Dynamics & Overview. Results. Outro. Models of Selfish Network Creation (1) b c d e f g h i j k a. ⢠n selfish agents want to create a connected undirected network ...
Introduction
Dynamics & Overview
Results
Outro
On Dynamics in Selfish Network Creation a a
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f
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d d
g
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g
f
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c
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d g
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b
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Bernd Kawald & Pascal Lenzner Humboldt-University Berlin a
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c
SPAA’13, Montr´eal
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d g
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d a
b
c
g d
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Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
j
• n selfish agents want to create a connected undirected
network G = (V , E )
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
j
• n selfish agents want to create a connected undirected
network G = (V , E )
• agents want to minimize cost for network usage while
maximizing connection quality
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
j
• n selfish agents want to create a connected undirected
network G = (V , E )
• agents want to minimize cost for network usage while
maximizing connection quality
• every edge costs α > 0, each edge has owner who pays for it
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
j
• n selfish agents want to create a connected undirected
network G = (V , E )
• agents want to minimize cost for network usage while
maximizing connection quality
• every edge costs α > 0, each edge has owner who pays for it • cost of an agent u in network (G , α):
cost(u) = edgecost(u) + distancecost(u) = α · (#edges bought by agent u) + distancecost(u)
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
j
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a
j
d
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
distancecost(u) =
(P
v ∈V (G )
∞,
dG (u, v ), if (G , α) is connected otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) 2 c
i 1 e
1 b 0
f 2
a
2
2 d
g
2 k 3
h 1 j
3
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
distancecost(u) =
(P
v ∈V (G )
∞,
dG (u, v ), if (G , α) is connected otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h
2α + 19 a
g j
d
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
distancecost(u) =
(P
v ∈V (G )
∞,
dG (u, v ), if (G , α) is connected otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) 2α + 17
c
α + 18 b
i 2α + 18 2α + 19
f e
2α + 19 a
α + 16
2α + 21
3α + 17 d
g
k h α + 19 j
α + 21
17
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
distancecost(u) =
(P
v ∈V (G )
∞,
dG (u, v ), if (G , α) is connected otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) 2 c
i 1 e
1 b 0
f 2
a
2
2 d
g
2 k 3
h 1 j
3
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
(P
v ∈V (G )
distancecost(u) =
Max-Version:
∞,
dG (u, v ), if (G , α) is connected otherwise
[Demaine et al., PODC’07]
distancecost(u) =
(
maxv ∈V (G ) dG (u, v ), if (G , α) is connected ∞, otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h
2α + 3 a
g j
d
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
(P
v ∈V (G )
distancecost(u) =
Max-Version:
∞,
dG (u, v ), if (G , α) is connected otherwise
[Demaine et al., PODC’07]
distancecost(u) =
(
maxv ∈V (G ) dG (u, v ), if (G , α) is connected ∞, otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) 2α + 3 c α+3
i 2α + 3 f
2α + 3
α+2
e
b
k α+3
h α+3
2α + 3 a
2α + 3 3α + 2 d
g j
3
• cost(u) = α · (#edges bought by agent u) + distancecost(u)
Sum-Version:
[Fabrikant et al., PODC’03]
(P
v ∈V (G )
distancecost(u) =
Max-Version:
∞,
dG (u, v ), if (G , α) is connected otherwise
[Demaine et al., PODC’07]
distancecost(u) =
(
maxv ∈V (G ) dG (u, v ), if (G , α) is connected ∞, otherwise
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f e
b
k h g
a d
• pure strategy Su of agent u: Su ⊆ V \ {u}.
Depending on the model we have:
• Su = set of neighbors of u • Su = set of neighbors to which u owns an edge
j
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f
Sa = {b, h}
e
b
k h g
a d
Sd = {b, f, j}
• pure strategy Su of agent u: Su ⊆ V \ {u}.
Depending on the model we have:
• Su = set of neighbors of u • Su = set of neighbors to which u owns an edge
j
Sj = ∅
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (1) c
i f
Sa = {b, h}
e
b
k h g
a d
Sd = {b, f, j}
j
• pure strategy Su of agent u: Su ⊆ V \ {u}.
Depending on the model we have:
• Su = set of neighbors of u • Su = set of neighbors to which u owns an edge
• S is vector of strategies of all agents • S and parameter α determines network (G , α) • network (G , α) with edge ownerships determines S
Sj = ∅
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2)
Outro
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG) [Alon et al. SPAA’10]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
Outro
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG) [Alon et al. SPAA’10]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium c e
b a d
Outro
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG) [Alon et al. SPAA’10]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium c 7 b
e 7
6
a 9 d
5
Outro
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG) [Alon et al. SPAA’10]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium c 6 b
e 7
5
a 8 d
6
Outro
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Outro
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq. c e
b a d
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq. c e
b a d
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq. c 7 e 7
b 6 a 9 d 5
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq. c 8 e 6
b 5 a 8 d 5
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
Greedy Buy Game (GBG) [L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Outro
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG) c
[L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
e
b a d
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG)
c 7
[L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
α+7 e
b 2α + 6 a 9
d 2α + 5
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG)
c 5
[L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
α+6 e
b 2α + 6 a α+6
d 2α + 5
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG)
c 5
[L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
α+6 e
b 2α + 6 a α+6
d 2α + 5
Introduction
Dynamics & Overview
Results
Outro
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG)
c 5
[L. WINE’12]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
α+8 e
b 2α + 7 a α+6
d α+6
Introduction
Dynamics & Overview
Results
Models of Selfish Network Creation (2) Swap Game (SG)
Asymmetric SG (ASG)
[Alon et al. SPAA’10]
[Mihal´ ak & Schlegel MFCS’12]
• • • •
no edge-owners no edge-cost only single edge-swaps both endpoints can swap ⇒ Swap Equilibrium
• • • •
edges have owners no edge-cost only single edge-swaps only owner can swap ⇒ Asymmetric Swap Eq.
Greedy Buy Game (GBG)
Buy Game (BG)
[L. WINE’12]
[Fabrikant et al. PODC’03]
• edges have owners • each edge costs α • agents can buy/swap/del
one own edge ⇒ Greedy Equilibrium
• edges have owners • each edge costs α • arbitrary strategy-changes
⇒ pure Nash Equilibrium
Outro
Introduction
Dynamics & Overview
Results
• previous work mainly focussed on structural properties
Outro
Introduction
Dynamics & Overview
Results
• previous work mainly focussed on structural properties
Open Problem: How can agents find equilibrium networks?
Outro
Introduction
Dynamics & Overview
Results
• previous work mainly focussed on structural properties
Open Problem: How can agents find equilibrium networks? ⇒ we focus on the network creation process
Outro
Introduction
Dynamics & Overview
Results
• previous work mainly focussed on structural properties
Open Problem: How can agents find equilibrium networks? ⇒ we focus on the network creation process • we analyze the most natural approach:
Distributed Local Search: • start with any connected network • at every step one agent is allowed to move
(agent chosen at random or random max cost agent) • moving agent performs move to best response strategy • iterate until no agent wants to change strategy
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP)
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP) • FIP ⇐⇒ generalized ordinal potential function exists • all sequences of improving moves are finite • no better response cycle
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP) • FIP ⇐⇒ generalized ordinal potential function exists • all sequences of improving moves are finite • no better response cycle • poly-FIPG: FIP + convergence in polyonmially many rounds
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP) • FIP ⇐⇒ generalized ordinal potential function exists • all sequences of improving moves are finite • no better response cycle • poly-FIPG: FIP + convergence in polyonmially many rounds
• possible convergence: • WAG: games which are weakly acyclic • some sequences of improving moves are finite
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP) • FIP ⇐⇒ generalized ordinal potential function exists • all sequences of improving moves are finite • no better response cycle • poly-FIPG: FIP + convergence in polyonmially many rounds
• possible convergence: • WAG: games which are weakly acyclic • some sequences of improving moves are finite • BR-WAG: games which are weakly acyclic under best response • some sequences of best response moves are finite
Outro
Introduction
Dynamics & Overview
Results
Classifying Games According to their Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• guaranteed convergence: • FIPG: games having the finite improvement property (FIP) • FIP ⇐⇒ generalized ordinal potential function exists • all sequences of improving moves are finite • no better response cycle • poly-FIPG: FIP + convergence in polyonmially many rounds
• possible convergence: • WAG: games which are weakly acyclic • some sequences of improving moves are finite • BR-WAG: games which are weakly acyclic under best response • some sequences of best response moves are finite
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle
[Brandes et al. WINE’08]
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11]
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11] • bounded-budget version is ASG (agents use up their budgets)
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11] • bounded-budget version is ASG (agents use up their budgets) • Sum-SG on trees ∈ poly-FIPG, ∈ / FIPG otherwise [L. SAGT’11]
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11] • bounded-budget version is ASG (agents use up their budgets) • Sum-SG on trees ∈ poly-FIPG, ∈ / FIPG otherwise [L. SAGT’11] • on trees: any improving sequence has length O(n3 ), speed-up to O(n) if max cost agents play best response
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11] • bounded-budget version is ASG (agents use up their budgets) • Sum-SG on trees ∈ poly-FIPG, ∈ / FIPG otherwise [L. SAGT’11] • on trees: any improving sequence has length O(n3 ), speed-up to O(n) if max cost agents play best response • Max-BG ∈ / FIPG via better response cycle
[Bil` o et al. WINE’12]
Outro
Introduction
Dynamics & Overview
Results
Previous Results on Dynamics poly-FIPG potential games
FIPG
BR-WAG WAG
• Sum-BG ∈ / FIPG via better response cycle [Brandes et al. WINE’08] • How fast converge bounded-budget versions? [Ehsani et al. SPAA’11] • bounded-budget version is ASG (agents use up their budgets) • Sum-SG on trees ∈ poly-FIPG, ∈ / FIPG otherwise [L. SAGT’11] • on trees: any improving sequence has length O(n3 ), speed-up to O(n) if max cost agents play best response • Max-BG ∈ / FIPG via better response cycle
[Bil` o et al. WINE’12]
⇒ for most variants nothing known for best response dynamics
Outro
Introduction
Dynamics & Overview
Our Results Max-Swap Game
Results
Outro
Introduction
Dynamics & Overview
Our Results Max-Swap Game • on trees: poly-FIPG,
at most O(n3 ) steps, speed-up to O(n log n)
Results
Outro
Introduction
Dynamics & Overview
Our Results Max-Swap Game • on trees: poly-FIPG,
at most O(n3 ) steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Results
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
at most O(n3 ) steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Asymmetric SG
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
Outro
Introduction
Dynamics & Overview
Results
Outro
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem
[SPAA’11]
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game • Sum: best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game • Sum: best response cycle
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game • Sum: best response cycle • Max: best response cycle
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game • Sum: best response cycle • Max: best response cycle
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game • Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
general host graphs
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game • Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
general host graphs
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game • Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
general host graphs
• extensive simulations show
convergence in < 8n steps
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game • Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
general host graphs
Outro
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
at most O(n3 ) steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
at most O(n3 ) steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
• assume improving swap uv → uw
Outro
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
• assume improving swap uv → uw T :
u
v
w
A
T0 :
B
u
A
v
w
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
u
v
w
A
T0 :
B
u
A
v
w
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
• x ∈ A, y ∈ B s.t. cT 0 (y ) = dT 0 (x, y )
u
w
v
A
B
x
T0 :
u
A
v
w y
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Remember: • cost(u) = max dG (u, v ) v ∈V (G )
• only single swap allowed
• both endpoints can swap
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
• x ∈ A, y ∈ B s.t. cT 0 (y ) = dT 0 (x, y )
⇒ cT (x) > cT 0 (y )
u
w
v
A
B
x
T0 :
u
A
v
w y
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
Definition: Sorted Cost Vector
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
• x ∈ A, y ∈ B s.t. cT 0 (y ) = dT 0 (x, y )
⇒ cT (x) > cT 0 (y )
u
w
v
A
B
x
T0 :
u
A
v
w y
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
Definition: Sorted Cost Vector
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
• x ∈ A, y ∈ B s.t. cT 0 (y ) = dT 0 (x, y )
⇒ cT (x) > cT 0 (y ) ⇒ − c→ > − c→0 T
lex
T
u
w
v
A
B
x
T0 :
u
A
v
w y
B
Introduction
Dynamics & Overview
Results
Outro
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
Definition: Sorted Cost Vector
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• assume improving swap uv → uw
⇒ ∀x ∈ A : cT (x) > cT 0 (x)
T :
• x ∈ A, y ∈ B s.t. cT 0 (y ) = dT 0 (x, y )
⇒ cT (x) > cT 0 (y ) ⇒ − c→ > − c→0 T
lex
T
⇒ diameter cannot increase
u
w
v
A
B
x
T0 :
u
A
v
w y
B
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Definition: Sorted Cost Vector
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• improving swap: − c→ G must decrease, diameter cannot increase
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Definition: Sorted Cost Vector
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• improving swap: − c→ G must decrease, diameter cannot increase • consider tree network T having diameter D ≥ 4:
Lemma After
n∗D−D 2 2
steps in T , diameter must decrease.
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Definition: Sorted Cost Vector
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• improving swap: − c→ G must decrease, diameter cannot increase • consider tree network T having diameter D ≥ 4:
Lemma After
n∗D−D 2 2
steps in T , diameter must decrease.
• Equilibria are stars or double-stars [Alon et al. SPAA’10]
Outro
Introduction
Dynamics & Overview
Results
Details for Max Swap Game on Trees Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Definition: Sorted Cost Vector
− 1 n c→ G = (γG , . . . , γG ), where γGi is cost of agent with i-th highest cost in network G .
• improving swap: − c→ G must decrease, diameter cannot increase • consider tree network T having diameter D ≥ 4:
Lemma After
n∗D−D 2 2
steps in T , diameter must decrease.
• Equilibria are stars or double-stars [Alon et al. SPAA’10]
⇒ process must converge after O(n3 ) steps.
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
swapped
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
swapped
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move:
Outro
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Outro
Details for Asymmetric Swap Games Asymmetric Swap Games
Remember: • cost(u) =
P
dG (u, v )
v ∈V (G )
• only single swap allowed • only own edges can be
• SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
swapped
Proof: Best response cycle, in every step only one agent unhappy, moving agent has only one improving move: b
b c
a d e
fd → fe
bf → ba
c
a
d e
f
b c
a
d e
f
b c
a
d e
f
fe → fd
f
ba → bf
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Open Problem
[Ehsani et al. SPAA’11]
Determine convergence speed of Sum and Max in bounded budget version.
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Open Problem
[Ehsani et al. SPAA’11]
Determine convergence speed of Sum and Max in bounded budget version.
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Solution: No convergence guarantee for Sum and Max!
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Open Problem
[Ehsani et al. SPAA’11]
Determine convergence speed of Sum and Max in bounded budget version.
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Solution: No convergence guarantee for Sum and Max! • best response cycle exists if B = α for all agents
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Open Problem
[Ehsani et al. SPAA’11]
Determine convergence speed of Sum and Max in bounded budget version.
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Solution: No convergence guarantee for Sum and Max! • best response cycle exists if B = α for all agents ⇒ sharp boundary between convergence and non-convergence
Outro
Introduction
Dynamics & Overview
Results
Details for Asymmetric Swap Games • each agent has budget B
Open Problem
[Ehsani et al. SPAA’11]
Determine convergence speed of Sum and Max in bounded budget version.
Asymmetric Swap Games • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG
• solve open problem [SPAA’11] • promising empirical results
Solution: No convergence guarantee for Sum and Max! • best response cycle exists if B = α for all agents ⇒ sharp boundary between convergence and non-convergence
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8:
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games We give best response cycle for 7 < α < 8: a
b
a
c
b
g
f
e
gf → gc
a
c
d
b
g
f
e
f buys f b
a
c
d
b
g
f
e
c rem. cb
Greedy Buy Game
a
c
d
b
g
f
e
gc → gf
a
c
d
b
c
d g
f
e
c buys cb
d g
f
e
f rem. f b
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents • either random move-policy or max cost move-policy
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents • either random move-policy or max cost move-policy
• connected random initial networks, always best responses
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents • either random move-policy or max cost move-policy
• connected random initial networks, always best responses
n n n • edge-range: n, 2n, 4n, α-range: 10 , 4 , 2 , n, 5000 runs each
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents • either random move-policy or max cost move-policy
• connected random initial networks, always best responses
n n n • edge-range: n, 2n, 4n, α-range: 10 , 4 , 2 , n, 5000 runs each
⇒ despite millions of runs: no cyclic instance found
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Details for (Greedy) Buy Games • we simulated Sum-GBG and Max-GBG with 10 to 100 agents • either random move-policy or max cost move-policy
• connected random initial networks, always best responses
n n n • edge-range: n, 2n, 4n, α-range: 10 , 4 , 2 , n, 5000 runs each
⇒ despite millions of runs: no cyclic instance found
⇒ suprisingly fast convergence: Sum < 7n moves, Max < 8n
Greedy Buy Game
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, SUM version 700
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost
600
500
Steps
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, SUM version 700
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost
600
500
Steps
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, SUM version 700
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random
600
500
Steps
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, SUM version 700
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random m=4n, a=n/10, random m=4n, a=n/4, random m=4n, a=n, random
600
500
Steps
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, SUM version 700
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random m=4n, a=n/10, random m=4n, a=n/4, random m=4n, a=n, random f(n)= 7n
600
500
Steps
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, MAX version 800
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost
700
600
Steps
500
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, MAX version 800
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost
700
600
Steps
500
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, MAX version 800
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random
700
600
Steps
500
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, MAX version 800
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random m=4n, a=n/10, random m=4n, a=n/4, random m=4n, a=n, random
700
600
Steps
500
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Outro
Details for (Greedy) Buy Games Max # of steps until convergence, MAX version 800
m=n, a=n/10, max cost m=n, a=n/4, max cost m=n, a=n, max cost m=4n, a=n/10, max cost m=4n, a=n/4, max cost m=4n, a=n, max cost m=n, a=n/10, random m=n, a=n/4, random m=n, a=n, random m=4n, a=n/10, random m=4n, a=n/4, random m=4n, a=n, random f(n)= 8n
700
600
Steps
500
400
300
200
100
0 10
20
30
40
50
60 Agents
70
80
90
100
Introduction
Dynamics & Overview
Results
Our Results Max-Swap Game • on trees: poly-FIPG,
O(n3 )
at most steps, speed-up to O(n log n) • in general: ∈ / FIPG via best response cycle
Greedy Buy Game
Asymmetric SG • SG-results on trees carry
over for Sum and Max
• in general: Sum ∈ / WAG,
Max ∈ / FIPG • solve open problem [SPAA’11] • promising empirical results
Buy Game
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• Sum: best response cycle • Max: best response cycle • Sum and Max ∈ / WAG on
• extensive simulations show
• bilateral Buy Game:
general host graphs
convergence in < 8n steps
general host graphs
Sum ∈WAG,Max / ∈FIPG /
Outro
Introduction
Dynamics & Overview
Open Problems
Results
Outro
Introduction
Dynamics & Overview
Results
Open Problems Conjecture The Sum-(G)BG and Max-(G)BG are not weakly acyclic.
Outro
Introduction
Dynamics & Overview
Results
Open Problems Conjecture The Sum-(G)BG and Max-(G)BG are not weakly acyclic. • give best response cycle where in every step exactly one agent
is unhappy and this agent has exactly one improving move
Outro
Introduction
Dynamics & Overview
Results
Open Problems Conjecture The Sum-(G)BG and Max-(G)BG are not weakly acyclic. • give best response cycle where in every step exactly one agent
is unhappy and this agent has exactly one improving move
Open Problem Why do dynamics in (Greedy) Buy Games converge so fast?
Outro
Introduction
Dynamics & Overview
Results
Open Problems Conjecture The Sum-(G)BG and Max-(G)BG are not weakly acyclic. • give best response cycle where in every step exactly one agent
is unhappy and this agent has exactly one improving move
Open Problem Why do dynamics in (Greedy) Buy Games converge so fast?
Open Problem Is convergence to approximate equilibrium guaranteed? If so, for which approx-factor?
Outro
Introduction
Dynamics & Overview
Results
e
u
s
Q
t i o n s
?
Outro
