Download Towards a Constructive Theory of Networked Interactions

Towards a Constructive Theory of
Networked Interactions
Constantinos Daskalakis
CSAIL, MIT
[email protected]
Based on joint work with Christos H. Papadimitriou
A Success Story of Game Theory
(and Mathematical Programming)
1928 Neumann: existence of min-max equilibrium in 2-player,
zero-sum games
proof uses Brouwer’s fixed point theorem;
+ Danzig ’57: equivalent to LP duality;
+ Khachiyan’79: 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.’’
What about multi-player or non zero-sum
Games?
1950 Nash: existence of an equilibrium in multiplayer,
general-sum games
Proof also uses Brouwer’s 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.
the Pavlovian reaction
“Is it NP-complete to find a Nash equilibrium?”
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 can’t 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 don’t exist?
the Pavlovian reaction
“Is it NP-complete to find a Nash equilibrium?”
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]
so, how hard is it to find a single
equilibrium?
- the theory of NP-completeness does not seem
appropriate;
NPcomplete
- in fact, NASH seems to lie below NP;
NP
- making Nash’s theorem constructive…
P
Complexity of the Nash Equilibrium
Theorem [Daskalakis, Goldberg, Papadimitriou ’06]:
If #players ≥ 4,
then finding a Nash equilibrium is PPAD-complete.
Computational Complexity
The hardest problems in NP
e.g.: quadratic programming
e.g.2: traveling salesman problem
NPcomplete
NP
PPAD
P
Solutions can be verified
in polynomial time
Solutions can be found
in polynomial time
e.g.: linear programming
e.g.2: zero-sum games
The PPAD Class [Pap. ’94]
PPAD = the class of all Brouwer fixed point computation
problems, where the function is piece-wise linear
NASH  PPAD
Nash’s Thm
[DGP 06]
:
NASH≥4 is PPAD-hard
[Chen, Deng ’06]
[Dask., Pap. ’06]
:
NASH3 is PPAD-hard
[Chen, Deng ’06]
:
NASH2 is PPAD-hard
N.B.
[CSVY ’06]
:
Ditto for Arrow-Debreu Equilibria in
markets with complementarities
In other words…
► Outside
of 2-player zero-sum games, the Nash
equilibrium is computationally broken.
► Recall Aumann’s
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.’’
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.
Approximations…
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. ‚‚
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?‚‚
Bottom line
► Arbitrarily close
approximation is the only
interesting question here…
Approximate Equilibria
Goal: compute mixed strategies so that no player has more than
an incentive to deviate, arbitrarily small
Approximation: Relative vs additive incentive
no player can improve payoff
by more than a factor of
by changing strategy
no player can improve payoff
by more than an additive
by changing strategy
(shift invariant)
(scale invariant)
[CDT ’06]: If
, then still PPAD-complete.
Larger epsilons?
Important Open Problem:
Is there an algorithm running in time
?
[Daskalakis ’09]:
Relative ε-NASH is PPAD-complete, even for constant ε’s.
So answer is No!
What about the additive ε-NASH, for constant ε’s?
An important open problem, at the boundary of intractability.
[N.B. a PPAD-completeness result is unlikely for additive ε’s…]
tractable special cases…
Networks of Competitors
- players are nodes
of a graph G
…
- edges are zero-sum
games
- player’s 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]
Networks of Competitors
The simplest case:
Networks of Competitors
The second simplest case:
LP duals
It was crucial that
such edge didn’t exist
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.
[ No-regret algorithms
• widely used game-playing algorithms
e.g. experts algorithm, (perturbed) fictitious play, etc.
• run at node u produces:
(mixed or pure)
no-regret property:
payoff received by u
in T periods
≥
payoff that u would have
received if she played any fixed
strategy xu at all time steps
]
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.
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.
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
Thank you for your attention