## Periodic Reporting for period 1 - MEDINI (Mechanism Design under Incomplete Information)

Reporting period: 2015-09-01 to 2017-08-31

The main focus of this project was on mechanism design with incomplete information. Mechanism design is a general term which, basically, includes all possible decentralised decision mechanisms, for instance, political voting, auctions, decision making within companies or governmental institutions. Besides, the project focuses on other but closely related aspects of decentralised public decision making.

Mechanism design is studied from many different angles: economists, political scientists, applied mathematicians, and computer scientists are interested in and contribute to this topic. The reason for this is the overall importance of the issue for society, both for society at large (politics, voting, public decision making) and on a smaller scale (decision making within governmental and nongovernmental institutions).

The objectives of the project are manifold:

• Mechanism design under incomplete information on the side of the individuals

• Mechanism design for matching problems, in particular pertaining to job markets

• The role of strategy-proofness: to which extent is it possible to extract the right information from individuals

In the section below it will be described what the project has contributed so far to these issues.

Mechanism design is studied from many different angles: economists, political scientists, applied mathematicians, and computer scientists are interested in and contribute to this topic. The reason for this is the overall importance of the issue for society, both for society at large (politics, voting, public decision making) and on a smaller scale (decision making within governmental and nongovernmental institutions).

The objectives of the project are manifold:

• Mechanism design under incomplete information on the side of the individuals

• Mechanism design for matching problems, in particular pertaining to job markets

• The role of strategy-proofness: to which extent is it possible to extract the right information from individuals

In the section below it will be described what the project has contributed so far to these issues.

(1) The main focus of the original project is on mechanism design with incomplete information. The usual situation is that the individuals (who are asked to vote, or more generally to provide information or input to the (public) decision mechanism have only partial and uncertain information about the characteristics and preferences of the others. This is modelled by assuming that these individuals have beliefs (i.e. probability assessments) about these characteristics. Within this setting, the project has resulted in the following working paper.

Dipjyoti Majumdar and Souvik Roy (2017): “Random dictatorship of OBIC decision schemes”, working paper

Summary: We consider the problem of characterizing the random social choice functions that are ordinal Bayesian incentive compatible (OBIC) with respect to some common prior belief. It is shown in Majumdar and Sen (2004) that, if the common prior belief is generic, then every

OBIC and ex-post efficient deterministic social choice function on the unrestricted domain is dictatorial. However, we show by means of an example that the same does not hold for OBIC and ex-post efficient random social choice functions. In view of this, we provide a sufficient

condition on the prior belief such that every ex-post efficient, neutral, and OBIC random social choice function is random dictatorial. We also show by means of examples that the ex-post efficiency and neutrality condition in our result are necessary.

Besides this, the following topics have been explored and results achieved:

(2) Strategy-proofness of a decision mechanism means that for each individual it is optimal to provide the correct information about its preferences. For instance, in voting this means that no individual can possibly gain by strategic voting, that is, not voting for its favourite candidate or alternative. Since in practice there is not always a clear-cut candidate or alternative resulting form the reported preferences (e.g. votes) of the individuals, in theory one may resort to probabilistic rules, meaning that one can attach weights to several candidates. This part of the project shows that under certain conditions such probabilistic rules are simply weighted combinations of deterministic rules, i.e. rules that assign exactly one candidate.

More technically, this is the title and short summary of the article that has already been published:

Peters, Hans, Souvik Roy, Soumyarup Sadhukhan, and Ton Storcken (2017): “An Extreme Point Characterization of Strategy-proof and Unanimous Probabilistic Rules over Binary Restricted Domains,” Journal of Mathematical Economics, 69, 84-90.

Summary: We show that every strategy-proof and unanimous probabilistic rule on a binary restricted domain has binary support, and is a probabilistic mixture of strategy-proof and unanimous deterministic rules. Examples of binary restricted domains are single-dipped domains, which are of interest when considering the location of public bads. We also provide an extension to infinitely many alternatives.

(3) This research focuses on matching between institutions and couples. A well-known area of application is the job market for medical doctors, where the jobs of possible partners of the doctors play an important role. So far, this part of the project has resulted in two working papers.

(a) Shashwat Khare and Souvik Roy (2017): “Stability in Matching with Couples having Responsive Preferences”, working paper

Summary: The paper studies matching markets where institutions are matched with possibly more than one individuals. The matching market contains some couples who view the pair of jobs as complements. We show that under responsive preferences, both for couples and hospitals, stable matching may fail to exist. Further we formulate restrictions on the preferences of couples, which are necessary and sufficient for the existence of a stable matching for any responsive preferences of the institutions. Finally, we formulate restrictions on the preferences of the institutions, which are necessary and sufficient for the existence of stable matching for any responsive preferences of the couples.

(b) Shashwat Khare and Souvik Roy (2017): “ Stability in Matching with Couples having Non-Responsive Preferences”, working paper

(4) This part of the project again focuses on strategy-proofness, see (2) above. The classical Gibbard-Satterthwaite result says that there are no strategy-proof rules if the preferences of the individuals can take any form, unless one allows for a dictator, which of course is undesirable. It is also well-known that under certain restrictions on preferences there are nondictatorial strategy-proof rules. This research adds to this literature. Possible applications include the following:

• Determining the geographical location for a public facility, like a hospital, town hall, school, sport facility

• Budget proposals for projects or public issues like education, defense, health care

• Locating obnoxious facilities like nuclear plants or windmills

• Problems concerning manipulation of preference aggregation, where a committee aggregates individual preferences into a collective preference or list, for instance a list of applicants for a vacancy or a list of contestants in a competition

Two papers deal with these issues:

(a) Souvik Roy_and Ton Storcken (2017): “Unanimity and strategy-proofness on short-path-connected domains”, working paper

(b) Gopakumar Achuthankutty and Souvik Roy (2017): “A domain characterization of min-max rules”, working paper

Dipjyoti Majumdar and Souvik Roy (2017): “Random dictatorship of OBIC decision schemes”, working paper

Summary: We consider the problem of characterizing the random social choice functions that are ordinal Bayesian incentive compatible (OBIC) with respect to some common prior belief. It is shown in Majumdar and Sen (2004) that, if the common prior belief is generic, then every

OBIC and ex-post efficient deterministic social choice function on the unrestricted domain is dictatorial. However, we show by means of an example that the same does not hold for OBIC and ex-post efficient random social choice functions. In view of this, we provide a sufficient

condition on the prior belief such that every ex-post efficient, neutral, and OBIC random social choice function is random dictatorial. We also show by means of examples that the ex-post efficiency and neutrality condition in our result are necessary.

Besides this, the following topics have been explored and results achieved:

(2) Strategy-proofness of a decision mechanism means that for each individual it is optimal to provide the correct information about its preferences. For instance, in voting this means that no individual can possibly gain by strategic voting, that is, not voting for its favourite candidate or alternative. Since in practice there is not always a clear-cut candidate or alternative resulting form the reported preferences (e.g. votes) of the individuals, in theory one may resort to probabilistic rules, meaning that one can attach weights to several candidates. This part of the project shows that under certain conditions such probabilistic rules are simply weighted combinations of deterministic rules, i.e. rules that assign exactly one candidate.

More technically, this is the title and short summary of the article that has already been published:

Peters, Hans, Souvik Roy, Soumyarup Sadhukhan, and Ton Storcken (2017): “An Extreme Point Characterization of Strategy-proof and Unanimous Probabilistic Rules over Binary Restricted Domains,” Journal of Mathematical Economics, 69, 84-90.

Summary: We show that every strategy-proof and unanimous probabilistic rule on a binary restricted domain has binary support, and is a probabilistic mixture of strategy-proof and unanimous deterministic rules. Examples of binary restricted domains are single-dipped domains, which are of interest when considering the location of public bads. We also provide an extension to infinitely many alternatives.

(3) This research focuses on matching between institutions and couples. A well-known area of application is the job market for medical doctors, where the jobs of possible partners of the doctors play an important role. So far, this part of the project has resulted in two working papers.

(a) Shashwat Khare and Souvik Roy (2017): “Stability in Matching with Couples having Responsive Preferences”, working paper

Summary: The paper studies matching markets where institutions are matched with possibly more than one individuals. The matching market contains some couples who view the pair of jobs as complements. We show that under responsive preferences, both for couples and hospitals, stable matching may fail to exist. Further we formulate restrictions on the preferences of couples, which are necessary and sufficient for the existence of a stable matching for any responsive preferences of the institutions. Finally, we formulate restrictions on the preferences of the institutions, which are necessary and sufficient for the existence of stable matching for any responsive preferences of the couples.

(b) Shashwat Khare and Souvik Roy (2017): “ Stability in Matching with Couples having Non-Responsive Preferences”, working paper

(4) This part of the project again focuses on strategy-proofness, see (2) above. The classical Gibbard-Satterthwaite result says that there are no strategy-proof rules if the preferences of the individuals can take any form, unless one allows for a dictator, which of course is undesirable. It is also well-known that under certain restrictions on preferences there are nondictatorial strategy-proof rules. This research adds to this literature. Possible applications include the following:

• Determining the geographical location for a public facility, like a hospital, town hall, school, sport facility

• Budget proposals for projects or public issues like education, defense, health care

• Locating obnoxious facilities like nuclear plants or windmills

• Problems concerning manipulation of preference aggregation, where a committee aggregates individual preferences into a collective preference or list, for instance a list of applicants for a vacancy or a list of contestants in a competition

Two papers deal with these issues:

(a) Souvik Roy_and Ton Storcken (2017): “Unanimity and strategy-proofness on short-path-connected domains”, working paper

(b) Gopakumar Achuthankutty and Souvik Roy (2017): “A domain characterization of min-max rules”, working paper

The project has resulted in six papers/publications so far, and will lead to more results in the near future. This way, it has contributed to the general body of knowledge in society. As is clear from the previous section there are numerous potential applications of the results of the project, whereas the project itself is of a theoretical nature.