Game Theory |

**In General** \ s.a. logic;
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).

@ __General references__: Berlekamp et al 82,
04;
Wells 88;
Bolt 90;
Gardner 90;
Berlekamp & Rodgers ed-99;
Bewersdorff 04.

@ __Rubik's cube__: Zassenhaus PhyA(82) [as illustration of group-theory concepts];
news BBC(10)aug [20 or fewer moves].

@ __Other games and puzzles__:
news PhysOrg(14)may [rock-paper-scissors, strategy];
Peterson 15 [prisoner's dilemma].

**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 interpretation.

**References**

@ __General__: von Neumann & Morgenstern 44;
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 & Sładkowski 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 & Sładkowski 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];
Sładkowski PhyA(03)cm/02;
Miakisz et al qp/04 [future];
Gutoski & Watrous proc(07)qp/06-in [general theory];
Nawaz PhD(07)-a1012 [quantization scheme, and information];
Bleiler a0808 [formalism];
Zhang a1012-in [Nash equilibria and correlated equilibria];
Phoenix & Khan a1202 [playable games].

@ __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,
Nawaz ChPL(13)-a1307 [prisoner's dilemma];
Ranchin a1603 [quantum Go game].

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send feedback and suggestions to bombelli at olemiss.edu – modified 1 aug 2018