Towards a Constructive Theory of Networked Interactions Constantinos Daskalakis CSAIL, MIT Based on joint work with Christos H. Papadimitriou

1928 Neumann: A Success Story of Game Theory (and Mathematical Programming) proof uses Brouwer’s fixed point theorem; + Danzig ’57: equivalent to LP duality; + Khachiyan’79: polynomial-time solvable; existence of min-max equilibrium in 2-player, zero-sum games ‘‘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.’’ Robert Aumann, 1987: + all no-regret learning algorithms converge to equilibria.

no efficient algorithm is known after 50+ years of research Nash: What about multi-player or non zero-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; existence of an equilibrium in multiplayer, general-sum games

“Is it NP-complete to find a Nash equilibrium?” the Pavlovian reaction

Why should we care about the complexity of equilibria? 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?” ‘‘[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.’’ Herbert Scarf writes… First, if we believe our equilibrium theory, efficient algorithms would enable us to make predictions: 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 Computation of Economic Equilibria, 1973

“Is it NP-complete to find a Nash equilibrium?” the Pavlovian reaction 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 a Nash equilibrium whose third bit is one, if any - find two Nash equilibria, if more than one exist [Gilboa, Zemel ’89; Conitzer, Sandholm ’03] two answers

- 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 Nash’s theorem constructive… NP NP- complete P

Complexity of the Nash Equilibrium Theorem [Daskalakis, Goldberg, Papadimitriou ’06]: If #players ≥ 4, then finding a Nash equilibrium is PPAD-complete.

Computational Complexity P e.g.: linear programming e.g.2: zero-sum games Solutions can be found in polynomial time Solutions can be verified in polynomial time NP NP- complete The hardest problems in NP e.g.: quadratic programming e.g.2: traveling salesman problem PPAD

The PPAD Class [Pap. ’94] PPAD = Nash’s Thm the class of all Brouwer fixed point computation problems, where the function is piece-wise linear NASH  PPAD [DGP 06]NASH ≥4 is PPAD-hard : [Chen, Deng ’06] NASH 3 is PPAD-hard : [Dask., Pap. ’06] [Chen, Deng ’06]NASH 2 is PPAD-hard : [CSVY ’06] Ditto for Arrow-Debreu Equilibria in markets with complementarities : N.B.

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. ► Special Classes of Games with tractable equilibria. ► Alternative Solution Concepts with better computational properties.

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: 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 compute mixed strategies so that no player has more than an incentive to deviate, arbitrarily small If, then still PPAD-complete.[CDT ’06]: (scale invariant) (shift invariant)

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. 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…] So answer is No!

tractable special cases…

Networks of Competitors - players are nodes of a graph G - player’s payoff is the sum of payoffs from all adjacent edges … … - edges are zero-sum games 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] - a Nash equilibrium can be found efficiently with linear-programming; - if every node uses a no-regret learning algorithm, the players’ behavior converges to a Nash equilibrium. In every network of competitors: - the Nash equilibria comprise a convex set;

[ No-regret algorithms run at node u produces: (mixed or pure) no-regret property: widely used game-playing algorithms e.g. experts algorithm, (perturbed) fictitious play, etc. payoff received by u in T periods ≥ payoff that u would have received if she played any fixed strategy x u at all time steps ]

Networks of Competitors Theorem [Daskalakis, Papadimitriou ’09] - a Nash equilibrium can be found efficiently with linear-programming; - if every node uses a no-regret learning algorithm, the players’ behavior converges to a Nash equilibrium. In every network of competitors: - the Nash equilibria comprise a convex set; N.B. but [+ Tardos ’09] the value of the nodes is not unique. strong indication that Nash eq. makes sense in this setting.

Another Tractable Case: Games with Symmetries In [DP 07, 08, 09] we solve multiplayer anonymous games w/ a few strategies per player, by exploiting symmetries through CLTheorems. 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. ‘‘The women of Cairo: Equilibria in Large Anonymous Games.’’ Blonski, Games and Economic Behavior, “Partially-Specified Large Games.” Ehud Kalai, WINE, ‘‘Congestion Games with Player- Specific Payoff Functions.’’ Milchtaich, Games and Economic Behavior, 1996.

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] need to characterize the classes of games where our predictions are reliable complexity of approximate equilibria + other solution concepts ditto for anonymous games [DP ’07, ’08, ’09]

Thank you for your attention