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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Weak dominance in second-price auction

Consider a sealed-bid second price auction with two bidders. Standard textbooks claim that bidding one's true valuation $v_i$ weakly dominates bidding lower than one's valuation $b_i<v_i$. But I am ...
PGupta's user avatar
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Determining bid price based on probability distribution

I’m doing a Game Theory Project but unsure how to continue with the mathematics. My idea was bid pricing strategies in sealed first-price reverse auctions e.g. government contracts, raw material ...
yang zhang's user avatar
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What is the expected payoff in an asymmetric first sealed bid price auction with uniform distributions?

A has an IID uniform distribution in [0,6] for item X. B has an IID uniform distribution in [2,4] for item X. There is an auction X, highest bidder wins and pays what they bid. The player's payoff is ...
Dio's user avatar
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Truthful budget partitioning

Suppose there is a budget $M$ that needs to be partitioned into two projects $A$ and $B$. There are two players, where the first player prefers to allocate $pM$ to A and $(1-p)M$ to $B$ (i.e., it ...
Doron's user avatar
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Conditional density of the second highest order statistics

Notation: $Y_1$: Highest order statistics of $(N-1)$ players' valuation. $F_n^M:$ The distribution function of the highest $n$th order statistics of $M$ players. $f_n^M:$ The density of the highest $n$...
Longye Tian's user avatar
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Bidding Theory and Probability of getting highest value in list after ignoring first half of list

You are offering an item for sale and there are n bidders. The bidders appear in sequence. When bidder i appears he or she makes a bid bi > 0. You must decide immediately whether to accept the bid ...
Luis Lascano's user avatar
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1 answer
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Expected revenue in an auction

Consider an auction in which two independent buyers bid for an antique object. If each of them draws their valuations from $\mathcal{U}[0,1]$ and the auction lets the highest bidder take the object by ...
Rick_Morty's user avatar
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Single Price Auction [closed]

I am self-studying the following question: there is a bag with unknown value (can be negative). Everyone bids a price and whoever bids the highest wins the bag. Design the optimal strategy. This is a ...
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optimal bid strategy for uniform distribution

Ten players take part in the following auction for a $\$$100 project from some company. If some player $i$ wins, the company will pay him his bid $b_i$. The ten players submit bids simultaneously, and ...
Craig Thone's user avatar
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98 views

Strategy in auction game given asymmetric information

Suppose that 100 coins are flipped, and for every head \$1 enters a bucket. You and a friend are bidding on the bucket in a first price sealed bid auction, where bids are limited to integer values (...
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All-pay auction question

Two players take part in the following auction for a £1000 prize. The two players submit bids simultaneously, and the higher bid wins the prize (if bids are identical each gets £500). Both the winner ...
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Equivalence of uniform distributions

If I have values $v$ uniformly distributed over $\left[1, 2\right]$ where $\operatorname{F}\left(v\right) = v - 1$, is this equivalent to a uniform distribution where $\operatorname{F}\left(v\right) = ...
user354621's user avatar
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Computing the optimal price from an auction with uniform distributed values

If I sell an item with value uniformly distributed between $0$ and $500$, and I value this item at $200$, then what is the ideal price to try to sell the item for? Note that once I set a price, I ...
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Most profitable multi unit open auction design

Imagine a seller have 300 apples to sell Bidders can offer their budget to buy apples. Bidders can buy multiple apples. Apple price must be same for everyone. Bidders can put bids like this: Karen: ...
Uchiha Itachi's user avatar
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144 views

Sealed bid first price auction

I need help with the following question: In a first-price sealed-bid auction with two players, both the player's type/valuations are taken from the set $\{10,20,30,40,50\}$ with the assumption that ...
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Solving a specific case of a VCG mechanism with quasi-linear functions

Consider the following allocation problem. One unit of a perfectly divisible good is allocated among three agents. Each agent has quasi linear utility, with valuations $v_{1}\left(x_{1}\right)=\sqrt{...
cem's user avatar
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Math formula for voting system that takes candidate age into account.

I have the following problem (not exactly but close): I have a website where people can post their cats. Its called bestcat.com. Everyday people submit cats and people can vote once on their favorite ...
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How to find an optimal strategy in an auction?

Consider an auction of sculptures by four artists: A, B, C and D. The auctioneer introduces each item to be sold in the auction room, and all bidders write their bids on a secretly sealed note that is ...
plpm's user avatar
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Handling Reserve in Vickrey Auction

I'm trying to learn more about auction theory and after studying Vickrey and first price auctions I wanted to try including a reserve price (in a Vickrey auction), but something seems to be going ...
Paul's user avatar
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Strategy for second-price auction with multiples

Suppose there is an auction website. The rules are that the winner pays second-highest bid, and if a bid is made with less than 3 minutes left on the timer, the timer is reset to 3 minutes. A seller ...
Erhannis's user avatar
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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 ...
nz_21's user avatar
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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 ...
nz_21's user avatar
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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'...
Luiz Peres's user avatar
1 vote
2 answers
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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 ...
user815048's user avatar
1 vote
0 answers
135 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 ...
AlexanderJI's user avatar
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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,...
Floris's user avatar
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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 ...
Tom's user avatar
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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 ...
avs's user avatar
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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 ...
Ponx's user avatar
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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 $x_i(\mathbf{b}) \in \{0,1\}$...
user306747's user avatar
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0 answers
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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 ...
Intelligent-Infrastructure's user avatar
1 vote
0 answers
490 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 $n$ ...
deb2014's user avatar
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4 votes
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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 ...
user107224's user avatar
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4 votes
1 answer
712 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 ...
TheMathBoi's user avatar
2 votes
1 answer
3k 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 ...
rkk1995's user avatar
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1 vote
0 answers
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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 ...
Sara Saletti's user avatar
2 votes
0 answers
921 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 ...
Christopher King's user avatar
1 vote
1 answer
140 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. ...
chillz's user avatar
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4 votes
1 answer
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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 $b_i$. ...
Jose Luis Montalvo Ferreiro's user avatar
-1 votes
1 answer
108 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,...
p2r3i5m7e's user avatar
1 vote
1 answer
28 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 ...
Jinsoo Bae's user avatar
2 votes
2 answers
143 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 ...
Jinsoo Bae's user avatar
2 votes
1 answer
80 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}...
Daniel Katzan's user avatar
1 vote
0 answers
316 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 ...
mnm123's user avatar
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0 votes
1 answer
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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 ...
zohaad's user avatar
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-1 votes
1 answer
450 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 ...
Simon Wilson's user avatar
2 votes
0 answers
35 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 \...
Jinsoo Bae's user avatar
2 votes
1 answer
80 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 $\...
Jinsoo Bae's user avatar
1 vote
0 answers
70 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 ...
Seppo420's user avatar
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0 votes
1 answer
407 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 ...
pafnuti's user avatar
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