Title: Microeconomics / Q1 / lecture10

Q: expected utility theory
A: an economic term summarizing the utility that an entity or aggregate economy is expected to reach under any number of circumstances. Typically seen as lotteries.
A mathematical function that represents a decision maker's preferences over uncertain outcomes. It is named after John von Neumann and Oskar Morgenstern, who developed the theory of expected utility in their seminal book "Theory of Games and Economic Behavior" (1944).
The VNM EU function is defined over a set of lotteries and assigns a real number, representing the expected utility, to each lottery. It allows decision makers to evaluate the attractiveness of different lotteries in a consistent and rational manner, taking into account their preferences and attitudes towards risk.
The VNM EU function is defined as:
EU(L) = Σ pi(u(xi))
where EU(L) is the expected utility of the lottery L, pi is the probability of outcome xi in the lottery L, and u(xi) is the utility or subjective value of outcome xi to the decision maker. The utility function u(xi) represents the decision maker's preferences over outcomes and is assumed to be strictly increasing and concave (i.e., the marginal utility of money decreases as the amount of money increases).
The VNM EU function satisfies four axioms of rational behavior:
1. Completeness: The decision maker has a preference between any two lotteries or is indifferent between them.
2. Transitivity: If the decision maker prefers lottery A to lottery B, and lottery B to lottery C, then they prefer lottery A to lottery C.
3. Independence: The decision maker's preferences over two lotteries depend only on the probabilities and outcomes of the lotteries, not on any other factors such as the order in which the outcomes are presented.
4. Continuity: If a sequence of lotteries becomes increasingly similar to a single lottery, then the decision maker's preference between the sequence and the single lottery becomes increasingly consistent.
The VNM EU function is widely used in decision theory and has important applications in economics, finance, and other fields where decision-making under uncertainty is a key issue.

Q: Types of lotteries
A: Simple and complex

simple lottery

Q: compound lotteries
A: The summation of propabilities of the simple lotteries and the corresponding alphas.

Q: consequentialist hypothesis
A: assume that the decision maker cares only about the reduced lottery over the N possible outcomes.
the assumption that a decision maker's preferences over lotteries depend solely on the consequences or outcomes of those lotteries, and not on any other factors such as intention, motive, or duty. In other words, the decision maker evaluates each lottery based on its expected consequences, and chooses the lottery that maximizes their expected utility.
For example, suppose a decision maker is offered a choice between two lotteries:
- Lottery A: a 60% chance of winning $100 and a 40% chance of winning $0
- Lottery B: a 100% chance of winning $50
If the decision maker would choose Lottery A because it has a higher expected value ($60 vs. $50) and therefore higher expected utility.
it is often used as a simplifying assumption in decision theory to focus on the evaluation of outcomes rather than the evaluation of other factors that may influence decision-making, such as moral considerations, social norms, or personal values.

Q: Assumptions of preferences within lotteries
A: Continuous and the independence axiom.
Continuous = that small changes in probabilities do not change the ordering between two lotteries.
idependence axiom = if we combine each of two lotteries with any third one, then the preference ordering of the two combined lotteries (which are compound lotteries) depends only on the preference ordering of the first two lotteries; that is, it does not depend on (is independent of) the third lottery.

Q: utility function of lotteries
A: Note: The “little u’s” can be interpreted as the utility of
their associated outcomes. For any n we could choose a
lottery that yields outcome n with probability 1 (all other
outcomes would have probability 0); for this type of
“degenerate” lottery, which we can call Ln, we have
U(Ln) = un, which implies that un is a “utility”.

Q: independence axiom
A: if we combine each of two lotteries with any third one, then the preference ordering of the two combined lotteries (which are compound lotteries) depends only on the preference ordering of the first two lotteries; that is, it does not depend on (is independent of) the third lottery.

Q: prospect theory
A: 1. Reference dependence. People derive utility from
gains or losses measured with respect to some reference
point (often their current situation) instead of deriving
utility from specific values of money (wealth). That is,
people are more sensitive to changes in their situations
than they are to levels/magnitudes. Hence no W in v( ).
2. Loss aversion. People are more sensitive to losses,
even very small ones, than they are to gains of the same
magnitude.
3. Diminishing sensitivity to gains and losses. With
respect to the “status quo” (no change in income/wealth),
the value function is convex for negative x but concave
for positive x. This is also seen in the diagram on p.18.
For example, the increase in value going from a $10 gain
to a $20 gain is larger than the increase in value going
from a $100 gain to a $110 gain. That is, people are risk
averse in gains; they prefer $500 for sure relative to a
50% change of gaining $1000. However, they are risk
seeking in losses: they prefer a 50% chance of losing
$1000 than a certain loss of $500.
4. Probability weighting. People do not weigh events by
their true probabilities but use probability weights that do
not equal actual probabilities. The probability weights are
systematic: they “overweight” rare events and under-
weight” likely events (seen next figure). An example of
this: even risk averse people prefer a 0.1% chance of
obtaining $5000 to a certain $5. This may explain why
people buy lottery tickets, or why we spend billions of
dollars to reduce death from rare events (plane crash,
terrorist attack) but not common events (car accidents).

Q: lottery space
A: set of all possible lotteries that a decision maker can face. A lottery is a probability distribution over a set of possible outcomes, where each outcome is associated with a certain amount of money or utility.
The lottery space includes all possible probability distributions over a set of nonnegative values of money or utility. Specifically, the lottery space can be defined as the set of all distribution functions that assign probabilities to the possible outcomes of a lottery, where the outcomes are nonnegative values of money.
For example, suppose a decision maker is offered a choice between two lotteries:
- Lottery A: a 50% chance of winning $100 and a 50% chance of winning $0
- Lottery B: a 100% chance of winning $50
The lottery space for this decision problem includes all possible probability distributions over the set of nonnegative values of money {0, $50, $100}. For example, one possible distribution function in this lottery space is:
- Probability of winning $0: 0.2
- Probability of winning $50: 0.3
- Probability of winning $100: 0.5
This distribution function represents a lottery that has a 50% chance of winning $100, a 30% chance of winning $50, and a 20% chance of winning $0.
The lottery space is a fundamental concept in decision theory because it allows decision makers to specify their preferences over lotteries and evaluate the tradeoffs between risk and expected value. By specifying their preferences over lotteries, decision makers can choose the lottery that maximizes their expected utility or subjective value.

Q: risk averse
A: A decision maker is said to be risk averse if they prefer a certain outcome over an uncertain outcome with the same expected value. In other words, they are willing to give up some expected value in exchange for certainty. This behavior is often observed in individuals who are more concerned about avoiding losses than achieving gains.

Q: Jensen's Inequality
A: Note that this inequality, called Jensen’s inequality, is
the defining property of a concave function. Thus for
expected utility theory risk aversion is equivalent to the
concavity of u( ).

Q: risk-neutral
A: They are indifferent between a certain outcome and an uncertain outcome with the same expected value. In this case, the decision maker is not concerned about the variability of the outcome and only cares about the expected value.

Q: strictly risk averse
A: they are willing to give up some expected value in exchange for certainty only when the uncertain outcome is worse than the certain outcome. If the uncertain outcome has the same expected value or is better than the certain outcome, the decision maker would prefer the uncertain outcome. This behavior is often observed in individuals who are highly risk-averse and place a high value on avoiding uncertainty.

Q: degenerate lottery
A: a lottery where one outcome has probability 1 (i.e., it is certain), and all other outcomes have probability 0. This means that the lottery reduces to a certain outcome.
For example, suppose a decision maker is offered the choice between two lotteries:
- Lottery A: a 100% chance of winning $100
- Lottery B: a 50% chance of winning $200 and a 50% chance of winning $0
Lottery A is a degenerate lottery because the outcome of winning $100 has a probability of 1, while all other outcomes have a probability of 0.
Degenerate lotteries are useful in decision theory as they help to simplify complex decision problems by reducing them to simpler ones that involve certain outcomes. By doing so, they allow decision makers to evaluate the tradeoffs between certain and uncertain outcomes in a more straightforward manner.

Q: reduced lottery
A: a way to simplify complex lotteries and make them more manageable for decision-making purposes. A reduced lottery is a simpler lottery that generates the same distribution of outcomes as a more complex lottery.
To obtain a reduced lottery from a compound lottery with K outcomes, we calculate the probability of each outcome n in the reduced lottery by adding up the products of the probability of each outcome in the original lotteries Lk multiplied by the probability that Lk occurs, αk. That is:
pn = α1p1n + α2p2n + ... + αKpKn
where pn is the probability of outcome n in the reduced lottery, pkn is the probability of outcome n in lottery Lk, and αk is the probability that lottery Lk occurs.
For example, suppose we have a compound lottery with two lotteries L1 and L2, and their corresponding probabilities α1 and α2. Each lottery has three possible outcomes with probabilities p11, p12, p13 for L1 and p21, p22, p23 for L2. We can obtain the reduced lottery by adding up the probabilities of each outcome:
- The probability of outcome 1 in the reduced lottery is: p1 = α1p11 + α2p21
- The probability of outcome 2 in the reduced lottery is: p2 = α1p12 + α2p22
- The probability of outcome 3 in the reduced lottery is: p3 = α1p13 + α2p23
The reduced lottery is now a simple lottery with three outcomes, each with a corresponding probability. The decision maker can use the probabilities of the reduced lottery to evaluate the expected value and make a decision based on their risk preference.
In summary, a reduced lottery is a simplified version of a complex lottery that retains the same distribution of outcomes. It can be obtained by combining the probabilities of each outcome in the original lotteries according to their respective probabilities.