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Introductory Microeconomics

More Formal Concepts of Game Theory and Evolutionary Game Theory

Prof. Wolfram Elsner Faculty of Business Studies and Economics

iino – Institute of Institutional and Innovation Economics

Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies

Chapter 08: More Formal Concepts of Game Theory and Evolutionary Game Theory

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Readings for this lecture

Mandatory reading this time:

More Formal Concepts of Game Theory and Evolutionary Game Theory, in: Elsner/Heinrich/Schwardt (2014): The Microeconomics of Complex Economies, Academic Press, pp. 193-226.

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/



Basic games of classical GT was introduced in Chapter 2

Now: Understand the formal structure of classical decision theory and GT

Classical GT relies on (boundedly) rational agents

Evolutionary GT allows to relax such assumptions and focuses on the dynamic performance of strategies

First: Understand most important formal concepts of classical GT

Then: Move to evolutionary GT

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Outline



A strategic game is described by

The rules of the game

The agents of the game (here: finite number)

The strategies of the agents (here: finite number)

The information available to the agents

Normal form game

Agents make decisions simultaneously

They do not know about the decision of the others

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Basic Concepts of Game Theory



Symbol Explanation

𝑠𝑖 A pure strategy of the ith agent

𝑆𝑖 Set of all pure strategies of agent i

𝑆 = 𝑆𝑖 , 𝑖 = 1, … , 𝑛 Set of strategies of all players

𝑠−𝑖 = 𝑠𝑗 𝑖≠𝑗 Strategies of all other agents than agent i

𝑠 = 𝑠𝑖 , 𝑠−𝑖 Feasible configuration of strategies

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Notation I



Symbol Explanation

Π𝑖 𝑆 = Π𝑖 𝑠 ∀𝑠 = Π𝑖 𝑠𝑖 , 𝑠−𝑖 ∀𝑠𝑖∀𝑠−𝑖 Set of payoffs for all possible combinations of strategies

𝐺 = 𝑆𝑖; Π𝑖 𝑆 ; 𝐼𝑖 , 𝑖 = 1, … , 𝑛 General description for a normal form game

𝐺 = 𝑆1, 𝑆2; Π1 𝑆 , Π2 𝑆 ; 𝐼1, 𝐼2 Description for a normal form game with two players

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Notation II



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Matrix notation

Strategy 1 Strategy 2

Strategy 1

Π𝐵(𝑠𝐴1 , 𝑠𝐵1)

Π𝐴(𝑠𝐴1 , 𝑠𝐵1)



Strategy 2





𝐺 = 𝑆1, 𝑆2; Π1 𝑆 , Π2 𝑆 ; 𝐼1, 𝐼2



In order to predict the outcome of an interaction, assumptions regarding the agents‟ behavior must be made

Utility maximizing agents with well-defined preference orderings, i.e. for any outcomes 𝑎 and 𝑏 the following holds:

Completeness: 𝑎 ≻ 𝑏 𝑜𝑟 𝑏 ≻ 𝑎 𝑜𝑟 𝑎 ~ 𝑏

Reflexivity: 𝑎 ≻ 𝑏 ⟺ 𝑏 ≺ 𝑎

Transitivity: 𝑎 ≻ 𝑏 𝑏 ≻ 𝑐 ⟹ 𝑎 ≻ 𝑐

Common Knowledge of Rationality

Every agent knows that all agents are rational, that all all other agents also know that all agents are rational, that they also are aware that all agents know that all are rational, etc.

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Assumptions regarding the agents



Consider a non-interactive decision situation (to bring or not to bring an umbrella) under risk (the state of the world is unknown, it may or may not rain).

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A Preliminary Consideration: Non-Interactive Concepts, Decision Theory

Rain No Rain

Bring Umbrella

4

5

No Umbrella

-10

10

P

l

a

y

e

r

S t a t e of the W o r l d

How will the player decide?

The obvious difference to game theory is that the State of the World is unknown and unconstrained by rationality



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Non-Interactive Concepts, Decision Theory

R No R

Max.

U 4

5

5

No U

-10

10

10

P

l

a

y

e

r


An optimistic concept: Maximax

Find the best possible payoff for every strategy and maximize

R No R

Min.

U 4

5

4

No U

-10

10

-10

P

l

a

y

e

r


An pessimistic concept: Minimax

Find the worst possible payoff for every strategy and maximize



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Non-Interactive Concepts, Decision Theory contd.

R No R

U 4

5

No U

-10

10

P

l

a

y

e

r


An opportunity cost based concept: Savage‟s Minimax Regret

Construct the regret matrix (how much would the player regret this decision in this state of the world compared to the other decision she could have taken), find the highest regret for every strategy and minimize

R No R

Max.

U 0

5

5

No U

6

0

6

S

t

r

a

t

.

R e g r e t



There are also parametric decision criteria (i.e. criteria that assign a priori probabilities to the states of the world) including, e.g. the Laplace and the Hurwicz criterion

Of course, all decision theory concepts may also be used to make predictions for strategic games (these predictions would be valid even without CKR, i.e. even if agents assume their opponents to be irrational)

However, for many problem structures, these concepts fail

GT allows for more advanced prediction methods by taking the agents‟ capability to make consistent, systematic, and rational decisions into account

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Non-Interactive Concepts, Decision Theory contd.



Players are rational, i.e. payoff maximizing and neither benevolent nor envious

They know that all other players think the same way

Expectations about the behavior of the other players can be formed

The players then play the strategy giving them the best outcome (highest payoff) given their expectations about the reasoning of the other players

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How to solve the game



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Dominance of strategies

and

The latter strategy 𝑠𝑖~ is said to be dominated by the first

If a strategy gives strictly higher payoffs regardless of the choice of the opponent, it is said to strictly dominate the other strategy

Rational players never play a strictly dominated strategy

Π𝑖 𝑠𝑖∗, 𝑠−𝑖 ≥ Π𝑖 𝑠𝑖

~, 𝑠−𝑖 ∀𝑠−𝑖

∃𝑠−𝑖: Π𝑖 𝑠𝑖∗, 𝑠−𝑖 > Π𝑖(𝑠𝑖

~, 𝑠−𝑖)

A strategy 𝑠𝑖∗ ∈ 𝑆𝑖 dominates a strategy 𝑠𝑖

~ ∈ 𝑆𝑖 iff



One way to predict the outcome of a game is therefore the successive elimination of strictly dominated strategies (SESDS)

SESDS does not require any assumptions about opponent‟s behavior

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SESDS

Strat. 1 Strat. 2

Stra-tegy

1

4 4

2 2

Stra-tegy

2

2 2

0 0

Strat. 1

Stra-tegy

1

4 4

Stra-tegy

2

2 2



SESDS does not always yield a solution:

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SESDS

P

l

a

y

e

r

A

P l a y e r B

Strategy 1 Strategy 2

Strategy 1 2

2

0 0

Strategy 2

0 0

2 2

There are no strictly dominated strategies to remove



SESDS does not always yield a solution:

But: Thanks to CKR, agents can form expectations about the choices made by other agents

They can choose the best possible response to the expected choice of their opponents

Due to CKR, the others will do so as well

The resulting situation is a combination of mutual best responses

This situation is called Nash Equilibrium (NE)

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Nash Equilibria



The formal definition is as follows:

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Nash Equilibria

If a game is solvable via SESDS the solution is also a NE

But: Not all NE can be explored via SESDS

Also, There are games that do not have a NE in pure strategies at all

Therefore: Introduce the distinction between pure and mixed strategies



How do you play Rock-Paper-Scissors?

To always play the same strategy is a bad idea

Mixed strategies capture the idea of playing different pure strategies with some probability

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Mixed Strategies

Rock Paper Scissors

Rock 0

0

1 -1

-1

1

Paper -1

1

0 0

1

-1

Scissors 1

-1

-1

1

0

0

P

l

a

y

e

r

A

P l a y e r B



A mixed strategy σi for player i is a vector in which every pure strategy is associated with a probability

For the two strategy case: 𝜎𝑖 =𝑝1 − 𝑝

The Nash Equilibrium is defined as a configuration of mixed strategies for the n players such that

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Nash Equilibrium and Mixed Strategies

Π𝑖 𝜎𝑖∗, 𝜎−𝑖∗ ≥ Π𝑖 𝜎𝑖 , 𝜎−𝑖

∗ ∀𝜎𝑖∀𝑖



Every finite n-person normal-form game with a finite number of strategies for each player has at least one NE in pure or mixed strategies (proven by John Nash in 1950)

Thanks to the NE and the concept of mixed strategies, all finite n-person normal-form games with a finite number of strategies can be solved in theory

How can we compute the NE in mixed strategies?

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Nash Equilibrium and Mixed Strategies



Define the 𝑥 × 𝑦–matrix 𝒜 with the payoffs for the first player:

𝒜 =2 13 0

Expected payoffs are given by:

Π1 = 𝜎1𝑡 ×𝒜 × 𝜎2

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Computation of NE in Mixed Strategies

Dove Hawk

Dove 2

2

3 1

Hawk 1

3

0 0

P

l

a

y

e

r

A

P l a y e r B



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Computation of NE in Mixed Strategies

Π1 = 𝑝 1 − 𝑝2 13 0

𝑞1 − 𝑞

Π1 = 𝜎1𝑡 ×𝒜 × 𝜎2

Π1 = −2𝑝𝑞 + 3𝑞 + 𝑝

𝜕Π1

𝜕𝑝= −2𝑞 +1

𝜕Π1𝜕𝑝= −2𝑞 + 1 = 0

q = 0.5 → 𝑝 = 0.5

This procedure can be employed for every (symmetric) game

Always take the first derivative of the expected payoff function with respect to the strategy parameter of the same player (𝑝 for 1, 𝑞 for 2) and solve it for 0

For symmetric games, the equations are identical for both (with 𝑝 and 𝑞 exchanged) and thus have to be solved only once



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Reaction Functions

𝑝 and 𝑞 can also be shown as the reaction functions p(𝑞) and 𝑞(𝑝) of the players‟ reaction to the opponent‟s choice

Intersections denote NE (the current Hawk-Dove game has 3)

Note: The caption in Figure 8.12 in Chapter 8 is incorrect (same Figure as on this slide; the Figure depicts Hawk-Dove, not Matching Pennies)



Mixed strategy NE are where players are indifferent between

both pure (and mixed) strategies (i.e. where 𝜕Π1

𝜕𝑝=0, 𝜕Π2

𝜕𝑞= 0)

Mixed strategy NE may seem oddly unstable but this is not necessarily the case (see Evolutionary Game Theory below).

Also, under CKR it is rational to deliberately choose mixed NE strategies in order to facilitate the emergence of the equilibrium and to avoid being exploited (Aumann‟s defense)

This is especially true if there is no pure strategy NE; consider the Matching Pennies game (or the Rock-Paper-Scissors game above)

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Reaction Functions



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Mixed Strategy NE

Heads Tails

Heads 1

-1

-1 1

Tails -1

1

1 -1

P

l

a

y

e

r

1

P l a y e r 2

In zero-sum games (like this Matching Pennies game), unequal payoff denotes one player being exploited by the other



Many extensions to the Nash Equilibrium have been developed

Some of these, the ones for extensive or repeated (subgame perfect Nash Equilibrium) as well as evolutionary games (ESS, … ), will be presented below

There are also refinements that ensure that the equilibrium is still valid under stochastic perturbations (the „trembling hand‟), e.g. Selten‟s Trembling Hand Perfect Equilibrium, Myerson‟s Proper Equilibrium (see textbook), or Harsanyi‟s Bayesian Nash Equilibrium.

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Extensions of the Nash Equilibrium



In normal-form games, the players make their decisions simultaneously but this is not the case for other types of games

For games in sequential form we use a new notation, the extensive-form notation

Note that this is necessary only if the agents have full information about the decisions made by the previous agents

Otherwise the game is equivalent to a normal-form game

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Extensive Form Games



Depending on the choice of the first player, the second player faces a different decision situation

It is therefore convenient to define complete strategies for the players

A complete strategy gives each player an instruction for all possible situations

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Extensive-Form Notation

Player 1

Player 2 Player 2

Result 1 Result 2 Result 3 Result 4



For an extensive game 𝐺𝐸 let 𝑉 be a set containing all possible states of 𝐺𝐸

𝑉𝐴 will contain all possible situations in which player A can possibly make a decision

A complete strategy for player A gives a an instruction for any element in 𝑉𝐴

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Complete Strategies

Player 1

Player 2

𝑉1

𝑉2 𝑉3

𝑉4 𝑉5 𝑉6 𝑉7

C

C C

D

D D

(0,0) (2,−1) (−1,2) (1,1)



Analytical derivation of NE cumbersome

Thanks to CKR we can rely on backward induction

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How to solve extensive games



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Backward Induction

Player 1

Player 2

𝑉1

𝑉2 𝑉3

𝑉4 𝑉5 𝑉6 𝑉7

C

C C

D

D D

The reasoning for player 1 is as follows:

What would player 2 do in situation 𝑉2?

Choose D

Result is would be 𝑉5

What would player 2 do in situation 𝑉3?

Choose D

Result would be 𝑉7

Since 1 prefers 𝑉7 (via 𝑉3) to 𝑉5 (via 𝑉2), she chooses D (which leads to 𝑉3 and then 𝑉7)

(0,0) (2,−1) (−1,2) (1,1)



Analytical derivation of NE cumbersome

Thanks to CKR we can rely on backward induction

Advantage: All NE found with backward induction are also subgame perfect

A NE is subgame perfect if it is also a NE of all the subgames that contain the NE

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How to solve extensive games



Up to this point every game we have investigated was played only once

The solutions we have obtained are called one-shot solutions

Now we consider repeated games

A supergame 𝒢 is a sequence of repetitions of the normal-form game 𝐺 that is infinite from the players‟ point of view, i.e. either really infinite or indefinite (with stochastic probability to end in each period)

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Repeated Games and Supergames



If 𝒢 is finite it can be expressed as an extensive game and solved via backward induction

If 𝒢 is infinite or indefinite, there is no last period, so backward induction will not work

Still, strategies can be characterized as complete (if they contain instructions for every possible situation) or not complete and equilibria can be characterized as subgame perfect or not

If both players follow complete strategies, all choices are predictable and the payoffs are thus known in advance (to rational players and observers); this allows to calculate a present value to be used in deciding between strategies in advance

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Solving Supergames



To obtain the present value for a future payoff 𝑎 one uses a discount parameter 𝛿 (with 0 ≤ 𝛿 ≤ 1) which denotes the player‟s valuation of future payoffs (the present value in time 0 of a payoff 𝑎 in time 0 is 𝑎, … of the same payoff in time 1 it is 𝛿𝑎, in time 2, 𝛿2𝑎, etc.)

The present value for an infinite sequence of payoff 𝑎 is

therefore given as Π = 𝑎 + 𝛿𝑎 + 𝛿2𝑎 + 𝛿3𝑎 +... =𝑎

1−𝛿

Agents make the decisions about their strategy plan based on the present values of the expected payoffs from the different strategy plans

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Present Values



Strategy plans can be very simple, e.g. always to play one strategy

In the PD, ALL-D is a strategy plan according to which the player always defects

Strategy plans can also be more complex, esp. when the choice of the player depends on the decisions made by other players in the past

In supergames, the players usually have a memory that allows expectation formation (which is known to the player so that the strategy plan can make use of the fact)

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Strategy Plans



Consider a supergame based on a social dilemma (a generalized PD), i.e.

a cooperative solution exists which constitutes the social optimum and a Pareto optimum of the underlying game

but the cooperative strategy is exploitable (i.e. the cooperative solution is not a NE)

The folk theorem states that it is possible to reach the cooperative solution in the supergame

This is achieved by using trigger strategies, positing a credible thread of punishment if the opponent should deviate from the cooperative strategy (Rubinstein‟s proof).

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Folk Theorem and Trigger Strategies



Define a strategy that guarantees a minimum of payoff for the opponent as the minmax strategy (analogous to maximin, only that her payoff is minimized by her opponents choice not maximized by her own choice)

If the opponent‟s payoff resulting from using the minmax strategy on her is less then her payoff from cooperation, the minmax strategy can serve as a threat of punishment for deviating from cooperation

The threat is said to be credible if the punishing player receives a higher or equal payoff from playing the minmax strategy than from allowing herself to be exploited (more generally: if the minmax strategy is not strictly dominated)

Credible threats can be used to construct a trigger strategy 𝑠𝑡𝑟𝑖𝑔𝑔𝑒𝑟𝒢

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Trigger Strategies and Rubinstein‟s Proof of the Folk Theorem



A strategy profile can include the instruction to employ 𝑠𝑚𝑖𝑛𝑚𝑎𝑥 always if a specific expectation is not met

By doing so, mutually beneficial agreements that do not constitute a NE in the one-shot case can be enforced

Example: The Prisoner’s Dilemma and tit-for-tat

𝑠𝑡𝑟𝑖𝑔𝑔𝑒𝑟𝒢

= 𝑠𝑇𝐹𝑇𝒢= 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒, 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 ℎ𝑎𝑠 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒𝑑 𝑙𝑎𝑠𝑡 𝑟𝑜𝑢𝑛𝑑 𝑑𝑒𝑓𝑒𝑐𝑡, 𝑖𝑓 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 𝑝𝑙𝑎𝑦𝑒𝑟 ℎ𝑎𝑠 𝑛𝑜𝑡 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑒𝑑 𝑙𝑎𝑠𝑡 𝑟𝑜𝑢𝑛𝑑

To defect is the minmax strategy that serves as credible threat, all players playing this trigger strategy is a NE in the repeated PD

Trigger strategies can be constructed in different ways; they could for instance also – less forgivingly – punish forever after one defection

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Trigger Strategies and Rubinstein‟s Proof of the Folk Theorem



Allows to conveniently construct microfounded economic models, thus closing gaps between micro, meso, and macro level

In Evolutionary Game Theory (EGT) we consider populations of agents, all continuously playing one and the same strategy according to their type

Agents are matched randomly to play again and again the same underlying game (which must be symmetric so that the positions of row player and column player are exchangable)

The population shares of these types (strategies) develop according to their performance in those games

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Evolutionary Game Theory



The changing share of types may be seen as reproduction with the offspring continuing the same strategies or as poorly performing agents consciously changing their strategy (type)

The most important solution techniques are

1. Analysis of evolutionary stability (evolutionary stable strategies, ESS)

2. Replicator dynamics

3. Simulation (see textbook Chapter 9)

The players are matched randomly

When they meet, they play the underlying game using their predetermined strategy

Composition of the population changes according to the agent‟s performance in playing the underlying game

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Evolutionary Game Theory



NE: Combination of mutual best answers

In the population context we consider only strategies:

A strategy is evolutionary stable if a population dominated by it is not invadable by any other strategy

If a population is dominated by an ESS, the situation will remain stable

What does “not invadable” mean?

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Evolutionary Stability



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The principle of evolutionary stability

Symbol Explanation

𝒫𝒢𝒱 An evolutionary population setting

𝐺𝒢𝒱 The underlying (symmetric) one-shot normal form game in this setting

The payoff matrix

Π𝜎1/𝜎2 = 𝜎1𝑇𝒜 𝜎2

Expected payoff of the first strategy against the second

𝒜



Consider a population of agents playing 𝜎∗

Now consider a very small group of players entering the group and playing 𝜎~ ≠ 𝜎∗

If the new strategy yields better payoffs than the old one, the

share of players playing 𝜎~ will increase

In this case the 𝜎~ has invaded 𝜎∗ and 𝜎∗cannot be said to be an

ESS

We will now formalize the concept of evolutionary stability

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The principle of evolutionary stability



Let 휀 be the (arbitrarily small) share of the invading group playing 𝜎~

A share of (1 − 휀) is therefore playing strategy 𝜎∗

𝜎∗ is an ESS if it yields a higher expected payoff than 𝜎~

Formally:

𝜎∗𝑇𝒜(1 − 휀)𝜎∗ + 𝜎∗𝑇𝒜휀𝜎~ > 𝜎~𝑇𝒜(1 − 휀)𝜎∗+ 𝜎~𝑇𝒜휀𝜎~

This rather complicated formula can conveniently be tested using two simple conditions,

The first is constructed from letting 휀=0 for which the above inequality must hold at least weakly, with ≥ (1st condition)

The second results additionally for the case that the first condition holds with equality (2nd condition).

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How to test for evolutionary stability



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(Weak) First con-dition of evolu-tionary stability:

Holds

Holds

𝝈∗ is evolutionary stable

Does not hold

𝝈∗is not evolutionary stable

Does not hold

Holds

Second condition of evolutionary stability

𝜎∗𝑇𝒜𝜎∗ ≥ 𝜎~𝑇𝒜𝜎∗

𝜎∗𝑇𝒜𝜎∗ > 𝜎~𝑇𝒜𝜎∗ 𝜎∗𝑇𝒜𝜎~ > 𝜎~𝑇𝒜𝜎~

Strict first condition of evolutionary stability:



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Test considering which candidate strategies: pure and mixed NE strategies, since non-NE-strategies are by definition not best answers to themselves (and this is required by the 1st condition of ESS).

Test against which invading strategies: Against all competing pure strategies (in the 2-strategy case – since mixed strategies are then only linear combinations combining properties of only two pure strategies)

In cases with more than 2 strategies, evolutionary stability must be tested against mixed strategies as well.



Consider as an example the Hawk Dove game above.

We have game matrix 𝒜 =2 13 0

and three NE strategies,

H =01, D =

10, and M =

0.50.5

Test H against D, 1st condition:

0 12 13 0

01≥ 1 0

2 13 0

01

0 ≱ 1

The condition does not hold, H is therefore not an ESS because it can be invaded by D-players

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Evolutionary Stability: Example



Test D against H, 1st condition:

1 02 13 0

10≥ 0 1

2 13 0

10

2 ≱ 3

The condition does not hold, D is therefore not an ESS because it can be invaded by H-players

M must be tested against both D and H; Test M against H, 1st condition:

0.5 0.52 13 0

0.50.5≥ 0 1

2 13 0

0.50.5

1.5 ≥ 1.5

Condition holds with equality, test of 2nd condition necessary

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Evolutionary Stability: Example contd.



Test M against H, 2nd condition:

0.5 0.52 13 0

01> 0 1

2 13 0

01

0.5 > 0

Condition holds, H cannot invade M

Test M against D, 1st condition:

0.5 0.52 13 0

0.50.5≥ 1 0

2 13 0

0.50.5

1.5 ≥ 1.5

Condition holds with equality, test of 2nd condition necessary

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Test M against D, 2nd condition:

0.5 0.52 13 0

10> 1 0

2 13 0

10

2.5 > 2

Condition holds, D cannot invade M

Since all mixed strategies are linear combinations of D and H, none of those (except M itself) can be able to invade M

M is therefore this game‟s only ESS

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The focus is not on single strategies but on stable compositions of populations

The population is modeled as a dynamical system

A dynamical system describes how state variables change over time

Consider a system with z state variables:

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Replicator Dynamics

The development path depends on the initial values and the development equations

𝜃𝑡 =

𝜃1,𝑡𝜃2,𝑡⋮𝜃𝑧,𝑡

= 𝜃𝑖,𝑡 𝑖=1,…,𝑧



Here the state variables represent the shares of specific agents playing a certain strategy (except for the last share which is the difference to 100%, thus determined by the others)

The shares might change according to a difference or differential equation:

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Replicator Dynamics

𝜃𝑡+1 = 𝐹𝐷~(𝜃𝑡)

𝑑𝜃(𝑡)

𝑑𝑡= 𝐹𝑑~(𝜃𝑡)



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Replicator Dynamics

Symbol Explanation

𝑓𝑖,𝑡 The evolutionary fitness of agent i at time t

𝜙 = 𝜃𝑖,𝑡𝑓𝑖,𝑡

𝑖

Average fitness in the population

For the evolutionary performance of the agents it is important how their fitness compares to the average fitness

One can now analyze the dynamical system and search for stable equilibria

𝜃𝑖,𝑡+1 = 𝐹𝐷~(𝜃𝑖,𝑡 , 𝑓𝑖,𝑡 , 𝜙𝑡)

𝑑𝜃𝑖(𝑡)

𝑑𝑡= 𝐹𝑑~(𝜃𝑖 𝑡 , 𝑓𝑖 𝑡 , 𝜙𝑖(𝑡))



Replicator Dynamics requires an explicit assumption on the form and speed of the replicator (this is implicit in ESS)

Typical forms set the dynamic as proportional to the relation of individual and average fitness with the most common canonical forms being

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Replicator Dynamics

𝜃𝑖,𝑡+1 = 𝜃𝑖,𝑡(𝑓𝑖,𝑡/𝜙𝑡) 𝑑𝜃𝑖(𝑡)

𝑑𝑡= 𝜃𝑖 𝑡 (𝑓𝑖 𝑡 − 𝜙(𝑡))



The conditions for equilibria are:

In order to test for the stability of the equilibria one calculates the eigenvalues of the development equation for the equilibria (see Chapters 10 and 11 of the textbook for details)

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How to determine (stable) equilibria

For difference equations, the equilibrium is stable if all eigenvalues have an absolute value smaller than unity

For differential equations, the equilibrium is stable if all eigenvalues are negative

𝜃𝑖,𝑡+1∗ = 𝐹𝐷

~ 𝜃𝑖𝑡, 𝑓𝑖,𝑡, 𝜙𝑡 = 𝜃𝑡∗

𝑑𝜃𝑖(𝑡)

𝑑𝑡= 𝐹𝑑~ 𝜃𝑖 𝑡 , 𝑓𝑖 𝑡 , 𝜙𝑖 𝑡 = 0



Consider again the Hawk-Dove game above with the

canonical replicator function 𝑑𝜃𝑖(𝑡)

𝑑𝑡= 𝜃𝑖 𝑡 (𝑓𝑖 𝑡 − 𝜙(𝑡)), call

the pure strategies 𝑥𝑖 and assume the 𝑓𝑖 = Π𝑖 = 𝑥𝑖𝑇𝒜 𝜃 (at any

point in time, leaving out the (𝑡) for convenience)

We further have 𝒜 =2 13 0

, 𝑥1 = 𝐷 =10

, 𝑥2 = 𝐻 =01,

and therefore also

𝑓1 = 1 02 13 0

𝜃11 − 𝜃1

= 1 + 𝜃1

𝑓2 = 0 12 13 0

𝜃11 − 𝜃1

= 3𝜃1

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Replicator Dynamics: Example



Since 𝜙 = 𝜃𝑖𝑓𝑖 𝑖 =𝜃1𝑓1+(1 − 𝜃1)𝑓2 , we can rearrange the replicator equation

𝑑𝜃1

𝑑𝑡= 𝜃1 (𝑓1 − 𝜙)

𝑑𝜃1

𝑑𝑡= 𝜃1 (𝑓1 − 𝜃1𝑓1 − (1 − 𝜃1)𝑓2)=𝜃1 ((1 − 𝜃1)𝑓1 − (1 − 𝜃1)𝑓2)

𝑑𝜃1

𝑑𝑡= 𝜃1 (1 − 𝜃1)(𝑓1 − 𝑓2)

Substituting 𝑓1 and 𝑓2 yields 𝑑𝜃1𝑑𝑡= 𝜃1 1 − 𝜃1 1 + 𝜃1 − 3𝜃1 = 𝜃1 1 − 𝜃1 1 − 2𝜃1

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Replicator Dynamics: Example contd.



Applying the equilibrium condition 𝑑𝜃1

𝑑𝑡= 0 yields three fixed

points 𝜃1,1 = 0, 𝜃1,2 = 1, and 𝜃1,3 = 0.5

To assess the stability, we bring the replicator equation into polynomial form

𝑑𝜃1𝑑𝑡= 2𝜃1

3 − 3𝜃12 + 𝜃1

and obtain the only element of the system‟s Jacobian 𝜕(𝑑𝜃𝑖/𝑑𝑡)

𝜕𝜃𝑖= 6𝜃1

2 − 6𝜃1 + 1

the linearization of which is the (dominant) eigenvalue.

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We obtain the three eigenvalues for the three equilibria

𝜆 𝜃1,1 = 1 (i.e. 𝜃1,1 is unstable),

𝜆 𝜃1,2 = 1 (i.e. 𝜃1,2 is also unstable),

𝜆 𝜃1,3 = −0.5 (i.e. 𝜃1,3 is stable)

The result is identical to that obtained from analysis of ESS: a composition of the population (or equivalently of each individual strategy) of both strategies with equal shares is the only stable equilibrium or fixed point. The pure strategy equilibria / fixed points exist but are unstable.

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This chapter

gave a comprehensive introduction to formal game theory including different notations (matrix, formal, extensive)

introduced advanced solution concepts of decision theory and game theory (including SESDS and Nash Equilibria in pure and mixed strategies, backward induction, …)

covered non-normal-form games (including extensive games, repeated games)

gave a formal introduction to Evolutionary Game Theory (ESS and replicator dynamics)

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Summary



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Readings for the next lecture

Compulsory reading:

Introduction to Simulation and Agent-Based Modeling, in: Elsner/Heinrich/Schwardt: Microeconomics of Complex Economies, pp. 227-247.

For further readings visit the companion website





introductory microeconomics - elsevier...introductory microeconomics more formal concepts of game...

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