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bayesian games
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______ online
• an example
• definitions
• applications
ganië’,exa m p le: a u ctio n theóiy’
online
• 2 bidders each bidding for a painting
• each player has a value 0 in 0 = {2,3} equally likely
• first price auction: highest bidder wins, pays bid
gani
a symmetric equilibrium:
online
• have to bid whole integers a1 = {o,1,2,3}
• let us look for a symmetric pure strategy bayesian equilibrium..
• so, we need to specify s(3)=?, s(2)=? so that no player wants to deviate if they both bid that way
—
. . gahi’
a symmetric equilibrium: meo
‘‘ online
• have to bid whole integers a1 = {o,1,2,3}
ii o
• let us look for a symmetric pure strategy bayesian equilibrium
• so, we need to specify s(3)=?, s(2)=? so that no player wants to deviate if they both bid that way
— %ntllr?t01 ,,
a symmetric equilibrium:
____________ _____ ‘““ online
• have to bid whole integers a. = {o 1 2 3}
0
• let us look for a symmetric pure strategy bayesian equilibrium
• so, we need to specify s(3)=?, s(2)=? so that no player wants to deviate if they both bid that way
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ile — %tflrlfl•ga, ;_
garne:
a symmetric equilibrium:
____________ “ online
• have to bid whole integers a, = {o,1,2,3}
• we can try various combinations and see
• if bid is equal to value, then payoff is zero no matter whether win or not, so let us try s(3)<3, s(2)<2
— %tvwiiflig.q a
a symmetric equilibrium:
“‘ online
• have to bid whole integers a, = {o,1,2,3}
• we can try various combinations and see
• if bid is equal to value, then payoff is zero no matter whether win or not, so let us try
s(2)<2
• • • • ci
a symmetric equilibrium:
online
• try: s(3)=2, s(2)=1
• if o = 3, and other uses this strategy, then
expected utility from
—bidding3=o
— bidding 2 = (32) (1/2 + 1/4) = 3/4
— bidding 1 = (31) (1/4) = 1/2
— biddingo=o
% fl, ijjtgq
garnee.
a symmetric equilibrium: me’’
‘“ online
• try: jj=2,)=1
—
• if 0. = 3, and other uses this strategy, then
expected utility from
—bidding3=o
— bidding 2 = (32) (1/2 + 1/4) = 3/4
— bidding 1 = (31) (1/4) = 1/2
—biddingo=o
—
(i. %fl,,ipfli, .
a symmetric_equilibrium:
______________ online
•
• if)and other uses this strategy, then
expected utility from
— bidding3=o
— bidding 2 = (32) (1/2 + 1/4) = 3/4
— bidding 1 = (31) (1/4) = 1/2
— biddingo=o
• try:
a symmetric equilibrium:
— ganiè .
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onime
• if) expected
ad other uses this strategy, utility from
then
— bidding = 0
— bidding 2=
— bidding 1 =
—biddingo=0
(32) (1/2 + 1/4) (31) (1/4) = 1/2
=3/4
— sfl.ii,ni_p •_
garne
a symmetric equilibrium:
online
• try:
• if 3)argi other uses this strategy, then expected utilityfrom
— bidding =g1
— bidding 1 = (31) (1/4) = 1/2
—biddingo=o
jp=
(32) (1/2 + 1/4)
=3/4
 — ‘nt,,,fla — game 
%ii !‘
 theory

online
a symmetric equilibrium:
• try:
2
• if ard other jisthis strategy, then
expected utility from
— bidding=.e ,j
— = i1( 1/+ 1/4) = 3/4
—biddingl= (31)(1/4)1/2
—biddingo=o
• try:
— bidding
42
—biddingo=o
—
a
.
symmetric
. .
equilibrium:
.
sw?.) pert4...
‘.jarr1e • • thëo::;11
online
•if
expected utility from
then
— bidding 1 =
=ce .l) ,,/
= i( i/+ 1/4)
(31) (1/4) = 1/2
=3/4
• try:
a symmetric equilibrium:
then
1/4)
— bidding 1 =(31) (1/4) = 1/2
—biddingo=o
— a
game
—theoiy
online
• if=.arqotherlse expected utility from
2
— bidding
gj
• try:
•if
expected utility from
=ü&/l)</ £
=jj (1/t+ 1/4)
— bidding 1 =(3i) (1/4) = 1/2
— % fl ii ittg a
game z
theory “
online
a symmetric equilibrium:
and other us
— bidding
42
then
— biddingo=o
— ganië
—theory
, online
— biddingo=o
=ce !?l)
= i( 1/ 1fi’
= jjl)qj3 =71
a symmetric equilibrium:
• try:
•if
expected utility from
— bidding
2
— bidding 1
then
çflhi —
game — online
then
4fr
a symmetric equilibrium:
expected
utility from
rategy,
— bidding=ec
i?=z’
— bidding 1
!,l) .,/‘ (1/+
1/4)
= j1)i3j3 =2v7
— biddingo=
(917
%ntiirtt,_. .._
. . . garne.
a symmetric equilibriu
online
• try: s(3)=2, s(2)=1
• if 01 = 2, and other uses this strategy, then
expected utilityfrom
— bidding3= (23) (1) =1
—bidding2= 0
—biddingl= (21)(1/4)=1/4
— biddingo=0
— — %tt1$’iflêq
garne
a symmetric equilibrium: meoiy
online
• try: s(3)=2, s(2)=1
• if 0 = 2, and other uses this strategy, then expected utiiitvfoim
—bidding3=( )(i) #i
— bidding2=
—biclciingi= (21)j=1/4
— biddingo=oj—
— %fliifli, a—
• • • • ‘...jarne
a symmetric equilibrium:
online
• try: s(3)=2, s(2)=1
• if 0 = 2, and other uses this strategy, then expected utiiiyfoim
—bidding3=( )(i)
—bidding2=
 i1n g4 = (21) [j :&y: l
— biddingo=oj—
expected u
— bidding 3
— bidding2=
—bing4= __
— biddingo=oj—
—
— %fl..iirti,
game
theory z11111,
nline
a symmetric equilibrium:
try:jj,
• 1f01=2,
and other uses this strategy, then
itvfipjil=c !)(:9
_wj
— % its’
game
a symmetric equilibrium:
online
• so, s(3)=2, s(2)=1 is a best response to the same strategy by the other player
• it is a (symmetric) pure strategy bayesian equilibrium