An Introduction to Game Theory and
Basic Elements of Noncooperative Games
Carlos Hurtado
Department of Economics
University of Illinois at Urbana-Champaign
[email protected]
Junel 7th, 2016
C. Hurtado (UIUC - Economics)
Game Theory
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
Introduction
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
Introduction
What do we do in Economic?
I What makes a theoretical model ”economics” is that the concepts we are
analyzing are taken from real life.
I Through the investigation of these concepts, we indeed try to understand reality
better, and the models provide a language that enables us to think about
economic interactions in a systematic way.
I We do not view economic models as an attempt to describe exactly the world, or
to provide tools for predicting the future.
I Although we will be studying formal concepts and models, they will always be
given an interpretation. An economic model differs substantially from a purely
mathematical model in that it is a combination of a mathematical model and its
interpretation.
I The word ”model” sounds more scientific than ”fable” or ”fairy tale”, but there is
not much difference between them. The author of a fable draws a parallel to a
situation in real life and has some moral he wishes to impart to the reader.
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Introduction
What do we do in Economic?
I The fable is an imaginary situation that is somewhere between fantasy and reality.
Any fable can be dismissed as being unrealistic or simplistic, but this is also the
fable’s advantage.
I Being something between fantasy and reality, a fable is free of extraneous details
and annoying diversions. In this unencumbered state, we can clearly discern what
cannot always be seen from the real world.
I On our return to reality, we are in possession of some sound advice or a relevant
argument that can be used in the real world. We do exactly the same thing in
economic theory.
I Thus, a good model in economic theory, like a good fable, identifies a number of
themes and elucidates them. We perform thought exercises that are only loosely
connected to reality and have been stripped of most of their real-life characteristics.
I However, in a good model, as in a good fable, something significant remains.
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Introduction
As if Rationality
I Rationality forms the basis of decision-making in the neoclassical school.
I Decision-makers are optimizers, given the constraints they find themselves in.
I Rationality assumes that decision-makers maximize things that give them
happiness and minimize things that give them pain.
I Implications:
- Narrows down the set of possible outcomes.
- Rational man is a clever individual.
- Rationality helps to predict the outcome of an economic system.
- Once economic agents have optimized their utility and reached a situation
where they do not want deviate, the economic system reaches a stable
outcome: ’equilibrium’.
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Introduction
Perfect Competition is a Benchmark
I Assumes the existence of many buyers and many sellers.
I Decision making is made independently and individually.
- Decisions are not made in coalitions (together/jointly)
- Decisions are not inter-dependent: Decision of one agent is neither
influenced by another agent, nor does it influence that of another agent.
I Independent and individual decision-making under perfect competition implies each
decision-maker tries to do the best they can irrespective of what other
decision-makers are doing. (really?)
I Perfect competition is a theoretical extreme. Like the ideal human body
temperature of 98.4 degrees Fahrenheit it almost never exists. It is used as a
benchmark to explain deviations from this ’perfect’ world.
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What is Game Theory?
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
What is Game Theory?
What is Game Theory?
I Branch of applied mathematics and economics that studies strategic situations
where there are several players, with different goals, whose actions can affect one
another.
I A game is any situation where multiple players can affect the outcome, a player is
a stakeholder, a move or option is an action a player can take and, at the end of
the game, the payoff for each player is the outcome.
I The value of game theory lies in understanding the interactions and likely
outcomes when the end result is dependent on the actions of others who have
potentially conflicting motives.
I Game theory is mainly used in economics, political science, and psychology, as well
as logic, computer science, and biology.
I The subject first addressed zero-sum games, such that one person’s gains exactly
equal net losses of the other participant or participants. Today, game theory
applies to a wide range of behavioral relations: the study of decision science,
including both humans and non-humans.
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A Brief History of Game Theory
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
A Brief History of Game Theory
A Brief History of Game Theory
1713
In a letter dated 13 November 1713 Francis Waldegrave provided the first known,
minimax mixed strategy solution to a two-person game. Waldegrave wrote the
letter, about a two-person version of a card game, to Pierre-Remond de Montmort
who in turn wrote to Nicolas Bernoulli, including in his letter a discussion of the
Waldegrave solution.
1838
The French economist Antoine Augustine Cournot discussed a duopoly where the
two duopolists set their output based on residual demand.
1871
In the first edition of his book The Descent of Man, and Selection in Relation to
Sex Charles Darwin gives the first (implicitly) game theoretic argument in
evolutionary biology: If births of females are less common than males, females can
expect to have more offspring. Thus parents genetically disposed to produce
females tend to have more than the average numbers of grandchildren, hence,
female births become more common. As the 1:1 sex ratio is approached, the
advantage associated with producing females dies away.
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A Brief History of Game Theory
A Brief History of Game Theory
Intellectual debate between the French Mathematician Emile Borel (1871-1956)
and the Hungarian Mathematician John Von Neumann (1903-1957):
Do zero-sum games have a solution?
1921
Emile Borel published four notes on strategic games and an erratum to one of
them. Borel gave the first modern formulation of a mixed strategy along with
finding the minimax solution for two-person games with three or five possible
strategies. Initially he maintained that games with more possible strategies would
not have minimax solutions, but by 1927, he considered this an open question as
he had been unable to find a counterexample.
1928
John von Neumann proved the minimax theorem. It states that every two-person
zero-sum game with finitely many pure strategies for each player is determined, ie:
when mixed strategies are admitted, this variety of game has precisely one
individually rational payoff vector.
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A Brief History of Game Theory
A Brief History of Game Theory
1934
R.A. Fisher independently discovers Waldegrave’s solution to the card game. Fisher
reported his work in the paper Randomisation and an Old Enigma of Card Play.
Intellectual debate between Hungarian Mathematician John Von Neumann
(1903-1957) and he Austrian economist Oskar Morgenstern (1902-1977):
Can utility be quantified?
1944
Theory of Games and Economic Behavior by John von Neumann and Oskar
Morgenstern is published. As well as expounding two-person zero sum theory this
book is the seminal work in areas of game theory such as the notion of a
cooperative game, with transferable utility (TU), its coalitional form and its von
Neumann-Morgenstern stable sets. It was also the account of axiomatic utility
theory given here that led to its wide spread adoption within economics.
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A Brief History of Game Theory
A Brief History of Game Theory
Von Neumann and Morgenstern formally laid the foundations of Game Theory as a
branch of applied mathematics. However, their effort failed to generate stir for two
reasons.
1. The Theory of Games and Economic Behavior was based on the notion of
zero-sum games. These are known as ’Games of Conflict’ or ’Non-Cooperative
Games’. Most games in social sciences are non-zero-sum games.
2. It did not establish how equilibrium in games of interdependent decision-making
would arise. The world did not have to wait too long for this solution.
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A Brief History of Game Theory
A Brief History of Game Theory
1950
Contributions to the Theory of Games I, H. W. Kuhn and A. W. Tucker eds.,
published.
1950
In January 1950 Melvin Dresher and Merrill Flood identified a game where it is in
the best interest for players to cooperate, but individual self-interest invokes them
to not cooperate. The game reaches a bad equilibrium that is inferior to a superior
outcome that could have been reached and was available. This phenomenon was
canonized by Albert Tucker.
In the summer of 1950 Tucker was at Stanford University. He was working on a
problem in his room when a graduate student of psychology knocked and asked
what he was doing. The answer was short: game theory. ”Why don’t you explain
to us in a seminar”? Tucker used his now famous example of two thieves who were
put into separate cells and asked the same question by the judge. Tucker
christened the phenomenon as The Prisoners’ Dilemma.
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A Brief History of Game Theory
A Brief History of Game Theory
1953
Extensive form games allow the modeler to specify the exact order in which players
have to make their decisions and to formulate the assumptions about the
information possessed by the players in all stages of the game. In two papers,
Extensive Games (1950) and Extensive Games and the Problem of Information
(1953), H. W. Kuhn included the formulation of extensive form games which is
currently used, and also some basic theorems pertaining to this class of games.
1953
In four papers between 1950 and 1953 John Nash (1928 - 2015) made seminal
contributions to both non-cooperative game theory and to bargaining theory.
Note: Nash arrived at Princeton in the Fall of 1948 to start a PhD. He came to
Princeton with one of the shortest reference letters. His professor Richard Duffin
(1909-1996) wrote just one sentence: ”This man is a genius.”
Just eighteen months later, Nash submitted a 28 page doctoral dissertation. The
number 28 went on to become a superstitious number at Princeton. Von
Neumann solved the mini-max theorem in 1928. Nash was born in 1928. Nash’s
doctoral dissertation was 28 pages.
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A Brief History of Game Theory
A Brief History of Game Theory
1953
I
I
I
I
I
In two papers, Equilibrium Points in N-Person Games (1950) and Non-cooperative
Games (1951), Nash proved the existence of a strategic equilibrium for
non-cooperative games -the Nash equilibrium- and proposed the ”Nash program”,
in which he suggested approaching the study of cooperative games via their
reduction to non-cooperative form.
This was the missing link to Von Neumann and Morgenstern’s magnum opus!!
These two papers established equilibrium where all players calculate their best
strategy based on what they assume is the best strategy of the other players.
Every finite game must have at least one solution such that once reached, no
player within the game will have an incentive to deviate from their chosen actions,
given the actions of the other players in the game.
The intuition behind Nash’s equilibrium is very simple: If all rational economic
agents in a system are trying to do the best they can, assuming the others are
doing the same, the economic system must be in equilibrium such that no single
agent will want to unilaterally deviate from their position.
A Nash equilibrium does not necessarily imply that a game will reach a solution
that is the best possible solution for all players in the game.
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A Brief History of Game Theory
A Brief History of Game Theory
1951
George W. Brown described and discussed a simple iterative method for
approximating solutions of discrete zero-sum games in his paper Iterative Solutions
of Games by Fictitious Play.
1953
The notion of the Core as a general solution concept was developed by L. S.
Shapley (Rand Corporation research memorandum, Notes on the N-Person Game
III: Some Variants of the von-Neumann-Morgenstern Definition of Solution, RM817, 1952) and D. B. Gillies (Some Theorems on N-Person Games, Ph.D. thesis,
Department of Mathematics, Princeton University, 1953). The core is the set of
allocations that cannot be improved upon by any coalition.
50’s
Near the end of this decade came the first studies of repeated games. The main
result to appear at this time was the Folk Theorem. This states that the
equilibrium outcomes in an infinitely repeated game coincide with the feasible and
strongly individually rational outcomes of the one-shot game on which it is based.
Authorship of the theorem is obscure.
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The Theory of Rational Choice
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
The Theory of Rational Choice
The Theory of Rational Choice
I This theory is that a decision-maker chooses the best action according to her
preferences, among all the actions available to her.
I Her ”rationality” lies in the consistency of her decisions when faced with different
sets of available actions, not in the nature of her likes and dislikes.
Actions:
- A set A consisting of all the actions that, under some circumstances, are
available to the decision-maker.
- In any given situation the decision-maker is faced with a subset of A, from
which she must choose a single element.
- The decision-maker knows this subset of available choices, and takes it as
given; in particular, the subset is not influenced by the decision-maker’s
preferences.
- The set A could, for example, be the set of bundles of goods that the
decision-maker can possibly consume; given her income at any time, she is
restricted to choose from the subset of A containing the bundles she can
afford.
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The Theory of Rational Choice
The Theory of Rational Choice
Preferences and payoff functions:
- We assume that the decision-maker, when presented with any pair of actions,
knows which of the pair she prefers, or knows that she regards both actions
as equally desirable (is ”indifferent between the actions”).
- Rational Choice (Transitivity): We assume further that these preferences are
consistent in the sense that if the decision-maker prefers the action a to the
action b, and the action b to the action c, then she prefers the action a to
the action c.
- Note that we do not rule out the possibility that a person’s preferences are
altruistic in the sense that how much she likes an outcome depends on some
other person’s welfare.
- We can ”represent” the preferences by a payoff function, which associates a
number with each action in such a way that actions with higher numbers are
preferred.
a ≺ b ⇐⇒ u(a) < u(b)
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The Theory of Rational Choice
The Theory of Rational Choice
A Digression on Transitivity:
- Aggregation of considerations as a source of intransitivity.
a 1 b 1 c;b 2 c 2 a; c 3 a 3 b ⇒ a b c a
- The use of similarities as an obstacle to transitivity.
An individual may express indifference in a comparison between two elements
that are too ”close” to be distinguishable.
I The theory of rational choice is that in any given situation the decision-maker
chooses the member of the available subset of A that is best according to her
preferences.
I The action chosen by a decision-maker is at least as good, according to her
preferences, as every other available action.
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The Theory of Rational Choice
The Theory of Rational Choice
I Exercise 1. (Altruistic preferences) Person 1 cares both about her income and
about person 2’s income. Precisely, the value she attaches to each unit of her own
income is the same as the value she attaches to any two units of person 2’s
income. How do her preferences order the outcomes (1, 4), (2, 1), and (3, 0),
where the first component in each case is person 1’s income and the second
component is person 2’s income? Give a payoff function consistent with these
preferences. Is that payoff unique?
I Exercise 2. (Alternative representations of preferences) Adecision-maker’s
preferences over the set A = a, b, c are represented by the payoff function u
forwhich u(a) = 0, u(b) = 1, and u(c) = 4. Are they also represented by the
function v for which v(a) = -1, v(b) = 0, and v(c) = 2? How about the function
w for which w(a) = w(b) = 0 and w(c) = 8?
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The Theory of Rational Choice
The Theory of Rational Choice
I Exercise 3. Deffine x ∼ y ⇐⇒ x y and y x . Denote the set of acctions by
X . Define I(x ) to be the set of all y in X for which y ∼ x . Show that the set (of
sets!) {I(x )|x ∈ X } is a partition of X . That is:
1. For all x and y , either I(x ) = I(y ) or I(x ) ∩ I(y ) = ∅.
2. For every x ∈ X , there is y ∈ X such that x ∈ I(y ).
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The Theory of Rational Choice
The Theory of Rational Choice
I Exercise 3 (Ans.). Deffine x ∼ y ⇐⇒ x y and y x .
- That is: x ∼ y ⇐⇒ ¬x y or ¬y x
- Let I(x ) = {y ∈ X |x ∼ y }.
2. For every x ∈ X , it is clear that x ∼ x . Then, for every x ∈ X , there is at
least y = x ∈ X such that x ∈ I(y ).
1. Let x , y ∈ X and assume that I(x ) 6= I(y ) and I(x ) ∩ I(y ) 6= ∅
There exist a z ∈ X , such that z ∈ I(x ) and z ∈ I(z). That is, x ∼ z and
y ∼ z.
Hence, x ∼ z ∼ y .
Therefore, I(x ) = I(y ) and I(x ) 6= I(y ) (Contradiction)
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Noncooperative Games
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
Noncooperative Games
Noncooperative Games
I There are two leading frameworks for analyzing games: cooperative and
noncooperative.
I This course focuses on noncooperative game theory, which dominates applications.
I But even if not, you should be aware that cooperative game theory exists, and is
better suited to analyzing some economic settings, e.g. where the structure of the
game is unclear or unobservable, and it is desired to make predictions that are
robust to it.
I Cooperative game theory assumes rationality, unlimited communication, and
unlimited ability to make agreements.
I Its goal is to characterize the limits of the set of possible cooperative agreements
that might emerge from rational bargaining.
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Noncooperative Games
Noncooperative Games
I Noncooperative game theory also assumes rationality.
I Noncooperative game theory replaces cooperative game theory’s assumptions of
unlimited communication and ability to make agreements with a fully detailed
model of the situation and a detailed model of how rational players will behave in
it.
I Its goal is to use rationality, augmented by the ”rational expectations” notion of
Nash equilibrium, to predict or explain outcomes from the data of the situation.
I As a result, noncooperative game theory is used for normative purposes in some
applications, such as mechanism design.
I Some applications of noncooperative game theory involve predicting which settings
are better for fostering cooperation.
I This is done by making behavioral assumptions at the individual level
(”methodological individualism”), thereby requiring cooperation to emerge (if at
all) as the outcome of explicitly modeled, independent decisions by individuals in
response to explicitly modeled institutions.
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Noncooperative Games
Noncooperative Games
I In game theory, maintaining a clear distinction between the structure of a game
and behavioral assumptions about how players respond to it is analytically as
important as keeping preferences conceptually separate from feasibility in decision
theory.
I We will first develop a language to describe the structure of a noncooperative
game.
I We will then develop a language to describe assumptions about how players
behave in games, gradually refining the notion of what it means to make a rational
decision.
I In the process we will illustrate how game theory can elucidate questions in
economics.
I As you learn to describe the structure, please bear in mind that the goal is to give
the analyst enough information about the game to formalize the idea of a rational
decision. (This may help you be patient about not yet knowing exactly what it
means to be rational.)
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What is a Game?
On the Agenda
1
Introduction
2
What is Game Theory?
3
A Brief History of Game Theory
4
The Theory of Rational Choice
5
Noncooperative Games
6
What is a Game?
C. Hurtado (UIUC - Economics)
Game Theory
What is a Game?
What is a Game?
I From the noncooperative point of view, a game is a multi-person decision situation
defined by its structure, which includes:
- Players: Independent decision makers
- Rules: Which specify the order of players’ decisions, their feasible decisions
at each point they are called upon to make one, and the information they
have at such points.
- Outcome: How players’ decisions jointly determine the physical outcome.
- Preferences: players’ preferences over outcomes.
I Assume that the numbers of players, feasible decisions, and time periods are finite.
I These can be relaxed, and it will be relaxed for decisions and time periods.
I Preferences over outcomes are modeled just as in decision theory.
I Preferences can be extended to handle shared uncertainty about how players’
decisions determine the outcome as in decision theory, by assigning von
Neumann-Morgenstern utilities, or payoffs, to outcomes and assuming that players
maximize expected payoff.
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What is a Game?
What is a Game?
I Assume for now that players face no uncertainty about the structure other than
shared uncertainty about how their decisions determine the outcome, that players
know that no player faces any other uncertainty, that players know that they know,
and so on; i.e. that the structure is common knowledge.
I It is essential that a player’s decisions be feasible independent of others’ decisions;
e.g. ”wrestle with player 2” may be not a well-defined decision, although ”try to
wrestle with player 2” can be well-defined.
I It is essential that specifying all of each player’s decisions should completely
determine an outcome (or at least a shared probability distribution over outcomes)
in the game.
I If a specification of the structure of a game does not pass these tests, it must be
modified until it does.
I Example: If you object to a game analysis is that players are not really required to
participate in the game, the (only!) remedy is to explicitly add a player’s decision
whether to participate to the game, and then to insist that it be explained by the
same principles of behavior the analysis uses to explain players’ other decisions.
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What is a Game?
Examples
I Matching Pennies (version A).
Players: There are two players, denoted 1 and 2.
Rules: Each player simultaneously puts a penny down, either heads up or tails up.
Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,
player 2 pays 1 dollar to player 1.
I Matching Pennies (version B).
Players: There are two players, denoted 1 and 2.
Rules: Player 1 puts a penny down, either heads up or tails up. Then, Player 2
puts a penny down, either heads up or tails up.
Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,
player 2 pays 1 dollar to player 1.
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What is a Game?
Examples
I Matching Pennies (version C).
Players: There are two players, denoted 1 and 2.
Rules: Player 1 puts a penny down, either heads up or tails up, without letting
player 2 know his decision. Player 2 puts a penny down, either heads up or tails up.
Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,
player 2 pays 1 dollar to player 1.
I Matching Pennies (version D).
Players: There are two players, denoted 1 and 2.
Rules: Players flip a fair coin to decide who begins. The looser puts a penny down,
either heads up or tails up. Then, the winner puts a penny down, either heads up
or tails up.
Outcomes: If the two pennies match, the looser pays 1 dollar to player 2;
otherwise, the winner pays 1 dollar to player 1.
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