Introduction to Game Theory

Introduction to Game Theory

Part 1. Static games of complete information

Chapter 1. Normal form games and Nash equilibrium

Ciclo Profissional 2^o Semestre / 2011

Gradua¸c˜ao em Ciˆencias Econˆomicas

Topics covered

1 Normal-form representation of games

2 Iterated elimination of strictly dominated strategies

3 Motivation and definition of Nash equilibrium

What is a game?

Definition

A game is a formal representation of a situation in which a number of individuals interact in a setting of strategic interdependence

Each individual’s welfare depends not only on his own actions but also on the actions of the other individuals

The actions that are best for an individual to take may depend on what he expects the other players to do

Normal-form representation of games

In the normal-form representation of the game each player simultaneouslychooses a strategy

the combination of strategies chosen by players determines a payoff for each player

Example

The prisoners’ dilemma

The prisoners’ dilema: the environment

Two suspects are arrested and charged with a crime

The police lack sufficient evidence to convict the suspects, unless at least one confesses

The suspects are in separate cells

The police explain the consequences that will follow from the actions they could take

The prisoners’ dilema: actions and payoffs

If neither confesses then will be convicted of a minor offense and sentenced to one month in jail

If both confess then both will be sentenced to jail for six months If one confesses but the other does not, then the confessor will be released immediately but the other will be sentenced to nine months in jail

I Six for crime

I Three for obstructing justice

The prisoners’ dilema: matrix representation

Each player has two strategies: ConfessorNot confess

We implicitly assume that each player does not like to stay in jail

Prisoner i₁

Prisoner i2

Not confess Confess Not confess −1,−1 −9, 0

Confess 0,−9 −6,−6

Prisoners’ dilemma

The normal form representation: Players

A finite setI of players

We write “player i” where iis the name of the player andI is the collection of names

We denote by nthe number of players, i.e.,n= #I The setI may denoted byI ={1,2, . . . , n}

We prefer the notation I ={i₁, i2, . . . , in}

The normal form representation: Strategies

The set of strategies available to player iis denoted bySi

An elements_i inS_i is called a strategy(or play or action) The setS_i is called strategy spaceand may have any structure:

finite, countable, metric space, vector space

The collection (s_i)i∈I= (s_i1, . . . , s_in) is called a strategy profile and denoted bys ors

Given an agentj and a profile s, we denote by (s⁰_j, s−j) the new profileσ = (σi)i∈I defined by

σi=

s⁰_j if i=j s_i if i6=j Ifj =ik for some 1< k < n then

(s⁰_j, s−j) = (s_i1, . . . , s_ik−1, s⁰_i

k, s_ik+1, . . . , s_in)

The normal form representation: Payoffs

The payoff of playeriis a function uⁱ : Q

j∈IS_j −→ [−∞,+∞]

s 7−→ uⁱ(s)

whereuⁱ(s) is the payoff of player iwhen he plays strategy s_i and any other playerj plays strategysj

We use alternatively the following notation

uⁱ(s) =uⁱ((s_j)j∈I) =uⁱ(s_i, s−i) =uⁱ(s_i1, s_i2, . . . , s_in) Sometimes, abusing notations we write

uⁱ(s1, s2, . . . , sn)

The normal form representation

Definition

A game in normal form is a family

G= (S_i, u_i)i∈I

where for each i∈I Si is a set

ui is a function from S =Q

k∈ISk to [−∞,∞]

Question?

We should know describe how to solve a game-theoretic problem Can we anticipate how a game will be played?

What should we expect to observe in a game played by rational players who are fully knowledgeable about the structure of the game and each others’ rationality?

Simultaneous moves

In a normal form game the players choose their strategies simultaneously

This does not imply that they actsimultaneously

It suffices that each choose his or her action without knowledge of the others’ choices

For the prisoners’ dilema, the prisoners may reach decisions at arbitrary times but it must be in separate cells

Bidders in an sealed-bid auction

Strictly dominated strategies

Definition

Consider a normal form game (Si, ui)i∈I

Lets⁰_i ands⁰⁰_i be two strategies inS_i

Strategys⁰_i is strictly dominatedby strategys⁰⁰_i if for each possible combination of the other players’ strategies, the player i’s payoff from playing s⁰_i is strictly less than the payoff playing s⁰⁰_i

Formally,

∀s_−i∈Y

k6=i

S_k, uⁱ(s⁰_i,s−i)< u_i(s⁰⁰_i,s−i)

Rationality

Rational players do not play strictly dominated strategies

Strictly dominated strategies: The prisoners’ dilemma

For a prisoner, playing Not confess is strictly dominated by playing Confess

Assume we are player i₁ If playeri2 chooses Confess

I We prefer to playConfess and stay 6 months in jail

I Then playingNot confessand stay 9 months in jail If playeri2 chooses Not confess

I We prefer to playConfess and be free

I Then playingNot confessand stay 1 month in jail

Prisoner i1

Prisoner i₂ Not confess Confess Not confess −1,−1 −9, 0

Confess 0,−9 −6,−6

Prisoners’ dilemma

Strictly dominated strategies: The prisoners’ dilemma

A rational player will choose to play Confess

The outcome reached by the two prisoners is(Confess,Confess) This results in a worse payoff for both players than would(Not confess,Not confess)

Thisinefficiency is a consequence of the lack of co-ordination This happens in many other situations

I the arms race

I the free-rider problem in the provision of public goods

Strictly dominated strategies: Iterated elimination

Can we use the idea that “rational players do not play strictly dominated strategies” to find a solution to other games?

Consider a game (in normal form) with two players Player i₁ has two available strategiesS_i1 ={Up,Down}

Player i2 has three available strategiesSi2 ={Left,Middle,Right}

The payoffs are given by the following matrix

Player i1

Playeri2

Left Middle Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Strictly dominated strategies: Iterated elimination

Playeri₁

Player i₂

Left Middle Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0 For Playeri₁

I Upis not strictly dominated byDown

I Downis not strictly dominated byUp

For Playeri₂ the strategy Right is strictly dominated byMiddle Player i2 will never playRight

If Playeri1 knows that Player i2 is rational

Then Player i1 can eliminate Rightfrom Player i2’s strategy set

Strictly dominated strategies: Iterated elimination

Both players can play the game as if it were the following game

Playeri₁

Player i₂ Left Middle

Up 1,0 1,2

Down 0,3 0,1

For Playeri₁ the strategy Down is strictly dominated byUp If Playeri₂ knows that Player i₁ is rational

And Playeri₂ knows that Playeri₁ knows that Playeri₂ is rational Then Player i2 can eliminate Downfrom Player i1’s strategy space

Strictly dominated strategies: Iterated elimination

Now the game is as follows

Playeri1

Playeri₂ Left Middle

Up 1,0 1,2

For Playeri₂ the strategy Left is strictly dominated byMiddle By iterated elimination of strictly dominated strategies The outcome of the game is(Up,Middle)

Strictly dominated strategies: Iterated elimination

Definition

This process is called iterated elimination of strictly dominated strategies

Proposition

The set of strategy profiles that survive to iterated elimination of strictly dominated strategies is independent of the order of deletion

Strictly dominated strategies: Iterated elimination

Drawbacks

Each step requires a further assumption about what the players know about each other’s rationality

To apply the process for an arbitrary number of steps, we need to assume that it is common knowledgethat players are rational

I All the players are rational

I All the players know that all the players are rational

I So on, ad infinitum

This process often produces a very imprecise prediction about the play of the game

Strictly dominated strategies: Limitations

Consider the following game

L C R

T 0,4 4,0 5,3 M 4,0 0,4 5,3 B 3,5 3,5 6,6

There are no strictly dominated strategies to be eliminated The process produces no prediction whatsoever about the play of the game

Question

Is there a stronger solution concept than iterated elimination of strictly dominated strategies which produces much tighter predictions in a very broad class of games?

Nash equilibrium: Motivation

Suppose that game theory makes a unique prediction about the strategy each player will choose

In order for this prediction to be compatible with incentives (or correct) it is necessary that each player be willing to choose the strategy predicted by the theory

Thus each player’s predicted strategies must be that player’s best response to the predicted strategies of other players

Such a prediction could be called strategically stableor self-enforcing

Because no single player wants to deviate from his or her predicted strategy

A solution of the game satisfying the previous property is called a Nash equilibrium

Nash equilibrium: Definition

Definition

Consider a game G= (Si, ui)i∈I

A strategy profiles^? = (s^?)i∈I is aNash equilibriumof G if for each player i, the strategy s^?_i is playeri’s best response to the strategies specified ins^? for the other players

In other words, s^?= (s^?_i)i∈I is aNash equilibrium if

∀i∈I, s^?_i ∈argmax{uⁱ(si, s^?_−i) : si ∈Si} i.e.,

∀s_i ∈Si, uⁱ(si, s^?_−i)6uⁱ(s^?_i, s^?_−i) Remark

The set

argmax{uⁱ(si, s^?_−i) : si∈Si}

Nash equilibrium: Interpretation

If the theory offers the profile s⁰ = (s⁰_i)i∈I that is not a Nash equilibrium then there exists at least one player that will have an incentive to deviate from the theory’s prediction

If a convention is to develop about how to play a given game then the strategies prescribed by the convention must be a Nash equilibrium, else at least one player will not abide the convention

Nash equilibrium: Examples

In a two-player game we can compute the set of Nash equilibria as follows:

For each player

I For each strategy for this player

F Determine the other player’s best response to that strategy

F Underline the corresponding payoff on the matrix

A pair of strategies (profile) is Nash equilibrium if both corresponding payoffs are underlined in the matrix

L C R

T 0,4 4,0 5,3 M 4,0 0,4 5,3 B 3,5 3,5 6,6

Nash equilibrium: Examples

Player i1

Playeri₂

Left Middle Right

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Prisoner i₁

Prisoner i₂ Not confess Confess Not confess −1,−1 −9, 0

Confess 0,−9 −6,−6

Prisoners’ dilemma

Nash equilibrium: a stronger solution

Consider a game G= (Si, ui)i∈I

Proposition

If iterated elimination of strictly dominated strategies eliminates all but the strategy profile s^?= (s^?_i)i∈I then s^? is the unique Nash equilibrium of the game

Theorem

If the strategy profile s^? is a Nash equilibrium then s^? survives iterated elimination of strictly dominated strategies

Nash equilibrium: a stronger solution

Remark

Nash equilibrium is a stronger solution concept than iterated elimination of strictly dominated strategies

Is it too strong? Can we be sure that a Nash equilibrium exists?

There can be strategy profiles that survive iterated elimination of strictly dominated strategies but which are not Nash equilibria

A classic example: The battle of sexes

A man (Pat) and a woman (Chris) are trying to decide on an evening’s entertainment

While at workplaces, Pat and Chris must choose to attend either the opera or a rock concert

Both players would rather spend the evening together than apart

Chris

Pat Opera Rock Opera 2,1 0,0

Rock 0,0 1,2

A classic example: The battle of sexes

There are two Nash equilibria: (Opera,Opera)and (Rock,Rock) We will see that in some games with multiple Nash equilibria one equilibrium stands out as the compelling solution: in particular a convention can be developed

In the example above, the Nash equilibrium concept loses much of its appeal as a prediction of play since both equilibria seem equally compelling: none can be developed as a convention