Table of Contents

1. Introduction to Behavioral Economics
   • 1.1 Definition and Scope
   • 1.2 Key Principles
   • 1.3 Relationship to Traditional Economics
2. Experimental Economics Methodology
   • 2.1 Types of Data
   • 2.2 Internal and External Validity
   • 2.3 Goals of Economic Experiments
   • 2.4 Experimental Design Principles
   • 2.5 Preparing and Running Experiments
   • 2.6 Analysis and Writing Up
3. Individual Decision Making Under Risk
4. Decision Making Under Ambiguity
5. Dynamic Decision Making
6. Game Theory Fundamentals
7. Advanced Game Theory
8. Prospect Theory
9. Sample Questions and Solutions

1. Introduction to Behavioral Economics

1.1 Definition and Scope

Behavioral Economics is a field that uses variants of traditional economic assumptions, often with psychological motivation, to explain and predict economic behavior, and to provide policy prescriptions. It is not a rejection of traditional economics, but rather a series of amendments that enhance our understanding of how people actually behave in economic contexts.

The field emerged from observations that people’s actual behavior often deviates systematically from the predictions of standard economic models. These deviations are not random but follow predictable patterns that can be studied and incorporated into economic theory.

1.2 Key Principles

The core principles of Behavioral Economics include:

1. Bounded Rationality: People try to choose the best feasible option but sometimes make mistakes due to cognitive limitations.
2. Reference Dependence: People care about how circumstances compare to reference points rather than absolute levels.
3. Self-Control Problems: Individuals often struggle with self-control, leading to time-inconsistent preferences.
4. Social Preferences: Although people mostly care about their own material payoffs, they also care about the actions, intentions, and payoffs of others, even people outside their family.
5. Market Limitations: Sometimes market exchange makes psychological factors cease to matter, but many psychological factors matter even in markets.
6. Choice Architecture: Limiting people’s choices could partially protect them from behavioral biases.

1.3 Relationship to Traditional Economics

Traditional economics assumes perfect rationality, where individuals maximize utility given their constraints. The most developed model of individual rational choice identifies the individual with a set of objectives and treats an action as rational because it will satisfy the objective, given the constraints. This is called instrumental rationality.

Behavioral economics modifies these assumptions:

• Instead of perfect information processing, it recognizes cognitive limitations
• Instead of consistent preferences, it allows for context-dependent choices
• Instead of pure self-interest, it incorporates social preferences
• Instead of perfect self-control, it acknowledges present bias and other time-inconsistencies

2. Experimental Economics Methodology

2.1 Types of Data

Experimental economics relies on different types of data, each with its own advantages and limitations:

Experimental Data

Data deliberately created for scientific purposes under controlled conditions. This includes:

1. Laboratory Data (LD): Gathered in an artificial environment designed for scientific purposes
   • Advantages: High level of control, ability to isolate specific variables
   • Disadvantages: Artificial setting may not reflect real-world behavior
2. Field Data (FD): Gathered in a naturally occurring environment
   • Advantages: More realistic context, external validity
   • Disadvantages: Less control over confounding variables

Happenstance Data

Also called observational data, these are a by-product of ongoing uncontrolled processes.

• Advantages: Real-world relevance, large datasets
• Disadvantages: Difficulty establishing causation, many confounding variables

2.2 Internal and External Validity

Internal Validity: Do the data permit correct causal inferences? This is a matter of proper experimental control. Key considerations include:

• Random assignment to treatments
• Control of confounding variables
• Proper measurement of outcomes
• Avoiding experimenter bias

External Validity: Can results be generalized from laboratory to non-laboratory conditions? This involves:

• Representative subject pools
• Realistic tasks and incentives
• Appropriate context and framing

The trade-off between internal and external validity is a fundamental challenge in experimental economics.

2.3 Goals of Economic Experiments

According to Roth (1986, 1988, 1995), economic experiments serve three primary purposes:

1. Speaking to Theorists: Testing theoretical predictions and assumptions
   • Examining whether behavior conforms to theoretical models
   • Identifying systematic deviations from theory
   • Refining theoretical assumptions
2. Searching for Facts: Discovering empirical regularities
   • Exploring areas where theory provides little guidance
   • Identifying behavioral patterns in complex environments
   • Documenting stylized facts for theory development
3. Whispering in the Ears of Princes: Informing policy decisions
   • Testing proposed market designs
   • Evaluating policy interventions
   • Providing evidence for regulatory decisions

2.4 Experimental Design Principles

Controlled Environment

The economic environment consists of:

• Individual economic agents
• Institutions through which agents interact
• Rules governing interactions

Direct Experimental Control

Key principles include:

• Constants: Variables held fixed across all treatments
• Treatments: Systematic variations in parameters of interest
• Randomization: Random assignment to eliminate selection bias

Within vs. Between-Subject Designs

• Within-subject design

  A single subject is observed in different treatments

  • Advantages: Controls for individual differences, requires fewer subjects
  • Disadvantages: Order effects, learning effects

• Between-subject design

  Different subjects are tested in different treatments

  • Advantages: No order effects, cleaner comparisons
  • Disadvantages: Requires more subjects, individual differences may confound results

Anonymity

When subjects interact, anonymity is necessary to:

• Prevent reputation effects
• Eliminate social pressures
• Ensure decisions reflect true preferences

2.5 Preparing and Running Experiments

Step-by-Step Process

1. Define Research Question
   • What specific hypothesis are you testing?
   • What is the theoretical framework?
   • What are the key variables of interest?
2. Simplify the Design
   • Reduce to the simplest possible version
   • Focus on essential elements
   • Eliminate unnecessary complexity
3. Choose Test Type
   • Absolute prediction testing
   • Comparative static analysis
   • Treatment effects
4. Implementation Decisions
   • Software choice (Z-tree, oTree, Visual Studio, Python)
   • Manual vs. computerized
   • Data collection methods
5. Parameter Selection
   • Pre-experimental simulations
   • Power calculations
   • Payoff calibration
6. Instructions and Materials
   • Clear, precise instructions
   • Comprehension questionnaires
   • Practice rounds
   • Demographic surveys
7. Pilot Testing
   • Small-scale test runs
   • Identify potential problems
   • Refine procedures
8. Main Experiment
   • Consistent protocols
   • Proper randomization
   • Data quality checks

Instructions

Instructions are crucial because:

• They define the experimental environment
• Unclear instructions invalidate results
• They must be replicable
• They should avoid loaded language

Best practices for instructions:

• Use neutral language
• Provide examples
• Test comprehension
• Keep consistent across treatments (except for treatment variables)

Incentives

Payment principles in experimental economics:

• Participants must be paid
• Payment must be linked to decisions (incentive compatibility)
• Avoid deception
• Use real monetary incentives

Payment considerations:

• Sufficient stakes to motivate careful decisions
• Avoid wealth effects that might distort behavior
• Consider show-up fees vs. performance pay
• Account for risk attitudes in payment schemes

2.6 Analysis and Writing Up

Data Analysis Approaches

1. Descriptive Statistics
   • Summary measures
   • Distribution analysis
   • Treatment comparisons
2. Econometric Analysis
   • Regression analysis
   • Panel data methods
   • Structural estimation
3. Non-parametric Tests
   • Mann-Whitney tests
   • Wilcoxon tests
   • Kolmogorov-Smirnov tests

Explanation vs. Prediction

The analysis approach depends on research goals:

• Explanation: Understanding mechanisms and causation
• Prediction: Forecasting behavior in new situations

Writing Up Results

Standard structure for experimental papers:

1. Introduction: Motivation and research question
2. Literature Review: Context and contribution
3. Experimental Design: Detailed procedures
4. Results: Analysis and findings
5. Discussion: Interpretation and implications
6. Conclusion: Summary and future directions

Key considerations:

• Transparency in reporting
• Complete procedural details
• Robustness checks
• Limitations acknowledgment

Research Areas for Experiments

The methodology applies across various domains:

1. Market Experiments
   • New market types (forward, spot, financial)
   • Trading mechanisms and market design
   • Regulatory effects on markets
   • Market interactions (simultaneous and sequential)
   • Macroeconomic experiments
2. Game Theory Applications
   • Two-person and multi-person games
   • Simultaneous and sequential interactions
   • Learning and adaptation
   • Coordination and communication
   • Social preferences and psychological factors
3. Individual Choice Studies
   • Risk and ambiguity preferences
   • Dynamic decision making
   • Emotional and demographic effects
   • Cognitive biases and heuristics

3. Individual Decision Making Under Risk

3.1 Economic Rationality and Expected Utility Theory

Economic Rationality

Economic rationality is based on instrumental rationality, where an action is considered rational if it satisfies the agent’s objectives given constraints. The most developed model assumes:

• Individuals have well-defined preferences
• They maximize utility subject to constraints
• They process information perfectly

Utility Theory

Utility represents the satisfaction derived from outcomes. Key concepts:

• Individuals can compare different outcomes
• Preferences can be ordered on a single scale
• Rational individuals maximize their utility
• Under risk, individuals maximize expected utility

Expected Value vs. Expected Utility

Expected Value: For a lottery \((x_1, ..., x_N; p_1, ..., p_N)\): \(EV = x_1p_1 + x_2p_2 + ... + x_Np_N = \sum_{i=1}^{N} x_i p_i\)

Expected Utility: For the same lottery: \(EU = u(x_1)p_1 + u(x_2)p_2 + ... + u(x_N)p_N = \sum_{i=1}^{N} u(x_i)p_i\)

Where \(u(x)\) is the utility function mapping monetary outcomes to utility values.

3.2 EU Axioms

Expected Utility Theory is built on four fundamental axioms:

1. Completeness (Comparability)

For any two lotteries \(A\) and \(B\), the decision maker can always say:

• \(A \succ B\) (A is preferred to B), or
• \(B \succ A\) (B is preferred to A), or
• \(A \sim B\) (indifference between A and B)

2. Transitivity

If \(A \succ B\) and \(B \succ C\), then \(A \succ C\)

3. Independence

If \(A \succ B\), then for any lottery \(C\) and probability \(p \in (0,1)\): \((A,C; p, 1-p) \succ (B,C; p, 1-p)\)

This is the crucial axiom that leads to the linearity of EU in probabilities.

4. Continuity

If \(A \succ B \succ C\), then there exists a probability \(p\) such that: \(B \sim (A,C; p, 1-p)\)

This ensures the existence of a utility function.

3.3 The Independence Axiom

The Independence Axiom deserves special attention as it’s the most controversial and frequently violated axiom.

Strong Independence

The axiom states that preferences between two lotteries should not be affected by mixing each with a common third lottery.

Betweenness Property

A weaker form states: If \(A \sim B\), then: \(A \sim (A,B; p, 1-p) \sim B\)

This is implied by independence but is less restrictive.

3.4 Utility Functions and Risk Attitudes

Risk Attitudes

Risk attitudes are characterized by the shape of the utility function:

1. Risk Averse: \(u''(w) < 0\) (concave utility function)
2. Risk Neutral: \(u''(w) = 0\) (linear utility function)
3. Risk Loving: \(u''(w) > 0\) (convex utility function)

Risk Aversion Measures

Arrow-Pratt Absolute Risk Aversion: \(R_A(w) = -\frac{u''(w)}{u'(w)}\)

Arrow-Pratt Relative Risk Aversion: \(R_R(w) = R_A(w) \cdot w = -\frac{wu''(w)}{u'(w)}\)

These measures indicate:

• Decreasing absolute risk aversion: Wealthier individuals take more risks in absolute terms
• Constant relative risk aversion: Risk-taking scales proportionally with wealth

3.5 Violations of Expected Utility Theory

The Allais Paradox

The Allais Paradox demonstrates a systematic violation of the independence axiom:

Problem 1:

• A: \(2,500 with probability 0.33;\)2,400 with probability 0.66; $$0 with probability 0.01
• B: $$2,400 with certainty

Problem 2:

• C: \(2,500 with probability 0.33;\)0 with probability 0.67
• D: \(2,400 with probability 0.34;\)0 with probability 0.66

According to EU theory, if someone prefers B to A, they should prefer D to C (by the independence axiom). However, most people choose B in Problem 1 and C in Problem 2, violating EU theory.

Common Consequence Effect

The Allais Paradox illustrates the common consequence effect: preferences are affected by common outcomes that should be irrelevant according to the independence axiom.

Explanations of the Allais Paradox

1. Kahneman and Tversky’s Explanation: People distort probabilities, overweighting small probabilities and underweighting large ones. The difference between 0.99 and 1.00 feels larger than between 0.33 and 0.34.
2. Allais’s Argument: Gambles must be appraised as wholes. The utility of one component cannot be defined independently of other components. The possibility of getting $$0 generates different levels of disappointment in different contexts.

Preference Reversal

Another violation discovered by Grether and Plott (1979):

Consider two gambles:

• Q: 7/36 chance of winning \(9; 29/36 chance of losing\)0.50
• R: 29/36 chance of winning \(2; 7/36 chance of losing\)1

Experimental findings:

• Many subjects choose R over Q in direct choice
• But assign a higher monetary value to Q when asked to price each gamble
• This creates a preference reversal: R preferred in choice, Q valued higher

Framing Effects

How choices are presented affects decisions, violating the assumption of description invariance:

Medical Treatment Example:

• Frame A: “90 out of 100 survive surgery”
• Frame B: “10 out of 100 die during surgery”

Same information, but Frame B leads to more choosing alternative treatments.

3.6 Alternative Theories to EU

Due to systematic violations, several alternative theories emerged:

1. Prospect Theory (Kahneman & Tversky, 1979)

• Replaces probabilities with decision weights
• Uses value function defined on gains/losses
• Incorporates reference dependence

2. Rank-Dependent Expected Utility (Quiggin, 1982)

• Transforms cumulative probabilities
• Maintains transitivity
• Allows for probability distortion

3. Disappointment Aversion Theory

• Incorporates anticipated disappointment
• Modifies utility based on expectations

4. Regret Theory (Loomes & Sugden)

• Incorporates anticipated regret
• Compares outcomes across choices
• Can explain preference reversals without violating transitivity

3.7 The Marschak-Machina Triangle

The Marschak-Machina triangle is a powerful tool for visualizing preferences over three-outcome lotteries.

Construction

For three outcomes \(x_1 < x_2 < x_3\) with probabilities \(p_1, p_2, p_3\):

• Horizontal axis: \(p_1\) (probability of worst outcome)
• Vertical axis: \(p_3\) (probability of best outcome)
• \(p_2 = 1 - p_1 - p_3\) (implied probability of middle outcome)

Indifference Curves in EU Theory

Under EU theory, indifference curves are straight lines with slope: \(\text{slope} = -\frac{u(x_2) - u(x_1)}{u(x_3) - u(x_2)}\)

Properties:

• Parallel straight lines
• Slope indicates risk attitude
• Steeper slopes indicate greater risk aversion

Indifference Curves in Non-EU Theories

Alternative theories produce different patterns:

• Curved indifference lines (Prospect Theory)
• Fan-shaped patterns (Rank-Dependent theories)
• Kinked curves (Disappointment Aversion)

3.8 Experimental Methods in Risk Research

Testing Axioms vs. Fitting Preference Functionals

Two main approaches:

1. Axiom Testing: Directly test whether behavior satisfies theoretical axioms
2. Functional Fitting: Estimate parameters of preference functionals

Experimental Design Considerations

1. Problem Selection:
   • Cover relevant parts of probability triangle
   • Include tests of key predictions
   • Balance between simple and complex choices
2. Presentation Format:
   • Verbal descriptions
   • Graphical representations
   • Probability wheels
   • Experience-based learning
3. Incentive Mechanisms:
   • Random lottery payoff mechanism
   • Pay all choices
   • Pay one randomly selected choice

Hey and Orme (1994) Study

Landmark study in preference estimation:

• 80 subjects, 100 pairwise choices
• All lotteries with outcomes \(0,\)10, \(20,\)30
• Tested multiple preference functionals
• Used stochastic choice models

Key findings:

• EU plus noise performs reasonably well
• More general theories fit better but have more parameters
• Individual heterogeneity is important

Stochastic Choice Models

Incorporating randomness in choice: \(P(L \succ R) = \Phi[\frac{V(L) - V(R)}{\sigma}]\)

Where:

• \(\Phi\) is cumulative normal distribution
• \(\sigma\) is noise parameter
• \(V(L), V(R)\) are valuations of lotteries

Statistical Considerations

Model comparison criteria:

• Akaike Information Criterion (AIC)
• Bayesian Information Criterion (BIC)
• Log-likelihood ratio tests

Key issues:

• Degrees of freedom
• Out-of-sample prediction
• Individual vs. aggregate analysis

3.9 Recent Developments

Heteroscedastic Error Specifications

Research shows noise is not constant:

• Varies with complexity of choice
• Depends on stake size
• Individual-specific patterns

Beyond Expected Utility

Modern approaches:

• Reference-dependent preferences
• Multiple reference points
• Context-dependent risk attitudes
• Cognitive limitations in probability processing

Implications for Economic Theory

The violations of EU theory have important implications:

• Market behavior may deviate from standard predictions
• Policy interventions need to account for behavioral biases
• Financial decisions exhibit systematic irrationalities
• Insurance and gambling behaviors need richer models

4. Decision Making Under Ambiguity

4.1 Risk vs. Ambiguity

Defining Ambiguity

While risk involves known probabilities, ambiguity refers to situations where:

• Probabilities do not exist
• Probabilities are unknown
• Decision makers feel unable to attach subjective probabilities

The Ellsberg Paradox

Classic demonstration of ambiguity aversion:

• Urn with 30 red balls and 60 black/yellow balls (unknown proportion)
• Choice 1: Bet on red vs. bet on black
• Choice 2: Bet on red/yellow vs. bet on black/yellow

Most people prefer red in Choice 1 and black/yellow in Choice 2, violating probability theory.

4.2 Theories of Behavior Under Ambiguity

1. Subjective Expected Utility (SEU) Theory

• Same as EU theory but with subjective probabilities
• Requires probabilities to be additive: \(p_1 + p_2 = 1\)
• Certainty equivalent: \(U(CE) = p_1u(x_1) + p_2u(x_2)\)

2. Choquet Expected Utility

Replaces probabilities with capacities (non-additive):

• Two events with capacities \(c_1\) and \(c_2\)
• \(c_1 + c_2 \neq 1\) (non-additivity)
• Evaluation: \(V = c_1u(x) + (1-c_1)u(y)\) if \(x > y\)

3. Maxmin Expected Utility

Decision maker specifies range of possible probabilities:

• Example: “Probability of heads is between 0.4 and 0.6”
• Choose action maximizing minimum expected utility
• Conservative approach to ambiguity

4. α-Maxmin Expected Utility

Weighted average of best and worst cases: \(V = \alpha \min_p EU(p) + (1-\alpha)\max_p EU(p)\)

Where \(\alpha\) represents pessimism level.

5. The Smooth Model

Allows for more flexible responses to ambiguity with multiple parameters.

4.3 Experimental Implementation of Ambiguity

Creating Ambiguity in the Lab

1. Ambiguous Urns: Unknown composition created by “disinterested person”

2. Bingo Blower Method

   :

   • Balls of different colors in motion
   • Varying number of balls changes ambiguity level
   • Natural uncertainty about exact proportions

Measuring Ambiguity Attitudes

• Compare choices under risk vs. ambiguity
• Elicit matching probabilities
• Use multiple ambiguity levels

Key Experimental Findings

• Widespread ambiguity aversion
• Ambiguity attitudes differ from risk attitudes
• Context matters for ambiguity perception

5. Dynamic Decision Making

5.1 Two Aspects of Dynamic Decisions

1. Passage of Real Time (Intertemporal Choice)

Decisions involving trade-offs across time periods

2. Sequentiality in Decision Making

Pattern of: Decision → Nature → Decision → Nature…

5.2 Dynamic Consistency

With Expected Utility Theory

EU theory implies coherent, intertemporally consistent decision makers:

• Plans made are executed
• Independence axiom ensures context irrelevance
• Backward induction yields unique solution

With Non-EU Theories

Alternative theories may lead to dynamic inconsistency:

• Plans may change over time
• Context matters for preferences
• Multiple solution methods may disagree

5.3 Methods for Solving Dynamic Problems

1. Strategy-Based Method

• Transform dynamic problem into choice of complete strategy
• Strategy specifies action at every decision node
• Compare expected utilities of all possible strategies

Example notation: \((X; YZ)\)

• Letter before semicolon: choice at first node
• Letters after: choices at second nodes (after nature moves up/down)

2. Backward Induction with Reduction

Process:

1. Work backward through decision tree
2. Eliminate suboptimal branches at each node
3. Apply reduction of compound lotteries
4. Transform to static problem

Steps:

• Identify optimal choice at final nodes
• Replace decision nodes with chosen outcomes
• Continue backward to initial decision

3. Backward Induction with Certainty Equivalents

• Replace decision branches with certainty equivalents
• Certainty equivalent: \(CE(P) = u^{-1}[EU(P)]\)
• Work backward substituting CE values

For lottery \((x,y;p,q)\): \(CE = u^{-1}[pu(x) + qu(y)]\)

5.4 Decision Trees

Notation and Structure

• Decision Nodes: Square nodes where player chooses
• Chance Nodes: Circle nodes where nature moves randomly
• Payoff Nodes: Terminal nodes with outcomes
• Information Sets: Show what player knows when deciding

Example: T(2,2) Tree

Simple tree with 2 decision nodes and 2 random nodes:

• First decision: Up or Down
• Nature moves: Random outcome
• Second decision: Based on nature’s move
• Final payoffs determined

5.5 Dynamic Inconsistency Example

Consider utilities:

With EU theory (all probabilities = 0.5):

• Strategy method → Optimal: (D;UU)
• Backward induction with reduction → Same result
• Backward induction with CE → Same result

With Rank-Dependent EU (γ = 0.9):

• Strategy method → Optimal: (U;UU)
• Backward induction with reduction → Optimal: (U;UU)
• Backward induction with CE → Optimal: (D;UU)

Dynamic inconsistency: Different methods yield different solutions!

5.6 Implications of Dynamic Inconsistency

Theoretical Issues

• Which method correctly represents behavior?
• How do people actually solve dynamic problems?
• Role of commitment and self-awareness

Practical Implications

• Time-inconsistent preferences
• Need for commitment devices
• Policy design considerations

6. Intertemporal Choice

6.1 The Basic Problem

Elements

• Stochastic income flow: \(y_t\)
• Savings at interest rate \(r\)
• Consumption: \(c_t = y_t - \text{savings}\)
• Lifetime utility maximization

Standard Discounted Utility Model

Maximize: \(U = u(c_t) + \rho u(c_{t+1}) + \rho^2 u(c_{t+2}) + ...\)

Where:

• \(u(·)\) is instantaneous utility function
• \(\rho\) is discount factor
• \(c_t\) is consumption at time \(t\)

Budget Constraint

Where \(W_t\) is wealth at time \(t\).

6.2 Present-Bias Discounting

Quasi-Hyperbolic Discounting

Where:

• \(\beta < 1\) represents present bias
• \(\rho = \frac{1}{1+\delta}\) is standard discount factor

Implications

• Time inconsistency in preferences
• Preference for immediate gratification
• Procrastination and self-control problems

6.3 Experimental Approaches

Measuring Time Preferences

1. Choice Tasks: Binary choices between sooner-smaller and later-larger rewards
2. Matching Tasks: Find amount making subject indifferent
3. Multiple Price Lists: Series of choices with varying delays/amounts

Key Findings

• Widespread present bias
• Hyperbolic discounting patterns
• Domain-specific discount rates
• Magnitude effects in discounting

6.4 Applications

Personal Finance

• Undersaving for retirement
• Credit card debt
• Investment decisions

Health Behaviors

• Exercise and diet choices
• Addiction
• Preventive healthcare

Policy Implications

• Default options in retirement plans
• Commitment savings products
• “Nudge” interventions

6.5 Advanced Topics in Dynamic Choice

Learning in Dynamic Settings

• Beliefs updating
• Experimentation vs. exploitation
• Adaptive decision making

Strategic Dynamic Interactions

• Repeated games
• Reputation effects
• Dynamic contracts

Computational Considerations

• Cognitive costs of planning
• Heuristics in dynamic decisions
• Bounded rationality in sequential choice

6. Game Theory Fundamentals

6.1 Strategic Interactions and Game Theory

Definition

Game theory analyzes individual decisions in situations of conflict or strategic interaction with other agents, aimed at maximizing each subject’s gain. In these situations, one player’s decisions affect the outcomes achievable by others and vice versa.

Key Characteristics

• Strategic Behavior: A decision maker behaves strategically when considering what they believe other agents will do
• Non-cooperative Games: Each party participates solely for their own benefit (though cooperation can emerge as a strategic outcome)
• Interactive Decision Making: Outcomes depend on all players’ choices

6.2 Elements of a Game

Every game consists of four fundamental elements:

1. Players

The decision makers involved in the game (individuals, firms, countries)

2. Actions

The possible moves each player can choose at any decision point

3. Strategies

Complete contingent plans specifying actions in all possible circumstances:

• A strategy must specify what to do in every possible situation
• Even in situations that may not occur during actual play
• Like an instruction manual for a representative to follow

4. Payoffs

The outcomes or utilities players receive from each combination of strategies

6.3 Types of Games

By Timing

1. Simultaneous Games: All players move at the same time
2. Sequential (Dynamic) Games: Players move one after another

By Information

1. Complete Information: Each player’s payoff function is common knowledge
2. Incomplete Information: Payoff functions are not common knowledge
3. Perfect Information: Players know entire history at each decision point
4. Imperfect Information: Players don’t know complete history when deciding

6.4 Game Representations

1. Normal Form (Strategic Form)

Represents games as payoff matrices:

• Rows: Player 1’s strategies
• Columns: Player 2’s strategies
• Cells: Payoffs (Player 1, Player 2)

Example - Prisoner’s Dilemma:

2. Extensive Form

Represents games as decision trees:

• Decision nodes (squares)
• Chance nodes (circles)
• Information sets (ovals)
• Branches showing possible moves
• Terminal nodes with payoffs

6.5 Solution Concepts

Dominant Strategies

A strategy that yields the best payoff regardless of opponents’ choices:

• Strictly Dominant: Always better than any other strategy
• Weakly Dominant: Never worse, sometimes better

Dominated Strategies

A strategy that is always worse than another available strategy:

• Can be eliminated from consideration
• Simplifies game analysis

Nash Equilibrium

A set of strategies where no player can benefit by unilaterally changing their strategy:

• Each player’s strategy is optimal given others’ strategies
• Self-enforcing: no incentive to deviate
• May not be unique or efficient

Finding Nash Equilibrium:

1. Identify best responses for each player
2. Find strategy profiles where all play best responses
3. These mutual best responses are Nash equilibria

Best Response Function (BRF)

Maps opponents’ strategies to a player’s optimal responses:

• Used when no dominant strategies exist
• Intersection of BRFs gives Nash equilibrium

6.6 Classic Games

1. Prisoner’s Dilemma

Setup: Two criminals can confess or stay silent

• Both silent: 1 year each (cooperate)
• Both confess: 6 years each (defect)
• One confesses: 0 years (7 for other)

Key Features:

• Dominant strategy: Confess
• Nash equilibrium: Both confess
• Pareto inefficient outcome
• Individual rationality leads to collective irrationality

2. Entry Game (Sequential)

Setup: Firm X decides whether to enter market with incumbent Y

• X choices: Enter or Not Enter
• Y choices (if X enters): Accommodate or Fight

Payoff Structure:

• X stays out: (0, 3) if Y produces much, (0, 2) if little
• X enters, Y accommodates: (1, 1)
• X enters, Y fights: (-1, -1)

Analysis:

• Backward induction solution
• Credibility of threats matters
• Subgame perfect equilibrium

3. Battle of the Sexes

Coordination game with conflicting preferences:

Features:

• Two pure strategy Nash equilibria
• Coordination problem
• Mixed strategy equilibrium exists

4. Matching Pennies

Zero-sum game with no pure strategy equilibrium:

6.7 Solution Methods

1. Dominance

• Eliminate strictly dominated strategies
• Iterated elimination of dominated strategies (IEDS)
• Requires belief in others’ rationality

2. Best Response Analysis

For each player and opponent strategy:

1. Calculate payoffs for all own strategies
2. Identify strategy yielding highest payoff
3. Mark best responses
4. Nash equilibria where all players use best responses

3. Backward Induction (Sequential Games)

1. Start at final decision nodes
2. Determine optimal choices
3. Work backward to initial decision
4. Results in subgame perfect equilibrium

6.8 Advanced Concepts

Information Sets

Group decision nodes where player cannot distinguish between them:

• Represents imperfect information
• Shown as ovals in extensive form
• Strategy must specify same action for all nodes in set

Subgame Perfect Equilibrium

Refinement of Nash equilibrium for sequential games:

• Strategy profile must be Nash equilibrium in every subgame
• Eliminates non-credible threats
• Found using backward induction

Common Knowledge

Information that:

1. All players know
2. All players know that all players know
3. And so on, infinitely

Required for:

• Nash equilibrium analysis
• Iterated elimination of dominated strategies

6.9 Applications

Economic Applications

1. Oligopoly Theory
   • Cournot competition (quantity)
   • Bertrand competition (price)
   • Stackelberg leadership
2. Auction Theory
   • First-price sealed bid
   • Second-price (Vickrey) auction
   • Common vs. private values
3. Bargaining Theory
   • Ultimatum game
   • Nash bargaining solution
   • Rubinstein bargaining

Social Applications

1. Public Goods Provision
   • Free-rider problem
   • Voluntary contribution mechanisms
2. Environmental Policy
   • Tragedy of commons
   • International agreements
   • Emission trading
3. Voting Theory
   • Strategic voting
   • Median voter theorem
   • Coalition formation

6.10 Behavioral Game Theory

Departures from Standard Theory

1. Social Preferences
   • Fairness concerns
   • Reciprocity
   • Inequality aversion
2. Bounded Rationality
   • Limited backward induction
   • Level-k thinking
   • Cognitive hierarchies
3. Learning and Adaptation
   • Reinforcement learning
   • Belief learning
   • Evolutionary game theory

Psychological Game Theory

Incorporates beliefs and emotions:

• Utility depends on beliefs about others
• Guilt, anger, reciprocity matter
• Psychological payoffs affect behavior

7. Advanced Game Theory

7.1 Mixed Strategies

Definition

A mixed strategy is a probability distribution over pure strategies. Instead of choosing a single action with certainty, players randomize their choices.

Notation

• Player A mixes between Up and Down with probabilities \((p_U, 1-p_U)\)
• Player B mixes between Left and Right with probabilities \((p_L, 1-p_L)\)

Finding Mixed Strategy Equilibria

Consider a game with no pure strategy Nash equilibrium:

Step 1: Make opponent indifferent

For Player B to mix, expected payoffs must be equal:

• If A plays Up with probability \(p_U\):
• B’s payoff from Left: \(2p_U + 5(1-p_U)\)
• B’s payoff from Right: \(4p_U + 2(1-p_U)\)

Setting equal: \(2p_U + 5(1-p_U) = 4p_U + 2(1-p_U)\)

Solving: \(p_U = 3/5\), so \((1-p_U) = 2/5\)

Step 2: Find other player’s mixing probabilities

For Player A to mix:

• A’s payoff from Up: \(1 \cdot p_L + 0 \cdot (1-p_L) = p_L\)
• A’s payoff from Down: \(0 \cdot p_L + 3 \cdot (1-p_L) = 3(1-p_L)\)

Setting equal: \(p_L = 3(1-p_L)\)

Solving: \(p_L = 3/4\), so \((1-p_L) = 1/4\)

Mixed Strategy Equilibrium: A plays (Up: 3/5, Down: 2/5), B plays (Left: 3/4, Right: 1/4)

Expected Payoffs in Mixed Equilibrium

Probability of each outcome:

• (1,2): \((3/5)(3/4) = 9/20\)
• (0,4): \((3/5)(1/4) = 3/20\)
• (0,5): \((2/5)(3/4) = 6/20\)
• (3,2): \((2/5)(1/4) = 2/20\)

Expected payoffs:

• Player A: \(1(9/20) + 0(3/20) + 0(6/20) + 3(2/20) = 15/20 = 3/4\)
• Player B: \(2(9/20) + 4(3/20) + 5(6/20) + 2(2/20) = 64/20 = 16/5\)

Nash’s Existence Theorem

Every finite game (finite players, finite strategies) has at least one Nash equilibrium (pure or mixed).

7.2 The Matching Pennies Game

Classic example requiring mixed strategies:

Analysis:

• No pure strategy equilibrium exists
• Mixed equilibrium: Both play 50-50
• Expected payoff: 0 for both players
• Zero-sum nature requires randomization

7.3 Information Sets and Signaling

Information Sets

Groups of decision nodes indistinguishable to the acting player:

• Represent imperfect information
• Player must choose same action at all nodes in set
• Shown as ovals in extensive form

Example: Simultaneous Moves in Extensive Form

Prisoner’s Dilemma can be represented with information sets showing simultaneity:

• Player 2’s decision nodes grouped in one information set
• Indicates Player 2 doesn’t know Player 1’s choice

Signaling Games

Sequential games where:

1. Sender has private information (type)
2. Sender chooses action (signal)
3. Receiver observes signal, updates beliefs
4. Receiver chooses response

Applications:

• Job market signaling (education)
• Product quality signaling (warranties)
• Financial signaling (dividends)

7.4 Refinements of Nash Equilibrium

Subgame Perfect Equilibrium (SPE)

• Strategies must form Nash equilibrium in every subgame
• Eliminates non-credible threats
• Found through backward induction

Perfect Bayesian Equilibrium (PBE)

For games with incomplete information:

1. Strategies are optimal given beliefs
2. Beliefs are updated using Bayes’ rule (where possible)
3. Off-equilibrium beliefs must be specified

Sequential Equilibrium

Stronger refinement requiring:

• Consistency of beliefs with strategies
• Sequential rationality at all information sets

7.5 Repeated Games

Finite Repetition

• With known endpoint, backward induction often leads to unraveling
• Cooperation difficult to sustain
• End-game effects

Infinite Repetition

• No final period for backward induction
• Cooperation sustainable through punishment strategies
• Folk theorems: Many outcomes supportable as equilibria

Trigger Strategies

Example in repeated Prisoner’s Dilemma:

• “Grim Trigger”: Cooperate until opponent defects, then defect forever
• “Tit-for-Tat”: Copy opponent’s previous move
• Conditions for cooperation depend on discount factor

8. Prospect Theory

8.1 Overview

Prospect Theory (Kahneman & Tversky, 1979) is a descriptive theory of decision making under risk that accounts for systematic violations of expected utility theory.

8.2 Two-Phase Choice Process

Phase 1: Editing

Preliminary analysis and reformulation of prospects:

1. Coding: Outcomes evaluated as gains/losses relative to reference point
2. Combination: Probabilities of identical outcomes are combined
3. Segregation: Riskless component separated from risky component
4. Cancellation: Common components across prospects eliminated
5. Simplification: Rounding of probabilities and outcomes
6. Detection of Dominance: Dominated alternatives eliminated

Phase 2: Evaluation

Edited prospects evaluated using:

• Value function for outcomes
• Probability weighting function

8.3 The Value Function

Key properties:

1. Reference Dependence: Defined on gains and losses, not final wealth
2. Loss Aversion: Steeper for losses than gains
3. Diminishing Sensitivity: Concave for gains, convex for losses
4. S-shaped: Creates risk aversion for gains, risk seeking for losses

Mathematical form: \(v(x) = \begin{cases} x^\alpha & \text{if } x \geq 0 \ -\lambda(-x)^\beta & \text{if } x < 0 \end{cases}\)

Where typically: \(\alpha = \beta \approx 0.88\) and \(\lambda \approx 2.25\)

8.4 Probability Weighting Function

Properties:

1. Non-linear transformation of probabilities
2. Overweighting of small probabilities
3. Underweighting of moderate/high probabilities
4. Certainty effect: Jump at probability 1

Functional form (Tversky & Kahneman, 1992): \(w(p) = \frac{p^\gamma}{(p^\gamma + (1-p)^\gamma)^{1/\gamma}}\)

Where typically \(\gamma \approx 0.6-0.7\)

8.5 Prospect Theory Value

For prospect \((x_1, p_1; ...; x_n, p_n)\): \(V = \sum_{i=1}^n w(p_i)v(x_i)\)

Note: Original version had issues with stochastic dominance.

8.6 Cumulative Prospect Theory

Addresses problems in original version:

• Uses cumulative probability weighting
• Preserves stochastic dominance
• Different weighting for gains and losses

For mixed prospects, separate evaluation of gains and losses: \(V = V^+ + V^-\)

Where positive and negative parts use different weighting functions.

8.7 Applications and Implications

Financial Markets

• Disposition effect (holding losers, selling winners)
• Equity premium puzzle
• Excessive trading

Insurance and Gambling

• Simultaneous gambling and insurance
• Preference for low deductibles
• Lottery ticket purchases

Policy Applications

• Framing of public policies
• Design of incentive schemes
• Default options in choice architecture

9. Sample Questions and Solutions

9.1 Expected Utility and Value Calculations

Question: Given lottery \((3, 0.25; 2.4, 0.5; 0, 0.25)\) with \(U(x) = x^2\), calculate: a) Expected Utility b) Expected Value c) Certainty Equivalent

Solution: a) \(EU = 0.25(3^2) + 0.5(2.4^2) + 0.25(0^2) = 0.25(9) + 0.5(5.76) + 0 = 5.13\)

b) \(EV = 0.25(3) + 0.5(2.4) + 0.25(0) = 0.75 + 1.2 + 0 = 1.95\)

c) \(CE = \sqrt{5.13} = 2.26\)

9.2 Game Theory Problems

Question: Find all Nash equilibria in:

Solution:

1. Check for dominant strategies: None exist
2. Find best responses:
   • If P2 plays C: P1’s best response is C (-7 > -6)
   • If P2 plays NC: P1’s best response is NC (0 > -1)
   • If P1 plays C: P2’s best response is NC (-1 > -7)
   • If P1 plays NC: P2’s best response is C (-6 > -7)
3. Nash equilibrium: (C, NC) with payoffs (-1, -1)

9.3 Decision Trees

Question: Explain the three methods for solving decision trees and when they might give different results.

Solution:

1. Strategy Method: Enumerate all complete strategies, calculate expected payoffs
2. Backward Induction with Reduction: Work backward, eliminate suboptimal branches
3. Backward Induction with Certainty Equivalents: Replace branches with CE values

These methods give same results under EU theory but may differ under non-EU theories due to:

• Different applications of probability weighting
• Context effects in evaluation
• Reference point shifts

9.4 Mixed Strategies

Question: Find the mixed strategy equilibrium in Matching Pennies.

Solution: Let \(p\) = probability Player 1 plays Heads, \(q\) = probability Player 2 plays Heads

For Player 2 to mix: \(1 \cdot p + (-1)(1-p) = (-1)p + 1(1-p)\) Solving: \(p = 1/2\)

For Player 1 to mix: \((-1)q + 1(1-q) = 1 \cdot q + (-1)(1-q)\) Solving: \(q = 1/2\)

Equilibrium: Both play 50-50, expected payoff = 0

Supplementary Material and Exam Preparation Guide

Additional Sample Problems

Problem Set 1: Marschak-Machina Triangle

Question: In a Marschak-Machina triangle with outcomes \(x_1 = 0\), \(x_2 = 10\), \(x_3 = 20\), draw and explain indifference curves for: a) A risk-neutral EU maximizer b) A risk-averse EU maximizer c) Someone following Prospect Theory

Solution: a) Risk-neutral: Parallel straight lines with slope = -1 b) Risk-averse: Parallel straight lines with slope > 1 (steeper) c) Prospect Theory: Curved lines due to probability weighting, possibly fanning out

Problem Set 2: Allais Paradox

Question: Explain why the following choices violate EU theory:

• Choice 1: B over A where A = (\(2500, 0.33;\)2400, 0.66; \(0, 0.01) and B = (\)2400, 1)
• Choice 2: C over D where C = (\(2500, 0.33;\)0, 0.67) and D = (\(2400, 0.34;\)0, 0.66)

Solution: Under EU, if B preferred to A, then: \(u(2400) > 0.33u(2500) + 0.66u(2400) + 0.01u(0)\) \(0.34u(2400) > 0.33u(2500) + 0.01u(0)\)

This implies D should be preferred to C, but most choose C over D, violating independence axiom.

Problem Set 3: Dynamic Games

Question: In a two-stage entry game, Firm 1 decides to enter or stay out. If enters, Firm 2 can fight or accommodate. Payoffs: (Out): (0,5), (Enter, Fight): (-3,-1), (Enter, Accommodate): (2,2). Find the subgame perfect equilibrium.

Solution: Using backward induction:

1. If Firm 1 enters, Firm 2 compares: Fight (-1) vs Accommodate (2)
2. Firm 2 chooses Accommodate
3. Firm 1 compares: Out (0) vs Enter (2)
4. SPE: Firm 1 enters, Firm 2 accommodates

Quick Reference Formulas

Expected Utility

Certainty Equivalent

Risk Aversion Measures

• Absolute: \(R_A(w) = -\frac{u''(w)}{u'(w)}\)
• Relative: \(R_R(w) = -\frac{wu''(w)}{u'(w)}\)

Prospect Theory Value

Probability Weighting (TK92)

Value Function

Mixed Strategy Equilibrium

Make each player indifferent between pure strategies

Key Concepts Summary Table

Common Exam Mistakes to Avoid

1. Confusing EU and EV: EU uses utility function, EV uses monetary values
2. Independence Axiom: Remember it’s about irrelevant alternatives, not statistical independence
3. Nash Equilibrium: Must check all players’ incentives to deviate
4. Mixed Strategies: Players must be indifferent to mix
5. Information Sets: Same strategy required at all nodes in set
6. Backward Induction: Work from end, consider credibility
7. Prospect Theory: Reference point crucial, separate coding of gains/losses

Exam Strategy Tips

1. For Calculations:
   • Show all steps clearly
   • Check arithmetic carefully
   • State assumptions explicitly
2. For Theory Questions:
   • Define key terms precisely
   • Give examples when possible
   • Explain intuition behind results
3. For Game Theory:
   • Draw payoff matrices clearly
   • Label players and strategies
   • Check all possible deviations
4. Time Management:
   • Read all questions first
   • Start with questions you know best
   • Leave time for checking

Practice Checklist

• Can calculate EU, EV, and CE
• Understand all axioms of EU theory
• Can identify violations of EU axioms
• Know how to draw in Marschak-Machina triangle
• Can find pure and mixed strategy Nash equilibria
• Understand backward induction
• Can distinguish game types (simultaneous/sequential, perfect/imperfect information)
• Know Prospect Theory components
• Can solve dynamic decision problems
• Understand experimental methodology

This supplementary material provides additional practice problems, quick reference guides, and exam preparation strategies to complement the main study guide. Good luck with your exam!