PHPE 400Individual and Group Decision Making

Eric PacuitUniversity of Maryland

pacuit.org

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First Steps

https://umd.instructure.com/courses/1321153/pages/first-steps?module_item_id=11107804

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First Steps

1. Make sure you are signed up and can login to Campuswirehttps://campuswire.com/p/GBDA07BBC.

2. Sign up for https://app.tophat.com/e/556838 with join code556838. You must purchase the pro-subscription.

3. Read the course policies (https://phpe400.org/policies) andsyllabus (https://umd.instructure.com/courses/1321153/assignments/syllabus).

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To Do

1. Answer the introductory quiz on Tophat:https://app.tophat.com/e/556838/page/344488501

2. Answer the mathematical notation quiz on Tophat:https://app.tophat.com/e/556838/page/344488510

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Grading

Participation 30%

Problem Sets 40%

Midterm 15%

Final Exam 15%

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Online Tools

Course Websitehttps://umd.instructure.com/courses/1321153

Online Discussionhttps://campuswire.com/c/GBDA07BBC/feed

Participation Questionshttps://app.tophat.com/e/556838

Readings and Course Noteshttps://umd.instructure.com/courses/1311302/modules

https://notes.phpe400.info

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Topics

1. Rational Choice Theory: Preference, Choice and Utility

2. Game Theory

3. Voting, Social Choice Theory, and Judgement Aggregation

4. Aggregating Utilities, Interpersonal Comparisons of Utility

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Practicalities: Math

The course is completely self-contained, but it does require that you becomecomfortable with some mathematical notation.

I Spend some time familiarizing yourself with the relevant mathematicalnotation: sets X = {a, b, c}, subset of X ⊆ Y, element of x ∈ X,cross-product X × Y = {(x, y) | x ∈ X, y ∈ Y}, relations R ⊆ X × X,functions f : X→ Y, . . .

https://notes.phpe400.info/mathematical-background/math-background.html

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Practicalities: Math

The course is completely self-contained, but it does require that you becomecomfortable with some mathematical notation.

I Spend some time familiarizing yourself with the relevant mathematicalnotation: sets X = {a, b, c}, subset of X ⊆ Y, element of x ∈ X,cross-product X × Y = {(x, y) | x ∈ X, y ∈ Y}, relations R ⊆ X × X,functions f : X→ Y, . . .

https://notes.phpe400.info/mathematical-background/math-background.html

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Practicalities: Math

The course is completely self-contained, but it does require that you becomecomfortable with some mathematical notation.

I Spend some time familiarizing yourself with the relevant mathematicalnotation: sets X = {a, b, c}, subset of X ⊆ Y, element of x ∈ X,cross-product X × Y = {(x, y) | x ∈ X, y ∈ Y}, relations R ⊆ X × X,functions f : X→ Y, . . .

https://notes.phpe400.info/mathematical-background/math-background.html

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Practicalities: Math

I Ask questions, especially about notation that you do not understand (nomatter how trivial).

I The participation questions are designed, in part, to make sure youunderstand the mathematical content.

I It is important to use the proper notation on the problem sets and theexams (otherwise I won’t understand your answer).

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Practicalities: Math

Economic models consist of clearly stated assumptions and behav-ioral mechanisms. As such, they lend themselves to the language ofmathematics. Flip the pages of any academic journal in economicsand you will encounter a nearly endless stream of equations andGreek symbols...

The reason economists use mathematics is typicallymisunderstood. It has little to do with sophistication, complexity, or aclaim to higher truth. Math essentially plays two roles in economics,neither of which is cause for glory: clarity and consistency.

(Rodrik, pgs. 22-23)

D. Rodrik. Economic Rules: The Rights and Wrongs fo the Dismal Science. W.W. Norton, 2015.

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Practicalities: Math

Economic models consist of clearly stated assumptions and behav-ioral mechanisms. As such, they lend themselves to the language ofmathematics. Flip the pages of any academic journal in economicsand you will encounter a nearly endless stream of equations andGreek symbols...The reason economists use mathematics is typicallymisunderstood. It has little to do with sophistication, complexity, or aclaim to higher truth. Math essentially plays two roles in economics,neither of which is cause for glory: clarity and consistency.

(Rodrik, pgs. 22-23)

D. Rodrik. Economic Rules: The Rights and Wrongs fo the Dismal Science. W.W. Norton, 2015.

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What is this course about?

1. Foundational assumptions in Economics and Political Science

“Rational choice” explanations of social phenomena

What does it mean (for an individual/group) to be rational (or reasonable)as opposed to irrational (or unreasonable)?

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(Faulty?) Assumptions

The main cause of the [2008 financial] crisis was the behavior of thebanks—largely the result of misguided incentives unrestrained by goodregulation. ...

There is one other set of accomplices—the economists who provide thearguments that those in the financial markets found so convenient andself-serving. The economists provided models—based on unrealisticassumptions of perfect information, perfect competition, and perfectmarkets—in which regulation was unnecessary. (Stiglitz, 333-4)

J. Stiglitz. The Anatomy of a Murder: Who Killed the American Economy. Critical Review: 21(2-3),pp. 329 - 339 (2009).

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(Useful?) Assumptions

In truth, simple models of the type that economists construct are absolutelyessential to understanding the workings of society. Their simplicity,formalism, and neglect of many facets of the real world are precisely whatmakes them valuable. These are a feature, not a bug. What makes a modeluseful is that it captures an aspect of reality. What makes it indispensable,when used well, is that it captures the most relevant aspect of reality in a givencontext. (Rodrik, 11)

D. Rodrik. Economic Rules: Why Economics Works, When it Fails and How to Tell the Difference.Oxford University Press, 2015.

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What is this course about?

1. Foundational assumptions in Economics and Political Science

2. Rational choice explanations of social phenomena

I What does it mean for an individual/group to choose rationally?

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Simple Choice Model

> >Menu

> >Preference

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Simple Choice Model

> >Choice

> >Preference

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Simple Choice Model

> >Rational Choice?

> >Preference

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Simple Choice Model

> >Rational Choice?

> >Preference

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Simple Choice Model

> >Rational Choice

> >Preference

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Simple Choice Model

> >Irrational Choice

> >Preference

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I Option uncertainty: What type of wine is it? Is the red wine sweet ordry? Is the white wine spoiled? Is the lemonade very sugary? . . .

[Options vs. Prospects]

I Menu uncertainty: Are there other drink choices that are available (e.g.,a beer or a soda)? . . .

I Context: What are we having to eat? What time of day is it? How manydrinks have you had? Are you driving home? . . .

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I Option uncertainty: What type of wine is it? Is the red wine sweet ordry? Is the white wine spoiled? Is the lemonade very sugary? . . .

[Options vs. Prospects]

I Menu uncertainty: Are there other drink choices that are available (e.g.,a beer or a soda)? . . .

I Context: What are we having to eat? What time of day is it? How manydrinks have you had? Are you driving home? . . .

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I Option uncertainty: What type of wine is it? Is the red wine sweet ordry? Is the white wine spoiled? Is the lemonade very sugary? . . .

[Options vs. Prospects]

I Menu uncertainty: Are there other drink choices that are available (e.g.,a beer or a soda)? . . .

I Context: What are we having to eat? What time of day is it? How manydrinks have you had? Are you driving home? . . .

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I Option uncertainty: What type of wine is it? Is the red wine sweet ordry? Is the white wine spoiled? Is the lemonade very sugary? . . .

[Options vs. Prospects]

I Menu uncertainty: Are there other drink choices that are available (e.g.,a beer or a soda)? . . .

I Context: What are we having to eat? What time of day is it? How manydrinks have you had? Are you driving home? . . .

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Decision Problems

Individual decision-making (against nature)

I E.g., Gambling

Individual decision making in interaction

I E.g., Playing chess

Collective decision making

I E.g., Carrying a pianoI E.g., Voting in an election

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Decision Problems

Individual decision-making (against nature)I E.g., Gambling

Individual decision making in interaction

I E.g., Playing chess

Collective decision making

I E.g., Carrying a pianoI E.g., Voting in an election

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Decision Problems

Individual decision-making (against nature)I E.g., Gambling

Individual decision making in interactionI E.g., Playing chess

Collective decision making

I E.g., Carrying a pianoI E.g., Voting in an election

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Decision Problems

Individual decision-making (against nature)I E.g., Gambling

Individual decision making in interactionI E.g., Playing chess

Collective decision makingI E.g., Carrying a piano

I E.g., Voting in an election

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Decision Problems

Individual decision-making (against nature)I E.g., Gambling

Individual decision making in interactionI E.g., Playing chess

Collective decision makingI E.g., Carrying a pianoI E.g., Voting in an election

whv{ axed

GAMINGTFI E VOTE

Ele ct io ns&Vi"at 1&i'e Cax:

Aren' t FairDo About I t )

WILLIAM POUNDBestsel l ing author of Fortun

STONEe's Formula14 / 22

Individual decision-making (against nature)

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Decision Problems (under certainty)A decision problem is defined by two sets:

1. A set of acts2. A set of outcomes

In a decision under certainty, each act is associated with a single outcome:

Acts Outcomesa o1

b o2

c o3

......

I A decision maker chooses an act.I The decision maker knows which

outcomes is associated with which act.I Two different acts may be associated

with the same outcome.

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Decision Problems (under certainty)A decision problem is defined by two sets:

1. A set of acts2. A set of outcomes

In a decision under certainty, each act is associated with a single outcome:

Acts Outcomesa o1

b o2

c o3

......

I A decision maker chooses an act.I The decision maker knows which

outcomes is associated with which act.I Two different acts may be associated

with the same outcome.

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Preferences

Preferring or choosing x is different that “liking” x or “having a taste for x”:one can prefer x to y but dislike both options

Preferences are always understood as comparative: “preference” is more like“bigger” than “big”

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Concepts of preference

1. Enjoyment comparison: I prefer red wine to white wine means that I enjoyred wine more than white wine.

2. Comparative evaluation: I prefer candidate A over candidate B means “Ijudge A to be superior to B”. This can be partial (ranking with respect tosome criterion) or total (with respect to every relevant consideration).

3. Favoring: Affirmative action calls for racial/gender preferences in hiring.

4. Choice ranking: In a restaurant, when asked “do you prefer red wine orwhite wine”, the waiter wants to know which option I choose.

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Concepts of preference

1. Enjoyment comparison: I prefer red wine to white wine means that I enjoyred wine more than white wine.

2. Comparative evaluation: I prefer candidate A over candidate B means “Ijudge A to be superior to B”. This can be partial (ranking with respect tosome criterion) or total (with respect to every relevant consideration).

3. Favoring: Affirmative action calls for racial/gender preferences in hiring.

4. Choice ranking: In a restaurant, when asked “do you prefer red wine orwhite wine”, the waiter wants to know which option I choose.

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Concepts of preference

1. Enjoyment comparison: I prefer red wine to white wine means that I enjoyred wine more than white wine.

2. Comparative evaluation: I prefer candidate A over candidate B means “Ijudge A to be superior to B”. This can be partial (ranking with respect tosome criterion) or total (with respect to every relevant consideration).

3. Favoring: Affirmative action calls for racial/gender preferences in hiring.

4. Choice ranking: In a restaurant, when asked “do you prefer red wine orwhite wine”, the waiter wants to know which option I choose.

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Concepts of preference

1. Enjoyment comparison: I prefer red wine to white wine means that I enjoyred wine more than white wine.

2. Comparative evaluation: I prefer candidate A over candidate B means “Ijudge A to be superior to B”. This can be partial (ranking with respect tosome criterion) or total (with respect to every relevant consideration).

3. Favoring: Affirmative action calls for racial/gender preferences in hiring.

4. Choice ranking: In a restaurant, when asked “do you prefer red wine orwhite wine”, the waiter wants to know which option I choose.

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Concepts of preference

1. Enjoyment comparison: I prefer red wine to white wine means that I enjoyred wine more than white wine.

2. Comparative evaluation: I prefer candidate A over candidate B means “Ijudge A to be superior to B”. This can be partial (ranking with respect tosome criterion) or total (with respect to every relevant consideration).

3. Favoring: Affirmative action calls for racial/gender preferences in hiring.

4. Choice ranking: In a restaurant, when asked “do you prefer red wine orwhite wine”, the waiter wants to know which option I choose.

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Mathematically describing preferences

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Mathematical background: Relations

Suppose that X is a set. A relation on X is a set of ordered pairs from X:R ⊆ X × X.

E.g., X = {a, b, c, d}, R = {(a, a), (b, a), (c, d), (a, c), (d, d)}

a b

c d

a R ab R ac R da R cd R d

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Representing Preferences

Let X be a set of outcomes. A decision maker’s preference over X isrepresented by relations on X.

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Strict Preference, I

A decision maker’s strict preference over a set X is represented as a relationP ⊆ X × X.

The underlying idea is that if P represents the decision maker’s strictpreference and xPy (i.e., the decision maker strictly prefers x to y), then thedecision maker would pay some non-zero amount money to trade y for x.

P is asymmetric: for all x, y ∈ X, if xPy, then it is not the case that yPx (writtennot-yPx).

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Strict Preference, I

A decision maker’s strict preference over a set X is represented as a relationP ⊆ X × X.

The underlying idea is that if P represents the decision maker’s strictpreference and xPy (i.e., the decision maker strictly prefers x to y), then thedecision maker would pay some non-zero amount money to trade y for x.

P is asymmetric: for all x, y ∈ X, if xPy, then it is not the case that yPx (writtennot-yPx).

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Strict Preference, I

A decision maker’s strict preference over a set X is represented as a relationP ⊆ X × X.

The underlying idea is that if P represents the decision maker’s strictpreference and xPy (i.e., the decision maker strictly prefers x to y), then thedecision maker would pay some non-zero amount money to trade y for x.

P is asymmetric: for all x, y ∈ X, if xPy, then it is not the case that yPx (writtennot-yPx).

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