Artificial Intelligence Chapter 16: Decision Making

Questions and Answers

What is the expected monetary value (EMV) of the gamble if the coin is fair in the given scenario?

$1,250,000

What are the expected utilities of accepting and declining the gamble based on the provided utilities?

EU (Accept) = 1/2 * U(Sk) + 1/2 * U(Sk+2,500,000), EU (Decline) = U(Sk+1,000,000)

What is the utility function for Mr. Beard as shown in Figure 16.2(a)?

U(Sk+n) = −263.31 + 22.09 log(n + 150,000)

What is the existence of a utility function based on?

Preferences Signup and view all the answers

What is the rational action to take if faced with the gamble scenario?

Decline the gamble Signup and view all the answers

Risk-averse agents prefer a sure thing with a payoff less than the expected monetary value of a gamble.

True Signup and view all the answers

Utility theory states that all rational agents must obey specific axioms.

True Signup and view all the answers

What is the utility of a lottery according to utility theory?

sum of the probability of each outcome times the utility of that outcome Signup and view all the answers

In utility theory, a utility function maps from lotteries to ___.

real numbers Signup and view all the answers

What is the basic principle of decision theory discussed in Section 16.1?

maximization of expected utility Signup and view all the answers

Which function captures the agent's preferences by assigning a single number to express the desirability of a state?

Utility function Signup and view all the answers

The Maximum Expected Utility (MEU) principle states that an agent should choose the action with the lowest expected utility.

False Signup and view all the answers

_______ is a random variable in decision theory that represents the possible outcome states given evidence observations.

Result (a) Signup and view all the answers

Match the following constraints with their descriptions in utility theory:

Orderability = Agent must either prefer one option to another or rate them as equally preferable. Transitivity = If agent prefers A to B and B to C, then the agent must prefer A to C. Continuity = If lottery B is between A and C in preference, then there is some probability p for which agent will be indifferent. Substitutability = If agent is indifferent between two lotteries A and B, then agent is indifferent between more complex lotteries with B substituted for A. Monotonicity = If agent prefers one outcome over another, they must prefer the lottery with a higher probability for that outcome. Decomposability = Compound lotteries can be reduced to simpler ones using the laws of probability. Signup and view all the answers

What does the optimizer's curse result from?

The ubiquity of utility-maximizing selection processes Signup and view all the answers

Decision theory is a descriptive theory that describes how actual agents act.

False Signup and view all the answers

What is the term used for the phenomenon where people are strongly attracted to gains that are certain?

certainty effect Signup and view all the answers

The Allais paradox involves people consistently preferring ______ over A and C over D.

B Signup and view all the answers

Match the following phenomenon with its description:

Framing Effect = The impact of the wording of a decision problem on choices Anchoring Effect = The bias towards known probabilities over unknown probabilities Ambiguity Aversion = Preferring certain gains over uncertain gains Signup and view all the answers

What is the term used to describe a situation where one option is of lower value on all attributes than some other option?

strict dominance Signup and view all the answers

What is the term for a situation where despite uncertainty, all possible concrete outcomes for S1 strictly dominate all possible outcomes for S2?

Stochastic dominance Signup and view all the answers

Stochastic dominance can always be determined by comparing the expected costs of two options.

False Signup and view all the answers

In deterministic preference structures, the basic regularity that arises is called preference __________.

independence Signup and view all the answers

Match the term with its description:

Preference independence = Two attributes X1 and X2 are preferentially independent of a third attribute X3 if the preference between outcomes does not depend on the particular value of X3. Mutual preferential independence = The set of attributes exhibits mutual preferential independence if each attribute is important but does not affect the trade-offs between other attributes. Stochastic dominance = Despite uncertainty, all possible concrete outcomes for S1 strictly dominate all possible outcomes for S2. Signup and view all the answers

What is postcoarctectomy syndrome?

A condition that occurs after coarctectomy surgery Signup and view all the answers

What is tachypnea?

Rapid breathing Signup and view all the answers

What is tachycardia?

Rapid heart rate Signup and view all the answers

What is paradoxical hypertension?

A condition where blood pressure increases despite treatment Signup and view all the answers

What is failure to thrive?

A condition where an individual fails to develop or grow at a normal rate Signup and view all the answers

What is aortic hypertension?

High blood pressure in the aorta Signup and view all the answers

What does the value of information derive from?

The value of information derives from the fact that with the information, one’s course of action can be changed to suit the actual situation. Signup and view all the answers

What are the two factors that determine the importance of tests according to the text?

Whether the test results would lead to a significantly better treatment plan Signup and view all the answers

The expected profit for the company, given survey information, is $C/n(n-1)$. Therefore, the company should be willing to pay the seismologist up to ____ dollars for the information.

C/n Signup and view all the answers

The expected value of information can be negative.

False Signup and view all the answers

What distinguishes sensing actions from ordinary actions in terms of order independence?

Order independence distinguishes sensing actions from ordinary actions. Signup and view all the answers

What is the design principle mentioned when discussing the implementation of an information-gathering agent?

Ask questions in a reasonable order Signup and view all the answers

Myopic control uses the VPI formula shortsightedly.

True Signup and view all the answers

DECISION ANALYSIS studies the application of decision theory to actual __________ problems.

decision Signup and view all the answers

What is a value function of the type V (noise, cost, deaths) called?

Additive value function Signup and view all the answers

Which type of function represents an additive value function?

Single-attribute utility function Signup and view all the answers

Utility independence extends preference independence to cover lotteries.

True Signup and view all the answers

For conciseness, in a multiplicative utility function, U = $k_1 U_1 + k_2 U_2 + k_3 U_3 + k_1 k_2 U_1 U_2 + k_2 k_3 U_2 U_3 + k_3 k_1 U_3 U_1$, an n-attribute problem exhibiting MUI can be modeled using _ single-attribute utilities and _ constants.

n; n Signup and view all the answers

Match the following node types with their representations in decision networks:

Chance nodes = Represent random variables Decision nodes = Represent points where decisions are made Utility nodes = Represent the agent's utility function Signup and view all the answers

Study Notes

Making Simple Decisions

Decision theory combines with probability theory to yield a decision-theoretic agent that can make rational decisions based on what the agent believes and wants.
The principle of maximum expected utility (MEU) says that a rational agent should choose the action that maximizes the agent's expected utility.

Combining Beliefs and Desires Under Uncertainty

Decision theory deals with choosing among actions based on the desirability of their immediate outcomes.
The agent's preferences are captured by a utility function, U(s), which assigns a single number to express the desirability of a state.
The expected utility of an action given the evidence, EU(a|e), is the average utility value of the outcomes, weighted by the probability that the outcome occurs.
The principle of maximum expected utility (MEU) says that a rational agent should choose the action that maximizes the agent's expected utility.

The Basis of Utility Theory

The principle of Maximum Expected Utility (MEU) seems like a reasonable way to make decisions, but it is by no means obvious that it is the only rational way.
There are six constraints that a rational preference relation should obey:
- Orderability: Given any two lotteries, a rational agent must either prefer one to the other or rate the two as equally preferable.
- Transitivity: Given any three lotteries, if an agent prefers A to B and prefers B to C, then the agent must prefer A to C.
- Continuity: If some lottery B is between A and C in preference, then there is some probability p for which the rational agent will be indifferent between getting B for sure and the lottery that yields A with probability p and C with probability 1 - p.
- Substitutability: If an agent is indifferent between two lotteries A and B, then the agent is indifferent between two more complex lotteries that are the same except that B is substituted for A in one of them.
- Monotonicity: Suppose two lotteries have the same two possible outcomes, A and B. If an agent prefers A to B, then the agent must prefer the lottery that has a higher probability for A (and vice versa).
- Decomposability: Compound lotteries can be reduced to simpler ones using the laws of probability.

Preferences Lead to Utility

From the axioms of utility, we can derive the following consequences:
- Existence of Utility Function: If an agent's preferences obey the axioms of utility, then there exists a function U such that U(A) > U(B) if and only if A is preferred to B, and U(A) = U(B) if and only if the agent is indifferent between A and B.
- Expected Utility of a Lottery: The utility of a lottery is the sum of the probability of each outcome times the utility of that outcome.

Utility Functions

A utility function is a function that maps from lotteries to real numbers.
An agent's preferences can be unusual, but we can't call it irrational.
Utility assessment involves presenting choices to the agent and using the observed preferences to pin down the underlying utility function.
A scale can be established by fixing the utilities of any two particular outcomes, such as a "best possible prize" and a "worst possible catastrophe".### Probability and Utility
The probability p is adjusted until the agent is indifferent between a standard lottery and a lottery involving a prize, thus determining the utility of the prize.
In medical, transportation, and environmental decision problems, people's lives are at stake, and a value is assigned to immediate death (u⊥).

Micromort and QALY

Micromort: a one in a million chance of death, used as a "currency" in medical and safety analysis.
People are willing to pay about $50 per micromort, based on studies.

Utility of Money

Utility theory has its roots in economics, and economics provides a candidate for a utility measure: money.
Money plays a significant role in human utility functions, and agents usually prefer more money to less.
The utility of money is not directly proportional to monetary value, but rather concave for positive wealth and convex for negative wealth.

Risk Aversion and Seeking

Agents with concave utility curves are risk-averse, preferring a sure thing over a gamble.
Agents with convex utility curves are risk-seeking, preferring a gamble over a sure thing.
The certainty equivalent of a lottery is the value an agent will accept in lieu of the lottery.

Insurance Premium and Risk Neutrality

The difference between the EMV of a lottery and its certainty equivalent is called the insurance premium.
Insurance companies take advantage of people's risk aversion by offering insurance at a premium.
For small changes in wealth relative to current wealth, almost any curve will be approximately linear, making agents risk-neutral.

Expected Utility and Post-Decision Disappointment

The rational way to choose the best action is to maximize expected utility.
Estimates of expected utility may be unbiased, but still lead to disappointment due to optimistic estimates.
The optimizer's curse: the tendency for the estimated expected utility of the best choice to be too high, leading to disappointment.

Human Judgment and Irrationality

Decision theory is a normative theory, describing how a rational agent should act, but humans may not act rationally.
The Allais paradox: people consistently prefer B over A (taking the sure thing) and C over D (taking the higher EMV), contradicting normative analysis.
The certainty effect: people are strongly attracted to gains that are certain, which may be due to reducing computational burden, distrust of probabilities, or accounting for emotional state.
The Ellsberg paradox: people's choices are influenced by underconstrained probabilities, leading to apparent irrationality.### Ambiguity Aversion
Ambiguity aversion is the tendency for people to prefer known probabilities over unknown ones.
Example: people prefer lottery A (100 dollars for a red ball) over B (100 dollars for a black ball) and also prefer D (100 dollars for a black or yellow ball) over C (100 dollars for a red or yellow ball), even though there is no state of the world where this is rational.

Framing Effect

The framing effect is the tendency for people to be influenced by the way a decision problem is framed.
Example: people prefer a medical procedure with a "90% survival rate" over one with a "10% death rate", even though they mean the same thing.

Anchoring Effect

The anchoring effect is the tendency for people to rely too heavily on the first piece of information they receive when making decisions.
Example: a restaurant offers a $200 bottle of wine to make the $55 bottle seem like a bargain.

Multiattribute Utility Theory

Multiattribute utility theory is used to handle decision problems with multiple attributes.
Attributes are characterized by multiple values, and a complete vector of assignments is a combination of these values.
Higher values of an attribute correspond to higher utilities.
Strict dominance occurs when one option is better than another on all attributes.
Stochastic dominance occurs when one option is better than another on all attributes, even when the outcomes are uncertain.

Stochastic Dominance

Stochastic dominance is a more general and useful concept than strict dominance.
It occurs when the cumulative distribution of one option is always to the right of the cumulative distribution of another option.
If an option A stochastically dominates an option B, then the expected utility of A is at least as high as the expected utility of B.
Qualitative probabilistic networks can be used to propagate stochastic dominance and make rational decisions without using numeric values.

Preference Structure and Multiattribute Utility

Multiattribute utility theory is based on the idea that preferences have regularity and structure.
Representation theorems are used to show that an agent with a certain preference structure has a utility function that can be decomposed into simpler functions.
Preference independence is a regularity that arises in deterministic preference structures.
Mutual preferential independence occurs when a set of attributes exhibits preference independence with respect to each other.
Mutual preferential independence leads to a simple form for the agent's value function, which can be derived using a theorem by Gérard Debreu.

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This chapter covers how an agent makes decisions based on utility theory and probability theory to get what it wants.

Artificial Intelligence Chapter 16: Decision Making

Questions and Answers

What is the expected monetary value (EMV) of the gamble if the coin is fair in the given scenario?

What are the expected utilities of accepting and declining the gamble based on the provided utilities?

What is the utility function for Mr. Beard as shown in Figure 16.2(a)?

What is the existence of a utility function based on?

What is the rational action to take if faced with the gamble scenario?

Risk-averse agents prefer a sure thing with a payoff less than the expected monetary value of a gamble.

Utility theory states that all rational agents must obey specific axioms.

What is the utility of a lottery according to utility theory?

In utility theory, a utility function maps from lotteries to ___.

What is the basic principle of decision theory discussed in Section 16.1?

Which function captures the agent's preferences by assigning a single number to express the desirability of a state?

The Maximum Expected Utility (MEU) principle states that an agent should choose the action with the lowest expected utility.

_______ is a random variable in decision theory that represents the possible outcome states given evidence observations.

Match the following constraints with their descriptions in utility theory:

What does the optimizer's curse result from?

Decision theory is a descriptive theory that describes how actual agents act.

What is the term used for the phenomenon where people are strongly attracted to gains that are certain?

The Allais paradox involves people consistently preferring ______ over A and C over D.

Match the following phenomenon with its description:

What is the term used to describe a situation where one option is of lower value on all attributes than some other option?

What is the term for a situation where despite uncertainty, all possible concrete outcomes for S1 strictly dominate all possible outcomes for S2?

Stochastic dominance can always be determined by comparing the expected costs of two options.

In deterministic preference structures, the basic regularity that arises is called preference __________.

Match the term with its description:

What is postcoarctectomy syndrome?

What is tachypnea?

What is tachycardia?

What is paradoxical hypertension?

What is failure to thrive?

What is aortic hypertension?

What does the value of information derive from?

What are the two factors that determine the importance of tests according to the text?

The expected profit for the company, given survey information, is $C/n(n-1)$. Therefore, the company should be willing to pay the seismologist up to ____ dollars for the information.

The expected value of information can be negative.

What distinguishes sensing actions from ordinary actions in terms of order independence?

What is the design principle mentioned when discussing the implementation of an information-gathering agent?

Myopic control uses the VPI formula shortsightedly.

DECISION ANALYSIS studies the application of decision theory to actual __________ problems.

What is a value function of the type V (noise, cost, deaths) called?

Which type of function represents an additive value function?

Utility independence extends preference independence to cover lotteries.

For conciseness, in a multiplicative utility function, U = $k_1 U_1 + k_2 U_2 + k_3 U_3 + k_1 k_2 U_1 U_2 + k_2 k_3 U_2 U_3 + k_3 k_1 U_3 U_1$, an n-attribute problem exhibiting MUI can be modeled using ___ single-attribute utilities and ___ constants.

Match the following node types with their representations in decision networks:

Study Notes

Making Simple Decisions

Combining Beliefs and Desires Under Uncertainty

The Basis of Utility Theory

Preferences Lead to Utility

Utility Functions

Micromort and QALY

Utility of Money

Risk Aversion and Seeking

Insurance Premium and Risk Neutrality

Expected Utility and Post-Decision Disappointment

Human Judgment and Irrationality

Framing Effect

Anchoring Effect

Multiattribute Utility Theory

Stochastic Dominance

Preference Structure and Multiattribute Utility

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For conciseness, in a multiplicative utility function, U = $k_1 U_1 + k_2 U_2 + k_3 U_3 + k_1 k_2 U_1 U_2 + k_2 k_3 U_2 U_3 + k_3 k_1 U_3 U_1$, an n-attribute problem exhibiting MUI can be modeled using _ single-attribute utilities and _ constants.