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Page 1: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game Theory

Wolfgang Frimmel

Introduction

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Page 2: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Organisational details - course

Lectures: Tuesdays, 13.45-15.15, K 269D

Material: www.econ.jku.at/frimmel

Literature:Robert Gibbons, A Primer in Game TheoryMartin Osborne, An Introduction to Game Theory (several copiesare available in the library)Muhamet Yildiz, Economic Applications of Game Theory, MITOpen Courseware (No. 14.12)Martin Osborne and Ariel Rubinstein, A Course in Game Theory(freely available at Rubinstein’s webpage)More advanced: Roger Myerson, Game Theory - Analysis ofConflict ; Drew Fudenberg and Jean Tirole, Game Theory

Course requirements:50% of possible exam points, counting for 95% of the gradeNo mandatory attendance, but up to 5% of the grade will becredited for attendanceExtra credits for active class participation

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Page 3: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Organisational details - Intensifying course

Intensifying course should be taken parallel or after the course.It is not useful to attend the intensifying course without the course!

Course requirements:

Problem sets for each unit

You prepare problem sets in advance and indicate which examplesyou have prepared – you need minimum 50% of feasible points

You receive extra credits for every additional point and for goodvoluntary presentations of solutions

We discuss the solutions to the problem sets in class

2 short tests during the semester – you need minimum 50% offeasible points

Grading is the sum of test scores and extra credits

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Page 4: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

What is Game Theory?

Game

A game is a description of strategic situations

Game Theory...

... is the study of multiperson decision problems (R. Gibbons)

... analyzes strategic behavior of individuals → analytical andnormative

... focuses on the structure of a strategic situation withoutdetailed description

... is the study of models of conflict and cooperation betweenrational decision-makers (R. Myerson)

We want to understand situations in which decision-makers interact

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Page 5: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game theory in economics

Firms competing for business

Mergers & acquisitions (corporate takeover bids,...)

Bidders competing in auctions (when should you bid in Ebay)

Bargaining (wages, contracts...)

Provision of public goods (tragedy of the commons,...)

Bi- or multilateral agreements (Greek bailout, TTIP,...)

Tax evasion

Finance (dividends as signals, stock market investments, ECBinterest policy,...)

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Page 6: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game theory in political science

Political candidates competing for votes (see Austrian elections,...)

Negotiations for government coalitions (e.g. grand coalition vs.Jamaica in Germany)

Bi-/Multilateral negotiations and agreements (EU-Turkey, EastUkraine, Iran Nuclear deal, Brexit...)

Allocation of refugees

Human trafficking in the Mediterranean

State attorneys vs. lawyers

Lobbying and legislator’s voting behaviour

Military conflicts (Cold War, North Korea,...)

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Page 7: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game theory and individual behavior

Team work (procrastination in group work at universities,housekeeping in shared appartments,...)

Criminal behavior

Network formation

Dating (Parship, Tinder,...)

Divorce

Parenting

How to avoid traffic jams?

Why are public toilets often dirty?

Why should you not learn French but only English?

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Page 8: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game theory in other disciplines

Biology

Fight over prey

Territorial fights

Darwinian competition (Evolutionary game theory)

Sports

Basically any sports that includes tactical elements

e.g. American football, soccer, Formula One, chess,...

Computer science

e.g. important for algorithms, modeling networks,...

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Page 9: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

The Godfathers of Game Theory

John von Neumann and Oskar Morgenstern published theirseminal book ”Theory of Games and Economic Behavior” (1944)

John Nash (Nobel Prize 1994) offered a generally applicablesolution concept for non-cooperative games (1950)

Reinhard Selten (Nobel Prize 1994) introduced the solutionconcept for subgame perfect equilibria (dynamic games) (1965)

John Harsanyi (Nobel Prize 1994) developed concepts forincomplete (Bayesian) games (1967)

L. Hurwicz, E. Maskin and R. Myerson were awarded theNobel prize for their foundations in mechanism design (2007)

In total, 11 game theorists were awarded Nobel prize (latestto Jean Tirole in 2014)

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Page 10: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Non-cooperative game theory

Timing of moves

Static games: All decision-makers decide simultaneously and move onceand at the same time

Dynamic games: Decision-makers move sequentially

Degree of information

In games of complete information, decision-makers have all relevantinformation about each other and about payoffs.

In games of incomplete information, decision-makers do not knowother agents’ characteristics, or past actions

Pure vs. mixed strategies

In pure strategies, decision-makers make their choices with certainty.

In mixed strategies, decision-makers randomize their choices.

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Page 11: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Non-cooperative game theory

What will happen if two (or more) rational players, who are aware ofthe structure of a game and other’s rationality, interact?

Two solution techniques to get prediction:

Dominance concepts: No strategic interaction

Equilibrium concepts: Strategic interaction

Static Dynamic(Simultaneous) (Sequential)

Games Games

Complete Nash Subgame perfectinformation equilibrium equilibrium

Incomplete Bayesian Perfect Bayesianinformation equilibrium equilibrium

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Page 12: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Outline of the course

1 Representation of games

2 Dominance

3 Static games of complete information:Pure strategy Nash equilibrium (NE)Mixed stategy NEApplications

4 Dynamic games of complete information:Subgame perfect NEApplications

5 Repeated games

6 Static games of incomplete Information - Bayesian Equilibrium

7 Dynamic games of incomplete Information - Perfect BayesianEquilibrium

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Page 13: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Game Theory

Wolfgang Frimmel

Theory of rational choice

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Page 14: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

How decisions are made

A decision-maker chooses the best action based on her preferencesamong all action available to her

Actions

a set A consists of all actions that are available to thedecision-maker

In any situation, the decision-maker is faced with a known subsetof A, from which she must choose one single action.

The subset is not influenced by decision-maker’s preferences

e.g. A could be a bundle of all goods available, where the subsetconstitutes a smaller set of affordable goods

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Page 15: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Theory of rational choice

Preferences

A decision-maker knows, when deciding between any pair ofactions, which one she prefers or when she is indifferent.

Preferences are assumed to be complete iff for any a,b ∈ A, eithera�b or b�a, and transitive, iff for any a,b,c ∈ A, a � b and b � cthen a � c

Given any preference relation �, we define strict preference � bya � b, and indifference ∼ by a ∼ b

Preferences are represented by a payoff function, which associatesa number to each action such, that actions with higher numbersare preferred: u(a) > u(b) iff a � b

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Page 16: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Theory of rational choice

A payoff function could be i.e. a utility function

Preferences can be represented by many different payoff functions;any increasing function of u represents the same preferences.

The decision-maker chooses an action of the available subset ofA that is best according to her preferences, which are expressedby her payoff function u.

The theory of rational choice also implies that the action chosenby a decision-maker is at least as good, according to herpreferences, as every other available action

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Page 17: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Decision-making under uncertainty

Consider a finite set A of prizes, and let P be the set of allprobability distributions p : A→ [0, 1] on A, where

∑a∈A

p(a) = 1

These probability distributions are called lotteries

Under certain conditions, a player’s beliefs can be represented by aprobability distribution (Savage, 1954)

Expected utility theory

A preference relation � on A is represented by avon Neumann-Morgenstern utility function u : A→ R iff

p � q ⇔ U (p) ≡∑a∈A

u(a)p(a) ≥∑a∈A

u(a)q(a) ≡ U (q)

for each p, q ∈ P

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Page 18: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Modeling strategic situations

In most real-world situations, those probabilities are not given todecision-makers ⇒ uncertainty about the other players’ strategies

The objective of a game theoretical analysis is to understand whatplayers believe about the other players’ strategies and what theywould play

We need to define player’s preferences under such uncertainty

Under standard assumptions (completeness, transitivity,independence axiom, continuity axiom), the vN-M utilityrepresentation allows us to easily describe these preferences in acompact way ⇒ vN-M utility function assigns real numbers to thecartesian product of players’ strategy sets (see next lecture)

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Page 19: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Assumptions

Rational behavior: Players are rational

Common knowledge: the rationality of every player is commonknowledge - if every player knows it, every play knows that everyother player knows it, every player knows that every other playerknows that every other player knows it...

Complete description of the game is common knowledge:players, actions, strategies, order of play, information and payoffsare common knowledge

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Page 20: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Too strong assumptions?

The predictions of game-theoretic models may be reasonable forsome applications, but also unrealistic for a lot of others

Hundreds of experiments show that game-theoretic predictions arenot confirmed

Assumptions may be systematically violated:

Homo oeconomicus vs. Homer Simpson

Social preferences

Cognitive limitations and limited strategic thinking

Optimism and wishful thinking

Overconfidence

False Consensus Effect: We tend to think others are just like us

Preferences may not be stable (Fehr, 2015)

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Page 21: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Behavioral Game Theory

Extensive use of experimental evidence and psychologicalintuition to generalize the standard assumptions of gametheory.

Use of standard game-theoretic models, but extend and improvethem to give more precise predictions:

relax assumptions

introduce non-standard preferences

extend models for learning or limited strategic thinking

connect theory to psychological and biological principles

extend rationality rather than abandoning it

The core of the solution concepts does not change by theseextension, but predictions will

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Page 22: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Example: rational agents

Dethrone homo oeconomicus by replacing him with someone whoacts ”more human” in economic models

Bounded rationality (Simon, 1955): rationality is limited byavailable information, the tractability of the decision problem,cognitive limitations of their minds, or time limitations

Prospect theory (Kahneman and Tversky, 1979):Introduction of nonstandard preferences, i.e. loss aversion andnon-linear probability weighting ⇒ losses weigh more than gains indecision making

Hyperbolic discounting (Laibson, 1997): tendency todiscount outcomes in the near future more than for outcomes inthe far future

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Page 23: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

Example: Social preferences

human reciprocity (Bolton and Ockenfels, 2000; Falk andFischbacher, 1998)

altruism (Fehr and Schmidt, 2006)

inequity aversion (Fehr and Schmidt, 1999)

guilt, envy (Fehr and Schmidt, 1999)

fairness (Rabin, 1993; Dufwenberg and Kirchsteiger, 1998, Fehrand Gachter, 2000)

trust (Berg et al., 1995; Guth et al., 1997; Cox, 2000)

...

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Page 24: Game Theory - Department Home | Johannes Kepler … · Martin Osborne, An Introduction to Game Theory ... A Course in Game Theory ... What is Game Theory? Game

This course

We study the core solutions concepts of game theory

We focus on our standard assumptions for rational agents

We relax our third assumption in Bayesian games

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