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Economic Issues in Shared Infrastructures Costas Courcoubetis - - PowerPoint PPT Presentation
Economic Issues in Shared Infrastructures Costas Courcoubetis - - PowerPoint PPT Presentation
Economic Issues in Shared Infrastructures Costas Courcoubetis Department of Computer Science, Athens University of Economics and Business Richard Weber Statistical Laboratory, University of Cambridge VISA 2009, August 17, Barcelona Resource
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Example: scheduling a server
▶ A single server is to be shared amongst 풏 agents.
Agent 풊 generates a jobs at rate 흀풊.
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Example: scheduling a server
▶ A single server is to be shared amongst 풏 agents.
Agent 풊 generates a jobs at rate 흀풊.
▶ Initially, agents contribute resource amounts 풚1, . . . , 풚풏,
building a server of rate ∑
풌 풚풌.
Under FCFS scheduling all jobs have mean waiting time 1/(∑
풌 풚풌 − ∑ 풌 흀풌).
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Example: scheduling a server
▶ A single server is to be shared amongst 풏 agents.
Agent 풊 generates a jobs at rate 흀풊.
▶ Initially, agents contribute resource amounts 풚1, . . . , 풚풏,
building a server of rate ∑
풌 풚풌.
Under FCFS scheduling all jobs have mean waiting time 1/(∑
풌 풚풌 − ∑ 풌 흀풌). ▶ Agent 풊 suffers a delay cost, so his net benefit is, say,
풏풃풊 = 흀풊풓 − 휽풊흀풊 1 ∑
풌 풚풌 − ∑ 풌 흀풌
− 풚풊 휽풊 is private information of agent 풊.
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The key issue in this talk
Agents (users) have private information (about the value of the tasks they wish to carry out). This creates a problem for efficiently sharing resources.
▶ Agents will attempt to free-ride. ▶ Obvious policies (like ‘internal market’, or ‘equal sharing’)
may not be suitable.
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The key issue in this talk
Agents (users) have private information (about the value of the tasks they wish to carry out). This creates a problem for efficiently sharing resources.
▶ Agents will attempt to free-ride. ▶ Obvious policies (like ‘internal market’, or ‘equal sharing’)
may not be suitable. How one chooses to share a facility’s resources will influence what agents reveal of their private information. We would like agents to truthfully reveal their privately held information since then we can operate the facility more efficiently.
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A model of a managed shared infrastructure
▶ An infrastructure as composed of resources.
(links, servers, buffers, etc.)
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A model of a managed shared infrastructure
▶ An infrastructure as composed of resources.
(links, servers, buffers, etc.)
▶ It can be operated in various ways, 흎1, 흎2, . . .
(by scheduling, routing, bandwidth allocation, etc.)
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A model of a managed shared infrastructure
▶ An infrastructure as composed of resources.
(links, servers, buffers, etc.)
▶ It can be operated in various ways, 흎1, 흎2, . . .
(by scheduling, routing, bandwidth allocation, etc.)
▶ The subset of agents who wish to use resources of the
infrastructure, say 푺, differs from day to day.
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A model of a managed shared infrastructure
▶ An infrastructure as composed of resources.
(links, servers, buffers, etc.)
▶ It can be operated in various ways, 흎1, 흎2, . . .
(by scheduling, routing, bandwidth allocation, etc.)
▶ The subset of agents who wish to use resources of the
infrastructure, say 푺, differs from day to day.
▶ If operated in manner 흎 on day 풕 then agent 풊 has utility
휽풊,풕풖풊(흎) 풖풊(⋅) is pubic knowledge, but only agent 풊 knows 휽풊,풕.
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A model of a managed shared infrastructure
▶ An infrastructure as composed of resources.
(links, servers, buffers, etc.)
▶ It can be operated in various ways, 흎1, 흎2, . . .
(by scheduling, routing, bandwidth allocation, etc.)
▶ The subset of agents who wish to use resources of the
infrastructure, say 푺, differs from day to day.
▶ If operated in manner 흎 on day 풕 then agent 풊 has utility
휽풊,풕풖풊(흎) 풖풊(⋅) is pubic knowledge, but only agent 풊 knows 휽풊,풕.
▶ 흎 is to be chosen as a function of 푺 and of the declared
휽풕 = (휽1,풕, . . . , 휽풏,풕).
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Agents pay for resources
Agents may contribute resources to a shared infrastructure, like 풚1(휽1), . . . , 풚풏(휽풏).
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Agents pay for resources
Agents may contribute resources to a shared infrastructure, like 풚1(휽1), . . . , 풚풏(휽풏). Other times agents pay fees. In this case, we should like them to cover some daily operating cost, 풄, 푬푺,휽 [ 풑1(푺, 휽) + ⋅ ⋅ ⋅ + 풑풏(푺, 휽) ] ≥ 풄
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Agents pay for resources
Agents may contribute resources to a shared infrastructure, like 풚1(휽1), . . . , 풚풏(휽풏). Other times agents pay fees. In this case, we should like them to cover some daily operating cost, 풄, 푬푺,휽 [ 풑1(푺, 휽) + ⋅ ⋅ ⋅ + 풑풏(푺, 휽) ] ≥ 풄 Agent 풊 wishes to maximize his expected net benefit : 풏풃풊(휽풊) = 푬푺,휽 [ 휽풊풖풊(흎(푺, 휽)) − 풑풊(푺, 휽)
- 휽풊
] He may be untruthful in declaring his 휽풊.
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The efficient frontier
We wish to find Pareto optimal points of the vector (풏풃1, . . . , 풏풃풏) = 푬휽 [ 풏풃1(휽1), . . . , 풏풃풏(휽풏) ]
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.6 0.5
full information private information 풄 = 0 풄 = 0.375 풏풃1 풏풃2
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Maximum social welfare
Suppose we wish to find the particular point that maximizes 풏풃1 + ⋅ ⋅ ⋅ + 풏풃풏 = 푬푺,휽 [휽1풖1(흎(푺, 휽)) + ⋅ ⋅ ⋅ + 휽풏풖풏(흎(푺, 휽))] − 풄
0.1 0.2 0.3 0.1 0.2 0.3
풄 = 0.375 풏풃1 풏풃2
We call this the ‘social welfare’.
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Our infrastructure optimization problem
Our infrastructure optimization problem:
▶ Say how the infrastructure will be operated for all possible
subsets of users 푺.
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Our infrastructure optimization problem
Our infrastructure optimization problem:
▶ Say how the infrastructure will be operated for all possible
subsets of users 푺.
▶ Say what fees will be collected from users.
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Our infrastructure optimization problem
Our infrastructure optimization problem:
▶ Say how the infrastructure will be operated for all possible
subsets of users 푺.
▶ Say what fees will be collected from users.
Do the above, as function of declared 휽풊, so that:
- 1. Users find it in their best interest to truthfully reveal their 휽풊.
- 2. Users will see positive expected net benefit from participation.
- 3. Expected total fees cover the daily running cost, say 풄.
- 4. Expected social welfare (total net benefit) is maximized
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Example: a scalar resource
▶ 2 participants, both present on all days.
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Example: a scalar resource
▶ 2 participants, both present on all days. ▶ On day 풕, agent 풊 has utility for resource of 휽풊,풕풖(풙),
assumed known to be distributed 푼[0, 1].
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Example: a scalar resource
▶ 2 participants, both present on all days. ▶ On day 풕, agent 풊 has utility for resource of 휽풊,풕풖(풙),
assumed known to be distributed 푼[0, 1].
▶ The infrastructure provides a single resource, parameterized
by a number (such as computing cycles), so operating methods correspond to allocations: {흎} ≡ {풙1, 풙2 : 풙1 + 풙2 ≤ 1}
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Example: a scalar resource
▶ 2 participants, both present on all days. ▶ On day 풕, agent 풊 has utility for resource of 휽풊,풕풖(풙),
assumed known to be distributed 푼[0, 1].
▶ The infrastructure provides a single resource, parameterized
by a number (such as computing cycles), so operating methods correspond to allocations: {흎} ≡ {풙1, 풙2 : 풙1 + 풙2 ≤ 1} Suppose 풖풊(풙) = 풙. Focus on one day 풕; with 휽풊 = 휽풊,풕.
푬휽1,휽2 ⎡ ⎢ ⎣ max
풙1,풙2 풙1+풙2≤1
{휽1풖1(풙1) + 휽2풖2(풙2)} ⎤ ⎥ ⎦ = 푬 [max{휽1, 휽2}] = 2
3
We call this the ‘first best’.
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The second-best solution
A ‘second-best’ is with fee structure: 풑풊(휽풊) = { 0 , 휽풊 < 휽0
1 2(휽2 풊 + 휽2 0) ,
휽풊 ≥ 휽0 Agent 풊 will not wish to participate if 휽풊 < 휽0, since his net benefit cannot be positive. The entire resource is allocated to the agent declaring the greatest 휽풊, provided this is > 휽0. Thus, the resource is given wholly to one agent, but perhaps to neither. But both agents may pay.
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The ‘solution’
This solution has the advantages that
▶ Agents are incentivized to truthful.
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The ‘solution’
This solution has the advantages that
▶ Agents are incentivized to truthful. ▶ The sum of the expected payments is
푬 [ 풑1(휽1) + 풑2(휽2) ] = 1/3 + 휽2
0 − (4/3)휽3 0 .
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The ‘solution’
This solution has the advantages that
▶ Agents are incentivized to truthful. ▶ The sum of the expected payments is
푬 [ 풑1(휽1) + 풑2(휽2) ] = 1/3 + 휽2
0 − (4/3)휽3 0 . ▶ The expected social welfare is decreasing in 휽0.
But by taking 1/3 + 휽2
0 − (4/3)휽3 0 = 풄 we maximize the
social welfare of 풏풃1 + 풏풃2 = 푬 [ 2 ∑
풊=1
휽풊풖풊(풙풊) − 풑풊(휽풊) ] subject to covering cost 풄.
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Second-best versus first-best
Expected social welfare as a function of 풄.
0.1 0.2 0.3 0.4 0.5 0.6 0.1 0.2 0.3 0.4 0.5 0.6
cost 풏풃1 + 풏풃2 = 풄 first-best second-best
For 풄 ∈ [0.333, 0.416] the second-best falls short of the first-best. There is no way to cover a cost greater than
5 12 = 0.416.
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Other mechanisms can also work.
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Other mechanisms can also work.
▶ We can ensure 풑1(휽1, 휽2) + 풑2(휽1, 휽2) = 풄.
풑1(휽1, 휽2) = 1
2풄+1 2(휽2 1+휽2 0)1{휽1>휽0}−1 2(휽2 2+휽2 0)1{휽2>휽0}
We call this ‘ex-post’ cost-covering.
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Other mechanisms can also work.
▶ We can ensure 풑1(휽1, 휽2) + 풑2(휽1, 휽2) = 풄.
풑1(휽1, 휽2) = 1
2풄+1 2(휽2 1+휽2 0)1{휽1>휽0}−1 2(휽2 2+휽2 0)1{휽2>휽0}
We call this ‘ex-post’ cost-covering.
▶ We can ensure ex-post incentive compatibility and rationality.
I.e., that an agent only pays if he gets resource, and is happy after-the-fact with the 휽풊 he declared. 풑1(휽1, 휽2) = max(휽0, 휽2)1{휽1>max(휽0,휽2)}
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A concave utility
Suppose 풖풊(풙) = √풙 Now the resource is shared differently. The optimal policy is found by solving a Lagrangian dual problem
min
흀≥0
⎧ ⎨ ⎩ 푬휽1,휽2 ⎡ ⎢ ⎣ max
풙1, 풙2≥0 풙1+풙2≤1 2
∑
풊=1
풉흀(휽풊)풖풊(풙풊) ⎤ ⎥ ⎦ − (1 + 흀)풄 ⎫ ⎬ ⎭ .
where 풉흀(휽풊) = (휽풊 + 흀(2휽풊 − 1)) and 풙풊(휽1, 휽2) = 풉흀(휽풊)2 ∑2
풋=1 풉흀(휽풋)2
As 흀 increases the fee structure changes, so that greater cost can be covered. The social welfare decreases, but is maximized subject to the constraint of covering the cost.
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The role of operating policy
The resource is not allocated in the ‘most efficient’ way.That would be 풙풊(휽1, 휽2) = 휽2
풊
휽2
1 + 휽2 2
( not 풉흀(휽풊)2 풉흀(휽1)2 + 풉흀(휽2)2 )
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The role of operating policy
The resource is not allocated in the ‘most efficient’ way.That would be 풙풊(휽1, 휽2) = 휽2
풊
휽2
1 + 휽2 2
( not 풉흀(휽풊)2 풉흀(휽1)2 + 풉흀(휽2)2 ) This is a key lesson: If one wishes to optimally incentivize participation in shared infrastructures, and to make the most of the resources available, then both the (i) fees, and (ii) policies for ‘ resource sharing, must play a part in providing the correct incentives to users.
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Conclusions
▶ In most realistic resource allocation problems there is private
information to participants.
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Conclusions
▶ In most realistic resource allocation problems there is private
information to participants.
▶ Resource management must give appropriate incentives.
To encourage agents who value the resource more to say so, and so be willing to contribute more towards the cost, we need to reward them better than an internal market would do. But figuring out exactly how to do this is not a simple task!
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Conclusions
▶ In most realistic resource allocation problems there is private
information to participants.
▶ Resource management must give appropriate incentives.
To encourage agents who value the resource more to say so, and so be willing to contribute more towards the cost, we need to reward them better than an internal market would do. But figuring out exactly how to do this is not a simple task!
▶ Simple-minded sharing policies (like proportional sharing) may
not to produce sufficient incentives for participants to contribute resources.
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Conclusions
▶ In most realistic resource allocation problems there is private
information to participants.
▶ Resource management must give appropriate incentives.
To encourage agents who value the resource more to say so, and so be willing to contribute more towards the cost, we need to reward them better than an internal market would do. But figuring out exactly how to do this is not a simple task!
▶ Simple-minded sharing policies (like proportional sharing) may
not to produce sufficient incentives for participants to contribute resources.
▶ Many new interesting problems!!!
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Optimal queue scheduling
Instead of declaring contributions they are willing to make, we can imagine that agents (equivalently) declare their 휽풊. Suppose 휽1 < 휽2 < ⋅ ⋅ ⋅ < 휽풏. As a function of these declarations we take contributions of the form 풚(휽풊) from some subset of agents 풊 = 1, . . . , 풋 (a set with smallest 휽풊).
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Optimal queue scheduling
Instead of declaring contributions they are willing to make, we can imagine that agents (equivalently) declare their 휽풊. Suppose 휽1 < 휽2 < ⋅ ⋅ ⋅ < 휽풏. As a function of these declarations we take contributions of the form 풚(휽풊) from some subset of agents 풊 = 1, . . . , 풋 (a set with smallest 휽풊). We employ a priority scheduling policy in which priority is always given to the current job belonging to the agent with greatest 휽풊.
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