Incentive Compatibility

In mechanism design theory, incentive compatibility is understood to mean the property of a mechanism that the best strategy of each participant is to follow the rules of the mechanism (i.e. in most applications: to truthfully report the requested private information) regardless of the strategy chosen by the other participants. Examples of such mechanisms, which are considered from the point of view of incentive compatibility, are, for example, auctions, incentive contracts and voting systems.

The concept goes back to Hurwicz (1972).

definition

Be a direct (sales) mechanism , that is, a mechanism with the property that the amount of possible actions of each participant corresponds to the amount of appreciation. Here and are two functions (with the buyer set with cardinality and the set of possible valuations of ) and . For every realization of an appreciation profile is given by the probability that the object receives; specifies the expected costs incurred for. It applies to everyone . ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$ ${\displaystyle \mathbf {Q} }$ ${\displaystyle \mathbf {M} }$ ${\displaystyle \mathbf {Q} :{\mathcal {X}}\rightarrow [0,1]^{n}}$ ${\displaystyle {\mathcal {I}}}$ ${\displaystyle n}$ ${\displaystyle {\mathcal {X}}_{i}}$ ${\displaystyle i}$ ${\displaystyle \mathbf {M} :{\mathcal {X}}\rightarrow \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle Q_{i}(\mathbf {x} )}$ ${\displaystyle i}$ ${\displaystyle M_{i}(\mathbf {x} )}$ ${\displaystyle i}$ ${\displaystyle \sum _{i\in {\mathcal {I}}}Q_{i}(\mathbf {x} )\leq 1}$ ${\displaystyle \mathbf {x} \in {\mathcal {X}}}$

Define

${\displaystyle q_{i}(z_{i})\equiv \int _{{\mathcal {X}}_{-i}}Q_{i}(z_{i},\mathbf {x} _{-i})f_{-i}(\mathbf {x} _{-i})\mathrm {\mathbf {d} } \mathbf {x} _{-i}}$

(with the common density of appreciations of all bidders except ) the likelihood that the property will receive if he reports the appreciation and all others truthfully report their appreciation. Further define ${\displaystyle f_{-i}(\cdot )}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle z_{i}}$

${\displaystyle m_{i}(z_{i})\equiv \int _{{\mathcal {X}}_{-i}}M_{i}(z_{i},\mathbf {x} _{-i})f_{-i}(\mathbf {x} _{-i})\mathrm {\mathbf {d} } \mathbf {x} _{-i}}$

than the expected cost of when reporting the appreciation and all other bidders truthfully express their appreciation. ${\displaystyle i}$ ${\displaystyle z_{i}}$

Definition: is incentive-compatible if: ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$

${\displaystyle q_{i}(x_{i})x_{i}-m_{i}(x_{i})\geq q_{i}(z_{i})x_{i}-m_{i}(z_{i})}$

for everyone , and . ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle x_{i}}$ ${\displaystyle z_{i}}$

Examples

Auctions

A second price auction with private valuations is an example of an incentive-compatible mechanism, provided the individual valuations are not interdependent. In a second-price auction, the highest bidder wins, but only has to pay the second-highest bid in the end. One can show - for which individual reference is made to the article second price auction . that it is a weakly dominant strategy for a participant in such an auction to bid in the amount of his actual appreciation, regardless of which bids other bidders submit.

Contract Theory: Principal-Agent Problems

In principal-agent constellations , the incentive tolerance of this contract is usually assumed for the identification of an optimal incentive contract. The following is an example of the contract-theoretical "basic model" with hidden information .

Consider a two-stage scenario in which the principal (here: an employer) first determines the desired amount of work . This is called and one assumes that the employer wants to get the employee to work, that is . The actual labor input cannot be observed by the principal; However, the project profit is verifiable and it is either low ( ) or high ( ). The probability that a high project profit ( ) will be achieved is if the agent is not working and if it is working. Be it . The costs of the agent from the work assignment are referred to as ( work suffering ). Since the labor input cannot be verified, the wage can only be conditioned on the project profit (and not on the labor input), so that the principal tries to maximize the following expression through the choice of the wage: ${\displaystyle q^{*}\in \{0,1\}}$ ${\displaystyle q^{*}=1}$ ${\displaystyle q}$ ${\displaystyle \pi }$ ${\displaystyle \pi ^{l}}$ ${\displaystyle \pi ^{h}}$ ${\displaystyle \pi ^{h}}$ ${\displaystyle p_{0}}$ ${\displaystyle p_{1}}$ ${\displaystyle p_{1}>p_{0}}$ ${\displaystyle C(q)}$ ${\displaystyle w(\cdot )}$

${\displaystyle p_{1}\cdot \left[\pi ^{h}-w(\pi ^{h})\right]+(1-p_{1})\cdot \left[\pi ^{l}-w(\pi ^{l})\right]}$,

and under the following conditions: On the one hand, it must be ensured that the agent accepts the contract at all, that is, his expected benefit must be at least as high as the benefit of his outside option, i.e. the benefit he experiences if he receives from does not consent to the contract in advance. This condition is known as the condition of participation. It ensures that the agent has a (weak) incentive to even enter into the contract. In addition, the incentive tolerance condition must be met. It reads here: ${\displaystyle p_{1}\cdot u\left(w(\pi ^{h})\right)+(1-p_{1})\cdot u(w(\pi ^{l}))-C(q=1)}$

${\displaystyle p_{1}\cdot u\left(w(\pi ^{h})\right)+(1-p_{1})\cdot u\left(w(\pi ^{l})\right)-C(q=1)\geq p_{0}\cdot u\left(w(\pi ^{h})\right)+(1-p_{0})\cdot u\left(w(\pi ^{l})\right)-C(q=0)}$.

That is, the agent's expected payoff when he is working must at least equal the expected payoff for non-work. The incentive compatibility condition ensures that the agent has a (weak) incentive to work, i.e. to do what the principal wants him to do.

Implications

Revenue equivalence

Revenue Equivalence Theorem : Consider two incentive-compatible direct (sales) mechanismsand. Then the expected costs under both mechanisms are identical except for a constant value and they amount to ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$ ${\displaystyle (\mathbf {Q} ,\mathbf {M} ')}$

${\displaystyle m_{i}(x_{i})=m_{i}(0)+q_{i}(x_{i})x_{i}-\int _{0}^{x_{i}}q_{i}(t_{i})\mathrm {d} t_{i}}$

Two auction formats that meet the requirements of this theorem may serve as an illustration. On the one hand, the second price auction with bidders. The appreciation of each bidder is independent and identically (iid) in accordance with a monotonically increasing distribution function F distributed. On the other hand, there is an auction format with the same distribution and participation requirements, in which every bidder submits a bid and the highest bidder receives the object, whereby the highest bidder pays nothing and all other (inferior) bidders lose the amount they bid. Please note that both auction formats are based on the same (efficient) allocation rule. ${\displaystyle n}$ ${\displaystyle i\in {\mathcal {I}}}$

Winning probability

Theorem: The following statements are equivalent:

1. ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$ is incentive compatible.
2. ${\displaystyle q_{i}}$is (weakly) monotonically increasing for all . ${\displaystyle i\in {\mathcal {I}}}$

Bayes-Nash incentive compatibility

Be a direct (sales) mechanism as defined above. Following d'Apremont and Gerard-Varet (1979), Bayes-Nash incentive-compatible is described if there is a Bayesian equilibrium in which all participants report their appreciations truthfully. ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$ ${\displaystyle (\mathbf {Q} ,\mathbf {M} )}$

literature

• Drew Fudenberg and Jean Tirole: Game Theory . The MIT Press, Cambridge, Massachusetts 1991, ISBN 978-0-262-06141-4 .
• Vijay Krishna: Auction Theory. 2nd ed. Academic Press, San Diego et al. a. 2010, ISBN 978-0-12-374507-1 .
• John O. Ledyard: Incentive compatibility. In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.0769 (online edition).

Remarks

1. ^ Leonid Hurwicz : On informationally decentralized systems. In: Roy Radner and CB McGuire (Eds.): Decision and Organization. A Volume in Honor of Jacob Marschak. North-Holland, Amsterdam 1972, pp. 297-336.
2. ↑ See Ledyard 2008.
3. ↑ See Krishna 2010, p. 62 f.
4. ↑ See Krishna 2010, p. 66.
5. ↑ See Krishna 2010, p. 64 f.
6. ^ Claude d'Apremont and Louis-André Gerard-Varet: On Bayesian incentive compatible mechanisms. In: Jean-Jacques Laffont (Ed.): Aggregation and Revelation Preferences. North-Holland, Amsterdam 1979, pp. 269-288.