Paranoid algorithm

In combinatorial game theory, the paranoid algorithm is a game tree search algorithm designed to analyze multi-player games using a two-player adversarial framework.^[1] The algorithm assumes all opponents form a coalition to minimize the focal player’s payoff, transforming an n-player non-zero-sum game into a zero-sum game between the focal player and the coalition.

The paranoid algorithm significantly improves upon the maxⁿ algorithm by enabling the use of alpha-beta pruning and other minimax-based optimization techniques that are less effective in standard multi-player game analysis.^[2] By treating opponents as a unified adversary whose payoff is the opposite of the focal player’s payoff, the algorithm can apply branch and bound techniques and achieve substantial performance improvements over traditional multi-player algorithms.^[3]

While the paranoid assumption may not accurately reflect the true strategic interactions in all multi-player scenarios—where players typically optimize their own payoffs—the algorithm has proven effective in practice for artificial intelligence applications in board games and other combinatorial multi-player games.^[3] The algorithm is particularly valuable in computer game AI where computational efficiency is crucial and the simplified opponent model provides adequate performance for real-time applications.

References

^ Sturtevant, Nathan; Korf, Richard (30 July 2000). "On Pruning Techniques for Multi-Player Games" (PDF). AAAI-00 Proceedings: 201–207.
^ Sturtevant and Korf, 2000
^ ^a ^b Sturtevant, Nathan (2003). "A Comparison of Algorithms for Multi-player Games". Lecture Notes in Computer Science. Vol. 2883. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 108–122. doi:10.1007/978-3-540-40031-8_8. ISBN 978-3-540-20545-6.

Game theory

Traditional game theory

Definitions	Asynchrony Bayesian regret Best response Bounded rationality Cheap talk Coalition Complete contract Complete information Complete mixing Conjectural variation Contingent cooperator Coopetition Cooperative game theory Dynamic inconsistency Escalation of commitment Farsightedness Game semantics Hierarchy of beliefs Imperfect information Incomplete information Information set Move by nature Mutual knowledge Non-cooperative game theory Non-credible threat Outcome Perfect information Perfect recall Ply Preference Rationality Sequential game Simultaneous action selection Spite Strategic complements Strategic dominance Strategic form Strategic interaction Strategic move Strategy Subgame Succinct game Topological game Tragedy of the commons Uncorrelated asymmetry Win–win game Zero-sum game
Equilibrium concepts	Backward induction Bayes correlated equilibrium Bayesian efficiency Bayesian game Bayesian Nash equilibrium Berge equilibrium Bertrand–Edgeworth model Coalition-proof Nash equilibrium Core Correlated equilibrium Cursed equilibrium Edgeworth price cycle Epsilon-equilibrium Gibbs equilibrium Incomplete contracts Inequity aversion Individual rationality Iterated elimination of dominated strategies Markov perfect equilibrium Mertens-stable equilibrium Nash equilibrium Open-loop model Pareto efficiency Payoff dominance Perfect Bayesian equilibrium Price of anarchy Program equilibrium Proper equilibrium Quantal response equilibrium Quasi-perfect equilibrium Rational agent Rationalizable strategy Satisfaction equilibrium Self-confirming equilibrium Sequential equilibrium Shapley value Strong Nash equilibrium Subgame perfect equilibrium Trembling hand equilibrium
Strategies	Appeasement Bid shading Cheap talk Collusion Commitment device De-escalation Deterrence Escalation Fictitious play Focal point Grim trigger Hobbesian trap Markov strategy Max-dominated strategy Mixed strategy Pure strategy Tit for tat Win–stay, lose–switch
Games	All-pay auction Battle of the sexes Nash bargaining game Bertrand competition Blotto game Centipede game Coordination game Cournot competition Deadlock Dictator game Trust game Diner's dilemma Dollar auction El Farol Bar problem Electronic mail game Gift-exchange game Guess 2/3 of the average Keynesian beauty contest Kuhn poker Lewis signaling game Matching pennies Obligationes Optional prisoner's dilemma Pirate game Prisoner's dilemma Public goods game Rendezvous problem Rock paper scissors Stackelberg competition Stag hunt Traveler's dilemma Ultimatum game Volunteer's dilemma War of attrition
Theorems	Arrow's impossibility theorem Aumann's agreement theorem Brouwer fixed-point theorem Competitive altruism Folk theorem Gibbard–Satterthwaite theorem Gibbs lemma Glicksberg's theorem Kakutani fixed-point theorem Kuhn's theorem One-shot deviation principle Prim–Read theory Rational ignorance Rational irrationality Sperner's lemma Zermelo's theorem
Subfields	Algorithmic game theory Behavioral game theory Behavioral strategy Compositional game theory Confrontation analysis Contract theory Drama theory Graphical game theory Heresthetic Mean-field game theory Negotiation theory Quantum game theory Social software
Key people	Albert W. Tucker Alvin E. Roth Amos Tversky Antoine Augustin Cournot Ariel Rubinstein David Gale David K. Levine David M. Kreps Donald B. Gillies Drew Fudenberg Eric Maskin Harold W. Kuhn Herbert Simon Herbert Scarf Hervé Moulin Jean Tirole Jean-François Mertens Jennifer Tour Chayes Ken Binmore Kenneth Arrow Leonid Hurwicz Lloyd Shapley Martin Shubik Melvin Dresher Merrill M. Flood Olga Bondareva Oskar Morgenstern Paul Milgrom Peyton Young Reinhard Selten Robert Aumann Robert Axelrod Robert B. Wilson Roger Myerson Samuel Bowles Suzanne Scotchmer Thomas Schelling William Vickrey

Combinatorial game theory

Core concepts	Combinatorial explosion Determinacy Disjunctive sum First-player and second-player win Game complexity Game tree Impartial game Misère Partisan game Solved game Sprague–Grundy theorem Strategy-stealing argument Zugzwang
Games	Chess Chomp Clobber Cram Domineering Hackenbush Nim Notakto Subtract a square Sylver coinage Toads and Frogs
Mathematical tools	Mex Nimber On Numbers and Games Star Surreal number Winning Ways for Your Mathematical Plays
Search algorithms	Alpha–beta pruning Expectiminimax Minimax Monte Carlo tree search Negamax Paranoid algorithm Principal variation search
Key people	Claude Shannon John Conway John von Neumann

Evolutionary game theory

Core concepts	Bishop–Cannings theorem Evolution and the Theory of Games Evolutionarily stable set Evolutionarily stable state Evolutionarily stable strategy Replicator equation Risk dominance Stochastically stable equilibrium Weak evolutionarily stable strategy
Games	Chicken Stag hunt
Applications	Cultural group selection Fisher's principle Mobbing Terminal investment hypothesis
Key people	John Maynard Smith George R. Price William Donald Hamilton Robert Axelrod

Mechanism design

Core concepts	Algorithmic mechanism design Bayesian-optimal mechanism Incentive compatibility Market design Myerson ironing Monotonicity Participation constraint Revelation principle Strategyproofness Vickrey–Clarke–Groves mechanism Virtual valuation
Theorems	Myerson–Satterthwaite theorem Revenue equivalence Border's theorem
Applications	Digital goods auction Knapsack auction Truthful cake-cutting

Other topics
Bertrand paradox Chainstore paradox Computational complexity of games Helly metric Multi-agent system PPAD-complete

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Paranoid algorithm

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