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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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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 ...
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1answer
37 views

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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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 ...
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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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bribing in second price auctions

In this paper, they analyse a second-price auction game with an option for the first player to bribe the second player to stay out of he game. In proposition 2, they show that for any $b\in(0, \Bbb{E}...
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Question on first and second price sealed bid auction

I’m studying on auction. But I couldn’t understand some questions about this topic. And I cannot solve such type questions. For example, I know that a set of symmetric and risk neutral bidders ...
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How to show that bids are in ex post equilibrium in a Vickrey auction?

There is one indivisible item. We have 2 bidders, 1 and 2, with valuations: $$ v_1 \in [0,2]\\ v_2\in [0,1] $$ They present their bids simultaneously, $b_1\in[0,2]$ and $b_2\in[0,2]$. The winner ...
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Game Theory: Auction Bidding

Imagine a situation where 4 players are playing in an auction game. There are 4 letters for sale A, B, C and D. A player "wins" if they are the first player to buy 3 of the same letter. Each player ...
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How to show that a selection $b_1(v)$ is always less than another selection $b_2(v)$

I am an economist and need some math help. My question is simply prove or disprove that $b_1(v) \leq b_2(v)$, which are selections of two maximization problems. I have two objective functions. $v \...
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1answer
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is $\frac{g(x)}{h(x)}>\frac{1-G(x)}{1-H(x)}$ true if $G(x) > H(x) > x$ on (0,1)?

In my research there are two CDFs, $G(x)$, $H(x)$ which support is [0,1]. The CDFs are twice differentiable and $G(x) > H(x) > x$ on (0,1). -> stochatical dominance. My crucial condition is $\...
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Tackling freeloader problem of a “group auction”

Not sure what to call these, which is a part of the problem, as I'd imagine there's plenty of literature on this. So, consider the following game: Three players, A, B and C. All of them having ...
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1answer
189 views

Minimum bids in first-price auction

Suppose there are two bidders $i=1,2$ who may either have a low valuation $V_L$ or a high valuation $V_H.$ Bidders do not know others' valuations. In a first-price auction, where ties are broken by ...
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GSP auctions - show that when Nash Equilibrium, the allocation is reasonable

I am working with Generalized Second Price Auction with $n$ players and $n$ slots to allocate. GSP auctions is defined with a utility function : $u_{i}(b) = \alpha_{i}(v_{i} - p_{i})$ where $p_{i}=b_{...
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145 views

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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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 ...
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1answer
78 views

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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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 ...
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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 ...
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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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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 ...
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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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481 views

All Pay Auction Revenue

I know that the bidding strategy for an all-pay auction is $ \frac{n-1}{n} v^n$, where $n$ is the number of bidders and $v$ is a bidder's value or type. Therefore, the expected revenue should be $\...
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Does Google AdWords traffic forecasts violate the properties of Second Price Auctions? [closed]

Google AdWords claim they use a "second price auction" strategy. One of the properties of this strategy is that bids are sealed. However, Google has a tool that lets you estimate your volumes. ...
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137 views

Incentive compatible revenue maximizing multiunit auction

The Vickrey-Clarke-Groves Auction is an example of incentive compatible (truthful reporting) multiunit auction, but it is only maximizing social utility, not the seller's utility. If my ...
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103 views

Second price auction when number of items and bidders are the same

For an advertising slot bidding problem, let's say there are two slots and two bidders. If bidder A bids \$10 and bidder B bids \$8, bidder A will win the first slot and pays \$8. How about bidder B? ...
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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 ...