Correlated Equilibrium and Seemingly-Irrational Behavior

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Tel-Aviv University

‫אביב‬-‫אוניברסיטת תל‬

Raymond and Beverly Sackler Faculty of Exact Sciences

‫הפקולטה למדעים מדויקים‬ ‫ע"ש ריימונד ובברלי סאקלר‬

Correlated Equilibrium and Seemingly-Irrational Behavior Thesis submitted for the degree of “Doctor of Philosophy” by

Yuval Heller

Under the supervision of

Prof. Eilon Solan

Submitted to the senate of Tel-Aviv University April 2011

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This work was carried out under the supervision of

Prof. Eilon Solan

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Acknowledgements First, I would like to express my deepest gratitude and appreciation to my supervisor Eilon Solan for his continuing support, guidance, encouragement and dedication throughout my doctoral studies.

I would like also to thank other professors at Tel-Aviv University for the kind and valuable help they gave me in my research: Eddie Dekel, Chaim Fershtman, Tzachi Gilboa, Ehud Lehrer, Zvika Neeman, Ariel Rubinstein, and David Schmeidler.

Special thanks for my friends and colleagues in our game theory lab: Shiri AlonEron, Galit Ashkenazi-Golan, Yaron Azrieli, Assaf Cohen, Yaarit Even, Ayala Mashiah-Yaakovi, Ron Peretz, and Roee Teper. It was a pleasure working together.

I would also like to thank the administrative staff in our school, especially Lily Brunstein, Sonia Hasday, Miriam Hercberg, Anat Koren-Dror, and Nurit Liberman for their willingness to help at any time.

Saving the most important for last, I want to thank my family for their love and support. I want to thank my dear parents and sisters for their constant encouragement over the many years of my education. I deeply thank my beloved children, Yaara and Lior, for being the light of my life. Above all, I give my heartfelt thanks to my beloved wife, Renana, for her unconditional love, endless patience, and total support.

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i γ i (x) ≤ αH − ǫ7

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stop

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(% /'*1%" (8 (" (% ǫ4/'*1%" 8&0 !:!01 ǫ > 06 H (% =&+'4(=$"!. (8 (" (% )&" %(+'4!, (6!69 (% )&" ǫ0 2%(+'4!6 7) "/$" =$%! 7 %$1 "/$" H (% =&+'4(=$"!. 5606"6 ǫ0 6

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H ∈ Hn ' 3).*4+3'%$2 5+(%)16 7!1!%! ǫ0 / ǫ <<

ǫ0 |I|·|D|

i / ',2 8)1 $'35 i ∈ I 4$% ai ≥ αF − ǫ! 95$, %5$1$ $:+(%(

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=>? @) (0&-'.$ )8 TU 5'( ' @'(5 ǫA$B0+4+&1+0. 7+%5 ' 3)11$(*),2+,- *'6)C +, = ? E+%5$1< ='? U = ∅ =() %5'% TU = T ?D )1 =&? x +( ' @'(5 9ǫA$B0+4+&1+0. +,

Q i i a ,a + ǫ D

i∈I

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8)1 $F$1" σ > n ≥ 0 ',2 H ∈ Hn / (035 %5'% cn,σ (H) +( ' # AF'40$2 F $3),(+(%$,% %! A80,3%+),! " i − ǫ ≤ ai ! G% +( )**( +8 %5$1$ $:+(%( ' M =56*$1?A1$3%',-4$ ai , ai + ǫ i∈I +( &'( +8 8)1 $F$16 i ∈ I / αH

i *4'6$1 i ∈ I (035 %5'% ai + ǫ ≤ αH − ǫ! #$% + &$ ' ;,+%$ 3)F$1+,- )8 [−1, 1] 7+%5 =,)% ,$3$(('1+46 2+(N)+,%? " i i 1$3%',-4$( a , a + ǫ i∈I / '44 )8 75+35 '1$ $+%5$1 -))2 )1 &'2! #$% B = {b1 , b2 , ..., bJ } &$ %5$ ($% )8 J &'2 |I|

+

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1$3%',-4$( +,

&)0,2$2 (%)**+,- %+.$/ ',2 H ∈ Hn ! G8 H +( (+.*4$ G 4$% cn,σ (H) = simple! O%5$17+($/ H +( 7!1!%! %) ().$ ǫ0 (H)! G, %5'% 3'($ G '((0.$ 7!4!)!-! %5'% ǫ <<

ǫ0 (H) |I|·|D| !

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95$ 3)4)1 cn,σ (H) +( 2$%$1.+,$2 &6

%5$ 8)44)7+,- *1)3$201$<

• H$% T (0) = Tn,σ (H)!

• P)1 1 ≤ j ≤ J '**46 J1)*)(+%+), !>I %) T (j−1) ',2 %5$ &'2 1$3%',-4$ hj = ' (0&-'.$ T (j) )8 T (j−1) ',2 (%1'%$-6 *1);4$ xj +, T (j) (035 %5'%<

Q i i aj , aj + ǫ %) )&%'+,

i∈I

=>? @) (0&-'.$ )8 T (j) 5'( ' (%'%+),'16 ǫA$B0+4+&1+0. 7+%5 ' 3)11$(*),2+,- *'6)C +, hj ! = ? E+%5$1 T (j) = T (j−1) )1 %5$ 8)44)7+,- %51$$ 3),2+%+),( 5)42< ='? P)1 $F$16 i ∈ I / aij − ǫ ≤ γ i (xj )!

i i =&? P)1 $F$16 i ∈ I ',2 $F$16 (%1'%$-6 y i < γ i (x−i j , y ) ≤ aj + 8ǫ!

=3? π (xj ) ≥ ǫ2 × pT (j) ,T (j−1) !

• G8 T (J) +( %1+F+'4 =%5$ ),46 ,)2$ +( %5$ 1))%?/ ($% cn,σ (H) = allbadD )%5$17+($ 20$ %) J1)*)(+%+), !>I ',2 .6 *1)3$201$ ),$ )8 %5$ 8)44)7+,- 5)42(<

=>? T (J) 5'( ' ($B0$,%+'4 (%'%+),'16 '&()1&+,- ǫA$B0+4+&1+0. 4/ 7+%5 ' *'6)C γ(x) +, ),$ )8 %5$ -))2 56*$1A1$3%',-4$(! #$% cn,σ (H) = (1, ol )/ 75$1$ ol +( %5$ -))2 1$3%',-4$ %5'% +,3402$( γx !

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