Efficiency of Nash Equilibria Maria Serna Fall 2016 AGT-MIRI, - - PowerPoint PPT Presentation

Contents Price of Anarchy/Stability Examples Load Balancing game

Efficiency of Nash Equilibria

Maria Serna Fall 2016

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

1 Price of Anarchy/Stability 2 Examples 3 Load Balancing game

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Efficiency at equilibrium

We have analyzed Braess’ paradox. The constructors’ goal was to improve traffic congestion, a global measure. Traffic congestion can be measured by the maximum travel time. The NE does not achieve optimal travel time.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Efficiency at equilibrium

We have analyzed Braess’ paradox. The constructors’ goal was to improve traffic congestion, a global measure. Traffic congestion can be measured by the maximum travel time. The NE does not achieve optimal travel time. How far are NE for optimal?

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Efficiency at equilibrium

We have analyzed Braess’ paradox. The constructors’ goal was to improve traffic congestion, a global measure. Traffic congestion can be measured by the maximum travel time. The NE does not achieve optimal travel time. How far are NE for optimal? To perform such an analysis for strategic games we have first to define a global function to optimize, this function is usually called the social cost or social utility.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Efficiency at equilibrium

We have analyzed Braess’ paradox. The constructors’ goal was to improve traffic congestion, a global measure. Traffic congestion can be measured by the maximum travel time. The NE does not achieve optimal travel time. How far are NE for optimal? To perform such an analysis for strategic games we have first to define a global function to optimize, this function is usually called the social cost or social utility. Society is interested in minimizing the social cost or maximizing the social utility.

AGT-MIRI, FIB-UPC NE Efficiency

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Social cost

Consider a n-player game Γ = (A1, . . . , An, u1, . . . , un). Let A = A1 × · · · × An. Let PNE(Γ) be the set of PNE of Γ. Let NE(Γ) be the set of NE of Γ.

AGT-MIRI, FIB-UPC NE Efficiency

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Social cost

Consider a n-player game Γ = (A1, . . . , An, u1, . . . , un). Let A = A1 × · · · × An. Let PNE(Γ) be the set of PNE of Γ. Let NE(Γ) be the set of NE of Γ. Let C : A → R be a social cost function. C can be extended to mixed strategy profiles by computing the average under the joint product distribution.

AGT-MIRI, FIB-UPC NE Efficiency

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Usual social cost functions

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Usual social cost functions

Utilitarian social cost : C(s) =

i∈N ui(s).

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Usual social cost functions

Utilitarian social cost : C(s) =

i∈N ui(s).

Egalitarian social cost: C(s) = maxi∈N ui(s).

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Usual social cost functions

Utilitarian social cost : C(s) =

i∈N ui(s).

Egalitarian social cost: C(s) = maxi∈N ui(s). Game specific cost/utility defined by the model motivating the game.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

The Price of anarchy of Γ is defined as PoA(Γ) = maxσ∈NE(Γ) C(σ) mins∈A C(S) .

AGT-MIRI, FIB-UPC NE Efficiency

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Price of Anarchy/Stability

The Price of anarchy of Γ is defined as PoA(Γ) = maxσ∈NE(Γ) C(σ) mins∈A C(S) . The Price of stability of Γ is defined as PoS(Γ) = minσ∈NE(Γ) C(σ) mins∈A C(S) .

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

The Price of anarchy of Γ is defined as PoA(Γ) = maxσ∈NE(Γ) C(σ) mins∈A C(S) . The Price of stability of Γ is defined as PoS(Γ) = minσ∈NE(Γ) C(σ) mins∈A C(S) . For social utility functions the terms are inverted in the definition.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

For games having a PNE, we might be interested in those values over PNE(Γ) instead of NE(Γ).

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

For games having a PNE, we might be interested in those values over PNE(Γ) instead of NE(Γ). For families of games, we might be interested in analyzing PoA and PoS as a function of some parameter. For example the number of players.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

For games having a PNE, we might be interested in those values over PNE(Γ) instead of NE(Γ). For families of games, we might be interested in analyzing PoA and PoS as a function of some parameter. For example the number of players. PoA measures the worst decentralized equilibrium scenario giving the maximum system degradation.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Price of Anarchy/Stability

For games having a PNE, we might be interested in those values over PNE(Γ) instead of NE(Γ). For families of games, we might be interested in analyzing PoA and PoS as a function of some parameter. For example the number of players. PoA measures the worst decentralized equilibrium scenario giving the maximum system degradation. PoS measures the best decentralized equilibrium scenario giving the best possible degradation.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

1 Price of Anarchy/Stability 2 Examples 3 Load Balancing game

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Braess’ Network

4000 drivers drive from A to B on x/100 45 45 x/100 A U R B

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Braess’ Network

4000 drivers drive from A to B on x/100 45 45 x/100 A U R B Set the social cost to be the maximum travel time.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Braess’ Network

4000 drivers drive from A to B on x/100 45 45 x/100 A U R B Set the social cost to be the maximum travel time. Optimal social cost is reached when half of the drivers take A − U − B and the other half A − R − B with social cost 65.

AGT-MIRI, FIB-UPC NE Efficiency

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Braess’ Network

4000 drivers drive from A to B on x/100 45 45 x/100 A U R B Set the social cost to be the maximum travel time. Optimal social cost is reached when half of the drivers take A − U − B and the other half A − R − B with social cost 65. In the NE all drivers take A − U − R − B with social cost 80.

AGT-MIRI, FIB-UPC NE Efficiency

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Braess’ Network

4000 drivers drive from A to B on x/100 45 45 x/100 A U R B Set the social cost to be the maximum travel time. Optimal social cost is reached when half of the drivers take A − U − B and the other half A − R − B with social cost 65. In the NE all drivers take A − U − R − B with social cost 80. PoA = PoS = 80/65 = 16/13

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing

Social cost: maximum travel time egalitarian

By the characterization of Nash flows all NE have the same cost. PoA=PoS = cost NE / opt

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Selfish routing

Social cost: maximum travel time egalitarian

By the characterization of Nash flows all NE have the same cost. PoA=PoS = cost NE / opt

Other social cost? A natural one is the total travel time utilitarian

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing: total routing time

The cost C(f ) of flow f is the sum of all delays incurred by traffic.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Selfish routing: total routing time

The cost C(f ) of flow f is the sum of all delays incurred by traffic. x f 1 1 − f 1 x 1 f A U R B C(f ) = f (f + 1) + (1 − f )2.

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing: total routing time

The cost C(f ) of flow f is the sum of all delays incurred by traffic. x f 1 1 − f 1 x 1 f A U R B C(f ) = f (f + 1) + (1 − f )2. Formally, if dP(f ) is the sum of latencies of edges in a path P : C(f ) =

• P

fP dP(f )

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Selfish routing: inefficiency of Nash flows

Nash flows do not minimize total latency

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing: inefficiency of Nash flows

Nash flows do not minimize total latency x 1 0.5 0.5 A B

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing: inefficiency of Nash flows

Nash flows do not minimize total latency x 1 0.5 0.5 A B Cost: f 2 + 1 − f + 0.5

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Selfish routing: inefficiency of Nash flows

Nash flows do not minimize total latency x 1 0.5 0.5 A B Cost: f 2 + 1 − f + 0.5 Optimal cost 0.25 + 0.5 + 0.5 = 1.25

AGT-MIRI, FIB-UPC NE Efficiency

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Selfish routing: inefficiency of Nash flows

Nash flows do not minimize total latency x 1 0.5 0.5 A B Cost: f 2 + 1 − f + 0.5 Optimal cost 0.25 + 0.5 + 0.5 = 1.25 Nash flow has cost 1.5

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Prisoner’s dilemma

Utility version Quiet Fink Quiet 2,2 0,3 Fink 3,0 1,1

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Prisoner’s dilemma

Utility version Quiet Fink Quiet 2,2 0,3 Fink 3,0 1,1 Cost version Quiet Fink Quiet 2,2 5,1 Fink 1,5 4,4

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Prisoner’s dilemma

Utility version Quiet Fink Quiet 2,2 0,3 Fink 3,0 1,1 Cost version Quiet Fink Quiet 2,2 5,1 Fink 1,5 4,4 Social cost?

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Contents Price of Anarchy/Stability Examples Load Balancing game

Prisoner’s dilemma

Utility version Quiet Fink Quiet 2,2 0,3 Fink 3,0 1,1 Cost version Quiet Fink Quiet 2,2 5,1 Fink 1,5 4,4 Social cost? Under the sum cost PoA = PoS = 8/4 = 2.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Matching pennies

Head Tail Head 1,-1

• 1,1

Tail

• 1,1

1,-1

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Matching pennies

Head Tail Head 1,-1

• 1,1

Tail

• 1,1

1,-1 This game has a NE ((1/2, 1/2)(1/2, 1/2)) But, it is unclear what is a natural social cost function.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

If there is no path from s to t the social utility is 0

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

If there is no path from s to t the social utility is 0

The social utility on every PNE is also 0 and PoA = PoS = 1

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

If there is no path from s to t the social utility is 0

The social utility on every PNE is also 0 and PoA = PoS = 1

If there is a path from s to t the social utility is 1.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

If there is no path from s to t the social utility is 0

The social utility on every PNE is also 0 and PoA = PoS = 1

If there is a path from s to t the social utility is 1.

There exists a PNE a in which us(a) = 1, so PoS = 1.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Sending from s to t

Recall that we were interested in sending a packet from s to t. The game has PNE, so we analyze PoA and PoS on the set of PNE. One natural social utility is value 1 if the packet is sent and 0

• therwise.

If there is no path from s to t the social utility is 0

The social utility on every PNE is also 0 and PoA = PoS = 1

If there is a path from s to t the social utility is 1.

There exists a PNE a in which us(a) = 1, so PoS = 1. If there exists a PNE a in which us(a) = 0, PoA = 1/0 = ∞,

• therwise PoA = 1.

Other social utility functions are left as exercises.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

1 Price of Anarchy/Stability 2 Examples 3 Load Balancing game

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game

There are m servers and n jobs. Job i has load pi. The game has n players, corresponding to the n jobs. Each player has to decide the server that will process its job. Ai = {1, . . . , m} The response time of server j is proportional to its load Lj(s) =

• i|si=j

pi. Each job wants to be assigned to the server that minimizes its response time: ci(s) = Lsi(s).

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Consider the best response dynamic Start with an arbitrary state. A node (or several) chooses a best strategy, one that maximizes its own payoff, given the current choices of the

• thers

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Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Consider the best response dynamic Start with an arbitrary state. A node (or several) chooses a best strategy, one that maximizes its own payoff, given the current choices of the

• thers

How to prove that such a process converges to a PNE?

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Consider the best response dynamic Start with an arbitrary state. A node (or several) chooses a best strategy, one that maximizes its own payoff, given the current choices of the

• thers

How to prove that such a process converges to a PNE? Seek for an adequate kind of potential function.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

BR-inspired-algorithm Order the servers with decreasing load (i.e., the decreasing response time): L1 L2 . . . Lm. Job i moves from server j to k, Lk + pi < Lj. We must have L1 . . . Lj . . . Lk . . . Lm. Thus, Lj − pi < Lk + pi

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

BR-inspired-algorithm Order the servers with decreasing load (i.e., the decreasing response time): L1 L2 . . . Lm. Job i moves from server j to k, Lk + pi < Lj. We must have L1 . . . Lj . . . Lk . . . Lm. Thus, Lj − pi < Lk + pi Reorder the servers by decreasing load and repeat the process until no job can move.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Does the algorithm converge?

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Does the algorithm converge? There are a finite number of (possibly exponential) assignments of jobs to servers.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Does the algorithm converge? There are a finite number of (possibly exponential) assignments of jobs to servers. At each step the sorted load sequence decreases!

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Does the algorithm converge? There are a finite number of (possibly exponential) assignments of jobs to servers. At each step the sorted load sequence decreases! So BR-inspired-algorithm terminates (although it can be rather slow).

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PNE?

Does the algorithm converge? There are a finite number of (possibly exponential) assignments of jobs to servers. At each step the sorted load sequence decreases! So BR-inspired-algorithm terminates (although it can be rather slow). The load balancing game has a PNE.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: Social cost

The natural social cost is the total finish time i.e., the maximum of the server’s loads c(s) =

m

max

j=1 Lj.

How bad/good is a PNE?

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoS

Let s be an assignment with optimal cost. Is s a PNE?

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoS

Let s be an assignment with optimal cost. Is s a PNE? not necessarily, no player in the worst server can improve, however other players can get a better benefit.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoS

Let s be an assignment with optimal cost. Is s a PNE? not necessarily, no player in the worst server can improve, however other players can get a better benefit. However, starting from an optimal solution the BR-inspired-algorithm terminates on a PNE with the same maximum load.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoS

Let s be an assignment with optimal cost. Is s a PNE? not necessarily, no player in the worst server can improve, however other players can get a better benefit. However, starting from an optimal solution the BR-inspired-algorithm terminates on a PNE with the same maximum load. Therefore, PoS(Γ) = 1.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA

Theorem The max load of a Nash equilibrium s is within twice the max load

• f an optimum assignment, i.e.,.

C(s) 2 min

s′ C(s′).

Which will give PoA(Γ) 2.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server,

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

The best possible algorithm is to evenly partition them among m servers (if possible), thus

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

The best possible algorithm is to evenly partition them among m servers (if possible), thus

k Lk/m ( ℓ pℓ)/m.

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

The best possible algorithm is to evenly partition them among m servers (if possible), thus

k Lk/m ( ℓ pℓ)/m.

We get C(s) = Lj (

k Lk)/m + pi

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

The best possible algorithm is to evenly partition them among m servers (if possible), thus

k Lk/m ( ℓ pℓ)/m.

We get C(s) = Lj (

k Lk)/m + pi ( ℓ pℓ)/m + pi

AGT-MIRI, FIB-UPC NE Efficiency

Contents Price of Anarchy/Stability Examples Load Balancing game

Load Balancing game: PoA bound

Let i be a job assigned to the max loaded server j.

Lj Lk + pi, for all other server k. Summing over all servers, we get Lj (

k Lk)/m + pi.

In an opt solution, i is assigned to some server, so C(s′) pi.

• k Lk is the total processing time for an assignment.

The best possible algorithm is to evenly partition them among m servers (if possible), thus

k Lk/m ( ℓ pℓ)/m.

We get C(s) = Lj (

k Lk)/m + pi ( ℓ pℓ)/m + pi

C(s′) + C(s′).

AGT-MIRI, FIB-UPC NE Efficiency