative $55,000
  • or $50,000.
  • So if BGD hired the consultant and he predicted approval of the variance, BGD should buy the land and apply for the variance for an expected profit of $58,000.
  • Now if BGD did hire the consultant and the consultant predicted denial of the variance application and BGD did nothing, this would result is a loss of $5000.
  • If it hired the consultant, and the consultant predicted denial of the variance and BGD bought the land and applied for the variance anyway, the expected profit would be .2 times $115,000 plus .8 times negative $75,000
  • or an expected loss of $37,000.
  • And if it hired the consultant, the consultant predicted denial of the variance and BGD purchased the option and applied for the variance, the expected profit would be .2 times $95,000 plus .8 times negative $55,000
  • or an expected loss of $25,000.
  • Thus if BGD hired the consultant and he predicted denial of the variance, BGD should not buy the land and suffer a loss of $5000.
  • Since the probability the consultant recommends approval of the variance is .4 and denial of the variance is .6, the net expected value of hiring the consultant is .4 times $58,000 plus .6 times negative $5000 or
  • $20,200

Slide 11

  • Summarizing,
  • The expected value to BGD if they do not hire the consultant is $10,000
  • And the expected value to BGD if they do hire the consultant is $20,200
  • So they should hire the consultant and if the consultant predicts approval of the variance, it should buy the land and apply for the variance; but if the consultant predicts denial of the variance, BGD should not buy the land at all.

Slide 12

  • Let’s review what we’ve discussed in this module.
  • We’ve shown how decision trees can be used to analyze a sequence of possible decisions.
  • In a decision tree, nodes are points in time where decisions are made or states of nature occur
  • Arcs in the decision tree give payoffs or Bayesian probabilities of the states of nature occurring
  • This allows for expected values to be calculated for each decision, which in turn enables us to determine the best sequence of decisions.

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