Manipulation with(out) money
in matching markets

Pallavi Jain, Indian Institute of Technology Jodhpur

The 8th International Conference on Algorithmic Decision Theory

Wed, 16th October 2024

Sushmita Gupta, The Institute of Mathematical Sciences

Manipulation with(out) money
in matching markets

Outline

I. Definitions & Setup

II. Contributions

III. Future Directions

Manipulation with(out) money
in matching markets

Outline

I. Definitions & Setup

II. Contributions

III. Future Directions

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

\(n\) men and \(n\) women

every man (woman) has a ranking over
a subset of the women (men)

a blocking pair in a matching \(\mu\) is a pair (m,w) such that
m prefers w over \(\mu(m)\) and w prefers m over \(\mu(w)\).

goal. find a matching that has no blocking pairs

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Truncation and permutation strategies have been studied quite extensively.

Constructive & destructive control
(when a centralized agency manipulates the outcome)
have been studied extensively in voting,
but not as much in matching markets

Manipulation by individual agents in stable matchings has a long history

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Reorder. reorder preferences arbitrarily.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Reorder. reorder preferences arbitrarily.

 Swap. swap a pair of agents who occupy consecutive positions.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Reorder. reorder preferences arbitrarily.

 Swap. swap a pair of agents who occupy consecutive positions.

Delete. leave the market (be removed from all lists)

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Reorder. reorder preferences arbitrarily.

 Swap. swap a pair of agents who occupy consecutive positions.

Delete. leave the market (be removed from all lists)

Delete Acceptability. a pair of agents delete each other from their lists

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Reorder. reorder preferences arbitrarily.

 Swap. swap a pair of agents who occupy consecutive positions.

Delete. leave the market (be removed from all lists)

Delete Acceptability. a pair of agents delete each other from their lists

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Add. 

A pre-specified subset of agents are “passive” in the sense that even though they are present in the preference list of other agents they are not considered while computing the matching.

The passive agents can be bribed to participate in the matching market.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Constructive-Partner Exists

manipulate so that a designated agent is saturated (i.e., matched)
in a stable matching of the manipulated instance

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Constructive Exists


manipulate so that a designated man-woman pair belongs to
any stable matching of the manipulated instance.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Destructive Partner Exists

 

manipulate so that a designated agent is
unsaturated in a stable matching of the manipulated instance.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Destructive Exists

 

manipulate so that a designated man-woman pair
does not belong to any stable matching of the manipulated instance.

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Exact Unique

 

manipulate so that a given matching is the unique stable matching 
of the manipulated instance

Manipulation with(out) money
in matching markets

I. Definitions & Setup

Given a particular manipulative action
and a stated goal,
can it be attained within the available budget?  

Manipulation with(out) money
in matching markets

Outline

I. Definitions & Setup

II. Contributions

III. Future Directions

Manipulation with(out) money
in matching markets

II. Contributions

Constructive-Partner Exists

Objective is to manipulate the given SM instance using a manipulative action such that a designated agent is matched in the new instance.  

NP-hard: Reorder, Swap, Add, Delete, Delete Acceptability

Parameters:
n’, the number of unmatched agents in a SM of the input instance,
b, the budget for manipulation

W[1]-hard with respect to b+ n’

Manipulation with(out) money
in matching markets

II. Contributions

Constructive-Exists

Objective is to manipulate the given SM instance using a manipulative action such that a designated man-woman pair  belongs to every stable matching in the new instance.

NP-hard:  Reorder , Swap, Add, Delete Acceptability | P: Delete

Parameters:
n’, the number of unmatched agents in a SM of the input instance,
b, the budget for manipulation

W[1]-hard with respect to b+ n’

Manipulation with(out) money
in matching markets

II. Contributions

Destructive-Partner Exists

Objective is to manipulate the given SM instance using a manipulative action such that a designated man-woman pair do not belong to any stable matching in the new instance.​

NP-hard:   Reorder , Swap, Add, Delete, Delete Acceptability

Parameters:
n’, the number of unmatched agents in a SM of the input instance,
b, the budget for manipulation

W[1]-hard with respect to b+ n’

Manipulation with(out) money
in matching markets

II. Contributions

Swap Exact-Unique

Objective is to manipulate the given SM instance by swapping adjacent pairs such that the given matching is the unique stable matching of the resulting instance.

Polynomial-time when preference list size is 2

NP-hard when preference list size is at least 4

XP with respect to budget b

Manipulation with(out) money
in matching markets

II. Contributions

Manipulation with(out) money
in matching markets

II. Contributions

Manipulation with(out) money
in matching markets

II. Contributions

Manipulation with(out) money
in matching markets

Outline

I. Definitions & Setup

II. Contributions

III. Open Problems

Manipulation with(out) money
in matching markets

III. Open Problems

Swap Exact-Unique

Objective is to manipulate the given SM instance by swapping adjacent pairs such that the given matching is the unique stable matching of the resulting instance.

Complexity when preference list size is 3

FPT with respect to b?