1
Risk Taking with Multiple Thresholds-Based
Payoffs

• Ioanna Tziri
• INSEAD

2
The Problem

• Performance level
• Greater than an upper threshold - promotion
• Btw lower and upper threshold - status-quo
• Below lower threshold - demoted
• What risk should the salesperson employ?

3
Example

• The new CEO of Siemens has introduced a traffic
  lights system to assess their performance based
  on profitability
• Red - poor margins
• Yellow - satisfactory margins
• Green - excellent figures
• FT, 4th July 2005

4
Literature

• Response to only two outcomes
• Gaba and Kalra (Mkt Science,1999)
• Lazear and Rosen (J. Political Econom, 1981)

5
The Model
payoff
P3
P2
P1
performance
T1
T2
6
Model Assumptions

• X represents performance with pdf
  g(x)(1/s)f(x/s), (s as a proxy for risk)
• f symmetric around zero
• f(x)/ f(x) weakly decreasing in x
• f decreasing in x for xgt0

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Analytical Results Optimal Strategies Analytical Results Optimal Strategies Analytical Results Optimal Strategies Analytical Results Optimal Strategies Analytical Results Optimal Strategies
Quotas Payoffs s 0 s Internal maximum s 8 s 0 or 8 Indifferent to risk level
0 lt T1 lt T2 for any M X
T1lt T2 lt 0 for any M X
T2 lt T1 M lt T2/(T2-T1) X
T2 lt T1 M gt T2/(T2-T1) X
T2 T1 0 lt M lt 0.5 X
T2 T1 M 0.5 X
T2 T1 0.5 lt M lt 1 X
T2 gt T1 M gt T2/(T2-T1) X
T2 gt T1 M lt T2/(T2-T1) X
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Intermediate Risk Level
payoff
P3
P2
P1
performance
T1
T2
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Example Normal Distribution

• If T2 lt - T1 and M gt T2 / (T2-T1) then our agent
  can choose an optimal risk level, which is given
  by

10
Visualization of Results for the Intermediate
Risk Level
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Pilot Study Example
Units of Gamma Sold Bonus
10 0
11,, 26 6
27 10
Segment Potential purchase
A 0 40
B 10 30
C 15 25
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Optimal and Real Life Choices
Intermediate Case
OPTIMAL CHOICE A B C
Expected Payoffs 5.76 6.48 6
REAL LIFE CHOICE A B C
Percentages 20 64 16
High Risk Case
OPTIMAL CHOICE A B C
Expected Payoffs 4.19 3.43 2
REAL LIFE CHOICE A B C
Percentages 36 50 14
13
Conclusions

• All situations are described by T1, T2 , M
• Optimal risk strategies are derived for each
  situation
• Intermediate risk level is optimal for
• T2lt-T1 and MgtT2 / (T2-T1)
• All results apply for rank-order contests with
  large number of contestants

14
Further Research

• Conduct further experiments
• Explore parallels with goal theory