Chapter 3 Emotion Index Model
3.1 Proposer
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立 政 治 大 學
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Chapter 3 Emotion Index Model
In this chapter, we will explain the detail about how to calculate and modify subjects’
emotion into index, and will justify the parameter setting. Basically, the concept of design how to calculate emotion between proposer and responder is quite similar. The basic structure of our model lists below, we would address more details in the first section (proposer section).
Proposers
nay-based + reference-based (bi-variant model)
Responders
nay-based (uni-variant model)
nay-based + reference-based (bi-variant model)
3.1 Proposer
As discussed in the introduction section, we consider two kinds of emotions which can be possibly triggered by the interactions between the opponents in the ultimatum game. In our assumption, the possible emotion-behavior models are composed of these two kinds of emotions. For the proposer, the first part of emotions-behavior model is that we consider the direct response, accept or reject, from responder. This nay-based emotion is based on the assumption that the acceptance decision from responder will lead to a
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positive feeling for proposer, whereas the rejection decision will lead to a negative feeling.
Thus, according to above description, we can define the reactions to events as a function
𝑋𝑋𝑡𝑡= �𝑋𝑋𝑡𝑡−1+ 𝛼𝛼𝑥𝑥+, 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑋𝑋𝑡𝑡−1− 𝛼𝛼𝑥𝑥−, 𝑖𝑖𝑖𝑖 𝑟𝑟𝑎𝑎𝑟𝑟𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎
( 1 )
Notice that, from Equation (1), emotions can accumulate over the courses of the game. 𝑋𝑋𝑡𝑡
is what we call the emotion index value from nay-based emotion at time t, 𝛼𝛼𝑥𝑥+ and 𝛼𝛼𝑥𝑥− represent the strength of the feeling when the proposal is accepted or rejected at one time, but this one-time stimulus will be added up over time. Because ultimatum game is so simple, although the time is not limit, subjects usually could finish a ten-period ultimatum game experiment in 10 minutes. We assume the nay-based stimulus would not total decade in such short time. This setting, for example, allows us to anticipate the case where the proposer may stay calm when a “fair” offer is rejected for the first time, but may feel irritated when it is rejected again, and may respond “angrily” when it is rejected for the third time.
The second part of our emotions-behavior model is the reference-based emotion.
This kind of emotions is based on the reference point that the proposer considers fair or reasonable in his mind. Despite so, the responder may actually accept/reject something different from the reference, be it reluctantly or aggressively. For example, the proposer may want to offer a share of 45 as his reference; nonetheless, being facing a “tough”
responder (Slembeck, 1999), he actually offered 50 reluctantly. Alternatively, he might actually offer only 35 to take an advantage of a soft responder. We consider two different setting in this reference-emotion index, equation ( 2 ) is a way to quantify the discrete
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type of emotions. A discrete type is that we consider reference as a threshold to determine whether the triggered emotion is positive or negative, similar to equation ( 1 ),
𝑌𝑌𝑡𝑡= �𝑌𝑌𝑡𝑡−1+ 𝛼𝛼𝑦𝑦+, 𝑂𝑂𝑡𝑡<R 𝑌𝑌𝑡𝑡𝑡𝑡1, 𝑂𝑂𝑡𝑡=R
𝑌𝑌𝑡𝑡−1− 𝛼𝛼𝑦𝑦−, 𝑂𝑂𝑡𝑡>R ( 2 )
𝑌𝑌𝑡𝑡 represents the emotion index value from reference-based emotion at time t, 𝛼𝛼𝑦𝑦+
and 𝛼𝛼𝑦𝑦− are the adjust power when proposer compares his/her offer (𝑂𝑂𝑡𝑡) and to his/her personal reference point (R). From Equation ( 2 ), emotions can also be accumulated over the courses of the game.
On the another hand, if the difference between offer and reference is matter, then one needs to take ( 3 ) into account
𝐷𝐷𝑡𝑡 = |𝑂𝑂𝑡𝑡− 𝑅𝑅| ( 3 )
in the emotion calculation. Here, we consider the possible significance of a relative
difference (
𝑎𝑎𝑡𝑡), i.e.,where F is the full range of offer. In next step, we modify Equation ( 2 ) as follows:
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relatively small when the discrepancy is small, and becomes increasingly larger when the discrepancy gets larger. Like the discrete type emotion, the continuous type emotion is also accumulative. Hence, a responder may be tolerant of an “inferior” offer (𝑂𝑂𝑡𝑡< R) for one time, for two times, but may suddenly lose the patience and reject the offer when coming to the third time.Actually, we had test different setting on 𝑎𝑎𝑡𝑡, not only square it, but proportion, square root and just random number (Appendix A). After checking the performance under different preferences for proposers, we find that we could usually get better results when using the “square” method. In such case, we decide to use square setting to operate our parameters in the following analysis.
Since we do not have a theory to tell us the ideal values of these emotional parameters and the reference (R), in the following analysis we shall simply assume a possible range for these parameters and then try various random combinations of them to see whether the significance of emotional influence on the offering decision can be found in any of these values. In other words, given a specific range, we are trying to examine whether there is a sub-domain in which the emotional influence can be identified. The range used in this research is given in Table 2.
Hence, in the later Monte Carlo Simulation, in each run, say run j, we shall randomly pick
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a quadruple 𝛼𝛼𝑗𝑗 ≡ �𝛼𝛼𝑥𝑥,𝑗𝑗+ , 𝛼𝛼𝑥𝑥,𝑗𝑗− , 𝛼𝛼𝑦𝑦,𝑗𝑗+ , 𝛼𝛼𝑦𝑦,𝑗𝑗− � from their given range (Table 2), place them into Equations ( 1 ) and ( 2 ), or ( 1 ) and ( 5 ), and use these two equations to trace the dynamics of emotions during the game. To demonstrate how the emotion index looks like, Figure 2 gives an illustration of the reference-based emotion index (Equation 2 ) of five subjects in their money experiments (the left panel), and that of another five subjects in their chocolate experiments (the right panel), with only subject 005 been chosen in both demonstrations. What shown in Figure 2 is based on one randomly generated parameters 𝛼𝛼𝑗𝑗 under R = 45 (for the money experiment) and R = 4 (for the chocolate experiment).
From these small samples, we can see that quite different patterns of emotions were developing during the game; some were up and down alternately, and some were persistently getting higher or lower.
Table 2 Range for Emotional Parameters and the Reference
Given that the applied numeracies differ in the money experiment and the chocolate experiment, the possible numbers used for reference differ accordingly. Hence, the notations R 𝑀𝑀 and R𝐶𝐶 are used to differentiate the references under the money experiments and the references under the chocolate experiment.
Parameter Range 𝜶𝜶𝒙𝒙+ [0,5]
𝜶𝜶𝒙𝒙− [0,5]
𝜶𝜶𝒚𝒚+ [0,5]
𝜶𝜶𝒚𝒚− [0,5]
𝑹𝑹𝑴𝑴 [0,10,15,20,25,30, … ,50,55,60,70, … ,100]
𝑹𝑹𝑪𝑪 [0,1,2,3, … ,10]
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Figure 2 Reference-Based Emotion Index, Equation ( 2 )
If emotion has an effect on decision of the proposer, we shall expect that influence can be observed through the change of the offer. Hence, we consider the following three possibilities: the offer is increased, decreased or remained the same. Equation ( 6 ) shows the three decision values.
𝑍𝑍𝑡𝑡 = �1, 𝑂𝑂𝑡𝑡> 𝑂𝑂𝑡𝑡−1
0, 𝑂𝑂𝑡𝑡= 𝑂𝑂𝑡𝑡−1
−1, 𝑂𝑂𝑡𝑡< 𝑂𝑂𝑡𝑡−1
( 6 )
To model the influence of emotions on the decision of the proposer, we first combine the two factors into one using a linear sum, as shown in Equation ( 7 ).
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𝑍𝑍𝑡𝑡∗ = 𝛽𝛽0+ 𝛽𝛽1𝑋𝑋𝑡𝑡+ 𝛽𝛽2𝑌𝑌𝑡𝑡+ 𝜖𝜖𝑡𝑡 ( 7 )
𝑍𝑍𝑡𝑡∗ can be broadly interpreted as the state of the mood. Notice that we also add a standard error term 𝜖𝜖𝑡𝑡 as an aggregation of other less systematic influences. We then use the threshold based decision rule ( 8 ) as the emotional model of decision making.
𝑍𝑍𝑡𝑡 = �1, 𝑍𝑍𝑡𝑡∗ ≥ 𝜃𝜃1 0, 𝜃𝜃1 > 𝑍𝑍𝑡𝑡∗ > 𝜃𝜃2
−1, 𝜃𝜃2 ≥ 𝑍𝑍𝑡𝑡∗
( 8 )
Where θ1 and θ2 are the two thresholds on which the decisions are based and these two parameters are endogenous given. Equations ( 6 ) to ( 8 ) lead to the familiar ordered logit model if we assume the cumulative function of 𝜖𝜖𝑡𝑡 is the logistic distribution, i.e.,
𝑃𝑃𝑟𝑟𝑃𝑃𝑃𝑃(𝜖𝜖𝑡𝑡≤ 𝑥𝑥) = 𝐹𝐹(𝑥𝑥) = 1
1 + 𝑎𝑎−𝑥𝑥 ( 9 )
By rearranging Equation ( 8 ) using Equation( 7 ) , we can derive the probability of each decision conditional on the given emotion indexes
X and
tY , as shown in
t Equation ( 10 ) and from there we can further derive the odds ratios, as shown in Equation ( 11 ).23
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Taking the logarithm of the odds ratio ( 11 ), one can have the following linear version of the threshold-based decision model, also known as logit ( 12 ).
1 0 1 2