Transcript EITM 2003 – TM Primer

Introduction to Formal Modeling
John Aldrich & Arthur Lupia
For EITM 2007
Outline
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The purpose of modeling
Structure
Relevance
Controversy
Elements of Logic
Your research design problem
 Where are they?
 Who is your target audience?
 What factual premises/truth claims will they accept.
 Where do they want to be?
 Which alternate conclusion will benefit them?
 What burden of proof and standard of evidence do they
impose?
Definitions
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Theory
Formal Theory
Rational Choice Theory
Rationality
Game Theory
 Cooperative
 Non-Cooperative*
Arguments
 The currency of scientific communication.
 The components of an argument are:
 The Conclusion
 The Premises
 Value comes from explaining as much as
possible with as little as possible.
Introduction to Logic
 Premise
 Conclusion
 Logical Validity
 Deductive
 Inductive
 Invalid
 Soundness
Introduction to Logic
 Logical Validity

Deductive



Inductive



If all of the premises are true, then the conclusion must be true.
The logical connection between premises and conclusion is one of necessity.
If all of the premises are true, then the conclusion may be true.
The logical connection between premises and conclusion is one of
possibility.
Invalid


If all of the premises are true, then the conclusion must be false.
The logical connection between premises and conclusion is one of
impossibility.
Examples
 George W. Bush is a man.
 George W. Bush is over 5’ 11’’ tall.
 All men who are over 5’ 11’’ tall are the
president.
 Therefore, George W. Bush is the
president.
Examples
 George W. Bush is a man.
 George W. Bush is over 5’ 11’’ tall.
 Some men who are over 5’ 11’’ tall are the
president.
 Therefore, George W. Bush is the
president.
Examples
 George W. Bush is a man.
 George W. Bush is over 5’ 11’’ tall.
 If a man is over 5’ 11’’ tall, then he is not
the president.
 Therefore, George W. Bush is the
president.
Examples
 Almost any random premise 1
 Almost any random premise 2
 Therefore, George W. Bush is the
president.
The value of logic in debate
 How to cast doubt on the reliability of a conclusion when an
argument has the following logical properties:
 Invalid:
 Reveal the logical relationship.
 Inductively valid:
 show that even if the premises are true the conclusion can be
false or demonstrate that one or more of the premises is
untrue.
 Deductively valid:
 demonstrate that one or more of the premises is untrue.
Standards for another time
 Soundness
 Waller (p 20), The argument “must be [deductively]
valid and all of its premises must actually be true.”
 Reliability
 Waller (p 21). “[A]n inductive argument with all true
premises, and whose premises strongly support its
conclsion ,will be a reliable inductive argument.”
 These standards are more subjective.
Logical Fallacy: Denying the Antecedent
 If it's raining, then the streets are wet.
 It isn't raining.
 Therefore, the streets aren't wet.
Logical Fallacy: Affirming the
Consequent
 If it's raining then the streets are wet.
 The streets are wet.
 Therefore, it's raining.
Logical Fallacy: Commutation of
Conditionals
 If James was a bachelor, then he was
unmarried.
 Therefore, if James was unmarried, then
he was a bachelor.
The two faces of “or”
 Most logic texts claim that "or" has two meanings:
 Inclusive (or "weak") disjunction: One or both of the disjuncts
is true, which is what is meant by the "and/or" of legalese.
 Exclusive (or "strong") disjunction: Exactly one of the
disjuncts is true.
 Example
 Today is Saturday or Sunday.
Today is Saturday.
Therefore, today is not Sunday
 Suppressed premise: Saturday is not Sunday.
Logical Fallacy: Denying a Conjunct
 It isn't both sunny and overcast.
 It isn't sunny.
 Therefore, it's overcast.
 Not both p and q.
 Not p.
 Therefore, not q.
Ockham’s Razor (14th c.)
 Arguments are most helpful to an audience
the extent that they actually bring clarity to
the phenomena you're studying.
 lex parsimoniae  entities should not be
multiplied beyond necessity
 In many cases, less is more.
What will you choose?
 All political scientists make assumptions about:
 Players, Actions, Strategies, Information, Beliefs,
Outcomes, Payoffs, and
 Method of inference (e.g., “I know it when I see it,” path
dependence, Nash Equilibrium, logit plus LLN).
 Some state their assumptions more precisely than
others.
 Conclusions depend on assumptions.
Questions to Ask Before You Begin
 Is the process static or dynamic?
 Do the actors have complete or incomplete
information?
 Are there many actors or just a few?
 How have others modeled the political
phenomena you're studying?
Starting to write down a formal
argument about politics
 Make it as simple as possible to start:
 Use discrete instead of continuous parameter spaces
 Use two actors instead of a zillion
 Use complete information rather than incomplete information
 Deriving Conclusions
 Uniqueness versus existence
 Interpreting Your Findings
Cooperative Game Theory
 Originally dominant, possibly the future?
Three Forms of Games
 Extensive Form
 Normal form
 Characteristic Function Form
 These differed by level of detail and generality
(most detail to most general)
 Before the “noncooperative revolution” game
theory was based on modeling based on
preferences only (not on probabilities a la
Harsanyi or on information)
Extensive Form
 Has the most details of moves
 Many of the central results developed in
the 1940s and 1950s (including what came
to be called sub game perfection and the
folk theorem)
 It was thought to be practically impossible
for studying any realistic setting – too few
moves, too few players
Normal (Matrix) Form
 Became the most used early (in both cooperative
and non-cooperative)
 Abstracted details of moves (and information
sets) to strategies, which seemed like a small loss
of detail
 A few results on one-to-one mappings between
normal form and extensive form equilibrium
 Served as the locus for the “debate” between
Nash and Von Neumann types.
Coalitions and
Characteristic Function Form
 The perceived weakness of both extensive and normal forms
is that each was unmanageable with any realistic number of
moves and players
 It was also thought to have ignored one of the most important
aspects of settings amenable to game theory – coalitions
 Political Science – Riker and minimal winning coalitions was a
huge literature
 Economics – Shubik used a cooperative solution concept, the
core, to demonstrate that the general equilibrium of ArrowDebreu was also the core of the game in economics terms
(see below)
 The first formal type publication in Political Science was the
Shapley- Shubik “value” applied to such as Congress and the
UN
N.B.: Shapely and Shubik were graduate students of Morganstern and of Nash
Payoff Configurations, Imputations, and
Characteristic Functions
 Payoff configuration – vector of utilities for any
given outcome.
 Imputations – Payoff configurations that are
 Individually rational, i.e., give every one their security
level 9what they can get playing rationally and alone)
 Pareto Optimal
 Characteristic Function – a form that assigns a
real-valued payoff to each coalition.
Coalitions and the Core, Value,
Bargaining Set
 Core –
 One definition is that the core of the game is the set of
imputations that give each coalition their security level
 Core is general eq. in economics;
 R. Wilson – Arrow’s theorem is the “observations that
the core of a voting game is generally empty”
 Is the eq. concept in use in most spatial modeling (of
the Davis-Hinich-Ordeshook-McKelvey sort)
Shapley Value
 Shapley-Shubik Power Index – Measures
“value” based on the incidence with which
a member or coalition is “pivotal” that is, by
its inclusion, changes a coalition from
losing to winning.
 Shapley Value is the more general version
looking at the marginal increase in payoff to
a coalition.
Bargaining Set and Competitive Solution
 A large class of games that provides a
solution to the game by which is meant a
coalition structure and a payoff
configuration
Win Set
 Perhaps the most important of the cooperative
style solution concepts still in use (social choice
theory, in general, has been approached as
cooperative game setting, with the win set one
concept used)
 Win set is defined (for a majority rule game) as
W(x) consists of the set of points that defeat x by
majority rule. If there exists a point, say, y for
which the win set is empty, it is a preference
induced equilibrium, according to Shepsle, and is
a point in the core.
Non-Cooperative Game Theory
The Dominant Form in Recent Decades
NC Game Theory Fundamentals
 Player goals are represented by utility functions with utility
defined over outcomes.
 Actions and Strategies
 A strategy is a plan of action.
 In games that can be modeled as if they are simultaneous,
actions and strategies are equivalent.
 In other games, strategies and actions are quite different with
strategies being the primary choice of interest.
 The combination of actions by all players determines a
payoff for each player.
Normal Form Games
A normal form game
•By convention, the payoff to the so-called row player is the
first payoff given, followed by the payoff to the column
player.
Graduate School Student B
Study
Loaf
Study
100,100
50,0
Loaf
0, 50
-10, -10
Graduate School Student A
Practical Description
 The normal form representation of a game specifies:
 The players in the game.
 The strategies available to each player.
 The payoff received by each player for each combination of
strategies that could be chosen by the players.
 Actions are modeled as if they are chosen simultaneously.
 The players need not really choose simultaneously, it is
sufficient that they act without knowing each others’ choices.
Components of a Normal Form
Game





Players
Actions
Strategies.
Information.
Outcomes.
A small number.
Define columns and rows.
Define columns and rows.
Complete.
Represented by vectors in
cells.
 Payoffs.
Elements of the vectors.
 Equilibrium concept. Nash.
Technical Definition




1 to n: players in an n-player game.
Si: player i’s strategy set.
Si: an arbitrary element of Si.
ui(si): player i’s payoff function.
 Definition: The normal-form representation of an
n-player game specifies the players’ strategy
spaces S1,…,Sn and their payoff functions
u1,…,un.
 We denote the game by G={S1,…Sn;u1,…un}.
Nash Equilibrium

For an equilibrium prediction to be correct, it is necessary that each
player be willing to choose the strategy described in the equilibrium.

Equilibrium represents the outcome of mutual and joint adaptation to
shared circumstances.

If the theory offers strategies that are not a Nash equilibrium, then at
least one player will have an incentive to deviate from the theory’s
prediction, so the theory will be falsified by the actual play of the
game.
Technical Definition
 In the n-player normal-form game G={S1,…Sn; u1,…un}, the
strategies (s1*,…sn*) are a Nash equilibrium if, for each player i,




s*i is (at least tied for) player i’s best response
to the strategies specified for the n-1 other players, (s*1,…s*i-1,
s*i+1,…s*n): ui(s*1,…s*i-1,s*i, s*i+1,…s*n)≥ ui(s*1,…s*i-1, si, s*i+1,…s*n)
for every feasible strategy si in Si;
that is, s*i solves max si Si ui(s*1,…s*i-1, si, s*i+1,…s*n).
 If the situation is modeled accurately, NE represent social
outcomes that are self-enforcing.
 Any outcome that is not a NE can be accomplished only by
application of an external mechanism.
Elimination of dominated strategies
Left
Middle
Right
Up
1,0
1,2
0,1
Down
0,3
0,1
2,0
Figure 1.1.1. Iterated domination produces a solution.
Left
Middle
Right
Top
0,4
4,0
5,3
Middle
4,0
0,4
5,3
Bottom
3,5
3,5
6,6
Figure 1.1.4. Iterated elimination produces no solution.
Requirements for Iterated
Domination
 To apply the process for an arbitrary number of steps, we
must assume that it is common knowledge that the players
are rational.
 We need to assume not only that all the players are rational,
but also that all the players know that all the players are
rational, and that all the players know that all the players know
that all the players are rational, and so on, ad infinitum.
 In the many cases where there is no or few strictly dominated
strategies, the process produces very imprecise predictions.
Example 1: A game with a dominated strategy.
Left
Right
Top
8, 10
-100, 9
Bottom
7, 6
6, 5
Example 2: A more complicated game: with dominated strategies.
Left
Middle
Right
Top
4, 3
5, 1
6, 2
Middle
2, 1
8, 4
3, 6
Bottom
3, 0
9, 6
2, 8
NE: Fun facts
 If iterated elimination of dominated strategies eliminates all
but one strategy for each player, then these strategies are
the unique NE.
 There can be strategies that survive iterated elimination of
strictly dominated strategies but are not part of any Nash
equilibrium.
 A game can have multiple Nash equilibria. The precision of
its predictive power at such moments lessens.
 Existence versus uniqueness
Solving for MS-NE



Row chooses “top” with probability p and bottom with probability 1-p.
Column chooses “left” with probability q and “right” with probability 1-q.
Right
Top
4, -4
1, -1
Bottom
2, -2
3, -3
Players choose strategies to make the other indifferent.



Left
4q+1(1-q)=2q+3(1-q)
-4p-2(1-p)=-1p-3(1-p)
The MS-NE is: p=.25, q=.5.

The expected value of either Row strategy is 2.5 and of either Column strategy is –2.5
Mixed strategy NE
 A mixed strategy Nash Equilibrium does not rely on an player
flipping coins, rolling, dice or otherwise choosing a strategy at
random.
 Rather, we interpret player j’s mixed strategy as a statement of
player i’s uncertainty about player j’s choice of a pure strategy.
 Individual components of mixed strategy equilibriums are chosen
to make the other player indifferent between all of their mixed
strategies.

To do otherwise is to give others the ability to benefit at your
expense. Information provided to another player that makes them
better off makes you worse off.
The last word.
 Theorem (Nash 1950): In the n-player
normal-form game G={S1,…Sn; u1,…un}, if
n is finite and Si is finite for every i then
there exists at least one Nash Equilibrium,
possibly involving mixed strategies.
Extensive Form Games
Extensive Form Games

Allows dynamic games – player moves can be treated as sequential
as well as simultaneous.

Complete information – games in which all aspects of the structure
of the game –including player payoff functions -- is common
knowledge.

Perfect information – at each move in the game the player with the
move knows the full history of the play of the game thus far.

Imperfect information – at some move the player with the move does
not know the history of the game.
The structure of a simple game of
complete and perfect information.
1.
2.
3.
Player 1 chooses an action a1 from the feasible set A1.
Player 2 observes a1 and then chooses a2 from the
feasible set A2.
Payoffs are u1(a1, a2) and u2(a1, a2).
1. Moves occur in sequence, all previous moves are observed,
player payoffs from each move combination are common
knowledge.
2. We solve such games by backwards induction.
The central issue is credibility.
Example 1 Here
3 legislators
Choices: Yes, No
Outcomes: Pass, Not.
Conceptual Advantage
 The central issue in all dynamic games is
credibility.
 Backwards induction outcomes.
 Subgame perfect outcomes.
 Repeated games – the main theme:
credible threats and promises about future
behavior can influence current behavior.
Structure of a EF Game

The structure of a simple game of complete and perfect information.

Player 1 chooses an action a1 from the feasible set A1.

Player 2 observes a1 and then chooses a2 from the feasible set A2.

Payoffs are u1(a1, a2) and u2(a1, a2).

Moves occur in sequence, all previous moves are observed, player payoffs from each
move combination are common knowledge.

We solve such games by backwards induction.
Backwards Induction
 At the second stage of the game, 2 faces the following problem,
given the previously chosen action a1, maxa2A2 u2(a1, a2).
 Assume for each a1A1, player 2’s optimization problem has a
unique solution denoted by R2(a1).
 Since player 1 can solve player 2’s problem as well as 2 can,
player 1 should anticipate player 2’s reaction to each action a1
that 1 might take, so 1’s problem at the first stage amounts to
maxa1A1 u1(a1, R2(a1)).
 (a*1, R2(a*1)) is the backward induction outcome of this game.
 Implies sophisticated rather than sincere behavior.
 The sequence of action can affect equilibrium strategies.
Requirements for BI
 The prediction depends on players knowing and reacting to what
would happen if the game was not played as the equilibrium
describes.
o
We must assume that decision makes are interested in and
capable of counterfactual reasoning.
o
o
In some cases, the amount of counterfactual reasoning required is
quite substantial.
If people reason “as if” they undertake such calculations, then the
theory’s validity is not imperiled.
 When can we assume that people are, or act as if they are,
capable of thinking through counterfactuals?
Subgame Perfect NE
A NE is subgame perfect if players’ strategies
constitute a Nash Equilibrium in every subgame.
 Player 1 chooses action a1 from feasible set A1.
 Player 2 observes a1 and then chooses action a2
from feasible set A2.
 Player 3 observes a1 and a2 and then chooses
action a3 from feasible set A3.
 Payoffs are ui(a1,a2,a3) for i=1,….,3.
 (a1, a2*(a1), a3*(a1, a2)) is the subgame-perfect
outcome of this two-stage game.
Example 2
 S-PNE on a Voting Tree. (Agenda: abcde)
 TYPE 1 D A B C E
 TYPE 2 A B C E D
 TYPE 3 C B E D A
 TYPE E e D A C B
Repeated Games
Repetitive Play Over Time &
The Folk Theorem
A general result.
 Definition: Given a stage game G, let G(T) denote the finitely
repeated game in which G is played T times, with the
outcomes of all preceding plays observed before the next play
begins. The payoff for G(T) are simply the sum of the payoffs
from the T stage games.
 Proposition: If the stage game G has a unique NE then, for
any finite T, the repeated game G(T) has a unique subgame
perfect outcome: the NE of G is played in every stage.
Cooperation from Repetition?
 Proposition; If G={A1,…An;u1,…un} is a static game of
complete information with multiple NE then there may be
subgame perfect outcomes of the repeated game G(T) in
which, for any t<T, the outcome in stage T is not a Nash
equilibrium of G.
The prisoners’ dilemma with one action added for each player.
Defect
Cooperate
Right
Defect
1, 1
5, 0
0, 0
Cooperate
0, 5
4, 4
0, 0
Bottom
0, 0
0, 0
3, 3
Defect
Cooperate
Right
Defect
1, 1
5, 0
0, 0
Cooperate
0, 5
4, 4
0, 0
Bottom
0, 0
0, 0
3, 3
o
o
Suppose players anticipate that (Bottom, Right) will be the second
stage outcome if the first stage outcome is (Cooperate, Cooperate),
but that (Defect, Left) will be the second-stage outcome otherwise.
The players, first stage interaction then amounts to the following
one-shot game:
Defect
Cooperate
Right
Defect
2, 2
6, 1
1, 1
Cooperate
1, 6
7, 7
1, 1
Bottom
1, 1
1, 1
4, 4
Implications
 Insights from one-shot games do not automatically transfer
to repeated interactions.
 Repeated games require special assumptions about time.
 Credible threats or promises about future behavior can
influence current behavior.
 For some situations, subgame perfection may not embody
a strong enough definition of credibility.
The Folk Theorem
 Let G be a finite, static game of complete information. Let
(e1,…en) denote the payoffs from a NE of G, and let
(x1,…xn) denote any other feasible payoffs from G. If xi>ei
for every player i and if  is sufficiently close to one, then
there exists a subgame-perfect NE of the infinitely
repeated game G(,) that achieves (x1,…xn) as the
average payoff.



Insights from one-shot games do not automatically transfer to
repeated interactions.
Repeated games require special assumptions about time.
Credible threats or promises about future behavior can influence
current behavior.
EF Games of Incomplete
Information
Reconciling Actions
and Uncertain Beliefs
“Hey, baby,… what’s your type?”
Key concepts

In a game of incomplete
information at least one player is
uncertain about another’s payoff
function.

i’s payoff function is ui(a1,…an;ti)
where ti is called player i’s type
and belongs to a set of possible
types.

Each type ti corresponds to a
different payoff function that i
might have.

t-i denotes others’ types and p(ti|ti) denote i’s belief about them
given ti.
Strategy
 In the game G={A1,…,An;T1,…tn; p1,…,pn;u1,…,un}, a strategy for i
is a function si(ti), where for each type ti  Ti, si(ti) specifies the
action from the feasible set Ai, that type ti would choose if drawn
by nature.
 Separating strategy: each type ti  Ti chooses a different action ai
 Ai.
 Pooling strategy, all types choose the same action.
 When deciding what to do, player i will have to think about what
he or she would have done if each of the other types in Ti had
been drawn.
Standard Assumptions
 It it is common knowledge that nature
draws a type vector t=(t1,…tn) according to
the prior probability distribution p(t).
 Each player’s type is the result of an
independent draw.
 Players are capable of Bayesian updating.
Bayes’ Theorem

A: state of the world. B: event.

Conditional probability p(B|A), is the likelihood of B given A.

We use Bayes’ Theorem to deduce the conditional probabilities of A given B.

Bayes Theorem. If (Ai)i=1,…,n is the set of states of the world and B is an
event, then p(Ai|B)=

Know:


The prior belief is p(A)
The posterior belief is p(A|B).
p ( Ai ) p ( B | Ai )
n

i 1
p ( Ai ) p ( B | Ai )
Sequential Rationality
 A pair of beliefs and strategies is sequentially
rational iff from each information set, the moving
player’s strategy maximizes its expected utility for
the remainder of the game given its beliefs and all
players’ strategies.
 Sequential rationality allows a process akin to
backwards induction on games of incomplete
information.
Perfect Bayesian Equilibrium
 A perfect Bayesian equilibrium is a belief-strategy
pairing such that
 the strategies are sequentially rational given the beliefs
 and the beliefs are calculated from the equilibrium
strategies by Bayes’ Theorem whenever possible.
 A defection from the equilibrium path does not
increase the chance that others will play
“irrationally.”
 Every finite n-person game has at least one
perfect Bayesian equilibrium in mixed strategies.
Draw a simple signaling game
 Given the receiver’s response, is the signal utility
maximizing for type 1?
 Given the receiver’s response, is the signal utility
maximizing for type 2?
 Given the sender’s strategy, does the response to
L maximize expected utility?
 Given the sender’s strategy, does the response to
R maximize expected utility?
 If a signal is off the equilibrium path, do there
exist off-the-path beliefs that can sustain the
equilibrium?
t1
t2
AR|L
AR |R
L
L
L
L
u
u
L
L
R
L
L
R
R
R
R
R
R
R
PBE?
t1
t2
AR |L
AR |R
u
d
R
R
L
L
u
u
u
d
d
d
u
u
d
u
R
R
L
L
L
R
d
d
u
u
d
u
u
d
d
d
u
d
L
L
L
R
R
R
u
d
d
d
u
d
PBE?
Requirements for PBE in ExtensiveForm Games
 An information set is on the equilibrium path if it
will be reached with positive probability  the
game is played according to the equilibrium
strategies.
 On the equilibrium path, Bayes’ Rule and
equilibrium strategies determine beliefs.
 Off the equilibrium path, Bayes’ Rule and
equilibrium strategies determine beliefs where
possible.
Implications
 In a PBE, players cannot threaten to play
strategies that are strictly dominated
beginning at any information set off the
equilibrium path.
 A single pass working backwards through
the tree (typically) will not suffice to
compute a PBE.
Morrow, Table 7.1
Concept
Replies
judged
Key
comparison
Beliefs
used?
Beliefs off
the equ
path?
Nash
Along the equ
path
Complete
Strategies
No
Irrelevant
Subgame
Perfect
In proper
subgames
Strategies within
proper subgames
No
Irrelevant
Perfect
Bayesian
At all
information
sets
Seq. Rationality
at all Info sets
Yes
Can be chosen.
Perfect
At all
information
sets
Against trembles.
No Weak. Dom.
S.
No
Irrelevant