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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 $...
Ubiquitous's user avatar
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9 votes
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What is the point of silent auctions?

But perhaps this is not always the case. The price might have been so high that you would not have been willing to bid above it. In fact it might have been so high that the no one but the highest ...
Giskard's user avatar
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9 votes
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How to auction shared item with a roommate?

If you think that the distribution of your value and your roommate's value are the same then I would suggest that you consider the following bidding protocol: each bidder submits a sealed bid, $b_i$. ...
Ubiquitous's user avatar
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9 votes
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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 ...
Giskard's user avatar
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8 votes
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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 ...
Michael Greinecker's user avatar
7 votes
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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 ...
Herr K.'s user avatar
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7 votes
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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 $x$ ...
Walrasian Auctioneer's user avatar
6 votes
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Optimal Random Bids

First I will just show that the 0.5 (or $\frac{1}{2}$) cut-off point does not work as a symmetric equilibrium, then you can decide for yourself if you want to think about the problem or read the ...
Giskard's user avatar
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6 votes
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How do penny auctions make money?

In a penny auction, each bid raises the going price of an object by just one penny. However, placing such a bid is costly whether or not you win. Usually it costs one dollar to place a bid -- ...
Shane's user avatar
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6 votes
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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 ...
Giskard's user avatar
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6 votes
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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 ...
Giskard's user avatar
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6 votes
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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 ...
Giskard's user avatar
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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 ...
superhulk's user avatar
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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 ...
Giskard's user avatar
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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 ...
tdm's user avatar
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5 votes

What are the advantages and disadvantages of second price auctions?

A lot of questions here. Might be helpful to split them into different posts, but let's take a crack at them. The reason why the second price sealed auction is set up this way is simple. Imagine you ...
Kitsune Cavalry's user avatar
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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,...
Herr K.'s user avatar
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5 votes
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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 ...
Michael Greinecker's user avatar
5 votes
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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 ...
Alecos Papadopoulos's user avatar
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 ...
superhulk's user avatar
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5 votes
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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 ...
Bayesian's user avatar
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5 votes
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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 $\int_x^z G(y) dy$.
Giskard's user avatar
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4 votes
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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 ...
Dave Harris's user avatar
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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 * \...
Maarten Punt's user avatar
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4 votes
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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 ...
ml0105's user avatar
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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 ...
Amit's user avatar
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