LECTURE 1:
INTRODUCTION TO
ECONOMIC DESIGN
PRESENTED BY TOM WILKENING
18 SEPTEMBER 2014
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
2
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.
3
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.
4
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?
• 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.
5
Why do we need
economic design?
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...
6
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
7
What distinguishes “Design
Economics” 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.
8
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.
9
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.
10
Important Lessons:
Incentives
• 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.
11
Important Lessons:
Incentives
• 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.
12
Important Lessons:
Incentives
• 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.
13
Important Lessons:
Incentives
14
Important Lessons:
Stability
• 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.
15
Important Lessons:
Stability
• 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.
16
Important Lessons:
Stability
• 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.
17
Important Lessons:
Stability
Market
1. National Residency Match
2. Edinburgh ('69)
3. Cardiff
4. Birmingham
5. Edinburgh ('67)
6. Newcastle
7. Sheffield
8. Cambridge
9. London Hospital
10. Medical Specialties
11. Dental Residencies
12. Osteopaths (< '94)
13. Osteopaths (>'94)
14. Pharmacists
15. Lab experiments
16. Kagel&RothQJE2000)
18
Stable?
yes
yes
yes
no
no
no
no
no
no
yes
yes
no
yes
yes
yes
no
Still in use?
yes (new design in ’98)
yes
yes
no
no
no
no
yes
yes
yes (~30 markets, 1 failure)
yes
no
yes
yes
yes
no
Important Lessons:
Information
• 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
?
19
Important Lessons:
Information
• 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?
20
Important Lessons:
Information
• 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.
$$
$$$$
?
21
$
Important Lessons:
Information
• 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.
22
Important Lessons:
Summing Up
• 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
23
Agenda
Agenda
• What distinguishes “design economics” from other engineering
disciplines?
• What tools are available to aide in economic design?
• Lessons from past designs
24
Game Theory
• 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.
25
Game Theory
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?
26
Game Theory
• 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
27
Game Theory
• Mechanism (also called Institution)
– Actions: What can each person do?
– Mechanism: Rules that govern how individual actions translate into
outcomes.
28
Game Theory
• 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.
29
Game Theory Analogy
Environment:
 Players
 Types
 Information
 Utility (Desires)
Market Mechanism
 Actions
 Timing
Strategic Interaction
Outcome
?
30
Experiments
• 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.
31
Laboratory Experiments
32
Laboratory Experiments
Environment
(Controlled by Experimenter)
Strategic Interaction
(Experiments Test This)
Mechanism
(Controlled by
Experimenter)
Predicted Outcome
 Is the mechanism robust to
changes in the environment?
 Do people behave as predicted?
 Do alternative mechanisms work
better?
33
Structural Estimation
• 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.
34
Structural Estimation
Environment
Unknown Information
Strategic Interaction
Old
Mechanism
Outcome
Ex-Post Information
35
Computational Economics
• 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.
36
Computational Economics
• 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.
37
Market Design
Environment
(Details of the Problem being
analysed. Informed by empirics)
Strategic Interaction
(Tested with Experiments)
Mechanism
(Designed)
Goal:
Design the
mechanism so
It leads to the best
possible
outcomes
Outcome
 Efficiency
 Revenue
 Secondary Objectives
38
Tools of Economic Design:
Summing Up
• 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
39
Agenda
Agenda
• What distinguishes “design economics” from other engineering
disciplines?
• What tools are available to aide in economic design?
• Lessons from past designs
40
Lessons: Both Theory and
Empirical Testing are essential
• 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.
41
Lessons: Political support is key
• 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.
42
Lessons: Be aware of competing
objectives
• 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.
43
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.
44
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.
45
LECTURE 2:
INTRODUCTION TO
GAME THEORY
PRESENTED BY TOM WILKENING
18 SEPTEMBER 2014
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.
47
Agenda
• Decision Theory
• Simultaneous Move Games
• Sequential Move Games
48
Decision Theory
• 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.
49
Decision Theory: An Example
• 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.
50
Decision Theory: Definitions
• 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.
51
Decision Theory: Decision Trees
• 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).
52
Decision Theory
• This example can be modelled as a tree:
1
Work
Home
1
1
Meeting
10
Coffee
6
TV
2
New Job
7
• To solve, we use Backward Induction
53
Decision Theory: Decision Trees
• 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.
54
Decision Theory
• Random events are represented by nodes corresponding to a move by
chance (or Nature, denoted by a square N in the game tree).
1
Take Umbrella
Home
N
Rain: 0.4
3
N
Sun: 0.6
8
Rain: 0.4
-10
Sun: 0.6
10
• 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
55
Decision Theory: Examples
• 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
56
Decision Theory: Examples
• Suppose your value is $60. What should you bid?
Your value
Candidate Bid 1
$30
Candidate Bid 2
$80
57
Decision Theory: Examples
• 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!
Your value
win
lose
Candidate Bid 1
$30
• Extending the logic – it is never in your interest to bid below your value.
58
Decision Theory: Examples
• 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!
Your value
win
lose
Candidate Bid 2
$80
• Extending the logic – it is never in your interest to bid above your value.
59
Decision Theory: Examples
• 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
Decisions versus Games
• 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.
61
Games are everywhere
•
•
•
•
•
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
62
Game Theory
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?
63
Game Theory
• 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)
64
Game Theory
• Mechanism (also called Institution)
– Actions: What can each person do?
– Mechanism: Rules that govern how individual actions translate into
outcomes.
65
Game Theory
• 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
66
Information
• 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,
67
Static vs. Dynamic
• 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
68
The Simplest Games
• 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.
69
The Simplest Games
• 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
70
The Simplest Games
• 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
71
The Simplest Games
• 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.
1
Give 0
Give 5
20,0
15,15
Give 10
10,30
Give 15
Give 20
5,45
0,60
u1 (m1,m2), u2 (m1,m2)
72
The Simplest Games
• 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.
1
Give 0
Give 5
20,0
15,15
Give 10
10,30
Give 15
Give 20
5,45
0,60
73
The Simplest Games
• 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:
u1(m1,m2) = m1+ m2 = (20 – x) + 3x
• This just changes the terminal payoffs of the game
1
Give 0
Give 5
20,0
30,15
Give 10
40,30
Give 15
Give 20
50,45
60,60
74
Key Assumptions in Game Theory
• 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.
75
Strategies
• 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.
76
Strategies
• 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}.
77
Strategies
• 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
78
Equilibrium
• 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.
79
Agenda
• Decision Theory
• Simultaneous Move Games
• Sequential Move Games
80
Simultaneous Move Games
• 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
81
Game Tables
• 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)
82
Game Tables
• Example: Investment Game
Player 1
Player 2
Invest
Don ' t Invest
Invest
15,15
50,0
Don’t
Invest
0,50
30,30
83
Nash Equilibrium
• 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.
84
Finding Nash Equilibrium
• 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!
85
Finding NE: cell-by-cell inspection
• 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
Player 1
Player 2
Invest
Don ' t Invest
Invest
15,15
50,0
Don’t
Invest
0,50
30,30
86
Finding NE: cell-by-cell inspection
• 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
Player 1
Player 2
Invest
Don ' t Invest
Invest
15,15
50,0
Don’t
Invest
0,50
30,30
87
Finding NE: cell-by-cell inspection
• 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
Player 1
Player 2
Invest
Don ' t Invest
Invest
15,15
50,0
Don’t
Invest
0,50
30,30
88
Finding NE: Elimination of Dominant
Strategies
• 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.
Player 2
Invest
Don ' t Invest
Invest
15,15
50,0
Don’t
Invest
0,50
30,30
Player 1
Player 1
Player 2
Invest
Invest
Don ' t Invest
15,15
50,0
Don’t
Invest
89
Finding NE: Iterated Elimination of
Dominant Strategies
• We can continue to do this by now looking at Player 2’s actions. This is
known as iterated deletion.
Invest
Don’t
Invest
Invest
Don ' t Invest
15,15
50,0
Player 2
Invest
Player 1
Player 1
Player 2
Invest
15,15
Don’t
Invest
90
Don ' t Invest
Finding NE: Iterated Elimination of
Dominant Strategies
• 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.
Player 1
A
B Player 2 C
D
A
3,1
2,3
10,2
100,2
B
4,5
3,1
6,4
100,2
C
2,2
5,4
12,3
100,2
D
5,6
4,5
9,7
105,2
91
Finding NE: Iterated Elimination of
Dominant Strategies
• For player 2, C dominates A. We can thus eliminate A
Player 2
A
B
C
D
3,1
2,3
10,2
100,2
C
2,2
5,4
12,3
100,2
D
5,6
4,5
9,7
105,2
Player 1
A
B
92
Finding NE: Iterated Elimination of
Dominant Strategies
• For Player 2, B dominates B. We can thus also eliminate D
Player 2
A
B
C
D
2,3
10,2
100,2
C
5,4
12,3
100,2
D
4,5
9,7
105,2
Player 1
A
B
93
Finding NE: Iterated Elimination of
Dominant Strategies
• For player 1, C dominates both A and D. We can thus eliminate these
Player 2
A
B
C
2,3
10,2
C
5,4
12,3
D
4,5
9,7
Player 1
A
D
B
94
Finding NE: Iterated Elimination of
Dominant Strategies
• Finally, for Player 2, B dominates C. We thus can eliminate C
Player 2
A
B
C
5,4
12,3
D
Player 1
A
B
C
D
95
Finding NE: Iterated Elimination of
Dominant Strategies
• Thus (C,B) is the Nash Equilibrium
Player 2
A
B
C
D
Player 1
A
B
C
5,4
D
96
Finding NE: Best Response Analysis
• 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.
Player 2
Player 1
A
B
C
D
A
3,1
2,3
10,2
100,2
B
4,5
3,1
6,4
100,2
C
2,2
5,4
12,3
100,2
D
5,6
4,5
9,7
105,2
97
Finding NE: Best Response Analysis
• 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
Player 2
Player 1
A
B
C
D
A
3,1
2,3
10,2
100,2
B
4,5
3,1
6,4
100,2
C
2,2
5,4
12,3
100,2
D
5,6
4,5
9,7
105,2
98
Finding NE: Best Response Analysis
• Nash equilibrium are going to be cells where both individuals are best
responding to each other. Again this is (C,B)
Player 2
Player 1
A
B
C
D
A
3,1
2,3
10,2
100,2
B
4,5
3,1
6,4
100,2
C
2,2
5,4
12,3
100,2
D
5,6
4,5
9,7
105,2
99
Nash Equilibrium:
Summing Up
• 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!
100
Nash Equilibrium are not
Necessarily Unique
• 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!
Player 1
• Example: Coordination Games.
Telstra
Virgin
Player 2
Telstra
Virgin
15,5
0,0
0,0
5,15
101
Nash Equilibrium may not
Be in pure strategies
• Some games do not have a Nash Equilibrium in pure strategies
• Example: Penalty Kick in soccer
Player 1
Player 2
Right
Left
Left
1,0
0,1
Right
0,1
1,0
102
Nash Equilibrium may not
Be in pure strategies
• 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.
103
Agenda
• Decision Theory
• Simultaneous Move Games
• Sequential Move Games
104
Sequential Move Games
• 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.
105
Sequential Move Games
• Lets take the coordination game that we thought about in the last
section.
Player 2
Telstra
Virgin
Telstra
15,5
0,0
Virgin
0,0
5,15
106
Sequential Move Games
• 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
Sequential Move Games
• We could specify the new game as a normal form game:
Player 2
T ,T
T ,V
V ,T
V ,V
T
15,5
15,5
0,0
0,0
V
0,0
5,15
0,0
5,15
108
Sequential Move Games
• 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
Sequential Move Games
• 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
Sequential Move Games
• The decision tree for the coordination game is as follows:
1
Telstra
Virgin
2
2
Telstra
15,5
Virgin
0,0
Telstra
Virgin
0,0
5,15
111
Backward Induction
• 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.
112
Sequential Move Games
• We first solve these subgames from the perspective of player 2 (the
person making the move).
1
Telstra
Virgin
2
2
Telstra
15,5
Virgin
0,0
Telstra
Virgin
0,0
5,15
113
Sequential Move Games
• We first solve these subgames from the perspective of player 2 (the
person making the move).
1
Telstra
Virgin
2
2
Telstra
15,5
Virgin
0,0
Telstra
Virgin
0,0
5,15
114
Sequential Move Games
• We then solve the next layer of the game using these payoffs
1
Telstra
15,5
Virgin
5,15
115
Sequential Move Games
• The easiest way to do this is to just draw arrows in each tree starting
from the bottom and working our way up.
1
Telstra
Virgin
2
2
Telstra
15,5
Virgin
0,0
Telstra
Virgin
0,0
5,15
116
Backward Induction
• 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.
117
Sequential Games:
Summing Up
• 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.
118
Using Game Theory
• 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?
119
Using Game Theory
• 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?
120
Using Game Theory
• 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.
121
Using Game Theory
• 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?
122