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Towards a Constructive Theory of Networked Interactions

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## Towards a Constructive Theory of Networked Interactions

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### Towards a Constructive Theory of Networked Interactions

Constantinos Daskalakis

CSAIL, MIT

costis@csail.mit.edu

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?

existence of an equilibrium in multiplayer, general-sum games

1950 Nash:

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

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;

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?

“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;NP-complete

- 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

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

The PPAD Class [Pap. ’94]

the class of all Brouwer fixed point computation problems, where the function is piece-wise linear

PPAD =

Nash’s Thm

NASH PPAD

:

[DGP 06]

NASH≥4is PPAD-hard

[Chen, Deng ’06]

:

NASH3is PPAD-hard

[Dask., Pap. ’06]

:

[Chen, Deng ’06]

NASH2is PPAD-hard

Ditto for Arrow-Debreu Equilibria in markets with complementarities

:

N.B.

[CSVY ’06]

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?

- Complexity of Approximate Nash Equilibria;

maybe players only find an approximate Nash Eq.

- Alternative Solution Concepts with better computational properties.

- Special Classes of Games with tractable equilibria.

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…

compute mixed strategies so that no player has more than an incentive to deviate, arbitrarily small

Goal:

Approximation: Relative vs additive incentive

no player can improve payoff by more than an additive by changing strategy

(shift invariant)

no player can improve payoff by more than a factor of by changing strategy

(scale invariant)

If , then still PPAD-complete.

[CDT ’06]:

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…]

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

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 that u would have received if she played any fixed strategy xuat all time steps

payoff received by u in T periods

≥

]

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

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