Game Theory Lecture Notes Long v1

The Nash Equilibrium Strategic and Extensive Form Information Sets Types of Information Summary

Introduction to Game Theory

Gregory Wheeler

CENTRIA

Center for Artificial Intelligence ResearchUniversidade Nova de Lisboa, Portugal

Lecture 2

Nash Equilibrium & Information


elements of a game

Rules of a Game

• players

• actions

• payoffs

• information

Descriptions

• strategies

• equilibria

• outcome

Types

• (non)cooperativegames

• strategic andextensive form

• (in)completeinformation

• (a)symmetricinformation


limited applicability thus far

Very few games have a dominant-strategy equilibrium.

The vast majority doesn’t even have iterated-dominance equilibria.

The Nash Equilibrium is the most important and mostwidespread equilibrium concept.


boxed pigs game

Food!

cc

cc

A button push costs a pig 2 units&

10 units of food are dispensed


boxed pigs game

Food!

cc

cc

Big Pig:

Small Pig:If Big gets to the dispenser first, Big gets 9 Small gets 1.

If Small is at the dispenser first, Small gets 4 Big gets 6. If both arrive at the same time, Small gets 3 Big gets 7.


boxed pigs game

Food!

cc

cc

Big Pig:

Small Pig:If Big gets to the dispenser first, Big gets 9 Small gets 1.

If Small is at the dispenser first, Small gets 4 Big gets 6. If both arrive at the same time, Small gets 3 Big gets 7.

Example: (Press, Press) = 5 for Big, 1 for Small5 = 10 dispensed, - 2 for effort, -3 for Small.1 = 10 dispensed, - 2 for effort, -7 for Big.


boxed pigs

Small PigPress Wait

Big PigPress (5, 1) (4, 4)Wait (9,−1) (0, 0)

No dominant-strategy equilibrium

Because what Big chooses depends on what he thinks the Small will

choose.

If Big thinks Small will press, Big will wait for the food.

If Big thinks Small will wait, Small will press the button.

(Press, Wait) is an iterated-dominant dominance equilibrium.


nash equilibrium

Definition

a) The strategy profile s∗ is a weak Nash equilibrium if noplayer has incentive to deviate from his strategy given that theother players do not deviate. Formally,

∀i , π(s∗i , s∗−i ) ≥ π(s ′i , s

∗−i ), ∀s ′i .

b) A strategy profile strategy profile s∗ is a strong Nashequilibrium if no player is indifferent between his equilibriumstrategy and some other strategy. Formally,

∀i , π(s∗i , s∗−i ) > π(s ′i , s

∗−i ), ∀s ′i .


boxed pigs

Small PigPress Wait

Big PigPress 5, 1 4 , 4

Wait 9 ,−1 0, 0

Test for Nash equilibrium

If Big picks Press, Small picks Wait.

If Small picks Wait, Big picks Press. ∴ (Wait, Press) is Nash

If Big picks Wait, Small picks Wait.

If Small picks Press, Big picks Wait.

(Wait, Press) is also unique.


remarks on nash equilibria

Every dominant-equilibrium strategy is a Nash equilibrium strategy.

But not every Nash-equilibrium strategy is a dominant-equilibrium

strategy.

A Nash strategy only needs to be a best response to the other Nash

strategies, not to all possible strategies.

If a strategy is dominant, it is a best strategy to any strategies the other

players pick, including their equilibrium strategies.

The difference between Nash- and dominant-equilibria is illustrated by

The Modeller’s Dillema.


modeller’s dilemma

ColumnDeny Confess

RowDeny (0, 0) (−10, 0)

Confess (0,−10) (−8,−8)

No Dominant Strategy

Notice that (Deny, Deny) is (0,0) rather than (-1,-1).



ColumnDeny Confess

RowDeny 0, 0 −10, 0

Confess 0,−10 -8 , -8

Confess!

There is a weak dominant strategy equilibrium

Confess is a weakly dominant strategy for each player.

So, (Confess, Confess) is an iterated dominance equilibrium

(Confess, Confess) is also a strong Nash equilibrium.



ColumnDeny Confess

RowDeny 0, 0 −10, 0

Confess 0,−10 -8 , -8

Be Quiet!

There is another Nash equilibrium: (Deny, Deny)

(Deny, Deny) is a weak Nash equilibrium.

(Deny, Deny) is Pareto-superior:

(0,0) is uniformly superior to (-8,-8)


battle of the sexes

WomanPrize Fight Ballet

ManPrize Fight 2,1 0,0

Ballet 0,0 1,2

Two Nash Equilibria

(Prize Fight, Prize Fight) and (Ballet, Ballet).

Each strategy is Pareto-efficient: no other increases the payoff to

one player without decreasing that of the other.

Remark: The prisoner’s dilemma has a unique Nash equilibroum but

is not Pareto-efficient.


battle of the sexes

remarksUnlike the previous games, the first player to move in Battle of theSexes has an advantage. If the woman was first to purchase ballettickets, this commitment (investment) would induce the man to goto the ballet. In many game the player who moves first enjoys afirst-mover advantage.


elements of a game

Rules of a Game

• players

• actions

• payoffs

• information

Descriptions

• strategies

• equilibria

• outcome

Types



• (in)completeinformation

• (a)symmetricinformation


learning from another player

• All of the games from lecture 1 assumed that the moves weresimultaneous. These are called normal form or strategicform games.

• In sequential move games, the Player 2 acquires informationabout Player 1’s opening move before he decides to move.These are called extensive form games

• Extensive form games allows players to learn about eachplayer’s private information by observing their behavior.


strategic form

Let n be the number of players, k the number of variables in theoutcome vector, p the number of strategy profiles, and q thenumber of action profiles.

Definition

a) The strategic form (or normal form) consists of:

1. All possible strategy profiles s1, . . . , sp

2. Payoff functions mapping s i onto the payoff n-vectorπi (i = 1, . . . , p).

b) The outcome matrix consists of

1. All possible action profiles a1, . . . , aq.2. Outcome functions mapping ai onto the outcome k-vector

z i = (i = 1, . . . , q).


coordination

electric light game

• Alpha Inc. and Omega Corporation are trying to decide whether todesign the electric lamps they sell to use Large light bulbs or Smalllight bulbs.

• Alpha and Omega will both sell more lamps if they use the samesized bulbs.

a) (Large, Large) and (Small, Small) are both Nash equilibria.b) Suppose that (Large, Large) Pareto-dominates (Small, Small).

• Alpha and Omega would then benefit from coordinating on (Large,

Large), since they have a shared ranking of equilibria.


follow the leader

The electric light game is normal form. Suppose now that Alpha isallowed to go first, thus committing to a particular bulb size. Thenew game, ‘follow the leader’, has an output matrix that isidentical to the electric light game. But its strategic form isdifferent because Omega’s actions are no longer single actions.The strategy set SΩ for Omega consists of the following 4strategies:

sΩ1 (If Alpha chose L, choose L; if Alpha chose S, choose L)

sΩ2 (If Alpha chose L, choose L; if Alpha chose S, choose S)

sΩ3 (If Alpha chose L, choose S; if Alpha chose S, choose L)

sΩ4 (If Alpha chose L, choose S; if Alpha chose S, choose S)


follow the leader

We may rewrite the strategy set SAlpha for Omega consists of:

sΩ1 (If Alpha chose L, choose L; if Alpha chose S, choose L)

sΩ2 (If Alpha chose L, choose L; if Alpha chose S, choose S)

sΩ3 (If Alpha chose L, choose S; if Alpha chose S, choose L)

sΩ4 (If Alpha chose L, choose S; if Alpha chose S, choose S)

as: (L|L, L|S),(L|L, S |S),(S |L, L|S),(S |L, S |S)


follow the leader

OmegaΩ1 Ω2 Ω3 Ω4

L|L, L|S L|L,S |S S |L, L|S S |L,S |SAlpha

α1:Large 2 , 2 (E1) 2 , 2 (E2) −1,−1 −1,−1

α2:Small −1,−1 1, 1 −1,−1 1 , 1 (E3)

Payoffs to: (Alpha, Omega).

Best-response payoffs are boxed; underlined, if weak.

Equilibrium Strategies OutcomeE1 Large, (L|L, L|S) Both pick LargeE2 Large, (L|L,S |S)E3 Small , (S |L,S |S)


follow the leader

Small

Small

Large

Large

Large

Small

Follow-the-leader: extensive form

!1

!2

!

(1,1)

(2,2)

(-1,-1)

(-1,-1)

Equilibrium Strategies OutcomeE1 Large, (L|L, L|S) Both pick LargeE2 Large, (L|L,S |S)

Both pick Large

E3 Small , (S |L,S |S)

Both pick Small


follow the leader

Small

Small

Large

Large

Large

Small


!1

!2

!

(1,1)

(2,2)

(-1,-1)

(-1,-1)

Equilibrium Strategies OutcomeE1 Large, (L|L, L|S) Both pick LargeE2 Large, (L|L,S |S) Both pick LargeE3 Small , (S |L,S |S)

Both pick Small


follow the leader

Small

Small

Large

Large

Large

Small


!1

!2

!

(1,1)

(2,2)

(-1,-1)

(-1,-1)

Equilibrium Strategies OutcomeE1 Large, (L|L, L|S) Both pick LargeE2 Large, (L|L,S |S) Both pick LargeE3 Small , (S |L,S |S) Both pick Small


follow-the-leader order of play

Order of play

1. α chooses bulb size to be either Large or Small.

2. Ω chooses bulb size to be either Large or Small.

NB: normal form lays out all possible strategies and then compares theirpayoffs. Order of play gives us a better example. Two other ways todescribe a game:

a) extensive form

b) the game tree


extensive form and the game tree

Definition

a) A node is a point in the game at which some player or Naturetakes an action, or the game ends.

b) A successor to node X is a node that may occur later in thegame if X has been reached.

c) A predessor to node X is a node that must be reached beforeX can be reached.

d) A starting node is a node with no successors.

e) An end node or end point is a node with no successors.

f) A branch is one action in a player’s action set at a particularnode.

g) A path is a sequence of nodes and branches leading from thestarting node to an end node.


extensive form and the game tree

DefinitionThe extensive form is a description of a game consisting of

1. A configuration of nodes and branches running without anyclosed loops from a single starting node to its end nodes.

2. An indication of which nodes belongs to which player

3. The probabilities of Nature uses to choose different branchesat its nodes.

4. The information sets into which each player’s nodes aredivided

5. The payoffs for each player at each end node.

The game tree is the same as the extensive form except that (5)is replaced by

5′. The outcomes at each end node.


follow the leader

Small

Small

Large

Large

Large

Small

Ranked Coordination in extensive form

!1

!2

!

(1,1)

(2,2)

(-1,-1)

(-1,-1)Omega's

InformationSet

Equilibrium Strategies OutcomeE1 Large, (L|L, L|S) Both pick LargeE2 Large, (L|L,S |S) Both pick LargeE3 Small , (S |L,S |S) Both pick Small


information sets

“How much did the President know, and when did heknow it?” –Senator Baker, during the Watergate hearings

• A game’s information structure, like the order of its moves, isoften obscured in the strategic form.

• But to uncover the information structure of a game requiressome technical definitions.


information sets

DefinitionPlayer i ’s information set ωi at any particular point of the gameis the set of different nodes in the game tree that he knows mightbe the actual node, but between which he cannot distinguish bydirect observation.


Top

Lower

Information sets and Information Partitions

!

(1,1)

(1,1)

!3

!4

(4,4)

(4,4)

(4,4)

(4,4)

(1,1)

(1,1)

(1,1)

Bottom

Middle

(1,1)(1,1)

2008 2009 2010

Omega's Information

set

!1

!2

Ω3 and Ω4 are in the same information set (cloud) for Omega, but not

for Alpha.


facts about information sets

(1) One node cannot belong to two different information sets.

Example

Suppose Ω3 belonged to Ω3,Ω4 and Ω2,Ω3 (unlike the diagram).

Then if the player reached Ω3, he wouldn’t know whether he was in

Ω3,Ω4 and Ω2,Ω3, which would imply that he was in an information

set Ω2,Ω3,Ω4.



(2) If the nodes in one of Omega’s information sets are nodes atwhich he moves, his action set must be the same at each node.

ExampleIn the diagram, Omega has the same action set at nodes Ω3 and Ω4.Because if he had some different action set at Ω3 (say), he would knowthat he is at Ω3 and his information set would reduce to Ω3.By similar reasoning, Ω1Ω2 is not an information set because Ω1 has 3

moves and the others have four moves, even though they both have the

same payoff.



(3) Rational deductions are not observations

Example

Look at the information set in the diagram. Alpha would clearly choose

Bottom, since the payoff of 8 dominates the payoff of 4. So, Omega

would have a reason to distinguish between Ω3 and Ω4. But deductions

are not observations, so the game tree does not split this information set.



(4) Information sets show the effects of unobserved moves byNature.

Example

Suppose that Nature moved before Alpha in the diagram. Omega’s

information sets would be depicted the same way.


information partition

DefinitionPlayer i ’s information partition is a collection of his informationsets such that:

1. Each path is represented by one node in a single informationset in the partition, and

2. The predecessors of all nodes in a single information set are inone information set.


information partition

Information partitions carve up the set of possible nodes into thosethat a player can distinguish.

Example

One of Alpha’s information partitions is (Ω1, Ω2, Ω3, Ω4). The

definition rules out including α in that partition, because the path

through α and Ω1 would be represented by two nodes. Instead, (α) is

a separate information partition.


information partitions

Nodes I II III IV

!1

!2

!3

!4

!1

!2

!3

!4

!1!2!3!4

!1!2!3

!4

!1

!2

!3

!4

Partition I is finer; Partition II is coarser.Partition I is a refinement of Parition II .Partition II is a coarsening of Partition INeither III nor IV are allowed in the Alpha/Omega game.

Exercise: Say why.


better information

• The formal definition of better information is a finerinformation partition.

• Not all information partitions are refinements or coarseningsof each other.

• Consider partitions II and IV . II separates nodes into 3information sets, IV into 2 information sets. IV is not acoarsening of II , however, and one cannot say for sure that aplayer with partition IV has worse information than a playerwith II .


advantages of ignorance

• Information quality is defined independently of the utility tothe player: it is possible for a player’s information to improveand for his equilibrium payoff to fall as a result. (Players mayrationally prefer to have worse information!)

• Coarse information can have several advantages:

a) It may permit a player to engage in trade because other playersdo not fear his superior information.

b) It may give a player a stronger strategic position because heusually has a strong position and is better off not knowing thata particular realization of the game is weak.

c) Poor information, like in the real economy, may permit playersto insure one another.


common knowledge

We have been implicitly assuming that the players know what thegame tree looks like. And we have been assuming that the playersalso know that the other players know what the tree looks like.Common knowledge avoids spelling out the infinite recursionthat this leads to.


common knowledge

DefinitionInformation is common knowledge if it is known to all theplayers, if each player knows that all the players know it, if eachplayer knows that all players know that all the players know it, andso forth ad infinitum.

NB: Because of the recursion, common knowledge is stronger thanmutual knowledge.

Example: We might both know Urdu, but we wouldn’t begin speaking to

one another in Urdu unless we knew that we each knew Urdu.


types of information

Information Meaning

Category

Perfect Each information set is a singleton

Certain Nature does not move after any player moves

Symmetric No player has information different from any other

player when he moves, or at the end nodes

Complete Nature does not move first, or her initial move

is observed by every player


perfect / imperfect information

DefinitionIf the game is of perfect information, each information set is asingleton. Otherwise the game is of imperfect information.

In games of perfect information, a player knows exactly where he is in the

game tree. No moves are simultaneous. All players observe Nature’s

move. Follow the Leader is a game of perfect information.


certainty / uncertainty

DefinitionA game of certaintly has no moves by Nature after any playermoves. Otherwise the game is one of uncertainty

Moves of Nature in a game may or may not be revealed to players

immediately. A game of certainty can be a game of perfect information if

there are no simultaneous moves.


symmetric / asymmetric information

DefinitionIn a game of symmetric information, a player’s information setat

a) any node where he chooses an action, or

b) an end node

contains at least the same elements as the information sets ofevery other player. Otherwise the game is one of asymmetricinformation.

Asymmetric games are kinds of imperfect information games, where some

player has useful private information: i.e., an information partition that

is different and not worse than another player’s. Symmetric information

can have moves by nature or simultaneous moves, so long as no player

has an information advantage.


incomplete information

DefinitionIn an incomplete information game, Nature moves first and isunobserved by at least one of the players. Otherwise the game isone of complete information.

A game with incomplete information also has imperfect information,

because some player’s information set includes more than one node.Two

kinds of games have complete but imperfect information: games with

simultaneous moves and games where Nature makes a move late in the

game which is not immediately revealed to all players. Many games of

incomplete information are asymmetric games. A symmetric incomplete

game: Nature assigns Students different abilities unknown to Dean. As

time passes, the abilities become common knowledge.


elements of a game

Rules of a Game

• players

• actions

• payoffs

• information

Descriptions

• strategies

• equilibria

• outcome

Types



• information:• perfect• certain• symmetric• complete

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