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__: news PhysOrg(14)may [rock-paper-scissors, strategy].

**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 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 23
mar 2016