To summarize, a strategy can be represented in a summary form as path in a sub problem graph; or in detailed form as a set of rules to trigger an action for every game configuration. With regard to Target the Two, the 422 and 442 strategies can be described as paths in the problem space (Figure 2); any possible path in the graph defines a sequence of sub-goals to be achieved in order to solve the problem. This representation of a strategy does not mention specific moves, but gives the players directions about the goal to be achieved at each step. Corresponding to every path in the graph, which defines a sequence of goals, are many different ways to realize the goals. For example, suppose that Colorkeeper plays a 422 strategy; he must search for his key-card (2§) and put it in Target area. In relation to the distribution of the cards, 2§ may be in many different (covered or clear) positions on the board, and therefore the sequences of actions corresponding to the same goal may differ considerably.

2.5 Coordination and rationality.

So far two competing, alternative strategies, i.e. 422 and 442, have been defined. Both require coordination and entail a division of labor that specifies two roles: one player must exchange his key-card in Target and the partner must end the game. In the 422 strategy these roles are played by Colorkeeper and Numberkeeper, respectively; in the 442 strategy the roles are reversed and they are played by Numberkeeper and Colorkeeper, respectively. With regard to coordination, it is to be stressed that the players are not allowed to communicate before and during the tournament; therefore the emerging division of labor is not the outcome of a verbal agreement, and the only source of coordinative information is the game situations: the position of the cards on the board and the partner's moves. Nevertheless, in some particular situations such information is not sufficient to coordinate the pair; at other times it is ambiguous and generates misunderstanding. For instance, consider the board shown in Figure 1: If Colorkeeper exchanges his card with the Up card, Numberkeeper receives ambiguous information, since he may think that Colorkeeper has taken his key-card, or that Colorkeeper has revealed Numberkeeper's key-card. In the first case they play the 422 strategy, in the second case they play the 442 one. Therefore, if a pair plays in a perfectly rational manner and it uses all available information, it always coordinates its action, but in ambiguous situations. Of course, playing in a perfectly rational way entails a high computational effort. We may therefore expect that , at least in the first stages of a tournament, novice players will try to find simpler strategies of behavior. This they can do by learning and using one only strategy, say the 422 , whatever the initial configuration of the game (II level) . Players therefore can avoid using all available information by adopting a strategy, like 442, which automatically coordinates their actions . Such not-fully rational behavior may paradoxically yield an advantage in terms of coordination because the ambiguity is reduced by reducing the source of relevant information. For instance, a pair of players that invariably adopt the 422 strategy will not perceive any ambiguity in the previous example. Colorkeeper sees his key card in Up and takes it; Numberkeeper looks for 2© and ends the game. This pair may be even more efficient than a rational one in situations in which too information produces ambiguity. Before exploring the problem involved, i.e. the relation between rationality and efficiency at a deeper level, better understanding is required of how repetitive behaviors, and in particular coordination, can be realized via sets of fixed rules of actions.

2.6 Routinized behaviors

Table 1. The two sub optimal sets (442 and 422) of Condition-action rules for Colorkeeper and Numberkeeper.

How can the condition-action representation of the behaviors be connected with the representation in terms of repeated actions? The answer is now quite simple: given the initial configuration of the game, by applying the two sets (one for Colorkeeper and one for Numberkeeper) of condition-action rules in Table 1 we generate theoretical sequences which represent the routines in terms of repeated actions. Table 2 reports some of the configurations previously considered (in Figure 4) to analyze the strategy played during the tournament in Egidi's experiment.

By applying the condition-actions rules reported in Table 1 to the configurations introduced in the left part of Table 2 we obtain the sequences of actions listed on the right part of the figure.

Table 2 illustrates of what was suggested in the introduction, i.e. that there are conditions whereby routinized behavior cannot be detected by looking for repeated sequences of actions: even though players blindly follow the same set of rules (instructions), the sequences of actions over time greatly differ because of the initial random distribution of the cards. Players who strictly adhere to the 422 strategy, for example, play sequences like UUTT, TUPT, TCPNPT, CPTT (Figure 4), which are very different. Therefore, the condition-action representation becomes necessary to detect all regularities in behaviors.

We can now finally give precise definition to "routinized behaviors". Routinized behaviors are characterized by the use of a fixed set of conditional rules automatically triggered by players .

Automaticity means that when subjects identify environmental conditions, they consider them as belonging to a repertoire of familiar conditions. This therefore triggers a familiar reaction without the players having to conduct further mental exploration of the problems involved.

Table 2. A sample of the sequences of moves made over time following the two alternative strategies.

This definition implies that players' actions may be sub optimal, because there is no a priori reason why the action rules should give rise to a fully rational behavior. But nor it there anything to prevent the contrary, i.e. that behaviors exist which are routinized and optimal: in fact suppose that there exists one optimal strategy which can be described by a set of action rules. A player able to learn and memorize all of these rules behave optimally. The problem is that normally the action rule system contains a huge number of rules and players have incomplete knowledge of them.

In our particular context, the complete system of rules with which to play optimally is composed of the two set of rules 442 and 422 and in addition a set of dynamic rules (Egidi, 1994 page 26) which prescribe the choice of the 442 or 422 strategy in relation to the information flowing from the board and from the partner's action. If a player fails to take account of the dynamic rules, his behavior will in consequence be sub optimal. The same happens a fortiori if he learns and uses only one of the available strategies.

Therefore it is clear that there are many different degrees of routinization, In our context we consider particularly relevant - and call strongly routinized - the behavior of subjects that always play the same strategy, say 422 (the rules are given by the Table 1) whatever the starting distribution of cards (II level).

This latter condition implies that players do not take into account information that may serve to improve their performance: new information does not induce these players to discover new rules of action. Therefore their mental activity does not entail a learning process (following March and Simon), or at least that their learning activity is not "organized" enough to allow the discovery of new rules of action even when they are clearly more efficient .

Summing up, the definition of routinization we have provided allows us to discriminate different degrees of routinization among players. We have two extreme situations: on the one hand there are pairs of players who learns to play only one strategy. Something prevents them from learning the alternative strategy available as well. They are locked in the set of rules defined by this strategy and they will use it even when it is not efficient. On the other hand we have fully rational players who know both strategies and use all available information to decide which strategy is the most efficient for every specific condition of the game. Why should we not consider the behaviors of these players routinized? In principle they can be routinized, because it is possible that they automatically execute the set of rules prescribed by the optimal strategy. Of course, using analysis at the behavioral level, it is impossible to discern whether the pairs of players who behave with full rationality are routinized or not. In order to discriminate whether they are learning and calculating their action, or whether they are not, we should ask players to verbalize their thinking, and use psychological tests as protocol analysis. This complementary research goes beyond the limits of our paper. More modestly in the next section we shall identify pairs of players who behave rationally , pairs of players who exhibit deviations from rational behavior, and we will classify these deviations in terms of different degrees of routinization.

2.7 Experimental evidence on routinization.

The data are plotted in Figure 4. For each second level run, the percentage of pairs playing the 442 strategy is reported. Almost all of players reacted by activating the same strategy in some runs, for instance 10, 16, 23, 24, 29 and 37. Therefore is advisable to verify whether there is some relationship between the strategy chosen by the players and the position of particular cards on the board at the beginning of the runs. A natural hypothesis is that players gradually increase their ability to discover their sub goals and therefore grow increasingly able to react to the key-cards which defines the subgoals, i.e. the cards 4©,2§, and 2©.

Figure 4. Percent of pairs that played the 442 strategy to solve the hands of the tournament.

A first step in verification of this assumption is to check the starting configurations of the tournament where one key card only is visible to one of the players, to ascertain whether all players choose the same strategy . Some of the hands are reported in Table 2, which also shows (middle of the figure) the positions of the visible key card on the board at the beginning of any hand. Comparison of Figure 4 with these features of the game configurations (Table 2), shows that the following hands fulfill our requirement: in hands 3, 6 and 28 Colorkeeper has his key-card (2§) in his hand; in hands 4 and 29, Colorkeeper has the double key-card (2©) in his hand. It is clear that in response to this pure information, players behave in a very regular manner: the majority of these choose the 422 path in the former cases, while the majority choose the 442 in the latter cases.

Having observed that the reaction of players to some of the pure configurations is stable, the next step is to explore how players react to all possible combinations of the elementary relevant information, thereby assessing whether they are able to generate a complete system of rules of action.

Consider this new simple rule (A): a player picks up his key card, when it is face up on the board, or uses it when he has the card in hand. If Colorkeeper follows this rule A, he must pick up his key card 2§ - and in consequence he starts to play the 422 strategy - when 2§ is visible on the board; this condition arises in hands 6, 7, 14, 28, 34, 36 (see Figure 4). We should therefore expect these hands to be played with 422 strategy. This hypothesis fits perfectly with the data in Figure 4, with the sole exception of hand n.7. But remember that 442 was played in hands 4 and 29, where Colorkeeper had the double key-card (2©) in his hand. We can therefore add a new rule: if a player has the 2© in his hand he passes, waiting until the partner has been able to find his key card. If we give priority to this latter rule, when also rule A can be applied, we can explain the choice of the strategy for all boards reported on Table 2.

The problem now is to determine whether these rules are "rationally complete", i.e. whether they prescribe choice of the best strategy for every board configuration. To do so we take a more careful look of the comparative efficiency of the two strategies (next section). But even if we limit discussion to the example in section 2.6 it is evident that there are many boards for which the rules are conflicting and many boards to which they are not applicable. The set of rationally complete rules exists, and it is more complex than the set considered above (see Egidi 1994, page 24-26). It can be straightforwardly shown, at least in the experiment in question, that very few players have discovered this set of rules (see section 4). As a consequence, the two rules we experimentally discovered, can be considered as "rules of thumb", which prescribe satisficing but not optimal behaviors.

This finding has some important consequences.

First, the players are strongly limited in their ability to explore the consequences of their move in order to decide which strategy to apply they mainly use the information directly visible, i.e. the cards in their hands or face up on the board. (This information is evidenced in the middle of Table 2). We thus have a first explanation for the fact that the initial conditions of the game trigger one solution or the one alternative to it: the great majority of players use the information available at the beginning of a hand to decide the first move, and this first move is crucial for the triggering of a strategy. They decide on the basis of the simple "rules of thumb" we cited, which are a clear example of boundedly rational behavior in the exploration of the problem.

Second, it is clear that the majority of players in the experiment are not strongly routinized because they are able to decide a different strategy according to the information on the board.

What we found out about routinization by inspecting the global behavior of players across time can be confirmed or disproved by analyzing the behaviors of individual pairs of players over time. Accordingly we now move to a discussion of the behavior of single pairs of players over time and try to establish whether there exists a typology of differently routinized pairs. If we take account of the fact that during first runs all the pairs must learn at least one strategy, and that in any case numerous errors are produced by lack of coordination, we may expect different behaviors to range between the two extremes of purely routinized and perfectly rational behavior.

For any run in a tournament we may expect a pair of fully rational subjects to play the most efficient of the 442 and 422 strategies. We may thus compute the proportions of games in the tournament which can be played more efficiently by using one strategy, say 442. In our case the proportion of times that the 442 strategy proved most efficient is 35%. We may therefore expect to find the same distribution in the data if players are fully rational.

On the other hand, we have players who are locked in a sub-optimal situation, and who are expected to use one strategy only for every initial configuration[5]. Let us now inspect the empirical data (Figure 5): from the small distribution over the tails it is clear that only a few pairs solved the games of the tournament playing only one strategy. The majority of them played both 422 and 442, and with the proportions we predicted, i.e. 35% and 65%. In consequence we have confirmation that the majority of players adopt behaviors which are not fully routinized.

Figure 5. Frequency distribution of 442 strategy.

3. Building cognitive traps

Table 3. Sample of second level configurations showing the strategy efficiency.

[[Delta]] is the difference between the number of moves required by the 442 and 422 strategies to achieve the goal. The key card distributions corresponding to the upper and lower values of delta on Table 3 are antithetical: the two extreme situations occur when Colorkeeper has his key card 2§ in his hand, and Numberkeeper has 2© (422's efficiency is maximum), and likewise when Numberkeeper has his key card 4© in his hand and Colorkeeper 2© (442's efficiency is maximum). This suggests that if players are exposed more frequently to one of these extreme configurations, they will learn and memorize it more easily.

Table 3 shows how many moves are required to solve the game playing the 442 and 422 strategies for some of the 120 possible initial configurations. If we order all the 120 strategies for increasing distance, and complete Table 3, we obtain the result plotted in Figure 6. The abscissa consists of all the different second level starting configurations, codified with integers. The vertical axis denotes the number of moves required to achieve the goal by using the 442 or 422 strategy. The right part of Figure 6 shows the starting configurations which can be solved more efficiently way by using 442. The left part of the figure depicts the opposite relation (422 is more efficient).

Figure 6. Landscape of strategy efficiency. The vertical axis consists of number of moves required by strategies 442 and 422 to solve the games.

Given the initial board of the game, what is the relation between the efficiency of the strategy and the complexity of the problem subspaces to be explored in order to discover the strategy? We assume that there is an inverse correlation between the two terms. In fact a strategy X is considered efficient, for a given starting configuration, if the number of moves required to solve the problem is small: but if, by using a strategy X, a hand can be played with a very small number of moves, it is reasonable to expect that a player exposed to this hand during the learning period should be able to discover it easily.

Vice versa, we expect that the higher the number of alternatives to be evaluated, the harder it will be to build a mental model of the situation and to understand how well this model fits with the actual situation. This assumption is complementary to the hypothesis of bounded rationality made in section 2.5 during discussion of Figure 4.

The working hypothesis of the experiment was therefore that by exposing a group of players to a set of preliminary runs characterized by starting configurations all easily solved by the same strategy, they would be "induced" to discover this solution more easily than the alternative one and to memorize it more deeply.

We selected a set of starting configurations of the board all of which required very few moves to be played with the same strategy 422, while the alternative strategy required a higher number of moves. Analogously we prepared a set of starting configurations which were easy to play using the 442 strategy. A tournament was organized as follows: in the first phase one group of players (called the 422 group) was exposed to the set of configurations that was more efficiently solved playing the 422 strategy. Another group (called the 442 group) was exposed to hands more easily played with the 442 strategy. Both groups were exposed to the same boards in the second part of the tournament, which consisted of 27 hands with randomly distributed cards. By comparing how the two groups exposed to the different training sets played the second part of the tournament, we can evaluate whether their behaviors were significantly different and therefore check if path-dependency occurred.[6]

4. Path dependent behaviors emerging from experimental data.

Figure 7. Frequency distribution of the 442 strategy, played in the first 15 runs of the tournament.

On the other hand, in order to judge whether or not rule-based behavior is a product of a path-dependent process of learning, analysis must focus on the last 27 games. In terms of the number of moves needed by the two groups to solve the 27 games, there is no difference: the average number of moves is around 165. In other words, there is no difference in the efficiency of the two groups. Nevertheless, the quality of the solutions played is clearly different as we shall see. Table 4 and Figure 8 report the percentage of pairs in each group that solved the games by playing a 422 strategy. The path-dependency effect is extraordinary strong. Analysis to assess the different use of the 422 and 442 strategies in the two groups can be made by using statistical tests. The Mann-Whitney-U test calculated for the two independent experimental groups is statistically significant (U =21,5 ; Z value -6,3375 , p < 0.0001 two tail).

As they were trained on two different sets of games, the subjects use more frequently the strategy that they learned first. Some pairs are able to discover and play the other strategy as well, but the difference between the two groups remains clear. In each group there is a large number of players who play the first learned strategy even when it is less efficient. This behavior implies that subjects stop to explore and discover the sub-goaling space and that they cope with new situations introduced in the second part of the tournament simply by using the strategies and the rules learned at the beginning.

Table 4. Percent of pairs within each group that played the two strategies

in the last 27 runs of the tournament

The path dependency effect is so strong that the reactions of players to almost all the starting configurations in the second part of the tournament are the opposite across groups: for example, when 2© is in Up position, and Colorkeeper has his key card 2§ in hand (see hands 17 and 32 in Table 4), almost all the players in the 422 group use the 422 strategy , while in the 442 group the majority of pairs play the other strategy.

In order to discriminate more sharply among different behaviors, it is convenient to check how many pairs behaved in a strongly routinized way and how many discovered the other strategy as they played the last 27 games. Figure 9 depicts this variety and provides a way to measure the strength of the routinized behavior in the last 27 games[7].

Figure 8. Percent of pairs within each group that played 442 strategy in the last 27 runs of the tournament.

An interesting feature of the distribution is that, between the two ext a system of condition-action rules (see Table 1) enabling the players to solve the game. The same system of conditions was used to classify the moves actually performed by the sixty pairs of subjects to solve the last 27 hands and to form clusters of pairs, based on Ward's method. Table 5 shows the pairs grouped into five clusters (rescaled distance of agglomeration = 2.6)[8]. The first cluster (the 422 routinized) consists of 12 pairs; they all belong to the 422 experimental group and they routinizedly play the 422 strategy. The second cluster (the 442 routinized) is made by 15 pairs all belonging to the 442 experimental group. Throughout the tournament these pairs learn only one strategy and play according to one single system of condition-action rules; they are identified in the tails of Figure 9 and their moves closely resemble the actions of routinized behavior reported in Table 1.

Cluster                Cas  Av.      Std    Av.mov  Std    
                       es   422%     Dev    es      Dev    
422 routinized         12   92       7.4    6.1     0.4    
behaviors                                                  
442 routinized         15   14       12.2   6.0     0.4    
behaviors                                                  
rational behaviors     11   42       15.8   6.2     0.7    
helper behaviors       12   60       8.6    6.6     1,4    
not well coordinated   8    24       20.2   6.1     0.3    
behaviors                                                  

During the second part of the tournament, the other pairs of subjects are able to explore the problem space further and to learn how to play a different strategy. As a consequence, the discovery of a new strategy dramatically raises a coordination problem, and the way pairs deal with it makes the difference among the next three clusters. The third cluster (rational behaviors) aggregates the pairs playing both strategies in a coordinate way. These subjects are able to change roles according to the strategy played and to use the information flowing from partner's action to coordinate their actions. Many pairs trying to use the two strategies find it extremely difficult to coordinate, in particular when the partner' actions can be interpreted in many different ways (see section 2.5 above): information is sometime ambiguous and players have great difficulties to decode it.

The fourth cluster (not well-coordinated behaviors) aggregates pairs in which only one subject discovered that there is another and different strategy with which to solve the games. Analysis of the condition-action rules of these pairs shows that one player is strongly routinized while his partner sometimes attempts to play the other strategy as well.

Finally, the fifth cluster (helper behaviors) consists of players behaving in a particular way: when one has the partner's key-card in his hand, he first offers the partner the card by playing Up, and only afterwards he looks for the complementary card (2) he needs. Even though this behavior is rather inefficient it may be the best solution when a player realizes that his partner has not understood how the game works.

Finally, a striking difference can be observed also between the present and the previous experiment. The difference can be evidenced by comparing the distribution in the tails (Figure 9 and Figure 5). In the new experiment (Figure 9) a larger number of pairs behaved in a routinized way and solved the games by using only one strategy. This difference can be ascribed to the path dependent character of the learning process .

5. Final remarks.

The experiment shows the onset of persistent differentiation in players' behavior. The group of players exposed to a set of configurations which led more easily to one strategy continued to use it more frequently in the second part of the tournament, and symmetric behavior arose in the other group. Moreover, in both groups there emerged a sub set of players with strongly routinized behaviors, i.e. groups of player which, after the training phase, adopted one strategy once and for all, and insisted on using it even when hands could not be efficiently played with the strategy adopted.

We have experimental evidence that these routinized players were locked into a sub optimal strategy insofar as they used the same set of rules of action even when they was inefficient, being unable or unwilling to find alternative rules of action. Furthermore a variety of different types of behavior, all of them imperfectly rational and routinized, have been revealed via application of cluster analysis methods.

These findings give rise to new problems and suggest new directions for future research, of which we shall try to draw up a short list before concluding .

First: methodological issues related to tacit knowledge. One of the most relevant features of our results is that they are independent of the verbal explanations that players may provide of their actions. We simply interpret the "logic of action" of the players by observing behaviors at a fine grain level: the elementary micro behaviors (actions corresponding to different board conditions) of the subject are observed , retrieved and compared with the environmental conditions. Therefore we can assume that behaviors are the consequence of mental models and more stringently that they are the outcome of the execution of a set of condition-action rules. This assumption is useful if we wish to extend our ability to explore the features of routinized behaviors. We would stress, however, that the differentiation we have shown is at the level of micro-behaviors. Therefore the results we have achieved (sub-optimality, path dependency, asymmetry in routinization) remain valid whatever model of thinking we assume.

This does not rule out an explanation of these behaviors which relies on a model of individual rationality and thought. We noted before that it is possible to use the experimental tests of cognitive psychology as complementary tools in verification of many aspects of the routinization process. We made some preliminary exploration in this direction by submitting players to a sequence of tests after the tournament which required them to verbalize their decisions in relation to some board configurations of Transform the Target.

The preliminary findings seem to indicate that even when following a fixed set of behavioral rules, players neither have a clear symbolic representation of these rules, nor store the complete set of them in long term memory. It seems to be confirmed that players are only partially aware of the set of rules they apply, and that their knowledge of the "logic of the game" is incomplete.

Second: are fully rational behaviors more efficient that routinized one? The previous results suggest that attention should be paid to the relationship between rationality and efficiency. We have shown that routinized behaviors imply that players' decisional links are disentangled: if they follow the same strategy for all runs, they must search for two complementary key-cards without interacting directly. The two goals can be achieved independently, and coordination is embodied in the strategy adopted via complementarity of the sub-goals. By contrast, fully rational behavior requires a dynamic coordination between the two players.

Therefore if we measure the efficiency of a pair by counting the number of moves they make to play the game, we find pairs playing with fully rational style but who are inefficient, because their errors are amplified by the dynamic rules of coordination and their performances are seriously impaired. Hence players who try to adopt the rational strategy , but are not fully familiar with it may be less efficient than purely routinized players .

On the contrary, expert players who are familiar with the two sub optimal strategies and use it without errors should play more efficiently than strongly routinized players. But this is not always true.

In fact, on the one hand, an expert player greatly reduces the computational effort required to play rationally by using the two sets of alternative rules of action (442 and 422) as routinized building blocks in his mental exploration of the problem. But, on the other hand , not all game configurations can be played efficiently by experts: as we have shown in section 2.4 , there are game configurations in which the actions generated by one player cannot clearly be interpreted by the partner; information in these situations is ambiguous, and routinized behavior is more efficient precisely because it does not use ambiguous information and therefore reduces errors.

Summing up, there are situations in which behaving in a routinized way is more efficient than using all available information. This happens when players cannot decode all available information, or to use a different expression, when players cannot the reduce Competence Gap (Heiner 1983).[9]

Third: knowledge incompleteness and persistency of differentiation. The above considerations suggest that in real organizations micro-learning activity is the fundamental force at work, insofar as it either enables actions to be stabilized into routinized behaviors or gives rise to a search for new alternative routines. Situations in which an individual's activity is fully routinized, i.e. when all possible contingencies are covered with memorized actions rules, are extreme cases, while in general memorization and routinization are partial and incomplete, and the learning process allows subjects to repair and complete the gaps in memory. This feature of human learning shows that different degrees of routinization can exist in human behaviors. In our experiments we observed that many players did not merely execute a set of condition - action rules. Their activity is supported by a persistent micro-learning activity. This activity enabled some players to escape a lock-in and move toward exploration of a new set of rules. It is intuitive that the more solid the sets of cooperative rules that the two players establish, the lower will be the incentive to escape the lock-in. In fact, a set of well consolidated rules can solve every situation, and even though there are configurations that can be solved more efficiently, by using the alternative strategy, the effort of jointly discovering the alternative strategy may be higher that the "price" of using a well known strategy under unfavorable circumstances. We suggest that this element, the "sunk costs" of the search process and of the accumulated knowledge, may explain how it happens that path dependency gives rise to persistent differentiation in mental models and behavioral rules (North and Denzau 1994).

The elements which prevent or activate the learning process are therefore keys in understanding more profoundly what characterizes the path dependent creation of different "mental models", and their persistence over time.

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