Three Recent Results in Algorithmic Game Theory

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Three Recent Resultsin Algorithmic Game Theory

• Christos H. Papadimitriou
• Joint work with Costis Daskalakis, Michael
  Schapira, Yaron Singer, and Greg Valiant

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The three main areas of AGT

• Algorithmic mechanism design
• Price of anarchy
• Computing equilibria
• This talk A recent result in each

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I The fundamental problemof algorithmic
mechanism design

• In what ways is the power of computation
  restricted when the inputs are provided by
  selfish agents?

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Shortest path VCG auction
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t
s
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pay e its declared cost c(e), plus a bonus equal
to dist(s,t)c(e) ?- dist(s,t)
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Shortest path VCG auction
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t
s
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Incentive compatible The selfish agents will
reveal their true inputs (costs)
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But what about TSP auction?(or combinatorial
auctions?)

• Must solve an NP-complete problem many times!
• Approximation?
• Approximation and incentive compatibility dont
  mix

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Fundamental question

• Is there an NP-hard problem that can be
  approximated well,
• but no polynomial-time incentive-compatible
  mechanism can yield a good approximation?
• (under some complexity assumptions, of course)

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To put it otherwise

• VCG implies that
• Pic P,
• NPic NP,
• NP-completeic NP-complete
• But is it also the case that
• APX APXic ?

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Answer No!

• The combinatorial public project problem (CPPP)
• Given n valuations on subsets of size k from a
  universe m
• Find the subset with k elements that has the best
  sum of valuations

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Answer No! (cont)

• If the valuations are (near-)submodular, the
  problem can be approximated within a factor of
  1-1/e
• Theorem (with M. Schapira and Y. Singer,
  2008) Unless NP is in BPP, no incentive
  compatible mechanism can achieve a constant
  approximation ratio.

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Sketch of proof

• Robertss theorem For unrestricted valuations,
  incentive-compatible mechanisms are affine
  maximizers
• It fails for most restricted domains (such as
  combinatorial auctions).
• But it works here! (by modification of
  Mualem-Nisan 2005).

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Sketch of proof (cont.)

• Exponential lower bound on the size of the affine
  maximizer via communication complexity
• Show that every exponential affine maximizer has
  a combinatorial core (Sauers Lemma)
• Embed an NP-hard problem in that core
• Make Sauers Lemma constructive
  (probabilistically, using Ajtai 1998)

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II The price of anarchySelfishness can hurt
you!
delays
Social optimum 1.5
x
1
0
Anarchical equilibrium 2
x
1
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This is the worst case!
Price of anarchy
4/3 Roughgarden and Tardos,
2000, Roughgarden 2002
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But is the model realistic?

• In the Internet, flows dont pick routes
• Nodes decide how to split incoming flows
• (Based on local information, but this is another
  story.)

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What if the nodes decide?
opt 31/20 p. of A 31/30 (cf 4/3) Only
problem scales down
b
x
1
a
a - b
0
x
1
1-a
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Big problem!

• Theorem (with G. Valiant, 2008)
• Price of anarchy is n a
• Proof Recursive construction

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But there are good news

• In series-parallel graphs, the price of anarchy
  is one
• In a model with prices, price of anarchy is also
  one

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III Computing Nash equilibria

• Shown to be PPAD-complete GDP06
• Even for 2-player games CD06
• Approximate Nash equilibrium? (i.e., no player
  can improve by more than e)
• Additive, with normalized utilities
• .75 ? .50 ? .39 ? .36 ? .34 ? ?

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The trouble with approximate Nash

• Algorithms expert to TSP user
• Unfortunately, with current technology we can
  only give you a solution guaranteed to be no more
  than 50 above the optimum

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The trouble with approximate Nash(cont.)

• Irate Nash user to algorithms expert
• Why should I adopt your recommendation and
  refrain from acting in a way that I know is much
  better for me? And besides, given that I have
  serious doubts myself, why should I even believe
  that my opponent(s) will adopt your
  recommendation?

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Bottom line

• PTAS is the only interesting question here

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Is there a PTAS?

• nlog n/e2 algorithm (?PTAS LMM03)
• PTAS for anonymous games DP07, DP08
• Basic idea Quantize probabilities to multiples
  of d
• Show distribution does not move much

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PTAS for sparse games

• Sparse games are known to be PPAD-complete CD07
• But they have an easy PTAS

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PTAS for linear supports

• If the game has an equilibrium with a small
  (constant) support, it is easy to find
• So, what it has an equilibrium with O(n) support?
• Theorem with C. Daskalakis PTAS
• Idea Pick a support of size O(log n / e2)
  uniformly at random
• With inverse poly probability, it works.

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Oblivious PTAS?

• All of these algorithms are oblivious
• Randomized algorithms that look at the game only
  to check if the found solution is an
  eapproximate Nash equilibrium
• Theorem with C. Daskalakis There is no
  oblivious PTAS for the general Nash equilibrium
  problem.

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So

• Many of the mysteries of Algorithmic Game Theory
  seem to be unraveling
• Finally, lower bounds for AMD
• Revisiting selfish routing
• PTAS for Nash equilibria? Much progress, but the
  general case seems hard
• Many new open problems

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Thank You!