Game Theory

In General [> s.a. mathematics; statistics.]
* Classical theory: The two most important theorems are the Minimax theorem and the Nash Equilibrium theorem.
* Prisoner's dilemma: A pair of captured criminals ponder strategy; If neither criminal confesses, both go free; If one confesses, the other receives a stiff sentence; If both confess, they each receive moderate sentences.

Games, Puzzles > s.a. geometry; number theory; Parrondo's Paradox; technology.
* Life: Cellular automaton invented by J Conway (inspired by von Neumann's project of a "universal constructor''), with 3 rules and 2 states per cell, birth (a dead cell becomes alive if 3 neighbors are alive), isolation (a cell dies if fewer than 3 neighbors are alive), and crowding (a cell dies if it has 4 or more live neighbors).
@ References: Berlekamp et al 82; Zassenhaus PhyA(82); Wells 88; Bolt 90; Gardner 90; Berlekamp & Rodgers ed-99.

Decision Theory
* Idea: The setup includes a set M of chance events, and for each m in M, a set S_m of possible outcomes, with the events set usually being E_m = 2^{S_m}, and a set R of rewards; Then a bet is a map P: S_m → R, and rational agents have to decide between different possible bets by establishiing an order (M, P) > (M', P') establishing which are the better bets; With a sufficient set of axioms, all of this is usually cast in terms of probability rules and "cash" values.
* Applications: In quantum mechanics, > see many-worlds.

References
@ General: Morgenstern & Von Neumann; Vajda 92; Wu qp/04, qp/04 [new mathematical representation], qp/05 [Hamiltonian formulation]; Hauert & Szabó AJP(05)may [and physics methods]; Hill AS(09)mar [the mathematics of optimal stopping].
@ Game of life: Fehsenfeld et al JPA(98) [scaling behavior]; Flitney & Abbott in(04)qp/02 [semi-quantum].
@ Murphy's law: SA(94)dec, p104 [toast].
> Online resources: Internet Encyclopedia of Science pages.

Quantum Games
@ Reviews: Lee & Johnson pw(02)oct; Piotrowski & Sladkowski IJTP(03)qp/02-in, qp/03-in; Flitney & Abbott FNL(02)qp; Iqbal PhD(04)qp/05; Grabbe qp/05 [intro for economists]; Szabó & Fáth PRP(07) [evolutionary, on graphs].
@ General references: Meyer PRL(99)qp/98 [strategy], qp/00-in; Eisert et al PRL(99) [strategy]; Eisert & Wilkens JMO(00)qp; Piotrowski & Sladkowski PhyA(02)qp/01 [application to market]; D'Ariano et al QIC(01)qp [quantum Monty Hall problem]; Lee & Johnson PRA(03)qp/02 [efficiency], qp/02 [non-cooperative]; van Enk & Pike PRA(02)qp [classical rules]; Sladkowski PhyA(03)cm/02; Miakisz et al qp/04 [future]; Gutoski & Watrous qp/06-in [general theory]; Bleiler a0808 [formalism].
@ And physics: Moraal JPA(00) [based on spin models]; Guevara a0803 [and quantum mechanics]; Kowalski & Plastino PhyA(08) [and matter-field interaction].
@ Specific games: Chen et al PLA(03), Wu qp/04 [prisoner's dilemma].

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send feedback and suggestions to bombelli at olemiss.edu – modified 17 jul 2009