How to construct a decision tree

1. List each decision nodes & its alternatives.2. List each chances nodes& its alternatives.3. Draw the nodes and links.4. Add costs & probabilities along links5. Calculate utilities for utility (leftmost) nodes6. Calculate expected utilities

Example

• You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

1. List each decision nodes & its alternatives.

• Which project?– Fancy– Mundane

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

2. List each chance nodes & its alternatives.

• Outcome?– Success– Failure

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

3. Draw nodes as tree

Project?

Outcome?

Outcome?

Fancy

Mundane

Success

Success

Failure

Failure

Decision nodes

are square

Chance nodes

are oval

Utility nodes are

like diamonds

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

4. Add costs and probabilities to links

Project?

Outcome?

Outcome?

Fancy

cost 1M$

Mundanecost 0.5M$

Success

p=0.1

Success

p=0.8

Failurep=0.2

Failurep= 0.9

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

5. Calculate utility values for utility nodes

Project?

Outcome?

Outcome?

Fancy

cost 1M$

Mundanecost 0.5M$

10M$

-0.5M$

1M$

-1M$

Success

p=0.1

Success

p=0.8

Failurep=0.2

Failurep= 0.9

Benefit: 0Cost: 0.5M$

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

Benefit: 11M$Cost: 1M$

6. Calculate expected utilities, moving leftward

Project?

Outcome?

Outcome?

Fancy

cost 1M$

Mundanecost 0.5M$

10M$

-0.5M$

1M$

-1M$

Success

p=0.1

Success

p=0.8

Failurep=0.2

Failurep= 0.9

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

EU = 0.1*10=1.0

EU = 0.9*(-1)=-0.9

EU = 0.8*1=0.8

EU = 0.2*(-0.5)=-0.1

6. Calculate expected utilities moving leftward

Project?

Outcome?

Outcome?

Fancy

cost 1M$

Mundanecost 0.5M$

10M$

-0.5M$

1M$

-1M$

Success

p=0.1

Success

p=0.8

Failurep=0.2

Failurep= 0.9

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

EU = 0.1*10=1.0

EU = 0.9*(-1)=-0.9

EU = 0.8*1=0.8

EU = 0.2*(-0.5)=-0.1

EU = 1.0-0.9= 0.1

EU = 0.8-0.1=0.7

6. Calculate expected utilities moving leftward

Project?

Outcome?

Outcome?

Fancy

cost 1M$

Mundanecost 0.5M$

10M$

-0.5M$

1M$

-1M$

Success

p=0.1

Success

p=0.8

Failurep=0.2

Failurep= 0.9

You need to decide whether to do a fancy project or a mundane project. Either project can succeed or fail. There is a 10% chance of success for the fancy project, and 80% chance of success for the mundane project. The fancy project costs 1M$ to execute, and if it succeeds, you get 11M$. If you lose, you get nothing. The mundane project costs 0.5M$ to execute, and you get 1.5M$ if it succeeds. If it fails, you get nothing. Which project should you do?

EU = 0.1*10=1.0

EU = 0.9*(-1)=-0.9

EU = 0.8*1=0.8

EU = 0.2*(-0.5)=-0.1

EU = 1.0-0.9= 0.1

EU = 0.8-0.1=0.7

Best choice

A more complex example• Your show dog has cancer. If you do nothing, there is a

90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

1. List each decision nodes & its alternatives.• Which treatment?

– surgery– chemo– Nothing

• After chemo fails, which treatment?– surgery– nothing

Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

2. List each chance nodes & its alternatives.

• Outcome?– Cancer cured– Died due to surgery (only for surgery node)– Cancer not cured

Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

3. Draw nodes

Treatment?

Outcome?

Outcome?

Surgery

chemo

Cured

Cured ?

Uncured

Surgery kills

•Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

Outcome?

Noth

ing

cost

0$

Die?uncured

Die

Live

Outcome?

Die

Live Die?

Die

Live

Treatment?Surgery

Nothing

4. Add costs & probabilities

Treatment?

Outcome?

Outcome?

Surgery

cost 10K$

chemocost 4K$

Cured

p=0.4

Cured

p=0.2

Uncuredp=0.8

Surgery killsp= 0.1

•Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

Outcome?

Noth

ing

cost

0$

Die?uncured

p=0.5

Die

p=0.9

Livep=0.1

Outcome?

Die

p=0.9Live

p=0.1Die?

Die

p=0.9

Livep=0.1

Treatment?Surgery

cost

10K$

Nothing

5. Calculate utility values for utility nodes

Treatment?

Outcome?

Outcome?

Surgery

cost 10K$

chemocost 4K$896K$

-10K$

Cured

p=0.4

Cured

p=0.2

Uncuredp=0.8

Surgery killsp= 0.1

•Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

Outcome?

Noth

ing

cost

0$

890K$

-10K$

Die?890K$

uncuredp=0.5

Die

p=0.9

Livep=0.1

Outcome?

0$

900K$

Die

p=0.9Live

p=0.1

-4K$

Die?896K$

Die

p=0.9

Livep=0.1

Treatment?Surgery

cost

10K$

Nothing

6. Calculate expected utilities

Treatment?

Outcome?

Outcome?

Surgery

cost 10K$

EU=444

chemocost 4K$EU=619.2896K$

-10K$

Cured

p=0.4

EU=365

Cured

p=0.2

EU=179.2

Uncuredp=0.8EU=440

Surgery killsp= 0.1EU=-1

•Your show dog has cancer. If you do nothing, there is a 90% chance she will die. If she has surgery, there is a 40% chance of curing the cancer, 10% chance of dying from the surgery, and a 50% chance that the cancer will survive, in which case, she has the usual 90% chance of dying. If she gets chemo, there is a 20% chance of curing the cancer, and an 80% that the cancer will remain, in which surgery can be performed, with the same risks and outcomes as mentioned above. Chemo cannot be done after surgery, by the way. The dog is worth 900K$ if she is alive, and nothing if she is dead. Surgery costs $10K and chemo costs 4K$. What should you do?

Outcome?

Noth

ing

cost

0$

EU=9

0890K$

-10K$

Die?890K$

uncuredp=0.5EU=80

Die

p=0.9

EU= -9Live

p=0.1EU=89

Outcome?

0$

900K$

Die

p=0.9

EU=0Livep=0.1

EU=90

-4K$

Die?896K$

Die

p=0.9

EU=-3.6Live

p=0.1EU=89.6

Treatment?

Surgery

cost

10K$

EU=440

NothingEU=86

My disk drive is flakey. Tech says that there is a 10% chance it will crash in the next week, and replacing the disk and data will cost me

$200. If I replace the disk, it will cost me $70 and it will not crash. If I save its contents to

DVD, reformat and restore it, then it costs me 4 hours (equivalent to $40) but reduces the

chance of crashing to 3%.

Save, reformat, restore?

Crash?

Crash?

No

Yes

-240

0

-200

-40

Yes

p=0.03

Yes

p=0.1

Nop=0.9

Nop= 0.97

Replace disk?

-70

Yes

No

EU = 0.03*(-240)+0.97*(-40) = -46

EU = 0.10*(-200)+0.90*(0) = -20

EU = -70

Quiz question• You will be drilling a water well in your backyard, and

you have to decide where to dig it. If you just dig where you think it is best, there is a 40% chance you’ll hit water. If you hire a seismologist, the chances increase to 60% but it costs you 10K$. If you hire a water witch, the chance of hitting water is 50% and the cost is 1K$. If you get 50K$ for hitting water, what should you do

• Draw the decision tree• Evaluate alternatives• Indicate best choice• Put your name on the paper and hand it in.

You will be drilling a water well in your backyard, and you have to decide where to dig it. If you just dig where you think it is best, there is a 40% chance you’ll hit water. If you hire a seismologist, the chances increase to 60% but it costs you 10K$. If you hire a water witch, the chance of hitting water is 50% and the cost is 1K$. If you get 50K$ for hitting water, what should you do?

Water?

Water?SeismologistCost 10K

EU=20

Nothing

Cost 0$

EU=20

49K$

-10K$

40K$

-1K$

Yes

p=0.4

EU=20

Yes

p=0.6

EU=24

Nop=0.4EU=-4

Nop= 0.5

EU=-0.5

Detector?

50K$

Water witch

cost 1K$

EU=24

Water?Nop= 0.6EU=0

Yes

p=0.5

EU=24.5

0