Mechanism Design Theory

The diagram shows an example of the application of the mechanism design theory to traffic management. If all mechanisms are missing, the pursuit of the optimum by the individual road users leads to chaos. The mechanism right in front of left can also only achieve half of the maximum possible total result. The largest possible overall result in this example is only achieved with the traffic light mechanism.

The mechanism design or mechanism design is a branch of game theory , the rules - and thus the incentives - for games sets out to achieve a desired overall result, even if the player only pursue their own interests. A mechanism is a set of rules used to control interactions between players.

This is achieved by implementing a higher-level structure (design) in which the players are given an incentive to behave according to these rules. The result of this mechanism is called the implementation of the desired overall result. The strength of this result depends on the solution concept, i.e. on the established rules. It is based on metagame analysis , which uses game theory methods to develop new rules for a game.

In the mechanism design is a recursive application of game theory analysis instead: There is not asked how the players will play a defined game, but like a game designed (designed, Design ) must be to obtain a certain result. The rules created in the game are called a mechanism . A classic area of ​​application for mechanism design theory is the design of rules in a market.

theory

The theory assumes that a market does not ensure an optimal allocation of resources through an invisible hand , but that an imperfect market prevails. A non-optimal market is to be optimized with the help of mechanisms .

The economic theory also shows that under certain conditions such as complete competition , no external effects , no individual players have market power , etc., the rule can be implemented by the market mechanism without state intervention. In this sense, the market mechanism can be seen as a form of implementation of the rule. In this case, implementation corresponds to the work of the invisible hand.

The principle of the mechanism design also works in other areas of life. The results and methods of the mechanism design are also applied in the field of economics and social sciences . It is then examined how the (mostly) legal framework can be changed in such a way that certain, desired behavior is encouraged or an unwanted one is prevented.

A practical application of mechanism design theory is the question of how relationships with business partners should be designed in order to achieve the desired results (the agreed rules are then the "mechanism" to be designed). In applied game theory, this rule design is popularly referred to as coopetition .

The scientists Leonid Hurwicz , Eric S. Maskin and Roger B. Myerson were awarded the Nobel Prize in Economics in 2007 for their research in the field .

definition

${\displaystyle N}$represents the number of players. Each player has a value called the type of player. In an auction, for example, this value would represent this player's reservation price for the goods on offer. Depending on his type, the player will choose the action , with the alternative courses of action made possible by the mechanism for the player . An example of an action in a closed auction would be a bid for a certain amount. Each player has the benefit , with the amount representing the possible outcomes of the mechanism. In an auction, the result would be the completed allocation of goods and payments that each player must make. The benefit for each player would be the reservation price of the goods allocated to him, minus the price to be paid. ${\displaystyle t_{i}\in T_{i}}$ ${\displaystyle s_{i}(t_{i})\in A_{i}}$ ${\displaystyle s_{i}(t_{i})\in A_{i}}$ ${\displaystyle i}$ ${\displaystyle u_{i}\colon T_{i}\times O\rightarrow {\mathbb {R} }}$ ${\displaystyle O}$

Accordingly, a mechanism is defined as a pair , representing the options for the player to act and the function that leads the player's actions to result . ${\displaystyle M}$ ${\displaystyle (A,g)}$ ${\displaystyle A=A_{1}\times \ldots \times A_{N}}$ ${\displaystyle g\colon A\rightarrow O}$ ${\displaystyle o}$

Direct mechanisms

A mechanism is direct if the options for action are equal to the number of values ​​of each player, for example for all . This is the case with auctions, where every bid by the player announces their preference value for the product. However, there is no need for actual evaluation if a different strategy produces better benefits. This leads to the thought of direct truthful mechanisms. ${\displaystyle A_{i}=T_{i}}$ ${\displaystyle i}$

In a direct mechanism, each individual involved is asked for his or her private information. The announcements are then assigned the result of a social selection function. A central result of the theory of mechanism design is that any social selection function that can be implemented can always also be achieved through a direct mechanism. This result is called the revelation principle. The direct mechanism takes over, so to speak, playing the balanced strategy belonging to the type of player. If it is optimal for a type to play a certain strategy under the indirect mechanism, it is optimal to advertise the type truthfully under the direct mechanism. A concrete example is the measurement of willingness to pay. A direct survey can be carried out to measure willingness to pay, but this direct mechanism has its limits because the direct survey is subject to validity and reliability deficiencies. In reality, the buyer always compares his benefits with the price, but here the price is viewed in isolation.

Direct truthful mechanisms

Also known as incentive-compatible mechanisms . A mechanism is said to be directly truthful with regard to a given game-theoretic solution concept if it applies to the strategy , i.e. the truthful disclosure of its own type, that it is an equilibrium strategy in the chosen solution concept. The Vickrey-Clarke-Groves Mechanism, for example, is directly truthful in dominant strategies. ${\displaystyle \!\ s_{i}(t_{i})=t_{i}}$

Dominant strategies only exist for a few mechanisms. Mechanism design problems are often modeled as Bayesian games , in which the player types are represented by random quantities and the outcome of the mechanisms in Bayesian Nash equilibrium is of interest.

Incentive-compatible mechanisms investigate which rules must be set so that both sides show a certain behavior in a certain situation. At the core is the connection between the outcome of the negotiation and the information provided by the actors about their respective private information. An example of an incentive-compatible solution for measuring willingness to pay would be the following scenario: The prospective buyer first specifies the price he would be willing to pay for a product. A random price is then drawn. If the price is below the stated willingness to pay, the interested party must buy the product at the drawn price; if the drawn price is higher, there is no purchase obligation. In this case, the mechanism is incentive-compatible because it is in the buyer's own interest to reveal his or her true willingness to pay.

Social choice

One function is called a social selection function . A mechanism implements a social selection function (with regard to a solution concept) if there is a tuple of strategies with the following properties: ${\displaystyle f\colon T_{1}\times \ldots \times T_{N}\rightarrow O}$ ${\displaystyle M}$ ${\displaystyle f}$ ${\displaystyle (s_{1},s_{2},\ldots ,s_{n})}$

• the tuple represents a balance in the chosen solution concept, and ${\displaystyle (s_{1},s_{2},\ldots ,s_{n})}$
• it applies , that is, the selection function applies in equilibrium . ${\displaystyle g(s_{1}(t_{1}),\ldots ,s_{N}(t_{N}))=f(t_{1},\ldots ,t_{N})}$ ${\displaystyle f}$

Revelation Principle

If there is a mechanism that implements a social selection function, then there is also a directly truthful (or incentive-compatible) mechanism that implements the same function.

Examples

basketball

An example of mechanism design is setting the rules of the game for a sport. In the basketball game, the team in charge of the ball must have completed its attack within 24 seconds, otherwise possession changes. There is no tie in basketball. If the score is equal after the end of the regular playing time, there is an extension of five minutes each time until a team has won by at least one point. This mechanism leads the two teams to compete faster and more aggressively when playing basketball.

Dividing the cake

There is a piece of cake for two children. How can this cake be divided for the two children so that the two children are satisfied? The satisfaction of the two children is here considered to be the intended result, and the distribution rule corresponds to the mechanism to be implemented in this distribution game. A good mechanism here is as follows: Child A should split the piece of cake into two parts, then child B will first select one part of it, and child A will receive the other part.

Vickrey auction

→ Main article : Vickrey auction

The Vickrey auction is an example of a mechanism for auctions. All bidders submit concealed bids at the same time and the bidder with the highest bid receives the goods to be auctioned. However, he only has to pay the price of the second highest bid. The rules here are designed in such a way that the best strategy for every bidder is to bid as much as the good is worth to him.

literature

• Hans Peter Grüner ; Economic Policy Allocation Theory Basics and Political-Economic Analysis; Pp. 24-30; 3. conditions; 2007; Jumper.
• Manfred J. Holler; Gerhard Illing; Introduction to game theory; Pp. 340-356; 5. requirements; 2003; Jumper.
• Bezalel Peleg; Peter Sudhöller; Introduction to the Theory of Cooperative Games; 2nd edition; 2007; Jumper.
• Steven J. Brams; Alan D. Taylor; The WIN / WIN Solution: Guaranteeing Fair Shares to Everybody; 1st edition; 1999; New York.
• Ingo Pies : Normative Institutional Economics. To rationalize political liberalism . JCB Mohr (Paul Siebeck) , Tübingen 1993.
• Sebastian Pickerodt; Information goods trade with the help of autonomous agents Profit maximization through price differentiation; Pp. 156-159; 1. Conditions; 2006; Wiesbaden.

Web links

• Background report of the Handelsblatt
• Game theory terms
• Professor Rieck's Game Theory - Page
• Explanation of the mechanism design and examples
• ipoint - the information portal of the University of Innsbruck
• Science for practice; Burkhard Hehenkamp; The basics of the mechanism design theory for the Nobel Prize to L.Hurwicz, ESMaskin and RBMyerson
• Science magazine for business and economics

Individual evidence

1. ↑ cf. Milgrom, Paul Robert: Putting Auction Theory to Work p. 21 (2004) Google Books
2. ↑ cf. Dutta, Prajit K .: Strategies and Games p. 349 (1999) Google Books
3. ↑ cf. Rieck, Christian: Professor Rieck's game theory page  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice.  
4. ↑ cf. Manfred J. Holler & Gerhard Illing: Introduction to game theory; Pp. 340-343; 5th edition; 2003; Jumper
5. ↑ Cf. Hans Peter Grüner, Green Economic Policy, Allocation Theory Basics and Political-Economic Analysis 3rd Edition p. 24 2007
6. ↑ ^{a b} See Archived copy ( memento of the original dated December 26, 2008 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. for a proof. 
7. ↑ See archived copy ( memento of the original dated February 7, 2009 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this notice. 
8. ↑ See STBrams, ADTaylor; The Win-Win Solution; Guaranteeing Fair Shares To Everybody; Norton 1999