Lecture 3. The KKT Conditions and the Price of Anarchy

30 Pages·2011·0.11 MB·English

IE 8534 1
Lecture 3. The KKT Conditions and the Price of Anarchy
Shuzhong Zhang
IE 8534 2
A noncooperative game:
Player k has a strategy set S (x−k), where x−k stands for the
k
strategies of all other players, and the payoff function f (xk; x−k),
k
with xk ∈ S (x−k), k = 1, 2, ..., K.
k
The social value of the joint strategies (x1, ..., xK) is
∑K
k −k
f (x ; x ).
k
k=1
The Price of Anarchy (Koutsoupias and Papadimitriou, 1999):
“The worst social value over all Nash equilibria”
PoA =
“The best social value”
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For minimization games, the PoA is more than 1: The larger PoA
value, the less efficient the Nash solution.
For maximization games, the PoA is less than 1: The smaller PoA
value, the less efficient the Nash solution.
Let us take a look at some examples.
Selfish Routing
A Wardrop type network
c1(f1) = 1
A B
c2(f2) = f2
Cost on arc ℓ: C (f ) = c (f )f .
ℓ ℓ ℓ ℓ ℓ
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Suppose infinitely and infinitesimal players, transporting 1 unit of
flow from A to B.
The Nash solution: (0, 1) for the two arcs, with total cost 1.
The social optimal solution (0.5, 0.5), with total cost 0.75.
Roughgarden (2001), Roughgarden and Tard¨os (2003): If c (f )’s are
ℓ ℓ
all affine (c (f ) = a f + b ), then
ℓ ℓ ℓ ℓ ℓ
4
≤
PoA .
3
How about a finite number of players K, transporting flows over a
network?
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Consider a directed graph G = (V ; L) with the set of nodes V , and the
| | | |
set of links L, with V = n and L = m.
Multiple parallel links are allowed but no self-loop exists.
Links are used to model the resources.
Let us denote A ∈ ℜn×m to be the node-to-arc incidence matrix.
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Suppose that there are K players in the game.
Each player wishes to use the network to transport commodities,
which are splittable.
Denote the origin-destination (OD) pairs for all the players to be
{s1, d1}, {s2, d2}, . . . , {sK, dK}. Let r denote a vector in ℜK, where
the component rk represents the amount of commodity that Player k
needs to transport. The transportation plan of Player k will be given
by a vector xk ∈ ℜm, which indicates the flow on each link. Clearly, a
feasible flow is given by the constraints Axk = rkδ − rkδ , where
sk dk
the notation δ signifies the unit vector in ℜn whose i-th component is
i
one while all others are zero.
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For each link l, we denote the total flow on the link to be
∑
f = K xk.
l k=1 l
→
the unit cost for the flow on the link l to be a function c : f , c (f ).
l l l l
Therefore, the data (G, r, c) specifies an instance of the
non-cooperative routing game under consideration.
Indeed, for Player k, given the decisions of other players
(conventionally denoted as x−k) is to minimize his/her own
transportation cost given as:
∑
k k −k k
C (x , x ) = x c (f ).
l l l
l∈L
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Naturally, given the decisions of all the players, the social cost is a
∑
simple summation: SC(x) = K Ck(xk, x−k).
k=1
Let xNash denote the flow when the game reaches a Nash equilibrium;
i.e. a solution at which no player will be able to improve his/her
situation unilaterally.
∗
At the same time, let us denote x to be the socially optimal solution,
namely the solution that minimizes the social cost function SC(x)
over all feasible solutions.
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The so-called Price of Anarchy (PoA) is defined as:
SC(xNash)
PoA = .
∗
SC(x )
We shall first consider the case where the unit cost function is affine
linear in the total flow; that is, the unit cost function on the link l is
≥
c (f ) = a f + b , where a , b 0. Then, each player k will face the
l l l l l l l
following optimization problem:
∑
(P ) min (a f + b )xk
k l∈L l l l l
s.t. Axk = rkδ − rkδ
sk dk
xk ≥ 0.
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∑
Replacing f with K xk, the above problem for Player k is a convex
l k=1 l
quadratic program, in which the decision vector is xk:
{ [( ) ]}
∑ ∑
(P′) min b xk + a xi xk + (xk)2
k l∈L l l l i̸=k l l l
s.t. Axk = rkδ − rkδ
sk dk
xk ≥ 0.
Shuzhong Zhang