Decision tree

# Decision tree

Overview
A decision tree is a decision support tool that uses a tree-like graph
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...

or model
Causal model
A causal model is an abstract model that describes the causal mechanisms of a system. The model must express more than correlation because correlation does not imply causation....

of decisions and their possible consequences, including chance
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

event outcomes, resource costs, and utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

. It is one way to display an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

. Decision trees are commonly used in operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, specifically in decision analysis
Decision analysis
Decision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...

, to help identify a strategy most likely to reach a goal. Another use of decision trees is as a descriptive means for calculating conditional probabilities
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

.
Discussion

Encyclopedia
A decision tree is a decision support tool that uses a tree-like graph
Diagram
A diagram is a two-dimensional geometric symbolic representation of information according to some visualization technique. Sometimes, the technique uses a three-dimensional visualization which is then projected onto the two-dimensional surface...

or model
Causal model
A causal model is an abstract model that describes the causal mechanisms of a system. The model must express more than correlation because correlation does not imply causation....

of decisions and their possible consequences, including chance
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...

event outcomes, resource costs, and utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....

. It is one way to display an algorithm
Algorithm
In mathematics and computer science, an algorithm is an effective method expressed as a finite list of well-defined instructions for calculating a function. Algorithms are used for calculation, data processing, and automated reasoning...

. Decision trees are commonly used in operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

, specifically in decision analysis
Decision analysis
Decision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...

, to help identify a strategy most likely to reach a goal. Another use of decision trees is as a descriptive means for calculating conditional probabilities
Conditional probability
In probability theory, the "conditional probability of A given B" is the probability of A if B is known to occur. It is commonly notated P, and sometimes P_B. P can be visualised as the probability of event A when the sample space is restricted to event B...

.

## General

In decision analysis
Decision analysis
Decision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...

, a "decision tree" — and the closely related influence diagram
Influence diagram
An influence diagram is a compact graphical and mathematical representation of a decision situation...

— is used as a visual and analytical decision support tool, where the expected value
Expected value
In probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...

s (or expected utility) of competing alternatives are calculated.

A decision tree consists of 3 types of nodes:-

1. Decision nodes - commonly represented by squares

2. Chance nodes - represented by circles

3. End nodes - represented by triangles

Drawn from left to right, a decision tree has only burst nodes (splitting paths) but no sink nodes (converging paths). Therefore, used manually, they can grow very big and are then often hard to draw fully by hand. Traditionally, decision trees have been created manually - as the aside example shows - although increasingly, specialized software is employed.

Analysis can take into account the decision maker's (e.g., the company's) preference
Preference
-Definitions in different disciplines:The term “preferences” is used in a variety of related, but not identical, ways in the scientific literature. This makes it necessary to make explicit the sense in which the term is used in different social sciences....

or utility function, for example:

The basic interpretation in this situation is that the company prefers B's risk and payoffs under realistic risk preference coefficients (greater than \$400K—in that range of risk aversion, the company would need to model a third strategy, "Neither A nor B").

## Influence diagram

A decision tree can be represented more compactly as an influence diagram, focusing attention on the issues and relationships between events.

The squares represent decisions, the ovals represent action, and the diamond represents results.

## Uses in teaching

Decision trees, influence diagrams, utility functions, and other decision analysis
Decision analysis
Decision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...

tools and methods are taught to undergraduate students in schools of business, health economics, and public health, and are examples of operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

or management science methods.

Amongst decision support tools, decision trees (and influence diagrams) have several advantages:

Decision trees:
• Are simple to understand and interpret. People are able to understand decision tree models after a brief explanation.
• Have value even with little hard data. Important insights can be generated based on experts describing a situation (its alternatives, probabilities, and costs) and their preferences for outcomes.
• Use a white box
White box (software engineering)
In software engineering white box, in contrast to a black box, is a subsystem whose internals can be viewed, but usually cannot be altered. This is useful during white box testing, where a system is examined to make sure that it fulfills its requirements....

model.
If a given result is provided by a model, the explanation for the result is easily replicated by simple math.
• Can be combined with other decision techniques. The following example uses Net Present Value calculations, PERT 3-point estimations (decision #1) and a linear distribution of expected outcomes (decision #2):

Decision trees:
• For data including categorical variables with different number of levels, information gain in decision trees
Information gain in decision trees
In information theory and machine learning, information gain is an alternative synonym for Kullback–Leibler divergence.In particular, the information gain about a random variable X obtained from an observation that a random variable A takes the value A=a is the Kullback-Leibler divergence DKL of...

are biased in favor of those attributes with more levels.

## Example

Decision trees can be used to optimize an investment portfolio. The following example shows a portfolio of 7 investment options (projects). The organization has \$10,000,000 available for the total investment. Bold lines mark the best selection 1, 3, 5, 6, and 7, which will cost \$9,750,000 and create a payoff of 16,175,000. All other combinations would either exceed the budget or yield a lower payoff.

## Example

In the game of "20 Questions", the querent tries to construct a short binary decision tree that isolates a specific item. The item's identity question is asked when the current decision tree node is considered reliable by the querent.

• Decision tables
• Decision tree complexity
• Decision tree model
Decision tree model
In computational complexity and communication complexity theories the decision tree model is the model of computation or communication in which an algorithm or communication process is considered to be basically a decision tree, i.e., a sequence of branching operations based on comparisons of some...

of computation
• Expectiminimax tree
Expectiminimax tree
An expectiminimax tree is a specialized variation of a minimax game tree for use in artificial intelligence systems that play two-player zero-sum games such as backgammon, in which the outcome depends a combination of the player's skill and chance elements such as dice rolls...

• Influence diagram
Influence diagram
An influence diagram is a compact graphical and mathematical representation of a decision situation...

• Markov chain
Markov chain
A Markov chain, named after Andrey Markov, is a mathematical system that undergoes transitions from one state to another, between a finite or countable number of possible states. It is a random process characterized as memoryless: the next state depends only on the current state and not on the...

• Morphological analysis
• Operations research
Operations research
Operations research is an interdisciplinary mathematical science that focuses on the effective use of technology by organizations...

• Topological combinatorics
Topological combinatorics
The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this gradually turned into the field of algebraic topology....

• Truth table
Truth table
A truth table is a mathematical table used in logic—specifically in connection with Boolean algebra, boolean functions, and propositional calculus—to compute the functional values of logical expressions on each of their functional arguments, that is, on each combination of values taken by their...