We submitted three models to the market entry prediction competition 2010. All three models are based on the inertia, sampling and weighting (I-SAW) model which will be explained in Section 2. In Section 3 we describe the four additional assumptions we examined throughout the three models, which we present in Section 4. In Section 5 we discuss the relative effect of each added assumption. Lastly, in Section 6 we summarize the analysis results and the theoretical conclusions.

Both the estimation experiment and the competition experiment are modeled as a series of M = 40 market entry games that are played by artificial agents. A market entry game G_m is characterized by different random values for its five parameters (k, H, pH, L, S). The I-SAW model [1] generates for each market entry game G_m a group of N = 4 agents that play repeatedly for R = 50 trials. Each agent i is characterized by five traits whose values differ between agents and are distributed uniformly with ε_i∼U[0, .24], π_i∼U[0, 6], ω_i∼U[0, .8], ρ_i∼U[0, .2], and µ_i∼U{1, 2, 3}. All agents have the same action space A = {enter, not enter} and each agent i has to choose in each round t ∈ T = {1, … R} an action a_i,t ∊ A without knowing how the other agents will decide.

The decision process of each agent i is divided into three stages: exploration, inertia, and exploitation. Exploration implies to enter the market with probability p^enter = 0.66 or otherwise not to enter. The probability for an agent to explore is given by p i explore = { 1 ε i ∼ U [ 0 , . 24 ] } i f { t = 1 t > 1 }

If an agent does not explore, then she enters the second stage. Inertia implies to repeat the last action a_i,t = a_i,t−1 with probability p i inertia = π i Surprise t − 1 ∈ [ π i ∼ U [ 0 , . 6 ] , 1 ] , with Surprise t − 1 ∈ [ 0 , 1 ]

All agents that have neither entered the exploration stage nor have decided in the inertia stage to repeat their last action, make their decision in the exploitation stage. In this stage each agent chooses the action a_i,t ∊ A with the highest estimated subjective value (ESV).

Given the set of payoffs for all past cases X(a_{i,past case}) = {x(a_i,1), …, x(a_i,t−1)} and the number of sample experiences or sample cases μ_i∼U{1, 2, 3}, the ESV of action a_i,t for an agent i is given by the sum of two terms: the average payoff from all past cases weighted by ω_i∼U[0, .8] and the average payoff from the set of sample cases {sample case¹, … sample case^μ_i} weighted by (1 – ω_i): ESV ( a i , t ) = ω i ∑ k = 1 t − 1 x ( a i , k ) t − 1 + ( 1 − ω i ) ∑ l = 1 μ i x ( a i , sample case l ) μ iwhere the sampling procedure for any sample case l is given by: sample case^l = t − 1 with probability ρ_i∼U[0, .2] and otherwise sample case^l∼U{1, …, t − 1}.

Because we added more than one assumption to the I-SAW model in each of our models, we cannot state the relative effect of each assumption individually. For this reason, we calculated the MSD scores after the competition by adding only one assumption to the I-SAW model (10,000 simulations) and summarized the relative effect of each assumption. The relative effect for the estimation and competition score is depicted in Table 2.

As depicted, each of our additional assumptions improved the estimation score. The first three assumptions (3.1, 3.2, and 3.3) also improved the competition score. Whereas the fourth assumption (3.4), while leading to the largest improvement for the estimation set, impaired the competition score, this clearly indicates over-fitting. Thus, we can conclude that the additional fourth assumption is responsible for the poor predictive performance of our second and third models.

In order to examine whether the very small improvement that resulted from adding the first assumption (3.1) was not obtained by chance, we conducted an additional analysis. One simple prediction of the decreasing exploration assumption is that in problems in which the best reply is relatively stable across trials, best reply behaviors are expected to become more common as time advances. On the other hand, constant exploration rate, as assumed by the original I-SAW model, predicts that in these cases, the frequency of best reply behaviors will remain constant over all trials. Problems 3 and 8 satisfy the relatively stable best reply requirement, since in these problems about 95% of the experiences yielded better payoffs for entering than staying out (obtained greater than forgone payoffs for entering and vice versa for staying out). The following table shows the percentages of best reply behaviors to previous trials for the first 12 trials:

Table 3 shows that the frequency of best reply behaviors increases with increasing numbers of trials, a result that cannot be explained by the original stable exploration assumption of the I-SAW model. Rather, these results can be captured by the assumption that the tendency to explore is higher in the first trials and decreases throughout the trials. Further support to the robustness of the decreasing exploration assumption can be found in the results of the following problem presented by Hochman and Erev (2007) [17]. In an experiment using the clicking paradigm, subjects were asked to choose repeatedly between unlabeled keys on the computer screen. Pressing on one of the keys always resulted in a payoff of eight points and the other always resulted in a payoff of nine. As in the market entry game, after each trial subjects received information about the forgone payoff, in addition to their obtained payoff. The surprising result was that the proportion of choosing the clearly better option increased gradually during the first 10 trials before reaching 90%–100% in later trials (see Figure 4 in [17]). Therefore, it seems that decreasing exploration over time is a robust phenomenon, even when collecting information actively is not needed and counterproductive.

In this paper, we examined four additional assumptions to the I-SAW model [1]. The first assumption implies that the tendency for exploration is higher at the beginning and decreases over time in the exploration stage. Although it improved the predictions only slightly, we showed that this assumption appears to be robust, even beyond market entry games. The second assumption suggests that the last surprising trial needs to be included with higher probability in the sampling of past cases in the exploitation stage. This minor change consistently improved the predictions slightly, and is in line with the von-Restorff-Effect [7-9] as well as with animal research on the disruptive effect of surprising events on memory recall [10]. In the third additional assumption, we proposed that very early trials are excluded with higher probability from the sample of experiences. We suggested that this can be a result of “doubt about experiences in very early trials”, though one can argue that it might result also from memory limitation. It is important to note, that this additional assumption yields a high relative effect in the competition and the estimation set, thus, we believe that future research should address its importance and its underling processes. The fourth assumption implies the revision of a reasonable alternative given an associated very bad experience in the previous trial. However, we did not find evidence to support this assumption; therefore, we concluded that the large improvement of the predictions for the estimated data set was the result of over fitting. We believe that the first three assumptions presented here address robust learning processes and are not only specific for market entry games. Future research is needed to determine the robustness and limitations of the above additional assumptions.