1
Lecture Notes for Economics 200C: Games and Information
Vincent Crawford, revised March 2000; do not reproduce except for personal use
1. Introduction
MWG 217-233; Kreps 355-384; Varian 259-265; McMillan 3-41
Robert Gibbons, "An Introduction to Applicable Game Theory,"
Journal of Economic
Perspectives
(Winter 1997), 127-149 (or you can substitute his book)
Noncooperative
game theory tries to explain outcomes (including cooperation) from the
basic data of the situation, in contrast to
cooperative
game theory, which assumes
unlimited communication and cooperation and tries to characterize the limits of the set of
possible cooperative agreements. In "parlor" games players often have opposed
preferences; such games are called
zero-sum
. But noncooperative game theory spans the
entire range of
interactive decision problems
from pure conflict to pure cooperation
(coordination games); most applications have elements of both.
A
game
is defined by specifying its
structure
: the players, the "rules" (the order of players'
decisions, their feasible decisions at each point, and the information they have when
making them); how their decisions jointly determine the outcome of the game; and their
preferences over possible outcomes. Any uncertainty about the outcome is handled by
assigning
payoffs
(von Neumann-Morgenstern utilities) to the possible outcomes and
assuming that players' preferences over uncertain outcomes can be represented by
expected-payoff maximization.
Assume game is a complete model of the situation; if not, make it one, e.g. by including
decision to participate. Assume numbers of players, decisions, and periods are finite, but
can relax as needed.
Something is
mutual knowledge
if all players know it, and
common knowledge
if all know
it, all know that all know it, and so on. Assume common knowledge of structure (allows
uncertainty with commonly known distributions modeled as "moves by nature"): games of
complete information
.
Can represent a game by its
extensive form
or
game tree.
E.g. contracting by ultimatum
(MWG uses Matching Pennies, Kreps has abstract examples): Two players, R(ow) and
C(olumn); two feasible contracts, X and Y. R proposes X or Y to C, who must either
accept (a) or reject (r). If C accepts, the proposed contract is enforced; if C rejects, the
outcome is a third alternative, Z. R prefers Y to X to Z, and C prefers X to Y to Z. R's
preferences are represented by
vN-M utility
or
payoff
function u(y)=2, u(x)=1, u(z)=0; and
C's preferences by v
2
(x)=2, v
2
(y)=1, v
2
(z)=0.
Draw game trees when C can observe R's proposal before deciding whether to accept,
and when C cannot. Order of
decision nodes
has some flexibility, but must respect timing
of information flows. Players assumed to have
perfect recall
of their own past moves and
other information; tree must reflect this. Each decision node belongs to an
information set
,