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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/.)

This paper describes the “Bounded Memory, Inertia, Sampling and Weighting” (BI-SAW) model, which won the

Studying and modeling human learning behavior in the laboratory has always been an important research topic in economics (See Chapter 6 of Camerer [

Erev

The competition organizers ran experiments to generate two sets of experimental data, the estimation game set and the competition game set. Seeing the experimental results of the estimation game set, participants of the competition are asked to predict the behavior of the competition game set when only several given game parameters were known, but not the actual responses of the experimental subjects.

Based on the results of the previous competition, Erev

However, we believe that the “perfect recall” assumption in the exploitation mode is not realistic. Specifically, the exploitation mode says that while sampling from the past, all past periods are considered regardless of the size of gap between now and a certain past. As Simon [

In fact, we find that the participants tend to choose the best response in the recent periods. Specifically, we calculate the percentages of choice in the current period (

Therefore, we introduce the concept of bounded memory into the I-SAW model to avoid this unlimited reminiscence problem. In particular, players will be able to recall exact payoffs only from some near past, though they should be able to have some vague idea or feelings about a general past of a choice. Hence, we set the sample mean to be sampled from some near past rather than sampling from all history, while maintaining the I-SAW assumption that the grand mean is the average of all past experiences. This modification captures the intuition that an elder may know that her birthday has been great all her life, but it is rare for an old lady to tell about what she got on her twelfth birthday, even though her recollection of last year's birthday present may be precise.

We estimate the modified model of Bounded memory, Inertia, Sampling and Weighting (BI-SAW) by grid search. The grid search method seeks the best fit parameter set under a particular chosen criterion and data. The criterion we utilize is exactly the one used in the prediction competition, the average of six normalized mean square deviations. This criterion focuses on the model's prediction ability in terms of game entry rate, efficiency rate and alternation rate. We estimated the BI-SAW model by using experimental data of the estimation game set provided by the competition organizers. For each parameter set of interest we perform ten thousand simulations of the estimation game set. This number was chosen to be large enough to eliminate the effect of randomness inherent in the BI-SAW population model which draws a set of parameters for each individual. We follow Erev

In the prediction competition, the BI-SAW model with a bounded memory limitation of 6 periods predicts data of the competition game set better than the benchmark I-SAW model, as well as other models submitted by 13 groups of contestants.

The remaining of the article is organized as follows: Section 2 describes the BI-SAW model, Section 3 describes the market entry game designed for the prediction competition, Section 4 presents our estimation method, Section 5 presents the results, and Section 6 concludes.

The Bounded Memory, Inertia, Sampling and Weighted (BI-SAW) model is a type of explorative sampler model that features strong inertia, recency effects, surprise triggers change, and a restriction on individual's ability of recalling past payoffs when sampling. To be specific, the BI-SAW model dictates that in every trial each individual enters one of the 3 response modes (Exploration, Exploitation or Inertia) based on randomness and her past experience such as surprise. The probability of choosing a specific action under each mode is then determined by specific predetermined rules that are also based on past experiences and randomness. The idea is that people would enter different mindsets when facing different payoff histories and the same histories will also determine the action people choose in each mindset.

Individual _{i}_{0}. For instance, if the individual faces a binary choice of 0 or 1 and _{0} = (_{0}, 1 − _{0}), she would choose 0 with probability _{0}, and choose 1 with probability (1 − p_{0}). Notice that _{0} is homogeneous across individuals.

If the exploration mode was not chosen, individual _{i}_{i}

To define this surprise, we shall first define the payoffs gap with respect to GrandM_{j}_{j}

The

Based on Gap(

Since _{i}_{i}

If neither the exploration mode nor the inertia mode were chosen, an individual will enter the exploitation mode. To be specific, the probability of entering this mode is 0 in the first trial, and in all other trials is

Under this mode, individual _{j}_{i}_{i}

We summarize our parameters and their interpretations as follow

_{i}

_{0} is the distribution of actions an individual follows when entering the exploration mode, and is the same for all individuals.

_{i}_{i}_{i}

_{i}_{i}_{i}

_{i}_{i}_{i}_{i}

_{i}_{i}_{i}_{i}

_{i}

We have thus defined an individual BI-SAW model where each individual is represented by a parameter vector (_{i}_{0}, _{i}_{i}_{i}_{i}_{0}, but use the average initial entry rate in the data instead. We then generate our outcome prediction by simulating 5000 times and take the average.

The market entry game we study in this paper exhibits environmental and strategic uncertainty in which four players each face a binary choice in each trial: entering a risky market or staying out (a safe prospect). After making their choices, each player will receive feedback including the payoff they just earned and the payoff they would have earned had they chosen the alternative.

The payoff of entering the risky market (_{t}_{t}

The probability that H will be realized in a trial is given by:
_{t}

The payoff of staying out depends on _{t}_{t}_{t}

Notice that the exact payoff structure described above is unknown to players. What they know is: their payoff in each trial depends on “their choices, the state of nature and on the choices of the other participants (such that the more people enter the less is the payoff from entry).” (Erev

Although parameters are the same in a game, _{t}

This market entry game is a stylized representation of a common economic problem: the utility of undertaking a particular activity depends on the environment, and decreases as the number of participants increases. For example, when choosing to go to the amusement park, one's utility depends on not only the weather, but also how many visitors there are. Therefore, both environmental and strategic uncertainty are taken into consideration when making decisions.

The competition organizers randomly draw 40 sets of game parameters (

The prediction competition uses mean normalized mean square deviation (MNMSD) scores as the criterion when comparing predictions of different models. This mean normalized MSD score is the average of mean square deviations (MSD) between experimental data and model prediction in three aspects: entry rate, efficiency level and alternation rate. Therefore, we need three MSDs to calculate the MNMSD score: entry MSD, efficiency MSD, and alternation MSD. Moreover, we divide each game's data into two blocks, block 1 (trials 1-25) and block 2 (trials 26-50), and calculate the three MSDs for each block. To obtain the MNMSD score of a certain model's prediction, we calculate the entry, efficiency and alternation MSD for block 1 and 2, normalize these six numbers to make them comparable, and take the average.

Here, the entry MSD for a certain model (in a given block of a particular game) is the squared difference of the predicted and actual entry rate (frequency of entry divided by the total number of decisions in the block). We then derive the overall entry MSD by taking the average of the entry MSD in every game. The alternation MSD is similarly defined. Thirdly, we calculate the efficiency MSD of each game by dividing total decision gain by total number of decisions, and average across games.

We do not estimate the probability of entering under exploration mode (_{0}). Instead, we set it equals to the average entry rate in all first trials of the estimation game set.

On the other hand, we estimate the population BI-SAW model through grid search to look for the best upper bounds:

We first report the results of our estimation and model fit, and then discuss the significance of the bounded memory assumption.

To obtain the 6 normalized MSD scores, we use the parameters estimated using the estimation game set, and simulate the experimental results of the competition game set for 5000 times. We then take the average of the 5000 simulation results, and normalize them.

Moreover, the BI-SAW model outperforms the I-SAW model in predicting out-of-sample data (the competition game set).

Since MNMSD is an abstract number, we use the following three ways to test the significance of the bounded memory assumption, which is the only difference between the BI-SAW and I-SAW model: outcome simulation on the competition game set, prediction on resampled game sets, and reversing the role of the estimation and competition game sets.

There is randomness in the 6 normalized MSD scores due to outcome simulation, so we have to make sure the BI-SAW model performs better not only because of this randomness. For this purpose, we repeat the simulation procedure described in Section 5.1 for 100 times.

To test whether the BI-SAW model's prediction power is robust to different game sets, we draw 40 games with replacement from the competition game set to form a new game set, and see if our estimated BI-SAW model still predicts well in the new game set. This resampling procedure is justified by the fact that the parameters of each game in both the estimation and competition game sets were also randomly drawn by the organizers of the competition. We repeat the resampling process for 100 times and use both the I-SAW and BI-SAW model to simulate results of these 100 resampled game sets.

Finally, to test whether our estimated BI-SAW model predicts well only in the estimation game set, we reverse the roles of the estimation and competition game sets. That is, we estimate the parameters using the original competition game set, and use it to predict the results of the original estimation game set. In other words, the original competition game set is now viewed as in-sample data, and the original estimation game set is now viewed as out-of-sample data.

In this paper, we propose the “Bounded Memory, Inertia, Sampling and Weighted (BI-SAW)” model in which the subjects' ability of recalling past experience is assumed to be limited. This assumption is crucial when modeling how people make decisions based on their past experience. We test if it improves models' prediction power in a market entry game setting with strategic and environmental uncertainty, in which each player receives feedback regarding earned and forgone payoffs after each decision.

To evaluate the significance of the bounded memory assumption, we verify that the prediction power of the BI-SAW model is consistently stronger than the benchmark I-SAW model by comparing model performance using the mean normalized mean square deviation (MNMSD) criterion in the following three settings. First of all, we repeatedly simulate the outcome of the two models for the competition game set for 100 times to see if the difference between MNMSD scores is significant. Secondly, we use 100 resampled game sets (by repeatedly drawing 40 new games from the competition game set) to check whether the prediction power of BI-SAW model is independent of game sets. Thirdly, we reverse the role of the estimation game set and the competition game set, and perform the same estimation-and-prediction exercise. In all three cases, the BI-SAW model outperforms the I-SAW model, by having lower out-of-sample MNMSD scores. These results confirm the robustness of the BI-SAW model performance. Thus, by incorporating the bounded memory assumption, the BI-SAW model integrates realistic limitations of the human brain into economic modeling, and commands a better ability in predicting subjects' choices.

There are still several open questions to be resolved in future work. The most obvious one is to generalize the BI-SAW model to cope with different information settings. For instance, it would be interesting to see if the BI-SAW model also outperform the I-SAW model in games in which forgone payoffs are unknown (such as those reported in Erev

Another area that deserves further investigation is exploring other possible specifications and extensions of the bounded memory assumption. In particular, we assume that all subjects recall payoffs of the last

Best response percentage.

The distribution of MNMSD scores for the I-SAW and BI-SAW model.

Resampling game set MNMSD scores for the I-SAW and BI-SAW model.

Parameter estimation range.

Parameters | Estimation Range | Precisions |
---|---|---|

[0.1,0.4] | 0.01 | |

[0,1] | 0.1 | |

[0,1] | 0.1 | |

[0,1] | 0.1 | |

[1,5] | 1 | |

[1,25] | 1 |

Estimated parameters and the normalized MSD scores of the I-SAW and BI-SAW model.

Model | I-SAW | BI-SAW | I-SAW | BI-SAW |
---|---|---|---|---|

Estimated with | Estimation game set | Competition game set | ||

| ||||

Entry Rate normalized MSD (block 1) | 1.5443 | 1.2763 | 1.1627 | 1.1510 |

Entry Rate normalized MSD (block 2) | 1.1495 | 1.1500 | 0.8621 | 0.8357 |

Efficiency normalized MSD (block 1) | 1.3106 | 1.0746 | 0.7105 | 0.7352 |

Efficiency normalized MSD (block 2) | 1.4899 | 1.3454 | 0.8218 | 0.8818 |

Alternation normalized MSD (block 1) | 1.4192 | 1.3802 | 0.7130 | 0.7118 |

Alternation normalized MSD (block 2) | 1.2913 | 1.2456 | 0.7437 | 0.8382 |

In-sample MNMSD | 1.3674 | 1.2454 | 0.8356 | 0.8589 |

Prediction on | Competition game set | Estimation game set | ||

| ||||

Entry Rate normalized MSD (block 1) | 1.7353 | 1.4009 | 1.6043 | 1.6133 |

Entry Rate normalized MSD (block 2) | 1.6431 | 1.3385 | 1.6608 | 1.6811 |

Efficiency normalized MSD (block 1) | 0.8878 | 0.7650 | 1.0480 | 1.0334 |

Efficiency normalized MSD (block 2) | 1.1714 | 1.0078 | 2.1256 | 2.1659 |

Alternation normalized MSD (block 1) | 0.7507 | 0.6808 | 1.8640 | 1.7951 |

Alternation normalized MSD (block 2) | 0.8571 | 0.8979 | 1.4367 | 1.3666 |

Out-of-sample MNMSD | 1.1742 | 1.0151 | 1.6232 | 1.6092 |

Estimated Parameters | ||||

| ||||

0.24 | 0.25 | 0.20 | 0.20 | |

0.8 | 0.8 | 0.6 | 0.6 | |

0.2 | 0.8 | 1.0 | 1.0 | |

0.6 | 0.6 | 0.6 | 0.6 | |

3 | 3 | 2 | 2 | |

- | 6 | - | 8 |

We are grateful to Joseph Tao-yi Wang for his encouragement, support and advice. We thank the suggestion of two referees, Yi-Ching Lee, the audience of the 2010 North American Meeting of the Economic Science Association. We acknowledge support from the Taiwan Social Science Experimental Laboratory of National Taiwan University and the NSC of Taiwan (NSC 98-2410-H-002-069-MY2, NSC 99-2410-H-002-060-MY3).

We use the data from the estimation game set.

In order to observe a large range of lags (

The uniformly distributed individual parameters and their implications can be reviewed in Erev

Even if we use a bounded memory limitation of 7 periods, the BI-SAW model still outperforms all other models.

Notice that, as we saw in the inertia mode,

Except for _{0}, which are the same for all individuals.

_{t}_{H}_{H}

See Erev

Reduce this sampling error is important as the MNMSD of competing models differ only by ±0.05 (

If