Iterative naïve best reply vs. elimination of
dominated strategy in a p-beauty contest
Andrea Morone
Università degli studi di Bari
a.morone@gmail.com
FUR
ROMA, 23/06/2006
Slide 1/39
P-beauty contest
The basic idea of the guessing game was first introduced by
Keynes (1936), in his famous metaphor about beauty contest:
There are traders who “devote [their] intelligences to anticipate
what average opinion expects average opinion to be. And there
are some, I believe, who practice the fourth, fifth and higher
degrees”,
Slide 2/39
Exactly like in a game where one is prompted
to choose the prettiest face from 6 faces;
One will not
choose the face
he/she really
likes, not even
the one he/she
thinks others
like, but the face
he/she thinks
the others think
the others
think… as being
the prettiest.
Slide 3/39
The rules of the game:
There are n players
They have to choose simultaneously a number g from a closed interval [0,100]
The winner is the player whose choice is closest to the target number G
n
1
G  p  gi
n i 1
Is a parameter that
captures the idea
that in a guessing
game, agents do not
act exactly as
described by Keynes’
beauty contest
game, but they want
to be a little bit away
from the mean
Slide 4/39
To solve the game we can employ iterated elimination of
dominated strategies:
g = p100 dominates all larger choices,
and then
g=p2100 dominates all remaining larger numbers, etc.
till only p∞100 remains.
This solution requires rationality in the sense of infinite
iterations and its common knowledge.
Slide 5/39
Nagel (1995) in her seminal paper suggested “that the
‘reference point’ or starting point for the reasoning
process is 50 and not 100.
The process is driven by iterative, naïve best replies
rather than by an elimination of dominated strategies”.
If we buy Nagel hypothesis we enter in the fascinating
path of bounded rationality and heterogeneous
subjects. Different subjects are characterised by
different cognitive levels.
Slide 6/39
Bosch et al. (2002) analysed ‘newspaper and lab
beauty-contest experiments’ and categorised
subjects according to their depth of reasoning.
They recognize that subjects are clustered at

zero-order belief

first-order beliefs

third-order beliefs

infinity-order beliefs.
Slide 7/39
Güth et al. (2002) analysed the ‘beauty contest’ from a different perspective.
They compare:
(a1) interior and boundary equilibria
(b1) homogeneous and heterogeneous players.
They find attractive results:
(a2) “… interior equilibria trigger more equilibrium-like behavior than boundary
equilibria”
(b2) “heterogeneity of players should trigger more thorough deliberations and, thus,
more equilibrium-like decisions.”
They reach fascinating conclusions:
(a3) “we find swifter convergence to the equilibrium when the equilibrium is interior”;
(b3) “more complexity by introducing heterogeneous players, however, is detrimental
for profit as well for convergence to the equilibrium”.
Slide 8/39
Now I was puzzled
•We have a simple story
•We have a nice Nash equilibrium
•We reach it using standard game theory tools
But if we look out of the window (or in the lab) we see
that people behave (a bit) differently
How can we “fix” this problem?
Let’s take a magic number (50)
But if we have this magic number why convergence is
faster toward an interior equilibrium than toward a
boundary one?
Slide 9/39
We have to study better the problem.
My targets:
(1) Analysing Güth et al.’s results concerning the
properties of interior equilibria in a more general setting.
(2) Generalising the naïve best reply strategy to the wider
class of games with interior equilibrium.
(3) Comparing the iterative naïve best reply strategy with
the elimination of dominated strategies for the
generalised p-beauty contest.
This more systematic study of naiveties will allow us to
address our main question:
(4) what speeds up convergence to equilibrium in a
beauty contest game: subjects naiveties or the presence
of an interior equilibrium?
Slide 10/39
p-beauty contest game with interior equilibria
1 n

G  p  g j  d 
 n j 1

We start comparing three cases (case 1 and 3 were first
analysed in Güth et al., 2002):
d = 0,
d = 25,
d = 50.
Slide 11/39
Let’s assume that a subject solves the game using the elimination of
dominated strategies.
…
…
1
2
3
4
pi = ½ , d = 0
gi > 50
gi > 25
gi > 12.5
gi > 6.25
Eliminated strategies
pi = ½ , d = 50
pi = ½ , d = 25
gi < 25, gi > 75
gi < 12.5, gi > 62.5
gi < 37.5, gi > 62.5
gi < 18.75, gi > 43.75
gi < 43.75, gi > 56.25
gi < 21.875, gi > 34.375
gi < 46.875, gi > 53.125
gi < 23.4375, gi > 29.6875
…
Step
…
The three problems are similar: in all cases (i.e. d = 0, 25, 50) he needs
an infinite number of iterations to reach the equilibrium.

gi = 0
gi = 50
gi = 25
Table 1:Repeated elimination of strictly dominated strategies
Slide 12/39
pi = ½ , d = 0
pi = ½ , d = 50
gi = 50
gi = 25
gi = 12.5
gi = 6.25
gi = 50
gi = 50
gi = 50
gi = 50
…
…
 -order
Guesses
gi = 0
gi = 50
Table 2: Iterative, naïve best reply
pi = ½ , d = 25
gi = 37.5
gi = 31.25
gi = 28.125
gi = 26.5625
…
Number of
iterative,
naïve best
reply
Zero-order
First-order
Second-order
Third-order
…
Now let’s assume that the subject is naive
gi = 25
Slide 13/39
Thus, replicating Güth et al.’s experiment with d = 25 allows us
to shed some light on what drives subjects’ behaviour in a pbeauty contest game.
If experimental results are not significantly different for d = 50
and d = 25, then we validate conclusion (a3) and refute the
iterative, naïve best reply story.
But, if we observe a significant difference between d = 50 and d
= 25, then we invalidate conclusion (a3) and may confirm the
iterative, naïve best reply story.
It is worthy to point out that using p = ½ could somehow
produce a bias in favour of the iterative, naïve best reply since it
is the only p in R for which the unique Nash equilibrium
coincides with d.
Slide 14/39
A generalization of p-beauty contest game with interior equilibria
Let m > 2 be the number of subjects in the game
gi  L, H 
1 n

u  g i   C  c g i  p  g j  d 
 n j 1

n
n


n
i
n
i
max  L, lim Lp   dp   g*  min  H , lim Hp   dp .
i 1
i 1
 n 

 n 

Slide 15/39
If we want to solve this generalized beauty contest game with the iterative naïve
best reply strategy we need to generalize it. As the guessing interval is [L, H], 50
is no longer a focal point. A good alternative candidate may be (H+L)/2.
The equilibrium using the iterative naïve best reply strategy is given by:

H L n n

i
p   dp 
if p  1 g i  max  L, lim
n 
2
i 1




n
H

L

n
i
if p  1 g  min  H , lim
p

dp
.

i

n 
2
i 1



Slide 16/39
The experimental Design
There are n = 32 subjects
Divided in 8 groups of 4 subjects
Subjects have to guess a number in [L, H]
1 n

The target number is: G  p  g j  d 
n

 j 1

1 n

The general form of the pay-off function is: u  g i   C  c g i  p  g j  d 
 n j 1

Where C is a positive (monetary) endowment
c (>0) is a fine subject i has to pay for every unit of deviation between his guess
and the target number
Slide 17/39
Treatment 1: d=25, p=2/3, L=0, H=100;
Treatment 2: d=50, p=2/3, L=0, H=100;
Treatment 3: d=25, p=1/2, L=0, H=100;
There is a natural
reference point: 50
Treatment 4: d=50, p=1/2 ,L=0, H=100;
Treatment 5: d=25, p=23, L=13, H=129;
Treatment 6: d=50, p=2/3, L=13, H=129;
Treatment 7: d=25, p=1/2, L=13, H=129;
Is (13+129)/2 a natural
reference point?
Treatment 8: d=50, p=1/2, L=13, H=129;
Treatment 9: d=50, p=1/4, L=0, H=100;
Treatment 10: d=50, p=1/4, L=13, h=129;
Slide 18/39
In treatment 1 the Nash equilibrium is 50.
If our ‘economic man’ approaches treatment 1 in a fully rational way using the
game theory tools he will find that this the game has a unique Nash equilibrium
in 50.
Working out that 50 is the Nash equilibrium need a long reasoning process.
Because he will first eliminate all the number greater than 83.33; than he will
realize that he can eliminate all the number greater than 72.22; he will repeat
this reasoning process and he will eliminate 64.81, 59.87, and after an infinity
number of iteration he will reach 50 i.e. the Nash equilibrium.
On the other hand if our ‘economic man’ is not fully rational and suffer of
Nagel’s naïveness he will choose 50 as reference point and then he will
immediately find out that 50 is the unique Nash equilibrium.
Slide 19/39
Treatment 1: d = 25, p =2/3, L = 0, H = 100
16
14
12
10
8
6
4
2
0
0
3
5
20
22
25
40
50
55
58
60
75
90
100
choice
Slide 20/39
In treatment 2 the Nash equilibrium is 100.
If in treatment 2 our ‘economic man’ is fully rational he will find that this
treatment has a unique Nash equilibrium in 100.
He will need only one elimination of dominated strategy to find the Nash
equilibrium.
On the other hand in this case if he is not fully rational using Nagel’s heuristic
he will choose 50 as reference point and if he performs one iteration he will
play 66.66; if he performs two iterations he will choose 77.77; if he is able to
perform three iterations his choice will be 85.18; finally if he can perform an
infinity number of iterations then he will reach the Nash equilibrium.
Slide 21/39
Treatment 2: d = 50, p =2/3, L = 0, H = 100
10
9
8
7
6
5
4
3
2
1
0
15.3
40
50
53.5
55
60
65
66
75
80
85
90
100
choice
Slide 22/39
In treatment 3 the Nash equilibrium is 25.
In treatment 3 full rationality requires playing the unique Nash equilibrium, i.e.
25.
Under both solving methods an infinity number of iterations will be needed in
order to detect the Nash equilibrium.
Slide 23/39
Treatment 3: d = 25, p =1/2, L = 0, H = 100
9
8
7
6
5
4
3
2
1
0
10
25
30
31
35
38
40
42
43
44
46
50
56
60
65
68
75
95
100
choice
Slide 24/39
In treatment 4 the Nash equilibrium is 50.
In treatment 4 full rationality requires playing the unique Nash equilibrium, i.e.
50.
If subjects adopted the elimination of dominated strategy an infinity number of
iterations are needed to reach the Nash equilibrium.
On the other hand if our ‘economic man’ is not fully rational and suffer of Nagel’s
naïveness he will choose 50 as reference point and then he will immediately find
out that 50 is the unique Nash equilibrium.
Slide 25/39
Treatment 4: d = 50, p =1/2, L = 0, H = 100
18
16
14
12
10
8
6
4
2
0
0
10
21
25
40
50
55
56
60
68
75
100
choice
Slide 26/39
In treatment 5 the Nash equilibrium is 50.
In treatment 5 in order to reach the unique Nash equilibrium, i.e. 50, an
infinity number of iterations are needed under both the assumptions.
Treatment 5: d = 25, p =2/3, L = 13, H = 129
6
5
4
3
2
1
0
13
15
20
26
40
48
50
52
55
60
63
70
71
73
81
90
99
100
choice
Slide 27/39
If in treatment 6 the unique Nash equilibrium in 100.
Subjects need only one elimination of dominated strategy to find the Nash
equilibrium.
On the other hand in this case if he is not fully rational using Nagel’s heuristic
he will choose 50 as reference point and if he performs one iteration he will
play 66.66; if he performs two iterations he will choose 77.77; if he is able to
perform three iterations his choice will be 85.18; finally if he can perform an
infinity number of iterations then he will reach the Nash equilibrium.
Slide 28/39
Treatment 6: d = 50, p =2/3, L = 13, H = 129
8
7
6
5
4
3
2
1
0
15
20
25
30
35
50
52
55
60
65
66
70
71
75
76
80
95
100
110
129
choice
Slide 29/39
In treatment 7 rationality requires playing the unique Nash equilibrium, i.e.
25.
Under both solving methods an infinity number of iterations will be needed in
order to detect the Nash equilibrium.
Slide 30/39
Treatment 7: d = 25, p =1/2, L = 13, H = 129
3.5
3
2.5
2
1.5
1
0.5
0
16
20
23
25
26
30
33
35
38
40
42
45
50
51
54
60
63
68
71
75
77
80
81
83
100
choice
Slide 31/39
In this treatment the Nash equilibrium is 50. If in treatment 8 full rationality
requires playing the unique Nash equilibrium, i.e. 50.
If subjects adopted the elimination of dominated strategy an infinity number of
iterations are needed to reach the Nash equilibrium.
Using iterated best replay Level-∞ subjects will play 50, i.e. their choices will
coincided with the Nash equilibrium.
Slide 32/39
Treatment 8: d = 50, p =1/2, L = 13, H = 129
6
5
4
3
2
1
0
13
25
25.56 28.5
30
40
45
50
54
56
58
60
64
65
68
70
71
75
80
90
100
120
123
129
choice
Slide 33/39
In this treatment the Nash equilibrium is 50/3.
To reach it a fully rational subject need an infinity number of iterations as well
as an infinity number of reasoning level are needed to a naïve subject to
detect the equilibrium.
Slide 34/39
Treatment 9: d = 50, p =1/4, L = 0, H = 100
6
5
4
3
2
1
0
10
13
15
16
17
18
20
25
27
30
33
33.3
35
36.26
50
60
65
66.5
74.8
75
100
choice
Slide 35/39
In this treatment the Nash equilibrium is 50/3.
To reach it a fully rational subject need an infinity number of iterations as well
as an infinity number of reasoning level are needed to a naïve subject to
detect the equilibrium.
Slide 36/39
Treatment 10: d = 50, p =1/4, L = 13, H = 129
3.5
3
2.5
2
1.5
1
0.5
0
13
15
20
22
25
30
30.3
31
33
40
45
47
48
50
58
58.3 62.5
66
70
71
80
81
82.3 100
121
choice
Slide 37/39
Preliminary conclusion
It seems that generalizing the p-beauty contest the Iterative naïve best reply
story is less convincing
Slide 38/39
Thank You
Slide 39/39