Towards a Constructive Theory of Networked Interactions

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Towards a Constructive Theory of Networked
Interactions

• Constantinos Daskalakis
• CSAIL, MIT
• costis_at_csail.mit.edu

Based on joint work with Christos H. Papadimitriou
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A Success Story of Game Theory (and
Mathematical Programming)
1928 Neumann
existence of min-max equilibrium in 2-player,
zero-sum games
proof uses Brouwers fixed point theorem
Danzig 57 equivalent to LP duality
Khachiyan79 polynomial-time solvable
all no-regret learning algorithms converge to
equilibria.
Robert Aumann, 1987
Two-player zero-sum games are one of the few
areas in game theory, and indeed in the social
sciences, where a fairly sharp, unique prediction
is made.
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What about multi-player or non zero-sum Games?
existence of an equilibrium in multiplayer,
general-sum games
1950 Nash
Proof also uses Brouwers fixed point theorem
intense effort for equilibrium algorithms
Kuhn 61, Mangasarian 64, Lemke-Howson 64,
Rosenmüller 71, Wilson 71, Scarf 67, Eaves
72, Laan-Talman 79, etc.
Lemke-Howson simplex-like, works with LCP
formulation

• no efficient algorithm is known after 50 years
  of research.

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Is it NP-complete to find a Nash equilibrium?
the Pavlovian reaction
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Why should we care about the complexity of
equilibria?

• First, if we believe our equilibrium theory,
  efficient algorithms would enable us to make
  predictions

Herbert Scarf writes
Due to the non-existence of efficient
algorithms for computing equilibria, general
equilibrium analysis has remained at a level of
abstraction and mathematical theoretizing far
removed from its ultimate purpose as a method for
the evaluation of economic policy.
The Computation of Economic Equilibria, 1973

• More importantly If we are to take equilibria
  seriously as models of behavior, computational
  tractability is an important modeling
  prerequisite.

If your laptop cant find the equilibrium, then
how can the market?
Kamal Jain, Microsoft Research
N.B. computational intractability implies the
non-existence of efficient dynamics converging to
equilibria how can equilibria be universal, if
such dynamics dont exist?
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Is it NP-complete to find a Nash equilibrium?
the Pavlovian reaction
two answers
1. probably not, since a solution is guaranteed
to exist
2. it is NP-complete to find a tiny bit more
info than just a Nash equilibrium e.g., the
following are NP-complete
- find two Nash equilibria, if more than one exist
- find a Nash equilibrium whose third bit is one,
if any
Gilboa, Zemel 89 Conitzer, Sandholm 03
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- the theory of NP-completeness does not seem
appropriate
so, how hard is it to find a single equilibrium?
- in fact, NASH seems to lie below NP
- making Nashs theorem constructive
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Complexity of the Nash Equilibrium

• Theorem Daskalakis, Goldberg, Papadimitriou
  06
• If players 4,
• then finding a Nash equilibrium is
  PPAD-complete.

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Computational Complexity
The hardest problems in NP
e.g. quadratic programming e.g.2 traveling
salesman problem
NP-complete
NP
Solutions can be verified in polynomial time
PPAD
P
Solutions can be found in polynomial time
e.g. linear programming e.g.2 zero-sum games
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The PPAD Class Pap. 94
the class of all Brouwer fixed point computation
problems, where the function is piece-wise linear
PPAD
Nashs Thm
NASH ? PPAD

DGP 06
NASH4 is PPAD-hard
Chen, Deng 06

NASH3 is PPAD-hard
Dask., Pap. 06

Chen, Deng 06
NASH2 is PPAD-hard
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In other words

• Outside of 2-player zero-sum games, the Nash
  equilibrium is computationally broken.

• Recall Aumanns quote

Two-player zero-sum games are one of the few
areas in game theory, and indeed in the social
sciences, where a fairly sharp, unique prediction
is made.
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Game Over?

• Alternative Solution Concepts with better
  computational properties.

• Complexity of Approximate Nash Equilibria
• maybe players only find an approximate Nash Eq.

• Special Classes of Games with tractable
  equilibria.

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Approximations
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The trouble with approximations

• 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 approximations(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

• Arbitrarily close approximation is the only
  interesting question here

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Approximate Equilibria
Goal
Approximation Relative vs additive incentive
(shift invariant)
(scale invariant)
CDT 06
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Larger epsilons?
Important Open Problem
Daskalakis 09
Relative e-NASH is PPAD-complete, even for
constant es.
So answer is No!
What about the additive e-NASH, for constant
es?
An important open problem, at the boundary of
intractability. N.B. a PPAD-completeness result
is unlikely for additive es
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tractable special cases
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Networks of Competitors
- players are nodes of a graph G
- edges are zero-sum games
- players payoff is the sum of payoffs from all
adjacent edges
N.B. finding a Nash equilibrium is PPAD-complete
for general games on the edges D, Gold, Pap 06
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Networks of Competitors
The simplest case
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Networks of Competitors
The second simplest case
LP duals
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Networks of Competitors
Theorem Daskalakis, Papadimitriou 09
In every network of competitors
- a Nash equilibrium can be found efficiently
with linear-programming
- the Nash equilibria comprise a convex set
- if every node uses a no-regret learning
algorithm, the players behavior converges to a
Nash equilibrium.
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No-regret algorithms

• widely used game-playing algorithms

e.g. experts algorithm, (perturbed) fictitious
play, etc.
no-regret property


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Networks of Competitors
Theorem Daskalakis, Papadimitriou 09
In every network of competitors
- a Nash equilibrium can be found efficiently
with linear-programming
- the Nash equilibria comprise a convex set
- if every node uses a no-regret learning
algorithm, the players behavior converges to a
Nash equilibrium.
strong indication that Nash eq. makes sense in
this setting.
N.B. but Tardos 09 the value of the nodes is
not unique.
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Another Tractable Case Games with Symmetries
Anonymous Games Every player is (potentially)
different, but only cares about how many players
(of each type) play each of the available
strategies.
e.g. symmetry in auctions, congestion games,
social phenomena, etc.
Congestion Games with Player- Specific Payoff
Functions. Milchtaich, Games and Economic
Behavior, 1996.
The women of Cairo Equilibria in Large
Anonymous Games. Blonski, Games and Economic
Behavior, 1999.
Partially-Specified Large Games. Ehud Kalai,
WINE, 2005.
In DP 07, 08, 09 we solve multiplayer anonymous
games w/ a few strategies per player, by
exploiting symmetries through CLTheorems.
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In Conclusion

• the Nash Equilibrium is broken for general games

• but not for zero-sum games vN-D-K

• ditto for networks of competitors DP 09

• ditto for anonymous games DP 07, 08, 09

• need to characterize the classes of games where
  our predictions are reliable

• complexity of approximate equilibria other
  solution concepts

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Thank you for your attention