We present an experimental game in the p-beauty framework. Building on the definitions of boundary and interior equilibria, we distinguish between ‘speed of convergence towards the game-theoretic equilibrium' and ‘deviations of the guesses from the game-theoretic equilibrium'. In contrast to earlier findings (GÃ¼th et al., 2002), we show that (i) interior equilibria initially produce smaller deviation of the guesses from the game-theoretic equilibrium compared to boundary equilibria; (ii) interior and boundary equilibria do not differ in the timeframe needed for convergence; (iii) the speed of convergence is higher in the boundary equilibrium.Guessing game, p-beauty contest, individual behaviour

Andrea Morone

Piergiuseppe Morone

English

Research Papers in Economics

     
 
 
  
  
Volume 30, Issue 3 
  
Boundary and interior equilibria: what drives convergence in a ‘beauty 
contest'? 
  
 
 
Andrea Morone  
University of Bari & University of Girona 
Piergiuseppe Morone  
University of Foggia
Abstract 
We present an experimental game in the p-beauty framework. Building on the definitions of boundary and interior 
equilibria, we distinguish between ‘speed of convergence towards the game-theoretic equilibrium' and ‘deviations of 
the guesses from the game-theoretic equilibrium'. In contrast to earlier findings (Güth et al., 2002), we show that (i) 
interior equilibria initially produce smaller deviation of the guesses from the game-theoretic equilibrium compared to 
boundary equilibria; (ii) interior and boundary equilibria do not differ in the timeframe needed for convergence; (iii) the 
speed of convergence is higher in the boundary equilibrium.
 
Citation: Andrea Morone and Piergiuseppe Morone, (2010) ''Boundary and interior equilibria: what drives convergence in a ‘beauty 
contest'?'', Economics Bulletin, Vol. 30 no.3 pp. 2097-2106. 
Submitted: May 04 2010.   Published: August 12, 2010. 
 
       1 
 
 
 
1. Introduction 
The p-beauty contest game is a well known and extremely simple game (Keynes, 1936; 
Nagel, 1995; Duffy and Nagel, 1997; Canerer et al., 1998; Weber, 2003) where n players 
are asked to choose a number from a closed interval [L, H]. The winning player will be the 
one that gets closer to a target number G. Such target number is defined as the average of 
all  guesses  plus  a  constant ( d),  multiplied  by  a  real  number  (p)  known  to  all  players. 
Formally, we have:  . In its simplest form the game parameterisation is 
set as follows:  ,  R is subject i’s guess and d is a constant set equal 
to 0.  
Under such definition of G the game-theoretic solution is a unique Nash equilibrium where 
all players choose 0. In fact, playing 0 is the only strategy that survives the procedure of 
iterated  elimination  of  dominated  strategies  (IEDS).  Moreover,  under  such  a  standard 
parameterisation the game converges to the same value, and within the same number of 
iterations, if players follow Nagel’s iterative naïve best replies (INBR) strategy.
1 
The game becomes more complicated if we set  ; in this case the game might well 
exhibit an interior equilibrium (i.e. different from 0 or 100) and for specific values of p, the 
solution of the game obtained using the two different strategies (IEDS and INBR) involves 
different numbers of iterations needed to reach the equilibrium. 
Güth et al. (2002) proposed a game where d was initially set equal to 0 and subsequently 
equal to 50. This allowed them to analyse the p-beauty contest from a different perspective, 
comparing, among other things, interior and boundary equilibria. They showed that the 
convergence toward the equilibrium is faster when the equilibrium is interior. 
In  this  paper w e   shall  confute  Güth  et  al.  (2002)  finding  showing  that  the  speed  of 
convergence is actually not necessarily faster for interior equilibria. We shall distinguish 
between ‘speed of convergence towards the game-theoretic equilibrium’ and ‘timeframe 
required for convergence’ (i.e. the number of iterations). These two concepts are indeed 
different as they crucially depend on the initial distance of a system from its equilibrium 
level. In the p-beauty contest game, such distance is captured by the initial ‘deviations of 
the guesses from the game-theoretic equilibrium’. We will prove that, for specific game’s 
parameterisations, the speed of convergence can be slower in interior equilibria than in 
boundary equilibria.  
The paper is structured as follows: in section 2 we present the experimental game and pose 
our research hypothesis. In section 3 we present the design of the experiment and in section 
4 our findings. We conclude the paper in section 5. 
 
2. Aim and setting of the experiment 
The aim of our experiment is testing the validity of Güth et al. (2002) finding that interior 
equilibria ( e.g.  d=50)  yield  smaller  deviations  of  the  guesses  from  the  game-theoretic 
                                                 
1 See Morone and Morone (2008) for a description of both strategies.   2 
equilibrium  when  compared  to  boundary  equilibria;
2  this  finding  leads  the  authors  to 
conclude that “swifter convergence to the equilibrium [is found] when the equilibrium is 
interior” (2002: 225). 
Schematically, we can summarise Güth et al. experiment’s parameterisations and equilibria 
(s*), in the following table: 
 
Table I: Güth et al. (2002) summary of parameters and results  
 
As we can see, the authors presented two comparable cases and showed how the treatment 
where the game-theoretic equilibrium is interior, requires less iterations for convergence. 
However, we can observe that for the interior equilibrium case there is the simultaneous 
coincidence, around the game-theoretic equilibrium, of two possible focal points: the d 
value as well as the salient point à la Schelling.
3 Moreover, in the case of d=50, if we 
believe  that  subjects  follow  Nagel’s  iterative  naïve  best  replies  strategy,  they  would 
converge towards the equilibrium after one iteration.  
The picture changes when we look at the boundary equilibrium treatment. Now, the salient 
point à la Schelling and the d value differ, as the salient point is always set in the middle of 
the interval [0,100]. Moreover, both theoretical strategies (i.e. IEDS and INBR) predict an 
infinite number of iteration for complete convergence to the equilibrium.
4 
Departing from these observations we intend to test whether the short timeframe required 
for convergence is an actual property of interior equilibria or whether it is just arising from 
the ad hoc specification chosen by Güth et al. (2002). We do so considering a new set of 
problems’ characterisation defined by different parameterisations of the game. Specifically, 
we shall compare the original parameterisation adopted by Güth et al. with a similar setting 
where we vary the value of p (set equal to 1/3) and the value of d (set equal to 0, 33 and 
50).  It  is  worth  noting  that,  like  in  the  original  experimental  setting,  these  new 
parameterisations produce both interior and boundary equilibria.  
If Güth et al.’s result is robust to different model parameterisations, we will always observe 
a  shorter  timeframe  required  for  convergence  in  the  game  with  interior  equilibrium; 
otherwise, we shall confute the validity of their results for problems’ parameterisations 
different from those originally selected by the authors. 
 
3. The design of the experiment 
In each treatment of the experiment there are n = 32 subjects divided into 8 groups, each of 
4 subjects. In each group subjects have to guess a number in the real interval [L, H]. The 
                                                 
2 Note that a plausible explanation of this finding is that experimental subjects often try to avoid extreme 
choices (see, for instance, Rubinstein et al., 1997). 
3 Nagel (1995) suggested that the salient point à la Schelling would be 50 (i.e. the middle of the interval). 
However, we believe there are other possible focal points; for instance d could be perceived as such by 
experimental subjects. 
4 Please see table A1 in the annex where we report the converging patterns generated by IEDS and INBR.   3 
closer their guess is to the target, the higher is the pay-off. The general form of the pay-off 
function is:  . 
The experiments were run in May 2008 at ESSE (Economia Sperimentale al Sud d’Europa) 
at the University of Bari. The software of the computerised experiment was developed in z-
Tree  (Fischbacher,  1998).  Groups  were  formed  randomly  at  the  beginning  of  the 
experiment and were kept invariant over the whole experiment (i.e. 10 periods). 
 
4. Results 
In this section we will analyse the results obtained in our experiments. However, before 
moving  to  our  new  findings  we  shall  present  the  results  obtained  when  running  the 
experiment using exactly the same parameterisation adopted by Güth et al. (2002). This 
will serve to cast away any doubt on the presence of any source of difference between our 
experimental design and the one adopted by Güth et al.  
In figure 1 we report the average values of treatments 1 and 2;
5 the converging patterns 
obtained in our preliminary set of experiments replicate exactly those obtained by Güth et 
al., as it shows smaller deviations of the guesses from the game-theoretic equilibrium and a 
shorter time frame for convergence in the treatment with boundary equilibrium.
6 In figure 2 
we report two further treatments where p is kept equal to 1/2 and d is set first equal to 33 
and,  subsequently,  to  100.  These  two  treatments  produce  very  similar  results  to  those 
reported in figure 1. Again, the system converges within a smaller number of iterations to 
the interior equilibrium, consistently displaying a smaller deviation of the guesses from the 
game-theoretic equilibrium.
7  
                                                 
5 As mentioned earlier, each treatment was repeated 10 times. Hence, all values reported in figures 1 and 2 are 
averages of 10 rounds.   
6 Deviation from the equilibrium are significantly smaller in treatment 2 than in treatment 1 (p < 0.01 for 
rounds 1-10 and 1-5, p < 0.1 for rounds 6-10). 
7 Deviation from the equilibrium are significantly smaller in treatment 3 than in treatment 4 (p < 0.01 for 
rounds 1-10, 1-5, and 6-10)   4 
   
Figure 1. Treatment averages (p=1/2; d=0 and d=50) 
 
 
 
Figure 2. Treatment averages (p=1/2; d=33 and d=100) 
 
We summarise the parameterisation and the results of these two last treatments in table II. 
As we can see, in this case the boundary equilibrium does not present the simultaneous 
coincidence of the two focal points around the equilibrium value (i.e. d and the salient point 
à la Schelling are different). 
   5 
 
Table II
9: Summary of parameters and results  
 
These first four treatments seem to confirm Güth et al. findings. We shall now move on to 
consider three other treatments where p is now set equal to 1/3 and d is set equal to 0, 33 
and 50, respectively.
8 We report a summary of these new treatments’ parameterisation and 
results in table III. 
 
 
Table III
9: Summary of parameters and results  
 
The picture emerging from these new treatments is rather different from what we obtained 
so far. First and foremost, we do not observe any significant difference in the timeframe of 
convergence  towards  the  equilibrium  across  treatments:  in  fact,  in  all  cases  (i.e.  one 
boundary  and t w o   internal)  it  takes  for  the  system  approximately  the  same  number  of 
rounds (about 6-7) to converge. However, we can easily observe that the initial deviation of 
the  guesses  from  the  game-theoretic  equilibrium  is  smaller  in  the  two  treatments  that 
converge towards interior equilibria. This finding suggests that the conclusions obtained by 
Güth  et  al.  are  only  partially  confirmed.  While  interior  equilibria  treatments  initially 
produce smaller deviation, the speed of convergence is higher in the boundary equilibrium 
treatment.  
 
                                                 
8 Note that we do not consider the treatment with p=1/3 and d=100 as it converges to an interior equilibrium 
(s*=50) and, therefore, is not comparable with p=1/2 and d=100. 
9 Please see table A2 in the annex where we report the converging patterns generated by IEDS and INBR.   6 
 
Figure 3. Treatment averages (p=1/3; d=0, d=33 and d=50) 
 
In fact, as observed above, all three treatments converge towards their theoretical equilibria 
in the same time frame;
10 therefore, the treatment which starts converging from a further 
point (i.e. the one which displays a higher deviation from the game-theoretic equilibrium) 
must converge faster, as it clearly emerges from figure 3.  
Hence, we can conclude that the initial deviation from the game theoretic equilibrium is 
always greater in boundary equilibria, independently from the initial parameterisation of 
the experiment. We share Güth et al.’s view that this finding might depend on the tendency 
to choose interior instead of extreme, boundary strategies. However, along this finding we 
also observe that under the new experiment parameterisation the timeframe required for 
convergence is the same for boundary and interior equilibria and, consequently, the speed 
of convergence is higher in boundary equilibria. In fact, if two runners reach the same 
target in the same timeframe, the one starting from further away must run faster in order to 
cover a larger distance in the same time. 
 
5. Conclusions 
The  experiment  presented  in  this  paper  follows  quite  closely  Güth  et  al.,  2002 a s  w e  
attempt  to  investigate  differences  in  the  speed  of  convergence  towards  boundary  and 
interior equilibria in the p-beauty contest game. In doing so, we distinguish between ‘speed 
of  convergence  towards  the  game-theoretic  equilibrium’  and  ‘timeframe  required  for 
convergence’.  These  two  concepts  crucially  depend  upon  the  initial  ‘deviations  of  the 
guesses from the game-theoretic equilibrium’. In contrast to earlier findings (Güth et al., 
2002), we obtain the following results: (i) interior equilibria treatments initially produce 
smaller deviation compared to boundary equilibria treatments; (ii) interior and boundary 
equilibria treatments do not differ in the timeframe needed for convergence; (iii) the speed 
of convergence is higher in the boundary equilibrium treatment. 
                                                 
10 Comparing treatment 5 and treatment 6, treatment 5 and treatment 7, treatment 6 and treatment 7 we can 
reject,  the  hypothesis  that  deviation  from  the  equilibrium  is  statistically  significantly  smaller  in  interior 
equilibrium treatments.   7 
 
 
References 
Bosch-Domènech A., Montalvo J.G., Nagel R., Satorra A.: (2002). One, Two, (Three), Infinity, ...: 
Newspaper and Lab Beauty-Contest Experiments. The American Economic Review, 92(5):1687-
1701.  
 
Camerer, C., Ho, T. H., and Weigelt, K.: (1998). Iterated dominance and iterated best response in 
experimental “p-beauty contest”. The American Economic Review, 88(4):947-969. 
 
Duffy, J., and Nagel, R.: (1997). On the robusness of behaviour in experimental “beauty contest’ 
games. The Economic Journal, 107(445):1684-1700. 
 
Keynes, J. M. (1936). The general theory of interest, employment and money. London: Macmillan. 
Fischbacher,  U.  (2007),  “Zurich  toolbox  for  readymade  economic  experiments”, 
Experimental Economics, 10, 171–178.  
 
Güth, W., Kocher, M., and Sutter, M.: (2002). Experimental ‘beauty contest’ with homogeneous 
and heterogeneous players and with interior and boundary equilibria. Economic Letters, 74:219-
228. 
 
Morone, A. and Morone, P. (2008). “Guessing game and people behaviours: What can we learn?”. 
In  M.  Abdellaboui  and  J.  D.  Hey ( E d . ) ,  “ A d v a n c e s  i n  D e c i s i o n  M a k i n g  u n d e r  R i s k  a n d  
Uncertainty” Springer. 
 
Nagel, R.: (1995). Unraveling in guessing games: an experimental study. The American Economic 
Review, 85(5):1313-1326. 
 
Rubinstein, A., Tversky, A., Heller, D., 1997. Naive strategies in competitive games. In: Albers, W. 
et al. (Ed.), Understanding Strategic Interaction: Essays in Honor of Reinhard Selten. Springer, 
Heidelberg, pp. 394–402. 
 
Weber, R., A.: (2003). Learning with no feedback in a competitive guessing game. Games and 
Economic Behavior, 44:134-144. 
 
 
 
   8 
Annex 
 
Table A1: IEDS and INBR game theoretical solution for treatments 1 and 2 
   9 
 
Table A2: IEDS and INBR game theoretical solution for treatments 3, 4, 5, 6 and 7 
 
 

Experimental ‘beauty contest’ with homogeneous and heterogeneous players and with interior and boundary equilibria.

Guessing game and people behaviours: What can we learn?”. In

Iterated dominance and iterated best response in experimental “p-beauty contest”.

Learning with no feedback in a competitive guessing game.

Naive strategies in competitive games.

On the robusness of behaviour in experimental “beauty contest’ games.

The general theory of interest, employment and money.

Unraveling in guessing games: an experimental study.

Zurich toolbox for readymade economic experiments”,

Boundary and interior equilibria: what drives convergence in a ‘beauty contest'?

http://www.accessecon.com/Pubs/EB/2010/Volume30/EB-10-V30-I3-P193.pdf

Boundary and interior equilibria: what drives convergence in a ‘beauty contest'?

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