An Introduction to Prospect Theory

Part Of: [Neuroeconomics] sequence
Content Summary: 1500 words, 15 min reading time

Preliminaries

Decisions are bridges between perception and action. Not all decisions are cognitive. Instead, they occur at all levels of the abstraction hierarchy, and include things like reflexes. 

Theories of decision tend to constrain themselves to cognitive phenomena. They come in two flavors: descriptive (“how does it happen”) and normative (“how should it happen”).

Decision making often occur in the context of imperfect knowledge. We may use probability theory as a language to reason about uncertainty. 

Let risk denote variance in the probability distribution of possible outcomes. Risk can exist regardless of whether a potential loss is involved. For example, a prospect that offers a 50-50 chance of paying $100 or nothing is more risky than a prospect that offers $50 for sure – even though the risky prospect entails no possibility of losing money.

Today, we will explore the history of decision theory, and the emergence of prospect theory. As the cornerstone of behavioral economics, prospect theory provides an important theoretical surface to the emerging discipline of neuroeconomics.

Maximizing Profit with Expected Value

Decision theories date back to the 17th century, and a correspondence between Pascal and Fermat. There, consumers were expected to maximize expected value (EV), which is defined as probability p multiplied by outcome value x.

EV = px

To illustrate, consider the following lottery tickets:

prospect-theory-interchangeable-expected-value-options-2

Suppose each ticket costs 50 cents, and you have one million dollars to spend. Crucially, it doesn’t matter which ticket you buy! Each of these tickets have the same expected value: $1. Thus, it doesn’t matter if you spend the million dollars on A, B, or C – each leads to the same amount of profit.

The above tickets have equal expected value, but they do not have equal risk. We call people who prefer choice A risk averse; whereas someone who prefers C is risk seeking.

Introducing Expected Utility

Economic transactions can be difficult to evaluate. When trading an apple for an orange, which is more valuable? That depends on a person’s unique tastes. In other words, value is subjective.

Let utility represent subjective value. We can treat utility as a function u() that operates on objective outcome x. Expected utility, then, is highly analogous to expected value:

EU = pu(x)

Most economists treat utility functions as abstractions: people act as if motivated by a utility function. Neuroeconomic research, however, suggests that utility functions are physically constructed by the brain.

Every person’s utility function may be different. If a person’s utility curve is linear, then expected utility converges onto expected value:

EU \rightarrow EV \mid u(x) = x

Recall in the above lottery, the behavioral distinction between risk-seeking (preferring ticket A) and risk-averse (preferring C). Well, in practice most people prefer A. Why?

We can explain this behave by appealing to the shape of the utility curve! Utility convexity produces risk aversion:

Prospect Theory- Utility Convexity & Risk Aversion

In the above, we see the first $50 (first vertical line) produces more utility (first horizontal line) than the second $50.

Intuitively, the first $50 is needed more than the second $50. The larger your wealth, the less your need. This phenomenon is known as diminishing marginal returns.

Neoclassical Economics

In 1947, von Neumann and Morgenstern formulated a set of axioms that are both necessary and sufficient for representing a decision-maker’s choices by the maximization of expected utility.

Specifically, if you assume an agent’s preference set accomodates these axioms…

1. Completeness. People have preferences over all lotteries.

\forall L_1, L_2 \in L either L_1 \leq L_2 or L_1 \geq L_1 or L_1 = L_2

2. Transitivity. Preferences are expressed consistently.

\forall L_1, L_2, L_3 \in L if L_1 \leq L_2 and L_1 \leq L_2 then L_1 \leq L_3

3. Continuity. Preferences are expressed as probabilities.

L_1, L_2, L_3 \in L then \exists \alpha, B  s.t. L_1 \geq L_2 \geq L_3 iff \alpha L_1 + (1-\alpha)L_3 \geq L_2 \geq BL_1 + (1 - B)L_3

4. Independence of Irrelevant Alternatives (IIA). Binary preferences don’t change by injecting a third lottery.

… then those preferences always maximize expected utility.

L_1 \geq L_2 iff sum(p_1u(x_1) \geq p_2u(x_2)

The above axioms constitute expected utility theory, and form the cornerstone for neoclassical economics.  Expected utility theory bills itself as both a normative and descriptive theory: that we understand human decision making, and have a language to explain why it is correct.

Challenges To Independence Axiom

In the 1970s, expected utility theory came under heavy fire for failing to predict human behavior. The emerging school of behavioral economics gathered empirical evidence that Neumann-Morgenstern axioms were routinely violated in practice, especially the Independence Axiom (IIA).

For example, the Allais paradox asks our preferences for the following choices:

prospect-theory-allais-paradox-4

Most people prefer A (“certain win”) and D (“bigger number”). But these preferences are inconsistent, because C = 0.01A and D = 0.01B. The independence axiom instead predicts that A ≽ B if and only if C ≽ D.

The Decoy effect is best illustrated with popcorn:

decoy2

Towards a Value Function

Concurrently to these criticisms of the independence axiom, the heuristics and biases literature (led by Kahneman and Tversky) began to discover new behaviors that demanded explanation:

  • Risk Aversion. In most decisions, people tend to prefer smaller variance in outcomes.
  • Everyone prefers gains over losses, of course. Loss Aversion reflects that losses are felt more intensely than gains of equal magnitude.
  • The Endowment Effect. Things you own are intrinsically valued more highly. Framing decisions as gains or as losses affects choice behavior.

Prospect Theory- Behavioral Effects Economic Biases (1)

Each of these behavioral findings violate the Independence Axiom (IIA), and cumulatively demanded a new theory. And in 1979, Kahneman and Tversky put forward prospect theory to explain all of the above effects.

Their biggest innovation was to rethink the utility function. Do you recall how neoclassical economics appealed to u(x) convexity to explain risk aversion? Prospect theory takes this approach yet further, and seeks to explain all of the above behaviors using a more complex shape of the utility function. 

Let value function \textbf{v(x)} represent our updated notion of utility.  We can define expected prospect \textbf{EP} of a function as probability multiplied by the value function

EP = pv(x)

Terminology aside, each theory only differs in the shape of its outcome function.

Prospect Theory- Evolution of Utility Function (3)

Let us now look closer at the the shape of v(x):

Prospect Theory- Value Function.png

This shape allows us to explain the above behaviors:

The endowment effect captures the fact that we value things we own more highly. The reference point in v(x), where x = 0, captures the status quo. Thus, the reference point allows us to differentiate gains and losses, thereby producing the endowment effect.

Loss aversion captures the fact that losses are felt more strongly than gains.  The magnitude of v(x) is larger in the losses dimension. This asymmetry explains loss aversion.

We have already explained risk aversion by concavity of the utility function u(x). v(x) retains convexity for material gains. Thus, we have retained our ability to explain risk aversion in situations of possible gains. For losses, v(x) concavity predicts risk seeking.

Towards a Weight Function

Another behavioral discovery, however, immediately put prospect theory in doubt:

  • The Fourfold Pattern. For situations that involve very high or very low probabilities, participants often switch their approaches to risk.

To be specific, here are the four situations and their resultant behaviors:

  1. Fear of Disappointment. With a 95% chance to win $100, most people are risk averse.
  2. Hope To Avoid Loss. With a 95% chance to lose $100, most people are risk seeking.
  3. Hope Of Large Gain. With a 5% chance to win $100, most people are risk seeking.
  4. Fear of Large Loss. With a 5% chance to lose $100, most people are risk averse.

Crucially, v(x) fails to predict this behavior. As we saw in the previous section, it predicts risk aversion for gains, and risk seeking for losses:

Prospect Theory- Fourfold Pattern Actual vs Expected (2)

Failed predictions are not a death knell to a theory. Under certain conditions, they can inspire a theory to become stronger!

Prospect theory was improved by incorporating a more flexible weight function.

EP = pv(x) \rightarrow EP = w(p)v(x)

Where w(p) has the following shape:

Prospect Theory- Weight Function (1)These are in fact two weight functions:

  1. Explicit weights represent probabilities learned through language; e.g., when reading the sentence “there is a 5% chance of reward”.
  2. Implicit weights represent probabilities learned through experience, e.g., when the last 5 out of 100 trials yielded a reward.

This change adds some mathematical muscle to the ancient proverb:

Humans don’t handle extreme probabilities well.

And indeed, the explicit weight function successfully recovers the fourfold pattern:

fourfold_pattern

Takeaways

Today we have reviewed theories of expected value, expected utility (neoclassical economics), and prospect theory. Each theory corresponds to a particular set of conceptual commitments, as well a particular formula:

EV = px

EU = pu(x)

EP = w(p)v(x)

However, we can unify these into a single value formula V:

V = w(p)v(x)

In this light, EV and EU have the same structure as prospect theory. Prospect theory distinguishes itself by using empirically motivated shapes:

Prospect Theory- Evolution of Both Functions

With these tools, prospect theory successfully recovers a wide swathe of economic behaviors.

prospect-theory-behavioral-explananda-2

Until next time.

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