3.4 EWA Learning Model
3.4.1 The Model
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3.4. EWA LEARNING MODEL
the level zero (d = 0) are grouped into “d < 0”. Similarly, the subjects with guesses smaller than the lower limit of the level 3 (d = 3) are grouped into
“d > 3”. The subjects, finally, are categorized into 6 classes: d < 0, d = 0, 1, 2, 3 and d > 3. In addition, the median reasoning level for each subject, denoted by di, is obtained by simply transforming these 6 ordinal classes to 6 numbers, from 0 to 5, and calculating the median of each subject’s 10 corresponding levels.
3.4 EWA Learning Model
3.4.1 The Model
We will apply different versions of EWA learning model to structurally char-acterize the learning behaviors in repeated BCG.
Camerer’s EWA Camerer and Ho (1999) provide the first attempt to calibrate the EWA parameters with the data in repeated BCG. They pro-posed some necessary details of design, in addition to the general structure described in Section 2.4.2, in order to implement the model, in particular, in the context of BCG. The strategy space for the subjects is in the interval [0, 100] with only integers allowed. These 101 strategies should be endowed with some initial attractions Aj(0) and being reinforced over time. If we treat all Aj(0) as the model parameters, we have to calibrate all of these 101 attrac-tions. By simply assuming that initial attractions are equal in ten-number intervals [0, 9], [10, 19], ... [90, 100], we can alleviate the problem of using too many degrees of freedom. Therefore, the first design detail is that 101 initial attractions will be reduced to only 10, denoted by A1(0), A2(0), ..., A10(0). We will call it as Assumption A1 later on. Second, we also assume that subjects know the winning number w = arg mingi[|gi− τ |] (Assumption A2) and they also neglect the effect of their own guesses on the target num-ber (Assumption A3). In fact, A2 is not true in both our and Camerer and Ho(1999)’s designs of experiment, because we only show the target number to the subjects and the winning number is unknown, unless they happen to be the winner of that particular round. Denote the distance between the winning number and the target number as d = |τ − w|, and also denote the prize for each round as nπ where n is the number of winners in a particular round. By the assumption A2 and A3, we can easily define the reinforcement intervals without introducing additional parameters. In particular, all sub-jects reinforce numbers in the intervals (τ − d, τ + d) by δnπ. The winners reinforce what they chose, which is one of the boundary number, either τ − d
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or τ + d, by π, and reinforce the other boundary number by δnπ. The losers reinforce both boundary numbers τ − d and τ + d by (δnπ)/(n + 1).
The assumptions A1 and A2 were later removed and replaced inCamerer, Ho and Chong (2002). First, instead of treating initial attractionsI Aj(0) as parameters and simultaneously calibrating them with other EWA parame-ters, Camerer, Ho and Chong (2002) empirically obtained Aj(0) from the choices data in first period. Formally, we recover initial attractions by the following equations
eλ·Aj(0) P10
k=1eλ·Ak(0) = fj, j = 1, ..., 10.
where fj represent the observed probability of strategy j in the first period.
We have 11 unknown variables, including 10 Aj(0) plus a λ, but only 10 conditions. The number of parameters are too many to be identified. There-fore, we fix the strategy j with the lowest fj to have Aj(0) = 0. Following Camerer, Ho and Chong (2002), we also remove the unrealistic assumption of A2 and introduce an additional parameter d to describe how losers com-pute forgone payoffs. We assume they reinforce numbers in the interval [τ −δnπd , τ +δnπd ]. The amount of reinforcement is δnπ at the target number, which is the maximum. Departing from the target number, the amount of reinforcement decrease at a rate of d, which is a parameter to be estimated.
The forgone payoff for the losers to be reinforced will be of a triangular form(see the panel (b) of Figure 3.2). Camerer, Ho and Chong (2002) also assign this parameter d to the winners and assume that they reinforce the numbers in the interval [τ − e, τ − e −δnπd ] and [τ + e, τ + e +δnπd ] with similar triangle form of reinforcement amount, if there is only one winner. Note that we didn’t incorporate the parameter d to the winners for the reason of simplicity.
We estimated both versions of EWA model described above.
EWA rule learning In both versions of EWA model in Camerer and Ho (1999) and Camerer, Ho and Chong (2002), subjects are assumed to be able to initialize 101 strategies attractions, maintain all of them in memory and then update over time. Considering the limited capacity of memory and attention in human brain, it is unlikely for subjects to learn in the way as Camerer’s EWA. One plausible approximation is to assume that subjects apply only few rules during the experiments. By the descriptive power of level-k model, revealed in repeated BCG (Duffy and Nagel, 1997), we as-sume that subjects apply and reinforce level-k rules instead of 101 numbers.
According to the level classification in Section 3.3, the strategy space
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3.4. EWA LEARNING MODEL
Figure 3.2: Reinforcement and the corresponding interval for winner and loser in beauty contest experiments
comes sj ∈ {d < 0, d = 0, d = 1, d = 2, d = 3, d > 3}. In this setting, subjects are only required to initialize, maintain and update 6 attractions Aji(t), where j ∈ {1, 2, 3, 4, 5, 6}. We define the payoff function as follows,
πi(sji, s−i(t)) =
( 100 if sji = target-d, 0 if sji 6= target-d.
where target-d denotes the level in which the target number located.4 Besides target-d, all other levels will not be reinforced. The amount of reinforcement for target-d will be 100 if it is chosen and will be 100δ if it is not chosen.
4Remember that, in our definition, a level is an interval including several numbers.
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3.4. EWA LEARNING MODEL
Table 3.1: Summary of Estimated Learning Models Parameters Strategy N(0), φ, ρ, δ,
λ
Aj(0) d
Model I Number Estimated Estimated
Model II Number Estimated Initialized
Model III Number Estimated Initialized Estimated Model IV Level-k rule Estimated Estimated
Model V Level-k rule Estimated Initialized
Model I is referred to Camerer and Ho (1999). Model II and Model III are referred to Camerer, Ho and Chong(2002)
Summary of estimated learning models Table3.1summarize the mod-els which were estimated in this study. Model I, II, III tried to replicate EWA learning proposed by Camerer and Ho (1999) and Camerer, Ho and Chong (2002). The common feature of these Camerer’s EWA learning is that sub-jects try to learn the attractiveness of all guessing numbers. On the contrary, Model IV and V assumed that subjects try to learn the attractiveness of level-k rules. We also tried some perturbations and modifications, including that initializing initial attractions Aj(0) by observed probability of strategy in the first period and introducing additional parameter d to replace the unrealistic assumption about the knowledge of winning number.