10
votes
Accepted
What is virtual valuation?
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 $...
- 17k
9
votes
Accepted
How are ties broken in a second price sealed bid auction?
The winner pays the second highest bid, which is 5 dollars, however who the winner is depends on the tie-breaking rule. There are multiple rules one could use. In most cases it is assumed that the ...
- 29.6k
8
votes
Accepted
Are spectrum auctions a tax?
Suppose you write some software that you can then freely sell at practically no cost per unit (the wonders of the internet). You want to make as much profit as possible. Since you have almost no per ...
- 13.5k
7
votes
Accepted
Auction models for tickets
First, you should note that there is a Revenue Equivalence Theorem that says under a set of conditions, the seller's revenue from using different forms of auctions will be the same. This same result ...
- 15.8k
7
votes
Accepted
Differential equation for first-price auction
For notational simplicity, let me define the distribution $G(s) = F^{N-1}(s)$ with density $g(s)$. Let $\underline{s} = 0$ (for simplicity).
We have
$$
b'(s)G(s)+b(s)g(s)=s g(s)
$$
Integrating to $x$ ...
- 1,924
6
votes
Accepted
In auction theory, why is my own valuation a random variable?
In most of the literature I have read on private value auctions you do actually know your own valuation in the auction, it is your private information. It has a distribution from the point of view of ...
- 29.6k
6
votes
Accepted
Bayesian-Nash equilibrium in a first-price auction
It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0.
A somewhat ...
- 29.6k
6
votes
Accepted
2 Player All-Pay Auction
This seems like a basic issue about probability calculus/theory.
The intuition behind the uniform distribution over $[a,b]$ is that all outcomes between and $a$ and $b$ are equally likely. Because of ...
- 29.6k
6
votes
What is virtual valuation?
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 ...
- 525
6
votes
Auctions and finding nash equilibrium of a dynamic game
You are only required to get Nash equilibria and not sequentially rational/subgame perfect equilibria. Hence Player 2's actions at information sets that do not occur (that do not reflect Player 1's ...
- 29.6k
6
votes
Auctions and finding nash equilibrium of a dynamic game
Let's first determine the sets of actions of the players.
An action of player 1 is simply a bid $x_1 \in \mathbb{R}_+$.
An action of player 2 is a function: $f_2: \mathbb{R}_+ \to \mathbb{R}_+$ that ...
- 12.5k
5
votes
Reverse Auction bidding strategy
Given the very general description of the problem, I can think of the following (also very general) way of formulating it mathematically.
Let $v_n$ be the buyer's value from owning $n$ plots of land,...
- 15.8k
5
votes
Accepted
Ex-post vs Ex-ante Budget Balance in Auctions
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 ...
- 13.5k
5
votes
Accepted
Virtual valuation when the distribution is discrete
Virtual valuation is the derivative of the expected revenue function with respect to the tail probability $q$ that is then evaluated at value $v$. The Revenue function is
$$R(q) = q\cdot v(q),\;\;\; q ...
- 33.9k
5
votes
Why are second price auctions preferred if they don't maximize expected revenue?
I don't think sellers would prefer SPA over FPA. In fact, the SPA is riskier than the FPA, if we look at it from a seller's perspective. The reason is because the distribution of prices in case of the ...
- 525
5
votes
Accepted
Auction with one buyer and multiple sellers
Such a setting is called a "procurement auction" or "reverse auction."
It does not fundamentally change anything. Instead of buyers with privately known valuations, the auctioneer ...
- 5,291
5
votes
Accepted
Auction Theory: Proving that the found equilibrium is indeed optimal
$$
G(z) (z-x) =
\int_x^z G(z) dy
$$
and since $G$ is increasing on $[x,z]$, the right hand side is larger than
$\int_x^z G(y) dy$.
- 29.6k
4
votes
Accepted
Vickrey Auction question
Optimal allocation refers to whom the seller awards the item, in order to maximize revenue. In a Vickrey auction, it is a Nash equilibrium for everyone to bid their valuations. This is stronger than a ...
- 1,246
4
votes
Accepted
Difference between double auction and continuous double auction?
So, if a distinction is made, as continuous double auctions are usually just called double auctions, then the difference has to do with frequency. It is easier to have an example.
The New York Stock ...
- 2,006
4
votes
Incentive compatibility: Weakly dominant strategy versus Nash equilibrium?
A Nash equilibrium that consists of weakly dominant strategies is a stronger solution concept than a NE itself.
Consider the following simple matrix game where best replies have been marked with *
\...
- 2,373
4
votes
Second Price Auction - adjusting PDF for reservation price
To find expected revenue of the seller in a second price auction with reserved price consisting of two bidders who bid their valuations in equilibrium, we do the following :
Given that valuations are ...
- 9,411
4
votes
Accepted
Auctions with affiliation
Roughly speaking, $X_1$ and $X_2$ are affiliated because they have the common component $T$. That is, if $X_1$ is large, $X_2$ tends to be large as well, because a large $X_1$ makes a large $T$ ...
- 5,291
4
votes
Accepted
Are there multiple equilibria in the second price auction?
Sure. An example: if both valuations are drawn from the $[0,1]$ interval then the strategies
$$
b_1(v_1) = v_1
$$
and
$$
b_2(v_2) =
\left\{\begin{array}{cc}
v_2 & \text{ if } v_2 < 1 \\
5 & ...
- 29.6k
4
votes
Multi-item Auctions in Mechanism Design
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 ...
- 15.8k
4
votes
Accepted
What's the definition of social welfare in a procurement auction?
You need to add $x_0 v_0$ to your social welfare, where $v_0$ is the value the seller assigns to keeping the good and $x_0=1$ in that case. Then, the efficient (social-welfare-maximizing) allocation ...
- 5,291
4
votes
Accepted
Is VCG mechanism applicable in reverse auction? If so, how?
In general, VCG is also applicable to reverse auction settings. VCG is not even restricted to auction settings and can be used quite generally, see wikipedia for an introduction. If you want a deeper ...
- 5,291
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