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?
ADT 2024 Talk - SMM
By Neeldhara Misra
