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1
Efficiency and Risk Trade-‐offs in Dynamic Oligopoly Markets
Laboratory for Informa?on and Decision Systems MassachuseCs Ins?tute of Technology
[1] Q. Huang, M. Roozbehani, M. Dahleh, “Efficiency and Risk Impacts of Market Architecture in Power Systems under Real-‐Time Pricing”, To Appear in IEEE CDC, 2012 [2] ____, “Efficiency and Risk Trade-‐offs in Dynamic Oligopoly Markets”, work in progress
Qingqing Huang, Mardavij Roozbehani, Munther Dahleh
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Motivation
2
• How does endogenous risk emerge in multi-agent dynamic system?
• What’s the impact of local incentives on system-wide efficiency and risk?
market architectures à agent behaviors à aggregate outcome
Efficiency VS Risk
Tradeoff
Exogenous shocks VS endogenous risk Tail risk, aggregate demand / price spikes
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Contribution
3
Efficiency
Robustness
Coopera3ve
Non-‐coopera3ve
Price of Anarchy
Value of Anarchy
coopera?on Reduce externality, more efficient More vulnerable to some exogenous shocks
system dynamics, uncertain?es, fric?ons (deadline constraints)
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4
• Previous works on endogenous risk • Heterogeneous beliefs [1] • Failure of the agents to rationalize feedback links [2]
• Our work • Agents are fully aware of pricing mechanism,
have perfect information of system state, form rational expectations about other agents in the market.
• Market architecture à Endogenous risk • Tradeoff
• Approach • Case study : Dynamic stochastic game, MDP • LTI reformulation and tradeoff analysis
[1] The leverage cycle (Geanakoplos 2009) [2] “Endogenous Risk” (Danielsson and Shin 2003)
Literature
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Setup
5
Agent arrival • L types (deadline constraint) • Uncertainty
• Bernoulli arrival • Workload distribu?on
(variance of aggregate demand process) (tail probability of aggregate demand process)
2
2
3
1
1
t
t+1
t+2
t+3
1 Decision • State informa?on • Load scheduling
Cost Pricing
System performance • Efficiency: expected average cost • Risk: aggregate demand spikes
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Setup
6
Market state: backlogged loads of existing agents
Price (marginal cost)
Supply Schedule
Dynamic demand scheduling
Market architectural properties Cooperative / non-cooperative Pricing method Risk sensitivity Information
u(t)
Exogenous demand shocks
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Setup
7
Applications: consumer response to real-time pricing in power grids load scheduling in cloud computing market multi-portfolio execution problem consumption risk sharing
Market state: backlogged loads of existing agents
Price (marginal cost)
Supply Schedule
Dynamic demand scheduling
u(t)
Exogenous demand shocks
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Non-cooperative Cooperative
Case study L=2 Solutions
t
t+1
t+2
• At most 1 decision maker at each period
• Linear quadra?c
Dynamic stochas?c game Markov perfect equilibrium
Infinite horizon average cost MDP
In coopera?ve scheme, agents respond more aggressively to other agents’ shock
Low efficiency Low risk
High efficiency High risk
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Case study L=2 Results
Cooperative Non-cooperative
Aggregate demand sample path!
small time scale large time scale
spikes
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Case study L=2 Results
Aggregate demand distribution!
Linear scale Log scale
risk
efficiency
Cooperative Non-cooperative
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• Jump linear system • No closed form solu?on
(a) (b)
General L analysis Modified system dynamics
• Cast to an LTI system • Mul?-‐objec?ve op?miza?on • Efficiency fron?er characteriza?on • Impact of market structure
Risk of spikes High backlog of load Absence of flexible load (Bernoulli)
Coopera?ve load scheduling less externality more rely on each other more heavily use of backlog
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• Focus on linear dynamics • Relax deadline constraints • Subs?tute risk measure
(a) (b)
General L analysis Modified system dynamics
• LTI system • Mul?-‐objec?ve op?miza?on
• Efficiency aggregate demand 2nd moment • Risk aggregate backlog 2nd moment • Aggregate unsa3sfied load 2nd moment
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General L analysis Results
Efficiency fron3er characteriza3on (op?mal F design with weighted outputs)
Three way tradeoff: • -‐ Efficiency aggregate demand 2nd moment • Risk aggregate backlog 2nd moment • Unsa?sfied load aggregate unsa?sfied load 2nd moment
-‐ Efficiency Risk
unsa?sfied load
feasible region
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Contribution
14
Efficiency
Robustness
Coopera3ve
Non-‐coopera3ve
Price of Anarchy
Value of Anarchy
Robust (low variance) yet fragile (fat tail)
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Conclusion
Efficiency
Robustness Other goals (e.g. complexity…)
Outcome in multi-agent
systems
System design Dynamic pricing rule?
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Thank you
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System state:
• Aggregate unshiftable loads
• Consumer arrival with flexible load
Load scheduling decision:
• At most 1 decision maker at : the new type 2 agent
• Split load into two periods based on state information
Case study L=2 System Dynamics
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Efficiency and Risk Implications of Architecture
18
Market architecture Welfare Consumer
behavior
Cooperative More aggressive Consumer More efficient
High risk
Non-cooperative
More conservative
Consumer Less efficient Low risk
absorbing exogenous uncertain3es
price of coordination?
value of anarchy?
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Efficiency
Risk
Producer surplus + Consumer surplus Variance
Tail probability Probability of aggregate unshiftable load peak
market architectures à agent behaviors à aggregate demand process
Case study L=2 Welfare measures
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Symmetric Markov Perfect equilibrium in dynamic stochastic game
Overlapping type 2 consumers Flavor of Stackelberg competition
t
t+1
t+2
Case study L=2 Solution: Non-Cooperative Case
• Full state (and everything else) informa?on • No coordina?on among strategic agents • Focus on steady state in symmetric equilibria
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Symmetric Markov Perfect equilibrium in dynamic stochastic game
Equilibrium strategy
There exists a unique symmetric MPE with linear equilibrium strategy: where are constants determined by .
Case study L=2 Solution: Non-Cooperative Case
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Bellman equation for infinite horizon average cost MDP
Optimal stationary policy
There exists an op?mal sta?onary load scheduling policy: where are constants determined by .
Case study L=2 Solution: Cooperative Case
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Non-cooperative Cooperative
market architectures à agent behaviors à aggregate demand process
Case study L=2 Solutions
t
t+1
t+2
A nc > a c, b nc < bc
(variance) (tail probability)
System performance Efficiency: expected average cost Risk: aggregate demand spikes
Low efficiency Low risk
High efficiency High risk
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x(t)
F
d(t)
u(t)
Bx(t)
y1(t) y2(t) y3(t)
General L analysis Modified system dynamics
When do spikes occur?
Absence of flexible load High backlog state