Algorithmic Mechanism Design
S Kameshwaran
Oct 16, 2002
Till now..
 Centralized  Decentralized
 Shortest Path Problem  Routing Problem
 Marriage Problem  Trading Problem
 Algorithms  Mechanisms
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Mechanism Design
Game Theory
Till now.. Mechanism Design
 Given:
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System comprising of self-interested, rational agents
Set of system wide goals
 Mechanism Design

Does there exist a mechanism that can implement the
goals?
 Implementation of the goals depends on the
individual behavior of the agents
Till now.. Game Theory
 Given a game (mechanism), predicts the outcome
by analyzing the individual behavior of the players
(agents)
 Interactive Decision Theory
 Game:
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N players
Rules of encounter: Who should act when and what are
the possible actions
Every possible outcome of the game
Game Theory
 Normal Form Games
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N players
Si=Strategy set of player i
Single simultaneous move: each player i chooses a
strategy siSi
Nobody observes others’ move
The strategy combination (s1, s2, …, sN) gives payoff (p1,
p2, …, pN) to the N players
All the above information is known to all the players and
it is common knowledge
Equilibrium
 An equilibrium s*= (s1*, s2*, …, sN*) is a strategy
combination consisting of a best strategy for each
of the N players in the game
 Equilibrium strategies are the strategies players
pick in trying to maximize their individual payoffs,
knowing that other players are also doing the same
 What is the best strategy? depends on the game
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I
Don’t
Confess
Confess
Don’t
Confess
-1, -1
-10, 0
Confess
0, -10
-8, -8
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I
Don’t
Confess
Confess
Don’t
Confess
-1, -1
-10, 0
Confess
0, -10
-8, -8
Strategy
Set
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I
Don’t
Confess
Confess
Don’t
Confess
-1, -1
-10, 0
Confess
0, -10
-8, -8
Strategy
Set
Strategy
Set
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I
Don’t
Confess
Confess
Don’t
Confess
-1, -1
-10, 0
Confess
0, -10
-8, -8
Strategy
Set
Strategy
Set
Payoffs
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
 Prisoner I’s Decision:
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner I’s Decision:

If II Don’t Confess then it is best to Confess
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner I’s Decision:
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If II Don’t Confess then it is best to Confess
If II Confess then it is best to Confess
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner I’s Decision:
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If II Don’t Confess then it is best to Confess
If II Confess then it is best to Confess
It is best to Confess for I, regardless of what II does: Dominant
Strategy
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
 Prisoner II’s Decision:
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner II’s Decision:
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If I Don’t Confess, then it is best to Confess
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner II’s Decision:
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If I Don’t Confess, then it is best to Confess
If I Confess, then it is best to Confess
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
 Prisoner II’s Decision:
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If I Don’t Confess, then it is best to Confess
If I Confess, then it is best to Confess
It is best to Confess for II, regardless of what I does: Dominant
Strategy
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
Prisoner II
Prisoner I Don’t Confess
Confess
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
Don’t Confess
Confess
-1, -1
-10, 0
0, -10
-8, -8
It is best for both I and II to Confess regardless of what other one does
Confess is a Dominant Strategy for both
(Confess, Confess) becomes the Dominant Strategy Equilibrium
Note: Its beneficial for both to Don’t Confess, but it is not an equilibrium
as both have incentive to deviate
Dominant Strategy Equilibrium:
Prisoner’s Dilemma
 Dominant Strategy Equilibrium is a strategy
combination s*= (s1*, s2*, …, sN*), such that si* is a
Dominant Strategy for each i
 Dominant Strategy is the best response to any
strategy of other players
 It is good for agent as it need not deliberate about
other agents’ strategies
 All games need not have dominant strategy
equilibrium
A Beautiful Mind: Nash
Equilibrium
 Nash Equilibrium is a strategy combination s*= (s1*,
s2*, …, sN*), such that si* is a best response to (s1*,
…,si-1*,si+1*,…, sN*), for each i
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(s1*, s2*, s3*) is a Nash Equilibrium (3 player game) iff
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s1* is the best response of 1, if 2 chooses s2* and 3 chooses s3*
s2* is the best response of 2, if 1 chooses s1* and 3 chooses s3*
s3* is the best response of 3, if 1 chooses s1* and 2 chooses s2*
Note: It is a simultaneous game and nobody knows what
exactly the choice of other agents
Nash Equilibrium assumes correct and consistent beliefs
Nash Equilibrium: Battle of the
Sexes
Woman
Man
Prize Fight
Ballet
Prize Fight
2, 1
0, 0
Ballet
0, 0
1, 2
Nash Equilibrium: Battle of the
Sexes
Woman
Man
Prize Fight
Ballet
Prize Fight
2, 1
0, 0
Ballet
0, 0
1, 2
 (Prize Fight, Prize Fight) is a NE: Best responses to
each other
Nash Equilibrium: Battle of the
Sexes
Woman
Man
Prize Fight
Ballet
Prize Fight
2, 1
0, 0
Ballet
0, 0
1, 2
 (Prize Fight, Prize Fight) is a NE: Best responses to
each other
 (Ballet, Ballet) is a NE: Best responses to each
other
Nash Equilibrium
 In a NE no agent can unilaterally deviate from its strategies
given others’ strategies as fixed
 Topologically it’s a fixed point
 There may be no, one or many NE
 Agent has to deliberate about the strategies of the other
agents
 If the game is played repeatedly and players converge to a
solution then it has to be NE
 Dominant Strategy Equilibrium  Nash Equilibrium (but
the converse is not always true)
Mechanism Design
 Games induced by mechanisms are different from
the previous games:
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The payoff/output matrix is not known to the players (i.e)
the players don’t know about the other players’ utilities
Game of Incomplete information
NE  Bayesian NE
 Dominant Strategy Equilibrium is used in StrategyProof Mechanism
 BNE is used in Bayesian Nash Mechanisms
Mechanism Design Problem: Type
of an Agent
 N agents, and each agent has some private
information called its type, tiTi (set of all possible
types)
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Agent i knows only its type but not the others’ types.
Other agents know agent i’s set of possible types is Ti
Auction Game:
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Each agent knows its value for the good but not others’ value.
The type of the agent is its value.
Ti=[75, 100]: The agents may value the good anywhere between
75 and 100 (known to all agents)
ti=80: Exact value of the good to the agent i (not known to other
agents)
Mechanism Design Problem:
Output Specification
 O is the set of outcomes
 Output Specification g: For a given set of
type configuration (t1, t2, …, tN) , it specifies
a valid outcome o
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Auction Game:
O: Different winners of the object
 g: arg maxi (t1, t2, …, tN) (allocate to the bidder with
highest value)
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Mechanism Design Problem:
Valuation and Utility
 If o is the outcome, ti is the type, then i’s valuation
is given by a real valued function: vi(o,ti)
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Auction Game: If agent i wins the good then its valuation
is equal to its value for the good, otw it is 0
 If pi is the payment made by the agent, then utility
of the outcome o, with type ti is ui= vi(o,ti)+pi
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Auction Game: If agent i’s value of the object is 100, and
if it pays 90, then the utility is 100-90=10
 Agent’s motive: Maximize (expected) Utility
Mechanism Design
Mechanism Design Problem
 Ti: Set of possible types of agent i, T = iTi
 Output Specification g:TO
 Valuation: vi(o,ti), Quasi-linear Utility: vi(o,ti)+pi
Mechanism
 M=<S, O, P>
 S= iSi, where Si is the strategy of agent i (Strategy is a
function of the type information)
 Mechanism specifies:
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Outcome o as a function of the strategy combination
Payment p as a function of the strategy combination
Strategy-Proof Mechanism Design
 If truth telling is the dominant strategy in a
mechanism then it is called as Strategy-Proof
Mechanism
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Agents report their true valuation function instead of
strategically manipulating it
 Utilitarian Mechanisms:
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A mechanism is utilitarian if its objective is to maximize
the overall value of the system: maxo i vi(o,ti)
The auction game and the marriage problem are
utilitarian
Strategy-Proof Mechanism Design
 Strategy-proof mechanism:
Mechanism: Utilitarian maxo i vi(o,ti)
 Utility: Quasi-Linear (vi(o,ti)+pi)
Then the following payment function ensures that the
mechanism is Strategy-proof (truth telling is dominant
strategy)
pi = j<>i vi(o,ti)+hi(t-i)
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(VCG Mechanisms)
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hi is an arbitrary function of types of other players
Strategy-Proof Mechanism Design
 Proof (Intuitive sketch):
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Payment made by agent i
pi = j<>i vi(o,ti)+hi(t-i)
Both the terms above are independent of the
type, strategy and valuation of i
So it is best for i to report its true value. Strategic
behavior may not lead to a beneficial outcome
(similar to Vickrey Auctions)
Strategy-Proof Mechanism
Design: Advantages
 For System Designer:
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The motive (maximizing the sum of the valuation of all
the agents) is achieved with certainty.
The outcome is pareto-efficient (in fact ex-post efficient)
 For Agents:
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Agents have truth telling as the dominant strategy, so
they need not require any computational systems to
deliberate about other agents strategies
Strategy-Proof Mechanism
Design: Disadvantages
 For System Designer:
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The payments may not be budget-balanced
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Budget balance ipi=0 (There is no external source of money. It
only gets exchanged among the agents)
In a market with several buyers and sellers, the total money
collected from the buyers is given to the sellers. If the total
money collected is less than that to be given, then market is at a
lost
VCG is the only strategy-proof pricing scheme for
utilitarian functions
Strategy-Proof Mechanism
Design: Disadvantages
 For System Designer (contd..):
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System has to calculate the utilitarian function N+1 times
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Once with all agents and the once for each agent removed from
the system
If the problem is hard to solve then the computational cost may
be very heavy
 For Agents:
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Agents may not like to tell the truth to the system
designer as it can be used in other ways.
Algorithmic Mechanism
Design
 Algorithmic issues in finding the solution to the
utilitarian function and the payments
 Research Problem: What is the complexity of
determining the payments to the agents?
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For shortest path problem, calculating a payment to the
agent requires to determine the shortest path without the
edge belonging to that agent
Calculate the shortest path n+1 times where n is the
number of edges in the optimal solution
Main result: The complexity of finding the payments to
all agents is same as the complexity of solving one
instance of the shortest path problem
Algorithmic Mechanism
Design
 What if finding the optimum solution to the
utilitarian mechanism is hard?
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Approximation schemes
If the problem can only be solved approximately, then
strategy-proofness breaks
Research Problems:
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What is the best approximation ratio for a given mechanism?
What are the payment functions for the approximate mechanism
so that it becomes strategy-proof?
Impossibility Result
No mechanism can be simultaneously efficient,
strategy-proof and budget-balanced
 One has to compromise on any of the above
 Budget-balance is mandatory for any real world
system (no system would like to run in loss)
 Efficiency is also mandatory as agents may not
prefer inefficient outcomes
 Bayesian Nash Mechanisms
Bayesian Nash Mechanisms
 Advantages:
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Lot of options in pricing: Any kind of pricing is fine as
long as it is budget-balanced
 Disadvantages:
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No dominant strategy: Computationally taxing for agents
Proof of existence of BNE for a mechanism is difficult
(proof heavily relies on fixed point theorems which
require lot of nice behavior from the system parameters)
Maybe more than one BNE: Choosing the best may not
be possible
Determining the BNE: No standard procedure
Algorithmic Aspects
 Open Problem:
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What are the algorithmic aspects for the
Bayesian Mechanisms?
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One has to consider the computational capabilities of
individual agents
Devising a method for proving the existence of
BNE without the use of fixed point theory
General method for determining at least one/all
of the BNE points
References
 Equilibrium Concepts:
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Games and Information: An Introduction to Game
Theory, Eric Rasmusen, Basil Blackwell Publishers,
1989
Any Game Theory Book
 Algorithmic Mechanism Design
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Algorithmic Mechanism Design, Noam Nisan and Amir
Ronen, 2001
Algorithms for Selfish Agents, Noam Nisan, 2001
Coming Up..
 18/10/02: Algorithms, Games and Internet

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Motivating the use of GT and Mechanism
Theory for modeling the Internet mathematically
by using several Internet applications like MultiCast Routing, Peer-Peer file sharing, etc
Several open problems in this area