The theory of drift (Binmore and Samuelson 1999) concerns equilibrium selection in which second order disturbances may have first-order effects in the emergence of one equilibrium over the other. We provided experimental evidence with human players supporting the model in Caminati, Innocenti and Ricciuti (2006). In this paper we test it with conditioning by computer players. When computers are removed and humans are matched against each other, the comparative static properties of the model are confirmed.drift, equilibrium selection, evolutionary games, experiments.

Alessandro Innocenti

Mauro Caminati

Roberto Ricciuti

English

Research Papers in Economics

Mauro Caminati 
Alessandro Innocenti 
Roberto Ricciuti 
  
Drift and Equilibrium Selection  
with Human and Computer Players 
12/2007 
 
UNIVERSITY OF SIENA 
WORKING PAPER  LABSI WORKING PAPER  
 
 
 
Drift and Equilibrium Selection with Human  
and Computer Players° 
 
 
 
Mauro Caminati
1,2, Alessandro Innocenti
1,2, Roberto Ricciuti
2,3 
 
 
1Department of Economic Policy, Finance and Development, University of Siena 
 
2LabSi, Experimental Economics Laboratory 
3Department of Government, University of Florence 
 
 
 
 
 
 
 
ABSTRACT 
 
The theory of drift (Binmore and Samuelson 1999) concerns equilibrium selection in which second-
order disturbances may have first-order effects in the emergence of one equilibrium over the other. 
We provided experimental evidence with human players supporting the model in Caminati, 
Innocenti and Ricciuti (2006). In this paper we test it with conditioning by computer players. When 
computers are removed and humans are matched against each other, the comparative static 
properties of the model are confirmed. 
 
 
 
 
Keywords: drift, equilibrium selection, evolutionary games, experiments. 
 
JEL Codes: C72, C92 
 
 
 
 
° We wish to thank Vito Moscato e Nikos Nikiforakis for research assistance, Brian Wallace for writing the 
software and the Strategic Research Fund of Royal Holloway, University of London for financial support. 
Roberto Ricciuti also thanks Icer for hospitality.    2 
 1.  Introduction 
 
A now standard argument used to select between multiple Nash equilibria in non-
cooperative game situations, amounts to introducing perturbations (Selten 1975). In the 
context of evolutionary game theory (Maynard Smith 1982), perturbations may not concern 
the game itself, but only the learning process which brings the players to play equilibrium 
strategies. Convergence outcomes and their speed come to depend on the way in which 
learning is affected by disturbances. Errors may take the form of small stochastic shocks 
(Young, 1993, Kandori, Mailath and Rob 1993, Ellison 1993), or the less innocent form of 
systematic mistakes arising from a pay-off dependent, superficial scrutiny of the game 
influenced by ‘similar’ game situations experienced in life. Specifically, the theory of drift 
(Binmore and Samuelson 1999) suggests that the out-of-equilibrium states that are 
observed in repeated plays of a game reflect the dynamic coupling of different factors: (i) 
initial conditions sustained by prior beliefs about the behaviour of opponents; (ii) 
population ‘learning’ reflecting the frequency increase in each subpopulation of players of 
the strategies that are more profitable, given the available information on the actual play of 
the game; (iii) drift; that is, the expected subpopulation-average ‘mistake’ in a given game 
situation; (iv) residual stochastic perturbations. 
This paper reports on experimental evidence concerning a modified version of the 
Dalek game in which the theory of drift predicts the conditions under which a strategy 
distribution in the population of players is stabilized near a Nash equilibrium component E 
that is not locally asymptotically stable in the learning dynamics (ii). Since stabilization 
requires that drift is compatible with (points to the interior of) E, the theory predicts that 
selection outcomes depend on the measure of E. A problem arising in this context is that 
the evidence for stabilization suggested by the data may not be entirely explained by the 
drift component (iii); it may be also caused by the fact that the same pay-off change 
designed to alter the size of E between experimental sessions, may also affect the initial 
conditions of the game and select them in the basin of attraction of a different equilibrium. 
To avoid the distortion brought by the dependence of initial conditions on pay-offs, 
in our experiment the proper play of the game is preceded by a phase in which each 
subpopulation of players is conditioned to play a specific set of equilibrium strategies 
through a repeated matching with a virtual opponent.  
   3 
  2. Experimental design 
 
  The modified Dalek game proposed by Binmore and Samuelson (1999) is shown in 
the Figure below.  
 
  Player 1 moves first choosing either to Exit or to Play. In the first case the game 
ends with payoffs 7 for player 1 and 5 for player 2. In the second case player 1 makes the 
second choice between Red and Black and player 2 chooses between Left and Right 
without knowing the choice taken by player 1. 
  We considered two different game versions: in the first one, called A, the payoff x 
of the strategy profile (Play-Red, Right) is 0; in the second one, called B, it is 6. Both 
games have the same two subgame perfect equilibria, (Play-Red, Left) and (Exit, Right). 
Binmore and Samuelson predict that when x is small the equilibrium (Exit, Right) will 
emerge, while when x is large (Play-Red, Right) will appear. The given reason is that with 
x = 6 the drift attracts more initial plays in the basin of attraction of (Play-Red, Right) than 
of (Exit, Right). 
  To test this prediction we introduced computers programmed to play two different 
strategies. In treatments denoted as A1 and B1 computers conditioned humans to play 
Red 
Exit 
22
1
1
7 
5 
Black 
Left Left  Right Right 
9 
3 
x 
0 
0 
0 
6 
6 
Play   4 
(Play-Red, Left), while in treatments A2 and B2 to play (Exit, Right). Computers always 
played the predetermined strategy (“full conditioning”) to make the conditioning process as 
effective as possible. We informed subjects that they would start playing against computers 
and that computers would be removed during the experiment in an indefinite period.  
  A bothersome feature of the experimental design was that players 1 could exit and 
stop the game at the first stage. To avoid that players 2 were inactive for all the experiment 
when matched with a computer programmed to Exit we decided to make subjects play both 
roles.
1     
Subjects were drawn from the undergraduate population at Royal Holloway, 
University of London during the academic year 2003/04. We ran five sessions of game A 
and five sessions of game B. Each session lasted 60 periods and included twelve people, six 
being directed to equilibrium (Play-Red, Left) and six to equilibrium (Exit, Right). 
Computers were actually removed after 25 periods. Each session lasted approximately an 
hour) and the average payoff was about £ 9.50. During the sessions the subjects were seated 
at computer terminals in separate seats to prevent communication or visual contact among 
them. The experiment was computerized using the software Z-tree (Fishbacher 1999).
2 
 
 3.  Results 
 
  Following Binmore and Samuelson’s theory, we tested the following two hypotheses: 
a)  in treatments A1 and A2, in which x is equal to 0, and in treatment B1, with x 
equal to 7, subjects continue to play the conditioned strategies even after computers are  
removed; 
  b)  in treatment B2, with x equal to 7, the conditioned strategy (Exit, Right) is 
displaced by (Play-Red, Left) even after computers are removed. 
  Experimental results are shown in Table 1. 
 
 
 
                                                 
1 This methods differs from Roth and Shoumaker (1983) and Binmore et al. (1993) in which subjects play the 
conditioned strategy with a probability lower than 1 (and fixed roles), but in these cases exit was not one of 
the strategies played by the computer. This problem was solved in Caminati et al. (2006) by imposing timing 
without observability, i.e., the two players chose simultaneously but the player 2 was not informed of player 
1’s choice.   
2 Instructions are available upon request.   5 
Table 1 Experimental results (percentage of plays) 
Treatment Players Conditioning Prediction  Plays  Period 1  Period 25  Period 60 
Exit 43.0  9.7  13.3 
Red  49.4 88.9 86.7  Player 1  Red  Red 
Black  7.6 1.4 0.0 
Left  30.8 58.3 63.6 
A1 
Player 2  Left  Left   Right  69.2 41.7 34.3 
Exit  84.0 96.0 70.7 
Red 11.3  2.0  26.7  Player 1  Exit  Exit 
Black  4.7 2.0 2.6 
Left  47.4 25.0 36.4 
A2 
Player 2  Right  Right  Right  52.6 75.0 63.6 
Exit  14.9 4.0 6.9 
Red  85.1 92.0 93.1  Player 1  Red  Red 
Black  0.0 4.0 0.0 
Left  85.1 98.8 98.7 
B1 
Player 2  Left  Left  Right  14.9 1.2 1.3 
Exit 68.0  83.6  3.0 
Red  30.5 16.4 97.0  Player 1  Exit  Red 
Black  1.5 0.0 0.0 
Left  33.2 27.3 98.2 
B2 
Player 2  Right  Left   Right 66.8  73.7  1.8 
 
  By inspection, the first result is that the conditioning process was quite effective in 
bringing humans towards the desired choice. The second result is that in the treatments in 
which we expected the conditioned outcome to be stable (A1, A2, B1), it actually was. 
Finally, in experiment B2 players 1 conditioned to play Exit drastically changed in favour 
of Red when computers were removed and players 2 did actually play Left, as required by 
our hypothesis (b). Figures 1 and 2 show the choices of human players grouped in 12 sets 
of 5 plays each. After computers’ removal indicated by the vertical line, subjects converge 
very quickly on the predicted strategy profile Left-Red. 
  A formal test of the comparative statics implied by our hypotheses can be done by 
applying a 
2 χ  test for independent samples. In this case we modify the previous 
hypotheses in the following way: 
  a*) there is a significant difference in the number of players 1 choosing Exit and 
Play-Red in treatments A1 and A2. There is no significant difference in the number of 
players 1 choosing Exit and Play-Red in experiments B1 and B2. 
 
 
 
   6 
Fig. 1 Choices of players 1 in treatment B2 
0
10
20
30
40
50
60
70
80
90
100
123456789 1 0 1 1 1 2
Groups of 5 periods
P
e
r
c
e
n
t
a
g
e
EXIT
RED
BLACK
 
 
 
Fig. 2 Choices of players 2 in treatment B2 
0
10
20
30
40
50
60
70
80
90
100
123456789 1 0 1 1 1 2
Groups of 5  periods
P
e
r
c
e
n
t
a
g
e
Left
Right
 
 
  b*) there is a significant difference in the number of players 2 choosing Left and 
Right in treatments A1 and A2. There is no significant difference in the number of players 
2 choosing Left and Right in experiments B1 and B2. 
In this case we have two groups (represented by a pair of treatments) and either 
three (for players 1) and two (for players 2) possible choices. The null hypothesis states that 
the two groups do not differ with respect to the relative frequency with which group 
members fall in several categories. To test this hypothesis we count the number of cases   7 
from each group which fall in the various categories, and compare the proportion of cases 
from one group in the various categories with the proportion of cases from the other group. 
The 
2 χ  has degrees of freedom equal df = (r - 1)(k - 1) where r is the number of rows and k 
is the number of columns. The number of columns is always equal to two, while the 
number of rows is equal to three (Exit, Red, and Black) for players 1, and equal to two 
(Left and Right) for players 2. Therefore the numbers of degrees of freedom are two and 
one, respectively.  
When we consider the full conditioning treatment, the statistics concerning the 
distribution of players 1 in treatments A1 and A2 is equal to 52.83. This is greater than the 
critical value associated with 0.001 for a
2 χ with 2 degrees of freedom (13.82). Therefore, 
the null of the test is rejected, in accordance to our hypothesis (a*). The opposite applies 
when comparing players 1 in treatments B1 and B2: we cannot reject the null of no 
difference between the two treatments since the test statistics is equal to 4.45, which is 
smaller than the critical value associated with the 0.1 significance level with 2 degrees of 
freedom (4.60). Again this support hypothesis (a*). 
The same applies when we test hypothesis b*: the test values for the difference 
between players 2 between treatments A1 and A2, on the one hand, and B1 and B2, on the 
other hand, are 1.08 and 1.93 respectively, which are smaller than the critical values of the 
2 χ with 1 degree of freedom associated with significance levels of 0.20 (1.64) and 0.10 
(2.71). Therefore in the first case the null of no difference is rejected, while in the second 
case it is accepted: in both cases our hypothesis (b*) is supported by data. 
 
4. Conclusions 
 
  This paper has tested in the laboratory the theory of drift that concerns equilibrium 
selection in which second-order disturbances may have first-order effects in the emergence 
of one equilibrium over the other. We have matched human players with computer players 
programmed to bring some players to one equilibrium and the remaining players to the 
other equilibrium in order to control for the distribution of initial conditions. Then 
computerised players have been removed and humans matched against each other. Our 
findings are consistent with the theory. When we slightly modify game payoffs so as not to 
change the set of subgame perfect equilibria but as to modify the basin of attraction of 
players’ initial plays, the drift becomes compatible only with one of the two equilibria that   8 
displaces the other equilibria even in presence of the divergent conditioning by computer 
players.  
 
 
 
References 
 
Binmore, K., Samuelson, L. (1999). Evolutionary Drift and Equilibrium Selection, Review 
of Economic Studies, 66, 363-393. 
Binmore, K., Swierzbinski, J., Hsu, S., and Proulx, C. (1993). Focal Points and Bargaining, 
International Journal of Game Theory, 22, 381-409. 
Caminati, M., Innocenti, A., Ricciuti, R. (2006). Drift Effect under Timing without 
Observability: Experimental Evidence, Journal of Economic Behavior and 
Organization, 61, 393-414.  
Ellison, G., (1993). Learning, Local Interaction and Coordination, Econometrica, 61, 1047–
1072. 
Fishbacher, U. (1999). Z-tree Manual, Institute for Empirical Research in Economics, 
University of Zurich. 
Kandori, M., Mailath, G., Rob, R., (1993). Learning, Mutation and Long-run Equilibria in 
Games, Econometrica, 61, 29–56. 
Maynard-Smith, J. (1982). Evolution and the Theory of Games, Cambridge University 
Press, Cambridge. 
Roth, A. E., Schoumaker, F. (1983). Expectations and Reputations in Bargaining: An 
Experimental Study, American Economic Review, 73, 362-372. 
Selten, R. (1975). Reexamination of the Perfectness Concept for Equilibrium Points in 
Extensive Games, International Journal of Game Theory, 4, 25-55. 
Young, P., (1993). The Evolution of Conventions. Econometrica, 61, 57–84. 
 LABSI WORKING PAPERS 
ISSN 1825-8131 (ONLINE VERSION) 1825-8123 (PRINT VERSION) 
 
LABSI EXPERIMENTAL ECONOMICS LABORATORY  
UNIVERSITY OF SIENA 
PIAZZA S. FRANCESCO, 7 53100 SIENA (ITALY) 
http://www.labsi.org    labsi@unisi.it 

Drift Effect under Timing without Observability: Experimental Evidence,

Evolution and the Theory of Games,

Evolutionary Drift and Equilibrium Selection,

Expectations and Reputations in Bargaining: An Experimental Study,

Focal Points and Bargaining,

Reexamination of the Perfectness Concept for Equilibrium Points in Extensive Games,

The Evolution of Conventions.

Drift and Equilibrium Selection with Human and Computer Players

Drift and Equilibrium Selection with Human and Computer Players

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