Title: Tea a
1Te???a ?a??????
?a???? ??ppa? ?µ?µa ??????s?? ?p??e???se?? ?a?ep?s
t?µ?? ?at??? pavlos.bma.upatras.gr
2?p?s??p?s?
- Ge???? ?e????af? t?? Te???a? ?a?????? ?a? t??
?fa?µ???? t??. - ?a????a µe ??ad?????? ????se??.
- ?a????a µe ?a?t?????e? ????se??.
- ???t?? St?at??????.
- ??d??? T?µata Te???a? ?a??????.
3?a?ade??µata
- ??a??t?ta ?a e??a? ?a?e?? ap??ß?ept??.
- ???a????? - S??e??as?a ?a? ???e???
eµp?st?s????. - S?????s? se ????? st?at????? (focal point).
- ???a??sµ?? a?ta??a??? p????f???a? (signals
screening). - ??ta-S??????st???...
4?st????? ??ad??µ?
? Te???a ?a?????? e??a? ? ep?st?µ? p?? µe?et? t??
?????????? s?µpe??f??? s ??a a?ta????st???
pe??ß?????.
- John von Neumann (1928, 1937).
- von Neumann Morgenstern (1944), Theory of
Games and - Economic Behaviour.
- John Nash (de?aet?a 50).
5?fa?µ????
- ??????µ???.
- ??????s? ?p??e???se??.
- ??e???t??? ???????a.
- ?????????? ?p?st?µe?.
- ????t???? ?p?st?µe?.
- ?????f????? (Internet, Intelligent Agents).
6?at?????e? ?a?????? (1)??ad?????? ? ?a?t?????e?
????se??
- S???? ??ad?????? ????se??
- ???sf???? ?a?t?????e? ????se??
- S??d?asµ?? ta?t??????? ?a? d?ad?????? ????se??
(p??p???t??) - ??e????t?µa p??t?? ?a? de?te??? ????s??.
7?at?????e? ?a?????? (2)????? ??t?pa???es? ?
?e???? S??e??as?a
- ?a????a µ?de????? a????sµat?? (zero sum games)
- - ?? ?fe??? t?? e??? pa??t? ap?te?e? ap??e?a
??a t?? ?????. - ?a????a µ?-µ?de????? a????sµat??.
- ?p?????? pe??????a s??e??as?a? µeta?? t??
a?ta????st??
8?at?????e? ?a?????? (3)?e???? ? ?pa?a?aµßa??µe?a
?a????a
- Sta te???? pa????a ? µ??ad???? st???? e??a? ?
µe??st?p???s? t?? - ?fe????.
- Sta epa?aµßa??µe?a pa????a ? s?µpe??f??? ???e
pa??t? - ?a?????eta? ?a? pa?aµ?t???? ?p?? ? a???p?st?a,
? e?t?µ?t?ta, ? - ap?fas?st???t?ta, ??p.
9?at?????e? ?a?????? (4)?????f???s? t?? ?a??t??
- ???? ?? pa??te? ????? p??sßas? st?? ?d?a
p????f???a (s????). - ?? pa??te? ????? e???p? p????f???a (pocker).
??apt?ss??ta? st?at?????? - ??a ep??e?t??? d??d?s? p????f???a? (signaling)
? ??a e?µa?e?s? - p????f????? (screening).
10?a????a µe ??ad?????? ????se??(??? ?a??te?)
????te?a??? ??????? st?? California - Smith
(?s????? ?p???f???) vs Taylor ?p?????? Smith
Ads / No Ads ?p?????? Taylor In / Out
?f???
Smith Taylor Ads In 1 1 Ads Out
3 3 No Ads In 2 4 No Ads Out 4
2
11?s????p?a ??ad?p??s??
Smith Taylor
In
1 1
Taylor
Ads
Out
3 3
Smith
In
2 4
No Ads
Taylor
4 2
Out
12???a?? st?? Se???
Taylor Smith
Ads
1 1
Smith
In
No Ads
4 2
Taylor
Ads
3 3
Out
Smith
2 4
No Ads
13????ap??? ????se??
0, 1
2, 3
4, 5
?
?
?
?
?
?
1, 0
?
?
?
S
S
S
?
?
?
?
?
?
2, 2
5, 4
3, 2
14?e??ss?te??? ?a??te?
?µp????? ???t?a - ??f?s?? (2 ??se??) -
?e??a??? (3 ??se??)
?ta??e?e? - ? (µe????) - ? (µesa?a)
- G (µ????)
???t?µ?se?? 5 ??f?s?? µa?? µe ????? 4
?e??a??? µa?? µe ????? 3 ????? st?? ??f?s??
2 ????? st?? ?e??a?? 1 ????? st??
?e??a?? µet? ap? ap?t???a st?? ??f?s??
15??s? ?a??????
1, 5, 5
?
?
?
?
5, 5, 2
?
?
5, 2, 5
?
?
?
?
3, 4, 4
G
?
2, 5, 5
?
?
?
?
4, 3, 4
?
?
4, 4, 3
?
?
?
4, 4, 4
16S????
?a????d? µ?de????? a????sµat?? µe d?ad??????
????se??. ?p?????? pe??ss?te??? ap? 10120 d??at??
t??p?? e??????? t?? pa????d???. ? ta??te???
?p?????st?? ??e???eta? 3?10103 ?????a ??a ?a
a?a??se? ???????? t? pa????d?.
?st????? - ?e?. 1992 ? Fritz2 ????se t??
Kasparov st? ??????? s?a??. - Feß. 1996 ?
Deep Blue (IBM) vs Kasparov 1-5. - ????
1997 O Deeper Blue (IBM) ?e?d??e? t?? Kasparv se
?a?????? p??t????µa.
17?e???µata µe ?a????d?a
?e??aµa-1 ??? pa??te? p??pe? ?a µ???ast???
100 Euro. ? pa??t?? ? p??te??e? µ?a ?ata??µ?, t?
pa??t?? ? µp??e? ?a t?? e?????e? ? ?a t??
ap??????. ?? ? ? ap?????e? t?? ?ata??µ?, de?
pa???e? ?a?e?? t?p?ta.
?e??aµa-2 ? pa??t?? ? pa???e? 1 Euro. ?p??e?
?a t? ??at?se? ? ?a t? d?se? st?? ?. A? t? d?se?
st?? ? t? p?s? a????eta? ?at? 1 Euro. Ge????te?a,
???e f??? p?? ta ???µata a??????? pa??t?
p??st??eta? 1 Euro, µ???? t?? te????? p?s?? t??
100 Euro..
18Prisoners Dilemma
- ????a??? St?at????? ?µ????? - S?µe??
?s????p?a? Nash ?µ????? - ?µ?????. - ??
s?µe?? ?e? ?µ????? - ?e? ?µ????? e??a?
?a??te?? ?a? ??a t??? d?? pa??te?.
19?s????p?a ?ata Nash
??a? s??d?asµ?? st?at?????? ß??s?eta? se
?s????p?a ?ata Nash, ?ta? ?a???a? pa??t?? de?
µp??e? ?a ße?t??se? t?? ??s? t?? a??????ta?
µ???µe??? st?at?????.
- ????a??e? St?at??????. - ?p?te???
St?at??????. - ????d?? Minimax. -
??a?t??t???? ??e????. - ???t?? St?at??????.
20????a??? St?at?????
??a st?at????? e??a? ????a??? ??a ??a? pa??t?,
ef?s?? d??e? t? µe?a??te?? ?fe??? ap? ?p??ad?p?te
???? st?at?????, a?e???t?ta ap? t? t? ?a ???e? ?
a?t?pa???.
21Battle of the Bismarck Sea (1943)
USA ?e? ????? ????a??? st?at?????. ??p??e?
? st?at????? ???e?a e??a? ????a???.
22????a??e? St?at?????? (?a??de??µa - 3)
23?p?te??? St?at??????
- ??a st?at????? e??a? ?p?te??? ??a ??a? pa??t?
?ta? d??e? µ????te?? ?fe??? ap? µ?a ????
st?at?????, a?e???t?ta ap? t? t? ???e? ?
a?t?pa???. - Se pa????d?a µe pe??ss?te?? ap?
d?? st?at?????? (??a ??a? t??????st??
pa??t?), e??a? d??at?? ?a ?p?????? ?p?te???
st?at?????? ????? ?a ?p???e? ?aµ?a ????a???
st?at??????. - ? d?ad????? d?a??af? t??
?p?te??? st?at?????? µe???e? t? µ??e??? t??
pa????d??? ?a? p????? f???? st? µ????te??
pa????d? ?p???e? ????a??? st?at?????.
24Domino vs Pizza Hut
- ???? ???at?? ??µ?? ????ß? (12 Euro) -
?a?????? (10 Euro) - F???? (5 Euro). - 3000
p?tse? t?? eßd?µ?da ap? p?st??? pe??te?. - 4000
pe??te? ep??????? t? f????te?? ?at?st?µa.
25?p?te??? St?at?????? (?a??de??µa - 2)
?? s?µe?? ???O-???S???? e??a? t? µ??? s?µe??
?s????p?a?, pa? ??? p?? t? ?????? - ??????
??e? t? ?d?? ?fe???.
26?s?e??? ?p?te??? St?at??????
?a?e??f???? µe ?pa??? 10 Euro. ???e pa??t??
µp??e? ? a????se? µ???? d?? e?s?t???a a??a? 5
Euro t? ?a???a.
?
0
1
2
0, 0
0, 5
0, 0
0
5, 0
0, 0
-5, 0
1
?
2
-5, -5
0, 0
0, -5
??µa-1 ??a???feta? ? as?e??? ?p?te??? st?at?????
2 t?? ?.
??µa-2 ??a???feta? ? as?e??? ?p?te??? st?at?????
2 t?? ?.
??µa-3 ??a???feta? ? as?e??? ?p?te??? st?at?????
0 t?? ? ?a? ?.
?a s?µe?a 0-1 ?a? 1-0 e??a? s?µe?a
?s????p?a? p?? de? ??t?p????ta? ap? t?? pa?ap???
d?ad??as?a.
27?s?e??? ?p?te??? St?at?????? (2)
? d?a??af? as?e??? st?at?????? e??a? p??a??? ?a
d?a????e? ??a ta s?µe?a ?s????p?a?.
28????d?? Minimax
?a??t?? - 2
?
?
?
4
8
15
?
10
9
13
?
?a??t?? - 1
G
13
7
3
?
12
5
6
??µa-1 ?p???t??µe ?t? ? a?t?pa??? µa?te?e?
t?? ep?????? µa?. ??µa-2 ?p??????µe t??
????te?? ?a?? st?at????? (maximin
minimax). ??µa-3 ?? maximin minimax t?te
????µe s?µe?? ?s????p?a?.
29??a?t??t???? ??e????
30??e?? ?a??te?
? ???a, ? ?ate???a, ?a? ? ?a??a ?a????ta? ?a
s??e?sf????? ??a t?? a?a?a???s? t?? p?????.
???te?a??t?te? 6 S??e?sf?????
?? ???e? d??. 5 S??e?sf????? ?a? ?? t?e??. 4
S??e?sf??e? µ??? ? µ?a ap? t?? ???e?. 3
S??e?sf??e? ? ?d?a ?a? µ?a ????. 2 ?e?
s??e?sf??e? ?a?e??. 1 S??e?sf??e? µ???? ? ?d?a.
31?a????d?a µe S??e??? ?etaß??t??
Pizza Hut ?a? Domino ?a???????? t?µ?? ??a t??
p?tse? t???. ?? d??st?µa t?µ?? e??a? s??e???.
??st?? p?tsa? 3 Euro.
SD(x, y) 12 - x 1/2 y SP(x, y) 12 - y
1/2 x PD(x, y) (x-3) SD(x, y) PP(x, y)
(y-3) SP(x, y)
PP(x,y) 15y - y2 1/2xy 3/2x - 36
ymax 7.5 0.25x
xmax 7.5 0.25y
32?a????d?a µe S??e??? ?etaß??t?? (2)
y
Domino
(13.5, 13.5)
Pizza Hut
y 7.5 0.25x
x 7.5 0.25y
x
33???t?? St?at??????
minimax ? maximin
???t? St?at????? ? ????s? ep????eta? t??a?a
µeta?? t?? ?a?a??? st?at??????,
ß?s? s???e???µ???? ?ata??µ??
p??a??t?ta?.
34???? St?at?????? ??sta (p-Mix)
?????
DL
CC
DL
50
80
??sta?
p- Mix
35???? ??sta
?fe??? ??sta
DL (?????)
CC (?????)
90
80
62
50
20
p-Mix
DL (?????) 50p 90(1-p) 90 - 40p CC
(?????) 80p 20(1-p) 20 60p
36???? St?at?????? ????? (q-Mix)
?????
q-Mix
DL
CC
50q 80(1-q)
DL
50
80
??sta?
90q 20(1-q)
CC
90
20
37???? ?????
?fe??? ??sta
90
80
62
50
DL (??sta?)
20
CC (??sta?)
q-Mix
DL (??sta?) 50p 80(1-q) 80 - 30q CC
(??sta?) 90q 20(1-q) 20 70q
38?s????p?a se ???t?? St?at??????
39?aµp??e? ???t?st?? ?p????s??
40????d?? ?a??p??sµ?? ??t?p????
?a??p??sµ?? ????? 50p 90(1-p) 80p
20(1-p) ? p 0.7 ?a??p??sµ?? ??sta 50q
80(1-q) 90q 20(1-q) ? q 0.6
?fe??? ??sta 62, ?fe??? ????? 38
41? ????? ?e?t???e? t?? ?e????? DL
?a??p??sµ?? ????? 30p 90(1-p) 80p
20(1-p) ? p 0.58 ?a??p??sµ?? ??sta 30q
80(1-q) 90q 20(1-q) ? q 0.5
42??-??de???? ?????sµa
Chicken
?a?a?t???st???
- ??? st?at?????? ?ST???S - ?S????S. - ???
s?µe?a ?s????p?a?. - ?? pa??te? p??t?µ???
d?af??et??? s?µe?a ?s????p?a?. - ? ep?????
?S????S - ?S????S e??a? p??? ?s??µ? ?a? ??a
t??? d?? pa??te?.
43Chicken se ???t?? St?at??????
Dean
St??f?
???e?a
q-Mix
0, 0
-1, 1
James
???e?a
1, -1
-2, -2
p-Mix
44?aµp??e? ???t?st?? ?p????s??
45Battle of the Sexes
?a?a?t???st???
- ??? st?at?????? ??a ???e pa??t?. - ??? s?µe?a
?s????p?a?. - ?? pa??te? p??t?µ??? d?af??et???
s?µe?a ?s????p?a?. - ?a? ta d?? s?µa?a
?s????p?a? e??a? p??t?µ?te?a ap? ?p???d?p?te ????
ap?t??esµa. - ?a???a ap?t??esµa de? e??a? t?s?
?atast??f??? ?s? st? Chicken.
46BOTS se ???t?? St?at??????
G??a??a
??d/a???
T?at??
q-Mix
??d/a???
2, 1
0, 0
??t?a?
T?at??
0, 0
1, 2
p-Mix
47?aµp??e? ???t?st?? ?p????s??
48??e?? St?at??????
?????
q-Mix
DL
CC
DL
50
80
CC
90
20
Lob
70
60
??sta?
p-Mix
49?fe??? ??sta st?? ???? ?????
?fe??? ??sta
90
80
70
60
50
Lob
DL
CC
20
q-Mix
0.5
50?e??µ??? ?a????d?
?a??p??sµ?? ????? 50p 70(1-p) 80p
60(1-p) ? p 0.25 ?a??p??sµ?? ??sta 50q
80(1-q) 70q 60(1-q) ? q 0.5
51???????? S?st?µata
52?a?ade??µata ????????? S?st?µ?t??
????d?? Condorcet (18?? a???a?) ?? a?t?pa???
d?a???????ta? a?? d??de? ?a? ? a?t?pa??? pe???e?
st?? ep?µe?? ????.
??e????f??? ????d?? ???? ?? a?t?pa???
d?a???????ta? ta?t?????a. ????t?? e??a? a?t?? p??
s???e?t???e? t??? pe??ss?te???? ??f???.
????d?? Borda (18?? a???a?) ?? ??f?f????
?atat?ss??? ????? t??? ?p???f????. ??????a µe t??
se??? ?at?ta??? ???e ?p???f??? ?aµß??e? µ?a
ßa?µ?????a. ?e?d??e? ? ?p???f??? µe t??
µe?a??te?? s???e?t??t??? ßa?µ?????a.
????d?? ?????s?? ???e ??f?f???? e?????e?
s???e???µ??? a???µ? ?p???f???. ?e?d??e? ?
?p???f??? µe t?? pe??ss?te?e? e????se??.
53Ge???? Te???s? ????????? S?st?µ?t??
???sµ??
- C 1, 2, . . . , n t? s????? t??
??f?f????. - ? x, y, z, . . . , q t? s????? t??
ep??????. - ???e ?p???f??? d?at?se? t?? ep?????? t??
s?????? ?. S?µß??????µe - µe R1, R2, . . . , Rn t?? d?at??e?? t??
??f?f???? st? t???? t?? - ??f?f???a?. ? n-?da P ltR1, R2, . . . , Rngt
???ete profile t?? - e???????? s?µat??.
- ??a e??????? s?st?µa ????eta? ?? µ?a s????t?s?
f µ?? ??a ???e - profile P ltR1, R2, . . . , Rngt t??
e???????? s?µat?? pa???e? µ?a - d??ta?? f(P) ?? ap?t??esµa t?? ??f?f???a?.
54Social Welfare Function
- Positive Association of Social and Individual
Values - ??? ? f ????e? p?? ? ?????t?ta µe profile P
p??t?µ? t? ? ??a?t? t?? ?, t?te ? p??t?µ?s? a?t?
t?? ?????t?ta? ?a p??pe? ?a pa?aµ??e? sta?e??
?ta? t? P µetaß??ete ?p?? t?? ?.
2. Independence of Irrelevant Alternatives
?st? profile P, ?a? A ? ?. ??? ? pe?????sµ?? t??
profile P st? ? s?µf??e? µe t??
pe?????sµ? t?? P st? ?, t?te t? ap?t??esµa t??
??f?f???a? ??a t?? ep?????? st? ?e??a? t?
?d?? ??a ta P, P.
3. Citizens Sovereignty G?a ?p???d?p?te
?e???? ep?????? ?, ?, ?p???e? t??????st?? ??a
profile P, ??a t? ?p??? ? f ????e? p?? t? ?
?pe?te?e? ?.
55Arrows Impossibility ResultSocial Choice and
Individual Values, 1951Nobel Prize 1972
???e e??????? s?st?µa p?? ??a??p??e? t?? s?????e?
(1) (3) ??e? d??t?t??a.
56Arrows Impossibility Result - ?p?de???
???sµ?? ??a ?µ?da ??f?f???? V ? C ???µ??eta?
decisive ??a t? ?e???? ep?????? ltx, ygt ??? ???e
f??? p?? ?? ??f?f???? st? V ??f?s??? ?µ?f??a xgty,
t? ap?t??esµa t?? ??f?f???a? ??a ta x,y e??a?
ep?s?? xgty, a?e???t?ta ap? t? t? ?a ??f?s??? ??
?p????p??. ??µµa-1 ??a ?µ?da ??f?f???? V ? C
e??a? decisive ??a t? ?e???? ep?????? ltx, ygt, iff
? ??f?f???a ?p?? ???? st? V ??f?s??? xgty ?a? ????
?? ?p????p?? ??f?s??? ygtx, ??e? ?? ap?t??esµa
xgty. ?p?de??? ?????pte? e????a ap? t?? s??????
(1). ??µµa-2 (Pareto Optimality) ?? s?????
???? t?? ??f?f???? C e??a? decisive ??a ???e
?e???? ep?????? ltx,ygt. ?p?de??? ?????pte?
e????a ap? t?? s?????e? (1) (3).
57Arrows Impossibility Result ?p?de??? (2)
?st? V t? µ????te?? ?p?s????? t?? C p?? e??a?
decisive ??a ??p??? ?e???? ep?????? ltx,ygt. ?st?
j t??a?? st???e?? t?? V, ?a? ?st? U, W ta s????a
U C-V, ?a? W V- j . ?f?? ????µe ?p???se?
p?? ?p?????? t??????st?? 2 ??f?f????, U ? W ?
?. ?st? t???? z µ?a ?p??ad?p?te t??t? ep?????,
?a? P t? a??????? profile
j W U x z y y x z z y x
- ?? ap?t??esµa t?? ??f?f???a? ??a t? profile P ?a
d?se? - xgty (t? V e??a? decisive ??a ta x,y)
- y?z (??µµa-1 ?a? ? ???sµ?? t?? V ?? t?
µ????te?? decisive s????? ??f?f????) - ??a, xgtz ?a? s??ep?? ap? ??µµa-1, t? j e??a?
decisive ??a t? ltx, zgt, ?a? ??a j V.
58Arrows Impossibility Result ?p?de??? (3)
G?a ?a ?????????e? ? ap?de??? ?a p??pe? ?a
de????µe p?? ??a t??a?a ep????? w ? x, t? j
e??a? decisive ??a t? ltw,zgt. ??a??????µe d??
pe??pt?se?? (i) z ? x, ?a?, (ii) z
x. ?e??pt?s? (i) ?st? P t? a??????? profile
j U w z x w z x
- ?? ap?t??esµa t?? ??f?f???a? ??a t? profile P ?a
d?se? - wgtx (??µµa-2)
- x?z (t? j e??a? decisive ??a t? ltx, zgt)
- ??a, wgtz ?a? s??ep?? ap? ??µµa-1, t? j e??a?
decisive ??a t? ltw, zgt.
59Arrows Impossibility Result ?p?de??? (4)
?e??pt?s? (ii) ?st? P t? a??????? profile
j U w q q x x w
- ?? ap?t??esµa t?? ??f?f???a? ??a t? profile P ?a
d?se? - wgtq (ap? ?e??pt?s? (i), t? j e??a?
decisive ??a t? ltw, zgt) - q?x (??µµa-2)
- ??a, wgtx ?a? s??ep?? ap? ??µµa-1, t? j e??a?
decisive ??a t? ltw, xgt. - S??ep??, t? j e??a? decisive ??a ?p???d?p?te
?e???? ep?????? lta,bgt ?a? ep?µ???? ? ??f?f???? j
e??a? d??t?t??a?.
60??µ?p?as?e?
- ???a ??????
- ???µat?st????
- e-Bay
- ???µ??e?e? ??µ?s???
- ?????pe? st?? ???a?d?a (Dutch Auction)
- ????a st?? ?ap???a
- ?ad??s????t?te? st?? ?µe???? (1994)
- 10 ?de?e?
- 46 ?????
- S??????? t?µ? 600 e?. (10 f???? p??? ap? t??
p??ß?e??)
61?a???e? ??µ?p?as???
- English Auction
- ? d?ad??as?a e??a? fa?e?? se ?????.
- ? t?µ? a?eßa??e? se ???e p??sf??? µ???? ?a
µe??e? ??a? a???ast??. - ? t?µ? p???s?? ????eta? ?? ? ?????te?? p??sf???.
- Dutch Auction
- ? d?ad??as?a e??a? fa?e?? se ?????.
- ? t?µ? ?ateßa??e? µ???? ?a ß?e?e? a???ast??.
- ? t?µ? p???s?? ????eta? ?? ? ?????te?? p??sf???.
- Vickrey Auction (Williams Vickrey ??aße??
Nobel 1996) - ?? p??sf???? e??a? sf?a??sµ??e?.
- ????t?? e??a? ? a???ast?? µe t?? ?????te??
p??sf???. - ? t?µ? p???s?? ????eta? ?? ? 2? ?????te??
p??sf???.
62?p?p???? ?a???e? - ?a?a??a???
- Reserve Price (fa?e??/???f?).
- ????ap??? p??sf????.
- Sta?e??? p??sa???se??.
- ??st?? s?µµet???? (p????/µetaßa???µe??).
63????ast??
- Perfect Information.
- Independent Private Values.
- Common Values.
64????ast?? µe ????? ?????f???s?
??µ?p?as?a ???a?a
- English Auction / ???f?
- ?? p??sf???? p??pe? ?a e??a? p???ap??s?a t?? 10.
- ?????? t?µ? 10.
- ???a ??a ??sta 54
- ???a ??a ?a??a 34
?a?at???s? ?s? µ????te?e? e??a? ??
p??sa???se??, t?s? ? t?µ? p?? p?????e? ? ??sta?
p??s???e? t?? a??a p?? ??e? ? p??a?a? ??a t??
?a??a.
65????ast?? µe ????? ?????f???s? (2)
??µ?p?as?a ???a?a
- Dutch Auction / fa?e??
- ? t?µ? ?e????e? ap? 100 ?a? p?fte? ?at? 10 t?
f???. - ???a ??a ??sta 54
- ???a ??a ?a??a 34
?a?at???s? ?s? µ????te?e? e??a? ?? µe??se??,
t?s? ? t?µ? p?? p?????e? ? ??sta? p??s???e? t??
a??a p?? ??e? ? p??a?a? ??a t?? ?a??a.
66????ast?? µe Independent Private Values
??µ?p?as?a S??????? ap? Comics
- English Auction / ???f?
- ???a ??a ??sta VA
- ???a ??a ?a??a VB
- ??t?µ?s? pa??t?? ??a a??a a?t?p???? ?µ???µ??f?
?ata??µ? st? (0, 1000).
S?µe?? ?s????p?a? bA VA/2, bB VB/2
F(x) 0 ? P x lt VB/2 (VA x) ? P x
gt VB/2 0.5 ?(VA x) ? P x VB/2
(VA x) ? P x gt VB/2
(VA x) ? P 2x gt VB (VA x) ?
min2x, 1
? xmax VA/2
67????ast?? µe Independent Private Values
??µ?p?as?a S??????? ap? Comics
- Vickrey Auction / ???f?
- ???a ??a ??sta VA
- ???a ??a ?a??a VB
- ??t?µ?s? pa??t?? ??a a??a a?t?p???? ?µ???µ??f?
?ata??µ? st? (0, 1000).
S?µe?? ?s????p?a? bA VA, bB VB
68Common Values Winners Curse
??µ?p?as?a S??????? ap? Comics (Superman ??????
???a?)
- Vickrey Auction / ???f?
- ???a s??????? VS VM
- ??t?µ?s? pa??t?? ??a a??a a???st?? ?µ???µ??f?
?ata??µ? st? (0, 500). - ??aµe??µe?? ???a ??a ??sta VS 250
- ??aµe??µe?? ???a ??a ?a??a VM 250
?p??es? ?? a???ast?? e??a? e???????e?? ?a? VM lt
VS 150.
? ??sta? p?????e? VM 250, e?? ? a??a t??
s??????? e??a? VS VM VM 150.
S?µe?? ?s????p?a? bA 2VA, bB 2VB
69??p??? ??????
70??p??? S?st?µata
??f?ß?t? p, q, - ?????? Strings
(????µata) -p-q--
?a???a? (?a?a?????) 1 ?? ta x, y, z e??a?
strings ap? pa??e? ?a? t? xpyqz e??a? ?e???µa,
t?te ?a? t? -xpyqz- e??a? ?e???µa.
?a???a? (?a?a?????) 2 ?? ta x, y, z e??a?
strings ap? pa??e? ?a? t? xpyqz e??a? ?e???µa,
t?te ?a? t? xpy-qz- e??a? ?e???µa.
71?a?at???se??
Logics main concern is with the soundness and
unsoundness of arguments.
????t?ta vs. ????e?a - Soundness ? Truth. -
If all premises are true and the conclusion is
false, then the argument is not sound.
72S??????sµ??
S??????sµ?? de est? ????? e? ? te???t?? t????
et???? t? t?? ?e?µ???? e? a?????? s?µßa??e?
t? ta?ta e??a?. ??a??t??? ???te?a
???st?t???? (384 p? 322 p?)
73?p? t?? ???st?t??? st?? Gödel
74?? ?a??d??? t?? Russell (1901)
Sets S 1, 3, 8 G 4, 2, 6,
11, 3, 5, 7
?? ?a??d???
? S S ? S
Contradiction
Yes
Is A in A ?
No
Contradiction
75?? ?a??d??? t?? Burry (1906)
Observations
- ?p???e? pepe?asµ???? a???µ?? s?µß???? st??
??????? ???ssa. - ?p?µ???? ?p???e? pepe?asµ???? a???µ?? a???????
p??t?se?? p?? pe??????? - µ???? 200 s?µß??a. ?st? ? t? s????? a?t?? t??
p??t?se??. - ?p?µ???? ?p???e? pepe?asµ???? a???µ?? a?e?a???
p?? µp??e? ?a - a?apa?asta?e? µe p??t?se?? t?? ?.
?? ?a??d???
The least positive integer that is not denoted
by an English expression containing fewer than
200 symbols.
76?p? ???t?? se ???a?a
Smith Taylor
In
Taylor
1 1
Ads
Out
3 3
Smith
In
2 4
No Ads
Taylor
4 2
Out
St?at?????? Smith Ads, No Ads St?at??????
Taylor In-In, In-Out, Out-In, Out-out
77?p? ???a?a se ???t??
78S??d?asµ?? ?a?t??????? ?a? ??ad?????? ????se??