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Mechanism design theory examples and complexity
1. Mechanism Design Theory:
How to Implement Social Goals
Examples and Algorithmic Design
Stathis Grigoropoulos
Ioannis Katsikarelis
Multi-Agent Systems Utrecht University 2012
3. Introduction
Game Theory : A Normal Game Form[1]
– A Set of Players: Who is involved?
– A Set of Rules (Institutions): What can the players do?
– A Set of Outcomes: What will happen when players perform a certain
action?
– A Set of Preferences: What players want the most of the possible outcomes?
The importance of Game Theory is established in numerous fields:
– Economic Phenomena
– Auctions, Bargains, Fair Division
– Social Phenomena
– Social Network Formation, Social Choice
– Political Sciences
– Election outcome, Political choices
4. Introduction
A closer look at a “real” case[2]:
• You are selling a rare painting for which you want to raise the
maximum revenue.
– Tyler, who values the painting at $100,000
– Alex who values it at $20,000
• You do not know their valuation, how to get max revenue?
– Auction!
– If you knew, you would set the price at $99,999 and be fine!
• Standard English open-cry Auction $20,001 (not max!)
• + Reserve Price say at $50,000 $50,001 (close, but..lucky!)
• But why stop at a reserve price? How about a reserve price
and an entry fee? But why stop at reserve prices and entry
fees?....
5. Context
The design of the institutions (Rules) through which individuals
(Players) interact can have a profound impact on the results
(Outcomes) of that interaction[3].
We saw:
• Auction is conducted with sealed bids versus oral ascending bids can have
an impact on what bidders learn about each other's valuations and
ultimately how they bid.
• Wanted: forecast the economic or social outcomes that these
institutions generate
6. Context
Theory of Mechanism Design[4]
“engineering” part of economic theory
• much of economic theory devoted to:
– understanding existing economic institutions
– explaining/predicting outcomes that institutions generate
– positive, predictive
• mechanism design – reverses the direction
– begins by identifying desired outcomes (goals)
– asks whether institutions (mechanisms) could be designed to achieve
goals
– if so, what forms would institutions take?
– normative, prescriptive - - i.e., part of welfare economics
7. Examples
Simple example[5], suppose
• mother wants to divide cake between 2 children, Alice and
Bob
• goal: divide so that each child is happy with his/her portion
– Bob thinks he has got at least half
– Alice thinks she has got at least half
call this fair division
• If mother knows that the kids see the cake in same way she
does, simple solution:
– she divides equally (in her view)
– gives each kid a portion
8. Examples
• But what if, say, Bob sees cake differently from
mother?
– she thinks she’s divided it equally
– but he thinks piece he’s received is smaller than Alice’s
• difficulty: mother wants to achieve fair division
– but does not have enough information to do this on her own
– in effect, does not know which division is fair
9. Examples
• Can she design a mechanism (procedure) for which
outcome will be a fair division?
(even though she does not know what is fair herself ?)
• Age-old problem
– Lot and Abraham dividing grazing land
10. Examples
Age-old solution:
– have Bob divide the cake in two
– have Alice choose one of the pieces
Why does this work?
• Bob will divide so that pieces are equal in his eyes
– if one of the pieces were bigger, then Alice would take that one
• So whichever piece Alice takes, Bob will be happy with other
• And Alice will be happy with her own choice because if she
thinks pieces unequal, can take bigger one
11. Examples
Example illustrates key features of mechanism design:
• mechanism designer herself doesn’t know in advance what
outcomes are optimal
• so must proceed indirectly through a mechanism
– have participants themselves generate information needed to identify
optimal outcome
• complication: participants don’t care about mechanism
designer’s goals
– have their own objectives
• so mechanism must be incentive compatible
– must reconcile social and individual goals
12. Examples
Example from the paper
Consider society with
• 2 consumers of energy – Alice and Bob
• Energy authority – must choose public energy source
gas
oil
nuclear power
coal
13. Examples
Two states of world
State 1 consumers weight future lightly (future relatively unimportant)
state 2 consumers weight future heavily (future relatively important)
Alice – cares mainly about convenience
In state 1: favors gas over oil, oil over coal, and coal over nuclear
In state 2: favors nuclear over gas, gas over coal, and coal over oil
− technical advances expected to make gas, coal, and
especially
nuclear easier to use in future compared with oil
Bob – cares more about safety
In state 1: favors nuclear over oil, oil over coal, and coal over gas
In state 2: favors oil over gas, gas over coal, and coal over nuclear
− disposal of nuclear waste will loom large
− gas will become safer
15. Examples
State 1
State 2
Alice
Bob
gas
nuclear
oil
oil
coal
coal
nuclear
gas
oil optimal
Alice
Bob
nuclear
oil
gas
gas
coal
coal
oil
nuclear
gas optimal
− authority could ask Alice or Bob about state
• but Alice has incentive to say “state 2” regardless of truth
always prefers gas to oil
gas optimal in state 2
• Bob always has incentive to say “state 1”
always prefers oil to gas
oil optimal state 1
So, simply asking consumers to reveal actual state too naive a mechanism
16. Examples
State 1
State 2
Alice
Bob
gas
nuclear
oil
oil
coal
coal
nuclear
gas
social optimum: oil
Alice
Bob
nuclear
oil
gas
gas
coal
coal
oil
nuclear
social optimum: gas
Authority can have consumers participate in the mechanism given by table
Bob
Alice
oil
coal
nuclear
gas
• Alice – can choose top row or bottom row
• Bob – can choose left column or right column
• outcomes given by table entries
• If state 1 holds
Alice will prefer top row if Bob plays left column
Bob will always prefer left column
so (Alice plays top, Bob plays left) is Nash equilibrium
neither participant has incentive to change unilaterally to another strategy
− so good prediction of what Alice and Bob will do
17. Examples
State 1
State 2
Alice
Bob
gas
nuclear
oil
oil
coal
coal
nuclear
gas
social optimum: oil
Alice
Bob
nuclear
oil
gas
gas
coal
coal
oil
nuclear
social optimum: gas
Bob
Alice
So, in state 1:
expect that
Alice will play top strategy
Bob will play left strategy
outcome is oil
oil is social optimum
Similarly, in state 2: gas is social optimum
oil
coal
nuclear
gas
18. Examples
State 1
State 2
Alice
Bob
gas
nuclear
oil
oil
coal
coal
nuclear
gas
social optimum: oil
Alice
Bob
nuclear
oil
gas
gas
coal
coal
oil
nuclear
social optimum: gas
Bob
Alice
oil
coal
nuclear
gas
• Thus, in either state, mechanism achieves social optimum, even though
− mechanism designer does not know the state herself
− Alice and Bob interested in own ends (not social goal)
• We say that mechanism implements the designer’s goals (oil in state 1,
gas in state 2)
19. Examples
State 1
State 2
Alice
Bob
gas
nuclear
oil
oil
coal
coal
nuclear
gas
optimum: oil
Alice
Bob
gas
nuclear
oil
oil
nuclear
coal
coal
gas
optimum: nuclear
Let us change the example a bit:
• Wrongly set nuclear as social optimum, observe that although oil is optimal
in state 1, it is not optimal in state 2, despite the fact that it falls in neither
Alice’s nor Bob’s rankings between states 1 and 2
• monotonicity is a property ensuring that the oil remain optimal in state 2
Theorem 1 (Maskin 1977): If a social choice rule is implementable, then it
must be monotonic.
Theorem 2 (Maskin 1977): Suppose that there are at least three individuals.
If the social choice rule satisfies monotonicity and no veto power, then it is
implementable.
20. Algorithmic Design
• Have shown you mechanisms in the cake, and
energy examples
• What about that auction?
• Examples raise questions (among others):
− How can we implement such a mechanism?
− How does Computer Science come into play?
21. Algorithmic Settings
●
●
●
●
An important part of Computer Science
research deals with distributed settings
These mainly focus on the connection of
several computers and the computations these
perform to achieve a common outcome
according to some protocol (algorithm)
It is generally assumed that the participants
follow the instructions of the protocol
This is obviously not always the case
(e.g. the Internet)
22. Two example applications
●
Load Balancing:
–
–
●
In a “perfect” world, the aggregate computational power of all
computers on the Internet would be optimally allocated online
among connected processors, which is in itself a difficult problem
In reality, all resources belong to individual entities that act in a
rationally selfish way, which implies the necessity of some form of
motivation for participation
Routing:
–
Information passes through several intermediate routers before
reaching its intended destination
–
Since routers are considered self-interested entities, this implies
that the protocols employed should take the router's potential
interests into consideration
23. Algorithmic Mechanism Design
●
●
●
●
Nisan and Ronen [6] used notions of mechanism
design to introduce a new framework for the
study of such problems
Theirs was not the first use of such notions in
Computer Science studies, but arguably one of the
most relevant and influential
This marriage of concepts has even more
interesting implications (e.g. complexity)
We only briefly mention the first steps of what is
now recognized as an important research direction
24. The model: Problems
●
Output specifications, defined algorithmically
–
–
●
Input is information about the setting (common) and the
participating agents (their types – private)
Output is the specific computed outcome, based on the
information above
Descriptions of agents' desires (preferences)
–
–
●
A valuation function for every agent, based on its type and
outcome
A payment from the mechanism to every agent for
participating, based on its type
The optimization version has an objective
function as an outcome
25. The model: Solutions
●
●
●
●
A problem is solved when the required output
is obtained, while agents try to achieve their
goals (maximize their utilities)
Agents have a specified family of strategies
and the outcome depends on all these choices
Depending on its strategy, the mechanism
provides a payment to each agent, which can
be used to make the agents' desires compatible
with the required outcome
Complexity matters
26. Desired characteristics of Mechanisms
●
Implementations with dominant strategies
–
–
●
Every agent has some dominant strategy
Every set of dominant strategies yields a desired outcome
Truthful Implementation
–
–
●
All agents want to report their type (no manipulation)
Correctly reporting one's type is a dominant strategy
The revelation principle states that given a
mechanism that implements a problem with
dominant strategies, there exists a truthful
implementation as well
27. Vickrey-Groves-Clarke Mechanisms
●
●
●
●
●
Originally defined for auctions (second-price)
VGC is a very useful type of mechanism,
applied to maximization problems, where the
objective is the sum of all agents' valuations
Intuitively, the payment defined by VGC for
an agent is its contribution to social welfare
VGC mechanisms have been famously proven
to be truthful (may even be the only truthful
implementations)
Complexity matters a lot
28. Shortest Paths Example
●
●
●
●
●
Consider a communication network, modeled
by a directed graph
There is one source and one sink node
Every edge is an agent, whose type (private
information) is te, the cost for sending a single
message across this edge
The goal is to find the cheapest path from the
source to the sink (single message)
Agent's valuation is 0 if its edge is not part of
e
the chosen path and -t otherwise
29. A Truthful Implementation
●
●
When all agents honestly report their types
(costs), the cheapest path can be calculated
The following mechanism ensures the above
is a dominant strategy for each agent:
Set the payment for each agent to be pe=0 if
its edge is not in the shortest path and
pe=dG-e-dG|e=0 otherwise (according to inputs)
●
●
This is a VGC mechanism, the computed
paths are the shortest paths and truth-telling is
a dominant strategy
Similar ideas are being used in today's Internet
30. Conclusion
•
Have seen some implemetations of the theory of
Mechanism Design
Many other potential applications
•
–
–
–
•
•
Policies to prevent financial crises
Sustainable gas emission policies
Elections Design
Combining Mechanism Design and Algorithmics is
very natural and yields interesting and useful
results
Although we merely mentioned a small part of
some introductory notions, the underlying theory is
itself plentiful and well-founded
32. References
1.
2.
3.
4.
5.
6.
Yoav Shoham, Kevin Leyton-Brown Multiagent Systems Algorithmic,
Game-Theoretic, and Logical Foundations, 2009
http://marginalrevolution.com/marginalrevolution/2007/10/mechanism-desig.html
, Mechanism Design for Grandma by Alex Tabarrok on October 15, 2007
Matthew O. Jackson, Mechanism Theory ,
https://www2.bc.edu/~unver/teaching/gradmicro/mechtheo.pdf revised
December 8 2003
Mechanism Design:How to Implement Social Goals (Eric S. Maskin)
Mechanism Design:How to Implement Social Goals , presentation by Eric S.
Maskin, Mechanism Design Theory(Harvard, Frankfurt, Shanghai, Prato,
Berlin).pdf, stifterverband.info/.../maskin_mechanism_design_theory.pdf,
visited 22-11-2012
Algorithmic Mechanism Design, Noam Nisan, Amir Ronen, Games and
Economic Behavior, Volume 35, Issues 1-2, April 2001, Pages 166-196
