1. CSCI 3210: Computational Game Theory
Inefficiency of Equilibria &
Routing Games
Ref: Ch 17, 18 [AGT]
Mohammad T. Irfan
Web:

2. Split or steal game $0+fr., $66K $33K, $33K $66K, $0+fr. $0, $0
NE outcome vs. socially best/optimal outcome
Lucy
Payoff matrix
Split
Steal
$33K, $33K
$0+fr., $66K
$66K, $0+fr.
$0, $0
Tony

3. Prisoner's "dilemma" game
Again: NE outcome vs socially optimal outcome
Suspect 2
Payoff matrix
Not Confess
Confess
1, 1
10, 0
0, 10
5, 5
Suspect 1
Costs (negative of payoffs)

4. Measuring the inefficiency of NE
What is the objective function to compare different outcomes?
Utilitarian
Egalitarian
How to deal with multiplicity of NE?
Inefficiency of which NE?
Price of anarchy vs. price of stability

5. Price of Anarchy (PoA) PoA =
Worst objective function value of a NE Objective function value of optimal outcome

6. Price of Stability (PoS)
Best objective function value of a NE Objective function value of optimal outcome

7. Example 1, 1 10, 0 0, 10 5, 5 Calculate PoA and PoS Suspect 2
Payoff matrix
Not Confess
Confess
1, 1
10, 0
0, 10
5, 5
Suspect 1
Costs (negative of payoffs)

8. Example 21, -1 10, 0 100, 10 7, 8 Calculate PoA and PoS Column player
Payoff matrix L R U
21, -1
10, 0 D
100, 10
7, 8
Row player
Costs (negative of payoffs)

9. PoA vs. PoS Consider costs PoA and PoS will be >= 1
PoA = PoS when all NE have the same cost (e.g., unique NE)
In general, PoA >= PoS
PoA: worst case guarantee in a system of independent agents
PoS: measures benefit of a protocol or proposed outcome

10. Pigou's example Assume one unit of traffic from s to t
Objective function: average travel time
NE: all traffic in lower edge. Avg cost = 1.
Optimal: ½ and ½ . Avg cost = ½ * 1 + ½ * ½.
PoA, PoS = ?
x = amount of traffic on edge

11. Routing Games

12. Model: nonatomic selfish routing
Multicommodity flow network
Directed network with multiple (source, sink) pairs
Each (source, sink) pair is called a commodity
ri amount of traffic for each commodity i
Each edge e has a delay or cost function ce
Every car going through an edge gets same delay
Cost of a path = sum of edge costs
Note: cost doesn't depend on identity of players
Congestion games
Nonatomic: Each player is a negligible fraction of the total traffic

13. Equilibrium flow Let f be a feasible flow (combining all commodities)
f is equilibrium flow if
A commodity uses a path P in its flow within f => All detours have higher (or equal) delay
In other words all paths in use by an equilibrium flow f have minimum-possible cost.
equilibrium flow

14. Cost of a flow
Total cost = sum of (delay on each edge * amount of traffic taking that edge)
Total delay faced by every car in traffic

15. Example: nonlinear Pigou
Consider large number p
Equilibrium: All down. cost = 1.
Optimal: ε up, (1-ε) down
Cost = ε * 1 + (1-ε) * (1-ε)p  0
equilibrium flow
PoA  +∞

16. Example: Braess' paradox
1 unit of traffic
equilibrium flow
PoA = 1
Equilibrium cost = ½ * ( ½ + 1) + ½ * (1 + ½) =3/2

17. Braess' paradox New super highway between v and w
Chongicchon restoration project
PoA = 4/3
Equilibrium cost = = 2
Optimal: ½ and ½ (ignore superhighway)