Value of information
Encyclopedia
Value of information is the amount a decision maker would be willing to pay for information prior to making a decision.
(VoC), and value of imperfect information. They are closely related to the widely known expected value of perfect information
and expected value of sample information
. Note that VoI is not necessarily equal to "value of decision situation with perfect information" - "value of current decision situation" as commonly understood.
assumption in standard individual decision-making philosophy states that what is made or known are not forgotten, i.e., decision-maker has perfect recall. This assumption translates into the existence of a linear ordering of these decisions and uncertainties such that;
Consider the case where the decision-maker is enabled to know the outcome of some additional uncertainties earlier in his/her decision situation, i.e., some ui are moved to appear earlier in the ordering. In such case, VoC is quantified as the highest price which the decision-maker is willing to pay for all those moves.
A special case is when the decision-maker is risk neutral
where VoC can be simply computed as;
This special case is how expected value of perfect information
and expected value of sample information
are calculated where risk neutrality
is implicitly assumed. For cases where decision-maker is risk averse or risk seeking, this simple calculation does not necessary yield correct result, and iterative calculation is the only way to ensure correctness.
Decision tree
and influence diagram
are most commonly used in representing and solving decision situation as well as associated VoC calculation. Influence diagram, in particular, is structured to accommodate team decision situation where incomplete sharing of information among team members can be represented and solved very efficiently. While decision tree is not designed to accommodate team decision situation, it can do so by augmenting it with information set
widely used in game tree
.
principle and thus extra works are needed to represent and solve for VoI properly.
Similar terms
VoI is sometimes distinguished into value of perfect information, also called value of clairvoyanceClairvoyance
The term clairvoyance is used to refer to the ability to gain information about an object, person, location or physical event through means other than the known human senses, a form of extra-sensory perception...
(VoC), and value of imperfect information. They are closely related to the widely known expected value of perfect information
Expected value of perfect information
In decision theory, the expected value of perfect information is the price that one would be willing to pay in order to gain access to perfect information....
and expected value of sample information
Expected value of sample information
In decision theory, the expected value of sample information is the expected increase in utility that you could obtain from gaining access to a sample of additional observations before making a decision. The additional information obtained from the sample may allow you to make a more informed,...
. Note that VoI is not necessarily equal to "value of decision situation with perfect information" - "value of current decision situation" as commonly understood.
Simple
A simple example best illustrates the concept. Consider a decision situation with one decision, for example deciding on a 'Vacation Activity'; and one uncertainty, for example what will the 'Weather Condition' be? But we will only know the 'Weather Condition' after we have decided and begun the 'Vacation Activity'.- The Value of perfect information on Weather Condition captures the value of being able to know Weather Condition even before making the Vacation Activity decision. It is quantified as the highest price the decision-maker is willing to pay for being able to know Weather Condition before making the Vacation Activity decision.
- The Value of imperfect information on Weather Condition, however, captures the value of being able to know the outcome of another related uncertainty, e.g., Weather Forecast, instead of Weather Condition itself before making Vacation Activity decision. It is quantified as the highest price the decision-maker is willing to pay for being able to know Weather Forecast before making Vacation Activity decision. Note that it is essentially the value of perfect information on Weather Forecast.
Formal
The above definition illustrates that the value of imperfect information of any uncertainty can always be framed as the value of perfect information, i.e., VoC, of another uncertainty, hence only the term VoC will be used onwards.Standard
Consider a general decision situation having n decisions (d1, d2, d3, ..., dn) and m uncertainties (u1, u2, u3, ..., um). RationalityRationality
In philosophy, rationality is the exercise of reason. It is the manner in which people derive conclusions when considering things deliberately. It also refers to the conformity of one's beliefs with one's reasons for belief, or with one's actions with one's reasons for action...
assumption in standard individual decision-making philosophy states that what is made or known are not forgotten, i.e., decision-maker has perfect recall. This assumption translates into the existence of a linear ordering of these decisions and uncertainties such that;
- di is made prior to making dj if and only if di comes before dj in the ordering
- di is made prior to knowing uj if and only if di comes before uj in the ordering
- di is made after knowing uj if and only if di comes after uj in the ordering
Consider the case where the decision-maker is enabled to know the outcome of some additional uncertainties earlier in his/her decision situation, i.e., some ui are moved to appear earlier in the ordering. In such case, VoC is quantified as the highest price which the decision-maker is willing to pay for all those moves.
Generalized
The standard definition is further generalized in team decision analysis framework where there is typically incomplete sharing of information among team members under the same decision situation. In such case, what is made or known might not be known in later decisions belonging to different team members, i.e., there might not exist linear ordering of decisions and uncertainties satisfying perfect recall assumption. VoC thus captures the value of being able to know "not only additional uncertainties but also additional decisions already made by other team members" before making some other decisions in the team decision situation.Characteristics
There are two extremely important characteristics of VoI that always hold for any decision situation;- Value of information can never be less than zero since the decision-maker can always ignore the additional information and makes decision as if such information is not available.
- No other information gathering/sharing activities can be more valuable than that quantified by value of clairvoyance.
Computation
VoC is derived strictly following its definition as the monetary amount that is big enough to just offset additional benefit of getting more information. In other words; VoC is calculated iteratively until;-
- "value of decision situation with perfect information while paying VoC" = "value of current decision situation".
A special case is when the decision-maker is risk neutral
Risk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...
where VoC can be simply computed as;
-
- VoC = "value of decision situation with perfect information" - "value of current decision situation"
This special case is how expected value of perfect information
Expected value of perfect information
In decision theory, the expected value of perfect information is the price that one would be willing to pay in order to gain access to perfect information....
and expected value of sample information
Expected value of sample information
In decision theory, the expected value of sample information is the expected increase in utility that you could obtain from gaining access to a sample of additional observations before making a decision. The additional information obtained from the sample may allow you to make a more informed,...
are calculated where risk neutrality
Risk neutral
In economics and finance, risk neutral behavior is between risk aversion and risk seeking. If offered either €50 or a 50% chance of each of €100 and nothing, a risk neutral person would have no preference between the two options...
is implicitly assumed. For cases where decision-maker is risk averse or risk seeking, this simple calculation does not necessary yield correct result, and iterative calculation is the only way to ensure correctness.
Decision tree
Decision tree
A decision tree is a decision support tool that uses a tree-like graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm. Decision trees are commonly used in operations research, specifically...
and influence diagram
Influence diagram
An influence diagram is a compact graphical and mathematical representation of a decision situation...
are most commonly used in representing and solving decision situation as well as associated VoC calculation. Influence diagram, in particular, is structured to accommodate team decision situation where incomplete sharing of information among team members can be represented and solved very efficiently. While decision tree is not designed to accommodate team decision situation, it can do so by augmenting it with information set
Information set
In game theory, an information set is a set that, for a particular player, establishes all the possible moves that could have taken place in the game so far, given what that player has observed. If the game has perfect information, every information set contains only one member, namely the point...
widely used in game tree
Game tree
In game theory, a game tree is a directed graph whose nodes are positions in a game and whose edges are moves. The complete game tree for a game is the game tree starting at the initial position and containing all possible moves from each position; the complete tree is the same tree as that...
.
Free Will
Special care is needed when the choice being made for a decision can influence how an uncertainty resolves in the future. Having a perfect or imperfect information on such uncertainty implies that the choice to be made can be inferred prior to making such choice. This circular logic is against free willFree will
"To make my own decisions whether I am successful or not due to uncontrollable forces" -Troy MorrisonA pragmatic definition of free willFree will is the ability of agents to make choices free from certain kinds of constraints. The existence of free will and its exact nature and definition have long...
principle and thus extra works are needed to represent and solve for VoI properly.
See also
- Decision analysisDecision analysisDecision analysis is the discipline comprising the philosophy, theory, methodology, and professional practice necessary to address important decisions in a formal manner...
- Decision treeDecision treeA decision tree is a decision support tool that uses a tree-like graph or model of decisions and their possible consequences, including chance event outcomes, resource costs, and utility. It is one way to display an algorithm. Decision trees are commonly used in operations research, specifically...
- Expected value of perfect informationExpected value of perfect informationIn decision theory, the expected value of perfect information is the price that one would be willing to pay in order to gain access to perfect information....
(EVPI) - Expected value of including uncertaintyExpected value of including uncertaintyIn decision theory and quantitative policy analysis, the expected value of including information is the expected difference in the value of a decision based on a probabilistic analysis versus a decision based on an analysis that ignores uncertainty....
(EVIU) - Expected value of sample informationExpected value of sample informationIn decision theory, the expected value of sample information is the expected increase in utility that you could obtain from gaining access to a sample of additional observations before making a decision. The additional information obtained from the sample may allow you to make a more informed,...
- Influence diagramInfluence diagramAn influence diagram is a compact graphical and mathematical representation of a decision situation...
- Value of controlValue of controlThe value of control is a quantitiative measure of the value of controlling the outcome of an uncertainty variable. Decision analysis provides a means for calculating the value of both perfect and imperfect control. The former value, informally known as the value of wizardry, is an upper bound for...