Introduction to Game Theory - Elsevier · Chapter 02: Introduction to Game Theory Best answer strategy A strategy that gives the highest possible payoff given the opponent’s choice

Introduction to Game Theory

Prof. Wolfram ElsnerFaculty of Business Studies and Economics

iino – Institute of Institutional and Innovation Economics

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies

Chapter 02: Introduction to Game Theory10.04.2014

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Readings for this lecture

Mandatory reading this time:

An Introduction to Game Theory, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 25-32.

The lecture and the slides are complements, not substitutes

An additional reading list can be found at the companion website

http://www.amazon.com/The-Microeconomics-Complex-Economies-Institutional/dp/0124115853/ref=sr_1_1?ie=UTF8&qid=1396453322&sr=8-1&keywords=Microeconomics+of+complex+economies


http://booksite.elsevier.com/9780124115859/


Chapter 02: Introduction to Game Theory

Social Interactions may be modeled as strategic games, i.e. Game Theory models.

The use of Game Theory requires a framework of assumptions translating the social setting into a simplified mathematical model.

Due to its abstract nature, underlying mechanisms valid for many situations can be identified.

Useful for prediction, explanation, and modeling in a wide range of fields (economics, sociology, …). Examples are given in later Chapters.

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Game Theory



In Game Theory, models are specified as strategic games, which are characterized by

A set of participants (agents) A set of behavioral options (strategies) A set of rules (including the sequence of decision making) The information sets available to the agents. Agents are assumed to be rational (pursuing maximum payoff,

not envious).

Now: Begin with an easy, but almost universally applicable form of a strategic game

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Strategic Games



Characterized by

2 agents

Each agent has two alternative strategies

Simultaneous choice of the strategies

Complete information (about game, strategies, payoffs, but not the opponent’s choice)

No collusion

Let us consider some examples:

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Simultaneous 2-Person-2-Strategy Normal-Form Games



Strategy 1 Strategy 2

Strategy 1

Strategy 2

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Social Optimum Game

P

l

a

y

e

r

A

P l a y e r B



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Social Optimum Game

P

l

a

y

e

r

A

P l a y e r B

Example / Interpretation:

• Invisible hand concept• The option an egoistic individual chooses is the same option chosen by a hypothetical social planner seeking to maximize the outcome for the entire society• Game will arrive at the social optimum• This point is Pareto-optimal• Model for conflict-free social interactions.

Strategy 1 Strategy 2

Strategy 14

4

2

2

Strategy 22

2

0

0

Payoffs given in the rows

Payoffs given in the columns



Pass Wait

Pass

Wait

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Anti-Coordination Game

P

l

a

y

e

r

A

P l a y e r B



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Anti-Coordination Game

P

l

a

y

e

r

A

P l a y e r B Example / Interpretation:

• Encounter in a narrow doorway• Agents cannot pass both at the same time• Pareto-efficient solution is that one waits and the other passes •Model for potential social conflict without actually conflicting agendas (necessity to coordinate)

Pass Wait

Pass0

0

1

1

Wait1

1

0

0



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Prisoners´ Dilemma Game

Example / Interpretation:

• Two burglars arrested after a robbery• They could give a guilty plea or be silent• If both remain silent they can only be held for a short time• If one defects she gets a shorter and the other a longer jail time•If both confess they are both jailed for a long time•Model for conflicting incentives (agendas) in social interactions



Be silent Confess

Be silent1

1

2

-1

Confess-1

2

0

0

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Prisoners´ Dilemma Game

P

l

a

y

e

r

A

P l a y e r B Interpretation/Example:

• What are the two individuals thinking?•If they choose to confess, they “do not make a mistake, no matter what the other one does”•If they choose to be silent, there is the huge risk of being exploited if the other one confesses•To confess is therefore strictly dominant



A comparison of different outcomes is not straightforward

In the Prisoners’ Dilemma, the agents prefer different outcomes (conflicting agendas, necessity to coordinate and to cooperate)

Also: Payoffs of different agents cannot be directly compared.

Solution: Concept of Pareto optimality

A situation is to be preferred over another if no player is worse of but at least one player is better of.

The concept avoids any direct comparison of payoffs of different agents and treats them as different dimensions to the problem.

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Pareto Optimal Outcomes



In the above Prisoners’ Dilemma there are 4 possible outcomes (W, V, X, and Y, see next slide): X is Pareto dominated by Y while W, V, and Y are Pareto optima.

Pareto dominance and likely outcomes of games? Although Pareto optimal are preferable to all agents, this is not

necessarily the likely outcome of social interactions. Consider the Prisoners’ Dilemma: Agents try to avoid being

exploited (Player B wants to avoid the lower left field, W, player A wants to avoid the upper right field, V) leading to a Pareto inferior outcome (lower right field, X).

The Pareto criterion evaluates outcomes, it does not predict them.

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Solving Games: Pareto Dominance



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Solving Games: Pareto Dominance

B1 B2

A11

Y1

2V

-1

A2-1

W 2

0X

0



If a strategy gives higher (or equal) payoffs than another, no matter what strategy the other player chooses, this strategy is said to dominate the other one (which is said to be dominated).

If a strategy always gives higher payoffs than another, it is said to strictly dominate the other (then called strictly dominated).

A strategy that (strictly) dominates all alternatives is called the (strictly) dominant strategy in this game.

Rational players would never choose a strictly dominated strategy.

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Solving Games: Dominant Strategies



Identifying dominant

strategies

2 > 0

0 > -1

Strategy „Confess“ therefore strictly dominates strategy „Be Silent“.

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Solving Games: Dominant Strategies

Be silent Confess

Be silent1

1

2

-1

Confess-1

2

0

0



Best answer strategy A strategy that gives the highest possible payoff given the

opponent’s choice is called the best answer strategy (corresponding too the opponent’s choice).

Nash equilibrium An outcome that is achieved as a result of every player playing

her best answer strategy. In a Nash equilibrium no player has incentive to deviate from her

choice (given that the opponent also sticks to her choice).

Nash equilibria (as well as strategic dominance) can therefore indeed serve as predictions of the outcome of the game.

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Solving Games: Nash Equilibrium



Be silent Confess

Be silent1

1

2

-1

Confess-1

2

0

0

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Solving Games: Nash Equilibrium

How to find a Nash equilibrium:

Underline the best answer payoffs given the strategy of the opponent (for both players) Mutual best responses are Nash equilibria



Extending 2-person normal-form games to n-person games is straightforward and usually does not change the basic results The agents have to choose between the same two strategies Payoffs depend on the choice of all other agents Consider two examples where n1 players choose the first and n2 players choose the second strategy. Resulting payoffs are:

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From 2-Person Games to N-Person Games

Social Optimum Game Social Dilemma Game



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Social Optimum vs. Prisoners Dilemma

Social Optimum Prisoners Dilemma

Nash equilibrium is the only Pareto-optimal outcome

Nash equilibrium is the only outcome, that is not Pareto-optimal

Resulting payoffs: Resulting payoffs:



Sequential games Players make their choices one after the other Second player knows the decision of the first one when making

her decision

Repeated games Agents remember choices and outcomes from previous periods Allows memorizing others behavior and learning

Supergames Indefinitely repeated games are called supergames

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Other Types of Games



This Chapter covered:

…an introduction to the theory of strategic games

…an introduction to matrix representation of normal-form games

…an introduction to common game structures

…an introduction to simple solution concepts for strategic games.

More will follow in Chapter 8.

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Conclusion



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Readings for the next lecture

Compulsory reading:

Problem Structures and Processes in Complex Economies, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 33-55.

For further readings visit the companion website





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Introduction to Game Theory - Elsevier · Chapter 02: Introduction to Game Theory Best answer strategy A strategy that gives the highest possible payoff given the opponent’s choice