Transcript ch09slides - Market Interactions and Market Power

Outline of Chapter 9: Using Simulation to
Solve Decision Problems
Real world decisions are often too complex to be analyzed effectively using influence
diagrams or decision trees. Monte Carlo simulation offers us a way to model these
decisions that contain a large amount of data.
What is a Monte Carlo Simulation?
Monte Carlo simulation is a method analyze a decision by repeatedly evaluating the
decision process using random values for uncertain values. The average outcome of the
simulation over many runs defines the probability distribution for decision.
Using Random Numbers to Simulate Reality
Monte Carlo simulations allow a decision model to become stochastic like the real
world. Evaluating the model numerous times can give an accurate probability
distribution of the various outcomes.
“A random number can be any number (x) from a group of
uniformly distributed numbers that falls within an established
boundary, usually between 0 and 1.” –D.S.
A random number can be used to choose a value from any random distribution- i.e.
to decide the outcome of an event in a simulation. Computers are a good way to
through random numbers.
The Power of Spreadsheets
Spreadsheets can be used to throw random numbers. A model can be built in a
spreadsheet and macros used to update the spreadsheet thousands of time and
cumulate possible outcomes. Example, see figure 9-4.
Variable
Gas price
Driving condition
Fuel efficiency
Miles driven/year
annual cost:
1.371922
0.597732
21.50049
10850.45
712.153
Generating Uniform Distributions
Given a random variable, x, from a uniform distribution on the interval [0,1]
we can compute a uniform random variable, y, on any continuous interval
[a,b]. The formula to do this is:
y= a+x*(b–a)
Using Discrete Distributions
Discrete, uniform distributions can be created from the uniform distribution [0,1] by
dividing the interval [1,0] into segments. i.e. if x < 0.50 then option A, else option B
gives a discrete, uniform distribution with 2 outcomes.
Lookup tables are useful to create distributions with more possible outcomes.
Using the Results of a Monte Carlo
A histogram and a cumulative distribution are two graphs that are effective at
displaying the results of a Monte Carlo simulation. Here is a simulation done in Excel.
Variable
Gas Price
Driving Condition
Fuel Efficiency
Miles Driven/Year
Formula
y=rand()+1
y=rand()+2
y=15+rand()*15
y=10000+rand()*10000
Gas Price
Driving Condition
1.268092961
2.844115341
1.705695704
2.576366395
1.944527456
2.985819489
1.528546827
2.59895298
1.973509533
2.049257935
1.065162598
2.463990819
1.868471602
2.132945438
1.739082644
2.118167982
1.027740771
2.356360494
1.709941178
2.098309603
1.898263876
2.657844103
1.855761126
2.960252972
1.95126606
2.71873943
1.291189285
2.672342925
1.573800481
2.053595609
1.851930805
2.949357561
1.279402958
2.012429292
Fuel
Annual
Efficiency
Miles Driven/Year Cost
26.12610649
19295.92752 1050.985 454.7155
21.68724932
11214.98268 1000.966 512.929
28.8174949
11110.58939 836.3703 526.7926
18.50736324
16408.73892 1576.621 575.0572
26.08823633
11962.65949 982.0893 592.1929
20.3842572
14049.63611 835.0962 592.1962
28.73066126
19057.83938
1338.8 600.4623
17.89263849
13556.64547 1494.575 619.1348
23.37309597
11552.13001 564.9115 619.547
16.25835503
12694.0969 1532.916 621.3155
24.23039383
16371.45304 1440.596 654.8278
22.64242276
19812.94997 1868.092 690.7983
22.77645975
17122.80532 1665.75 724.4211
18.7854511
14709.33819 1178.701 738.8519
28.37467
18381.71298 1099.087 739.8709
20.50565833
16008.71166 1688.683 757.9861
29.0099803
15461.48322 732.7134 765.0104
A histogram shows the relative frequency of each of the
possible outcomes.
Cumulative probability distributions are useful to
determine the risk profile of the possible outcomes.
Commercial Software
Spreadsheets like Microsoft Excel are useful in doing Monte Carlo simulations.
There are also software packages available such as:
• Crystal Ball by Decisioneering
•@Risk by Palisade.
Both software packages can create models with many variable inputs and outcomes
and create histograms and cumulative distributions over numerous runs.
The Role of Monte Carlo
Monte Carlo simulation is not a replacement for DA, but is useful to find
distributions of uncertain quantities. The thought and effort put into creating the
simulations should be just as rigorous as that put into DA.
“Monte Carlo simulation provides another approach to dealing with uncertain
quantities. We can use this method to learn about the interactions of many
uncertainties… with the use of a computer… (but) the same amount of care
must be taken as when building any decision model.”