MULTIOBJECTIVE CONSUMER
CHOICE MODELLING BASED ON
DEPENDING-ON-CONTEXT
PREFERENCE
Presenting author: Evgenij Ozhegov
NRU HSE - Perm
Dept. of Applied mathematics and social system modelling
MCDM 2011, 14.06.2011
Outline
• Generalization of the knowledge about recent consumer
choice modelling approaches;
• Consideration of particular models;
• Formulating of the multiobjective problem of consumer
choice;
• Empirical issue of using such model.
2
Structure of consumer choice models
• By the number of consumers models are divided on:
• Individual demand models
• Aggregated demand models
• In terms of time factor:
• Static models
• Dynamic models
• By the number of consumer choice criteria:
• Singlecriterion models
• Multicriteria models
• By the homogeneity of the products:
• Models with homogeneous products
• Models with heterogeneous products
3
Classical rational demand
• 𝑥 = (𝑥1 , … , 𝑥𝑛 ) – set of consumer products, where 𝑥𝑖 –
amount of consumed product i.
• 𝑥 ≻ 𝑥′ means that 𝑥 better (strongly preffered) than 𝑥′
• 𝑥 ≽ 𝑥′ - 𝑥 not worse (weakly preffered) 𝑥′
• 𝑥 ∼ 𝑥′ - 𝑥 equvivalent 𝑥′
• Rational strong preference ≻
coherent, antireflexive,
transitive, asimmetric.
• Weak preference ≽ reflexive, coherent, transitive.
4
Classical rational demand
• The utility function introduced as the function which:
• 𝑥 ≻ 𝑥′ ⟺ 𝑢 𝑥 > 𝑢 𝑥′
• 𝑥 ≽ 𝑥′ ⟺ 𝑢 𝑥 ≥ 𝑢 𝑥′ .
• Consumer choice problem is choosing the most prefered set
for each budget constraint B, defined by the pair < 𝑝, 𝐼 > as
• 𝐵 = {𝑥 ∈ 𝑋 𝑥𝑖 ≥ 0, 𝑝𝑥 ≤ 𝐼} – budget constraint
• 𝑥 ∗ = 𝑎𝑟𝑔 𝑚𝑎𝑥 𝑢(𝑥) – Classical Marshallian demand
𝑥∈𝐵(𝑝,𝐼)
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Why this model cannot be used?
Because of the non-rational preferences! There are:
1.
Problems with equvivalence (lack of knowledge about the
alternatives);
2.
Problems with transitiveness (cup of tea with 1 mg of sugar ∼ cup
of tea with 2 mg of sugar ∼…∼ cup of tea with 20 g of sugar, but
not cup of tea with 1 mg of sugar ∼ cup of tea with 20 g of sugar);
3.
Dependence of the preferences on context (fashion, external
conditions);
4.
Dependence on the question preposition [Canheman, Tversky,
1979];
5.
Dependence of the preferences on the time.
6
Particular models
(2) Using of the generalized utility function [Shafer, 1974];
(5) Using of the discounted utility in the problem of intertemporal choice (Exponential,
hyperbolical discounting) [Shane, Loewenstein, O'Donoghue, 2002];
(3) Models of price setting for new product (innovators - followers) [Bass, 1989];
(3) MAUT (multiattributive utility theory) - dividing the utility of product on attributes [Dyer,
2005];
(3) Using of the preference which is depended on context [Ozhegov, 2010]:
𝑠 = (𝑠1 , … , 𝑠𝑚 ) – external factors and if
∃𝑠, 𝑠 that 𝑥 ≻𝑠 𝑥′ and 𝑥′ ≻𝑠 𝑥 then
𝑥 ≻𝑠 𝑥 ′ ⟺ 𝑢 𝑥, 𝑠 > 𝑢 𝑥 ′ , 𝑠
𝑠
𝑥 ′≻ 𝑥 ⟺ 𝑢 𝑥 ′ , 𝑠 > 𝑢 𝑥, 𝑠
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Multicriteria consumer choice problem (*)
• 𝑥 = (𝑥1 , … , 𝑥𝑛 ) – product set
• 𝑢 = (𝑢1 , … , 𝑢𝑚 ) – utility criteria
• 𝑢𝑗 : 𝑋 ⟶ 𝑅
• 𝐵 = 𝑥 ∈ 𝑋 𝑥𝑖 ≥ 0, 𝑝𝑥 ≤ 𝐼
• 𝑢𝑗 ⟶
max
𝑥∈𝐵 𝑝,𝐼
∀𝑗 = 1. . 𝑚
8
Solution concept using the method of
criteria weighting
∀𝑗 = 1. . 𝑚, ∃𝛼𝑗 ≥ 0,
𝑚
𝑗=1 𝛼𝑗
=1
If 𝛼𝑘 > 𝛼𝑙 than the k-th criterion is more important than the
l-th.
Then the problem (*) will be:
𝑈=
𝑚
𝑗=1 𝛼𝑗 𝑢𝑗
⟶ max
𝑥∈𝐵(𝑝,𝐼)
𝑥 ∗ = 𝑎𝑟𝑔 𝑚𝑎𝑥 𝑈
𝑥∈𝐵(𝑝,𝐼)
9
How to use that model?
• Why in different context we see different behavior (except
for different <p,I>)?
• Because of the different weights of the criteria during the
time!
10
Dynamical case (**)
• 𝑥(𝑡) = (𝑥1 (𝑡), … , 𝑥𝑛 (𝑡)) – product set, 𝑥𝑖 (𝑡) – i-th product
consumption density in the t time moment;
• 𝑢 = (𝑢1 (𝑡), … , 𝑢𝑚 (𝑡)) – utility criteria;
• 𝑢𝑗 : 𝑋[0. . 𝑇] ⟶ 𝑅;
• 𝛼𝑗 𝑠 𝑡
– weight of j-th criterion importance depending on
the context 𝑠 𝑡 = (𝑠1 (𝑡), … , 𝑠𝑘 (𝑡));
•
𝑚
𝑗=1 𝛼𝑗
𝑠 𝑡
= 1;
11
Dynamical case (**)
• I – consumer expenditure for the period;
• [0..T] – time interval;
•𝑈 𝑡 =
𝑚
𝑗=1 𝛼𝑗
𝑠 𝑡 𝑢𝑗 (𝑥(𝑡)) – weighted utility criterion;
𝑇
• 0 𝑈 𝑡 𝑑𝑡 ⟶ max ;
𝑥∈𝐵(𝑝,𝐼)
• 𝐵 𝑝, 𝐼 = {𝑥 𝑡 ∈ 𝑋 𝑡 𝑥𝑖 (𝑡) ≥ 0,
𝑇
0
𝑛
𝑖=1 𝑝𝑖
𝑡 𝑥𝑖 𝑡 𝑑𝑡 ≤ 𝐼} –
budget constraint.
12
Fixing the consumtion effects
• Consumption effect is constantly observed dependence of the utility
of some product on the external factors.
• How can we fix the existence of that effect and its functional form?
• Denote that product as 𝑥1 (𝑡) and 𝑥−1 (𝑡) will be all other goods (a
quantity that characterizes the change in amount of the consumer
basket, with the exception of the investigated product).
• 𝑢𝑗 𝑡 = 𝑓 𝜆𝑗 , 𝑥−1 𝑡 , 𝑥1 𝑡
, 𝑗 = 1. . 𝑚
• 𝜆𝑗 - j-th criterion parameters vector
13
Fixing the consumption effects
• 𝑈 𝑡 =
𝑚
𝑗=1 𝛼𝑗 (𝑡)𝑢𝑗 (𝑡)
𝑚𝑎𝑥 for each t
• 𝑝1 𝑡 𝑥1 𝑡 + 𝑝−1 𝑡 𝑥−1 𝑡 ≤ 𝐼(𝑡)
• 𝑥 𝑡 ≥0
• Denote as 𝛼1 𝑡 = 𝑓(𝜃, 𝑠 𝑡 )/(𝑓 𝜃, 𝑠 𝑡
+
𝑚
𝑗=2 𝑐𝑗 )
the value of
relative importance of the criteria which describes the
investigated effect
• 𝛼𝑗 𝑡 = 𝑐𝑗 /(𝑓 𝜃, 𝑠 𝑡
+
𝑚
𝑗=2 𝑐𝑗 ) ,
𝑗 = 2. . 𝑚
will be the fixed
relative importance of other criteria
14
Empirical issue: Giffen effect to
buckwheat
• The prices for the buckwheat rose from 45 rubles per kg to 140
in summer 2010;
• There was unreasonably high demand for the buckwheat under
the Giffen effect;
• The sample contained 888 observations of the consumers from
25.04.2010 to 13.08.2010 (111 days, 8 random consumers in
each day) in big retail store;
• Known the amount of buckwheat bought, the price for 1 kg,
expenditure and the weighted price level for each product set.
15
Empirical issue: Giffen effect to
buckwheat
• At each time t each consumer performs a consumption of
product set which is close to optimal
• 𝐹 𝑡 =0=
𝜕
𝜕𝑥1 (𝑡)
𝑚
𝑗=1 𝛼𝑗
𝑡 𝑢𝑗 𝑡
=
𝐹(𝜆(𝑚), 𝑐(𝑚), 𝜃, 𝑠 𝑡 , 𝑝1 𝑡 , 𝑝−1 𝑡 , 𝑥1 𝑡 , 𝐼 𝑡 )
𝜕𝐹(𝑡)
•
𝜕𝑥1 (𝑡)
<0
• 𝑥1 (𝑡) ≥ 0
• Where m is the amount of criteria which he takes in account
16
Empirical issue: Giffen effect to
buckwheat
• 𝑢𝑗 𝑡 = 𝑎0𝑗 + 𝑎1𝑗 𝑥1 𝑡 𝑏1𝑗 + 𝑎−1𝑗 𝑥−1 (𝑡)𝑏−1𝑗 ;
• 𝑓 𝜃, 𝑠 𝑡
= 𝜆0 + 𝜆1 𝑝1 𝑡 + 𝜆2 Δ𝑝1 𝑡 ;
• Number of estimated params for each m is 6m+2;
• Estimated by the method of maximum likelihood;
• The best model with the minimum AIK compared with best
model
(𝑓 𝜃, 𝑠 𝑡
without
+
criterion
with
𝛼1 𝑡 = 𝑓(𝜃, 𝑠 𝑡 )/
𝑚
𝑗=2 𝑐𝑗 )
17
11.08.2010
08.08.2010
05.08.2010
02.08.2010
30.07.2010
27.07.2010
24.07.2010
21.07.2010
18.07.2010
15.07.2010
12.07.2010
Importance of 𝑢1 (𝑡)
09.07.2010
06.07.2010
03.07.2010
30.06.2010
27.06.2010
24.06.2010
21.06.2010
18.06.2010
15.06.2010
12.06.2010
09.06.2010
06.06.2010
03.06.2010
31.05.2010
Importance of 𝑢2 (𝑡)
28.05.2010
25.05.2010
22.05.2010
19.05.2010
16.05.2010
13.05.2010
10.05.2010
07.05.2010
04.05.2010
01.05.2010
28.04.2010
25.04.2010
The best model: 2 criteria
Price for buckwheat
1
160
0.9
140
0.8
0.7
120
0.6
100
0.5
80
0.4
60
0.3
0.2
40
0.1
20
0
0
18
-0.5
25.04.2010
28.04.2010
01.05.2010
04.05.2010
07.05.2010
10.05.2010
13.05.2010
16.05.2010
19.05.2010
22.05.2010
25.05.2010
28.05.2010
31.05.2010
03.06.2010
06.06.2010
09.06.2010
12.06.2010
15.06.2010
18.06.2010
21.06.2010
24.06.2010
27.06.2010
30.06.2010
03.07.2010
06.07.2010
09.07.2010
12.07.2010
15.07.2010
18.07.2010
21.07.2010
24.07.2010
27.07.2010
30.07.2010
02.08.2010
05.08.2010
08.08.2010
11.08.2010
The best model: predicting power
Predicted 𝑥1 (𝑡)
𝑥1 (𝑡)
Price for buckwheat
4
160
3.5
140
3
120
2.5
100
2
1.5
80
1
60
0.5
40
0
20
0
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THANK YOU FOR THE ATTENTION!
Your questions
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