lecture 1: introduction to economic design presented by tom wilkening 18 september 2014

LECTURE 1:INTRODUCTION TO ECONOMIC DESIGN

PRESENTED BY TOM WILKENING 18 SEPTEMBER 2014

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Introduction

• Design economics is a relatively new discipline:• Traditionally, the role of economists has been to analyse outcomes

taken the rules as given.• The Design of institutions have typically been left to the discretion of

Legislators and regulators, lawyers and judges, managers, or others

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Introduction

• Since the 1990s, however, economists have taken a substantial role in design, especially in the development of auctions and clearing houses.• Spectrum Auctions• Matching systems for medical students, schools, and transplants• Internet Advertisement Auctions

• These developments suggest an emerging discipline of design economics, the part of economics intended to further the design and maintenance of markets and other economic institutions.

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Why do we need economic design?

Economic environments have been evolving for millennium. Why do we all the sudden need to transition from economist to engineer?

• Complexity• For simple commodity exchanges, the rules are quite straight forward.

Price, quality, and quantity evolve quite naturally as the primitives of exchange.

• As we study more complicated environments, however, the rules are much less straight forward.

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Economic environments have been evolving for millennium. Why do we all the sudden need to transition from economist to engineer?

• Market Failures• Decentralized markets have a long tradition of solving market failures.

• Warrants, guarantees, and reputation mitigate the lemons problem• Many markets, however, break down over time suggesting that

evolutionary pressures are not leading to an efficient outcome.• In these cases, trial and error has not solved the market failure.

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Analogy: Physics and Bridge Building

• A Standard Auctions with 20 goods requires 20 bids.• A Combinatorial Auctions with 20 goods requires 220 bids. This is just

over 1,000,000...

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Introduction to Design Economics

Agenda• What distinguishes “design economics” from other engineering

disciplines?• What tools are available to aide in economic design?• Lessons from past designs

Further Reading:• Alvin E. Roth “The Economist as Engineer: Game Theory,

Experimentation and Computation as Tools for Design Economics”. Econometrica, Vol. 70, No. 4, 1341-1378

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What distinguishes “DesignEconomics” from other engineering disciplines?

Systems engineering is an established discipline with a long tradition of success in the design of systems. Why can’t we just adopt their toolkit?

Unlike an engineering system, decisions in markets and economies are made by distributed decision makers. Individual actors are autonomous: There are multiple individuals

interacting with one another, each of whom makes his or her own decisions. Interaction matters: The outcome is determined by the confluence of

decisions, each from a typically uncoordinated source. Individuals have freedom: Individuals have the ability to walk away from a

situation. Participation in the system is typically voluntary.

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An Experiment

• Everyone in the room who wishes to participate will write down the word “Invest” or “Don’t Invest” on the index cards along with their name and a random 6 digit number. Cards will be collected.

• I will draw two index cards from the pack and pay the recipients based on their choices.

• If both people write “Don’t Invest”: Each person will receive $30.

• If one person writes “Invest” and the other writes “Don’t Invest”: The person who wrote “Invest” will receive $50, the other gets $0.

• If both people choose “Invest” Each person will receive $15.

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Why does individual autonomy matter?

• The Investment Game is an example of a prisoner’s dilemma. This environment is heavily studied by economists in understanding a variety of settings such as investment, research and development, contests, cheating, and war.

• Prisoner’s dilemmas are often used in game shows due to the tension that arises between individual and group incentives

Golden Balls Extremely popular British TV show from 2008 – 2009 Four part game show which involves lying and voting other contestants off

until their are only two players left. They then play the “golden balls” game.

goldenballs.flv

• What can we learn from the Golden Balls Game?

• Lesson 1: System where the incentives of individuals are at odds with the objectives of the designer are likely to be unstable and lead to inefficiencies.

• For good or ill, the incentives generated in an economic system influence the decisions of the individual.– It doesn’t take very many professional rent seekers to destroy an

economic institution– Aligning incentives of the individual and society leads to better

outcomes.

Important Lessons: Incentives

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• Example: Arizona’s Alternative Fuel Program– Objective: Increase the amount of vehicles using alternative fuel

systems in the Phoenix area. Desire to reach a critical mass for LTG fuel stations.

– Approach: Lump-sum rebate of up to 40% of final price of vehicles which use converted to alternative fuels. Exemption from sales taxes, vehicle licensing, and emissions.

• Problem– No requirement of usage. SUV owners bought a $1,000 dollar

secondary fuel tank and disabled it after delivery. Purchasers saved $22,000 without actually using secondary fuel.


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• Outcome– Program expanded from an expected $3 million to $600 million. This

was roughly 10% of the state budget.– Law repealed, cost $200 million.– Political disaster – legislators removed from office who wrote bill or

purchased cars using the scheme– Environmental outcome: Decrease in overall vehicle fuel efficiency due

to large vehicle sales.


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• Lesson 2: Imperfect systems may work in the short run but often collapse in the long run.

• Individuals who act in good faith in a bad economic environment are exploited over time. This leads to increasingly inefficient outcomes.– Sarah cooperated in her first game but defected in her second game.

She learned to behave in a socially manipulative way.– Golden Balls: discontinued after the 2009 season, partly due to an

increase in defections over time.

Important Lessons: Stability

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• Example: The Medical Match– In the early 1900s, the United States had a decentralized hiring system

for newly trained doctors.– However, by the 1940s, students were being hired two years before

finishing medical school. (Unwinding) – In 1945, the hospitals established a limit on the amount of time prior to

graduation a student could be contacted. • Students tended to wait on offers to see if better offers arose.• Schools often would miss out on their second best candidate.

– In response, schools began to make exploding offers. Students responded by reneging on accepted offers.


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• The problem of pad designs highlighted the two major problems of a two-sided match– Students preferred to wait as long as possible to accept an offer hoping

for a better one.– Hospitals wanted to take advantage of their position by making

exploding offers• In 1950, a new (and very successful) centralized matching system was

established. The central premise of this clearing house is stability.– Stability requires that a student and hospital cannot remain unmatched

if both the student and the hospital prefer one another than their current assignments.


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Market Stable? Still in use? 1. National Residency Match yes yes (new design in ’98)2. Edinburgh ('69) yes yes3. Cardiff yes yes4. Birmingham no no5. Edinburgh ('67) no no6. Newcastle no no7. Sheffield no no8. Cambridge no yes9. London Hospital no yes10. Medical Specialties yes yes (~30 markets, 1 failure)11. Dental Residencies yes yes12. Osteopaths (< '94) no no13. Osteopaths (>'94) yes yes14. Pharmacists yes yes15. Lab experiments yes yes16. Kagel&RothQJE2000) no no

• Lesson 3: Information plays a major role in the efficiency of decentralized systems. The inability of the system designer to know everything makes system design a challenge

• Golden Balls

Important Lessons: Information

Sarah

?

Good Bad

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• Example: In my last trip to Europe I had a stop over in Milan on a Luftansa flight:– At the check in line, first class had more idle staffed counters than the

coach line had open counters. The difference in check-in time was about 30 minutes.

– At the security gate, Lufthansa had a special first class security line. The difference in security was roughly 45 minutes.

– In the plane, economy had 9 seats across with limited leg room. There was extra unused space on the plane in front of coach and behind which in most other configurations would have been used to space the seats.

• Why is the quality of economy so low? Is it a desire for the firm to compete on price? For cost savings?


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• Most of the inefficiencies were intentionally designed!• Lufthansa can’t restrict the ticket I buy – it must make economy

unattractive to business customers and the rich.


First Class

Me

?

Rich Poor

Efficient Economy

$$$$

$$

Distorted Economy

$

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• The inability to observe the characteristics of the individual actors or businesses often forces us to embed inefficiency into the systems.

• Markets and auctions are frequently advocated as a way to reduce information asymmetries and generate information.– Auctions are the most efficient way to allocate goods when valuations

are private– Markets are the fastest known mechanisms for aggregating large

amounts of information.– Within the economic profession “Economic Design” and “Market

Design” are interchangeable terms. Information is at the centre of both.


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• What distinguishes “design economics” from other engineering disciplines?– Autonomous Decision Makers– Interactions matter – Individuals have Freedom

• These three differences cause economists to focus on:– Incentives– Stability– Information

Important Lessons: Summing Up

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Agenda



• Game Theory: Discipline of economics which analyses how individuals interact. It is interested in how the `rules of the game’ affect human decision making.

• Two Branches:– Cooperative Game Theory – Used extensively in matching theory to

understand when people want to leave the system– Non-Cooperative Game Theory – The main workhorse in economics.

Used to study auctions, contracts, and markets.

Game Theory

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Non-cooperative game theory models are “Shakespearean” in nature. We start by splitting the economic problem into three parts:

• Environment: Who are the actors influencing decisions, what information do they have, what are their goals?

• Market Mechanism: What are the rules? • Strategic Actions: How are individuals responding to one another?

Game Theory

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• Environment:– Players (Agents): Who are the actors making decisions? How many?– Types: Is there possibility for heterogeneity?– Information (Beliefs): What do we know about the others?– Possible Outcomes: What can happen?– Utility: How much people value each of the outcomes

Game Theory

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• Mechanism (also called Institution)– Actions: What can each person do?– Mechanism: Rules that govern how individual actions translate into

outcomes.

Game Theory

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• Strategic Interaction– The main tenant of economic models is that individuals are active in

their decision process• Optimization• Best Response • Nash Equilibrium• Bounded Rationality

– Economic models typically look at the “Planning Reward” system of cognition and assume that emotional responses to stimuli do not dominate cognitive choices.

Game Theory

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Game Theory AnalogyEnvironment: Players Types Information Utility (Desires)Market Mechanism Actions TimingStrategic InteractionOutcome

HamletClaudius(Uncle)

Mother

?

Bad

Play within

the Play

Good

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• Market design involves a commitment to detail, a need to deal with all of a market’s complications, not just its principle features.– Many design details do not feature in general theoretical models and

thus there is no a priori way to distinguish between many similar implementations.

– Individuals who are new to an economic environment are also unlikely to behave exactly as predicted by theory. Complex mechanisms are confusing and learning occurs over time.

• In many cases we want to know just how robust mechanisms are to real individuals and to small changes in environments. For this we use lab experiments.

Experiments

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Laboratory Experiments

http://mercury.ecom.unimelb.edu.au/orsee/images/largepic1.jpg

Laboratory Experiments

Environment(Controlled by Experimenter)

Mechanism(Controlled by Experimenter)

Strategic Interaction(Experiments Test This)

Predicted Outcome Is the mechanism robust to

changes in the environment? Do people behave as predicted? Do alternative mechanisms work

better?

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• The best economic institution is going to depend intimately on the actual environment, which is often not fully known. We thus need a way to uncover information about the real environment.

• To uncover underlying information about the environment, economists use structural estimation. This approach takes historical data and combines it with assumptions about how individuals act to learn more about the environment.

• Structural estimation is often combined with lab and field experiments to improve the precision of estimates and rule out other forces.

Structural Estimation

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Structural Estimation

EnvironmentUnknown Information

OldMechanism

Strategic Interaction

OutcomeEx-Post Information

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• Many times, it can be shown that in small scale environments, individuals have incentives to take inefficient actions under some conditions. – In matching systems, for instance, individuals can often do better by

lying about their preferences.– With full information, inefficient outcomes often exist in auctions.

Computational Economics

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• We often wish to know whether these same actions are distortionary as things scale up. Since we cannot typically run lab experiments with the same number of players as the real thing, we must resort to simulation.– Computational Economics is often used to simulate a market with a

large number of players and determine to what extent incentives change as the number of individuals grow large.

Computational Economics

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Market DesignEnvironment(Details of the Problem being analysed. Informed by empirics)

Mechanism(Designed)

Strategic Interaction(Tested with Experiments)

Outcome Efficiency Revenue Secondary Objectives

Goal:Design the mechanism so It leads to the best possibleoutcomes

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• Economic designs main tool of analysis is non-cooperative game theory. Game theory divides the problem into three parts – Environment– Mechanism or Institution– Strategic Interaction

• Design also turns to a variety of empirical methods to ensure that the design is correctly tailored to the problem at hand– Experimental Economics– Structural Estimation– Computational Economics

Tools of Economic Design: Summing Up

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Agenda



• Many of the most spectacular design failures have come from a lack of theory or testing before implementation– Insufficient Theoretical Analysis: The first Australian spectrum auction

was heavily gamed by individuals who used withdrawal rules to their advantage, a phenomenon easily predicted by theory.

– Insufficient Experimentation: A German spectrum auction closed in the first round due to using the smaller digits of bids to signal a division of licenses across the major players.

Lessons: Both Theory andEmpirical Testing are essential

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• Design economics is often interested in redesigning systems in which there is a group with vested interest in the older rules. Redesign typically requires strong efficiency gains to overcome inertia.

• Design economics also leads to measurable outcomes. While this is seen as a strength in improving a system, it is dangerous politically.

• Design also tends to enter into new territories where laws are grey. This often requires legislative intervention.

Lessons: Political support is key

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• The outcome of economic systems typically have more than one dimension. There is usually a tradeoff between different objectives.– Revenue vs Efficiency: Auctions have a fundamental tradeoff between the

amount of money that can be collected and the predicted level of efficiency. – Downstream competition: Auctions for licenses often will have an impact on

downstream competition. Granting monopolies will generate more revenue since the winning company will be able to charge higher prices.

– Secondary measures: Auction rules are often used to try to address secondary objectives such as supporting small businesses or local businesses. These rules often have large implications on outcomes.

• The more objectives that are included the greater scope there is for external lobbying and gaming. Simple goals typically lead to better designs.

Lessons: Be aware of competingobjectives

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An Experiment: Ultimatum Game• Two Roles: Proposer and Responder• The Proposer starts with 20 tokens. Responder starts with 0

• Actions: 1. The Proposer will offer any number of tokens {0,1,2,…,20} to the

responder.2. The responder has two options:• Accept – Each party earns $1 per token that they have.• Reject – Both parties end with $0

• I will randomly draw two players from the class and pay based on their actions.

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An Experiment: Dictator Game• Two Roles: Proposer and Recipient• The Proposer starts with 20 tokens. Recipient starts with 0

• Actions: 1. The Proposer will offer any number of tokens {0,1,2,…,20} to the

recipient.2. Each token kept yields $1 to the Proposer. Each token given yields

$3 to the recipient.

LECTURE 2:INTRODUCTION TO GAME THEORY

PRESENTED BY TOM WILKENING 18 SEPTEMBER 2014

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Game theory

• Learning about decision theory and game theory is an important foundation for design economics (the rest of the course) and public policy generally.

• A sound understanding of how people are likely to react to the rules a system, is vital for us to predict and design, and to diagnose and correct various policies.

• The next three lectures will cover a range of tools to examine how people are likely to make decisions and respond to each other, which is an important basis for future weeks looking at auctions, markets, contracts and matching tools in mechanism design.

Sharon Oxlade

Suggested new slide - just something outlining the context of this in the broader course, and why it's important.

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Agenda

• Decision Theory• Simultaneous Move Games• Sequential Move Games

• To understand game theory, we need to first understand some basic decision theory

• Decision theory is applied to single agent problems, in which the agent’s decisions do not influence on the payoffs and the decisions of others.

• The agents choose a feasible action x, and her payoff u(x) depends on her actions x.

• Decision theory assumes that individuals optimize and choose what is best for them.– For continuous choices, we typically use calculus: u’(x) = 0– For discrete problems, we often use decision trees to simplify the

analysis.

Decision Theory

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• Example: Consumers– Choose what flavour of ice cream to eat when offered three choices:

Chocolate, Vanilla, and Strawberry• u(Chocolate) = 5• u(Vanilla) = 3• u(Strawberry) = 2

– Decision theory assumes that agents optimize – thus Chocolate is chosen.

Decision Theory: An Example

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• For each possible outcome, the payoff is the number that indicates how much the player values this outcome.– Payoffs may represent many things such as profit, income, happiness,

etc.– They also don’t only have to depend on my outcome.

• If the outcome is random, the expected payoff is the weighted average of the numbers associated with the different possible outcome.– Example:

• It rains 40% of the time. Your payoff is 3 if it rains.• It is sunny 60% of the time. Your payoff is 8 if it is sunny.

– Expected Payoff = .4 x 3 + .6 x 8 = 1.2 + 4.8 = 6.

Decision Theory: Definitions

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• Discrete maximisation can also be used for more complicated decisions. Consider the following example:– Each morning when your alarm goes off you decide whether to go to

work or sleep in– If you get up for work, you need to decide whether to go to the morning

briefing (where you get a payoff of 10) or to go for coffee (which gives you a payoff of 6).

– If you sleep in and get up at midday, you need to decide whether to watch TV (which gives you a payoff of 2) or to apply for a new job (which gives a payoff of 7).

Decision Theory: Decision Trees

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• This example can be modelled as a tree:

• To solve, we use Backward Induction

Decision Theory

Work Home

Meeting Coffee TV New Job

10 6 2 7

1

1 1

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• Random events can also be incorporated into decision problems and decision trees.

• Consider the following problem– You must decide whether to “Take an Umbrella” or “Not Take an

Umbrella”– If you take an umbrella:

• If it rains, your payoff is 3. It rains 40% of the time.• If it is sunny, your payoff is 8. It is sunny 60% of the time.

– If you don’t take an umbrella• If it rains, your payoff is -10. It rains 40% of the time.• If it is sunny, your payoff is 10. it is sunny 60% of the time.

Decision Theory: Decision Trees

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Sharon Oxlade

edited - 69 to 60

• Random events are represented by nodes corresponding to a move by chance (or Nature, denoted by a square N in the game tree).

• To solve, we calculate the expected payoffs and then use backward induction. Comparing the options with and without the umbrella:

0.4 x 3+ 0.6 x 8 = 6 > 2 = 0.4 x -10 + 0.6 x 10

Decision Theory

Take Umbrella Home

Rain: 0.4 Sun: 0.6 Rain: 0.4 Sun: 0.6

3 8 -10 10

1

N N

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• After staying home from your job and being fired you are interested in enrolling in a work program. Enrollment in the work program is under study and uses the following rules for enrollment.

– You announce a price you are willing to pay – your ‘bid’ {0,1,…,100}– The coordinator picks a random number out of a bingo cage {0,1,…,100}– If your bid is greater than or equal to the bingo ball

• You are enrolled• You pay the amount indicated on the bingo ball that was picked

– If your bid is less than the bingo ball• You are not enrolled• You pay nothing

• This is known as a Becker-Degroot-Marshack or BDM Mechanism

Decision Theory: Examples

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• Suppose your value is $60. What should you bid?


Your value

Candidate Bid 1$30

Candidate Bid 2$80

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• Suppose your you bid $30.• If the bingo ball comes up at $45, you lose but would gain $15. You are

better off bidding higher!

• Extending the logic – it is never in your interest to bid below your value.


Your value

Candidate Bid 1$30

win

lose

58

Sharon Oxlade

how do you gain anything? You value the course at $60 and do not participate, so you are no better (or worse) off than currently.

• Suppose your you bid $80.• If the bingo ball comes up at $70, you win but pay $70. You are losing

$10 with this bid!

• Extending the logic – it is never in your interest to bid above your value.


Your value

win

lose

Candidate Bid 2$80

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• The Becker-Degroot-Marshack or BDM Mechanism generates a decision problem where it is always in the best interest of the decision maker to reveal their true valuation.– We will revisit similar mechanisms when looking at auctions.

• We can determine the best action by sequentially eliminating dominated strategies. This is a useful technique in lots of game theory.

• What are your incentives if:– Enrollment was random and you paid the amount you announced?– Enrollment was based on how much you announce but you always paid

0?


60

• In decision theory, the actions of others do not affect your payoffs (and vice versa).

• In a game, the actions of others do affect the payoff of your actions (and vice versa)

• To make optimal choices in a game we often need to anticipate the strategies of the other players.

Decisions versus Games

61

• Purchasing at a store• Penalty kicks in soccer, tax audits• Hiring decisions• Retail: Coles vs. Woolworths, BP vs. Shell• Politics: Labour vs. Liberal, Republicans vs. Democrats

Games are everywhere

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As was noted in the first lecture, games are split into three parts:

• Environment: Who are the actors influencing decisions, what information do they have, what are their goals?

• Market Mechanism: What are the rules? • Strategic Actions: How are individuals responding to one another?

Game Theory

63

• Environment:– Players (Agents): Who are the actors making decisions? How many?– Types: Is there possibility for heterogeneity?– Information (Beliefs): What do we know about the others?– Possible Outcomes: What can happen?– Utility: How much people value each of the outcomes. (Maps

Outcomes into Payoffs)

Game Theory

64

• Mechanism (also called Institution)– Actions: What can each person do?– Mechanism: Rules that govern how individual actions translate into

outcomes.

Game Theory

65

• Strategic Interaction– The main tenant of economic models is that individuals are active in their

decision process– For much of our analysis we assume people are perfect at forming their

strategy and carrying out their plans.• Optimization• Best Response • Nash Equilibrium• Subgame-Perfect Nash Equilibrium

Game Theory

66

• Games are often defined by information– Complete information: Situations where the timing, feasible moves and

payoffs of the game are all common knowledge• Examples: Chess, Rock-Paper-Scissors

– Private information: Situations where individuals have information that is not available to others.• Example: Poker, Negotiation with used car dealers,

Information

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• Games are also defined based on whether they are static or dynamic– Static Games: All individuals make their decision once at the beginning

of the game without observing any other actions• Examples: Rock-Paper-Scissors

– Dynamic Games: Games have a sequence of moves with new information being transmitted about past actions.• Example: Chess

Static vs. Dynamic

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• The simplest games we can analyze are those that look just like decision problems. These are games where only one person makes an action that influences the payoffs of both players.

• Example: The Dictator Game– Player 1 is given 20 Tokens– Player 1 Gives x (0, 5,10,15,20) tokens to player 2– Player 1 receives $1 for each token kept. Player 2 receives $3 for each token

received.

The Simplest Games

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• Environment– Players: {Player 1}, {Player 2}– Possible Outcomes: {$20,$0}, {$15, $5}, {$10, $30}, {$5, $45}, {$0, $60}– Utility Over Outcomes: u1 (m1,m2), u2 (m1,m2)– Information: Complete Information– Types: None

• Actions:– Player 1: Choose x in {0, 5, 10, 15, 20}

• Mechanism– Giving x leads to m1 =20-x and m2=3x

The Simplest Games

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• Games where a single person makes a decision are solved in a manner equivalent to a decision problem.

• We determine the outcome from each outcome and assign a payoff to each potential outcome. For example, if individuals only care about their own payoff: – Player 1’s utility: u1(m1,m2) = m1 = (20 – x)

– Player 2’s utility: u2 (m1,m2) = m2 = 3x

The Simplest Games

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• Games where a single person makes a decision are solved in a manner equivalent to a decision problem.

• Payoffs at end nodes are ordered by player number.

The Simplest Games

Give 0

1

Give 5 Give 10 Give 15Give 20

20,0 15,15 10,30 5,45 0,60

u1 (m1,m2), u2 (m1,m2)

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• When looking at the solution, we look for the action that gives an individual the highest payoff. We ignore the other players payoff as this isn’t relevant.

The Simplest Games

Give 0

1


20,0 15,15 10,30 5,45 0,60

73

• This isn’t to say that we can’t incorporate a preference for giving into the analysis. For example, if player 1 cares about the total amount of money given away, we can rewrite his utility as:

• This just changes the terminal payoffs of the game

The Simplest Games

Give 0

1


20,0 30,15 40,30 50,45 60,60

u1(m1,m2) = m1+ m2 = (20 – x) + 3x

74

• As this example illustrates, the utility function doesn’t just have to be the money I receive. Game theory is flexible in what we put into the problem.

Key Assumptions• Rationality: Players aim to maximize their payoffs (whatever these are)

– Players are perfect calculators and flawless followers of their best strategy

• Common Knowledge– Each player knows the (underlying) mechanism and rules– Each player knows that each player knows the rules– Each player knows that each player knows that each player knows the

rules– Etc. etc. etc.

Key Assumptions in Game Theory

75

• A strategy of a player is a complete plan of action that specifies an action at each of her possible moves (even if she never gets to this move).– If you write your strategy down properly, you can give it to anyone and

they can play just like you would no matter what has happened in the game.

Strategies

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• Strategies denote an action in every possible situation in which you might act. For example:– I flip a coin:– If Heads you play the dictator game– If Tails, you can choose A or B

• If you choose A you both receive $20• If you choose B you both receive $0

Strategies here is an amount x you give in the dictator game and your choice of A or B if Tails arises. For instance, if purely selfish, the optimal strategy would be {0,A}.

Strategies

77

• Strategies are pure when players choose their actions with certainty• Strategies are mixed when players choose their actions with some

randomness (for example, flipping a coin)• Strategies are discrete when there is a (small) finite number of possible

strategies– Example: Accept/Reject, Vote A or B

• Strategies are continuous when there is an (almost) infinite number of possible strategies– Examples: Price, Proportion, Quantity, Investment

Strategies

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• An equilibrium is a set of strategies that are stable so that no one prefers to change their actions based on the actions of others.

• A Nash equilibrium looks for strategy profiles for all individuals such that no individual has an incentive to change any of their actions given the strategies of the others.

• We will use some refinements of Nash equilibrium as we go along– Subgame Perfect Nash Equilibrium eliminate Nash equilibrium that are

supported by non-credible threats and do not satisfy backward induction.

– Bayes-Nash equilibrium is the solution concept we use when thinking about games of incomplete information.

Equilibrium

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Agenda


• Simultaneous (move) games can represent the following situations:– Players choose their actions simultaneously– Players choose their actions without knowing what the other players

have done or will do

• The Prisoner Dilemma Game Given in lecture 1 is a simultaneous move game.– Actions: “Invest” or “Don’t Invest”– Outcome was based on the combination

• Both “Don’t Invest”: Each received $30• One “Invest”, One “Don’t Invest”: Investor received $50, Other $0• Both “Invest”: Each receive $15

Simultaneous Move Games

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• Simultaneous move games with discrete strategies are most often depicted in game tables:– These games are called Normal or Strategic Form Games– By convention, the first player is the row player – each row

corresponds to a strategy of her.– The second player is the column player – each column corresponds

to a strategy of her.– Each cell corresponds to an outcome and lists the payoffs. Payoffs

are (first player payoff, second player payoff)

Game Tables

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• Example: Investment Game

0,50 30,30

Game Tables

Invest

Invest 'Don t Invest

2Player

Play

er 1 15,15 50,0

Don’t Invest

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• A Nash Equilibrium is a list of strategies, one for each player, such that no player can get a higher payoff by switching to some other strategy given the strategies of the other players.

• A best response is a player’s best strategy given the strategies of the other players.

• In a Nash Equilibrium, each player is thus playing her best response to the other players’ best responses

• Nash Equilibrium is seen as a good solution concept because the predicted outcomes are stable. No player would want to deviate unilaterally.– There is nothing here about fairness or maximizing total welfare – these

are encoded in the payoffs.

Nash Equilibrium

84

• There are a few techniques for finding Nash Equilibrium– Cell-by-cell inspection– Iterated elimination of dominated strategies– Best-response analysis

• Finding solutions can be a bit of an art. Use whatever works best for you!

Finding Nash Equilibrium

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• For each cell of the normal form game, look to see if a player can increase her payoff by unilaterally switching her strategy

• If no player can increase her payoff, it is a Nash Equilibrium

Finding NE: cell-by-cell inspection

0,50 30,30

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

86




0,50 30,30

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

87




0,50 30,30

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

88

• We can also start by considering a strategy for each player and see if there is a better strategy regardless of what the other player does. If such a superior strategy exists we can abandon the original one.

Finding NE: Elimination of DominantStrategies

0,50 30,30

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

89

• We can continue to do this by now looking at Player 2’s actions. This is known as iterated deletion.

Finding NE: Iterated Elimination of Dominant Strategies

Invest


2Player

Play

er 1 15,15 50,0

Don’t Invest

Invest


2Player

Play

er 1 15,15

Don’t Invest

90

• The strategy of iteration deletion can work on much more complicated games and can often help us solve complicated problems

• For every payoff to player 1 in row B, there is another row that offers a higher payoff. This means that we can eliminate B.


2Player

Play

er 1

3,1 2,3 10,2 100,2

4,5 3,1 6,4 100,2

2,2 5,4 12,3 100,2

5,6 4,5 9,7 105,2

A B C D

A

B

C

D

91

Sharon Oxlade

Can you walk through these a little more slowly - just providing the intuition behind where you start and why each one is eliminated at each point?

• For player 2, C dominates A. We can thus eliminate A


2Player

Play

er 1

3,1 2,3 10,2 100,2

2,2 5,4 12,3 100,2

5,6 4,5 9,7 105,2

A B C D

A

B

C

D

92

• For Player 2, B dominates B. We can thus also eliminate D


2Player

Play

er 1

2,3 10,2 100,2

5,4 12,3 100,2

4,5 9,7 105,2

A B C D

A

B

C

D

93

Sharon Oxlade

D?

• For player 1, C dominates both A and D. We can thus eliminate these


2Player

Play

er 1

2,3 10,2

5,4 12,3

4,5 9,7

A B C D

A

B

C

D

94

• Finally, for Player 2, B dominates C. We thus can eliminate C


2Player

Play

er 1

5,4 12,3

A B C D

A

B

C

D

95

• Thus (C,B) is the Nash Equilibrium


2Player

Play

er 1

5,4

A B C D

A

B

C

D

96

• In best-response analysis, we hold player 2’s action fixed and look at what the best choice for Player 1 is. We highlight these best responses.

Finding NE: Best Response Analysis

2Player

Play

er 1

3,1 2,3 10,2 100,2

4,5 3,1 6,4 100,2

2,2 5,4 12,3 100,2

5,6 4,5 9,7 105,2

A B C D

A

B

C

D

97

• We do the same thing for Player 2: Holding Player 1’s action fixed, we look to see what action of Player 2 is best


2Player

Play

er 1

3,1 2,3 10,2 100,2

4,5 3,1 6,4 100,2

2,2 5,4 12,3 100,2

5,6 4,5 9,7 105,2

A B C D

A

B

C

D

98

• Nash equilibrium are going to be cells where both individuals are best responding to each other. Again this is (C,B)


2Player

Play

er 1

3,1 2,3 10,2 100,2

4,5 3,1 6,4 100,2

2,2 5,4 12,3 100,2

5,6 4,5 9,7 105,2

A B C D

A

B

C

D

99

• Nash Equilibrium are a set of strategy profiles where each person – given the actions of the others – is choosing actions that maximize their payoff.

• No individual will choose a strictly dominated strategy in a Nash Equilibrium!

• Each individual takes into account that their opponents also never use strictly dominated strategies: they too play the best response to your strategy!

Nash Equilibrium:Summing Up

100

• One of the key results that John Nash proved is that every game has at least one Nash Equilibrium.

• However, Nash Equilibrium are not necessarily unique!

• Example: Coordination Games.

Nash Equilibrium are not Necessarily Unique

0,0 5,15

Telstra

Telstra Virgin

2Player

Play

er 1 15,5 0,0

Virgin

101

• Some games do not have a Nash Equilibrium in pure strategies

• Example: Penalty Kick in soccer

Nash Equilibrium may notBe in pure strategies

0,1 1,0

Left

Right Left

2Player

Play

er 1 1,0 0,1

Right

102

• In such games, a player should not systematically pick the same strategy because her opponent could take advantage of this.– Players should randomize their actions – i.e. they should play mixed

strategies– Mixed strategies again require stability – both parties mix so that the other

party is indifferent between mixing.

Nash Equilibrium may notBe in pure strategies

103

104

Agenda


• Sequential move games occur when actions occur through time and individuals can condition their actions based on events that happened in the past.– Timing now plays an important role in determining what happens in a game

• The Ultimatum Game is an example of a sequential game:– Player 1 offers an amount x to player 2.– Player 2 accepts or rejects the offer. The accept/rejection is conditional on

the offer.

Sequential Move Games

105

• Lets take the coordination game that we thought about in the last section.


0,0 5,15

Telstra

Telstra Virgin

2Player

15,5 0,0

Virgin

106

• Suppose that instead of it being a simultaneous game, player 1 gets to choose her phone carrier first.

• For Player 1, then, a strategy is simply choosing Telstra and Virgin• For Player 2, we need to specify an action at each potential decision

point.– Strategies will thus be a pair of actions that are conditional on whether

Player 1 has chosen Telstra or Virgin. We are thus specifying an action for each potential history.


107

• We could specify the new game as a normal form game:


0,0

T

,T T ,T V

2Player

15,5

V

,V T ,V V

15,5

5,15

0,0 0,0

0,05,15

108

• One issue with this normal form approach is that it doesn’t give us any sense of timing. We can’t fully reconstruct the game from the normal form.

• A bigger issue is that some of the Nash Equilibrium of this game are somewhat strange. For example, if Player one chooses T, Player 2 using the strategy {V,V} will be getting 0. She could do better at this point in time by abandoning her strategy and switching to strategy {T,V}!– Thus, the Nash Equilibirum V, {V,V} exists in part because player 2 is not

switching their strategy when they potentially should.• We call this type of equilibrium non-credible because the equilibrium is

based on a strategy profile that the individual would like to change when they arrive at a future decision.


109

• To address these issues, we typically use game trees to analyse sequential games.

• Game trees are often referred to as extensive form games.• Just like decision trees, we will write down a tree that specifies all the

decisions across each of the potential decision nodes.– Game trees are joint decision trees for all the players in the game.


110

• The decision tree for the coordination game is as follows:


Telstra Virgin

Telstra Virgin Telstra Virgin

15,5 0,0 0,0 5,15

1

2 2

111

• To find equilibrium that are credible we want to make sure no one wants to change their mind later on in the game. We do this through backward induction.

• A subgame is a game comprising a portion of a larger game, starting at a non-initial node of the larger game.

• Backward induction starts at the final subgames, and finds the Nash equilibrium for these subgames.

• It then moves up the tree using these choices as the predicted actions in the later subgames.

Backward Induction

112

• We first solve these subgames from the perspective of player 2 (the person making the move).


Telstra Virgin


15,5 0,0 0,0 5,15

1

2 2

113

• We first solve these subgames from the perspective of player 2 (the person making the move).


Telstra Virgin


15,5 0,0 0,0 5,15

1

2 2

114

• We then solve the next layer of the game using these payoffs


Telstra Virgin

15,5 5,15

1

115

• The easiest way to do this is to just draw arrows in each tree starting from the bottom and working our way up.


Telstra Virgin


15,5 0,0 0,0 5,15

1

2 2

116

• An equilibrium found through backward induction is called a Subgame Perfect Nash Equilibirum (SPNE)

• Such an equilibrium is a set of strategies that is an Nash Equilibrium at every single subgame of the larger game. This includes subgames that are off the equilibrium path of play.

• The solution predicts stable outcomes, as no player wants to deviate from her equilibrium strategies.

Backward Induction

117

• Sequential games occur when individuals can condition their play on events that happen in the past.

• Strategies get more complicated in sequential games since we must specify an action for every single history.

• We often use a refinement of Nash Equilibrium that eliminates equilibrium that are based on non-credible threats.

• Subgame-Perfect Nash Equilibrium satisfy this refinement by requiring that the strategy profile is a Nash equilibrium in every subgame.

Sequential Games:Summing Up

118

• Example:– Consider the Ultimatum Game discussed earlier

• What is the SPNE of this game?• What are some of the other Nash Equilibrium of the Game?

Using Game Theory

119

• Consider the following game– A monopolist can produce a good that costs $5 to make.– The monopolist has two potential types of consumers:

• Low Type Consumers: values the good at $20.• High Type Consumers: values the good $50.

– The consumer is a High type with probability p

• The monopolist can offer a single take-it-or-leave it offer to the consumer. The consumer may accept or reject it. – What price should be set for each p?

Using Game Theory

120

• The government is trying to buy land to make a railroad. There are two owners of the land.– Owner 1 values the land at $5– Owner 2 values the land at $5

• The government values both pieces of land at $15. It values one piece of land at $0.

• The game is as follows:– Owner 1 offers to sell his land at price p1 to the government– The government accepts or rejects the offer.– Owner 2 offers to sell his land at price p2 to the government– The government accepts or rejects the offer.

Using Game Theory

121

• What price will owner 2 charge?• Will the government ever buy from owner 1?• What are some alternative rules that will lead to better outcomes?

Using Game Theory

122

lecture 1: introduction to economic design presented by tom wilkening 18 september 2014

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