Price of stability explained

In game theory, the price of stability (PoS) of a game is the ratio between the best objective function value of one of its equilibria and that of an optimal outcome. The PoS is relevant for games in which there is some objective authority that can influence the players a bit, and maybe help them converge to a good Nash equilibrium. When measuring how efficient a Nash equilibrium is in a specific game we often also talk about the price of anarchy (PoA), which is the ratio between the worst objective function value of one of its equilibria and that of an optimal outcome.

Examples

Another way of expressing PoS is:

PoS=

valueofbestNashequilibrium
valueofoptimalsolution

,PoS\geq0.

In particular, if the optimal solution is a Nash equilibrium, then the PoS is 1.

In the following prisoner’s dilemma game, since there is a single equilibrium

(B,R)

we have PoS = PoA = 1/2.
Prisoner's Dilemma
Left Right
Top(2,2) (0,3)
Bottom(3,0) (1,1)
On this example which is a version of the battle of sexes game, there are two equilibrium points,

(T,L)

and

(B,R)

, with values 3 and 15, respectively. The optimal value is 15. Thus, PoS = 1 while PoA = 1/5.
Left Right
Top(2,1) (0,0)
Bottom(0,0) (5,10)

Background and milestones

The price of stability was first studied by A. Schulz and N. Stier-Moses while the term was coined by E. Anshelevich et al. Schulz and Stier-Moses focused on equilibria in a selfish routing game in which edges have capacities. Anshelevich et al. studied network design games and showed that a pure strategy Nash equilibrium always exists and the price of stability of this game is at most the nth harmonic number in directed graphs. For undirected graphs Anshelevich and others presented a tight bound on the price of stability of 4/3 for a single source and two players case. Jian Li has proved that for undirected graphs with a distinguished destination to which all players must connect the price of stability of the Shapely network design game is

O(logn/loglogn)

where

n

is the number of players. On the other hand, the price of anarchy is about

n

in this game.

Network design games

Setup

Network design games have a very natural motivation for the Price of Stability.In these games, the Price of Anarchy can be much worse than the Price of Stability.

Consider the following game.

n

players;

i

aims to connect

si

to

ti

on a directed graph

G=(V,E)

;

Pi

for a player are all paths from

si

to

ti

in

G

;

ci

;

ne

players choose edge

e

, the cost

stylede(ne)=

ce
ne
is split equally among them;

styleCi(S)=

\sum
e\inPi
ce
ne

styleSC(S)=\sumiCi(S)=\sumene

ce
ne

=\sumece

.

Price of anarchy

The price of anarchy can be

\Omega(n)

. Consider the following network design game.

Consider two different equilibria in this game. If everyone shares the

1+\varepsilon

edge, the social cost is

1+\varepsilon

. This equilibrium is indeed optimal. Note, however, that everyone sharing the

n

edge is a Nash equilibrium as well. Each agent has cost

1

at equilibrium, andswitching to the other edge raises his cost to

1+\varepsilon

.

Lower bound on price of stability

Here is a pathological game in the same spirit for the Price of Stability, instead.Consider

n

players, each originating from

si

and trying to connectto

t

. The cost of unlabeled edges is taken to be 0.

The optimal strategy is for everyone to share the

1+\varepsilon

edge, yieldingtotal social cost

1+\varepsilon

. However, there is a unique Nash for this game.Note that when at the optimum, each player is paying
style1+\varepsilon
n
, and player 1 can decrease his cost by switching to the
style1
n
edge. Once this has happened, it will be in player 2's interest to switch to the
style1
n-1
edge, and so on. Eventually, the agents will reach the Nash equilibrium of paying for their own edge. This allocation has social cost

style1+

1
2

++

1
n

=Hn

, where

Hn

is the

n

th harmonic number, which is

\Theta(logn)

. Even though it is unbounded, the price of stability is exponentially better than the price of anarchy in this game.

Upper bound on price of stability

Note that by design, network design games are congestion games.Therefore, they admit a potential function

style\Phi=\sume

ne
\sum
i=1
ce
i
.

Theorem. [Theorem 19.13 from Reference 1] Suppose there exist constants

A

and

B

such that for every strategy

S

,

ASC(S)\leq\Phi(S)\leqBSC(S).

Then the price of stability is less than

B/A

Proof. The global minimum

NE

of

\Phi

is a Nashequilibrium, so

SC(NE)\leq1/A\Phi(NE)\leq1/A\Phi(OPT)\leqB/ASC(OPT).

Now recall that the social cost was defined as the sum of costs over edges, so

\Phi(S)=\sume

ne
\sum
i=1
ce
i

=\sumece

H
ne

\leq\sumeceHn=HnSC(S).

We trivially have

A=1

, and the computation above gives

B=Hn

, so we may invoke the theorem for an upper bound on the price of stability.

See also

References

  1. A.S. Schulz, N.E. Stier-Moses. On the performance of user equilibria in traffic networks. Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), 2003.
  2. E. Anshelevich, E. Dasgupta, J. Kleinberg, E. Tardos, T. Wexler, T. Roughgarden. The Price of Stability for Network Design with Fair Cost Allocation. SIAM Journal on Computing, 38:4, 1602-1623, 2008. Conference version appeared in FOCS 2004.
    1. L. Agussurja and H. C. Lau. The Price of Stability in Selfish Scheduling Games. Web Intelligence and Agent Systems: An International Journal, 9:4, 2009.
  3. Jian Li. An

O(logn/loglogn)

upper bound on the price of stability for undirected Shapely network design games. Information Processing Letters 109 (15), 876-878, 2009.