Questions & explanations
1. Give an example of choices that violate WARP.
Consider a consumer with $10. With prices ($2,$1), they choose (3,4) costing $10. With prices ($1,$2), they choose (4,3) costing $10 too. Check first: (3,4) costs $10; (4,3) would cost $11, so not affordable. Second: (4,3) costs $10; (3,4) would cost $11, not affordable. Actually this does not violate. A violation needs a directly revealed preference reversal. For example: At prices ($1,$1) and income $5, choose (3,2). At prices ($2,$1) income $8, choose (1,4). At first, (1,4) costs $5, affordable but not chosen, so (3,2) is preferred. At second, (3,2) costs $8, affordable but not chosen, that would violate. But careful: need to ensure both are affordable in each other's budgets. A classic violation: choose (2,2) when income $6 and prices ($2,$1); then choose (1,4) when income $8 and prices ($1,$2) – check if (1,4) was affordable in first and not chosen, etc. But for simplicity, we can say: if at first you choose A when B is affordable, and later choose B when A is affordable, that violates WARP. Example: with $10, prices ($1,$1) choose (6,4). With same income and prices ($1,$1) but
2. Give an example of a dataset that satisfies GARP but violates SAREP.
Consider two goods and three observations: Observation 1: budget line budget1, choose bundle A. Observation 2: choose A at budget2 (same bundle). Observation 3: choose B at budget3. Suppose that A is strictly preferred to B (since A was chosen over B in observation 1 when both affordable), and B is indirectly revealed preferred to A because B is chosen when A is also affordable? Actually need a cycle. Example: Let prices and incomes be such that in obs1, A chosen, B also affordable; in obs2, B chosen, A affordable; in obs3, both affordable but choose A. Then direct strict: A>B (obs1), B>A (obs2 strict), so direct cycle violates WARP and hence SAREP. But if obs2 had indifference? Actually GARP allows: A strictly > B, B indirectly > A? Need careful. Simpler: Suppose A strictly preferred to B, and B is revealed preferred to A only indirectly through a chain where one step is indifference. For instance, obs1: choose A over B (A > B). obs2: choose B over C (B > C). obs3: choose C and A is affordable? If C is indifferent to A? Then we have A > B > C = A, so indirect cycle but no strict dir
3. Give an example of a Vickrey-Clarke-Groves mechanism for a public project.
Imagine three neighbors deciding on a park bench that costs $60. Each reports how much they value it: Alice $30, Bob $20, Carol $10. Sum = $60, so the bench is built. Alice pays a tax: without her, Bob+Carol = $30, which is less than cost, so the project fails. Alice's tax is the harm: the others would have saved $30 if no bench, but now they get no benefit? Actually, Clarke tax: Alice pays (Bob+Carol's values) minus (cost without her)? Let's correct: Without Alice, total reported = $30, under cost, so project rejected. With Alice, project accepted. Alice's tax is the reduction in others' surplus: Bob+Carol reported $30, they get the bench, but they would have preferred no bench? They value it positively. Standard formula: Clarke tax = sum of others' net benefits without you minus sum of others' net benefits with you. Without Alice, no bench, so others' net benefit = 0. With Alice, bench, others enjoy $30 but pay nothing? Actually, they also pay taxes? Let's simplify: In reality, Clarke tax can be complex. Best to give a simple intuition: If your report changes the decision, you pay
4. What is the Slutsky matrix?
The Slutsky matrix is a table of numbers that shows how a change in the price of one good affects the demand for another good, after removing the income effect. Each entry in the matrix is the substitution effect, which measures how consumers switch between goods when prices change, holding their real income constant. The matrix is important because it must have two specific properties for a rational consumer: negative semidefiniteness and symmetry. Negative semidefiniteness means that when you multiply the matrix by a vector of price changes, the result is always non-positive, implying that the own-price substitution effect is negative. Symmetry means that the cross-price substitution effects are equal, like the effect of good A's price on good B's demand equals the effect of good B's price on good A's demand. These properties come from the consumer's utility maximization problem.
5. Why is the Slutsky matrix negative semidefinite?
The Slutsky matrix is negative semidefinite because it comes from a utility-maximizing consumer who faces a budget constraint. If you take any vector of price changes, the change in the cost of the original consumption bundle is zero when you adjust income to keep the consumer's real income constant. The consumer can only be as well off as before, so the new bundle must cost at least as much, leading to the negative semidefinite property. Mathematically, for any price change vector v, the quadratic form v' * S * v is less than or equal to zero, where S is the Slutsky matrix. This ensures that demand is consistent with utility maximization. If it were not negative semidefinite, the consumer could choose a bundle that costs less and is better, which would violate rationality.
6. Why do we need sequential equilibrium instead of just Perfect Bayesian equilibrium?
Perfect Bayesian equilibrium does not restrict beliefs off the equilibrium path strongly enough. In some games, there can be many Perfect Bayesian equilibria, some of which rely on arbitrary or unreasonable beliefs after unexpected actions. Sequential equilibrium adds a consistency requirement: beliefs must be the limit of beliefs from a sequence of completely mixed strategies that make every action possible. This ensures that beliefs are based on some small chance of mistakes, making them more logical. For example, in a game where a player takes an action that should never happen in equilibrium, Perfect Bayesian equilibrium lets you choose any belief, but sequential equilibrium ties it to the structure of the game. This yields more believable predictions.
7. Give an example of a game where a Nash equilibrium exists but no sequential equilibrium exists.
A simple game: player 1 can go up or down; if up, game ends. If down, player 2 chooses left or right. Payoffs: up (2,2); down-left (3,1); down-right (0,0). A Nash equilibrium is: player 1 chooses up, player 2 chooses left (since if player 1 goes down, left is better). But is it sequentially rational? At the node after down, player 2's best is left, so that is fine. Actually here a sequential equilibrium exists. For a game without one, consider a game where player 2's off-path belief cannot be consistent because player 1 never goes down. Usually, if there is a requirement that beliefs are consistent, some Nash equilibria are ruled out. In many signaling games, pooling equilibria may fail to be sequential due to unreasonable off-path beliefs.
8. How is Perfect Bayesian equilibrium different from sequential equilibrium?
Perfect Bayesian equilibrium and sequential equilibrium are both used for dynamic games with incomplete information. The main difference is that sequential equilibrium has a stricter requirement for beliefs off the equilibrium path. In sequential equilibrium, there must be a sequence of completely mixed strategies that converge to the equilibrium strategies, and the beliefs must be the limit of the beliefs from those strategies. This ensures beliefs are consistent even for actions that never happen in equilibrium. Perfect Bayesian equilibrium only requires that beliefs are consistent with Bayes' rule on the equilibrium path, but they can be arbitrary off the path. So sequential equilibrium is a refinement of Perfect Bayesian equilibrium.
9. Give an example where two rational people with common priors can agree to disagree without violating the theorem.
Actually, the theorem says that with common priors and common knowledge of the posteriors, they cannot agree to disagree. So any example of disagreement must violate one condition. For instance, if Alice has private information and Bob has different private information, and they only tell each other their final beliefs without revealing their private information, then those beliefs are not common knowledge because they don't know how the other arrived at that belief. So they can disagree. In legal settings, two experts might have different opinions because they started from different models (different priors). The theorem is often misinterpreted; it doesn't say people always agree, just that if the conditions hold, they must.
10. Give an example of a signaling game and identify the perfect Bayesian equilibrium.
Consider a job market where a worker has high or low ability (private info). The worker chooses education level (signal). The employer sees education and offers a wage. In a separating equilibrium, high-ability workers get education and low-ability workers do not. The employer believes that any worker with education is high ability and pays a high wage, and without education is low ability and pays a low wage. This is a PBE if the high-ability worker's cost of education is low enough and the low-ability worker's cost is high enough. Both strategies are optimal given beliefs, and beliefs are updated by Bayes' rule: after seeing education, probability of high ability is 1; after no education, probability of high ability is 0.
11. Give an example showing the application of the Revelation Principle to a non-auction setting.
Consider a public good provision problem where citizens have private valuations for a project. The government wants to decide whether to build the project and how to share costs. By the Revelation Principle, the government can design a direct mechanism where each citizen reports their valuation. Then, based on reports, the project is built if total reported values exceed cost, and payments are made according to a rule that ensures truth-telling. For instance, the Vickrey-Clarke-Groves mechanism is a direct truthful mechanism. Without the principle, the government might consider complex voting or bargaining processes. The principle shows that any outcome from those processes can be replicated by a truthful direct mechanism.
12. How does the symmetry of the Slutsky matrix relate to the integrability conditions?
The symmetry of the Slutsky matrix is a key part of the integrability conditions, which are mathematical requirements for a demand system to be derived from a utility function. If a demand system has a symmetric Slutsky matrix, then there exists a potential function that can be integrated to find the underlying expenditure function. This means that the cross-price substitution effects are consistent with each other, like the effect of good A's price on good B's demand matches the effect of good B's price on good A's demand. Without symmetry, no utility function exists that can generate that demand system. Symmetry together with negative semidefiniteness gives the full set of conditions for rational consumer behavior.