Distributed control and game designFrom strategic agents to programmable machines

Dario PaccagnanPhD Defense

Coordination of multiagent systems

2

Coordination of multiagent systems

Competitive Cooperative

2

PhD research overview

Aggregative games

. Large population, algorithms[TAC18a]

. Equilibrium efficiency[L-CSS18], [CDC18]

. Algorithms and applications[CDC16], [ECC16], [CPS18]

. Traffic and Inertial equilibria[IFAC17], [CDC17]

Combinatorial allocation

. Optimal utility design[Submitted, J18a][Submitted, J18b]

. Role of information[TAC18b][Allerton17]

. Worst vs best perf. tradeoff[Submitted, J18c]

Others [CDC15], [PLANS14]

3

PhD research overview

Aggregative games

. Large population, algorithms[TAC18a]

. Equilibrium efficiency[L-CSS18], [CDC18]

. Algorithms and applications[CDC16], [ECC16], [CPS18]

. Traffic and Inertial equilibria[IFAC17], [CDC17]

Combinatorial allocation

. Optimal utility design[Submitted, J18a][Submitted, J18b]

. Role of information[TAC18b][Allerton17]

. Worst vs best perf. tradeoff[Submitted, J18c]

Others [CDC15], [PLANS14]

3

PhD research overview

Aggregative games

. Large population, algorithms[TAC18a]

. Equilibrium efficiency[L-CSS18], [CDC18]

. Algorithms and applications[CDC16], [ECC16], [CPS18]

. Traffic and Inertial equilibria[IFAC17], [CDC17]

Combinatorial allocation

. Optimal utility design[Submitted, J18a][Submitted, J18b]

. Role of information[TAC18b][Allerton17]

. Worst vs best perf. tradeoff[Submitted, J18c]

Others [CDC15], [PLANS14]

3

Aggregative games

- Introduction

- Convergence between Nash and Wardrop

- Efficiency of equilibria

4

Aggregative games

players: i ∈ {1, . . . ,M}

constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , )

σ(x) =1

M

M∑i=1

x i

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Aggregative games

players: i ∈ {1, . . . ,M}constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , )

σ(x) =1

M

M∑i=1

x i

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Aggregative games

players: i ∈ {1, . . . ,M}constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , x−i )

σ(x) =1

M

M∑i=1

x iAggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Aggregative games

players: i ∈ {1, . . . ,M}constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , σ(x))

σ(x) =1

M

M∑i=1

x iAggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Aggregative games

players: i ∈ {1, . . . ,M}constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , σ(x)) σ(x) =1

M

M∑i=1

x i

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Aggregative games

players: i ∈ {1, . . . ,M}constraints: x i ∈ X i ⊆ Rn

cost: J i (x i , σ(x)) σ(x) =1

M

M∑i=1

x iAggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

Aggregative game:

Nash equilibrium

J i (x̂ i ,�(x̂)) J i (x i ,�(x i , x̂�i ))

Wardrop equilibrium

J i (x̄ i ,�(x̄)) J i (x i ,�(x̄))

Strong monotonicity condition:

rx [

rxi J i (x i ,�(x))

]Ni=1 � ↵INn

Under strong monotonicity and uniform Lipschitzianity:kx̂ � x̄k constp

N

3/ 7

N agentsx i 2 Xi ⇢ Rn

J i (x i ,�(x))

demand-response energy markets smart urban mobility

identical replicas [1] heterogeneous new agents

[1] Haurie, Marcotte “On the relationship between Nash-Cournot and Wardrop equilibria” (1985).

5

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j

= 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

easier to study

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

easier to study

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

easier to study

6

Two equilibrium notions

x̂ Nash equilibrium

J i (x̂ i , σ(x̂)) ≤ J i (x i , σ(x i , x̂−i ))

x̄ Wardrop equilibrium

J i (x̄ i , σ(x̄)) ≤ J i (x i , σ(x̄))

= x i

M + 1M

∑j 6=i x̂

j = 1M

∑j x̄

j

What is the relation between x̂ and x̄?

Nash operator

F̂ (x) = [∇x i Ji (x i , σ(x))]Mi=1

Wardrop operator

F̄ (x) = [∇x i Ji (x i , z)|z=σ(x)]Mi=1

easier to study

6

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i

, ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:

– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)

– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Main result I

Theorem (Convergence for large M) [TAC18a]

Lipschitzianity of J i , boundedness of X i , ∇x F̂ (x) � αI or ∇x F̄ (x) � αI ,

||x̂ − x̄ || ≤ const/√M

0 100 200 300 400 500 600 700 8000

0.03

0.06

0.09

0.12

0.15

Population size M

Distance

‖σ(x̂)− σ(x̄)‖1/√M

Consequences:– equilibrium computation (algorithms)– equilibrium efficiency

7

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}

cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}

cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion

minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)

≥ 1

8

Equilibrium efficiency: electric vehicle charging

- A fleet of EVs to recharge

- Each vehicle min bill in [1, n]

- Charging requirements

players: i ∈ {1, . . . ,M}cost of i : p(σ(x) + d)>x i

constr: x i ∈ X i

System level objective

- Minimize congestion minx∈X

Js(x) = p(σ(x)+d)>(σ(x)+d)

How much does selfish behaviour degrade the performance?

PoA =maxx∈NE(G) Js(x)

Js(xopt)≥ 1

8

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial

=⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial

=⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial

=⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial =⇒ WE are efficient for any M

=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial =⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial =⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

Main result II

Theorem (Equilibrium efficiency) [L-CSS18]

Assume sufficient regularityAssume p(z + d) = [g(z1 + d1); . . . ; g(zn + dn)], g : R≥0 → R≥0

. If g is a pure monomial =⇒ WE are efficient for any M=⇒ NE are efficient for large M

1 ≤ PoA ≤ 1 + const/√M

. If g is not a pure monomial =⇒ there exists inefficient instances(both NE/WE)

3 10 20 30 40 50 60 70 80 90100 120 1501

1.051.1

1.151.2

1.251.3

1.35

M

PoA

monomialnon monom

9

PhD research overview

Aggregative games

. Large population, algorithms[TAC18a]

. Equilibrium efficiency[L-CSS18], [CDC18]

. Algorithms and applications[CDC16], [ECC16]

. Traffic and Inertial equilibria[IFAC 17], [CDC17]

Combinatorial allocation

. Optimal utility design[Submitted, J18a][Submitted, J18b]

. Role of information[TAC18b][Allerton17]

. Worst vs best perf. tradeoff[Submitted, J18c]

Others [CDC15], [PLANS14]

10

PhD research overview

Aggregative games

. Large population, algorithms[TAC18a]

. Equilibrium efficiency[L-CSS18], [CDC18]

. Algorithms and applications[CDC16], [ECC16]

. Traffic and Inertial equilibria[IFAC 17], [CDC17]

Combinatorial allocation

. Optimal utility design[Submitted, J18a][Submitted, J18b]

. Role of information[TAC18b][Allerton17]

. Worst vs best perf. tradeoff[Submitted, J18c]

Others [CDC15], [PLANS14]

10

Combinatorial allocation

- Introduction

- GMMC problems are intractable

- Utility design approach and performance guarantees

11

Combinatorial resource allocation

. a set of resources

. a set of agents

Goal: assign resources to agents to maximize a given welfare function

About 548.000.000 results (0,59 seconds)

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Men's Running Shoes. Nike.comhttps://store.nike.com/us/en_us/pw/mens-running-shoes/7puZ8yzZoi3Dominate your run and find the right fit for your running style with the latest men's running shoes fromNike. Enjoy free shipping and returns with NikePlus.

Shoe

Sneakers are shoes primarily designed for sports or other forms ofphysical exercise, but which are now also often used for everyday wear.Wikipedia

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12

Combinatorial resource allocation

. a set of resources

. a set of agents

Goal: assign resources to agents to maximize a given welfare function

About 548.000.000 results (0,59 seconds)

SponsoredShop for running shoes

Hoka One One -Speed Goat 2 ...CHF 99,33€ 86,03mrporter.comBy Google

BROOKSRunningschuh ...CHF 159,00intersport.chFree shippingBy Google

Hoka One One -Mach Mesh ...CHF 99,33€ 86,03mrporter.comBy Google

ROA - ObliqueRippy Mesh- ...CHF 259,04€ 224,35mrporter.comBy Google

Gucci, Platformsneakers - ...CHF 840,00mytheresa.com/...Free shippingBy Google

Running Shoes - Runner's Worldhttps://www.runnersworld.com/running-shoes/Black Running Shoes You'll Want to Wear On the Run and Beyond. Black is always in style, and thesekicks combine top-notch performance with street-savvy ...The Best Running Shoes · The 7 Best Running Shoes ... · Pronation, Explained

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Rating Hours

Och Sport4.1 (44) · Running StoreBahnhofstrasse 56 · 044 215 21 21Open ⋅ Closes 7PM

On AG3.5 (8) · ManufacturerPfingstweidstrasse 106 · 044 225 15 55

"Best running company in the world based in Zurich!"

Ochsner Shoes4.3 (6) · Shoe ShopShoes & trainers for adults & childrenStadelhoferstrasse 8 · 044 252 50 77Open ⋅ Closes 8PM

On | Schweizer Performance Laufschuhe & Bekleidung - On Runninghttps://www.on-running.com/en-ch/Test shoes or Performance Running Gear commitment free. Love them ... Your perfect partner in theworld's lightest fully-cushioned shoe for Running Remixed.

Mens Running Shoes & Running Clothing for Men | Onhttps://www.on-running.com/en-ch/t/mensShop On's collection of men's running shoes & running clothes. Made with innovative technology forunparalleled comfort. Free shipping & returns.

Men's Running Shoes | Running Warehousehttps://www.runningwarehouse.com/mens-running-shoes.htmlFind your favorite running shoes at Running Warehouse! Huge selection and top-rated customerservice. Free 2 day shipping & free returns on all orders.

Top Men's Running Shoes | Road Runner Sportshttps://www.roadrunnersports.com/rrs/mensshoes/mensshoesrunning/Welcome to our premium collection of men's running shoes where you can browse through a selectionfrom the best brands and top styles available on the ...

Running | Clothing, Shoes, Trainers | Accessories, Watches ...https://www.sportsdirect.com/runningBuy all your running kit including men's and ladies running shoes, clothes and accessories from worldleading brands like Nike and Karrimor. Order today!

616 Best Running Shoes (June 2018) | RunRepeathttps://runrepeat.com/ranking/rankings-of-running-shoesAll 618 running shoes ranked by the best – based on reviews from 2351 experts & runners. Theultimate list. Updated June 2018!

Men's Running Shoes. Nike.comhttps://store.nike.com/us/en_us/pw/mens-running-shoes/7puZ8yzZoi3Dominate your run and find the right fit for your running style with the latest men's running shoes fromNike. Enjoy free shipping and returns with NikePlus.

Shoe

Sneakers are shoes primarily designed for sports or other forms ofphysical exercise, but which are now also often used for everyday wear.Wikipedia

Usage: Running

Product type: Shoe

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Sneakers

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12

Combinatorial resource allocation

. a set of resources

. a set of agents

Goal: assign resources to agents to maximize a given welfare function

About 548.000.000 results (0,59 seconds)

SponsoredShop for running shoes

Hoka One One -Speed Goat 2 ...CHF 99,33€ 86,03mrporter.comBy Google

BROOKSRunningschuh ...CHF 159,00intersport.chFree shippingBy Google

Hoka One One -Mach Mesh ...CHF 99,33€ 86,03mrporter.comBy Google

ROA - ObliqueRippy Mesh- ...CHF 259,04€ 224,35mrporter.comBy Google

Gucci, Platformsneakers - ...CHF 840,00mytheresa.com/...Free shippingBy Google

Running Shoes - Runner's Worldhttps://www.runnersworld.com/running-shoes/Black Running Shoes You'll Want to Wear On the Run and Beyond. Black is always in style, and thesekicks combine top-notch performance with street-savvy ...The Best Running Shoes · The 7 Best Running Shoes ... · Pronation, Explained

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Rating Hours

Och Sport4.1 (44) · Running StoreBahnhofstrasse 56 · 044 215 21 21Open ⋅ Closes 7PM

On AG3.5 (8) · ManufacturerPfingstweidstrasse 106 · 044 225 15 55

"Best running company in the world based in Zurich!"

Ochsner Shoes4.3 (6) · Shoe ShopShoes & trainers for adults & childrenStadelhoferstrasse 8 · 044 252 50 77Open ⋅ Closes 8PM

On | Schweizer Performance Laufschuhe & Bekleidung - On Runninghttps://www.on-running.com/en-ch/Test shoes or Performance Running Gear commitment free. Love them ... Your perfect partner in theworld's lightest fully-cushioned shoe for Running Remixed.

Mens Running Shoes & Running Clothing for Men | Onhttps://www.on-running.com/en-ch/t/mensShop On's collection of men's running shoes & running clothes. Made with innovative technology forunparalleled comfort. Free shipping & returns.

Men's Running Shoes | Running Warehousehttps://www.runningwarehouse.com/mens-running-shoes.htmlFind your favorite running shoes at Running Warehouse! Huge selection and top-rated customerservice. Free 2 day shipping & free returns on all orders.

Top Men's Running Shoes | Road Runner Sportshttps://www.roadrunnersports.com/rrs/mensshoes/mensshoesrunning/Welcome to our premium collection of men's running shoes where you can browse through a selectionfrom the best brands and top styles available on the ...

Running | Clothing, Shoes, Trainers | Accessories, Watches ...https://www.sportsdirect.com/runningBuy all your running kit including men's and ladies running shoes, clothes and accessories from worldleading brands like Nike and Karrimor. Order today!

616 Best Running Shoes (June 2018) | RunRepeathttps://runrepeat.com/ranking/rankings-of-running-shoesAll 618 running shoes ranked by the best – based on reviews from 2351 experts & runners. Theultimate list. Updated June 2018!

Men's Running Shoes. Nike.comhttps://store.nike.com/us/en_us/pw/mens-running-shoes/7puZ8yzZoi3Dominate your run and find the right fit for your running style with the latest men's running shoes fromNike. Enjoy free shipping and returns with NikePlus.

Shoe

Sneakers are shoes primarily designed for sports or other forms ofphysical exercise, but which are now also often used for everyday wear.Wikipedia

Usage: Running

Product type: Shoe

People also search for

Sneakers

View 10+ more

Shoe Boot Footwear Clothing Sandal

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12

Combinatorial resource allocation

. a set of resources

. a set of agents

Goal: assign resources to agents to maximize a given welfare function

About 548.000.000 results (0,59 seconds)

SponsoredShop for running shoes

Hoka One One -Speed Goat 2 ...CHF 99,33€ 86,03mrporter.comBy Google

BROOKSRunningschuh ...CHF 159,00intersport.chFree shippingBy Google

Hoka One One -Mach Mesh ...CHF 99,33€ 86,03mrporter.comBy Google

ROA - ObliqueRippy Mesh- ...CHF 259,04€ 224,35mrporter.comBy Google

Gucci, Platformsneakers - ...CHF 840,00mytheresa.com/...Free shippingBy Google

Running Shoes - Runner's Worldhttps://www.runnersworld.com/running-shoes/Black Running Shoes You'll Want to Wear On the Run and Beyond. Black is always in style, and thesekicks combine top-notch performance with street-savvy ...The Best Running Shoes · The 7 Best Running Shoes ... · Pronation, Explained

More places

Rating Hours

Och Sport4.1 (44) · Running StoreBahnhofstrasse 56 · 044 215 21 21Open ⋅ Closes 7PM

On AG3.5 (8) · ManufacturerPfingstweidstrasse 106 · 044 225 15 55

"Best running company in the world based in Zurich!"

Ochsner Shoes4.3 (6) · Shoe ShopShoes & trainers for adults & childrenStadelhoferstrasse 8 · 044 252 50 77Open ⋅ Closes 8PM

On | Schweizer Performance Laufschuhe & Bekleidung - On Runninghttps://www.on-running.com/en-ch/Test shoes or Performance Running Gear commitment free. Love them ... Your perfect partner in theworld's lightest fully-cushioned shoe for Running Remixed.

Mens Running Shoes & Running Clothing for Men | Onhttps://www.on-running.com/en-ch/t/mensShop On's collection of men's running shoes & running clothes. Made with innovative technology forunparalleled comfort. Free shipping & returns.

Men's Running Shoes | Running Warehousehttps://www.runningwarehouse.com/mens-running-shoes.htmlFind your favorite running shoes at Running Warehouse! Huge selection and top-rated customerservice. Free 2 day shipping & free returns on all orders.

Top Men's Running Shoes | Road Runner Sportshttps://www.roadrunnersports.com/rrs/mensshoes/mensshoesrunning/Welcome to our premium collection of men's running shoes where you can browse through a selectionfrom the best brands and top styles available on the ...

Running | Clothing, Shoes, Trainers | Accessories, Watches ...https://www.sportsdirect.com/runningBuy all your running kit including men's and ladies running shoes, clothes and accessories from worldleading brands like Nike and Karrimor. Order today!

616 Best Running Shoes (June 2018) | RunRepeathttps://runrepeat.com/ranking/rankings-of-running-shoesAll 618 running shoes ranked by the best – based on reviews from 2351 experts & runners. Theultimate list. Updated June 2018!

Men's Running Shoes. Nike.comhttps://store.nike.com/us/en_us/pw/mens-running-shoes/7puZ8yzZoi3Dominate your run and find the right fit for your running style with the latest men's running shoes fromNike. Enjoy free shipping and returns with NikePlus.

Shoe

Sneakers are shoes primarily designed for sports or other forms ofphysical exercise, but which are now also often used for everyday wear.Wikipedia

Usage: Running

Product type: Shoe

People also search for

Sneakers

View 10+ more

Shoe Boot Footwear Clothing Sandal

Feedback

All Images Maps Shopping News More Settings Tools

running shoes Sign in

running shoes - Google Search https://www.google.com/search?client=firefox-b&ei=YYojW-...

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12

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}

allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Generalized Multiagent Maximum Coverage (GMMC)

resources: r ∈ R, vr ≥ 0

agents: i ∈ {1, . . . ,M}allocations: ai ∈ Ai ⊆ 2R

welfare: W (a) =∑

r∈∪iaivrw(|a|r ) w : N→ R≥0

System-level objective: maxa∈AW (a)

Hardness and approximability

. Reduces to max-cover for w(j) = 1, Ai = Aj

. NP-hard

. If w is concave and Ai = Aj , best poly-algorithm achieves 1− c/eand is centralized, c = 1− [w(M)− w(M − 1)]

13

v1

v2

v3

v4

v5

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:

distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed

, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient

, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

1− c/e

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

GT

1− c/e

max-cover1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

GT

1− c/e

max-cover

1− 1/e

14

Main result III

Game theory can be used to produce algorithms that are:distributed, efficient, match/improve existing approximations

Example: - # agents ≤ 40- w(j) = jd , d varies in [0, 1]

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

Approx.ratio

GT

1− c/e

max-cover1− 1/e

14

The game-theoretic approach

Game design

15

maxW (a)

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

maxW (a)

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilities

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilitiesgame

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilitiesgame

equilibria

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilitiesgame

equilibria

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilitiesgame

equilibria

maxW (a)

Design a game(agents, constraints, utilities)

Requirement:equilibria have high welfare

Use existing algorithms tofind an equilibrium

The game-theoretic approach

Game design

15

. Distributed algorithm

. Good approximation

. Polytime

utilitiesgame

equilibria

Utility design and approximation ratio

ui (ai , a−i )

=∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ?

Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f

→ game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f }

→ NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) =

infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) = infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Utility design and approximation ratio

ui (ai , a−i ) =∑r∈ai

vrw(|a|r )f (|a|r ) f : N→ R≥0 (distributed)

How to design f ? Maximize worst-case performance

Given instance I, fix f → game Gf = {I, f } → NE(Gf )

PoA(f ) = infGf : #agents ≤M

mina∈NE(Gf ) W (a)

W (aopt)≤ 1

PoA(f ) is the approx. ratio of any equilibrium-computing algorithm

16

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? =

minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? =

minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? =

minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? =

minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? =

minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? = minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Theorem (Characterizing PoA) [J18a], [J18b]

PoA(f ) is the solution to a tractable LP in 2 variables, O(M2) constraints

PoA(f ) =1

W ?

W ? = minf ∈RM

≥0

infλ∈R≥0, µ∈R

µ

s.t. 1{b+x≥1}w(b + x)− µ1{a+x≥1}w(a + x)+

+ λ[af (a + x)w(a+x)− bf (a + x + 1)w(a + x + 1)] ≤ 0

∀(a, x , b) ∈ I ⊂ {0, . . . ,M}3

Corollary Optimizing PoA

Determining f ∈ RM≥0 maximizing PoA(f ) is a tractable linear program

17

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0.2

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0.4

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0.6

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 0.8

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 1

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 1

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

GT

1− c/e

18

Back to the main result

Example:

- # agents ≤ 40

- w(j) = jd , d = 1

1 2 3 4 5

12345

j

w

0 0.2 0.4 0.6 0.8 10.5

0.6

0.7

0.8

0.9

1

d

PoA(fopt)

Solution of LP

GT

1− c/e

1− c/e

18

Conclusions and Outlook

Aggregative games

. Convergence between Nash and Wardrop

. Equilibrium efficiency

. Stochasticity and data

Combinatorial allocation

. GMMC problems and utility design approach

. Characterization and optimization of PoA

. Submodular maximization

19

Conclusions and Outlook

Aggregative games

. Convergence between Nash and Wardrop

. Equilibrium efficiency

. Stochasticity and data

Combinatorial allocation

. GMMC problems and utility design approach

. Characterization and optimization of PoA

. Submodular maximization

19

Conclusions and Outlook

Aggregative games

. Convergence between Nash and Wardrop

. Equilibrium efficiency

. Stochasticity and data

Combinatorial allocation

. GMMC problems and utility design approach

. Characterization and optimization of PoA

. Submodular maximization

19

Conclusions and Outlook

Aggregative games

. Convergence between Nash and Wardrop

. Equilibrium efficiency

. Stochasticity and data

Combinatorial allocation

. GMMC problems and utility design approach

. Characterization and optimization of PoA

. Submodular maximization

19

Conclusions and Outlook

Aggregative games

. Convergence between Nash and Wardrop

. Equilibrium efficiency

. Stochasticity and data

Combinatorial allocation

. GMMC problems and utility design approach

. Characterization and optimization of PoA

. Submodular maximization

19

Acknowledgment to collaborators

Prof. J. Lygeros Prof. J.R. Marden Prof. M. Kamgarpour

Dr. B. Gentile Dr. F. Parise Dr. V. Ramaswamy

R. Chandan G. Burger B. Ogunsola

20

Publications - part 1 of 2

[L-CSS18] D. Paccagnan, F. Parise and J. Lygeros. “On the Efficiency of Nash Equilibria inAggregative Charging Games”. IEEE Control Systems Letters, 2018.

[TAC18a] D. Paccagnan?, B. Gentile?, F. Parise?, M. Kamgarpour, and J. Lygeros. “Nashand Wardrop equilibria in aggregative games with coupling constraints”. IEEETransactions on Automatic Control, 2018.

[CPS18] B. Gentile?, F. Parise?, D. Paccagnan?, M. Kamgarpour and J. Lygeros. “Agame theoretic approach to decentralized charging of plug-in electric vehicles”.Challenges in Engineering and Management of Cyber-Physical Systems, RiverPublishers, 2018.

[CDC17] B. Gentile, D. Paccagnan, B. Ogunsola and J. Lygeros. “A Novel Concept ofEquilibrium Over a Network”. IEEE Conference on Decision and Control, 2017.

[IFAC17] G. Burger, D. Paccagnan, B. Gentile, and J. Lygeros. “Guarantees ofconvergence to a dynamic user equilibrium for a network of parallel roads”. IFACWorld Congress, 2017.

[CDC16] D. Paccagnan?, B. Gentile?, F. Parise?, M. Kamgarpour, and J. Lygeros.“Distributed computation of generalized Nash equilibria in quadratic aggregativegames with affine coupling constraints”. IEEE Conference on Decision andControl, 2016.

[ECC16] D. Paccagnan, M. Kamgarpour, and J. Lygeros. “On Aggregative and Mean FieldGames with Applications to Electricity Markets”. European Control Conference,2016.

21

Publications - part 2 of 2

[TAC18b] D. Paccagnan and J.R. Marden. “The Importance of System-Level Information inMultiagent Systems Design: Cardinality and Covering Problems”. IEEETransactions on Automatic Control, 2018.

[J18a] D. Paccagnan, R. Chandan and J.R. Marden. “Distributed resource allocationthrough utility design - Part I: optimizing the performance certificates via theprice of anarchy”. Submitted; arXiv:1807.01333, 2018.

[J18b] D. Paccagnan and J.R. Marden. “Distributed resource allocation through utilitydesign - Part II: applications to submodular, supermodular and set coveringproblems”. Submitted; arXiv:1807.01343, 2018.

[J18c] V. Ramaswamy, D. Paccagnan and J.R. Marden. “Multiagent CoverageProblems: The Trade-off Between Anarchy and Stability”. Submitted;arXiv:1710.01409, 2017.

[ALL17] D. Paccagnan and J.R. Marden. “The Risks and Rewards of ConditioningNoncooperative Designs to Additional Information”. Allerton Conference onCommunication, Control, and Computing, 2017.

[CDC15] D. Paccagnan, M. Kamgarpour, and J. Lygeros. “On the Range of FeasiblePower Trajectories for a Population of Thermostatically Controlled Loads”. IEEEConference on Decision and Control, 2015.

[PLANS14] M.J. Joergensen, D. Paccagnan, N.K. Poulsen, and M.B. Larsen. “IMUCalibration and Validation in a Factory, Remote on Land and at Sea”. IEEEPosition Location and Navigation Symposium, 2014.

22