104
ecs289m Spring, 2008 Non-cooperative Games S. Felix Wu Computer Science Department University of California, Davis [email protected] http://www.cs.ucdavis.edu/ ~wu/
ecs289m Spring, 2008
Non-cooperative Games
S. Felix WuComputer Science DepartmentUniversity of California, Davis
[email protected]://www.cs.ucdavis.edu/~wu/
05/06/2008 Non-cooperative Game Theory 2
Non-cooperative Game Theory
• The Structure of the Game– Four elements– Extensive versus Strategic/Normalized
forms– The Structure of the Game
• Strategies– Mixed, Rationalizable, Dominance
• Nash Equilibrium• Dynamic Game
05/06/2008 Non-cooperative Game Theory 3
Non-cooperative Game Theory
• The Structure of the Game– Four elements– Extensive versus Strategic/Normalized
forms– The Structure of the Game
• Strategies– Mixed, Rationalizable, Dominance
• Nash Equilibrium• Dynamic Game
05/06/2008 Non-cooperative Game Theory 4
Four Elements
• A game: Formal representation of a situation of strategic interdependence
– Set of agents, I (|I|=n)• AKA players
– Each agent, j, has a set of actions, Aj• AKA moves
– Actions define outcomes• For each possible set of actions there is an
outcome.
– Outcomes define payoffs• Agents’ derive utility from different outcomes
05/06/2008 Non-cooperative Game Theory 5
Matching Pennies
• Agents: {Alice, Bob}• Actions: {Head, Tail}• Outcomes: {Matched, Not}• Payoffs: {(-1, 1), (1, -1)}
Alice gives 1 dollar to Bob!
05/06/2008 Non-cooperative Game Theory 6
Matching Pennies
• Agents: {Alice, Bob}• Actions: {Head, Tail}• Outcomes: {Matched, Not}• Payoffs: {(-1, 1), (1, -1)}
Bob gives 1 dollar to Alice!
05/06/2008 Non-cooperative Game Theory 7
Matching Pennies
• Simultaneous moves• Sequential moves
05/06/2008 Non-cooperative Game Theory 8
Extensive Form (or Game Tree)
Agent Alice
Agent Bob
H
H H
T
TT
Action
Terminal node (outcome)
Payoffs
(-1,+1) (-1,+1)(+1,-1) (+1,-1)
05/06/2008 Non-cooperative Game Theory 9
Tick-Tack-Toe
05/06/2008 Non-cooperative Game Theory 10
Matching Pennies
• Sequential moves• Simultaneous moves
– “Bob doesn’t know Alice’s move”– Not Perfect Information
05/06/2008 Non-cooperative Game Theory 11
Bob can not distinguish…
Agent Alice
Agent Bob
H
H H
T
TT
Action
Terminal node (outcome)
Payoffs
(-1,+1) (-1,+1)(+1,-1) (+1,-1)
05/06/2008 Non-cooperative Game Theory 12
Agent Alice
Agent Bob
H
H H
T
TT
Action
Terminal node (outcome)
Payoffs
Another representation …
(-1,+1) (-1,+1)(+1,-1) (+1,-1)
05/06/2008 Non-cooperative Game Theory 13
Information Sets
Agent Alice
Agent Bob
H
H H
T
TT
(-1,+1) (-1,+1)(+1,-1) (+1,-1)
Action
Terminal node (outcome)
Payoffs
05/06/2008 Non-cooperative Game Theory 14
Matching Pennies
• Sequential moves• Simultaneous moves
– “Bob doesn’t know Alice’s move”– Not Perfect Information
• Assumption: “Perfect Recall”– “remember the history”
05/06/2008 Non-cooperative Game Theory 15
Example #1: Perfect Recall?
p q
x x yy
a b ba
05/06/2008 Non-cooperative Game Theory 16
Example #2: Perfect Recall?
p q
x x yy
a b ba baba
05/06/2008 Non-cooperative Game Theory 17
Example #2: Perfect Recall?
p q
x x yy
a b ba baba
Felix likes that for sure!
05/06/2008 Non-cooperative Game Theory 18
Matching Pennies
• Sequential moves• Simultaneous moves
– “Bob doesn’t know Alice’s move”– Not Perfect Information
• Assumption: “Perfect Recall”• Perfect Information: each information
set contains a single decision node.
05/06/2008 Non-cooperative Game Theory 19
Matching Pennies
• Sequential moves• Simultaneous moves
– “Bob doesn’t know Alice’s move”
• Assumption: “Perfect Recall”• Perfect Information: each information
set contains a single decision node.• Random moves
– Flip a coin to decide Head or Tail
05/06/2008 Non-cooperative Game Theory 20
Common Knowledge
• Structure of the Game• All the players know about the structure
of the game, all players know that their rivals know it, and,…
05/06/2008 Non-cooperative Game Theory 21
Strategy
• Let HX denote the collection of agent X’s information sets, A the set of possible actions in the game, and C(h) A the set of actions possible at information set h.
• A strategy for agent X is a function:– €
⊂
€
sX : HX → A
∀h ∈ HX ,sX (h)∈ C(h)
Per player (Info Set => Action)
05/06/2008 Non-cooperative Game Theory 22
Strategies for Matching Pennies
• Bob’s four possible strategies:
– S1: Play H if Alice plays H; play H if Alice plays T
– S2: Play H if Alice plays H; play T if Alice plays T
– S3: Play T if Alice plays H; play H if Alice plays T
– S4: Play T if Alice plays H; play T if Alice plays T
Information Sets
05/06/2008 Non-cooperative Game Theory 23
Strategies imply…
• A sequence of moves actually taken• A probability distribution over the
terminal nodes of the game
• Strategies ~ Outcomes ~ Payoffs
05/06/2008 Non-cooperative Game Theory 24
Strategies imply…
• A sequence of moves actually taken• A probability distribution over the
terminal nodes of the game
• Strategies ~ Outcomes ~ Payoffs– “Strategic/Normal Forms”
€
ΓN = [I,{Si},{ui(•)}]
05/06/2008 Non-cooperative Game Theory 25
Matching Pennies
Alice
Bob
H
H
T
T
-1, +1
-1, +1
+1, -1
+1, -1
ActionOutcome
Payoffs
05/06/2008 Non-cooperative Game Theory 26
Bob doesn’t know Alice’s move…
Alice
Bob
H
H
T
T
-1, +1
-1, +1
+1, -1
+1, -1
ActionOutcome
Payoffs
Strategies
05/06/2008 Non-cooperative Game Theory 27
Strategies for Matching Pennies
• Bob’s four possible strategies:
– S1: Play H if Alice plays H; play H if Alice plays T
– S2: Play H if Alice plays H; play T if Alice plays T
– S3: Play T if Alice plays H; play H if Alice plays T
– S4: Play T if Alice plays H; play T if Alice plays T
Information Sets
05/06/2008 Non-cooperative Game Theory 28
Bob does know Alice’s move…
Alice
Bob
H
S1
T
-1, +1
+1, -1
+1, -1
S2 S3 S4
-1, +1
-1, +1
+1, -1
+1, -1
-1, +1
05/06/2008 Non-cooperative Game Theory 29
Alice
Bob
H
S1
T
-1, +1
+1, -1
+1, -1
S2 S3 S4
-1, +1
-1, +1
+1, -1
+1, -1
-1, +1
S2: Play H if Alice plays H; play T if Alice plays T
05/06/2008 Non-cooperative Game Theory 30
Non-cooperative Game Theory
• The Structure of the Game– Four elements– Extensive versus Strategic/Normalized
forms– The Structure of the Game
• Strategies– Mixed, Rationalizable, Dominance
• Nash Equilibrium• Dynamic Game
05/06/2008 Non-cooperative Game Theory 31
Strategies
• Strategy:– A strategy, sj, is a complete contingency plan; defines
actions agent j should take for all possible information sets of the world
• Strategy profile: s =(s1,…,sn )– s-i = (s1,…,si-1,si+1,…,sn)
• Utility function: ui(s)– Note that the utility of an agent depends on the strategy
profile, not just its own strategy– We assume agents are expected utility maximizers
05/06/2008 Non-cooperative Game Theory 32
Mixed Strategy
€
agent[i] : sij ∈ Si
σ i : Si →[0,1]
σ i(sij ) ≥ 0, σ i(sij
j
∑ ) =1
S = Si
1≤ i≤n
∏ , [s∈S
∑ σ 1(s1)σ 2(s2)...σ n (sn )]ui(s)
Randomization over a set of pure and deterministic strategies for each agent
05/06/2008 Non-cooperative Game Theory 33
Mixed Strategy
€
agent[i] : sij ∈ Si
σ i : Si → [0,1]
σ i(sij ) ≥ 0, σ i(sij
j
∑ ) =1
S = Si
1≤ i≤n
∏ , [s∈S
∑ σ 1(s1)σ 2(s2)...σ n (sn )]ui(s)
Δ(Si) = {(σ 1i,...,σ Mi)∈ ℜM : (σ mi ≥ 0,∀m)∧( σ mi
m=1
M
∑ =1)}
ΓN = [I,{Δ(Si)},{ui(•)}]
Randomization over a set of pure and deterministic strategies for each agent
05/06/2008 Non-cooperative Game Theory 34
Dominant/Dominated Strategy
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) > ui( ′ s i,s−i)
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) < ui( ′ s i,s−i)
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≤ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 35
Dominant/Dominated Strategy
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) > ui( ′ s i,s−i)
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) < ui( ′ s i,s−i)
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≤ ui( ′ s i,s−i)
Strictly dominant
Strictly dominated
Weakly dominated
05/06/2008 Non-cooperative Game Theory 36
Dominant Strategies
• Agents’ will play best-response strategies– Rationalizable
• A dominant strategy is– a best response for all s-i
– They do not always exist– Inferior strategies are called dominated
€
ΓN = [I,{Si},{ui(•)}]
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 37
Alice
Bob
H
S1
T
-1, +1
+1, -1
+1, -1
S2 S3 S4
-1, +1
-1, +1
+1, -1
+1, -1
-1, +1
S2: Play H if Alice plays H; play T if Alice plays T
Assuming Alice and Bob simultaneously choose a strategy..
05/06/2008 Non-cooperative Game Theory 38
Iterated Elimination of “Dominated”
• Let RiSi be the set of removed strategies for agent i
• Initially Ri=Ø• Choose agent i, and strategy si such that siSi\Ri
and there exists si’ Si\Ri such that
• Add si to Ri, continue
ui(si’,s-i)>ui(si,s-i) for all s-i S-i\R-i
05/06/2008 Non-cooperative Game Theory 39
Prisoner’s Dilemma
• Art and Bob been caught stealing a car: sentence is 2 years in jail.
• DA wants to convict them of a big bank robbery: sentence is 10 years in jail.
• DA has no evidence and to get the conviction, he makes the prisoners play a “game”.
05/06/2008 Non-cooperative Game Theory 40
Rules of the Game
Players cannot communicate with one another.
If both confess to the larger crime, each will receive a sentence of 3 years for both crimes.
If one confesses and the accomplice does not, the one who confesses will receive a sentence of 1 year, while the accomplice receives a 10-year sentence.
If neither confesses, both receive a 2-year sentence.
05/06/2008 Non-cooperative Game Theory 41
Another one
05/06/2008 Non-cooperative Game Theory 42
Strategies
The strategies of a game are all the possible outcomes of each player.
The strategies in the prisoners’ dilemma are:
Confess to the bank robbery
Deny the bank robberyFour outcomes:
Both confess.
Both deny.
Art confesses and Bob denies.
Bob confesses and Art denies.
05/06/2008 Non-cooperative Game Theory 43
How would Bob play?
05/06/2008 Non-cooperative Game Theory 44
Iterated Elimination of “Dominated”
• Let RiSi be the set of removed strategies for agent i
• Initially Ri=Ø• Choose agent i, and strategy si such that siSi\Ri
and there exists si’ Si\Ri such that
• Add si to Ri, continue
ui(si’,s-i)>ui(si,s-i) for all s-i S-i\R-i
05/06/2008 Non-cooperative Game Theory 45
Iterated Elimination of Dominated Strategies
• Let RiSi be the set of removed strategies for agent i
• Initially Ri=Ø• Choose agent i, and strategy si such that siSi\Ri
and there exists si’ Si\Ri such that
• Add si to Ri, continue
• We might not be able to eliminate many!
ui(si’,s-i)>ui(si,s-i) for all s-i S-i\R-i
05/06/2008 Non-cooperative Game Theory 46
Art versus Bob
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
05/06/2008 Non-cooperative Game Theory 47
Can Bob eliminate ONE?
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
05/06/2008 Non-cooperative Game Theory 48
Dominant Strategy for Bob
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) > ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 49
Dominant Strategy for Bob
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) > ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 50
Best Response for Bob
€
∀(s−Bob ∈ S−Bob )∧( ′ s Bob ≠ sBob )
uBob (sBob,s−Bob ) ≥ uBob ( ′ s Bob ,s−Bob )
05/06/2008 Non-cooperative Game Theory 51
Dominant Strategy for Bob
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) > ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 52
Dominant Strategy for Art
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 53
Dominant Strategy for both
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 54
Dominant Strategy for both
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 55
Stable, Equilibrium
Can we converge into a “stable” strategy such that we satisfy all the players to some degree?
Can we converge into a “stable” strategy such that none of the players wanted to change or deviate “unilaterally”?
05/06/2008 Non-cooperative Game Theory 56
Unilateral Change/Deviate
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 57
Nash Equilibrium
05/06/2008 Non-cooperative Game Theory 58
Nash Equilibrium
Given
€
∀(i)∧( ′ s i ≠ si) ∋ui(si,s−i) ≥ ui( ′ s i,s−i)€
ΓN [I,{Si},{ui(•)}],s = {s1,...,sI }
The strategy s is a Nash Equilibrium if
05/06/2008 Non-cooperative Game Theory 59
Dominant Strategy for both
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 60
3-player Game (1-B, 2-R, 3-G)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 61
Nash Equilibrium?
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 62
(u1,s2,s3)?
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 63
(u1,s2,s3)
0,0,0 3,3,0
3,0,3 3,0,3
3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
3,3,0
3,0,3
4,1,1
05/06/2008 Non-cooperative Game Theory 64
(u1,s2,s3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 65
(u1,s2,s3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
4,1,1 3,3,0
4,1,1
05/06/2008 Non-cooperative Game Theory 66
(u1,s2,s3) (s1,u2,t3)?
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 67
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 68
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 69
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Which one will the players converge into?
05/06/2008 Non-cooperative Game Theory 70
The Prediction Problem
• Given a game, how can we describe or specify the mechanism(s) behind it such that we will have a better prediction?
– Social network formation– Economical outcomes– Many other applications
– The basic Nash Equilibrium may still be too loose…
05/06/2008 Non-cooperative Game Theory 71
Nash Equilibrium Extensions
• Undominated (uNE)• Strong (SNE)• Coalition-Proof (CPNE)
05/06/2008 Non-cooperative Game Theory 72
Undominated Nash Equilibrium
• Nash Equilibrium– Strategy s given (and so was s-i)
• Undominated Nash Equilibrium (uNE)– Consider all possible s-i
05/06/2008 Non-cooperative Game Theory 73
Undominated Nash Equilibrium
If all agents/players choose the “best response”…
€
ΓN [I,{Si},{ui(•)}]
∀(i)∧(s−i ∈ S−i)∧( ′ s i ≠ si) ∋ui(si,s−i) ≥ ui( ′ s i,s−i)
• A dominant strategy equilibrium is a strategy profile where the strategy for each player is dominant
• Agents do not need to counter-speculate!
05/06/2008 Non-cooperative Game Theory 74
Iterated Elimination of Dominated Strategies
• Let RiSi be the set of removed strategies for agent i
• Initially Ri=Ø• Choose agent i, and strategy si such that siSi\Ri
and there exists si’ Si\Ri such that
• Add si to Ri, continue
• Thm: If a unique strategy profile, s*, survives then it is a Nash Eq.
ui(si’,s-i)>ui(si,s-i) for all s-i S-i\R-i
05/06/2008 Non-cooperative Game Theory 75
Dominant Strategy for both
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
€
∀(s−i ∈ S−i)∧( ′ s i ≠ si),ui(si,s−i) ≥ ui( ′ s i,s−i)
05/06/2008 Non-cooperative Game Theory 76
Microsoft versus Yahoo
Y
6-4
N
(7,3) (0,0)
5-5 split7-3
Y N Y N
(6,4) (0,0) (5,5) (0,0)
05/06/2008 Non-cooperative Game Theory 77
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Which ONES are Nash Equilibrium?
05/06/2008 Non-cooperative Game Theory 78
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Seven!Which ONES are Undominated Nash Equilibrium?
05/06/2008 Non-cooperative Game Theory 79
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Seven!Which ONES are Undominated Nash Equilibrium?
05/06/2008 Non-cooperative Game Theory 80
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Seven!Which ONES are Undominated Nash Equilibrium?Only YYY for player 2!
05/06/2008 Non-cooperative Game Theory 81
Strong Nash Equilibrium
• Nash Equilibrium (NE)– No Coalition allowed
• Strong Nash Equilibrium (SNE)– Works for ALL possible coalitions
05/06/2008 Non-cooperative Game Theory 82
Coalition
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
Not Strong NE (SNE)
Not stable/NE at all!!
05/06/2008 Non-cooperative Game Theory 83
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Seven!Which ONES are Strong Nash Equilibrium?
05/06/2008 Non-cooperative Game Theory 84
The Splitting Game
7,3 7,3 7,3
6,4 6,4 0,0
5,5 0,0 5,5
YYY YYN YNY
7-3
6-4
5-5
7,3 0,0 0,0
0,0 6,4 6,4
0,0 5,5 0,0
YNN NYY NYN
0,0 0,0
0,0 0,0
5,5 0,0
NNY NNN
Seven!Which ONES are Strong Nash Equilibrium?ALL of them!
05/06/2008 Non-cooperative Game Theory 85
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 86
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Coalition:2&3
05/06/2008 Non-cooperative Game Theory 87
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Coalition:1&3
05/06/2008 Non-cooperative Game Theory 88
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
05/06/2008 Non-cooperative Game Theory 89
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Is any of them “Strong”? No!!
05/06/2008 Non-cooperative Game Theory 90
Coalition-Proof Nash Equilibrium
• NE is too loose & SNE is too restrictive, and CPNE is somewhere in between…
• Under SNE, a coalition can move from a NE to any other cell, but that cell might not be stable…
• Under CPNE, a coalition can be only allowed to move a “self-enforcing” cell (I.e., no further deviation from that cell).
05/06/2008 Non-cooperative Game Theory 91
(-2,-2) is not Self-Enforcing
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
Not Strong NE (SNE)
Not stable/NE at all!!
In Prisoner’s Dilemma, therefore, (-3,-3) is NE, uNE, also CPNE(?), but only not SNE!!
05/06/2008 Non-cooperative Game Theory 92
(u1,s2,s3) is not SNE, how about CPNE?
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Deviation to [0,3,3] disallowed under CPNE!
05/06/2008 Non-cooperative Game Theory 93
(u1,s2,s3) is not SNE, how about CPNE?
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
Deviation to [2,2,2] is, however, allowed, &, therefore [4,1,1]
(u1,s2,s3) is not
CPNE!
05/06/2008 Non-cooperative Game Theory 94
The game of
1,1 3,0 1,1
0,3 3,3 1,4
1,1 4,1 2,2
s3 t3 u3
s2
t2
u2
u1
€
Γ(u1) = Γ(sN \{2,3}* ) = [T = {2,3};(Si)i∈T ;(ui
*)i∈T ]
€
Γ(u1)
05/06/2008 Non-cooperative Game Theory 95
The game of
1,1 3,0 1,1
0,3 3,3 1,4
1,1 4,1 2,2
s3 t3 u3
s2
t2
u2
u1
€
Γ(u1) = Γ(sN \{2,3}* ) = [T = {2,3};(Si)i∈T ;(ui
*)i∈T ]
€
Γ(u1)
05/06/2008 Non-cooperative Game Theory 96
SNE versus CPNE
• SNE: regardless of any possible coalitions, the NE will survive.– Assuming the agents within the coalition are
selfless
• CPNE: considering any possible coalitions, but also consider the relation between the bigger game and the inner-circle game.– Agents in a coalition might move away from
a NE if the new state will be better for ALL members of the coalition, but then, we need to consider the sub-game.
– Assuming the agents within the coalition are non-cooperative
05/06/2008 Non-cooperative Game Theory 97
SNE versus CPNE
• SNE: regardless of any possible coalitions, the NE will survive.– Assuming the agents within the coalition are
selfless
• CPNE: considering any possible coalitions, but also consider the relation between the bigger game and the inner-circle game.– Agents in a coalition might move away from
a NE if the new state will be better for ALL members of the coalition, but then, we need to consider the sub-game.
– Assuming the agents within the coalition are non-cooperative
05/06/2008 Non-cooperative Game Theory 98
(u1,s2,s3) (s1,u2,t3) (t1,t2,u3) (u1,u2,u3)
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
4 NEs0 SNE
05/06/2008 Non-cooperative Game Theory 99
(u1,u2,u3) is the unique CPNE!
3,3,0 0,0,0 3,3,0
3,0,3 3,0,3 3,0,3
4,1,1 3,0,3 4,1,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
0,0,0 0,3,3 0,3,3
3,3,0 0,3,3 1,4,1
s2 t2 u2
s1
t1
u1
3,3,0 0,3,3 1,4,1
3,0,3 1,1,4 1,1,4
4,1,1 1,1,4 2,2,2
s2 t2 u2
s1
t1
u1
s3 t3
u3
4 NEs0 SNE1 CPNE
05/06/2008 Non-cooperative Game Theory 100
Coalition in Prisoner’s Dilemma
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-1, -10
-10, -1
Not Strong NE (SNE)
Not stable/NE at all!!
05/06/2008 Non-cooperative Game Theory 101
Updated Prisoner’s Dilemma
Art
Bob
Confess
Confess
Deny
Deny
-3, -3
-2, -2
-2, -10
-10, -2
NE, ~CPNE, ~SNE
NE, CPNE, SNE
05/06/2008 Non-cooperative Game Theory 102
Definition of CPNE“Coalition-Proof Nash Equilibria: I. Concept” Berheim/Peleg/Whiston, J. of Economic Theory, 42, 1-12 (1987).
In a single player game is a Coalition-Proof Nash Equilibrium if and only if maximizes .
Let n > 1 and assume that Coalition-Proof Nash Equilibrium has been defined for games with fewer than n players. Then,
• For any game with n players, is self-enforcing if, . , is a Coalition-Proof Nash Equilibrium in the game of .
• For any game with n players, is a Coalition-Proof Nash Equilibrium if it is self-enforcing and if there does not exist another self-enforcing strategy vector such that .
€
Γ,s* ∈ S
€
ui(s) > ui(s*),∀ i =1,...,n
€
s*
€
u1(s)
€
s* ∈ S
€
Γ
€
∀j ∈ J,s j*
€
Γ /s− j*
€
Γ
€
s* ∈ S
€
s∈ S
05/06/2008 Non-cooperative Game Theory 103
What’s next?
• Many mathematical tools, definitions, and theories…
• Now, let’s try to apply them to online social networks…
05/06/2008 Non-cooperative Game Theory 104
Materials Mainly Covered…
Chapter 5Chapters 7~9
