The Price of Anarchy
Shan Li
Justin Azadivar
Price of Anarchy-the Idea
Motivation: equilibria of noncooperative
games typically inefficient
Prisoner’s Dilemma
¾
Congestion of the highway
¾
„
„
Price of anarchy: quantify inefficiency
w.r.t some objective function
Example: Braess’s Paradox
„
„
Setup: traffic (e.g.,many cars or
packets) pick s-t paths to minimize
travel time
Travel time increases with congestion
C(x) = x
C(x) = 1
t
s
C(x) = 1
C(x) = x
Example: Braess’s Paradox
C(x) = x
½ of traffic
C(x) = 1
t
s
C(x) = 1
C(x) = x
½ of traffic
At “equilibrium”, travel time = 3/2 for all
Example: Braess’s Paradox
„
Initial Network:
x
1/2
1/2
1
t
s
1 1/2
„
Travel time = 1.5
1/2
x
Example: Braess’s Paradox
„
Augmented Network:
x
s
1
1
„
„
t
0
Travel time = 2
All traffic worse off!
x
The Definition
„
„
Price of anarchy: worst-case ratio
between “social cost” of equilibrium
and of optimum
Definition:
Price of
Anarchy
of a
game
Obj fn value of a Nash equilibrium
=
---minimization:
---maximization:
Optimal obj fn value
>=1
<=1
Pigou’s Example
„
1 unit of traffic travels from s to t
c(x) = x
s
Cost depends on congestion
t
c(x) = 1
No congestion effects
Note: top edge is a dominant strategy
¾
¾
¾
In equilibrium: all traffic on top edge
All traffic incurs cost 1
Due to [Pigou 1920]
Could we do better?
Consider instead : traffic split equally
c(x) = x
s
t
c(x) = 1
™
™
™
Flow = 1/2
Flow = 1/2
Cost of Nash flow = 1*1+0*1 = 1
Cost of optimal flow = ½*½ + ½*1 = 3/4
Price of anarchy = 4/3
xd
Nonlinear Pigou’s Example
„
Bad Example:
xd
1
(d large)
1-e
s
t
0
„
„
1
e
Equilibrium has cost 1, min cost
0
Nash flow can cost arbitrarily more than the
optimal (min-cost) flow
---even if cost functions are polynomials
Resource Allocation Games
„
„
„
Based on model of [Kelly 97]
Players submit bids, price-based mechanism
responds by allocating capacity to players
Kelly’s assumption: players are “price-takers”, don’t
anticipate how their bid affects price computed by
mechanism
--standard assumption when many players
Thm: selfish behavior
allocation
optimal
Resource Allocation Games
‰ [Johari
/ Tsitsiklis 04]: but what if
players are price-anticipating?
-- selfish behavior no longer achieves optimal
solution
‰ Main
4/3
result: price of anarchy is exactly
Supply Chain Games
„
„
Quantifying the efficiency of a
decentralized system vs. the
performance of a centralized system
Inefficiencies arise from decentralizing
operations (bullwhip effect, double
marginalization), competition, and
information asymmetries.
Supply Chain Games
„
The Price of Anarchy in Supply
Chains: Quantifying the Efficiency
of Price-Only Contracts
[Gergia Perakis , Guillaume Roels
MIT, 2005]
Supply Chain Games
„
Price-only Contracts:
™A buyer and a seller agree only on a
constant transaction price
™Without specifying the amount that will
be transferred
Supply Chain Games
„
Price-only Contracts:
™It is well known that it does not provide
incentives to the parties to coordinate
their inventory/capacity decisions.
™The level of inventory in the decentralized
supply chain is less than the optimal
inventory level that would be held if the
supply chain was integrated.
Supply Chain Games
„
„
Example of Loss of efficiency due to
decentralization ---double marginalization
Limited performance of price-only
contracts
¾Retailer bears demand risk vs. Supplier
¾Assembly system
¾Stochastic demand vs. Deterministic demand
Supply Chain Games
„
Improve coordination in supply chain
™Buy-back
™Revenue sharing
™Quantity flexibility
™Sales rebate
™Quantity discount contracts
Supply Chain Games
„
„
Back to price-only contract
Important to quantify the loss of
efficiency
Price of Anarchy
Model Framework
„
Consider a supply chain facing the
newsvendor problem.
The supply chain has to build its
inventory Q before observing the
demand realization
Model Framework
„
We assume:
™ Costs are linear
™ c per-unit purchasing cost
™ p per-unit selling cost
™ Salvage value is zero
(What if nonzero salvage cost?)
Model Framework
„
We assume (cont.)
™ Demand D is random and has a
cumulative distribution function F(x) that
is strictly increasing and differentiable,
with probability density function f(x).
Model Framework
„
We assume (cont.)
™ The demand distribution has an increasing
generalized failure rate (IGFR)
h(x) = f (x) / F (x) is the hazard rate
g ( x ) = xh ( x ) is the generalized failure rate
™ The IGFR assumption is sufficient to guarantee a
well-behaved (concave) problem for the contract
initiator in a decentralized setting
Model Framework
„
Centralized Supply Chain
max pE[min{Q, D}] − cQ
Q≥0
It is well known that it is newsvendor
problem – concave – has unique optimal
solution
c
Q = F (c / p )
Model Framework
„
„
Decentralized Supply Chain
( Price of Anarchy)
Since local optimization is always
dominated by global optimization,
decentralizing operations usually
results in a loss of efficiency
Model Framework
„
Defination:
The Price of Anarchy (PoA) is the worstcase ratio of the profit achieved with the
centralized supply chain over the profit
achieved with the decentralized supply chain
− cQ c + pE[min{Q c , D}]
PoA = max
F ∈Φ − cQ d + pE[min{Q d , D}]
Where Φ is the set of nonnegative demand
distribution with IGFR property
Model Framework
Model Framework
Model Framework
„
Pull Mode:
The upstream
partner bears the
whole risk of the
supply chain
„
Push Mode:
The downstream
partner bears the
whole risk of the
supply chain
Model Framework
1. The leader offers the follower a contract
specifying the per-unit whole sale price w
2. The follower accept the contract if his
expected profit is above his reservation
profit, assumed to be zero.
Otw, no transaction between parties.
3. The manufacturer chooses his/her level of
inventory at a per-unit cost c
Model Framework
Pull:
4. Demand D is realized.
5. The retailer orders what is needed to meet
demand, at a per-unit cost w. Each unit of
satisfied demand generates a revenue p. If
the manufacturer does not have enough
inventory, the excess demand is lost at no
cost.
Model Framework
Push:
4. The retailer places an order, at a per-unit
cost w.
5. Demand D is realized. Each unit of
satisfied demand generates a revenue p. If
the manufacturer does not have enough
inventory, the excess demand is lost at no
cost.
Serial Push Supply Chain
The inventory is held at the retailer’s
site.
ie. the retailer orders before observing
the demand
„ Consider two Stackelberg games.
„
Serial Push Supply Chain
„ Manufacturer
„
„
is the Leader
The incentive compatibility (I.C.) constraint states that the retailer
maximizes his expected profit
The individual rationality (I.R.) constraint ensures that the retailer
earns at least his reservation profit.
Serial Push Supply Chain
„
Theorem:
In a two-stage push supply chain,
when the manufacturer is the initiator
of the price-only contract,
Serial Push Supply Chain
„ Retailer
„
„
is the Leader
It is easy to see that the retailer proposes a price
w=c.
Therefore, full coordination is achieved, i.e.PoA =
1.
Serial Pull Supply Chain
The inventory is held at the
manufacturer’s site.
ie. the retailer orders after observing
the demand
„ Consider two Stackelberg games.
„
Homework
„
1)
2)
Consider the Serial Pull Supply Chain
structure, derive the optimization
formulation when
Retailer is the leader
Manufacturer is the leader