Treatment Role (ˆλsender,λˆreceiver) Log Likelihood
(1) Pooled Pooled 2.2 -5860.8
(2) Pooled Separate (2.4,1.3) -5753.1
(3) Separate Pooled
N=1: 2.1
-5858.8
N=2: 2.3
N=5: 2.2
(4) Separate Separate
N=1: (3.0,1.3)
-5728.9
N=2: (2.7,1.2)
N=5: (2.2,1.4)
Table 4.1: QRE Estimates of Three Embedded Models
tion.
Null Model (1) (2) (3) Alternative Model
(2) 0.00 – –
(3) 0.13 – –
(4) 0.00 0.00 0.00
Table 4.2: p-Values of Likelihood Ratio Tests for Embedded QRE Models. Model numbers corre- spond to the numbered rows of Table 4.1
Treat. Role (ˆλsend.,ˆλrec.) ( ˆχsend.,χˆrec.) LL
(5) Pooled Pooled 2.5 0.57 -5685.2
(6) Pooled Separate (2.9, 1.3) (0.61, 0.04) -5518.1 (7) Separate Pooled
N=1: 2.1 N=1: 0.06
-5613.4
N=2: 2.6 N=2: 0.36
N=5: 2.8 N=5: 0.96
(8) Separate Separate
N=1: (3.0,1.3) N=1: (0.00,0.04)
-5446.3
N=2: (3.3,1.2) N=2: (0.41,0.07)
N=5: (2.9,1.4) N=5: (0.97,0.00) Table 4.3: Cursed-QRE Estimates of Four Embedded Models
λˆsender and ˆλreceiver in all three treatments, can reject that the data was generated by each of the other three models, at all standard levels of significance.
The final four models allowχandλto vary along parallel dimensions to the previous four models (the first model assumes the data is pooled, the second splits the data by player-roles, etc.). The results of these estimations can be found in Table 4.3, which mirrors Table 4.1 in its presentation.
As with the uncursed models, we tested the fit of the models using likelihood ratio tests. Each of the embedded models reject the hypothesis that its parent model(s) generated the empirical data at all standard levels of significance. The results of the two sets of tests therefore imply that the best model is model 8, the model that specifies separate levels of cursedness and separate level of quantal responsiveness for both roles, for each treatment.
The estimated behavior of models 4 (fully separate QRE) and 8 (fully separate CE-QRE) are graphed in Figures 4.3 through 4.5. In addition, Figures 4.3 and 4.4 display the empirical rates of play and the rational best response. Figure 4.5 displays experimental messaging behavior as box plots that are a function of the sender’s signal. For almost every signal in every treatment, all quartiles were equal to the sincere message. In order to visualize the variance in the data, the box
Average Sent-Message
<100 100 >100 aΦ
Data CE-QRE Data CE-QRE Data CE-QRE Data CE-QRE
N= 1 2.2 2.7 37.5 44.6 99.3 92.0 6.7 5.1
N= 2 1.4 1.7 33.3 32.1 97.2 80.5 8.9 5.1
N= 5 0.9 0.4 14.3 5.9 99.5 88.2 6.5 1.5
Table 4.4: Percentage Receivers Chose the Risky Action, Conditional on Binned Average Message
plots of Figure 4.5 therefore display sextiles (6-quantiles). Consistent with this, we present the QRE and CE-QRE estimates for first and fifth sextile messages as well as for the median message, for each signal.
One of the most striking aspects of the figures and of Table 4.3 is the variance in the maximum likelihood estimate of ˆχ across roles and treatments. Estimates of ˆχ for receivers are all nearly zero. Recall, that a fully cursed receiver plays based on his prior. Therefore, for receivers in every treatment, χ-cursedness amounts simply to a dampening of beliefs towards the receiver’s prior.
Every level of cursedness less than fully cursed will therefore result in the same best response. When combined with QRE, for any given positive value of λ, increasing χ results in the receiver mixing across all feasible actions more evenly at each information set. That is, for receivers, χ and λ have the same effect (but in opposite directions). MLE estimates of ˆχ greater than 0 for receivers therefore represent extremely small gains in likelihood. Note that in Figure 4.3, which displays predicted receiver behavior, we’ve plotted estimates from both fully separate models, and there is no discernible difference in predicted rates of play. Table 4.4 displays the predicted likelihood that receivers chose the risky product as a function of which side of 100 was the average message and as a function of the null message, and shows that the models predict receiver behavior well, on average.
The predicted level of cursedness increases in the number of senders. Senders in the single-sender treatment are completely uncursed. Senders in the multiple-sender treatments, however, appear to be strongly cursed, with ˆχ = 0.41 in the two-sender case, and with senders almost fully cursed in the five-sender case; ˆχ = 0.97. There are no clear trends in the estimations of λ, however each ˆλ must be taken in the context of the estimation of χ. We estimate ˆλto be lower in the one-sender treatment than in the five-senders treatments, however because of the respective estimates of χ,
% Sincere % Exaggeration
Data QRE CE-QRE Random Data QRE CE-QRE Random
N=1 91.6 23.9 23.9 11.1 7.7 35.6 35.6 16.1
N=2 84.4 25.7 25.8 11.1 15.0 30.5 38.8 16.7
N=5 77.1 19.7 21.3 11.1 21.3 26.1 41.0 16.5
Table 4.5: Select Message Frequencies/Likelihoods. QRE and CE-QRE data generated by maximum likelihood parameters estimated separately for each role and each treatment.
the appropriate interpretation is that the single senders are playing slightly more noisily around a rational best response, than multiple senders are playing around a highly cursed, suboptimal response.
Both the QRE and CE-QRE models pick up the theoretically expected and empirically demon- strated comparative static of increasing rates of participation as sender’s signals move away from 100. The cursed model improves upon the fit of the QRE predictions substantially by predicting higher rates of participation at higher values. QRE estimates for the five sender treatment predict the average rate of participation well, but provides a poor fit as a function of players’ types, pre- dicting that players participate more frequently given signals below 100 than they do with signals above.
Recall from the previous chapter that subject behavior was strongly anchored to the sincere messaging strategy, with increasing rates of exaggerated messaging as a function of the number of senders. This trend is reproduced in Table 4.5 along with parallel estimates generated by the QRE and CE-QRE models. As with participation, QRE alone does poorly in predicting messaging behavior. As is evident in Figure 4.5, QRE predicts too high a level of variance in messaging behavior. In particular, it predicts a large amount of dampened messages (in the direction of 100 from their signal), and even predicts messaging on the side of 100 opposite to a sender’s signal.
Both of these predictions are rarely observed in the data. Moreover, as is evident from Table 4.5, QRE significantly under-predicts sincerity, significantly over predicts exaggeration, and predicts a comparative statics in exaggeration across treatments opposite to what is observed in the data.
CE-QRE is not immune to these shortcomings. However it does improve upon them in many ways.
The CE-QRE messaging predictions are in narrower bands around the observed data, and predict
far fewer messages on the side of 100 opposite to a sender’s signal. It predicts slightly higher rates of sincerity and exaggeration, and correctly estimates the comparative static on exaggeration across treatments.