IT CAN BE SMART TO BE STUPID

1
IT CAN BE SMART TO BE STUPID David H.
Wolpert NASA Ames Research Center,
ti.arc.nasa.gov/people/dhw/ Michael
Harre University of Sydney Julian
Jamison University of Southern California NASA-AR
C-05-007
2
ONLY IDEA IN THIS TALK
If before playing a game, we choose what
persona to adopt for that game, then bounded
rationality is optimal behavior.
3
ROADMAP
1) Review game theory
2) Illustrate rationality games
3) Explain travelers dilemma and ultimatum game
4) Illustrate altruism games explain prisoners
dilemma
5) Applications and mathematical details
4
REVIEW OF GAME THEORY
N independent players, each with possible
moves, xi ? Xi Each i has a distribution
qi(xi) q(x) ?iqi(xi) N utility functions
ui(x) player i wants maximal Eq(ui) Eq(ui)
depends on q but i only sets qi
Equilibrium concept mapping from ui ? q
5
REVIEW OF GAME THEORY - 2
Ex. 1 Nash Equilibrium (NE) q For all
players i, Eq(ui) cannot rise by changing qi
while keeping q-i fixed. Ex. 2 Quantal
Response Equilibrium (QRE) q Simultaneously
for all i, qi(xi) ? exp?iE(ui xi) Crude
model of bounded rationality. In fair
agreement with experiment. Phase transitions
for finite systems.
6
ROADMAP
1) Review game theory
2) Illustrate rationality games
3) Explain travelers dilemma and ultimatum game
4) Illustrate altruism games explain prisoners
dilemma
5) Applications and mathematical details
7
PRISONERS DILEMMA (PD)

• Col player
• L R
• T 6, 0 4, 4
• Row player
• D 5, 5 0, 6
• ? Red (T, R) is NE
• ? Aqua (D, L) is Pareto superior point
• Experimentally, people often play (D, L) in
  non-repeated games.
• WHY?

Row utility, Col utility
8
PRISONERS DILEMMA FOR IRRATIONAL COL
1) Say that Col is irrational ? She plays L
with probability .5, no matter the effect on
her utility. ? And in particular no matter
what move Row makes. L R T
6, 0 4, 4 T 5, 2 D 5, 5
0, 6 D 2.5, 5.5 2) So Col hasnt helped
herself by being irrational (2 lt 4).
?
9
A GAME WHERE IRRATIONALITY HELPS COL
1) Again have Col be irrational, i.e., play L
with probability .5, no matter the effect on her
utility. 2) However do this for a different game
L R T 0, 0 6,
1 T 3, 0.5 D 5, 5 4,
6 D 4.5, 5.5 3) So by being irrational,
Col induces Row to flip her move. 4) The result
is that Cols utility increases. (5.5 gt 1)
?
10
IRRATIONALITY HELPS COL - 2
L R T 0, 0 6,
1 T 3, 0.5 D 5, 5 4,
6 D 4.5, 5.5 5) Crucial that Col not
exploit Rows playing D (by making move R). Col
must be truly irrational, not just fake it. 6)
If Rows move were fixed (to D), Col should be
rational. But not if Rows rationality is fixed
(to full). 7) Note Row is hurt by Cols
irrationality.
?
Being an airhead can benefit you.
11
EVEN ANTI-RATIONALITY CAN HELP
1) Now consider a game where irrationality wont
help. L R T 0, 0
6, 1 T 3, 0.5 D 5, 5 0,
6 D 2.5, 5.5 2) However have Col be
antirational ? She does whatever is
worst for her. ? That means she plays L no
matter what Row does L R T
0, 0 6, 1 T 0, 0 D 5, 5
0, 6 D 5, 5
?
?
12
ANTI-RATIONALITY HELPS COL - 2
L R T 0, 0 6,
1 T 0, 0 D 5, 5 0, 6 D
5, 5 3) So anti-rationality helps Col (5 gt
1). Again, crucial that Col not exploit Rows
playing D. Col must be truly anti-rational. 4)
Again, Row is hurt by Cols antirationality. 5) N
.b., antirationality is equivalent to replacing
utility uCol (x) with -uCol(x).
?
Being your own worst enemy can benefit you.
13
PROPERTIES OF QRE
i) If ?i ?, then we have full rationality
q?i(xi) ? exp?iEq?(ui xi)
delta function about argmaxxi iEq?(ui
xi) ii) If ?i 0, then we have
irrationality q?i(xi) ? exp?iEq?(ui
xi) constant, independent of xi iii) If
?i -?, then we have anti-irrationality
q?i(xi) ? exp?iEq?(ui xi)
delta function about argminxi iEq?(ui xi)
14
RATIONALITY GAMES
1) Given A concrete game (X, u)
(X, u) Xi, ui(x) i ? 1, ..., N where x
(x1, ..., xN) ? X A rationality set for
each player, Bi i ? 1, ..., N, where each
Bi is a set of real numbers. 2) This specifies a
nested sequence of two games i) First each
player i independently chooses rationality ?i ?
Bi ii) That specifies a QRE ? i, q?i(xi) ?
exp?iEq?(ui xi) iii) The payoff for
player i in the first, rationality, game
is Eq?(ui)
15
RATIONALITY GAMES - 2

• 3) So the first rationality game is (B, v)
• (B, v) Bi, vi(?) i ? 1, ..., N,
• where ? (?1, ..., ?N), and each vi maps ? to
  Eq?(ui).
• 4) Player is choice of ?i
• ? Affects the mixed strategy q?-i of the
  others
• ? But also forces her to use q?i
• ? Optimal choice of ?i trades off those
  effects in light of the rationality
  choices of the other players.
• 5) N.b., in examples above, only Col had a
  rationality set with more than one element.

16
RATIONALITY GAMES EXAMPLE
1) Recall the game L R T
0, 0 6, 1 T 0, 0 D 5, 5
0, 6 D 5, 5 2) Change this by
providing both players i with Bi -?, ?
L R -? ? T 0, 0
6, 1 -? 0, 0 0, 6 D 5, 5
0, 6 ? 5, 5 6, 1
?
?
17
RATIONALITY GAMES EXAMPLE - 2
L R -? ? T 0, 0
6, 1 -? 0, 0 0, 6 D 5, 5
0, 6 ? 5, 5 6, 1 3) Note that
the rationality game matrix is just the concrete
game matrix with rearranged entries (not always
true). 4) The NE of the rationality game is for
Col to be anti-rational, and for Row to be
rational. 5) Introducing rationality choices has
helped Col, but hurt Row.
18
WHAT LIMITS ON RATIONALITY GAMES?
L R -? ? T 0, 0
6, 1 -? 0, 0 0, 6 D 5, 5
0, 6 ? 5, 5 6, 1 1) ?
rationality games where all players want to be
anti-rational if their opponent(s) are rational?
2) ? rationality games where all players want to
be anti-rational if their opponent(s) are
anti-rational? ? Would mean (-?, -?) is a NE
of the rationality game. 3) ? rationality games
where both (1) and (2) hold? 4) ? rationality
games where the rationality game NE is Pareto
superior to the concrete game NE? Even where (2)
holds?
19
YES, YES, YES, and YES
0,6 4,7 -1,5 4,4 -1,6 5,5
2,3 7,4 -2,1 3,2 0,0 5,-1
1,1 6,0 1,-2 6,-1 ? Reds
are the four NE for the four rationality
pairs i) Bottom-left is NE for rationality
pair (?, ?) ii) Next one up and over is for
(-?, ?) iii) Next one up and over is for (?,
-?) iv) Top-right is for (-?, -?).
20
YES, YES, YES, and YES - 2
0,6 4,7 -1,5 4,4 -?
? -1,6 5,5 2,3 7,4 -? 4, 4
3, 2 -2,1 3,2 0,0 5,-1
? 2, 3 1, 1 1,1 6,0 1,-2
6,-1 1) Each player prefers to be
anti-rational if her opponent is rational. 2)
Each player prefers to be anti-rational if her
opponent is anti-rational rationality game NE is
(-?, -?). 3) The rationality game NE is
Pareto-superior to the concrete game NE (4 gt 1,
and 4 gt 1).
21
NOT ALWAYS THOUGH
3,3 6,2 -1,0 4,1 -?
? 2,-1 5,0 -2,-2 3,-3 -? 1, 1
5, 0 1,8 8,7 0,5 2,6
? 0, 5 3, 3 0,3 7,4
-3,2 1,1 1) The rationality game matrix is
the PD. 2) So the rationality game NE is (-?,
-?). 3) This is Pareto inferior to the concrete
game NE (1 lt 3, 1 lt 3) ? Both players are hurt
if they play a rationality game before the
concrete game.
?
22
COMMENTS
1) What is the distribution of concrete games
among humans? 2) Playing a rationality game is
computationally harder than playing the
underlying concrete game ? In real world,
cant have rationality sets be too large. 3)
Examples above all had pure strategy rationality
game NE ? Rationality mixed strategy is
capriciousness, randomly choosing how
rational to be from one game to the next. 4)
Social intelligence (primates, corvids,
cetaceans) is ability to play rationality games.
23
ROADMAP
1) Review game theory
2) Illustrate rationality games
3) Explain travelers dilemma and ultimatum game
4) Illustrate altruism games explain prisoners
dilemma
5) Applications and mathematical details
24
TRAVELERS DILEMMA (TD)

• 1) Two travelers fly with an identical antique
  in their baggage
• 2) The airline accidentally destroys both
  antiques
• 3) The airline asks them separately for the
  antiques value,
• allowing answers 2, 3, ..., 101
• 4) To induce honesty, airline tells travelers it
  will compensate both with lower of their two
  claims, with bonus R to person with smaller
  claim, and a penalty R to the other person
• ., . ., . ., . ., .
• 50 ., . 50,50 52,48 ., .
• 51 ., . 48,52 51,51 ., .
• ., . ., . ., . ., .
• 5) NE is (2, 2).

Example (R 2)
25
TRAVELERS DILEMMA (TD) - 2

• 1) In experiments, subjects play close to 100,
  not 2.
• 2) Even true for game-theoretician subjects, if
  real money used.
• 3) If R increases, behavior is closer to NE.
• 4) Creator of TD ... behavior generated by
  rationally rejecting rational behavior ... (is
  necessary) to solve the paradoxes of rationality
  that plague game theory.
• ., . ., . ., . ., .
• ., . 50,50 52,48 ., .
• ., . 48,52 51,51 ., .
• ., . ., . ., . ., .

26
TRAVELERS DILEMMA (TD) - 3
1) Associated rationality game with rationality
sets ? ? 0, ? ., . ., . ., .
., . 0 ? ., .
50,50 52,48 ., . 0 34.8, 34.8
53.3, 49.6 ., . 48,52 51,51 ., .
? 49.6 53.3 2, 2 ., . ., .
., . ., . 2) Rationality game has 3 NE
(one is mixed P(?i 0) .78 ?i). 3) Resultant
P(x) has 25 mass uniformly distributed over
all x, and the rest evenly divided between 97 and
98. 4) As R grows, mixed NE puts more mass on ?.
Eventually only have 1 NE, at (?, ?). Exactly
like in experiments.
?
27
ULTIMATUM GAME (UG)

• 1) First player (F) chooses an offer x ? 0, 1,
  ..., 100.
• 2) Second player (S) chooses from accept,
  reject.
• 3) Both players have utility 0 if S rejects.
• 4) S gets x if she accepts, and F gets 1 - x
• 5) NE is (1, accept)
• 6) In experiment, F usually offers x 30, and S
  usually rejects if x much smaller.
• WHY?

28
ULTIMATUM GAME - 2
1) Associated rationality game with rationality
sets ? ? ? 2) Subtlety Ss move is choice
of map x ? accept, reject 3) Pure strategy NE
of rationality game associated concrete game
pure strategy (21, accept). 4) If S is
irrational enough, then for her P(accept) to be
gtgt 1/2, must have x gtgt 0. So if F knows S is
irrational enough, she offers x gtgt 0. So it
benefits S to be irrational enough. It benefits
S to be spiteful
29
ROADMAP
1) Review game theory
2) Illustrate rationality games
3) Explain travelers dilemma and ultimatum game
4) Illustrate altruism games explain prisoners
dilemma
5) Applications and mathematical details
30
PERSONA GAMES

• 1) Can have persona sets other than
  rationality.
• 2) Example is altruism games
• L R
• T 2, 0 1, 1 Selfish Saint
  
• D 3, 2 0, 3 1, 1 3,
  2
• Col can either choose persona uCol, which is
  being Selfish, or choose persona uRow, which is
  being a Saint. If shes a Saint, she cares only
  about Row, not about herself.
• 4) So Col benefits by being a saint.
• 5) Row also benefits by Cols sainthood.

?
31
PERSONA GAMES - THE PD
1) Consider an altruism game where both persona
sets are Selfish, Charity, where Charity
persona .5 uCol .5 uRow 2) Have the concrete
game be the PD L R C
S T 6, 0 4, 4 C 5, 5 4,
4 D 5, 5 0, 6 S 4, 4 4,
4 3) The persona game NE is for both players to
be charitable. ? That persona NE causes
cooperation in the concrete game. ? That persona
game NE is Pareto superior to the concrete
game NE.
32
PERSONA GAMES - GENERALIZED PD
1) Consider an altruism game where both persona
sets are Selfish, Charity, where Charity
persona sume (1 - s)uyou 2) Have the
concrete game be the generalized PD L
R T ?, 0 ?, ? where ? gt ?
gt ? D ?, ? 0, ? 3) The persona game
NE is for both players to be charitable iff s
? 1 - ? / ? , ? /? ? That persona NE causes
cooperation in the concrete game. ? That persona
NE is Pareto superior to the concrete game NE.
33
PERSONA GAMES - GENERALIZED PD

• Define B ? ? - ?, R1 ? ? - s?, R2 ? ? - (1 -
  s)?
• Intuitively, B is the benefit of cooperation over
  defection.
• R1 gt 0 for joint cooperation to be NE of the
  realized game
• So intuitively, R1 is how robust joint
  cooperation is.
• 4) Given (2), R2 gt 0 for joint charitable to be
  NE of persona game.
• So intuitively, R2 is how robust joint
  charitable is.
• 5) R1 R2 B lt 1
• Unavoidable tradeoff between robustness of
  cooperation and benefit of cooperation

34
ROADMAP
1) Review game theory
2) Illustrate rationality games
3) Explain travelers dilemma and ultimatum game
4) Illustrate altruism games explain prisoners
dilemma
5) Applications and mathematical details