Slide 1/39

Iterative naïve best reply vs. elimination of dominated strategy in a p-beauty contest

FUR

ROMA, 23/06/2006

FUR

ROMA, 23/06/2006

Andrea Morone

Università degli studi di Baria.morone@gmail.com

Slide 2/39

P-beauty contestThe 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 3/39

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 4/39

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

iig

npG

1

1Is 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 5/39

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 6/39

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 7/39

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 8/39

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 9/39

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 10/39

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 11/39

p-beauty contest game with interior equilibria

dgn

pGn

jj

1

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 12/39

Let’s assume that a subject solves the game using the elimination of dominated strategies.

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.

Eliminated strategies Step

pi = ½ , d = 0 pi = ½ , d = 50 pi = ½ , d = 25 1 gi > 50 gi < 25, gi > 75 gi < 12.5, gi > 62.5 2 gi > 25 gi < 37.5, gi > 62.5 gi < 18.75, gi > 43.75 3 gi > 12.5 gi < 43.75, gi > 56.25 gi < 21.875, gi > 34.375 4 gi > 6.25 gi < 46.875, gi > 53.125 gi < 23.4375, gi > 29.6875

…

…

…

…

gi = 0 gi = 50 gi = 25 Table 1:Repeated elimination of strictly dominated strategies

Slide 13/39

Now let’s assume that the subject is naive

Guesses Number of iterative,

naïve best reply

pi = ½ , d = 0 pi = ½ , d = 50 pi = ½ , d = 25

Zero-order gi = 50 gi = 50 gi = 37.5 First-order gi = 25 gi = 50 gi = 31.25 Second-order gi = 12.5 gi = 50 gi = 28.125 Third-order gi = 6.25 gi = 50 gi = 26.5625

…

…

…

…

-order gi = 0 gi = 50 gi = 25 Table 2: Iterative, naïve best reply

Slide 14/39

Thus, replicating Güth et al.’s experiment with d = 25 allows us to shed some light on what drives subjects’ behaviour in a p-beauty 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 15/39

A generalization of p-beauty contest game with interior equilibria

Let m > 2 be the number of subjects in the game

HLgi ,

dgn

pgcCgun

jjii

1

1

.lim,min*lim,max11

n

i

in

n

n

i

in

ndpHpHgdpLpL

Slide 16/39

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:

.2

lim,min1

2lim,max1

1

1

n

i

in

ni

n

i

in

ni

dppLH

Hgpif

dppLH

Lgpif

Slide 17/39

The experimental Design

There are n = 32 subjects

Divided in 8 groups of 4 subjects

Subjects have to guess a number in [L, H]

The target number is:

The general form of the pay-off function is:

dgn

pgcCgun

jjii

1

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

dgn

pGn

jj

1

1

Slide 18/39

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;

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;

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;

There is a natural reference point: 50

Is (13+129)/2 a natural reference point?

Slide 19/39

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 20/39

Treatment 1: d = 25, p =2/3, L = 0, H = 100

0

2

4

6

8

10

12

14

16

0 3 5 20 22 25 40 50 55 58 60 75 90 100

choice

Slide 21/39

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 22/39

Treatment 2: d = 50, p =2/3, L = 0, H = 100

0

1

2

3

4

5

6

7

8

9

10

15.3 40 50 53.5 55 60 65 66 75 80 85 90 100

choice

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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 24/39

Treatment 3: d = 25, p =1/2, L = 0, H = 100

0

1

2

3

4

5

6

7

8

9

10 25 30 31 35 38 40 42 43 44 46 50 56 60 65 68 75 95 100

choice

Slide 25/39

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.

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Treatment 4: d = 50, p =1/2, L = 0, H = 100

0

2

4

6

8

10

12

14

16

18

0 10 21 25 40 50 55 56 60 68 75 100

choice

Slide 27/39

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

0

1

2

3

4

5

6

13 15 20 26 40 48 50 52 55 60 63 70 71 73 81 90 99 100

choice

Slide 28/39

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 29/39

Treatment 6: d = 50, p =2/3, L = 13, H = 129

0

1

2

3

4

5

6

7

8

15 20 25 30 35 50 52 55 60 65 66 70 71 75 76 80 95 100 110 129

choice

Slide 30/39

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 31/39

Treatment 7: d = 25, p =1/2, L = 13, H = 129

0

0.5

1

1.5

2

2.5

3

3.5

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

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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 33/39

Treatment 8: d = 50, p =1/2, L = 13, H = 129

0

1

2

3

4

5

6

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 34/39

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 35/39

Treatment 9: d = 50, p =1/4, L = 0, H = 100

0

1

2

3

4

5

6

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 36/39

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 37/39

Treatment 10: d = 50, p =1/4, L = 13, H = 129

0

0.5

1

1.5

2

2.5

3

3.5

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 38/39

Preliminary conclusion

It seems that generalizing the p-beauty contest the Iterative naïve best reply story is less convincing

Slide 39/39

Thank You