Implementation theory: Difference between revisions

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'''Implementation theory''' is an area of research in [[game theory]] closelyconcerned relatedwith towhether a class of [[mechanismMechanism design|mechanisms (or institutions) can be designed]] wherewhose anequilibrium attemptoutcomes isimplement madea togiven addset intoof a[[normative]] gamegoals aor mechanism[[Welfare sucheconomics|welfare]] thatcriteria.<ref thename equilibrium= of"palfrey">Palfrey, theThomas gameR. conforms"Chapter to61 someImplementation Theory." conceptHandbook of socialGame optimalityTheory (forwith instanceEconomic [[ParetoApplications, optimality]])2002. {{doi|10.1016/S1574-0005(02)03024-2}}.</ref>

There are two general types of implementation problems: the economic problem of [[Production (economics)|producing]] and [[Resource allocation|allocating]] [[Public good (economics)|public]] and [[private good]]s and choosing over a finite set of alternatives.<ref name="maskin">Maskin, Eric and Sjöström, Tomas. "Implementation Theory." Handbook of Social Choice and Welfare, 2002. {{doi|10.1016/S1574-0110(02)80009-1}}.</ref> In the case of producing and allocating public/private goods, [[solution concept]]s are focused on finding [[Dominant Strategy|dominant strategies]].

In a game where multiple agents are to report there preferences (or their type), it may be in the best interest of some agents to lie about their preferences. This may improve their [[outcome (game theory)|payoff]], but it may not be seen as a fair outcome to other agents. In order to implement a more "fair" outcome, in a repeated game, the other players may choose to punish any "cheaters".

In his paper "Counterspeculation, Auctions, and Competitive Sealed Tenders", [[William Vickrey]] showed that if preferences are restricted to the case of quasi-linear utility functions then the mechanism dominant strategy is dominant-strategy implementable.<ref>Vickrey, William. "Counterspeculation, Auctions, and Competitive Sealed Tenders." The Journal of Finance 16, no. 1 (1961): 8–37. {{doi|10.1111/j.1540-6261.1961.tb02789.x}}. {{JSTOR|2977633}}.

The conditions of a repeated game where an arbitrary outcome may be enforced are set out in theorems often known as [[folk theorem (game theory)|folk theorems]]. If a game is not repeated, it may only be possible to implement outcomes which are [[Nash equilibrium|Nash equilibria]] or satisfy some other [[equilibrium selection|equilibrium]] concept.

</ref> "A [[social choice]] rule is dominant strategy [[incentive compatible]], or [[Strategy proof|strategy-proof]], if the associated [[revelation mechanism]] has the property that honestly reporting the truth is always a dominant strategy for each agent."<ref name="maskin" /> However, the payments to agents become large, sacrificing budget neutrality to incentive compatibility.

In a game where multiple agents are to report theretheir preferences (or their type), it may be in the best interest of some agents to lie about their preferences. This may improve their [[outcome (game theory)|payoff]], but it may not be seen as a fair outcome to other agents.<ref>Jackson, Matthew InO. order"A toCrash implementCourse ain moreImplementation "fairTheory." outcome,Social inChoice aand repeatedWelfare game18, theno. other4 players(2001): may655–708. choose{{doi|10.1007/s003550100152}}. to punish any "cheaters"{{JSTOR|41106420}}.</ref>

Although largely theoretical, implementation theory may have profound implications on policy creation because some [[Social choice theory|social choice]] rules may be impossible to implement under specific game conditions.<ref name="palfrey" />

== Implementability ==

In [[mechanism design]], implementability is a property of a [[social choice function]]. It means that there is an [[incentive-compatible]] mechanism that attains ("implements") this function. There are several degrees of implementability, corresponding to the different degrees of incentive-compatibility, including:

* A function is '''dominant-strategy implementable''' if it is attainable by a mechanism which is dominant-strategy-incentive-compatible (also called [[strategyproof]]).

* A function is '''Bayesian-Nash implementable''' if it is attainable by a mechanism which is Bayesian-Nash-incentive-compatible.

See for a recent reference. In some textbooks, the entire field of mechanism design is called '''implementation theory'''.<ref>Martin J. Osborne & Ariel Rubinstein: A Course in Game Theory (1994).</ref>

== See also ==

[[Incentive Compatibility]]

==References==

{{reflist}}

{{game theory}}

[[Category:Game theory]]