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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1answer
30 views

(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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2answers
40 views

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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188 views

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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53 views

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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33 views

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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1answer
53 views

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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44 views

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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43 views

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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1answer
36 views

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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1answer
55 views

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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189 views

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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197 views

Strategy for this auction on 100 coin flips question?

Got an interesting interview/game question today: You have the opportunity to participate in an auction for a treasure chest. The seller anonymously flips 200 coins, and for each head he adds \$1 ...
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1answer
243 views

Hard Auction Game Theory Question: Optimal Strategy with Asymmetric Info

There is a box containing $X$ dollars. $X$ is a random variable which is uniformly distributed between $0$ and $1000$. The box is auctioned via a first price sealed bid auction. There are two ...
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1answer
319 views

Expected value of $2$nd highest draw from uniform dist out of n draws

Jane wants to auction off an item, but does not know where to go to find bidders. David offers to find bidders for her, but will charge her $\$10$ per bidder he gets to show up. Each bidder will ...
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69 views

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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342 views

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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1answer
59 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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1answer
5k views

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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1answer
90 views

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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1answer
25 views

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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2answers
130 views

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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1answer
70 views

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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208 views

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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437 views

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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1answer
167 views

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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31 views

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
66 views

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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65 views

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
261 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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113 views

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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1answer
186 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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305 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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25 views

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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318 views

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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1answer
88 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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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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2answers
452 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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1answer
4k views

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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1answer
1k 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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622 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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1answer
148 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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1answer
137 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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60 views

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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2answers
286 views

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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86 views

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 ...
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1answer
661 views

Computing a revenue for VCG auction

I would like your help with the following question regarding computing a revenue for a seller of an VCG (vickrey clarke groves) auction, I'm really new to this auctions\game theory so I'd really ...
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1answer
279 views

Generalized Mechanism Design, Stanley Reiter diagram and Vickrey Auction

I am trying to learn this new topic, Mechanism Design and stumbled upon the "Stanley Reiter" diagram (see the top-right side on the page). I have also learned that the Second Price Sealed Bid ...