9

Firstly, it's right to say that the hard ending rule on ebay seems to be behind sniping. Here's a figure from Roth & Ockenfels' AER paper: On eBay (fixed end time) bids arrive much later than on the now defunct Amazon auction platform (which used an extendable end time). One explanation (see here and here) for this is that people want to hide their ...


9

If you think that the distribution of your value and your roommate's value are the same then I would suggest that you consider the following bidding protocol: each bidder submits a sealed bid, $b_i$. the highest bidder, $i$, received the console and makes a payment of $\frac{b_i+b_j}{2}$ to the loser, where $b_j$ is the loser's bid. Cramton, Gibbons, and ...


9

But perhaps this is not always the case. The price might have been so high that you would not have been willing to bid above it. In fact it might have been so high that the no one but the highest bidder would have given that bid. Knowing this the highest bidder could decrease his bid. In fact in English auctions (non-silent rising bid auctions) the final ...


9

Suppose you face a single buyer whose willingness to pay, $v$, is distributed according to $F(v)$. If you charge a price $p$, he will buy if and only if $v>p$, leaving you with expected revenue of $$r(p)=\Pr(v>p)p=[1-F(p)]p.$$ Let's maximise revenue by computing an FOC: $$r'(p)=1-F(p)-F'(p)p=0.$$ We can rearrange this as $$\phi(p)\equiv p-\frac{1-F(...


8

Suppose you write some software that you can then freely sell at practically no cost per unit (the wonders of the internet). You want to make as much profit as possible. Since you have almost no per unit cost, maximizing your profit will amount to maximizing the revenue, the price per unit times the number of units sold. Note what did not enter the ...


7

First, you should note that there is a Revenue Equivalence Theorem that says under a set of conditions, the seller's revenue from using different forms of auctions will be the same. This same result holds also in the multi-object case. Thus, if you have $n$ identical objects, you can sell them using a ($n+1$)th price sealed bid auction, where each bidder ...


7

For notational simplicity, let me define the distribution $G(s) = F^{N-1}(s)$ with density $g(s)$. Let $\underline{s} = 0$ (for simplicity). We have $$ b'(s)G(s)+b(s)g(s)=s g(s) $$ Integrating to $x$ gives us $$ \int_0^x b'(s)G(s)+b(s)g(s) ds = \int_0^x s g(s)ds $$ Notice that $\frac{\partial }{\partial s}(b(s)G(s)) = b'(s)G(s)+b(s)g(s)$, so $$ b(x)G(x) = \...


6

First I will just show that the 0.5 (or $\frac{1}{2}$) cut-off point does not work as a symmetric equilibrium, then you can decide for yourself if you want to think about the problem or read the complete answer. Let us denote the cut-off points by $c_x,c_y$. Suppose both players use the strategy $c = \frac{1}{2}$. Let us denote the numbers of player $x$ and ...


6

Generally, and surprisingly, yes, lotteries are similar to auctions. What do I mean by that? In an auction, you increase the likelihood of winning by increasing your bid. In many auctions, if you increase the bid sufficiently much, that probability will go to one. A similar thing is true for lotteries: the more tickets you buy, the higher is the likelihood ...


6

In a penny auction, each bid raises the going price of an object by just one penny. However, placing such a bid is costly whether or not you win. Usually it costs one dollar to place a bid -- sometimes this is obfuscated by making you purchase bids (or bidding credits) before you participate. In any case, if the opening bid of an auction for something is 0....


6

If there is a single item for sale, bidders are risk neutral, and bidders have independently and identically distributed private values then any two auctions that allocates the item to the highest value bidder in equilibrium offers the same surplus to the lowest value bidder yield the same revenue (in expectation). This is the celebrated revenue ...


6

A good article to read about this is by Susan Athey, Jonathan Levin and Enrique Seira. It has the title: "Comparing Open and Sealed Bid Auctions: Theory and Evidence from Timber Auctions," was published in the Quarterly Journal of Economics, vol. 126(1), 2011, 207-257, and is available online at: http://faculty-gsb.stanford.edu/athey/documents/...


6

It is actually assumed that $b_i(v_i)$ is of the form $\alpha_i+\beta_i \cdot v_i$. So it is an affine function. Linearity only works if the bottom of the uniform distribution is 0. A somewhat intuitive reasoning is that if the valuations are uniformly distributed over $[c,d]$ and $b_i(v_i)$ is a symmetric equilibrium, then $b_i(v_i)+k$ should define a ...


6

In most of the literature I have read on private value auctions you do actually know your own valuation in the auction, it is your private information. It has a distribution from the point of view of others, who can only guess at your valuation. Another interpretation would be that your valuation is actually random before getting detailed information on the ...


6

The winner pays the second highest bid, which is 5 dollars, however who the winner is depends on the tie-breaking rule. There are multiple rules one could use. In most cases it is assumed that the winner is selected randomly from among the people who placed the highest bids. But you could also select the winner from this group alphabetically or by checking ...


6

You are only required to get Nash equilibria and not sequentially rational/subgame perfect equilibria. Hence Player 2's actions at information sets that do not occur (that do not reflect Player 1's actual strategy) do not need to be best responses. All you have to make sure is that no one is better off by deviating. In case 4., regardless of what Player 2 ...


6

Let's first determine the sets of actions of the players. An action of player 1 is simply a bid $x_1 \in \mathbb{R}_+$. An action of player 2 is a function: $f_2: \mathbb{R}_+ \to \mathbb{R}_+$ that determines for every action $x_1$ of Player $1$ an action $x_2 = f_2(x_1) \in \mathbb{R}_+$. Let us denote by $F_2$ the set of all actions of player 2. Let's now ...


5

If bidder valuations are dependent on the information revealed by other bids (particularly increasing in their number and level), then we would expect too low bids because bidders have to internalize the effects of their bids on others and reduce them in response. Allowing sniping fixes this problem because last second bids don't give others an opportunity ...


5

A first price standard and reverse auction are formally equivalent to each other, and the same method can be used to solve both: First Price Auction In a first price auction, $n$ bidders choose their bid, $b_i$, as a function of their value $v_i$ (distributed according to $F$. They seek to maximise their expected payoff: $$[v_i-b_i(v_i)]\Pr(b_i\geq\max_j ...


5

It is a known result in Mechanism Design that both first-price as well as second-price auctions yield the same expected revenue, under certain conditions (like independence of valuations, private information, etc). Consult Jehle, G. A., & Reny, P. J. (2011). Advanced Microeconomic Theory (3d ed.), ch.9 on Auctions and Mechanism Design, for a very ...


5

A lot of questions here. Might be helpful to split them into different posts, but let's take a crack at them. The reason why the second price sealed auction is set up this way is simple. Imagine you have a set of individuals/bidders, thinking of what $b_i$ to bid. They all have independent values, $v_i$ of a single object up for auction (their enjoyment ...


5

This seems like a basic issue about probability calculus/theory. The intuition behind the uniform distribution over $[a,b]$ is that all outcomes between and $a$ and $b$ are equally likely. Because of this, one can get an intuitive grasp about the probability that the outcome $x$ falls in a subinterval $[a_1,b_1] \subset [a,b]$. As the ratio of their length ...


5

Given the very general description of the problem, I can think of the following (also very general) way of formulating it mathematically. Let $v_n$ be the buyer's value from owning $n$ plots of land, with $v_3>v_2>v_1$. Let $c_i$ be seller $i$'s private valuation of the plot she owns. Let $b_i$ be seller $i$'s bid, and we assume the buyer will ...


5

These are widely used technical terms with a precise mathematical meaning: Ex-ante budget balance means that the expected sum of all transfers is zero. Ex-post budget balance means that the sum of all transfers is zero with probability one. Some authors require that the sum of transfers is always zero.


5

I don't think sellers would prefer SPA over FPA. In fact, the SPA is riskier than the FPA, if we look at it from a seller's perspective. The reason is because the distribution of prices in case of the SPA is a mean preserving spread of the distribution of prices in FPA(you can refer to Vijay Krishna's Auction Theory for the proof). However, the VCG ...


5

The idea is simple: the seller wants to target that individuals who's ready to pay him the highest amount, thus targets the person with the highest virtual valuation. To target the individual who's ready to pay the most, we appeal to the concept of stochastic dominance(specifically, we are talking about first order stochastic dominance). The term $\...


5

Such a setting is called a "procurement auction" or "reverse auction." It does not fundamentally change anything. Instead of buyers with privately known valuations, the auctioneer faces sellers with privately known costs. Hence, bidders underbid each other. An efficient auction is then (typically) one where the bidder with lowest costs ...


5

$$ G(z) (z-x) = \int_x^z G(z) dy $$ and since $G$ is increasing on $[x,z]$, the right hand side is larger than $\int_x^z G(y) dy$.


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