# 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.

36 questions
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### English Auction & Second-Price Sealed-Bid Auction

I am currently new to studying auction theory and I came across the following web page which explains the different mechanisms and strategies used in the two auctions (See title) I do not understand ...
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### Third-price sealed bid auctions with n bidders

Contect: There are 𝒏 bidders for a painting in a sealed bid third-price auction (ie the winner is the bidder who makes the highest bid, and she pays the price equal to the third highest bid). Bidder �...
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### Pure Strategy Equilibrium in Second Price Action

Bayesian Nash Equilibrium is to always bid the true value. Is this the only equilibrium? Are there any other pure strategy bayesian nash equilibria for second price auction?
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### Finding mixed strategy Nash Equilibrium in First Price Sealed Bid Auction with Complete Information

Let we think about $\textbf{First Price Sealed Bid Auction}$ in which bidder who offers highest bid wins the object. There are only two bidders in this auction for the simplicity, $I_1$ and $I_2$. ...
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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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### game theory-Are second priced bids always a nash equilibrium?

I'm trying to understand the basics of game theory and the topic of auctions have come up. What I want to know is second priced bids always a nash equilibrium? Suppose we have this question. We ...
1answer
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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. ...
1answer
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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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### Does $b(v_i) = b_i^* = arg \max_{b} E_{-i} [v_i \frac{b_i}{\sum_{j\in N/i}b(v_{j}) +b_i} -b_i]$ have a unique equilibrium?

I am solving a symmetric equilibrium in an auction. In the auction, there is a set of bidders $N = (1,2,3,...,n)$. Each bidder is assigned a value, for example, bidder $i$'s value is $v_i$. Bidder $i$ ...
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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 ...
2answers
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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 $v_{i}$ ...
0answers
281 views

### 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 ...
1answer
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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 ...
1answer
867 views

### 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 ...
2answers
339 views

### 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 ...
1answer
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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 ...
1answer
820 views

### 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 ...
0answers
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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 $v_i \in \mathbb{R}^{+}.$ ...
2answers
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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 ...