In economics and game theory, mechanism design is the study of designing rules of a game or system to achieve a specific outcome, even though each agent may be self-interested. Economics is the social science that studies the production distribution, and consumption of goods and services. Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, Game theory is a branch of Applied mathematics that is used in the Social sciences (most notably Economics) Biology, Engineering, In Economics, an agent is an actor in a model that (generally solves an optimization problem This is done by setting up a structure in which agents have an incentive to behave according to the rules. The resulting mechanism is then said to implement the desired outcome. The strength of such a result depends on the solution concept used in the rules. In Game theory, a solution concept is a formal rule for predicting how the game will be played It is related to metagame analysis, which uses the techniques of game theory to develop rules for a game. Metagame analysis involves framing a problem situation as a strategic Game in which stakeholders try to realise their objectives by means of the options available

## Design Goals

Mechanism designers commonly try to achieve the following basic outcomes: truthfulness, individual rationality, budget balance, and social welfare. Honesty is the human quality of communicating and acting Truthfully related to Truth as a value Homo economicus, or Economic man, is the concept in some Economic theories of man (that is a Human) as a rational, perfectly informed and Microeconomics is a branch of Economics that studies how individuals households and firms and some states make decisions to allocate limited resources typically in markets "Social welfare" redirects here For other uses see Welfare A social welfare provision refers to any program which seeks to provide However, it is impossible to guarantee optimal results for all four outcomes simultaneously in many situations, particularly in markets where buyers can also be sellers[1], thus significant research in mechanism design involves making trade-offs between these qualities. Other desirable criteria that may be achieved include fairness (minimizing variance between participants' utilities), maximizing the auction holder's revenue, and Pareto efficiency. Pareto efficiency, or Pareto optimality, is an important concept in Economics with broad applications in Game theory, Engineering and the More advanced mechanisms sometimes attempt to resist harmful coalitions of players. A coalition is an alliance among individuals during which they cooperate in joint action, each in their own Self-interest.

A common exercise in mechanism design is to achieve the desired outcome according to a specific solution concept. The celebrated Gibbard-Satterthwaite theorem shows that any outcome that can be implemented as a dominant strategy equilibrium is necessarily dictatorial. In Game theory, dominance (also called strategic dominance) occurs when one strategy is better than another strategy for one player no matter how that A dictator is an Authoritarian ruler (eg Absolutist or autocratic) who assumes sole and absolute power without hereditary ascension such as an Absolute This is similar to Arrow's Impossibility Theorem. In Social choice theory, Arrow’s impossibility theorem, or Arrow’s paradox, demonstrates that no voting system can convert the ranked preferences of individuals By contrast, implementation in Nash equilibrium is possible for a much wider range of social choice rules. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which Social choice theory studies voting rules that govern and describe how individual preferences are aggregated to form a collective preference

The 2007 Swedish Bank prize in Memory of Alfred Nobel was awarded to Leonid Hurwicz, Eric Maskin, and Roger Myerson "for having laid the foundations of mechanism design theory". The Nobel Memorial Prize in Economic Sciences, officially named The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel (Sveriges riksbanks pris i ekonomisk Leonid “Leo” Hurwicz ( August 21, 1917 June 24, 2008) was an American Economist and Mathematician. Eric Stark Maskin (born December 12, 1950) is a American Economist and Nobel laureate recognized with Leonid Hurwicz Roger Bruce Myerson (born March 29 1951) is an American Economist and Nobel laureate recognised with Leonid Hurwicz

## Definition

Let N be the number of players/participants. Each player i can have a type/signal/valuation $t_i\in T_i$, e. g. in an auction the type of a player would be his valuation/reservation price for the good(s) offered. "Auctioneer" redirects here For the DC Comics supervillain see Auctioneer (comics. In Microeconomics, the reservation (or reserve) price is the maximum price a buyer is willing to pay for a good or service; or conversely the Depending on his type, the player will pick an action $s_i(t_i) \in A_i$, where Ai is the set of possible actions for player i offered by the mechanism, e. g. an action in a sealed-bid auction would be a bid of a certain amount. A sealed first-price auction is a form of Auction where bidders submit one bid in a concealed fashion Each player has utility $u_i:T_i \times O \rightarrow {\mathbb R}$, where O is the outcome generated by the mechanism. In an auction the outcome would be the final allocation of goods and the payments each player has to make.

Consequently, a mechanism M is defined to be a pair (A,g), where $A=A_1 \times \ldots \times A_N$ is the set of action offered to the players/participants and $g:A \rightarrow O$ is the function that maps the player's actions to an outcome o.

### Direct Mechanisms

Say a mechanism is direct, if the set of actions equals the set of types for each player, i. e. $\forall i \leq N: A_i=T_i$. This is true for auctions, where each player's action is to announce their valuation of the product. However, there is no need to announce the true valuation if a different strategy yields better utility. This leads to the notion of

### Direct Truthful Mechanisms

Also known as incentive compatible mechanisms. In Mechanism design, a process is said to be incentive compatible if all of the participants fare best when they truthfully reveal any private information asked for Say a mechanism is direct truthful, if it is the dominant strategy to take si(ti) = ti, i. e. to truthfully announce ones own type/valuation. As an example the Vickrey-Clarke-Grove (VCG) Mechanism is direct truthful[2]. A Vickrey auction is a type of sealed-bid Auction, where bidders submit written bids without knowing the bid of the other people in the auction

### Social Choice

We call a function $f:T_1 \times \ldots \times T_N \rightarrow O$ a social choice function. Say a mechanism M implements a social choice function f in dominant strategies, if the set of strategies that lets M generate the same output as f is a dominant Nash Equilibrium. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which In other words, there is a dominant Nash Equilibrium $(s_1,\ldots ,s_N)$ such that $g(s_1(t_1), \ldots , s_N(t_N))=f(t_1, \ldots , t_N)$. In Game theory, the Nash equilibrium (named after John Forbes Nash, who proposed it is a Solution concept of a game involving two or more players in which

### Revelation Principle

If there is a mechanism that implements a social choice function in dominant strategies, then there is also a direct truthful (or incentive compatible) mechanism implementing the same function[3]. In Mechanism design, a process is said to be incentive compatible if all of the participants fare best when they truthfully reveal any private information asked for

## Applications

One branch of mechanism design is the creation of markets, auctions, and combinatorial auctions. Sao Paulo Stock Exchangejpg|thumb| Virtual market arena where buyer and seller are not present and trade via intemediates and electronical information "Auctioneer" redirects here For the DC Comics supervillain see Auctioneer (comics. Another is the design of matching algorithms, such as the one used to pair medical school graduates with internships. In Mathematics, the stable marriage problem (SMP is the problem of finding a stable matching &mdash a Matching in which no element of the first matched Medical education A medical school or faculty of medicine is a Tertiary educational institution—or part of such an institution—that teaches Medicine A medical intern is a term used for a Physician in training who has completed Medical school. A third application is to the provision of public goods and to the optimal design of taxation schemes by governments. In Economics, a public good is a good that is non-rivaled and non-excludable. Optimal tax theory is the study of how best to design a tax to avoid distortion and inefficiency