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Contracts are a subset of all mechanisms where agreements are enforcable. An example of a mechanism that is not a contract: A second price auction (or Vickrey auction) is a truth-telling mechanism where the enforcability of contracts is not required. In the truth-telling equilibrium no one has any incentive to change their bid, no matter the outcome. This ...


11

While answering a comment, I realized I had a post-worth response. R has become the "default language" for a lot of computational research statistics (for a number of reasons; nice NYT article here). It's high level, free and open-source, and has a closely-related journal for publishing statistical algorithms. Citations and peer review are key for ...


9

In contract theory The first-best refers to the best you could do if you knew agents' preferences over labor an income (i.e., if you did not have to impose the incentive compatibility constraint), and the second-best is the best you can do if agents have to reveal their preferences themselves. In mechanism design A useful reference is Galichon, Alfred, Ex-...


8

Suppose you face a single buyer whose willingness to pay, $v$, is distributed according to $F(v)$. If you charge a price $p$, he will buy if and only if $v>p$, leaving you with expected revenue of $$r(p)=\Pr(v>p)p=[1-F(p)]p.$$ Let's maximise revenue by computing an FOC: $$r'(p)=1-F(p)-F'(p)p=0.$$ We can rearrange this as $$\phi(p)\equiv p-\frac{1-F(...


6

Consider the incentive and participation constraints. That is, look at which buyer types would choose which option. A buyer does not buy anything if $$v_a < p_a \quad v_b < p_b \quad v_a +v_b < p_{ab}.$$ A buyer chooses only good $i \in \{a,b\}$ with $j\neq i$ if $$v_i - p_i \geq 0 \quad v_i - p_i \geq v_i + v_j - p_{ab}.$$ Otherwise she chooses ...


5

It is a known result in Mechanism Design that both first-price as well as second-price auctions yield the same expected revenue, under certain conditions (like independence of valuations, private information, etc). Consult Jehle, G. A., & Reny, P. J. (2011). Advanced Microeconomic Theory (3d ed.), ch.9 on Auctions and Mechanism Design, for a very ...


5

These are widely used technical terms with a precise mathematical meaning: Ex-ante budget balance means that the expected sum of all transfers is zero. Ex-post budget balance means that the sum of all transfers is zero with probability one. Some authors require that the sum of transfers is always zero.


5

The idea is simple: the seller wants to target that individuals who's ready to pay him the highest amount, thus targets the person with the highest virtual valuation. To target the individual who's ready to pay the most, we appeal to the concept of stochastic dominance(specifically, we are talking about first order stochastic dominance). The term $\...


5

By definition we have $$ \begin{align*} u(\theta) & = \theta q(\theta)-t(\theta) \\ \\ u(\theta + \delta) & = (\theta + \delta) q(\theta + \delta)-t(\theta + \delta) \end{align*} $$ By incentive compatibility (where $\theta + \delta$ is the true type, $\theta$ is the false type) we have $$ u(\theta + \delta) \geq (\theta + \delta) q(\theta)-t(\theta) ...


5

There’s a large literature on this in both CS and Economics. For economics, a good starting point would be Vijay krishna’s Auction Theory book (2009 edition). It has a whole section on multi-object auctions. For CS, check out some of the papers by Jason Hartline. He may have lecture notes as well. Naom Nisan and others have a book called algorithmic game ...


4

First of all, the general form of the problem you got there is extremely demanding. In multidimensional screening problems, analytically often all hell breaks loose in a sense that it is just not tractable. One way out might be this recent approach by Gabriel Carrol: Robustness and Separation in Multidimensional Screening (also provides an introduction that ...


4

Other methods or mechanisms by which goods are traded in a market economy: 1. Seller posts price. Probably the most familiar method to most people. Examples: McDonald's posts a price of \$1 for a hamburger, an airline posts a price of \$400 for a ticket, etc. Anyone who's willing and able to pay the posted price gets the good. 2. Buyer posts price. ...


4

As the previous answer mentions Contract Theory is a subset of Mechanism Design. Contract Theory is the study of Mechanism Design restricted to a single agent, i.e how to incentivize a single agent to achieve a desirable outcome, when you allow for multiple agents, the same problem becomes a mechanism design problem.


4

A social choice function presumes the individuals' preference parameters $\theta_i$'s are observable, whereas in a mechanism, such knowledge is not presupposed. Therefore, in a mechanism, the arguments of the outcome function are strategies of the players, which are observable, not their preference parameters, which, although indirectly determine the players'...


4

In mechanism design you are free to choose the rules of the game. The designer can determine $(S, g)$, i.e., what players can do and what happens when players played some strategy profile $s \in S := \times S_i$. In a direct mechanism, players are simply asked to report their type. Hence, every player $i$ must have a strategy that corresponds to "I am type $...


4

In addition to @Tomcat's suggestions, you may also want to check out the literature on matching markets. Easley and Kleinberg have an introductory textbook* on the subject. Chapter 10 covers the basic model of matching markets. Chapter 15 goes over the auction of ad slots as an application. *Easley, David, and Jon Kleinberg (2010) Networks, Crowds, and ...


3

We are given that $u$ is increasing and concave, and $u(0) = 0$. This implies that $\dfrac{u(t)}{t}$ is decreasing in $t$, and also, $\dfrac{u(t)}{t} > u'(t)$ for all $t$. Nicole's maximization problem : \begin{eqnarray*} \max_{x} \ q(x)(w-t(x))\end{eqnarray*} FOC : $q'(x)(w-t(x)) = q(x)t'(x)$ Suppose $x_N$ solves Nicole's problem. Therefore, it ...


3

A Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept than a NE itself. Consider the following simple matrix game where best replies have been marked with * \begin{array}{c|cc} P1/P2&\text{left}&\text{right}\\ \hline \text{Up}&1^*,1^*&0^*,0\\ \text{Down}&0,0^*&0^*,0^* \end{array} Both Up and ...


3

The solution to an optimal contract problem is called "first best" if it maximizes the principal's objective function subject to all constraints except the incentive constraints. The solution to an optimal contract problem is called "second best" if it maximizes the principal's objective function subject to all constraints, including the incentive ...


3

I think it is the chain rule. Let $w'(w) = w$, since we are looking for revealing mechanisms. The condition $$ \frac{\partial V}{\partial w'} (w'(w),w) = 0 $$ holds for all $w$ because the mechanism is revealing for all types. As the (not partial) differentiate w.r.t. $w$ of the right hand side is 0, the same goes for the differentiate of the left hand size, ...


3

Why are you doing $\frac{\partial^2 V}{\partial w'^2}$ ? Even if it is said that $w^{'}=w$ at the optimum, it should be taken different when you differentiate it for first order conditions. So, you differentiate it according to $w^{'}$ and $w$.


3

The simple answer is they estimate the demand curves for each product and, using their cost structure and market characteristics (competition structure, etc.) set price to maximize profits. This is standard for any firm, though. How Google in particular and these big firms in general (Amazon, Microsoft, etc.) estimate demand curves is somewhat different ...


3

Incentive compatible here means that you have no incentive to lie. A direct mechanism that is not incentive compatible would entice some agents to report a false type.


3

Most prominently, Wilson criticized the role of common priors in game theory. Starting from the "Wilson doctrine", some work in mechanism was done in that direction. Note that this research endeavor is not hopeless: in a popular auction format, the second-price auction, it is a (weakly) dominant strategy to report your true value. That is, your belief about ...


3

Mechanism design with multi-dimensional types (here: the willingness-to-pay for each object) is a notoriously difficult problem. Even when you abstract from bundling as you do by assuming that the seller only wants to buy one of the goods. Your problem is studied by John Thanassoulis in "Haggling over substitutes", JET 2004. Unfortunately, there is ...


2

First Best is the ideal optimal solution of a given problem, i.e. the mathematical solution of the model with "no imperfections". If that solution is not attainable then the solution must be binding to some constraint, in which case we call it a Second Best solution. 'Not attainable' here means that there's a discrepancy between the theoretical predictions ...


2

The matchingMarkets package in the R software now implements two constraint encoding functions to find all stable matchings in the three most common matching problems: hri: college admissions problem (including the student and college-optimal matchings) and stable marriage problem (including men and women-optimal matching) sri: stable roommates problem. ...


2

Patrick Prosser has some great java code at http://www.dcs.gla.ac.uk/~pat/roommates/distribution/ which, among other things, can compute all the stable matchings in roommate problems. The code is for roommates problems, but Patrick's code allows preferences over roommates to include unacceptable roommates. To implement a two-sided market, just make sure any ...


2

This analogy is due to Bulow and Roberts. Think of a diagram from Econ 101 with price on the vertical axis and quantity on the horizontal axis and consider a monopolist with constant marginal cost of zero. Given a demand curve, you can draw the marginal revenue and find the monopoly quantity where this function intersects the axis. Now take that same ...


2

You may need to provide more information for a clear answer. How does the mechanism look like in detail? What is the goal you want to achieve? Regarding your first question, it is not "necessary" to show that a mechanism is prior-free but it is a nice-to-have feature. Whether it is of first-order importance, as always, depends on the circumstances. For ...


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