Introduction to Kahneman and Tversky’s Prospect Theory
kahneman and tversky prospect theory math example pdf refers to the comprehensive mathematical framework developed by psychologists Daniel Kahneman and Amos Tversky to describe how individuals make decisions under risk. Unlike classical expected utility theory, which assumes rational decision-making, prospect theory accounts for observed human biases and behaviors, such as loss aversion and probability distortion. This theory has profound implications in economics, psychology, and behavioral finance, providing a more accurate depiction of real-world decision processes.
Background and Significance of Prospect Theory
Limitations of Expected Utility Theory
- Assumes individuals are rational actors maximizing expected utility.
- Fails to explain common irrational behaviors like risk-seeking in losses or risk aversion in gains.
- Overlooks psychological biases influencing decision-making.
Development of Prospect Theory
In response to these limitations, Kahneman and Tversky introduced prospect theory in 1979. Their goal was to model how people actually evaluate potential gains and losses, leading to more accurate predictions of decision-making behavior under risk.
Core Components of Prospect Theory
Value Function
The value function in prospect theory is defined on deviations from a reference point (often the current wealth level). It exhibits key features:
- Concave for gains – diminishing sensitivity
- Convex for losses – diminishing sensitivity
- Steeper for losses than for gains – loss aversion
This asymmetry captures the phenomenon that losses feel more painful than equivalent gains feel pleasurable.
Probability Weighting Function
Instead of objective probabilities, individuals transform probabilities through a weighting function:
- Overweight small probabilities
- Underweight large probabilities
This accounts for behaviors like lottery participation and insurance purchase.
Mathematical Formulation of Prospect Theory
Value Function Equation
The value function \( v(x) \) is typically modeled as:
v(x) =
\begin{cases}
x^\alpha, & \text{if } x \geq 0 \\
-\lambda (-x)^\beta, & \text{if } x < 0
\end{cases}
where:
- \( x \) = deviation from reference point (gain or loss)
- \( \alpha, \beta \) = parameters controlling curvature (usually \( 0 < \alpha, \beta \leq 1 \))
- \( \lambda \) = loss aversion coefficient (>1, indicating losses loom larger)
Probability Weighting Function
The probability weighting function \( w(p) \) can be modeled as:
w(p) = \frac{p^\gamma}{(p^\gamma + (1 - p)^\gamma)^{1/\gamma}}
where \( \gamma \) controls the curvature of the weighting function:
- \( \gamma < 1 \) = overweight small probabilities and underweight large ones
- \( \gamma = 1 \) = linear weighting (no distortion)
Example: Calculating a Prospect Theory Value
Setup of the Example
Suppose an individual faces a choice between:
- A sure gain of \$100
- A 50% chance to win \$200 and a 50% chance to win nothing
Using prospect theory, we want to evaluate which option the individual perceives as more valuable, considering the value and probability weighting functions.
Step-by-Step Calculation
- Set parameters (example values):
- \( \alpha = 0.88 \), \( \beta = 0.88 \)
- \( \lambda = 2.25 \) (loss aversion coefficient; not relevant here since no loss, but important in other contexts)
- \( \gamma = 0.61 \) (probability weighting parameter)
- Calculate the subjective value of the sure gain (\$100):
- Calculate the weighted probability for the 50% chance:
- Calculate the subjective value of the lottery:
- For the 50% chance to win \$200:
- Weighted probability:
- Compute the overall subjective value of the lottery:
- Compare to the sure gain:
- Modeling investor behavior and market anomalies
- Understanding how people value potential gains and losses
- Designing better risk communication strategies
- Developing policies that account for human biases
- Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica.
- Kahneman, D. (2011). Thinking, Fast and Slow. Farrar, Straus & Giroux.
- Thaler, R. H., & Sunstein, C. R. (2008). Nudge: Improving Decisions About Health, Wealth, and Happiness.
v(100) = 100^{0.88} ≈ 100^{0.88} ≈ 75.9
w(0.5) = \frac{0.5^{0.61}}{(0.5^{0.61} + (1 - 0.5)^{0.61})} ≈ \frac{0.5^{0.61}}{2 \times 0.5^{0.61}} = 0.5
In this case, with \( \gamma=0.61 \), the weighting is close to the objective probability, but generally, it would differ more.
v(200) = 200^{0.88} ≈ 200^{0.88} ≈ 136.1
w(0.5) ≈ 0.5 (from above)
V_{lottery} = w(0.5) v(200) + (1 - w(0.5)) v(0) = 0.5 136.1 + 0.5 0 = 68.05
v(100) ≈ 75.9
Since 75.9 > 68.05, the individual perceives the sure \$100 as more valuable than the lottery, aligning with risk aversion in gains.
Graphical Representation of Prospect Theory
Value Function Graph
The typical shape of the value function is an S-curve that is steeper for losses than for gains, illustrating loss aversion. It is concave for gains (diminishing returns) and convex for losses.
Probability Weighting Graph
The probability weighting function often shows an inverse S-shape, overweighting small probabilities and underweighting large ones. This explains behaviors like buying lottery tickets (overweighting small chances) or avoiding insurance (underweighting high probabilities of loss).
Applications of Prospect Theory and Mathematical Examples
Behavioral Economics and Finance
Decision-Making in Business and Policy
Conclusion
Kahneman and Tversky’s prospect theory revolutionized our understanding of decision-making under risk by incorporating psychological insights into a mathematical framework. The core components—value function and probability weighting—are expressed through specific equations that can be applied to real-world scenarios. The example provided illustrates how to quantitatively assess choices, revealing why individuals often deviate from classical rational models. Their work continues to influence diverse fields, emphasizing the importance of accounting for human biases in decision-making models.
References and Further Reading
For detailed mathematical derivations and more complex examples, downloading the relevant PDF files on prospect theory math examples can provide additional insights and practice problems, aiding in mastering the application of this influential theory.
Frequently Asked Questions
What is the core idea behind Kahneman and Tversky's Prospect Theory as explained in their math examples?
Kahneman and Tversky's Prospect Theory suggests that people value gains and losses differently, leading to decision-making that deviates from expected utility theory. Their math examples illustrate how individuals overweight small probabilities and exhibit loss aversion, influencing choices under risk.
How does the math example in the PDF illustrate the concept of loss aversion in Prospect Theory?
The math example shows that the subjective value of potential losses weighs more heavily than equivalent gains, demonstrating that people tend to prefer avoiding losses over acquiring similar-sized gains, which is a key aspect of loss aversion modeled mathematically in the theory.
What equations are commonly used in the Prospect Theory PDF examples to represent value and weighting functions?
The main equations include the value function v(x), which is typically concave for gains and convex for losses, often modeled as v(x) = x^α for gains and -λ(-x)^β for losses, and the probability weighting function w(p), which captures how people overweight small probabilities and underweight large ones, often represented as w(p) = p^γ / (p^γ + (1 - p)^γ)^{1/γ}.
Can you explain the significance of the 'value function' in the Prospect Theory math examples from the PDF?
The value function quantifies how individuals perceive gains and losses relative to a reference point. It is typically S-shaped—concave for gains, convex for losses—and steeper for losses, reflecting loss aversion. The math examples demonstrate how this function influences decision weights and choices under risk.
How do the math examples in the PDF demonstrate probability distortion in Prospect Theory?
The examples show how people tend to overweight small probabilities and underweight large probabilities using the probability weighting function w(p). This distortion explains behaviors like gambling or insurance purchasing, where actual probabilities are perceived differently in decision calculations.
What are typical parameter values used in the Prospect Theory math examples to fit experimental data?
Parameters such as α and β (often around 0.88) for the value function, λ (around 2.25) for loss aversion, and γ (around 0.61) for probability weighting are commonly used. These values are derived from empirical data and help the math examples accurately model observed decision behaviors.
Where can I find a comprehensive PDF with Kahneman and Tversky's Prospect Theory math examples?
A highly recommended resource is Kahneman and Tversky’s original papers, such as 'Prospect Theory: An Analysis of Decision under Risk,' which include detailed mathematical examples. Many academic websites and university courses also provide PDFs and lecture notes explaining the math behind Prospect Theory.
