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We report experimental results on the minority of threegame, where three players choose one of two alternatives and the most rewarding alternative is the one chosen by a single player. This coordination game has many asymmetric equilibria in pure strategies that are nonstrict and payoffasymmetric and a unique symmetric mixed strategy equilibrium in which each player's behavior is based on the toss of a fair coin. This straightforward behavior is predicted by equilibrium selection, impulsebalance equilibrium, and payoffsampling equilibrium. Experimental participants rely on various decision rules, and only a quarter of them perfectly randomize.
Many games have multiple (perfect) equilibria so that the equilibrium concept alone does not answer or explain how the game will be played. One such class of games are market entry games (Selten and Güth [
In this paper, we present experimental evidence on the minority of threegame, which in one important aspect differs from previously studied market entry games: all its asymmetric equilibria in pure strategies are nonstrict and imply earning discrepancies. Thus, the unique symmetric mixed strategy equilibrium appears the natural benchmark to which we will compare observed behavior. According to the unique symmetry invariant equilibrium of the minority of threegame, each player's behavior is based on the toss of a fair coin. Such straightforward behavior is also predicted by alternative solution concepts such as the impulsebalance equilibrium and the payoffsampling equilibrium. Thus, deviations from the mixed strategy hypothesis result most likely from heterogeneous decision rules.
Our experimental setup provides an adequate environment to identify alternative decision rules. First, we endow experimental subjects with a mixing device to directly elicit mixed strategies and to generate i.i.d choice sequences. Indeed, unlike in games with many interacting parties where (population) shares of different strategies may be interpreted as a mixed population strategy, triadic interaction is better studied by directly eliciting individual mixing. Of course, mixing may be due to ambiguous expectations rather than indifference. Controlling for information retrieval could allow to disentangle the hypotheses of ambiguous expectations (to be correlated with more retrievals) and of indifference (previous choices of others render both choices equally good).
Eliciting individual mixing is not new (see, e.g., Ochs [
Second, we implement a
In Section 2 we provide a thorough theoretical analysis of the minority of threegame. Section 3 describes the experimental protocol and Section 4 presents the experimental results. Section 5 discusses previous related research and Section 6 concludes.
In this section, we first introduce the minority of threegame and derive its standard gametheoretical predictions. Second, we theoretically analyze a uniformly and an asymmetrically perturbed version of the game, and we discuss the implications of Harsanyi and Selten's [
Three players have to choose one of two alternatives independently, and the most rewarding alternative is the one chosen by a single player. Hence the two alternatives are perfectly symmetric and players' payoffs are solely based on how players distribute the payoffs between them. Formally, we denote the minority of threegame by
There exist 6 pure strategy equilibria: (
Interestingly enough, the standard gametheoretical predictions remain unchanged if one considers a slightly modified version of the minority of threegame where player
In view of the theories of equilibrium selection, the minority of threegame is quite pathologic since it is one of the rare applications where one encounters a minimal formation containing multiple equilibria (see Harsanyi and Selten [
The minority of threegame
We now demonstrate that this pathology is fundamental in the sense that (i) it is noise persistent; and (ii) asymmetries will not question the solution. The idea of “noise” is to solve the unperturbed game as an idealization. Asymmetry of “noise” appears to be realistic but is rather arbitrary from a normative perspective.
If we neglect that pure strategy equilibria are represented by equilibria in extreme mixed strategies (all freely disposable probability is put on one choice), the same multiplicity of equilibria exists also in the
For all
Instead of assuming uniform trembles, one can consider asymmetric trembles. More precisely, let us introduce a minor asymmetry in the sense that no two players have the same minimal choice probabilities, which we assume to be the same for both their pure strategies. Let
The minority of threegame with the asymmetric trembles
When one tries to select a unique solution for the (
In the asymmetric (trembles) case, the two “pure strategy equilibria” qualify as primitive formations since both equilibria are strict. Since neither of these two solution candidates can (payoff or) risk dominate the other, the theory of Harsanyi and Selten [
Below, we establish that the completely mixed equilibrium is also predicted by alternative solution concepts which, arguably, rely on less stringent assumptions regarding the knowledge and understanding of players.
Contrary to the common approach, which is based on the dynamics of evolution and learning, Osborne and Rubinstein [
In the minority of threegame, if player
As already mentioned, a payoffsampling equilibrium of a symmetric game corresponds to the steady state of a dynamic process involving a large
According to Osborne and Rubinstein's [
For each
Impulsebalance equilibrium is based on a simple principle of ex post rationality. It applies to all games in which players repeatedly decide on one parameter and in which the feedback environment allows conclusions about what would have been the better choice in the last interaction. Ockenfels and Selten [
Note that to incorporate loss aversion, the impulses are calculated with respect to transformed payoffs. Impulsebalance equilibrium weights losses are doubled,
The set of impulsebalance equilibria is identical to the set of Nash equilibria of the minority of threegame: it consists of (1/2,1/2,1/2) and all permutations of (
In addition to the prediction of the stationary concepts described above, we test the predictions of a series of learning models against our experimental data at the individual level. We consider the two varieties of learning which have received the most scrutiny in experiments, belief learning models and reinforcement learning models, as well as learning models which formalize the dynamic processes that might lead to the alternative equilibrium concepts. Partly, the studies not only suggest learning models but also test them.
One widely used model of learning is the process of fictitious play (FP), see Fudenberg and Levine [
As a special case, we also consider the Cournot adjustment model (Cournot), where players choose a pure strategy that is a best response to the belief formed from the immediate past.
An alternative, very elementary type of learning, is reinforcement learning (RL), which has recently become the subject of ongoing experimental research in economics (see, Erev and Roth [
Camerer and Ho's [
The response sensitivity is a parameter which needs to be calibrated before predictions can be compared to individual behavior. We allow the response sensitivity to take any value in the interval [0, ∞].
Impulsebalance learning (IBL) relates to the concepts of impulsebalance equilibrium and learning direction theory (Selten [
Impulsematching learning (IML) is identical to impulsebalance learning except that the difference between the bestreply payoff and the actual payoff is
Payoffsampling learning (PSL) (see originally Chmura, Selten, and Goerg [
The sample size is a parameter which needs to be calibrated before predictions can be compared to individual behavior. We allow the sample size to take any value between 2 and 7. Allowing for wider ranges of sample sizes as in (Selten and Chmura [
The experiment consisted of four sessions, with 27 subjects in each session,
In each round, subjects were asked to give a probability distribution over the two alternatives (
In each round, subjects had the possibility to collect some information about the five previous rounds. Concretely, subjects had access to: (i) their choice and their earnings; (ii) the percentage of their interacting opponents who chose alternative “X” and the percentage of their interacting opponents who chose alternative “Y”; and (iii) the average earnings of their interacting opponents who chose “X” and the average earnings of their interacting opponents who chose “Y”. These informationgathering data will illuminate the behavioral rules subjects used and enable an indirect test of whether they are bestreplying (beliefbased learning).
The four sessions of the computerized experiment were conducted at the Experimental Laboratory of the Max Planck Institute of Economics in Jena, Germany. The experiment was programmed and conducted with the software zTree (Fischbacher [
At the beginning of each session, subjects randomly drew a cubicle number. Once all subjects were seated in their cubicles, instructions were distributed. Cubicles were visually isolated from each other and communication between the subjects was strictly prohibited. Subjects first read the instructions silently and then listened as the monitor read them aloud (the monitor was a native German speaker). Questions were answered privately. A short control questionnaire and two training rounds followed [
In this section, we attempt to characterize the decision rules used by participants. First, we provide some aggregate statistics of our experimental data. At the aggregate level, participants? Behavior seems based on the toss of a fair coin. Second, we investigate individual behavior. Our individualdata analysis strongly indicates the existence of heterogeneous decision rules among participants.
Finally,
To assess the descriptive power of a given learning model
Our analysis of the goodness of fit indicates heterogeneity of decision rules and low predictive power of belief learning models. The latter result is confirmed by the analysis of participants' information retrieval. Participants mainly retrieved information about their own payoff as well as their opponents' payoffs in the previous round. Few information retrievals were made concerning earlier rounds than the previous round. Thus, any attempt to account for the heterogeneity of decision rules by different kinds of information retrieval would have been futile due to no, or very minor, differences in retrieving information.
Though the minority game is related to market entry games, its equilibrium structure is different and favors the completely mixed equilibrium as a natural benchmark. Thus, the numerous evidence gathered by experimental economists on behavior in market entry games [
Few experimental studies have been conducted by economists on the minority game. Chmura and Pitz [
Market entry games are prototypical of coordination problems arising from a newly emergent profit opportunity that can be exploited only by a limited number of individuals. Many experimental studies have been conducted in an effort to find out which type of equilibrium participants are likely to coordinate upon. However, none of these experimental studies has yielded evidence to suggest that participants consistently play equilibrium strategies. Pursuing this line of research, we have conducted an experiment on the minority of threegame.
While this game has multiple asymmetric equilibria in pure strategies that are nonstrict and payoffasymmetric, it also has a unique symmetric mixed strategy equilibrium, in which each player selects the two actions with equal probability. We show that such straightforward behavior is predicted by Harsanyi and Selten's [
Our results indicate that participants rely on various decision rules and that a quarter of them [
In conclusion, heterogeneity of behavioral rules seems a persistent fact of games with many equilibrium outcomes, and this considerable heterogeneity in behavior does not result only from participants' inability to generate random sequences of actions or from repeated game considerations.
Distribution of the changes in
Temporal paths of
Mean deviation scores averaged over all 108 subjects.
The minority of threegame.

Payoffs and mixed strategies at the matching group level.
 

group  Mean  Std. Dev.  Mean  Std. Dev. 
1  0.258  0.438  49.042  31.081 
2  0.256  0.437  49.769  33.050 
3  0.247  0.432  52.900  37.387 
4  0.229  0.421  48.049  32.430 
5  0.242  0.429  47.958  40.439 
6  0.253  0.435  49.573  36.756 
7  0.269  0.444  49.613  41.347 
8  0.251  0.434  48.776  36.404 
9  0.264  0.441  53.271  38.950 
10  0.255  0.437  44.042  37.381 
11  0.260  0.439  47.380  42.629 
12  0.260  0.439  50.198  33.481 
Mean  0.254  0.435  49.214  36.778 
STEWAL  n.s.  
SES  n.s.  n.s.  
IML  n.s.  n.s.  n.s.  
IBL  < 0.01  < 0.05  < 0.05  n.s.  
PSL  < 0.01  < 0.01  < 0.01  < 0.01  < 0.01  
FP  < 0.01  < 0.01  < 0.01  < 0.01  < 0.01  < 0.05  
Cournot  < 0.01  < 0.01  < 0.01  < 0.01  < 0.01  < 0.01  n.s. 
The

Impulses in the direction of the alternative not chosen

We thank two anonymous referees for useful comments and suggestions. Sebastian Goerg, Ming Jiang, and Christoph Göring provided valuable research assistance. We especially thank Anthony Ziegelmeyer and Thomas Pitz for their massive contribution to this paper.
A
If
If 
At least one of the two sets
If
If
For a given
where
The payoff structure of the minority of threegame with the considered asymmetric trembles is given by
As already mentioned, an
Below, we rewrite this equality. First,
Second,
Finally, (1 −
With
and
Clearly, for a given
We now show by induction that for each
Assuming that, for each
Finally, we show that for each
First, for each
Second, we show that the third term of the function also decreases on the interval [0, 1]. Let
To summarize, for each
Impulsebalance equilibrium requires that player
The minority of threegame belongs to the class of winloss games. For the sake of completeness, one can denote by π the monetary payoff associated with the less rewarding alternative,
This payoff structure is the one underlying a threeplayer market entry game with two markets, each market having a unitary capacity, and a payoff function decreasing in a linear way.
One might assume different trembles also for different choices, which appears even more arbitrary, however.
Each player's expected payoff at the symmetric equilibrium equals 1/4 which leads to an expected total payoff of 3/4 for the three players.
Actually, each player needs to know only her own set of actions.
Points were converted to euros in the calculation of subjects' final earnings at a conversion rate of 1 point to 1 euro.
We took subjects through two training rounds to familiarize them with the software, especially the mixed strategy device. During the two trial rounds, subjects were not able to freely choose the composition of the urn. Indeed, in the first trial round the urn had to consist of 99 “X” balls and 1 “Y” ball, whereas in the second trial round the urn had to consist of 1 “X” ball and 99 “Y” balls. Subjects whose questionnaire results indicated that they had not sufficiently understood the rules of the game were replaced and paid 5 euros for answering the questionnaire (35 subjects were invited for each session).
Since the minority of threegame is an exceptional case, this share is hardly comparable to experimental games with unique pure or mixed strategy equilibria.