# Questions tagged [auction-theory]

For questions about auction theory, an applied branch of economics which deals with how people act in auction markets and researches the properties of auction markets.

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### Revenue Equivalence in Auction Theory: how does an English auction generate the same revenue as First Price Sealed Bid?

The theroem: The revenue equivalence theorem states that, if all bidders are risk-neutral bidder and have independent private value for the auctioned items, then all four of the standard single unit ...
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### (Wikipedia Bug?) Equilibrium in natural ordering of Auction prices

According to wikipedia: Natural ordering Order the buyers in decreasing order of their bid: b1≥b2≥...≥bn. Order the sellers in increasing order of their bid: s1≤s2≤...≤sn. Let k be the largest index ...
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### The expected value of the lower of two values in a uniforme distribuition in [0,1], one of which is known to exceed $\frac{1}{2}$

I am reading a paper about Auction Theory, named "Auctions Versus Negotiations" by Jeremy Bulow and Paul Klemperer - 1996. Anyway, I just saw two affirmations about probabilities that I don'...
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### Dice Auction Optimal Strategy

Suppose two dice are rolled, and Alice and Bob are playing an auction-style game on the sum of the two dice. The first dice roll is shown to Alice, and the second one is shown to Bob. Then, Alice ...
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### All pay auction with dissimilar distributions

I'm trying to find Nash equilibrium for all-pay auction with two bidders. Values of bidders are $v_1$ and $v_2$ and they bid $b_1 = f_1 (v_1)$ and $b_2 = f_2 (v_2)$ correspondingly. Values have ...
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### Equilibria of first-price auctions with discrete valuation

Assume we have a first-price auction with discrete and independent private values and identical distributions of all bidders. It is known that there exists a Bayes-Nash equilibrium where bidders bid ...
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### Sealed bid first price auction with 2 players.

I would like help with the following question; Consider a sealed bid first price auction with 2 players in which the valuation of each of the players is best described by a uniform distribution on [10,...
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### First-price sealed-bid auction with information advantage

Consider two bidders $A$ and $B$ bidding for a box of 10 coins. The auction is performed in a first-price sealed-bid manner. The winner gets $k$ dollars if there are $k$ heads in the box. The ...
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### How to calculate expected payoff and optimal bid in this auction?

There is a seller $S$ with an object with quality $x$. The seller knows this quality; the buyer $B$ does not but beliefs that $x$ is uniformly distributed on $[0,M]$ with $0<M\leq2$. With true ...
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### Expected value of 2nd-bigger number out of n independent random variables in uniform distribution [r,1], r>0

I am having a problem on which I have 4 players bidding in second price auction with reserved price (r). I need to find the expected value of 2nd bigger number when there are 3 players or 4 bidding ...
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### VCG mechanism for a reverse auction

Consider a simple reverse auction for some work. Bidder (potential worker) $i$ submits a cost $b_i$ for the work based on their valuation (true cost) $\theta_i$ of it. Let $x_i(\mathbf{b}) \in \{0,1\}$...
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### Algorithm to solve 'user optimum' 'polygamic' stable marriage problem: Optimally assign travellers to shared rides.

I am looking for inspiration to reformulate the classical assignment problem into something behaviourally richer (closer to Nash equilibrium, or a stable marriage problem). I find it tricky to ...
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### Explanation on the auction algorithm

I am currently studying the auction algorithm developed by Bertsekas and other people. Indeed, auction algorithms can be used to solve the optimal assignment problem. Say you have $n$ agents and $n$ ...
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### How to solve a VCG auction game?

So i have this question: And these are the 5 highest marginal valuations according to the answer sheet: This is how the answer sheet defines the vcg mechanism How is the answer sheet chosing these ...
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### What is the Nash Equilibrium of the Dollar Auction?

The dollar auction is a type of auction in which players bid money for a dollar. Whoever bids the most pays what they bid and gets the dollar. Whoever bids the second most pays what they bid and get ...
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### Expectation of $x^n$ when $x$~$U[0,1]$ [closed]

I'm trying to follow along with my textbook on Auction Theory, but don't understand one of the steps. I don't see how they got $E[x^N] = 1/(N+1)$ in the final step. Please see the linked image below. ...
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### what is the Nash equilibrium in a Third price auction?

Consider an auction in which the winner is the highest bidder, and he pays the third highest bid. Suppose there are three players and each player $i\in\{1,2,3\}$ has a valuation $v_i$ and bids $b_i$. ...
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### Confused about what exactly constitutes an optimal auction

When it comes to Bayesian single-item auctions the condition that is usually given for optimality is that the auction maximizes the seller's expected revenue in a Bayes-Nash equilibrium of the auction,...
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### If $H(b) ≤ b$ then $\int_0^k(v−b)h(b)−[1−H(b)]db≤0$?

$H$ is a CDF over $[0,1]$ and $v$ is a constant I want to prove that if $H(b)≤b$ for all $b \in [0,1]$ then : $$\int_0^k(v−b)h(b)−[1−H(b)]db≤0$$ for all $k∈[0,1]$ I asked a similar question, but ...
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### If H(b) $\leq$ b then $\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db$?

$H$ is a CDF over $[0,1]$ I want to prove that if $H(b) \leq b$ then : $$\int_0^k (v-b)h(b) - [1-H(b)]db \leq \int_0^k (v-b) - [1-b]db$$ for all $k \in [0,1]$ I think this is true, but a rigorous ...
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### GSP Auctions prove dominant strategy [closed]

I need to show that considering a GSP auction for $n$ players, any bid $b_{i} > v_{i}$ is dominated by bid $b_{i} = v_{i}$. Where $i$ is the player, $b_{i}$ is the bid for player $i$ and $v_{i}$ ...
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### Nash Equilibrium in All Pay Auction 2 bidders

In my example we have 2 bidders, one with a higher valuation for the object than the other. In order to find a N.E. we first made some observations: 1) There cannot be atoms at $b_i>0$ 2) It cannot ...
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### First Price Auctions

Please tell me where I am wrong? I am asking this because sub-question a) didn't went as expected. I hope I did the derivates well for b) Furthermore, should I add more reasoning at c)? My answers ...
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### Example of VCG mechanism

I have some problems in understanding the VCG (Vickrey-Clarke-Groves) mechanism in game theory. Actually the bid for player $j$ is defined as: $p_j$ = (optimal welfare for the other players if player ...
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### Efficiency in Mechanism Design - Myerson-Satterthwaite

I was reading some online notes on the 4 conditions that cannot all hold given the Myerson-Satterthwaite theorem and this definition of efficiency confused me: An efficient mechanism selects the ...
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### Expected Revenue of 2nd Price Auction with Reserve Price

For a second price auction with two bidders, $n=2$, whose valuations are i.i.d on $U[0,1]$ and a reserve price $r$, I am trying to solve this problem using order statistics rather than the classic way ...
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### Common Value Second Price Auction - Winner's Expected Payment

How would you find the winner's expected payment in a second price auction with common values? For example, suppose we have the case where two players are random variables $x_1$ and $x_2$ and are ...
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### Second Price Auction with Reserve Prices - Expected Payment of Winner

I am having trouble understanding a second price auction with a reserve price, i.e. a second price auction where each player’s valuation is uniformly distributed on $[0, 1]$, and the two valuations ...
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### Uniform Distribution and First-Price Sealed Bid

For the First-Price Sealed Bid, I know that the optimal bid is $$(n-1)/n * v_i$$ However, I am confused about a step in finding this value. We are told that there are $n$ players each with a ...
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### Is it possible to use some kind of auction algorithm to determine payment percentages?

Me and my friends decided to rent a 3-bedroom apartment together to save money. The rooms have different areas and we all probably value those areas differently. Rent per month for the whole apartment ...
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### Brown Bears Bidding on Honey

I'm dealing with a problem where Brown Bears are bidding on a pot of honey. Imagine that we have $n$ Brown Bears $B_1,B_2,...,B_n,$ and brown bear $B_i$ values the honey pot $v_i \in \mathbb{R}^{+}.$ ...
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### System of Differential Equations- Asymmetric First-Price Auction

I am working on a problem in my Auction Theory textbook regarding a two-player asymmetric first price auction. Assume the bidders are risk neutral. The problem statement is as follows: Suppose ...
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### VCG auction — polynomial-time algorithm when bidders are unit demand

Is there a polynomial time algorithm to run a VCG when bidders are unit-demand? I though to look at the Bipartite graph when the left side is the bidders the right is the items and the edges are the ...