— game
bayesian games
online
• an example
• definitions
• applications

— gani
bayesian games
online
• players n={1,...,n}
• action sets: a a — a1 x a2 x ... a
• typesets: o—o1xo2x...o
• payoff/utility functions: u1: a x 0 - r

— % a..) lflij •
game
bayesian games med
online
• players n={1,...,n}
• action sets: a a — a1 x a2 x ... a
• type sets: 0 0 — o x °2 x ... o
• payoff/utility functions: u1 : r.

• ganfé
our conflict exam pie
—“‘ online
• players n={1,2}
• action sets: a1 = {fight, not}
• type sets: 01 ={strong, weak) 02 ={02)
• payoff/utility functions: u: a x 0 - r as given by the matrices

— —
a— ‘jarne
our conflict example meoiy1
online
• players n={1,2}
• action sets: a1 = fight, not}
• type sets: 01 ={strong, weak} ={02}
• payoff/utility functions: u1: a x 0 - r

——
— %fliifli,4 ,_
game
uncertainty the&y
‘“-‘ online
• for now, take 0 to be finite
• beliefs: p1(o1 i 0) conditional probability that i gives to others’ types being 0 given 0
s

— sflwiiflt,_ •
game
uncertainty
online
• fornow,takeotobefinite
• beliefs: p1(o i o) conditional probabilitythat i gives to others’ types being o.. given 0
• in our conflict example:
p2(01= strong i 02) = p p2(01= weak i 02) = 1-p

— %flrilflt,g ,._
game
strategies theóry
online
• a plan of which action to take for every player and each type for that player
• s.:o->a (pure or s1:01->a(a1) mixed)
• s(9) is the action type e, of player i takes

—
a— game
strategies
online
• a plan of which action to take for every player and each type for that player
• se-a (pure or : o-j mixed)
• (e,) is the action type o of player i takes

— %iflhifli,1 a—
game
strategies -‘theory
nhine
• a plan of which action to take for every player and each type for that player
• s:o1—a
• in the conflict example, whether to fight or not as a function of type

— %flwilfli,.140_
arne
expected utility
online
• for each i and 0 given the strategies s, the expected payoff is:
p (o... i 0) u• (s (0), s_i (0); 0, 0.)

expected utility
• for each i and 8. given the strategies expected payoff is:
— %flwi “ta.ç a— game “theory sjr
online
s, the
cjl
p(u. i 0,) u(s,(0),
i -
s (0); 0, 0)

— %flviitti,, •
game
—the&y
online
expected utility
• for eachlan given the strategiess the expected payoff is:
pj(o-j
i
0,)
u,
(s (0),
s1 (0. ‘
‘i,
0
j,
0
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________ —
ganié’
bayesian (nash) equilibrium theo’
____ — online
strategies, s= ..., sn), such that for each i and s1(81) maximizes player i’s expected utility, given
the strategies of other players.
• for each land 0 given the strategiess:
0-i r1 (0..., i o) u (s1 (ei), s... (o.); 0 , 0)
e- pi (8-i i 9) u1 (a1, s (0); 0, 0) for all a1 in a1

1k %fl&lrti,.g
uarne
bayesian (nash) equilibrium
n1ine
• strategies, such that for each i and
0: s1(91) maximizes player i’s expected utility, given the strategies of other players.
• for each i and 0. given the strategiescd 0-i i (0... i 0) u (id s. (9...); 0, o.,)
p1 (9 i 9) u1 (a’, s (9); o, 9) for all a1 in a1

— sn i ire.,...
garne.
existence of equilibrium
online
• finite number of pure strategies, finite set of types:
• equilibrium exists (corollaryto nash’s theorem—john harsanyi (1967-68))

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Professor Matthew O. Jackson & Yoav Shoham are offering a free online course on Game Theory starting in March 23, 2012. https://class.coursera.org/gametheory/

Offered by Coursera: https://www.coursera.org/