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Complexity class
computer science, PPAD ("Polynomial Parity Arguments on Directed graphs") is a complexity class introduced by Christos Papadimitriou in 1994. PPAD is a subclass
PPAD_(complexity)
Complexity class
that introduced PPAD and PPA. PPP contains both PPAD and PWPP (polynomial weak pigeonhole principle) as subclasses. These complexity classes are of particular
PPP_(complexity)
This is a list of PPAD-complete problems. Sperner's lemma Brouwer fixed-point theorem Kakutani fixed-point theorem Nash equilibrium Core of Balanced Games
List of PPAD-complete problems
List_of_PPAD-complete_problems
Complexity class
is reducible to that problem. PPAD is defined in a similar way to PPA, except that it is defined on directed graphs. PPAD is a subclass of PPA. This is
PPA_(complexity)
Type of cryptographic software obfuscation
Additionally, if iO and one-way functions exist, then problems in the PPAD complexity class are provably hard. However, indistinguishability obfuscation
Indistinguishability obfuscation
Indistinguishability_obfuscation
Complexity class
Goldwasser. The complexity of decision versus search. SIAM Journal on Computing, Vol. 23, No. 1, February 1994. Daskalakis, Costis (2015). "22. PPAD". MIT OpenCourseWare
FNP_(complexity)
Hollender, Alexandros; Savani, Rahul (2022-12-19). "The Complexity of Gradient Descent: CLS = PPAD ∩ PLS". Journal of the ACM. 70 (1): 7:1–7:74. arXiv:2011
FIXP
Notion in combinatorial game theory
Combinatorial game theory measures game complexity in several ways: State-space complexity (the number of legal game positions from the initial position)
Game_complexity
Algorithm analysis method
smoothed complexity polynomial in n and 1/s, where s is the input perturbation size, unless PPAD ≤ RP. In particular, the smoothed complexity of the Lemke-Howson
Smoothed_analysis
Class of theorems about Nash equilibrium payoff profiles in repeated games
equilibria for one-shot finite games, a problem which lies in the PPAD complexity class. The practical consequence of this is that no efficient (polynomial-time)
Folk_theorem_(game_theory)
Complexity class
W.; Hollender, Alexandros; Savani, Rahul (2023). "The Complexity of Gradient Descent: CLS = PPAD ∩ PLS". Journal of the ACM. 70: 1–74. arXiv:2011.01929
TFNP
Complexity class
that a complexity class called CLS (Continuous Local Search) is equal to the intersection of PPAD and PLS. Equilibria, fixed points, and complexity classes:
PLS_(complexity)
of complexity classes in computational complexity theory. For other computational and complexity subjects, see list of computability and complexity topics
List_of_complexity_classes
Computing the fixed point of a function
Sperner's lemma), and therefore it is PPAD-complete. This implies that computing an approximate fixed-point is PPAD-complete even for very simple functions
Fixed-point_computation
Study of algorithms in strategic environments
algorithms for finding equilibria. Of special importance is the complexity class PPAD, which includes many problems in algorithmic game theory. Mechanism
Algorithmic_game_theory
Economical computational problem
smoothed complexity polynomial in n and 1/s, where s is the input perturbation size, unless PPAD ≤ RP. In particular, the smoothed complexity of the Lemke-Howson
Nash_equilibrium_computation
Game in algorithmic game theory
in such a sparse game is PPAD-hard, and that there does not exist a fully polynomial-time approximation scheme unless PPAD is in P. In symmetric games
Succinct_game
utilities, computing a CE is PPAD-hard. Their proof shows also that this market-equilibrium problem does not have an FPTAS unless PPAD is contained in P. When
Arrow–Debreu_exchange_market
Every graph has evenly many odd vertices
He defined the complexity class PPA to encapsulate problems such as this one; a closely related class defined on directed graphs, PPAD, has attracted
Handshaking_lemma
polynomial (the problem is PPAD-hard even with goods), it runs fast on random instances. It also proves that the problem is in PPAD, the solutions are rational-valued
Fisher_market
Logical paradox in decision-making theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Paradox_of_tolerance
Hypothesis in computational complexity theory
hard or even complete for some complexity class C {\displaystyle C} , in particular NP-hard (but often also PSPACE-hard, PPAD-hard, etc.). This means that
Computational hardness assumption
Computational_hardness_assumption
Theorem on triangulation graph colorings
was first studied by Christos Papadimitriou. He introduced a complexity class called PPAD, which contains this as well as related problems (such as finding
Sperner's_lemma
Finding an optimal algorithm for playing chess
solved at least weakly. Calculated estimates of game-tree complexity and state-space complexity of chess exist which provide a bird's eye view of the computational
Solving_chess
Paper-and-pencil game for two players
positions (the state space complexity) or the 26,830 possible games up to rotations and reflections (the game tree complexity) on this space. If played
Tic-tac-toe
Type of fair division
(does not depend on n). Then, finding an ε-approximate consensus-halving is PPAD-hard, which is theoretically weaker than PPA-hard. The proof is by reduction
Consensus_splitting
Economical computational problem
unless PPAD is in P. Chen and Teng proved PPAD-hardness for a Fisher market with SPLC utilities. Chaudhury, Garg, McGlaughlin and Mehta proved PPAD-hardness
Market equilibrium computation
Market_equilibrium_computation
Situation where total gains match total losses
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Zero-sum_game
Game whose outcome can be correctly predicted
Chess Fully solving chess remains elusive, and it is speculated that the complexity of the game may preclude it ever being solved. Through retrograde computer
Solved_game
Standard example in game theory
tournament. The programs that were entered varied widely in algorithmic complexity, initial hostility, capacity for forgiveness, and so forth. Axelrod discovered
Prisoner's_dilemma
problem does not have a fully polynomial-time approximation scheme, unless PPAD ⊆ P. On the other hand, there are algorithms for finding an approximate equilibrium
Leontief_utilities
English saying meaning "equivalent retaliation"
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Tit_for_tat
bundle given the prices. The A-CEEI allocation is hard to compute: it is PPAD complete. However, in realistic-size problems, A-CEEI can be computed using
Approximate Competitive Equilibrium from Equal Incomes
Approximate_Competitive_Equilibrium_from_Equal_Incomes
Process for fairly dividing resources
explicitly by polynomial-time algorithms, the envy-free cake-cutting problem is PPAD-complete. Value-based approximation uses the following definitions: A partition
Envy-free_cake-cutting
Search algorithm
; Wigderson, A. (1986). "Probabilistic Boolean Decision Trees and the Complexity of Evaluating Game Trees". 27th Annual Symposium on Foundations of Computer
Alpha–beta_pruning
Decision rule used for minimizing the possible loss for a worst-case scenario
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Minimax
Generalization of the normal-form game
in a concave game is PPAD-complete. In fact, they prove that the problem is in PPAD even for general concave games, and it is PPAD-hard even in the special
Concave_game
identifying concise representations of hedonic games and on the computational complexity of finding stable outcomes. Formally, a hedonic game is a pair ( N , (
Hedonic_game
Concept in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Shapley_value
Concept in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Focal_point_(game_theory)
Political model of international conflict resolution
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Two-level_game_theory
Concept in conflict studies
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Conflict_escalation
Game theory concept
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Game_form
Problem in process of sharing surplus
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Cooperative_bargaining
Combinatorial game theory theorem
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Sprague–Grundy_theorem
Military strategy during the Cold War with regard to the use of nuclear weapons
Lindsay, Jon R. (2019). Cross-Domain Deterrence: Strategy in an Era of Complexity. Oxford University Press. doi:10.1093/oso/9780190908645.001.0001. ISBN 978-0-19-090960-4
Deterrence_theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Implementation_theory
Branch of game theory about two-player sequential games with perfect information
greater weight on theoretical results, including the analysis of game complexity and the existence of optimal strategies through methods like the strategy-stealing
Combinatorial_game_theory
Theorem in topology
Daskalakis and Demaine prove that a computational version of Jordan's theorem is PPAD-complete. As a corollary, they show that Jordan's theorem implies the Brouwer
Jordan_curve_theorem
Concept in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Incentive_compatibility
Making of satisfactory, not optimal, decisions
than a strict rigid rule of optimization. They do this because of the complexity of the situation. An example of behaviour inhibited by heuristics can
Bounded_rationality
Israeli-American psychologist and economist (1934–2024)
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Daniel_Kahneman
Overuse of a shared resource
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Tragedy_of_the_commons
Class of games where players choose their actions sequentially
chess, backgammon, tic-tac-toe, and Go, with decision trees varying in complexity—from the compact tree of tic-tac-toe to the vast, unmappable tree of chess
Sequential_game
Model of conflict for two players in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Chicken_(game)
Israeli psychologist (1937–1996)
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Amos_Tversky
Point in the convex hull of a set P in Rd, is the convex combination of d+1 points in P
problem of finding the colorful set lies in the intersection of the complexity classes PPAD and PLS. Shapley–Folkman lemma Helly's theorem Kirchberger's theorem
Carathéodory's theorem (convex hull)
Carathéodory's_theorem_(convex_hull)
Solution concept in Game Theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Cursed_equilibrium
Preference of known risks to unknown risks
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Ambiguity_aversion
Concept in game theory involving long-term strategic planning
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Farsightedness_(game_theory)
Condition in economics and game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Perfect_information
Game theory scenario
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Win–win_game
Level of information in economics and game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Complete_information
Italian economist (born 1961)
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Pierpaolo_Battigalli
Game theory case weighing own/others' sacrifice
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Volunteer's_dilemma
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Strategic_move
Tendency to overestimate in auctions
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Winner's_curse
Chemical compound of DNA and RNA
of biological uses that include a range of functions ranging in both complexity and versatility. These include camouflage, display, and vision among other
Guanine
Iterated game for peace and conflict studies
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Peace_war_game
Solution concept in cooperative game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Myerson_value
Decrease in severity of conflicts
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
De-escalation
Modelling evolution using differential equations
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Evolutionary_dynamics
Economic model of competition
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Bertrand_competition
Dynamical system
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Replicator_equation
Concept in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Non-credible_threat
Search heuristic for combinatorial games
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Aspiration_window
Solution to the fair division problem
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Moving-knife_procedure
Variation of minimax game tree search
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Negamax
Paradox of combining strategies
irrelevant for utility theory, stock buying, and the emergence of life". Complexity. 9 (1): 23–27. doi:10.1002/cplx.10112. Winning While Losing: New Strategy
Parrondo's_paradox
Conflict between safety and cooperation
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Stag_hunt
Solution concept in Game Theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Bayes_correlated_equilibrium
Term in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Move_by_nature
Hungarian-American economist and philosopher (1920–2000)
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
John_Harsanyi
Situation where all parties are worse off
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
No-win_situation
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Transferable_utility
the envy-free cake-cutting problem has the same complexity as finding a Brouwer fixed-point, i.e. PPAD-complete. In this problem, n housemates have decided
Simmons–Su_protocols
Algorithmically defined graph
way may not necessarily be NP-complete, as it is unknown whether PPA = NP. PPAD is an analogous class defined on implicit directed graphs that has attracted
Implicit_graph
Solution concept of a non-cooperative game
because, even though the equilibrium must exist, it is unknown due to the complexity of the game, for instance in Chinese chess. Or, if known, it may not be
Nash_equilibrium
Complete plan on how a game player will behave in every possible game situation
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Strategy_(game_theory)
Analog of Pareto efficiency for situations with incomplete information
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Bayesian_efficiency
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Coalition-proof Nash equilibrium
Coalition-proof_Nash_equilibrium
Game that repeats a base game
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Repeated_game
Theorem in game theory
satisfy certain restrictions on their variation. Scott Aaronson studied the complexity and rate of convergence of various types of dialogues with more than two
Aumann's_agreement_theorem
Algorithm in game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Paranoid_algorithm
Incomplete-information coordination game
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Electronic_mail_game
Economic Model
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Bertrand–Edgeworth_model
Hand game for two players or more
gestures has the effect of reducing the odds of a tie, while increasing the complexity of the game. The probability of a tie in an odd-number-of-weapons game
Rock_paper_scissors
Class of strategies employed in a repeated non-cooperative game
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Trigger_strategy
Weakly optimal allocation of resources
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Pareto_efficiency
Concept in economics and game theory
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete Mathematics portal Game theory WikiProject
Price_of_anarchy
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