Post on 19-Jul-2020
Outline Introduction Game Representations Reductions Solution Concepts
Game Theory
Enrico Franchi
May 19, 2010
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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].
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
Battle of Sexes
1.a
2.b
2.b
WL
LW
WL
LW
WL
LW
(3,1)
(0,0)
(0,0)
(1,3)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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 <)
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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.
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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 )
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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 ).
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
Nash Theorem
Theorem (Nash Theorem)
Given any finite game Γ in strategic form, there exists at least oneequilibrium in×i∈N ∆(Ci ).
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
The Prisoner’s Dilemma
C2C1 g2 f2g1 5,5 0,6f1 6,0 1,1
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
Battle of Sexes
C2C1 LW WLLW 3,1 0,0WL 0,0 1,3
Enrico Franchi Game Theory
Outline Introduction Game Representations Reductions Solution Concepts
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
Enrico Franchi Game Theory