Choice under Uncertainty (Lecture 1c)

.title[
# Choice under Uncertainty (Lecture 1c)
]
.subtitle[
## EC404; Spring 2024
]
.author[
### Prof. Ben Bushong
]
.date[
### Last updated January 23, 2024
]

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# Today

###  **Our Outline:**

(1) [Guiding Principles](#Guide)

(2) [Introduction to Reference Dependence](#section1)

(3) [A New Model: Prospect Theory](#section2)

(4) [Loss Aversion](#section3)

(5) [Diminishing Sensitivity](#section4)

(6) [The Value Function](#section5)

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# Guiding Principle

## \#2: Use Extreme Cases to Clarify Your Thinking

Some concepts are hard to get your head around. It can be easier to think about things in the extreme limit.

- E.g., "risk aversion" will continue to trip you up throughout this section.
- Think about the limit case: a person who is **infinitely** risk averse.
- Now think about the other limit case: a person who is risk neutral.
- In between those lies reality. The **definition** of the term only offers that a person is a teensy tiny itty bitty bit above risk neutrality. How she actually behaves depends on *how* risk averse she is.

**Your task:** think about limit cases.

---

**Moe:** " If you want to signal me, use this bird call."

*[Moe whistles like a bird. An eagle swoops down and pecks him on the face.]*

`$\quad \quad$` "Ow! Not the face!"

*[The eagle switches to pecking Moe in the groin.]*

`$\quad \quad$` "Ooh! Ooh! Okay, the face!

*[The eagle switches back.]*

`$\quad \quad$` "Ooh! Whoa, that actually feels good after the crotch!"

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# Reference-Dependent Feelings

### A Simple Truth

In virtually all physiological and psychological reactions, people's responses tend to reflect adaptation, change, and contrast, rather than solely absolute levels of outcomes.

- Feelings (and, just as importantly, choice) are reference-dependent.

- This suggests a modifictation to the models that we use.

> We should consider a modified utility function `$u(x;r)$` rather than `$u(w+x),$` where `$r$` is some reference point or reference level.

We'll explore this idea for the next few lectures. There are deep implications for economics in this simple observation.

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# Prospect Theory

![](graphics/kahnemantversky.jpg)
Amos Tversky and Daniel Kahneman worked on this in the 1970s. Kahneman won the Nobel Prize in Economics in 2002 for their joint work.

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# Prospect Theory

Propect Theory proposes two phases of choice process:

1. Editing

2. Evaluation

We begin our discussion with the former, but our focus today will be on the latter.

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# Prospect Theory: Editing Stage

## Editing Stage

The psychology of the editing stage is straightforward: a person needs to organize & reformulate some complex situation into a simplified problem.

- More concretely: a choice problem is described to you, and then you transform it into the lotteries that you will evaluate.

### Some Examples:

- **Coding:** code outcomes as gains (or losses) relative to some reference point.
- **Cancellation:** discard shared components.
- **Simplification:** rounding off probabilities.
- Eliminating dominated alternatives.

This is an example of the type of thing we won't spend a lot of time on in this course, but it was important to early pioneers.

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# Prospect Theory: Evaluation Stage

### Evaluation in a Nutshell

Of course, once we face a decision problem we must evaluate it.

Kahneman and Tversky (1979, p. 277) stress that attending to changes from reference points is a basic aspect of human nature:

> Our perceptual apparatus is attuned to the evaluation of changes or differences rather than to the evaluation of absolute magnitudes ... The same principle applies to non-sensory attributes such as health, prestige, and wealth.

The two key features of evaluation emphasized by Kahneman and Tversky (1979) and subsequently by many others:

1. Loss Aversion
2. Diminishing Sensitivity

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# Prospect Theory: Loss Aversion

### Loss Aversion: A "Definition"

(Note: this is one of the few times where you can just rely on your intuitive response to the terms. It means exactly what you think it means.)

**People dislike losses more than they like same-sized gains.**

- Vast majority of people turn down 50/50 lose $500, gain $550 bet

- As highlighted last time, this is *not* due to curvature in utility function.

- Not discussed last time: the strongest such aversion appears to involve mixes of gains and losses.

---

# Prospect Theory: Loss Aversion

Loss aversion is an absolutely central component of prospect theory.

It's existence or importance remains a source of scholarly debate.

> Note that I say scholarly. I suspect the average person would immediately agree with the assertion that losses > gains.

### My View of Loss Aversion

It is central in a number of everyday activities:

- Moral considerations (e.g., Hippocratic Oath)
- "Endowment Effect" or "Status Quo Bias" in financial trades
- "Disposition Effects", in investments and houses
- Aversion to (nominal) wage and consumption declines
- Income-targeting

---

# Prospect Theory: Diminishing Sensitivity

### Diminishing Sensitivity: A "Definition"

In the following pairs, which "feel" like a bigger difference?

| Option A | Option B |
|:---------|---------:|
| visually 101 ft. away vs. 100 ft. away|  1 ft. v. 0 ft. |
| carrying a suitcase 21 v. 20 blocks | 2  v. 1 block |
| gain 100 days from now v. 101 days | gain 0 days v. 1 day |
| 19% chance v. 18% chance | 1% chance v. 0% |
| gaining $101 v. gaining $100 | gaining $1 v. gaining $0 |
| losing $101 v. losing $100 | losing $1 v. losing $0 |
| losing $101 v. losing $100 | losing $2 v. losing $1 |

---

### Diminishing Sensitivity: A Better "Definition"

People pay less attention to incremental differences when changes are further away from the reference point.

- Prefer $420 for sure or 50/50 chance at $900?

- Prefer losing $420 for sure or 50/50 chance to lose $900?

Reflects big and general fact about human psychology:

**We most often think in terms of proportions rather than absolutes.**

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# Prospect Theory: Evaluation Stage

A person evaluates a prospect `$\left( x,p;y,q\right)$` according to:

`$$V\left( x,p;y,q\right) \quad =\quad \pi \left( p\right) v\left( x\right)+ \pi \left( q\right) v\left( y\right)$$`

Reminder: EU theory says evaluate according to:

`$$U\left( x,p;y,q\right) =pu\left( w+x\right) +qu\left( w+y\right) +\left(1-p-q\right) u\left( w\right)$$`

### What's new?

`$\pi\left(\cdot\right)$` is the **probability-weighting function**.

`$v\left( \cdot \right)$` is the **value function**.

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# Prospect Theory: Evaluation Stage

Put in a different notation, a person evaluates a prospect `$\left( x_1,p_1;\dots; x_n,p_n\right)$` according to:

`$$V\left( x_1,p_1;\dots; x_n,p_n\right) \quad =\sum_{i=1}^N \pi \left( p_i\right) v\left( x_i\right).$$`

Contrast this with the Expected Utility definition

`$$EU\left( x_1,p_1;\dots; x_n,p_n\right) = \sum_{i=1}^N p_i u\left(w + x_i\right)$$`

and that of Expected Value

`$$EV\left( x_1,p_1;\dots; x_n,p_n\right) = \sum_{i=1}^N p_i  x_i$$`

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# Prospect Theory: Value Function

### Breaking Down the Components of the Theory

Three key features of the value function `$v\left( \cdot \right)$`:

(1) The carriers of value are **changes** in wealth. Thus: `$v\left( 0\right) =0$`.

--
- Thus: `$v\left( 0\right) =0$`.
- Implicit in this assumption is that the reference point is current wealth.
- There are lots of examples where this is a bad assumption.

(2) *Diminishing sensitivity* to the magnitude of changes.

- Formally: `$v'(x)>0$` for all `$x$`, and `$v^{\prime\prime}\left(x\right)<0$` for `$x>0$`, while `$v^{\prime \prime }\left(x\right) >0$` for `$x<0$`.

(3) *Loss aversion*, or losses loom larger than gains.

- Sloppy formality: `$v(x)<v(-x)$` for all `$x>0$`.
- Formally: `$v(x)+ v(-x) < v(y) + v(-y)$` for all `$x>y$`.

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# Prospect Theory: Value Function

These assumptions lead to the following visual form of the value function:

(see board; or just google it if you're not in class.)

A functional form that's often used:

`$$v(x)=\{ x^{\alpha}\quad \text{ if }\quad x\geq 0$$`
`$$\quad \quad \quad \lambda (x)^{\beta}\quad \text{if }\quad x\leq 0$$`

An even easier functional form that we will mostly use eliminates the exponents.

**Note:** This second functional form removes diminishing sensitivity and isolates the effect of loss aversion on decision-making. In lots of settings this will greatly simplify the problem while leaving the fun stuff intact.

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# Implications (and non-Implications)

If a person maximizes her preferences meeting the assumptions above, she...

(i) ...will turn down any 50/50 lose $X, gain $X bets.

-Implication (i) is implied by Loss Aversion.

- Non-Implication (i). "...is necessarily averse to all fair bets."

- The assumptions do *not* guarantee a person will turn down all fair bets.

(ii) ...is risk averse among bets involving only gains.

- Implication 2 is implied directly by Diminishing Sensitivity.

(iii)... is risk-*loving* among bets involving only losses.

Implication (iii) is also implied directly by Diminishing Sensitivity.

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# Implications (and non-Implications)

(iv) ... is " first-order risk-averse."

- Implication (iv) requires important additional assumption.

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# An Aside

Let `$x$` be a **random variable** with distribution `$F(x)$`.

Let `$\mathbb{E}(x)$` denote the expectation of `$x$` and `$\sigma^2_x$` the variance.

Consider the lottery `$k + x$` as the lottery that pays `$k$` plus the realization of the random variable `$x$`.

**Claim:** Let `$\mathbb{E}(x) = 0$` and consider an expected utility maximizer. 
Suppose `$t >0$` such that `$-\pi \sim t\cdot x + k$`.  Then

`$$\pi \approx \frac{-t^2\sigma^2_x }{2}~ \frac{u''(w+k)}{u'(w+k)}$$`
--

Put another way: the "risk premium" decreases at rate `$t^2$`, while the "size" of the risk decreases at rate `$t$`.

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# First-Order Risk Aversion

Thus for small risks, a person must be almost *risk neutral*: the "premium" required to take on that risk would go to zero as the size of the risk goes to zero.

--
**Assumption:** A decision maker is *first-order risk averse* if for prospect theory value function `$v(\cdot)$`:

`$$\lim_{x\to 0} \frac{v'(-x)}{v'(x)} \equiv L > 1$$`

when approached from the `$x>0$` direction.

We will carry this assumption through many of our functional forms.

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# Prospect Theory: Probability-Weighting Function

Recall: 
`$$\quad V\left( x,p;y,q\right) \quad =\quad \pi \left( p\right) v\left( x\right)\quad +\quad \pi \left( q\right) v\left( y\right)$$`

We turn to key features of the probability-weighting function `$\pi \left( \cdot\right)$`:

[EU theory says `$\pi \left( p\right) =p$`.]

Natural assumptions:

- `$\pi \left( 0\right) =0$`, `$\pi \left( 1\right) =1$`, and `$\pi$` is increasing.

- Subcertainty: `$\pi \left( p\right) +\pi \left( 1-p\right)<1.$`

- Subproportionality: 
`$$\frac{\pi ( pq)}{\pi( p)} \leq \frac{\pi (pqr)}{\pi ( pr)}$$` 
for `$p,q,r\in \left(0,1\right)$`.

- For small `$p$`, `$\pi \left( p\right) >p.$`

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# Probability-Weighting Function

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# Probability-Weighting Function

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# Probability-Weighting Function