Humans are not rational. See (Dostoevsky 1864), the parable of the destroying free will. In this note we will describe some behavioural economics ideas that could be interesting in this side.

Economical Games

The rules of the Ultimate game

The proposer makes an offer as to how this money should be split between the two. The second player (the risponder) can either accept or reject this offer. If it is accepted, the money is split as proposed, but if the risponder rejects the offer, then neither player receives anything.

From a purely rational stand, you should always accept. But humans tend to reject offers that are unfair, even if it means they will get nothing. This is a clear example of how humans are not purely rational.

Analysis of Activations in the brain

ACC, conflict detection, attention, performance monitoring, impulse working memory. The insula is emotions, self-awareness, social, and negative emotions, it answers very strongly for unfair parts, probably insula detects the disgust part.

St.Petersburg Paradox

A fair coin is tossed at each stage. The initial stake begins at 2 dollars and is doubled every time tails appears. The first time heads appears, the game ends and the player wins whatever is the current stake. Thus, the player wins 2 dollars if heads appears on the first toss, 4 dollars if tails appears on the first toss and heads on the second, 8 dollars if tails appears on the first two tosses and heads on the third, and so on.

$$E = \sum_{k=1}^{\infty} \frac{1}{2^{k}} 2^{k} = \sum_{k=1}^{\infty} 1 = \infty$$

Yet, most people would not be willing to pay more than 10 dollars to play this game. This is an example of how humans are not purely rational, or the rationality part is not capturing human reasoning well.

Expected Utility Theory

Main idea of the theory

The value of an item is not based on the price, but rather on the utility it yields (100$ is more to a poor person than to a rich person).

There is also an example in the scriptures that like the widow that gives a small amount, but it is a lot for her. This is an example of how humans are not purely rational, or the rationality part is not capturing human reasoning well.

$$U(x) = \sum_{i=1}^{n} p_i u(x_i)$$

Where $u(x)$ is the utility function, and $p_i$ is the probability of each outcome.

$$\sum_{i=1}^{n} p_i u(x_i) - c = 0$$

Marginal Utility

See Budget and Preferences for more info on the economical side.

Individuals make economic decisions by weighing the benefits of consuming an additional unit of a good or service against the cost of acquiring it. In other words, value is determined by the additional utility of satisfaction provided by each extra unit consumed. (source Wikipedia)

$$\frac{dU}{dg} = \frac{dU}{dx} \cdot \frac{dx}{dg}$$

When the second derivative is negative, then it is a diminishing marginal utility.

Monkey’s Utility function in risk-based settings

TODO

On the high end they were risk-averse, while on the low end they were usually risk seeking, this means that marginal utility theory is able to explain the behaviour of the animal in this settings. Dopamine reward prediction error responses reflect marginal utility.

Neurons encode utility and not absolute reward. But other parts are not well explained by marginal utility.

Cumulative Prospect Theory

Framing problem

This is something similar with the copy machine study: (Langer et al. 1978). This was discovered by Kahneman and also discussed in (Kahneman 2011), basically even if you express the same thing, the framing is dramatically different.

•Program A: “200 people will be saved” •Program B: “there is a 1/3 probability that 600 people will be saved, and a 2/3 probability that no people will be saved” -> 72 percent preferred program A •Program C: “400 people will die” •Program D: “there is a 1/3 probability that nobody will die, and a 2/3 probability that 600 people will die” -> 78% preferred program D

The idea of the theory

$$V(x) = \sum_{i=1}^{n} p_i v(x_i)$$

Where $x_i$ is the outcome of the gamble, $p_i$ is the probability of each outcome, and $v(x)$ is the value function.

usually the value function is concave for gains and convex for losses, which means that people are risk averse for gains and risk seeking for losses.

Decision weights

Risk seeking and risk averse. Attempts to solve the linearity of expected utility theory.

References

[1] Kahneman “Thinking, Fast and Slow” {Farrar, Straus and Giroux} 2011

[2] Langer et al. “The Mindlessness of Ostensibly Thoughtful Action: The Role of “Placebic” Information in Interpersonal Interaction” Journal of Personality and Social Psychology Vol. 36(6), pp. 635–642 1978

[3] Dostoevsky “Notes from Underground” Wm. B. Eerdmans Publishing 1864