🧠 Behavioral Economics & Choice Architecture

Neoclassical economic theory models agents as hyper-rational, forward-looking Homo economicus with unlimited cognitive capacity. Behavioral economics incorporates empirical psychology into formal economic models to explain systematic, predictable deviations from standard rationality.

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1. 🔍 Bounded Rationality & Heuristics

Herbert Simon introduced bounded rationality: humans face computational limitations, incomplete information, and finite time, relying on cognitive heuristics (rules of thumb) that lead to systematic cognitive biases.

1.1 Canonical Cognitive Biases

Bias / Heuristic	Psychological Mechanism	Market Consequence
Anchoring & Adjustment	Over-reliance on initial numeric anchor, adjusting insufficiently.	High sticker prices inflate perceived discount value.
Availability Heuristic	Over-weighting events easily recalled from memory (recent/vivid).	Spikes in flood insurance purchases only after a disaster.
Representativeness	Judging probability based on similarity to mental stereotypes.	Gambler's Fallacy, extrapolating short stock trends.
Hyperbolic Discounting	Present bias: preference for immediate rewards over future rewards.	Low personal savings rates, procrastination, Gym memberships.

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2. 📉 Kahneman-Tversky Prospect Theory (1979)

Prospect Theory replaces expected utility theory with an empirical model of decision-making under risk.

       Value V(x)
           ▲               Gains (Concave: Risk-Averse)
           │              .-------
           │           .-'
           │         .'
───────────┼───────.'────────────────► Outcomes x ($)
          │    Reference Point (Status Quo)
         / │
        /  │
       /   │
      /    │  Losses (Convex: Risk-Seeking, 2x as Steep!)
     /     │

2.1 Three Foundational Axioms of Prospect Theory

1. Reference Dependence: Well-being is evaluated as gains or losses $\mathrm{\Delta }x=x-x_0$ relative to a subjective reference point $x_0$, rather than absolute wealth levels.
2. Diminishing Sensitivity: The value function $V(x)$ is concave for gains ($V^{″}(x)<0⟹$ risk aversion) and convex for losses ($V^{″}(x)>0⟹$ risk-seeking behavior).
3. Loss Aversion: "Losses loom larger than gains." Losing $\mathrm{\$}100$ produces more psychological pain than the pleasure of winning $\mathrm{\$}100$:

$$\lambda \approx 2.25⟹V(-x)\approx -2.25\cdot V(x)\text{for }x>0$$

2.2 Probability Weighting Function $w(p)$

Humans systematically overweight small probabilities ($p<0.10⟹w(p)>p$, driving lottery tickets and catastrophic insurance demand) and underweight moderate-to-high probabilities.

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3. 🎯 Choice Architecture & Nudge Theory (Thaler & Sunstein)

A Nudge alters choice architecture without forbidding any options or significantly altering financial incentives.

• Default Options (Opt-out vs. Opt-in): Automatic enrollment in retirement 401(k) plans raises participation from $<40\mathrm{\%}$ to $>90\mathrm{\%}$.
• Salience & Framing: Displaying calorie counts prominently or framing green energy as the default option.
• Commitment Devices: Pre-committing future pay raises to savings (Save More Tomorrow program).

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4. 🎯 Olympiad-Level Worked Master Problem

Master Problem: Prospect Theory vs Expected Utility Evaluation

Problem: An investor with reference point $W_0=\mathrm{\$}0$ has a piecewise power value function:

$$V(x)=\{\begin{array}{ll}{x}^{0.8}& x\ge 0\\ -2.25(-x{)}^{0.8}& x<0\end{array}$$

The investor is offered a lottery: $50\mathrm{\%}$ chance to win $\mathrm{\$}1000$ and $50\mathrm{\%}$ chance to lose $\mathrm{\$}X$.

1. Calculate the minimum potential gain required to accept a 50/50 bet with a loss of $\mathrm{\$}500$.
2. Determine whether a standard risk-neutral expected value maximizer would accept this gamble.

Step-by-Step Rigorous Solution:

1. Calculate Subjective Prospect Value $V(\text{Lottery})$:

   $$V(\text{Lottery})=0.5\cdot V(G)+0.5\cdot V(-500)$$$$V(-500)=-2.25(500{)}^{0.8}=-2.25\times 144.22=-324.50$$

2. Set $V(\text{Lottery})\ge 0$ for acceptance:

   $$0.5\cdot G^{0.8}+0.5(-324.50)\ge 0⟹G^{0.8}\ge 324.50$$$$G\ge (324.50{)}^{1/0.8}=(324.50{)}^{1.25}\approx \mathrm{\$}1376.50$$

3. Compare with Standard Expected Value:

   • Risk-neutral agent requires $G\ge \mathrm{\$}500$ (EV $\ge 0$).
   • Prospect theory agent requires $G\ge \mathrm{\$}1376.50$ due to loss aversion ($\lambda =2.25$). Insight: Loss aversion creates strong status quo bias and equity premium anomalies in financial markets.