Mechanism Design

2 The Institutional View on Markets

2.2 Mechanism Theory

2.2.1 Incentival Mechanism Theory

2.2.1.1 Mechanism Design

The theory of mechanism design⁵⁸ provides a theoretical toolbox for designing institutions with a particular emphasis on incentives (Maskin and Sjöström 2002). Basically, the problem of designing a mechanism, i.e. game form, is to implement a mechanism (M,y) such that the equilibrium outcome satisfies the social choice function y^SCF. Here the problem arises that the agents may have an incentive to misrepresent their preferences: “The basic design prob-lem can be stated simply. The “gaming” behavior that could undermine price discovery, and thereby efficiency, is the strategy called »hiding in the grass«” (Wilson 2001, par. 6). Even abstracting from communication costs, the mechanism designer may attempt to elicit the true preferences. But if the agents know the outcome rule they improve their situation by simply report false preferences. Ideally, the task of the mechanism designer is to devise a mechanism such that (1) the agents announce their true preferences and that (2) the “right” allocation is chosen. The critical design question is accordingly whether there exists such a mechanism that implements a specific social choice function (Maskin 1999).

Some authors distinguish implementation theory from mechanism design by referring to the multiplicity problem: Mechanism design literature merely focuses on the question, whether a specified outcome can be attained as an equilibrium of some mechanism. This also implies that it ignores other equilibria than the desired. Implementation theory also accounts for those undesired, multiple outcomes, by requiring that all equilibrium outcomes satisfy the desired properties (Jackson 2001). In situations, where mechanism design theory comes to a negative result, i.e. no mechanism can attain a given social choice function as an equilibrium, the im-possibility is strict. However, in situations, where mechanism design comes to a positive re-sult, in a way that there exists a mechanism that can implement a given objective in equilib-rium, the possibility should be handled with care. The reason is that there might exist more equilibria, which do not satisfy the demanded objectives (Jackson 2001). Implementation the-ory is, however, aggravated by its natural devotion to highly abstract mechanisms “with little or no concerns for practical application” (Palfrey 2001, 2274). The degree of abstraction hinges on the objective of implementation theory: Usually mechanisms are constructed in a way that they apply to arbitrary social choice functions. For example, an equilibrium concept, say a Bayesian-Nash, equilibrium, is selected and analyzed under which conditions a social

57 The term informational environment is taken from auction theory (Krishna 2002). In our terminology the informational environment refers to the preference structure (cf. chapter 2.1.2.2).

58 “Mechanism design in general, in the spirit laid out in Hurwicz (1973) has become a recognized subject in the theoretical literature, and even boasts a specialized journal, the Review of Economic Design”(Roth 2002).

choice function is implementable. Thereby, the domain restrictions upon the environment should be as minimal as possible. In other words, a single game form is identified that imple-ments all arbitrary social choice functions in equilibrium (assumed the specified equilibrium concept). In order to do so, the mechanism must be of abstract nature.

In the following, the mechanism design literature is reviewed as it provides more practical insights what can and cannot be achieved by the mechanism designer. Mechanism design is supposed to bridge the gap between theoretic microeconomic implications and practical appli-cability.⁵⁹

The results of mechanism design theory are easier to understand if one has a thorough under-standing about the exogenous and endogenous variables of the mechanism design problem.

Usually, mechanism design takes the environment e, the set of outcomes X, the behavior m, the (quasi-linear) utility function and the distribution functions as given. Now, the mechanism designer has to choose a mechanism (M, y) such that the utility of the society is maximized.

However, the problem is hardly tractable as the message space M can be extremely huge. A very valuable shortcut has been developed to restrict to a certain set of mechanisms. The shortcut, known as the revelation principle⁶⁰, relies on game-theoretic reasoning on behavior and states that for any equilibrium of any (indirect) mechanism there exists an equivalent in-centive compatible direct mechanism that attains the same outcome.

Remark 2.2-1: Revelation Principle

For the understanding of the revelation principle the notion of direct mechanisms and in-centive compatibility are essential.

• A direct mechanism is defined as mechanism, where the message space M is the type space of the agents Θ . The agents may only announce claims about their true prefer-ence. Those announcements can but must not be truthful. Note that a direct mechanism also represents a social choice function.

• An incentive-compatible direct mechanism is a mechanism, where the agents truth-fully report their preferences, which are private information. Incentive compatibility accordingly implies that the agents put their selfishness behind. Truthful reporting the preferences can either be a dominant strategy or equilibrium of a mechanism. If truth-ful announcement is a dominant strategy, the utility drawn out of truth telling is as least as good as any other arbitrary strategy. This – so-called strategy proofness – is a quite strong requirement: regardless of the other agents’ strategies, truth telling is al-ways the most profitable strategy (Jackson 2002a). Dominant strategies coincide with the removal of game-theoretic reasoning; agents need not to form conjectures about the other agents’ reactions.⁶¹ Naturally, theorists strive for an implementation of social choice functions in dominant strategies. However, implementation in dominant strate-gies imposes a very strong demand on the design problem and is accordingly not al-ways possible. A weaker formulation of incentive compatibility requires truth telling as equilibrium behavior.

59 As such, mechanism design founds the basis for a computerized mechanism design, a fruitful application area for electronic markets.

60 The revelation principle was initially developed by the works of Gibbard, Myerson, and others (Gibbard 1973; Myerson 1982). See (Myerson 1989) for a survey on mechanism design literature.

61 This truth telling property extends the mechanism beyond those, where only “honest men” are taking part.

As previously mentioned, the revelation principle says that for any mechanism there exits an equivalent incentive-compatible direct mechanism that implements the same social choice function.

The underlying intuition of the revelation principle is as follows. Suppose the microeco-nomic system can be totally simulated in the laboratory. This simulation comprises the strategies of the participating agents, the choice, transfer and adjustment process rules of a complex (indirect) mechanism. The simulated mechanism will compute the agent’s opti-mal strategy faithfully based on the preferences. For an agent, it is an optiopti-mal strategy to truthfully report his preferences to the new (simulated) direct mechanism, because the program optimizes the agent’s strategy faithfully based upon this report. Hence, it does not make sense to lie to the simulator (Matthews 1995). Evidently, this new direct incen-tive-compatible mechanism implements the same social choice function as the indirect mechanism.

This shortcut allows – without loss of the designer’s objective – restricting one’s attention to direct incentive compatible mechanisms.

With the device of the revelation principle mechanism design can derive a number of impos-sibility theorems that reveal the set of properties that cannot be attained by any mechanism under a specific environment (Sen 1999). The reasoning is now straightforward, if no direct mechanism exists, which satisfy some properties, there is no mechanism (including iterative and other indirect mechanisms) that satisfies these set of properties.

2.2.1.1.1 Impossibility results

In the following the most discussed impossibility theorems are summarized. Those theorems state for which combination of properties no mechanism does exist. Table 2 sketches the im-possible combinations of properties dependent on the preference domain⁶², i.e. the restrictions on possible orderings of the alternatives, and the used equilibrium concept.

Environment Performance Theorem

Preference Domain Resources Equilibrium

Con-cept Impossible

Gibbard-Satterthwaite Rich Discrete set of

Commodi-ties Dominant non-dictatorial

Hurwicz-Green-Laffont Quasi-linear Single units of the same

resource Dominant Allocative efficient

and Budget Balanced Myerson-

Sattterthwaite Quasi-linear Single units of the same

resource Bayesian-Nash Allocative efficient, Budget Balanced, Individual rational Green-Laffont Quasi-linear Single units of the same

resource

Coalition-proof Allocative efficient

Table 2: Impossibility Results (Parkes 2001)

As the impossibility theorems demonstrate the feasibility of mechanisms satisfying some properties, they are important for design. In the following, the details of the impossibility theorems are outlined.

• Gibbard-Satterthwaite Theorem

The Gibbard-Satterthwaite theorem reveals that it is impossible to implement a non-dictatorial social choice function in dominant strategies, if the preferences are sufficiently

62 For a detailed formulization of preference domains see Dasgupta et. al. (Dasgupta, Hammond et al.

1979).

rich⁶³ (Gibbard 1973; Satterthwaite 1975). A dictatorial social choice function denotes those social choice functions that are dependent on the utility of a single agent, i.e. the dic-tator. Changes in the individual utilities other than the dictator do not influence the social choice function. Unfortunately, the theorem states that it is impossible to implement a truthful non-dictatorial social choice function in dominant strategies if there are more than three agents and the preference domain is rich. A rich preference domain simply requires that all strict orderings must be possible (Maskin and Sjöström 2002). If the preference domain is restricted this strong impossibility theorem no longer holds.

• Hurwicz-Green-Laffont Theorem

Parkes points in his dissertation at the – as he calls it – “Hurwicz Impossibility Theorem”

(Parkes 2001). Basically the theorem states that it is impossible to implement an incentive compatible mechanism in dominant strategies that is allocative efficient and budget-balanced when the preferences are quasi-linear and single units of the same good are allo-cated. It can be shown that only the so-called Groves mechanisms are strategy-proof, i.e.

incentive compatible in dominant strategies. However, no Groves mechanism achieves budget balance in this restricted environment (Green and Laffont 1977; Jackson 2002a).

• Myerson-Satterthwaite Theorem

The Myerson-Satterthwaite theorem extends the results of the Hurwicz-Green-Laffont theorem to Bayesian implementation problems. Accordingly, truth telling is no longer re-quired to be a dominant strategy but Bayesian-Nash equilibrium behavior. Bayesian in-centive compatible demands that truthful reporting is a Bayesian-Nash equilibrium. Any deviation from truth telling reduces the expected utility of any agent. Even with this re-laxed (less strict) equilibrium concept, the Hurwicz-Green-Laffont impossibility holds if additionally individual rationality is also required (Myerson and Satterthwaite 1983;

Fudenberg and Tirole 2000). In summary, it is impossible to achieve all three properties allocative efficiency, individual rationality and budget balance in markets with quasi-linear agent preferences as a Bayesian-Nash equilibrium (Parkes 2001).

• Green-Laffont Theorem

Another impossibility theorem from Green and Laffont reveals that there does not exist a strategy-proof mechanism that is allocative efficient and simultaneously robust to manipu-lations by coalitions. This impossibility still occurs in environments where agents have quasi-linear preferences (Green and Laffont 1979; Parkes 2001).

2.2.1.1.2 Possibility results

Mechanism design not only provides impossibility but also possibility results. Those possibil-ity results must, however, be used with great care. As previously mentioned, those possibilpossibil-ity results demonstrate what is principally possible. Due to the chance of other (bad) equilibria it is not guaranteed that the mechanisms always provide those desirable properties. Possibility results can be distinguished into two groups optimality and efficiency theorems. Optimality theorem(s) seek to identify mechanisms that maximize the revenue a selling agent receives, whereas efficiency tries to find mechanisms that maximize total utility of the society.

Optimality Theorem

In his salient paper, Myerson developed an approach to derive a revenue maximizing mecha-nism (Myerson 1981). Basically, he transformed the mechamecha-nism design problem to an optimi-zation problem. His objective is to construct a choice rule that maximizes the expected reve-nue under three additional constraints. The first constraint, allocative feasibility requires that resources can only be allocated if they do exist. The second constraint, incentive

63 Reny points at the similarity between the Gibbard-Satterthwaite theorem and Arrow’s celebrated impos-sibility theorem in a voting setting (Reny 2001).

ity, demands agents to truthfully announce their true preferences. The third constraint, indi-vidual rationality, presupposes the expected utility of participation to be higher than non-participation.

Myerson subsequently developed conditions on the choice and corresponding transfer rules without explicitly formulating them. Moreover, Myerson derived an optimal auction for the single-item case. Overall, optimal auctions are exclusively a theoretical construct without practical application (Wolfstetter 1999). Satterthwaite characterizes this claim in a more de-tailed way: “optimal mechanism depends critically on the agents and mechanism designer sharing a common knowledge prior of the ex ante distribution of each other's preferences.

Common knowledge among the agents who participate in the mechanism is a strong assump-tion; for this common knowledge to extend to the mechanism designer is arguably untenable”

(Satterthwaite 2001). Thus, the description of optimal mechanisms ends here with the refer-ence to Bulow and Roberts for further depiction (Bulow and Roberts 1989).

Efficiency Theorems

The Gibbard-Satterthwaite theorem projects a dismal chance for mechanism design: Only dictatorial social choice functions are strategy proof and together efficient. Nonetheless, the negative results of the Gibbard-Satterthwaite theorem can be easily circumvent by either re-stricting the preference domains or by using a less stringent equilibrium concept. The most important results are presented in Table 3. Similar to Table 2, Table 3 demonstrates which properties a mechanism can attain under a specific environment.

• Groves Mechanism

If only quasi-linear preferences are permitted there exists a class of mechanisms that are allocative efficient, and for which truth telling is a dominant strategy (Groves 1973).

Those mechanisms usually called Groves-mechanisms are characterized by an efficient

choice rule: ⁼ _∈

∑

i

i

^ X i

h

^ ) argmax v (h, )

(

*

h θ θ

and a transfer rule for the i-th agent

ˆ ) ( q ) ), ˆ , (

* h ( v ˆ)

(

t _{i i}

i j

j i i j

i −

≠ − +

=

∑

^{θ θ θ θ}

θ

Note that the transfers depend on two components:

The first term is sum of the valuation for all agents j≠ , where agent i announces some i value θ⁾_i and all other agents are faithfully reporting their preference. This component of the transfers accounts for the effects agent i places on the society by his announcement of his preferences. Those externalities are internalized, as the effects posed on the other agents are incorporated in the transfer function. This way, the society goals can be recon-ciled with the individual goals.

The second term qi is an arbitrary function, which depends on all but agent i’s preferences.

Thus, there exist a number of mechanisms that belong to the class of Groves-mechanisms.

If the preferences are quasi-linear, the Groves mechanisms are the only class of mecha-nisms for which allocative efficiency and strategy proofness holds. Accordingly, one can restrict one attention to those class of mechanisms (Green and Laffont 1977). Holmstrom additionally shows that further restricting of the preference domain does not sway this re-sult; the class of Groves-mechanisms remains the only class, which attains those

proper-ties (Holmstrom 1979).⁶⁴ Nonetheless, Groves mechanisms have also undesirable proper-ties as they usually do not balance budget.

• VCG Mechanism

One version of the Groves scheme is the so-called VCG (Vickrey-Clarke-Groves), pivotal or Clarke mechanism. What makes the VCG mechanism powerful in mechanism design are the nice properties associated with it. The VCG mechanism denotes a special Groves mechanism where the arbitrary part of the transfer schedule is specified as

∑ ( )

∈ ≠

−

=

i

j j j

H i h

i( ) max v h,

q θ⁾ θ .

The total transfers amounts to

∑ ∑

∈ ≠

≠ − −

=

i

j j j

H i h

j j i i j

i( ˆ) v (h*( ˆ , ), ) max v (h, )

t θ θ θ θ θ .

As the transfers are always negative (i.e. the agents have to pay), the mechanism is always feasible. Furthermore, the interpretation of the transfers is instructive (Jackson 2002a): if agent i’s presence does not make a difference in the maximizing problem (viz. agent i is not part of the optimal allocation), the transfers are zero. Otherwise i’s presence is pivotal, as the social welfare, i.e. the sum of all agents, is affected by the participation of agent i.

The transfers exactly reflect the loss in valuation of the other agents, which is incurred by the participation of agent i. Accordingly, the VCG mechanism incorporates the marginal impact on the other valuations by the announcement of θ⁾_i into the transfer function inter-nalizing this external effect. At the bottom-line the individual agent is thus forced to con-sider also social welfare when making his choice. Altogether, the VCG mechanism is the only mechanism that achieves allocative efficiency, individual rationality and is also fea-sible, as the transfers – although they do not balance – are negative.⁶⁵

• AGV Mechanism

The AGV mechanism basically represents the “expected Groves” mechanism.

d’Aspremont and Gerard-Varet, and independently Arrow show that one can weaken the requirement of dominant strategy incentive compatible to Bayesian strategy incentive compatible as long as the agents have probabilistic beliefs over the type distribution (d’Aspremont and Gérard-Varet 1979; Jackson 2002a). The choice rule is exactly the same as for the Groves mechanism. Only the transfer rule differs in a way that not the ac-tual valuations are used but their expected value. That is:

ˆ ) ( q ) ), ˆ , (

* h ( v E

ˆ) (

t _{i i}

i j

j i i j

i i −

≠ − ⎥+

⎦

⎢ ⎤

⎣

= ⁻ ⎡

∑

^{θ θ θ θ}

θ _θ

Again, the second term qi denotes an arbitrary function independent of agent i’s valuation.

The first term reflects the expected value of the other valuations, provided that agent i an-nounces some arbitrary valuation θ⁾_i, and all other agents correctly report theirs.

Comparable with the Groves mechanism the AGV mechanism can achieve an allocation efficient and, different than the Groves mechanisms, one can construct the qi in such a way that budget is balanced (Arrow 1979; d’Aspremont and Gérard-Varet 1979). Follow-ing Krishna and Perry’s instructive presentation (Krishna and Perry 1998), function qi is given by

∑ ( )

≠ −

⎟⎠

⎜ ⎞

⎝

⎛

= −

i

j j j

i SW

1 I

q 1 θ⁾ ,

64 This holds when the domain is smoothly connected. For the notion of smoothly connectedness see Holmstrom (Holmstrom 1979).

65 Only in the special case Groves and Loeb showed that the VCG balance budget if the valuations are quadratic (Groves and Loeb 1975).

where SW denotes the “expected social welfare” or in the terminology used here the ex-pected sum of all j≠ individual valuations when the i-th agent is reporting i θ⁾_i. That is

( )

⎥

⎦

⎢ ⎤

⎣

= ⎡

∑

≠ −

−

−

i j

j i i j

i i

i E v (h*(ˆ , ), )

SW θ⁾ _θ θ θ θ

The AGV mechanism thus internalizes the “expected externalities” that arise with agent i’s announcement of θ⁾_i. In this case, the budget balances: however the mechanism does not satisfy individual rationality (Krishna and Perry 1998).⁶⁶ Recalling Myerson-Satterthwaite’s impossibility theorem this is not astonishing, as it rules out all three prop-erties allocative efficiency, budget balance, and individual rationality (see Table 2).

• GVA mechanism

The Generalized Vickrey Auction⁶⁷ was developed by Nobel Laureate William Vickrey (Vickrey 1961) and denotes an application of the VCG mechanism to combinatorial (re-source) allocation problems, henceforth CAP. Actually, the VCG mechanism is imple-mented by a sealed bid combinatorial auction.

CAP can be easily formulated as follows (Parkes 2001; de Vries and Vohra 2003). Sup-pose there are n agents and K∈ resources that are to be allocated. The agents can re-H port a valuation for every subset S of H. CAP is thus concerned with the computation of the allocative efficient allocation, i.e. the maximization of the sum of individual valua-tions:

j , i , 0 S S . t.

s

) , S ( v max

arg

* S

j i

i i i

) S ,..., S , S (

S 1 2 n

∀

=

∩

= ₌

∑

^θ

Note that the constraint of this simple maximization problem is assuring a feasible alloca-tion, in a way that a resource is only allocated once and each agent receives only a single subset Si.

The GVA transfers reduces to

∑ ∑

≠

≠ − −

=

i

j j j j

i

j j i j

i(ˆ) v (S* , ) v (S *, )

t θ θ θ ,

where S*-i denotes the “second best allocation”, i.e. best allocation without the i-th agent being present (Parkes 2001). The intuition of the VCG mechanism remains the same; the transfers equal the marginal effect on the valuations of the other agent that agent i induces by his participation.

Even in the combinatorial setting, the GVA attains allocative efficiency, strategy proof-ness, and individual rationality. The GVA mechanism does not balance budget, but at least, it generates a surplus. In auction settings this surplus is not considered harmful as the transfers are given to the seller.

• GL mechanism

The Groves-Ledyard (GL) mechanism assumes quasi-linear and quadratic preferences.

Quadratic preferences basically reflect the public nature of the resource (Groves and Led-yard 1977). In such a setting, Groves and LedLed-yard developed a choice and transfer scheme such that truth telling is a Nash equilibrium. The resulting allocation is efficient and, moreover, individual rationality is assured.

66 A nice presentation of the proofs is given by Parkes (Parkes 2001).

67 Note that the distinction into mechanisms and auctions is dropped at this point.

Environment Performance Mechanism

Preference Domains Resources Equilibrium Concept Possible

Groves Quasi-linear Exchange Dominant Allocative efficient and

(Budget Balanced or Individual rational)

AGV Quasi-linear Exchange Bayesian-Nash Allocative efficient and

Budget Balanced

VCG Quasi-linear Exchange Dominant Allocative efficient and

individual rational

GVA Quasi-linear Combinatorial Dominant Allocative efficient,

Bu-dget Balanced* and indi-vidual rational

GL* Quasi-linear and Quadratic Exchange Nash Allocative efficient, and Individual rational

Table 3: Possibility Results concerning Efficiency (cf. Parkes 2001)