Comparison of Lottery Choices and MEU

Question

Given these two lottery choices:

A: [0.75, $3k;    0.25, $0]

B: [1.0, $2k;    0.0, $0]

and

C: [0.3, $3k;    0.7, $0]

D: [0.4, $2k;    0.6, $0]

People seem to prefer B over A, and C over D

What are the implications of this observation?

What is the contradiction here?  What assumption did you use to arrive at your contradiction?

Answer 1

1.First, we could analyze the expected value of these four options:

Option A:  

Expected value = 0.75*3,000+ 0.25*0=2,250 

Option B:

Expected value = 1.0*2,000=2,000 

Option C:

Expected value = 0.3*3,000+ 07*0= 900 

Option D:

Expected value = 0.4*2,000+ 0.6*0=800  

2.Implications of the observation: 

 According to expected value theory and risk aversion, people should prefer the option with higher expected value and probability. But this observation means people are not choosing choices with higher expected value and lower risk.

This shows two important principles of psychoeconomics:

One is risk aversion, where people avoid risk when faced with certainty and high-risk choices. For example, people prefer option B even though A has a higher expected value.

The other is probability weighting, when faced with a lower probability of a high payoff, people tend to overestimate the likelihood of that low probability event occurring. For example, although the probability of D is higher, people prefer C because it offers the possibility of winning a larger prize. 

3.Contradictions and Assumptions:

The paradox of this observation is that it contradicts the traditional theory of expected value in economics. People may not follow the principles of rationality and maximizing expected value when making decisions.

  The assumption that leads to this contradiction is that people are rational economic people, and when faced with uncertainty, they tend to choose the option with the highest expected value and follow the principle of maximizing expected utility.

Answer 2

The lottery preferences you stated emphasize a widespread observation in behavioral economics and decision theory, which is typically tied to the ideas of risk aversion and anticipated utility theory. Let us examine the lottery and the consequences of these preferences:

Step 1: Analyze the Lotteries

Lottery A vs. Lottery B:

A: There is a 75% probability of earning $3,000 and a 25% chance of getting nothing.
B: There is a 100% possibility of winning $2,000.
Value Expected:

A:0.75×$3,000+0.25×$0=$2,250

B:1.0×$2,000=$2,000

Lottery C vs. Lottery D:

C: There is a 30% chance of earning $3,000 and a 70% chance of winning nothing.
D has a 40% chance of earning $2,000 and a 60% chance of getting nothing.
Value Expected:

C:0.3×$3,000+0.7×$0=$900

D:0.4×$2,000+0.6×$0=$800

Step 2: Consequences and Contradictions

Preference for B over A demonstrates risk aversion. People would rather have a guaranteed $2,000 prize than a greater predicted value but unknown $3,000.
Preference for C over D: This demonstrates a desire for risk. People choose the lottery with a lesser expected value but a larger probability of winning ($3,000).

Step 3: The Disagreement

Expected Utility Theory holds that people will always select the choice with the highest expected value. The preferences described, however, contradict this. People choose a lower predicted value in C vs. D, indicating risk-seeking behavior, and a certain but lower value in B vs. A, indicating risk aversion.
Inconsistency: The preferences for B over A and C over D are contradictory. One exhibits risk aversion, whereas the other exhibits risk seeking.


Step 4: Assumption that leads to a contradiction

The assumption here is that people behave consistently in terms of risk preferences across various contexts. These preferences, however, suggest that people's attitudes toward risk can shift depending on the context, framing of options, or size of prospective outcomes.

This discovery is a famous example of a divergence from standard economic theory, namely anticipated utility theory, and is frequently investigated in the field of behavioral economics. It demonstrates how complex human decision-making can be and how it can be influenced by factors other than quantitative anticipated values.