Networks and Games
Christos H. Papadimitriou
UC Berkeley
christos
• Goal of TCS (1950-2000):
Develop a mathematical understanding of the
capabilities and limitations of the von
Neumann computer and its software –the
dominant and most novel computational
artifacts of that time
(Mathematical tools: combinatorics, logic)
• What should Theory’s goals be today?
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The Internet
• Huge, growing, open, end-to-end
• Built and operated by 15.000 companies in
various (and varying) degrees of competition
• The first computational artefact that must be
studied by the scientific method
• Theoretical understanding urgently needed
• Tools: math economics and game theory,
probability, graph theory, spectral theory
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Today:
•
•
•
•
•
Nash equilibrium
The price of anarchy
Vickrey shortest paths
Congestion games
Collaborators: Alex Fabrikant,
Joan
Feigenbaum, Elias Koutsoupias,
Eli
Maneva, Milena Mihail, Amin Saberi,
Rahul Sami, Scott Shenker
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Game Theory
strategies
strategies
3,-2
payoffs
(NB: also, many players)
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e.g.
prisoner’s dilemma
matching pennies
1,-1 -1,1
3,3
0,4
-1,1 1,-1
4,0
1,1
chicken
0,0
0,1
1,0
-1,-1
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concepts of rationality
• undominated strategy
(problem: too weak)
• (weakly) dominating srategy (alias “duh?”)
(problem: too strong, rarely exists)
• Nash equilibrium (or double best response)
(problem: may not exist)
• randomized Nash equilibrium
.
.
.
Theorem [Nash 1952]: Always exists.
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is it in P?
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The critique of mixed
Nash equilibrium
• Is it really rational to randomize?
(cf: bluffing in poker, tax audits)
• If (x,y) is a Nash equilibrium, then any y’
with the same support is as good as y
(corollary: problem is combinatorial!)
• Convergence/learning results mixed
• There may be too many Nash equilibria
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The price of anarchy
cost of worst Nash equilibrium
“socially optimum” cost
[Koutsoupias and P, 1998]
Also: [Spirakis and Mavronikolas 01,
Roughgarden 01, Koutsoupias and Spirakis 01]
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Selfishness can hurt you!
delays
x
1
Social
optimum: 1.5
0
x
1
Anarchical
solution: 2
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Worst case?
Price of
anarchy
= “2” (4/3 for linear delays)
[Roughgarden and Tardos, 2000,
Roughgarden 2002]
The price of the Internet architecture?
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Simple net creation game
(with Fabrikant, Maneva, Shenker PODC 03)
• Players: Nodes V = {1, 2, …, n}
• Strategies of node i: all possible subsets of
{[i,j]: j  i}
• Result is undirected graph G = (s1,…,sn)
• Cost to node i:
ci[G] =  | si | + i distG(i,j)
delay costs
cost of edges
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Nash equilibria?
• (NB: Best response is NP-hard…)
• If  < 1, then the only Nash equilibrium is
the clique
• If 1 <  < 2 then social optimum is clique,
Nash equilibrium is the star (price of
anarchy = 4/3)
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Nash equilibria (cont.)
•  > 2? The price of anarchy is at least 3
• Upper bound: 
• Conjecture: For large enough , all Nash
equlibria are trees.
• If so, the price of anarchy is at most 5.
• General wi : Are the degrees of the Nash
equilibria proportional to the wi’s?
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Mechanism design
(or inverse game theory)
• agents have utilities – but these utilities are
known only to them
• game designer prefers certain outcomes
depending on players’ utilities
• designed game (mechanism) has designer’s
goals as dominating strategies (or other
rational outcomes)
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e.g., Vickrey auction
• sealed-highest-bid auction encourages gaming and
speculation
• Vickrey auction: Highest bidder wins,
pays second-highest bid
Theorem: Vickrey auction is a truthful mechanism.
Theorem: It maximizes social benefit and
auctioneer expected revenue.
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e.g., shortest path auction
6
s
3
5
4
t
6
10
3
11
pay e its declared cost c(e),
plus a bonus equal to dist(s,t)|c(e) = - dist(s,t)
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Problem:
1
1
1
1
s
1
10
t
Theorem [Elkind, Sahai, Steiglitz, 03]: This is
inherent for truthful mechanisms.
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But…
• …in the Internet (the graph of
autonomous systems) VCG overcharge
would be only about 30% on the
average [FPSS 2002]
• Could this be the manifestation of
rational behavior at network creation?
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• Theorem [with Mihail and Saberi, 2003]:
In a random graph with average degree d,
the expected VCG overcharge is constant
(conjectured: ~1/d)
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Question:
• Are there interesting classes of games
with pure Nash equilibria?
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e.g.: the party affiliation game
• n players-nodes
• Strategies: +1, -1
3
• Payoff [i]:
sgn(j s[i]*s[j]*w[i,j])
3
-2
1
-9
Theorem: A pure Nash equilibrium exists
Proof: Potential function i,j s[i]*s[j]*w[i,j]
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PLS-complete
(that is, as hard as any problem in which
we need to find a local optimum)
[Schaeffer and Yannakakis 1995]
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Congestion games
[joint work with Alex Fabrikant]
•
•
•
•
•
n players
resources E
delay functions Z
Z
strategies: subsets of E
-payoff[i]:  e in s[i] delay[e,c(e)]
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delay fcn: 10, 32, 42, 43, 45, 46
2, 3
1, 4
5, 6
1
3
2, 4, 5, 6
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Theorem [Rosenthal 1972]:
Pure equilibrium exists
Proof:
Potential function =
e j = 1c[e] delay[e,j]
(“pseudo-social cost”)
Complexity?
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Special cases
• Network game vs Abstract game
• Symmetric (single commodity)
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Abstract, non-symmetric
Abstract, symmetric
Network, non-symmetric
PLS-complete
polynomial
Network, symmetric
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Algorithm idea:
1, 45
delay fcn
1, 42
1, 31
10,31,42,45
1, 10
capacity cost
min-cost flow finds equilibrium
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Also…
• Same algorithm approximates equilibrium
in non-atomic game (as in [Roughgarden
2003])
• “Price of anarchy” is unbounded, and
NP-hard to compute
• Other games with guaranteed pure
equilibria?
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