lecture1.2
TRANSCRIPT
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
§ 1.2 SymmetrySalon: Cloud vis-a-vis Clock
Preston S. Fan
School of Management, New York Institute of Technology
April, 2014
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Table of Contents
1 Recycling Topos in Music and NarrativesGoldberg VariationsRecycled Narratives
2 SymmetryThe Universe of Clocks
3 Asymmetric World of CloudsExploitation of SymmetrySymmetry in Auction Strategy
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Goldberg Variations
G222 ˘ ˘2
ˇ ˇ ˇIJIJ
ˇ ˇŰŰ 3
ˇ ˇ 6IJIJ6ˇ
G222 ˇ ˇ ˇ
èèˇ
2
2 ˇ ˇÑÑ
ˇ ˇ ˇŃŃ
G222 ˇ ˇ ˇ ˇ ˇ ˇ ˇŒŒ
ˇ2
ˇ ˇŮŮ
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Goldberg Variations
G222 2 ˇ ˇŒŒ
ˇ2
6 6ĚĚ ˇ ˇ 6ËË
6ˇ
G222 6 ˇĚĚ ˇ ˇˇ
ÅÅˇ 2 ˇ ˇĚĚ
G222ˇ ˇ ˇÅÅ
6 2 ˇˇ ˇ ˇ
(Bach, J.S. 1885. Musikalisches Opfer, ed. A. Drffel. Leipzig,Germany: Breitkopf Hartel)
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Recycled Narratives
A story goes:Once upon a time, there lived an old man in a cabinet on a
hill. He was telling a story to his dog: ”Once upon a time, therelived an old man in a cabinet on a hill. He’s telling a story tohis dog...”
Similar to the paintings of M.C. Escher (1898−1972).
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
The Universe of Clocks
Symmetry
Regular (irregular?)
Symmetry is to simplify (copy or lose information)
Symmetry is then short of information
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
The Universe of Clocks
Human behaviors
In a strictly competitive game, imitation is a winingstrategy
No chance moves: Chess, Tic-Tac-Toe, etc.With chance moves: Nim, Hex, Poker, etc.
Make every cloud a clock: measure is what we measure
Security strategy: iterated rationality
I think whether you think I think you think I think youthink I think... (Fixed point theory)No (rational) one agrees to disagree... (Asymptotic beliefs)
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Symmetry is more like a result asymptotically than anautomaton?
Survival of smallest information set
Iterated expectationE[E(E(y |x1, x2, x3) |x1, x2) |x1] = E(y |x1).
A result of what?
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
6 1 87 5 32 9 4
Transpose =
”Flip left− side− right”
+
”Rotate 90◦counter − clockwise”
2
7
69
5
14
3
8
6
1
8
7
5
3
2
9
4
83
4
15
9
67
2
Linearized world of clocks
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
In a symmetric game with no equilibrium in pure strategy,invite Chance moves.
A randomization device or mixed strategy short-circuitsthe vicious circle of ”I think you think I think...”
Player IIHead Tail
I
Head 1−1
−11
Pla
yer
Tail −11
1−1
q
p
0.5
0.5
1
1
q = R1(p) p = R2(q)
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Exploitation of Symmetry
In political science, for only two candidates,
H. Hotelling(1929): Median Voter Theorem
If there are three or more candidates, what result would youexpect?
(Backward induction)
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Exploitation of Symmetry
In the rational wonderland of economic science, does timingmatter?
In the game of Chess, Nim or Skittles, timing determineswho is the winner.
However, for a lot of games, their sequential version would differa lot from their simultaneous version.
ex: Stackelberg’s with respect to Cournot’s
Preston S. Fan § 1.2 Symmetry
Recycling Topos in Music and Narratives Symmetry Asymmetric World of Clouds
Symmetry in Auction Strategy
Sequential rationality (Bayesian)
Symmetry might exist in either of
TimezoneSpace
How to avoid a Dutch book?
Bayesian updatingSeparating beliefs from preferences
To utilize symmetry in the auction design:
Duality in optimization and mini-max saddle point;
Nash’s strategy-stealing argument on Hex game;
Tit-for-tat automaton
Questions: Reciprocity or symmetry
Preston S. Fan § 1.2 Symmetry
Relevant Readings
Manfred Eigen, Ruthild WinklerLaws of the Game: How the Principles of Nature GovernChancePrinceton University Press. 1993.
Ken BinmorePlaying for RealOxford University Press. 2007.
Preston S. Fan § 1.2 Symmetry