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In mechanism design, we do just the opposite by starting with the outcomes. We identify the outcomes we would like to have. And then we work backwards to see whether we can engineer institutions (mechanisms) that will lead to those outcomes. This is the normative or prescriptive part of economics.
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- Introduction
- i max {bk} + γ bj q∗ ( = i b) .
- Type Spaces
- i ti ∈ Ti i
- = ( )I Ti,θi,πi =1.
- Θ p( ). θ−i∈ θi,θ−i
- i i → /∅.
- Robust Foundations for Dominant and Ex Post Incentive Compatibility
- f θ
- f T
- T f
- Full Implementation
- i θ)
- f θ f θ i,qi
- i γ
- i − i > γ
- Open Issues
This essay brings together and presents a number of results on the theme of robust mechanism design and robust implementation that we have been working on in the past decade. This work examines the implications of relaxing the strong informational assumptions that drive much of the mechanism design literature. It discusses joint work of the two of ...
k i = j i We observe that if γ = 0, then the payment rule of the “general-ized” second price sealed bid auction reduces to the familiar rule of the second price sealed bid auction. If agents bid “truthfully,” setting their bid equal to their payoff bi type θi , then the generalized second price auction leads to the realization of the social choice ...
We will be interested in situations where there is common knowledge of the structure of the environment described in the previous section, but the planner may not know much about each agent’s beliefs or higher-order beliefs about other agents’ types. Thus rather than making the usual “Bayesian” assumption that the planner knows some true com-mon pr...
of his payoff type. Thus there is a function θi : Ti Θ → i, with ( ) being agent is payoff type when his type is . A type of agent θi ti ti must also include a description of his beliefs about the types of the i other agents. Writing ∆( ) for the space of probability distributions Z on Z, there is a function : πi Ti → ∆( T−i , ) with ( ) being agen...
The standard approach in the mechanism design literature is to assume a common knowledge prior, p ∈ ∆(Θ), on the set of payoff types Θ. This standard approach can be modelled in our language by identifying the set of types with the payoff types Θ and defining Ti i beliefs by ( )[ ] πi θi θ−i p( ) θi,θ−i
−i It is useful to distinguish two distinct, critical and strong, assumptions embedded in the standard approach. First, it is assumed that there is a unique belief type associated with each payoff type. More precisely, we will say that a type space is a payoff type space if each is a
others’ payoff types are implicitly defined and by writing ( ) ψi ti for those beliefs, we have that: ( )[ ] = ψi ti θ−i ( )[ ] πi ti t−i . : ( )= {t−i θ−i t−i θ−i} Now suppose we restrict attention to type spaces with the property that ψi ti ( ) ∈ Ψ ( ( )) for all agents θi ti i and types . If we require each payoff ti type to have only a single p...
In Bergemann and Morris (2005), we ask whether a planner can design a mechanism with the property that for any belief and higher-order beliefs that the agents may have, there exists a Bayesian equilibrium of the corresponding incomplete information game where an acceptable outcome is chosen. If we can find such a mechanism, then we say that we have...
( q( θ),y(θ)) specifying the allocation and transfers in our single good environment. For a given (large) type space and a given social choice T function , interim incentive compatibility on a type space requires
that: θi ( ) + ti γ ( θj tj ) qi ( θ( t)) ( − yi θ( t))πi ( )[ ti t−i ] t−i∈T−i = j i ≥ θi ( ) + ( ) qi ( θ( )) yi θ( ( )) ti γ θj tj ti,t−i − ti,t−i j=i t−i∈T−i ( ×πiti )[ t−i ] for all i, t ∈ T and ti ∈ Ti . We refer here to “interim” rather than “Bayesian” incentive com-patibility to emphasize that the beliefs o...
The above discussion applied to social choice functions. Does it extend to social choice correspondences, where multiple outcomes are acceptable for the planner for any given profile of payoff types? Suppose that the planner wanted to implement an allocation rule but did q not care about transfers — i.e., the usual setting in which efficient alloca...
All of the above results are phrased in terms of incentive compatibility, and by use of the revelation principle, are therefore statements about the existence of a truthtelling equilibrium in the direct mechanism. The construction of the truthtelling equilibrium of course presumes that when we verify the truthtelling constraint of agent that the ot...
strictly dominant strategies, that is = forms a strictly dominant bi θi strategy in this mechanism. The strictness is established by making the allocation responsive to the bid of agent even if agent is not
and yi , θi + θj qi yi >θi + qi ( γ − γ θ) ( − yi θ) , θj = = j i j i and ( ) ( ) + θi γ θj qiθ − yi θ > θi γ θj qi − yi. j=i j= i The latter condition requires that if the socially desired alternatives differ in state and , then there must exist an θ θ agent and a reward allocation ( ) such that if the true state were qi,yi θ and a...
minΨ > minΨ + γ maxΨ j. j= i j=i We can rewrite the inequality as maxΨ minΨ
(maxΨ − minΨ ) . j= i Thus Ψ separates Ψ if and only if the difference between the smallest −i i and the largest element in the set Ψ is larger than the weighted sum of i the differences of the smallest and the largest element in the remaining sets Ψ for all = does not separate Ψ if the above j j i. Conversely, Ψ −i i inequality is reversed, i.e., ...
In most of the work discussed, we defined the allocation problem in terms of social choice function or correspondence, which specified for each profile of payoff types a specific allocation. Importantly, the θ social choice function was defined independent of the beliefs of the agents and/or the principal. While this specification accommodates many...
18 Oct 2007. Mechanism design theory is a major breakthrough in the modern economic analysis of institutions and markets. It revolutionalised the way economists think about optimal institutions and regulation when governments don't “know it all.”. It has had a major impact on current policy-making and will continue to do so in the future.
1.2 Examples of Mechanism Design Problems in Network Economics 1.2.1 Mechanism Design and Algorithms for Selfish Agents In distributed settings, there are certain algorithmic problems where the agents cannot be assumed to follow the algorithm but are driven by selfish goals [16]. In such situations, the agents are capable
Mechanism design sits at the intersection of economics, game theory and artificial intelligence. It takes an engineering approach to solving a wide range of problems where there are interactions among individuals, markets and institutions. Its theoretical foundations were formed in the 1960s by University of Minnesota professor Leonid Hurwicz ...
anism design can go beyond the classical approach, and, for example, explicitly model how prices are set. In some cases, the game-theoretic approach has led to a new appre-ciation of the market mechanism. The theory shows, for example, that so-called double auctions (where buyers and sellers post their bid- and ask-prices) can be e fficient trad-
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1 Mechanism Design Framework Mechanism design has been referred to as the engineering branch of game theory. It is concerned with designing mechanisms (i.e., games) such that the outcomes that arise when the games are played by ra-tional agents (i.e., the equilibria) achieve some desiderata. The mech-anism design framework thus consists of ...
