Math auctions. Lecture in Yandex
Auctions are used to bid on virtually any product — fish and flowers, 3G licenses, government securities, shares of private companies, advertisements on Yandex. The basic rules of some auctions have remained unchanged over the 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 are the same as practice. For the study of auctions and in general, economic mechanisms were awarded several Nobel Prizes.
As part of the Small School of Data Analysis, a lecture on auctions was delivered by Sergey Izmalkov, Professor of Economics at the New Economic School, Chairman of the NES Academic Council. It was auctions that became the subject of his doctoral dissertation. In the lecture, Sergey Borisovich talks about how the trade in wives in Babylon is similar and the placement of advertisements on the Internet. Under the cut you will find the text version of the lecture, but we would recommend that you still watch the video - the lecturer talks about everything in an accessible and very exciting way.
An auction is a method (procedure) for selling a product, perhaps a purchase of a product. An auction usually has explicit rules. For example, I will look at the received offers, 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 to one seller, but expect that someone else may come and make a better offer. And you will start a conversation with a new seller or tell the old one: here, they say, a new buyer has appeared, so if you want to win, do you want to increase your offer? In the end, it’s all about making a deal. What are the auctions, as the rules of the game, as the rules of the sale, can be goals? Goals: to make it happen quickly and easily; and still it is necessary to determine at what price and who will buy.
')
How to buy fish? In the world, fresh fish - almost everywhere - is not sold in stores, but from auctions. There is a huge market in Marseille, there is a huge market in Tokyo, New York, and there is a market in Hawaii. Looking through the video about the fish auction in Hawaii, we ask ourselves the following questions: how does this auction work, why was the auction chosen as the selling method, and not the store, why are the selling rules exactly like that?
So, there is a row on which the fish lies, there is an auctioneer who lists prices. He can enumerate them from greater to lesser or vice versa, and he does it quickly. When listing from larger to smaller, the price of one specific fish is found out. If the auction goes from smaller to larger, the auctioneer raises the price until there are at least two people willing to pay for the fish, and does so until one is left who buys it for a larger amount. Everything happens very quickly, and the rules themselves are aimed at selling everything quickly.
It seems that selling at auction is much cheaper than in a store. In fact, this is not the case if there are a lot of goods. But if the item is unique or there are only a few items, then the auctions work fine - and especially in situations where you do not know exactly how many people are willing to pay. And this shows the dignity of the auction. If you have a lot of goods, the market will turn out: 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 both in the store - and the store will sell much faster .
Another example: how are cows sold in the USA? The cows are let in, the fast recitative begins (this is the language of the auctioneers), the sale is completed, the cows are driven out and they start to wind up others. Direct observation: everything happens quickly - it took about a minute to sell the lot. Here in one place many such “things” are sold: one person sells 36 cows, the other - a whole herd, while at a fish auction they sell one fish. It means that it is not necessary to sell one thing at a time;
How to sell flowers in Holland? Here is an example of the 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. Through this auction 21 million flowers are sold, 80% of them are exported.
How to sell paintings in England? Everything happens more slowly, the auctioneer makes movements with his hands, which suggest who will give more for this picture. Explicit rules are such that the auctioneer stands and waits: if someone somehow gives a sign, he raises the price by 100,000 automatic and considers that this person is ready to buy and this is the current rate. The auction will end when there remains one person who has named the highest price that no one can interrupt. and the auctioneer will say: "Sold."
What are the auctions? From simple auctions there is English, with a rising price, and Dutch, with a falling price (at that moment when someone says “stop”, the buyer sells the goods at the price that was when it was said “stop”). There are other types of auctions - closed: the broker receives bids, reviews them, determines the winner and the price based on the results of bids. If, ranking the bids, we find the bid with the highest price and sell it, it will be the auction of the 1st price. Auction of the 2nd price is used less often, but very interesting. This is the auction in which we received bids, chose the highest bid and sell 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 make bets (in fact, they put money on the table), the organizer determines who put 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 that participate are limited in funds, then, as it seems, this is the best auction sale. The winner is the one who spent the most. As an auction of this type, one can consider (and investigate) a political game — an election campaign in which each participant spends money on an election campaign, in which only one candidate wins.
1. Consider the English auction for a raise.
Options: “scream louder”, “recitative”, “thermometer”.
Participant
Theorem : the best (dominant) strategy is to stand up to the level of its value, stay at its own price.
The one who has more value wins, but he will pay the previous value, which was marked before him. An important property of the English auction - efficiency.
2. Dutch auction (and auction 1st price).
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 to games. The more you bet, the greater the probability of a win, but the less the win itself, respectively, you need to balance these two pieces.
What is better: Dutch or English auction? Vickrey's theorem (for which William Vickrey received the Nobel Prize in Economics) states: they are the same.
In 1981, Roger Myerson wrote an article that answers the question: “How to sell your house with maximum benefit,” and for that, he also received the Nobel Prize (of course, not for an answer, but for a 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 of Roger Myerson is the principle of candor, which says that, no matter how complicated your sales scheme is, there is a simple scheme for this complex scheme, a direct mechanism when you directly ask people: “What is your value?” Mechanism is that there are two rules: the function of placing the goods and the function of payments. According to the principle of frankness, whatever the game, there is a direct game, a direct mechanism (where people communicate values), such that everyone says values are true, and the result of this game is exactly the same as in the original.
The income equivalence theorem says: in two mechanisms with the same placement function, each participant of each specific type (pending) 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 wins the one with the greatest value. This means that the function of placing goods in the English auction considers which of the reports is the largest, and gives this person the entire product. In the Dutch auction, if the assumptions of all participants are the same, then the greater the value, the greater will be the bid. The person who put the most will have the highest value. If so, then the function of placing the goods in the Dutch auction will be exactly the same: who has the most value, he won.
If we take an auction in which everyone pays, there, too, the function of product placement will be such that the one with the greatest value wins. This means that all auctions are the same in terms of seller income. This is not the answer to the question about the optimal mechanism. The optimal mechanism in the case of identical representations is a simple auction with a correctly chosen reserve price, the minimum price at which the seller is ready to sell the goods. And all auctions are the best mechanisms: among all possible sales methods, they all maximize the seller’s income.
The income equivalence theorem can be applied in another way. If I know that the 1st price auction or the Dutch auction is equivalent to the English auction (and in the English auction I know how much a person pays, if he wins: he pays the expected second bid), respectively, I know what strategy then in auction 1- nd prices: I bet equal to the maximum of the competitors' bids (provided that all the bids of the competitors are lower than mine).
Auction properties
An example of a popular auction is eBay. Rules of the auction - behind the scenes: in fact it is an English auction. It is not obvious, not rising prices, but you make bets. The biggest price (this is the current winner) is what it pays before the auction ends. This is a static and dynamic auction: any person can make a bid at any time - if the rate is the highest, a new person will be a possible winner.
More complex auctions
As part of the Small School of Data Analysis, a lecture on auctions was delivered by Sergey Izmalkov, Professor of Economics at the New Economic School, Chairman of the NES Academic Council. It was auctions that became the subject of his doctoral dissertation. In the lecture, Sergey Borisovich talks about how the trade in wives in Babylon is similar and the placement of advertisements on the Internet. Under the cut you will find the text version of the lecture, but we would recommend that you still watch the video - the lecturer talks about everything in an accessible and very exciting way.
An auction is a method (procedure) for selling a product, perhaps a purchase of a product. An auction usually has explicit rules. For example, I will look at the received offers, 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 to one seller, but expect that someone else may come and make a better offer. And you will start a conversation with a new seller or tell the old one: here, they say, a new buyer has appeared, so if you want to win, do you want to increase your offer? In the end, it’s all about making a deal. What are the auctions, as the rules of the game, as the rules of the sale, can be goals? Goals: to make it happen quickly and easily; and still it is necessary to determine at what price and who will buy.
')
How to buy fish? In the world, fresh fish - almost everywhere - is not sold in stores, but from auctions. There is a huge market in Marseille, there is a huge market in Tokyo, New York, and there is a market in Hawaii. Looking through the video about the fish auction in Hawaii, we ask ourselves the following questions: how does this auction work, why was the auction chosen as the selling method, and not the store, why are the selling rules exactly like that?
So, there is a row on which the fish lies, there is an auctioneer who lists prices. He can enumerate them from greater to lesser or vice versa, and he does it quickly. When listing from larger to smaller, the price of one specific fish is found out. If the auction goes from smaller to larger, the auctioneer raises the price until there are at least two people willing to pay for the fish, and does so until one is left who buys it for a larger amount. Everything happens very quickly, and the rules themselves are aimed at selling everything quickly.
It seems that selling at auction is much cheaper than in a store. In fact, this is not the case if there are a lot of goods. But if the item is unique or there are only a few items, then the auctions work fine - and especially in situations where you do not know exactly how many people are willing to pay. And this shows the dignity of the auction. If you have a lot of goods, the market will turn out: 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 both in the store - and the store will sell much faster .
Another example: how are cows sold in the USA? The cows are let in, the fast recitative begins (this is the language of the auctioneers), the sale is completed, the cows are driven out and they start to wind up others. Direct observation: everything happens quickly - it took about a minute to sell the lot. Here in one place many such “things” are sold: one person sells 36 cows, the other - a whole herd, while at a fish auction they sell one fish. It means that it is not necessary to sell one thing at a time;
How to sell flowers in Holland? Here is an example of the 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. Through this auction 21 million flowers are sold, 80% of them are exported.
How to sell paintings in England? Everything happens more slowly, the auctioneer makes movements with his hands, which suggest who will give more for this picture. Explicit rules are such that the auctioneer stands and waits: if someone somehow gives a sign, he raises the price by 100,000 automatic and considers that this person is ready to buy and this is the current rate. The auction will end when there remains one person who has named the highest price that no one can interrupt. and the auctioneer will say: "Sold."
What are the auctions? From simple auctions there is English, with a rising price, and Dutch, with a falling price (at that moment when someone says “stop”, the buyer sells the goods at the price that was when it was said “stop”). There are other types of auctions - closed: the broker receives bids, reviews them, determines the winner and the price based on the results of bids. If, ranking the bids, we find the bid with the highest price and sell it, it will be the auction of the 1st price. Auction of the 2nd price is used less often, but very interesting. This is the auction in which we received bids, chose the highest bid and sell 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 make bets (in fact, they put money on the table), the organizer determines who put 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 that participate are limited in funds, then, as it seems, this is the best auction sale. The winner is the one who spent the most. As an auction of this type, one can consider (and investigate) a political game — an election campaign in which each participant spends money on an election campaign, in which only one candidate wins.
1. Consider the English auction for a raise.
Options: “scream louder”, “recitative”, “thermometer”.
Participant
- he needs to determine the value: how much he is willing to pay
- he needs to define a strategy: when to stop him.
Theorem : the best (dominant) strategy is to stand up to the level of its value, stay at its own price.
The one who has more value wins, but he will pay the previous value, which was marked before him. An important property of the English auction - efficiency.
2. Dutch auction (and auction 1st price).
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 to games. The more you bet, the greater the probability of a win, but the less the win itself, respectively, you need to balance these two pieces.
What is better: Dutch or English auction? Vickrey's theorem (for which William Vickrey received the Nobel Prize in Economics) states: they are the same.
In 1981, Roger Myerson wrote an article that answers the question: “How to sell your house with maximum benefit,” and for that, he also received the Nobel Prize (of course, not for an answer, but for a 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 of Roger Myerson is the principle of candor, which says that, no matter how complicated your sales scheme is, there is a simple scheme for this complex scheme, a direct mechanism when you directly ask people: “What is your value?” Mechanism is that there are two rules: the function of placing the goods and the function of payments. According to the principle of frankness, whatever the game, there is a direct game, a direct mechanism (where people communicate values), such that everyone says values are true, and the result of this game is exactly the same as in the original.The income equivalence theorem says: in two mechanisms with the same placement function, each participant of each specific type (pending) 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 wins the one with the greatest value. This means that the function of placing goods in the English auction considers which of the reports is the largest, and gives this person the entire product. In the Dutch auction, if the assumptions of all participants are the same, then the greater the value, the greater will be the bid. The person who put the most will have the highest value. If so, then the function of placing the goods in the Dutch auction will be exactly the same: who has the most value, he won.
If we take an auction in which everyone pays, there, too, the function of product placement will be such that the one with the greatest value wins. This means that all auctions are the same in terms of seller income. This is not the answer to the question about the optimal mechanism. The optimal mechanism in the case of identical representations is a simple auction with a correctly chosen reserve price, the minimum price at which the seller is ready to sell the goods. And all auctions are the best mechanisms: among all possible sales methods, they all maximize the seller’s income.
The income equivalence theorem can be applied in another way. If I know that the 1st price auction or the Dutch auction is equivalent to the English auction (and in the English auction I know how much a person pays, if he wins: he pays the expected second bid), respectively, I know what strategy then in auction 1- nd prices: I bet equal to the maximum of the competitors' bids (provided that all the bids of the competitors are lower than 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 expected income of the seller, they do not require the seller special knowledge about how much people are willing to pay, the very competition between the participants will play its role.
- Explicitly and quickly determine the price.
- Used almost everywhere. popular area of economic research.
An example of a popular auction is eBay. Rules of the auction - behind the scenes: in fact it is an English auction. It is not obvious, not rising prices, but you make bets. The biggest price (this is the current winner) is what it pays before the auction ends. This is a static and dynamic auction: any person can make a bid at any time - if the rate is the highest, a new person will be 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 conduct purchases from auctions (for developed countries, this is almost 20% of their budgets).
- Privatization auctions (sale of broadcast frequency licenses, public transport routes).
- Exchange, (sale of government bonds).
- Contextual advertising (in Yandex and Google).
Source: https://habr.com/ru/post/246399/
All Articles