Mathematics auctions. Lecture in Yandex

    Auctions are used for trading almost any goods - fish and flowers, licenses for 3G-communications, government securities, shares of private companies, advertisements on Yandex. The basic rules of some auctions have remained unchanged for centuries. At the same time, auctions are a dynamically developing area of ​​modern economic theory. This is primarily due to their practical importance, as well as the availability of data and the ability to formally analyze auctions as games whose rules coincide with practice. For the study of auctions and in general economic mechanisms, several Nobel prizes were awarded.

    As part of the Small School of Data Analysis, a lecture on auctions was delivered by Sergey Izmalkov, professor of economics at the Russian School of Economics, chairman of the NES academic council. It was auctions that became the subject of his doctoral dissertation. In a lecture, Sergei Borisovich talks about what trade in wives in Babylon and how advertisements are posted on the Internet are similar. Under the cut you will find a text version of the lecture, but we would still recommend that you watch the video - the lecturer talks about everything quite accessible and very exciting.

    An auction is a way (procedure) of selling a product, maybe buying a product. An auction usually has explicit rules. For example, I will look at the proposals received, choose the best one and sell for this best offer - this is an obvious rule of the game. In life, these rules are not always explicitly formulated, for example, in the situation when you are talking with one seller, but expect that someone else will come and make a better offer. And you will start a conversation with a new seller or say to the old one: here, they say, a new buyer has appeared, so if you want to win, would you like to increase your offer? In the end, all this is aimed at making a deal. What auctions, like rules of the game, like rules of sale, can have goals? Objectives: to make this happen quick and easy; and it’s still necessary to determine at what price and who will buy it.

    How to buy fish? In the world, fresh fish - almost everywhere - is not sold in stores, but at auctions. There is a huge market in Marseille, there is a huge market in Tokyo, New York, and there is a market in Hawaii. Watching a video about a fish auction in Hawaii, we ask ourselves the following questions: how is this auction arranged, why was the auction selected, and not the store, as the method of sale, why are the rules for selling these?

    So, there is a row on which the fish lies, there is an auctioneer who lists prices. He can list them from larger to smaller, or vice versa, and he does it quickly. When transferring from larger to smaller, the price of one specific fish is determined. If the auction goes from smaller to larger, the auctioneer raises the price until there are at least two people who are ready to pay for the fish, and does this until there is only one left to buy it for a large amount. Everything happens very quickly, and the rules themselves are aimed at quickly selling everything.

    It seems that selling at auction is much more profitable than at the store. In fact, this is not so if there is a lot of goods. But if the product is unique or there are only a few items, then auctions work great - and especially in situations where you do not know exactly how much a person is willing to pay. And this is the virtue of the auction. If you have a lot of goods, you will get a market: there are a huge number of people who want to buy, there are a large number of people who want to sell, and if you want, you can accumulate statistics and sell like in a store - and in the store selling will be much faster .

    Another example: how are cows sold in the USA? Cows are allowed into the corral, a quick recitative begins (this is the language of the auctioneers), the sale ends, the cows are driven out and others start to turn on. Direct observation: everything happens quickly - it took about a minute to sell the lot. Here in one place there are many such “things” for sale: one person sells 36 cows, the other a whole herd, while at the fish auction they sell one fish each. So, it’s not necessary to sell one thing at a time, you can sell as a group.

    How to sell flowers in Holland? Here is an example of a famous Dutch auction. The price here goes down. There are five small rooms in which people sit and press buttons in order to buy flowers. 21 million flowers are sold through this auction, 80% of them are exported.

    How to sell paintings in England? Everything happens more slowly, the auctioneer makes hand movements that suggest who will give more for this picture. The explicit rules are that the auctioneer stands and waits: if someone somehow gives a sign, he raises the price automatically by 100 thousand and believes that this person is ready to buy and this is the current bid. The auction will end when there is only one person who has named the highest price that no one can kill. and the auctioneer will say: "Sold."

    What are the auctions? Of the simple auctions, there is English, with an increasing price, and Dutch, with a decreasing price (at the moment when someone says “stop”, the buyer sells the goods at the price that was when “stop” was said). There are other types of auctions - closed: the broker receives applications, considers them, and determines the winner and price based on the results of the applications. If, by ranking applications, we find the application with the highest price and sell it, it will be an auction of the 1st price. The 2nd price auction is used less often, but is very interesting. This is an auction in which we received bids, selected the highest bid and sold to the one who offered it, but the selling price is the second largest bid, that is, the maximum bid from a competitor. There is an auction where "everyone pays." All people place bets (in fact, they put money on the table), the organizer determines who bet the most, and this person becomes the winner and takes all the money. You pay, but you can both win and lose. If the players who participate are limited in funds, then this seems to be the best auction to sell. The one who spent the most wins. As an auction of this type, one can consider (and explore) the political game - the election campaign, where each participant spends money on the election campaign, in which only one of the candidates wins.

    1. Consider an English raise auction.
    Options: “shout louder”, “recitative”, “thermometer”.
    • he needs to determine the value: how much is he willing to pay
    • he needs to determine a strategy: when he should stop.

    Theorem : the best (dominant) strategy is to stand at the level of your value, to dwell on your own price.
    The one who has the most value will win, but he will pay the previous value that was indicated before him. An important feature of an English auction is efficiency.

    2. Dutch auction (and 1st price auction).
    Strategy: Bet. There is value, and you need to determine the value at which you raise your hand and say “stop.” The participant with the highest value wins, and he will pay this value. Thus, the Dutch auction and the 1st price auction are strategically equivalent as games. The more you bet, the greater the probability of winning, but the less the winning itself, respectively, you need to balance these two pieces.

    Which is better: Dutch or English auction? Wickrey's theorem (for which William Wickrey received the Nobel Prize in economics) states: they are the same.

    imageRoger Myersonin 1981, he wrote an article that answers the question: “How to sell your house with the maximum benefit”, and for this he also received the Nobel Prize (of course, not for the answer, but for the description of internal mechanics). Myerson asked the question: "What is the best way to sell a product?" The problem is that there are a huge number of sales methods. We have already seen a set of simple ways, but they can be combined. The first statement by Roger Myerson is the principle of frankness, which says that no matter how complicated your sales scheme is, there is a simple scheme, a direct mechanism for this complex scheme, when you directly ask people: “What is your value?” The mechanism is that there are two rules: the product placement function and the payment function. According to the principle of frankness, whatever the game, there is a direct game,

    The income equivalence theorem says: in two mechanisms with the same placement function, each participant of each particular type (in anticipation) will pay the same, which means that if we aggregate all payments, the seller’s expected income will be the same.

    In the English auction, the one with the highest value wins. This means that the function of placing goods in an English auction considers which of the reports is the largest, and gives this person all the goods. In a Dutch auction, if the assumptions of all bidders are the same, then the greater the value, the greater the bid. The person who bet the most will have the highest value. If so, then the function of placing goods in a Dutch auction will be exactly the same: whoever has the most value, he won.

    If we take an auction in which everyone pays, there too the product placement function will be such that the one with the highest value wins. This means that all auctions are the same in terms of sales revenue. This is not an answer to the question of the optimal mechanism. The optimal mechanism in the case of identical representations is a simple auction with the right reserve price, the lowest price at which the seller is ready to sell the goods. And all auctions are optimal mechanisms: among all possible methods of sale, they all maximize the seller’s income.

    The income equivalence theorem can be applied in another way. If I know that a 1st price auction or a Dutch auction is equivalent to an English auction (and in an English auction, I know how many people pay if they win: he pays the expected second bid), respectively, I know what strategy is then in the 1- auction 1st price: I make a bid equal to the maximum of the competitors 'bets (provided that all competitors' bids are below mine).

    Auction Properties
    • Effective. They place the goods in the best way, creating the maximum value from the transaction, the goods go to the one who is willing to pay the most.
    • Optimal. They maximize the seller’s expected income, they don’t require the seller any special knowledge about how much people are willing to pay, the competition itself between the participants will play a role.
    • Explicitly and quickly determine the price.
    • Used almost everywhere. A popular area of ​​economic research.

    An example of a popular auction is eBay. The auction rules are behind the scenes: in fact, this is an English auction. It is not explicit, not rising prices, but you are betting. The highest price (this is the current winner) is that he pays until the auction is over. This is a static and dynamic auction: anyone can bid at any time - if the bid is the highest, a new person will become a possible winner.

    More complex auctions
    • If you sell a lot of goods, you can sell sequentially, in parallel or together.
    • States and large companies make purchases from auctions (for developed countries, this is almost 20% of their budgets).
    • Privatization auctions (sale of licenses for broadcast frequencies, public transport routes).
    • Exchange, (sale of government bonds).
    • Contextual advertising (in Yandex and Google).

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