More Like this

1. Smooth Nash Equilibria: Algorithms and Complexity

   https://doi.org/10.4230/LIPIcs.ITCS.2024.37  

   Daskalakis, Constantinos; Golowich, Noah; Haghtalab, Nika; Shetty, Abhishek (January 2024, Leibniz International Proceedings in Informatics (LIPIcs):15th Innovations in Theoretical Computer Science Conference (ITCS 2024))

   Guruswami, Venkatesan (Ed.)

   A fundamental shortcoming of the concept of Nash equilibrium is its computational intractability: approximating Nash equilibria in normal-form games is PPAD-hard. In this paper, inspired by the ideas of smoothed analysis, we introduce a relaxed variant of Nash equilibrium called σ-smooth Nash equilibrium, for a {smoothness parameter} σ. In a σ-smooth Nash equilibrium, players only need to achieve utility at least as high as their best deviation to a σ-smooth strategy, which is a distribution that does not put too much mass (as parametrized by σ) on any fixed action. We distinguish two variants of σ-smooth Nash equilibria: strong σ-smooth Nash equilibria, in which players are required to play σ-smooth strategies under equilibrium play, and weak σ-smooth Nash equilibria, where there is no such requirement. We show that both weak and strong σ-smooth Nash equilibria have superior computational properties to Nash equilibria: when σ as well as an approximation parameter ϵ and the number of players are all constants, there is a {constant-time} randomized algorithm to find a weak ϵ-approximate σ-smooth Nash equilibrium in normal-form games. In the same parameter regime, there is a polynomial-time deterministic algorithm to find a strong ϵ-approximate σ-smooth Nash equilibrium in a normal-form game. These results stand in contrast to the optimal algorithm for computing ϵ-approximate Nash equilibria, which cannot run in faster than quasipolynomial-time, subject to complexity-theoretic assumptions. We complement our upper bounds by showing that when either σ or ϵ is an inverse polynomial, finding a weak ϵ-approximate σ-smooth Nash equilibria becomes computationally intractable. Our results are the first to propose a variant of Nash equilibrium which is computationally tractable, allows players to act independently, and which, as we discuss, is justified by an extensive line of work on individual choice behavior in the economics literature. 

   more » « less
2. Nash Equilibrium Problems of Polynomials

   https://doi.org/10.1287/moor.2022.0334  

   Nie, Jiawang; Tang, Xindong (May 2024, Mathematics of Operations Research)

   This paper studies Nash equilibrium problems that are given by polynomial functions. We formulate efficient polynomial optimization problems for computing Nash equilibria. The Moment-sum-of-squares relaxations are used to solve them. Under generic assumptions, the method can find a Nash equilibrium, if there is one. Moreover, it can find all Nash equilibria if there are finitely many ones of them. The method can also detect nonexistence if there is no Nash equilibrium. Funding: J. Nie was supported by the National Science Foundation [Grant DMS-2110780]. 

   more » « less
3. On the existence of Nash Equilibrium in games with resource-bounded players

   Halpern, Joseph Y; Pass, Rafael; Reichman, Daniel (September 2019, Proceedings of the 12th International Symposium on A Game Theory (SAGT)})

   We consider computational games, sequences of games G = (G1,G2,...) where, for all n, Gn has the same set of players. Computational games arise in electronic money systems such as Bitcoin, in cryptographic protocols, and in the study of generative adversarial networks in machine learning. Assuming that one-way functions exist, we prove that there is 2-player zero-sum computational game G such that, for all n, the size of the action space in Gn is polynomial in n and the utility function in Gn is computable in time polynomial in n, and yet there is no ε-Nash equilibrium if players are restricted to using strategies computable by polynomial-time Turing machines, where we use a notion of Nash equilibrium that is tailored to computational games. We also show that an ε-Nash equilibrium may not exist if players are constrained to perform at most T computational steps in each of the games in the sequence. On the other hand, we show that if players can use arbitrary Turing machines to compute their strategies, then every computational game has an ε-Nash equilibrium. These results may shed light on competitive settings where the availability of more running time or faster algorithms can lead to a “computational arms race”, precluding the existence of equilibrium. They also point to inherent limitations of concepts such as “best response” and Nash equilibrium in games with resource-bounded players. 

   more » « less
4. Game Theoretic Liquidity Provisioning in Concentrated Liquidity Market Makers

   https://doi.org/10.1145/3711700  

   Tang, Weizhao; El-Azouzi, Rachid; Lee, Cheng Han; Chan, Ethan; Fanti, Giulia (March 2025, Proceedings of the ACM on Measurement and Analysis of Computing Systems)

   Automated marker makers (AMMs) are decentralized exchanges that enable the automated trading of digital assets. Liquidity providers (LPs) deposit digital tokens, which can be used by traders to execute trades, which generate fees for the investing LPs. In AMMs, trade prices are determined algorithmically, unlike classical limit order books. Concentrated liquidity market makers (CLMMs) are a major class of AMMs that offer liquidity providers flexibility to decide not onlyhow muchliquidity to provide, butin what ranges of pricesthey want the liquidity to be used. This flexibility can complicate strategic planning, since fee rewards are shared among LPs. We formulate and analyze a game theoretic model to study the incentives of LPs in CLMMs. Our main results show that while our original formulation admits multiple Nash equilibria and has complexity quadratic in the number of price ticks in the contract, it can be reduced to a game with a unique Nash equilibrium whose complexity is only linear. We further show that the Nash equilibrium of this simplified game follows a waterfilling strategy, in which low-budget LPs use up their full budget, but rich LPs do not. Finally, by fitting our game model to real-world CLMMs, we observe that in liquidity pools with risky assets, LPs adopt investment strategies far from the Nash equilibrium. Under price uncertainty, they generally invest in fewer and wider price ranges than our analysis suggests, with lower-frequency liquidity updates. In such risky pools, by updating their strategy to more closely match the Nash equilibrium of our game, LPs can improve their median daily returns by $116, which corresponds to an increase of 0.009% in median daily return on investment (ROI). At maximum, LPs can improve daily ROI by 0.855% when they reach Nash equilibrium. In contrast, in stable pools (e.g., of only stablecoins), LPs already adopt strategies that more closely resemble the Nash equilibrium of our game. 

   more » « less
5. From Nash Equilibria to Chain Recurrent Sets: An Algorithmic Solution Concept for Game Theory

   https://doi.org/10.3390/e20100782  

   Papadimitriou, Christos; Piliouras, Georgios (October 2018, Entropy)

   In 1950, Nash proposed a natural equilibrium solution concept for games hence called Nash equilibrium, and proved that all finite games have at least one. The proof is through a simple yet ingenious application of Brouwer’s (or, in another version Kakutani’s) fixed point theorem, the most sophisticated result in his era’s topology—in fact, recent algorithmic work has established that Nash equilibria are computationally equivalent to fixed points. In this paper, we propose a new class of universal non-equilibrium solution concepts arising from an important theorem in the topology of dynamical systems that was unavailable to Nash. This approach starts with both a game and a learning dynamics, defined over mixed strategies. The Nash equilibria are fixpoints of the dynamics, but the system behavior is captured by an object far more general than the Nash equilibrium that is known in dynamical systems theory as chain recurrent set. Informally, once we focus on this solution concept—this notion of “the outcome of the game”—every game behaves like a potential game with the dynamics converging to these states. In other words, unlike Nash equilibria, this solution concept is algorithmic in the sense that it has a constructive proof of existence. We characterize this solution for simple benchmark games under replicator dynamics, arguably the best known evolutionary dynamics in game theory. For (weighted) potential games, the new concept coincides with the fixpoints/equilibria of the dynamics. However, in (variants of) zero-sum games with fully mixed (i.e., interior) Nash equilibria, it covers the whole state space, as the dynamics satisfy specific information theoretic constants of motion. We discuss numerous novel computational, as well as structural, combinatorial questions raised by this chain recurrence conception of games. 

   more » « less