Game Theory - unipr.it · Game Theory Game theory can be de ned as the study of mathematical models of con ict and cooperation between intelligent rational decision-makers. (R. Myerson)

Outline Introduction Game Representations Reductions Solution Concepts

Game Theory

Enrico Franchi

May 19, 2010

Enrico Franchi Game Theory


1 IntroductionScope of Game TheoryAgent preferencesUtility Functions

2 Game RepresentationsExample: Game-1Extended FormStrategic FormEquivalences

3 ReductionsBest ResponseDomination

4 Solution ConceptsNash EquilibriumPareto OptimalityExamplesImportance of Nash Equilibria



Game Theory

Game theory can be defined as the study of mathematicalmodels of conflict and cooperation between intelligent rationaldecision-makers. (R. Myerson)

A decision-maker is rational if he makes decisions consistentlyin pursuit of his own objectives

A player is intelligent if he knows everything that we knowabout the game and he can make inferences about thesituation that we can make



Utility

The concept of “utility” is grounded in the concept of“preference”

Let O denote a finite set of outcomes. For any o1, o2 ∈ O:

o1 o2 if the agent weakly prefers o1 to o2

o1 ∼ o2 if the agent is indifferent between o1 and o2

o1 o2 if the agent strongly prefers o1 to o2

A lottery is a probability distribution [p1 : o1, . . . , pk : ok ]where each oi ∈ O, each pi ∈ [0, 1] and

∑ki=1 p1 = 1

We extend to lotteries.



Preference relation axioms

Completeness ∀o1, o2 ∈ O o1 o2 or o2 o1 or o1 ∼ o2.

Transitivity If o1 o2 and o2 o3, then o1 o3.

Substitutability If o1 ∼ o2 then for all sequences of one or moreoutcomes o3, . . . , ok and sets of probabilitiesp, p3, . . . , pk for which p +

∑ki=3 pi = 1

[p : o1, p3 : o3, . . . pk : ok ] = [p : o2, p3 : o3, . . . pk : ok ]

Decomposability If ∀oi ∈ O, Pl1(oi ) = Pl2(oi ) then l1 ∼ l2

Monotonicity If o1 o2 and p > q, then[p : o1, 1− p : o2] [q : o1, 1− q : o2]

Continuity If o1 o2 and o2 o3, then ∃p ∈ [0, 1] suchthat o2 ∼ [p : o1, 1− p : o3].



Expected Utility Theorem

Theorem (von Neumann-Morgenstern, 1944)

If a preference relation satisfies the axioms completeness,transitivity, substitutability, decomposability, monotonicity andcontinuity, then there exists a function u : O 7→ [0, 1] satisfying:

u(o1) > u(o2) iff o1 p2 (1)

u([p1 : o1, . . . , pk : ok ]) =k∑

i=1

pi · u(oi ) (2)

Proof.

See R. Myerson or Y. Shoham&K. Leyton-Brown.



Description of Game-1

At the beginning of the game, player 1 and player 2 put 1euro in the pot

Player 1 draws a card from a shuffled deck of cards in whichhalf the cards are red and half are black

Player 1 looks the card privately and decides whether to raiseor fold

If he folds, the game ends. The money goes to 1 if the card isred or to 2 if it is black

If he raises, he puts another euro in the pot and player 2decides whether to meet or pass

If player 2 passes, game ends and money goes to player 1

If player 2 meets, he adds another euro to the pot and gameends. Player 1 takes the money if the card is red and player 2takes the money if it is black



An (in?)adequate graph of Game-1

(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1 2

21



Comments on the graph of Game-1

Each terminal node reports “utilities” for both players

Nodes have a label: 0 for “chance nodes”, i for nodes playedby the i-th player

Chance nodes edges are labelled with the probability of thatedge to be taken

Player edges are labelled with the action the player chooses

Lacks information. E.g., from the graph, we do notunderstand player 2 does not know the color of the card

If the game is complex, the representation is very unpractical



Extended Form of Game-1

(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b



Description of the Extended Form(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b

Each nonterminal node has a player label1 . . . n. Nodes assigned a player label 0 arecalled chance nodes. N = 1, . . . , n is theset of players in the game. Nodes labelledwith i are decision nodes for player i

Every alternative at a chance node has alabel that specifies its probability. The sumof the probabilities sum to 1

Every node that is controlled by a player hasa second label that specifies the informationstate that the player would have if the pathof the play reached this node. The playerknows only what specified by this label



Description of the Extended Form(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b

Each alternative at a node that is controlledby a player has a move label. For any twonodes x and y with the same player labeland the same information label, and for anyalternative at node x there must be exactlyone alternative at node y that has the samenode label

Each terminal node has a label that specifiesa vector of n numbers (u1, . . . , un). Theseare the payoff to player i in some outcome ofthe game



On extended form (again...)

If two nodes belong to the same player and have the sameinformation state, the player is unable to distinguish themduring the game

It is also true that if two nodes belong to the same player andhe is unable to distinguish them, then they should have thesame information state

A game satisfies perfect recall if whenever a player moves, heremembers all the information that he knew earlier in thegame, including all his past moves. We often assume thiscondition to hold

A game has perfect information if no two nodes have thesame information state. This entails that each player exactlyknows where he is in the graph



Strategies

Definition (Strategy)

A strategy for a player in an extensive-form game is any rule fordetermining a move at every possibile information state in thegame

Let Si be the set of all information states for player i

Let s ∈ Si , Ds is the set of moves available to i when hemoved in a node with information state s

A strategy ci is a function mapping an information state to anaction available to i in that information state

With an abuse of notation, the set of strategies for i is

×s∈Si Ds (why is this acceptable?)

A strategy is a complete rule that specifies a move for theplayer for all possible cases, even though only one will arise



Example

(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b

Strategies for player 1:

a : raise, b : raise,a : fold, b : raise,a : raise, b : fold,a : fold, b : fold

Strategies for player 2:pass, meet



A game

(2,2)

(4,0)

(1,0)

(3,1)

1.1

2.2

2.2

L

R

L

R

T

B

Player 2 does not observeplayer 1’s move

Player 1 would be better offchoosing T against bothplayer 2’s strategies



Another game

(2,2)

(4,0)

(1,0)

(3,1)

1.1

2.2

2.3

L1

R1

L2

R2

T

B

Player 2 observes 1’s actualchoice before choosing

Player 1 can influence player2

L1 is the best response to T,R2 is the best response to B

Strategies for player 2 are:

T : L1,B : L2, T : R1,B : L2,T : L1,B : R2, T : R1,B : R2

If the game is played onlyonce, we could not discoverplayer 2 strategy from hismove



The Battle of Sexes (Example)

Example (The Battle of Sexes)

A wife and a husband have to decide what to do the evening.They both want to go to the cinema, however, they disagree onwhich movie to see. The husband prefers to see Lethal Weapon(LW), while the wife prefers to see Wondrous Love (WL). Luckily,they both agree that going to the cinema alone is far worse thanseeing an uninteresting movie.

Unluckily, they are a strange couple and both are very stubborn.Once they have decided where they intend to go, they won’tchange their minds, no matter what the partner does. They justhope to “guess” the right choice, since they have no idea whattheir parter has in her/his mind.



Battle of Sexes

1.a

2.b

2.b

WL

LW

WL

LW

WL

LW

(3,1)

(0,0)

(0,0)

(1,3)



Battle of Sexes (another point of view...)

2.b

1.a

1.a

WL

LW

WL

LW

WL

LW

(3,1)

(0,0)

(0,0)

(1,3)



Games in strategic form

Definition (Strategic Form)

A game in strategic form is a tuple Γ:

Γ = (N, (Ci )i∈N , (ui )i∈N) (3)

N is a nonempty set of players (usually finite)

For each i ∈ N, Ci is the nonempty set of (pure) strategiesavailable to i

A strategy profile is a combination of strategies

C =×j∈N Cj is the set of strategy profiles

For each i , ui : C 7→ R is a utility function specifying thepayoff for player i



Conversion to Strategic Form

Given a game in extended form, automated procedures toconvert it in strategic form exist

von Neumann and Morgenstern procedure to get the normalrepresentation in strategic form of an extended form game:

N is the same set of players of the extended formFor any i ∈ N, Ci is the same set of strategies available to i inthe extended formLet wi (x) the payoff for player i in terminal node x in theextended form. Let Ω∗ the set of terminal nodes in theextended form game. The utility of a strategy profile c , forplayer i is:

ui (c) =∑x∈Ω∗

P(x |c)wi (x)



Probability P(x |c) of a node

If x is the root of the game, P(x |c) = 1

If x immediately follows a chance node y and q is the chanceprobability associated to the branch from y to x , thenP(x |c) = qP(y |c)

If x immediately follows a player node y belonging to player iin the information state r , then P(x |c) = P(y |c) if ci (r) isthe move label on the alternative from y to x and P(x |c) = 0otherwise.



Card game in strategic form

(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b

C2

C1 meet pass

a : raise, b : raise 0,0 1, -1a : raise, b : fold 0.5, -0.5 0, 0a : fold, b : raise -0.5, 0.5 1, -1a : fold, b : fold 0,0 0, 0



Another game in strategic form

(2,2)

(4,0)

(1,0)

(3,1)

1.1

2.2

2.2

L

R

L

R

T

B

C2

C1 L R

T 2, 2 4, 0B 1, 0 3, 1



Another game in strategic form

(2,2)

(4,0)

(1,0)

(3,1)

1.1

2.2

2.3

L1

R1

L2

R2

T

B

C2

C1 L1L2 L1R2 R1L2 R1R2

T 2, 2 2, 2 4, 0 4, 0B 1, 0 3, 1 1, 0 3,1

L1R2 means player 2 answers L1 to T and R2 to B.



Notation

N−i is the set of all players except i

Let C−i =×j∈N−iCj

Let e−i = (ej)j∈N−i∈ C−i and di ∈ Ci , (e−i , di ) ∈ C is the

strategy profile such that the i-th component is di and allother components are in e−i .

∆(X ) is the set of probability distributions over X

The set of points in X that maximize function f is:

arg maxy∈X

=

y ∈ X

∣∣∣∣f (y) = maxz∈X

f (z)



Best Response Equivalence

If a player i believes that some distribution η ∈ ∆(C−i )predicts the behaviour of other players, then i chooses hisstrategy in Ci to maximize his payoff.

The set of “best responses” to η is:

G ηi (C−i ) = arg max

di∈Ci

∑e−i∈C−i

η(e−i )uI (e−i , di )

Two games Γ and Γ are best-response equivalent iff the set ofbest responses for every player and every possible probabilitydistribution over the others’ strategies are the same



Payoff equivalence

Two strategies ei and di are payoff equivalent iff∀c−i ∈ C−i , ∀j ∈ N uj(c−i , di ) = uj(c−i , ei )

Payoff equivalent strategies can be merged and substitutedwith a single strategy

If all payoff-equivalent strategies are substituted, we say thegame in in purely reduced normal representation



Randomized Strategies

Definition

A randomized strategy for player i is any probability distributionover the set of Ci . ∆(Ci ) is the set of randomized strategies forplayer i . Conversely, strategies in Ci are called pure strategies.

Let σi ∈ Ci , σi (ci ) is the probability that i plays ci is he isimplementing σi .

A strategy di is randomly redundant iff there is σi ∈ ∆(Ci )such that σi (di ) = 0 and

uj(c−i , di ) =∑ei∈Ci

σi (ei )uj(c−i , ei ) ∀c−i ∈ C−i , ∀j ∈ N

The [fully] reduced normal representation is derived from thepurely reduced normal representation by eliminating allrandomly redundant strategies.



Domination

Definition (Domination)

Let Γ = (N, (Ci )i∈N , (ui )i∈N) a game in strategic-form. A strategydi ∈ Ci is strongly dominated for player i iff there exists somerandomized strategy σi ∈ ∆(Ci ) such that:

ui (c−i , di ) <∑ei∈Ci

σ(ei )ui (c−i , ei ) ∀−i ∈ C−i (4)



Elimination of Dominated Strategies

It can be proved that if di is strongly dominated for player i ,than di is never a best response, no matter what he believesabout other players’ strategies.

Removing dominated strategies does not change the analysisof the game

It is possible that a previously not strongly dominated strategybecomes strongly dominated after removal of some stronglydominated strategy

The order in which dominated strategies are removed does notmatter: it is possible to develop an algorithm to eliminate allstrongly dominated strategies

It is better not to remove simply dominated strategies(defined as strongly dominated, with ≤ instead of <)



Elimination of Dominated Strategies

Let Γ = (N, (Ci )i∈N , (ui )i∈N) a strategic form game and let

C(1)i denote the set of all strategies in Ci that are not strongly

dominated for i

Consider Γ(1) = (N, (C(1)i )i∈N , (ui )i∈N)

By induction, for every positive integer k we can define the

strategic-form game Γ(k) = (N, (C(k)i )i∈N , (ui )i∈N), where

C(k)i is the set of all strategies in C

(k−1)i that are not strongly

dominated for i in Γ(k−1)

Ci ⊇ C(1)i ⊇ C

(2)i ⊇ . . . ∀i ∈ N

Since every set is finite, there is K such that

C(K)i = C

(K+1)i = C

(K+2)i = . . . ∀i ∈ N



Residual Game

Definition

Given the number K , we let Γ(∞) = Γ(K) and C(∞)i = Ci (K ) for

every player i . The strategies in C(∞)i are iteratively undominated

in the strong sense. Γ(∞) is called the residual game generatedfrom Γ by iterative strong domination.

Since every player is rational, no player would use a dominated

strategy. So he would choose in C(1)i . Moreover, since he is

intelligent, he knows no player j would choose a strategy outside

C(1)j . As a consequence, he should choose a strategy in C

(2)i .

Repeatedly using the assumptions of rationality and intelligence, itis clear that only iteratively undominated strategies are played.



Notation

Let σ ∈×i∈N ∆(Ci ) a randomized strategy profileWe define:

ui (σ) =∑ci∈C

∏j∈N−i

σj(cj)

ui (c)

ui (σ−i , τi ) =∑ci∈C

∏j∈N−i

σj(cj)

τi (ci )ui (c)

We denote with [di ] the randomized strategy which givesprobability 1 to di

Consider σi =∑

ci∈Ciσi (ci )[ci ]

ui (σ−i , [di ]) =∑

c−i∈C−i

∏j∈N−i

σj(cj)

ui (c−i , di )



Nash Equilibrium

Definition (Nash Equilibrium)

A randomized-strategy profile σ is a Nash equilibrium of Γ iff itsatisfies condition:

σi (ci ) > 0⇒ ci ∈ arg maxdi∈Ci

ui (σ−i , [di ]) (5)

for every player i and every ci ∈ Ci

Lemma

A condition equivalent to Equation (5) is that:

ui (σ) ≥ ui (σ−i , τi ) (6)

for every player i and for every τi ∈ ∆(Ci ).



Nash Theorem

Theorem (Nash Theorem)

Given any finite game Γ in strategic form, there exists at least oneequilibrium in×i∈N ∆(Ci ).



Card game in strategic form

(red)

fold

raisemee

t

pass

meet

passraise

fold

(black)

.5

.5

(2,-2)

(-2,2)

(1,-1)

(1,-1)

(1,-1)

(-1,1)

0

1.a 2.0

2.01.b

C2

C1 meet pass

a : raise, b : raise 0,0 1, -1a : raise, b : fold 0.5, -0.5 0, 0a : fold, b : raise -0.5, 0.5 1, -1a : fold, b : fold 0,0 0, 0



Pareto Optimality

A game may have multiple equilibria

Some equilibria are inefficient

An outcome is (weakly) Pareto Efficient iff there is no otheroutcome that would make all players better off



The Prisoner’s Dilemma

C2C1 g2 f2g1 5,5 0,6f1 6,0 1,1



Battle of Sexes

C2C1 LW WLLW 3,1 0,0WL 0,0 1,3



Importance of Nash Equilibria

We are trying to predict the behaviour of rational intelligentplayers

Their behaviour should tend to equilibrium because otherwisethey would change their strategies

Equilibria should be self-fulfilling prophecies

The focal-point effect, equity and efficiency


Documents

Game Theory - unipr.it · Game Theory Game theory can be de ned as the study of mathematical models of con ict and cooperation between intelligent rational decision-makers. (R. Myerson)