Introduction to Game Theory and Mechanism Design - Userweb

Introduction to Game Theory and Mechanism Design Paul Harrenstein and Mathijs de Weerdt ¨ Munich and Delft University of Technology, Delft Ludwig-Maximilians-Universitat,

EASSS’08, Lisbon, 6th May 2008

Harrenstein and De Weerdt (LMU and TU Delft)

Game Theory and Mechanism Design


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Outline

1

Game Theory

2

Social Choice

3

Mechanism Design: Theory

4

Mechanism Design: Applications




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Game Theory

Introduction

Game Theory

1

Game Theory Introduction Strategic Games Elementary Concepts Nash Equilibrium Bayes-Nash Equilibrium Conclusion

2

Social Choice

3


4





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Game Theory

Introduction

What is This Tutorial Trying to Accomplish?

What is the subject matter of game theory and which phenomena does it help us understand? What is the problem of game theory? What are the elementary concepts of game theory? What is the relevance of game theory to agent research? How can game-theoretic concepts be put to use so as to design better systems?




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Game Theory

Introduction

Example: Defence-Attack Situation: Attacker (Red, column player) can attack either target A or target B, but not both. Defender (Blue, row player) can defend either of two targets but not both. Target A is three times as valuable as Target B.

A

B

A

B

A

4, 0

3, 1

B

1, 3

4, 0

jdfkjd

Question: Which target is Red to attack and which target is Blue to defend?




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Game Theory

Introduction

Battle of the Sexes

Fight

Ballet

Fight

2, 1

0, 0

Ballet

0, 0

1, 2


jdfkjd



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Game Theory

Introduction

Game Theory versus Decision Theory

expected utility

possible courses of action

Issue: Find the course of action that maximizes expected utility given particular stochastic parameters.




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Game Theory

Introduction

What is Game Theory Trying to Accomplish?

Point of Departure: Game theory as interactive decision theory. Issue: Assume many agents operating in the same environment each faced with a different optimization problem. Observation I: Dependence of the outcome on all the players’ actions Hence, the optimality of an action depends on the optimality of the other players’ actions As this holds for all players, a circularity threatens. Observation II: Yet, the other players’ decisions cannot be considered stochastic parameters in a straightforward way. Conclusion: New (mathematical) concepts required to take over the role of the optimum (solution concepts).




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Game Theory

Introduction






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Game Theory

Introduction

What is Game Theory Trying to Accomplish? “...But we must plan what we are to do about Moriarty now.” “As this is an express, and as the boat runs in connection with it, I should think we have shaken him off very effectively.” “My dear Watson, you evidently did not realize my meaning when I said that this man may be taken as being quite on the same intellectual plane as myself. You do not imagine that if I were the pursuer I should allow myself to be baffled by so slight an obstacle. Why, then, should you think so meanly of him?” “What will he do?” “What I should do.” “What would you do, then?” “Engage a special.” “But it must be late.” “By no means. This train stops at Canterbury; and there is always at least a quarter of an hour’s delay at the boat. He will catch us there.” “One would think that we were the criminals. Let us have him arrested on his arrival.”

Dover

Dieppe

Dover

0, 1

1, 0

Dieppe

1, 0

0, 1


jdfkjd



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Game Theory

Introduction

What is Game Theory Trying to Accomplish? “It would be to ruin the work of three months. We should get the big fish, but the smaller would dart right and left out of the net. On Monday we should have them all. No, an arrest is inadmissible.” “What then?” “We shall get out at Canterbury.” “And then?” “Well, then we must make a cross-country journey to Newhaven, and so over to Dieppe. Moriarty will again do what I should do. He will get on to Paris, mark down our luggage, and wait for two days at the depot. In the meantime we shall treat ourselves to a couple of carpet-bags, encourage the manufactures of the countries through which we travel, and make our way at our leisure into Switzerland, via Luxembourg and Basle.” At Canterbury, therefore, we alighted, only to find that we should have to wait an hour before we could get a train to Newhaven.

Dover

Dieppe

Dover

0, 1

1, 0

Dieppe

1, 0

0, 1


jdfkjd



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Game Theory

Introduction


     Harrenstein and De Weerdt (LMU and TU Delft)



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Game Theory

Introduction






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Game Theory

Introduction






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Game Theory

Introduction






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Game Theory

Introduction

What is Mechanism Design Trying to Accomplish?

Fix a set of possible outcomes X . Given a preference profile (θ1 , . . . , θn ) certain outcomes X ∗ ∈ X are more desirable than others from an outsider’s point of view (e.g., assigning an object to the agent that values it most). However, preferences of the players are unknown to the designer, or hard to obtain. Issue: Design a game (mechanism) such that the outcome given a particular (fixed) solution concept of this game generates one of the desired outcomes in X ∗ for all (relevant) preference profiles (θ1 , . . . , θn ).




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Game Theory

Introduction

Nobel Prizes for Game Theory

1972

Arrow

Welfare theory

1978

Simon

Decision making

1994

Nash, Harsanyi, Selten

Equilibria

1996

Vickrey

Incentives

1998

Sen

Welfare economics

2005

Aumann and Schelling

Conflict and cooperation

2007

Hurwicz, Maskin and Myerson

Mechanism design




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Game Theory

Strategic Games

Games and Game Forms: The Strategic Form Players: Who is involved? Rules: What can the players do? What do they know when they move? Outcomes: What will happen when the players move in a particular way? Preferences: What are the players’ preferences over the possible outcomes?

Fight

Ballet

Fight

2, 1

0, 0

Ballet

0, 0

1, 2


jdfkjd



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Game Theory

Strategic Games

Games and Game Forms: The Strategic Form Definition: A game form G is a quadruple (N , S , X , ω) where: N, a set of n players S=

i ∈N

Si , an n-dimensional space of strategy profiles

Let Si denote the set of strategies of player i X , a set of outcomes

ω : S → X , an outcome function Definition: A strategic game Γ is a quintuple (N , S , X , ω, θ), where:

(N , S , X , ω) is a game form θ = (θ1 , . . . , θn ) is a preference profile over X




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Game Theory

Strategic Games

Preferences Let X be a set of outcomes.

Θ ⊆ 2X ×X θi ⊆ X × X , reflexive, transitive and complete θ = (θ1 , . . . , θn ) ∈ ΘN L (X ), set of linear orders over X (+ anti-symmetry) P (X ) = 2X ×X , set of weak orders over X Notations: x %θi y and x %i y if (x , y ) ∈ θi x ∼i y if both x %i y and y %i x x i y if both x %i y and not y %i x s %i s 0 if ω(s ) %i ω(s 0 )




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Game Theory

Strategic Games

Utilities

Definition: A utility function ui : X → R represents preferences θi over outcomes X so that: ui (x ) ≥ ui (y ) iff x %i y

Fact: All preference relations over a countable set X are representable by a utility function. These utility functions are invariant under monotonically increasing functions. Fact: Let X = R × R and θ be the lexicographic order on X :

(x , x 0 ) % (y , y 0 ) iff x > y or both x = u and x 0 ≥ y 0 Then, θ cannot be represented by a utility function.




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Game Theory

Strategic Games

Security Level Definition: The pure security level of a player is the least payoff he can guarantee himself, no matter what strategies the other players play, i.e.: max min(ui (t1 , . . . , si , . . . , tn )). s ∈S


t ∈S

1, 0

0, 1

0, 1

1, 0



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Game Theory

Strategic Games

Mixed Strategies and Expected Utility Definition: Let (N , S , u) be a strategic game. Then:

Σi = ∆(Si ), set of mixed strategies Σ = Σ1 × · · · × Σn , set of mixed strategy profiles P Expected utility lotteries u∗ (λ) = x ∈X λ(x ) · u(x ) P Q Expected utility mixed strategies u∗ (σ) = s ∈S i ∈N σi (si ) · ui (s ) Von Neumann-Morgenstern Utilities Outcomes of mixed strategy profiles are lotteries λ over X Qualitative preferences over lotteries represent attitudes towards risk Question: Given a i b i c, does b %i [ 12 a ; 21 c ] or [ 21 a ; 21 c ] %i b hold? Conclusion: Preferences θ should be defined over ∆(X )




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Game Theory

Strategic Games

Mixed Strategies and Expected Utility Definition: Let (N , S , u) be a strategic game. Then:

Σi = ∆(Si ), set of mixed strategies Σ = Σ1 × · · · × Σn , set of mixed strategy profiles P Expected utility lotteries u∗ (λ) = x ∈X λ(x ) · u(x ) P Q Expected utility mixed strategies u∗ (σ) = s ∈S i ∈N σi (si ) · ui (s ) Von Neumann-Morgenstern Utilities Provided certain conditions hold for θ ⊆ ∆(X ) × ∆(X ), a utility function U : X → R exists such that for all λ, λ0 ∈ ∆(X ):

λ % λ0 iff

X x ∈X


λ(x ) · U (x ) ≥

X

λ0 · U (x )

x ∈X



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Game Theory

Elementary Concepts

The Prisoner’s Dilemma

Two suspects are taken into custody and separated. The district attorney is certain that they are guilty of a specific crime, but he does not have adequate evidence to convict them at a trial. He points out to each prisoner that each has two alternatives: to confess to the crime the police are sure they have done, or not to confess. If they will both do not confess, then the district attorney states he will book them on some very minor trumped up charge such as petty larceny and illegal possession of a weapon, and they will both receive minor punishment; if they both confess they will be prosecuted, but he will recommend less than the most severe sentence; but if one confesses and the other does not, then the confessor will receive lenient treatment for turning state’s evidence whereas the latter will get “the book” slapped on him. (Luce and Raiffa, 1957, p. 95)




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Game Theory

Elementary Concepts

The Prisoner’s Dilemma

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1



jdfkjd


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Game Theory

Elementary Concepts

Pareto Efficiency

Definition: An outcome x is (weakly) Pareto efficient if there is no outcome that is strictly better for all players, i.e., if there is no y ∈ X such that for all i ∈ N: y i x

Definition: A lottery λ ∈ ∆(X ) is (weakly) Pareto efficient if there is no lottery that is strictly better for all players, i.e., if there is no λ0 ∈ ∆(X ) such that for all i ∈ N: u∗ (λ0 ) i u∗ (λ)




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Game Theory

Elementary Concepts

Pareto Efficiency

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1

jdfkjd

Which are the Pareto efficient outcomes?




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Game Theory

Elementary Concepts

Pareto Efficiency

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1

jdfkjd

Which are the Pareto efficient outcomes?




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Game Theory

Elementary Concepts

Dominance

Definition: A strategy si for player i (strongly) dominates another strategy si0 if for any choice of action of the opponents, si leads to a more preferable outcome than si0 , i.e., if: for all t ∈ S :


(t1 , . . . , si , . . . , tn ) i t1 , . . . , si0 , . . . , tn .



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Game Theory

Elementary Concepts

Dominance

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1

jdfkjd

Which are the strongly dominant strategy profiles?




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Game Theory

Elementary Concepts

Dominance

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1

jdfkjd





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Game Theory

Elementary Concepts

Dominance

conceal

inform

conceal

2, 2

0, 3

inform

3, 0

1, 1

jdfkjd





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Game Theory

Elementary Concepts

Dominance

Definition: A mixed strategy σi for player i (strongly) dominates a pure strategy si0 if for any choice of action of the opponents, σi has a greater expected utility than si0 , i.e., if: for all t ∈ S :


ui∗ (t1 , . . . , σi , . . . , tn ) > ui∗ t1 , . . . , si0 , . . . , tn .



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Game Theory

Elementary Concepts

Best Responses

Definition: A strategy si∗ is pure best response of a player i to in a pure strategy profile (s1 , . . . , sn ) if for all t ∈ Si :

(s1 , . . . , si∗ , . . . , sn ) %i (s1 , . . . , ti , . . . , sn )

Definition: A mixed strategy σ∗i is (mixed) best response of a player i to in a mixed strategy profile (σ1 , . . . , σn ) if for all τ ∈ Σi : ui∗ (σ1 , . . . , σ∗i , . . . , σn ) ≥ ui∗ (σ1 , . . . , τi , . . . , σn )




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Game Theory

Elementary Concepts

Best Responses

Definition: A strategy si∗ is pure best response of a player i to in a pure strategy profile (s1 , . . . , sn ) if for all t ∈ Si :

(s1 , . . . , si∗ , . . . , sn ) %i (s1 , . . . , ti , . . . , sn )

Definition: A mixed strategy σ∗i is (mixed) best response of a player i to in a mixed strategy profile (σ1 , . . . , σn ) if for all τ ∈ Σi : ui∗ (σ1 , . . . , σ∗i , . . . , σn ) ≥ ui∗ (σ1 , . . . , τi , . . . , σn )




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Game Theory

Nash Equilibrium

Nash Equilibrium Definition: A pure strategy profile s ∗ is a pure Nash equilibrium if no player has an incentive to unilaterally deviate from s, i.e., if for all players i: for all t ∈ S :

s ∗ %i s1∗ , . . . , ti , . . . , sn∗

2, 2

0, 3

1, 0

0, 1

2, 1

0, 0

3, 0

1, 1

0, 1

1, 0

0, 0

1, 2




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Game Theory

Nash Equilibrium

Nash Equilibrium Definition: A mixed strategy profile σ∗ is a Nash equilibrium if no player has an incentive to unilaterally deviate from σ, i.e., if for all players i: for all τ ∈ Σ :

ui∗ (σ∗ ) ≥ ui∗ σ∗1 , . . . , τi , . . . , σ∗n

2, 2

0, 3

1, 0

0, 1

2, 1

0, 0

3, 0

1, 1

0, 1

1, 0

0, 0

1, 2




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Game Theory

Nash Equilibrium

Nash’s Theorem

Theorem (Nash 1950): Every strategic game with a finite number of pure strategies has a Nash equilibrium in mixed strategies.

Remark: The proofs are non-constructive and use Brouwer’s or Kakutani’s fixed point theorems.




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Game Theory

Nash Equilibrium

Properties of Nash Equilibrium

Nash equilibrium is perhaps the most important solution concept for non-cooperative games, for which numerous refinements have been proposed. Any combination of dominant strategies is a Nash equilibrium. Nash equilibria are not generally Pareto efficient. Existence in (pure) strategies is not in general guaranteed. Nash equilibria are not in general unique (equilibria selection, focal points). Nash equilibria are not generally interchangeable. Payoffs in different Nash equilibria may vary.




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Game Theory

Nash Equilibrium

Alternative Characterization of Nash Equilibria

Lemma: A mixed strategy profile σ is a Nash equilibrium iff for all players i Given (σ1 , . . . , σi −1 , σi +1 , . . . , σn ), all actions in the support of σi yield the same expected utility Given (σ1 , . . . , σi −1 , σi +1 , . . . , σn ) no action not in the support of σi yields a higher expected utility than any action in the support of σ.




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Game Theory

Nash Equilibrium

Alternative Characterization of Nash Equilibria

Lemma: A mixed strategy profile σ is a Nash equilibrium iff for all players i ui∗ (σ1 , . . . , s , . . . , σn ) = ui∗ (σ1 , . . . , t , . . . , σn ), support of σi

for all actions s , t ∈ Si in the

ui∗ (σ1 , . . . , s , . . . , σn ) ≥ ui∗ (σ1 , . . . , t , . . . , σn ), but t not in the support of σi

for all actions s , t ∈ Si with s in




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Game Theory

Nash Equilibrium

Exercise

2, 1

0, 0

0, 0

1, 2

Compute the Nash equilibria of the Battle of the Sexes.




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Game Theory

Bayes-Nash Equilibrium

Complete and Incomplete Information

Complete information: the structure of the game is common knowledge among the players Incomplete information: uncertainties among the players about the game Uncertainty modeled as probabilities over possible games with a common prior Restriction: only uncertainty about other players’ preferences




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Game Theory


Bayesian Games and Bayes-Nash Equilibrium

Definition:

A Bayesian game is a quintuple (N , A , Θ, ρ, u), where:

N, a set of n players A = A1 × · · · × An , a set of action profiles

Θ = Θ1 × · · · × Θn , a set of type profiles ρ : Θ → [0, 1], a common prior over type profiles u = (u1 , . . . , un ), a utility profile ui : S × Θ → R is a utility function for i Remark: Observe that ui (a , θ1 , . . . , θn ) can also depend on the types θ1 , . . . , θi −1 , θi +1 , . . . θn of the other players.




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Game Theory


Bayesian Games and Bayes-Nash Equilibrium (θ1 , θ2 )

L

R

T

2, 0

0, 2

B

0, 2

2, 0

(θ1 , θ20 )

L

R

T

2, 2

0, 3

B

3, 0

1, 1

ρ(θ1 , θ2 ) = .3 (θ10 , θ2 )

L

R

T

2, 2

0, 0

B

0, 0

1, 1

ρ(θ1 , θ20 ) = .1 (θ10 , θ20 )

L

R

T

2, 1

0, 0

B

0, 0

1, 2

ρ(θ10 , θ2 ) = .2



ρ(θ10 , θ20 ) = .4 EASSS’08, Lisbon, 6th May 2008

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Game Theory


Bayesian Games and Bayes-Nash Equilibrium Definition:

For (N , A , Θ, ρ, u) a Bayesian game:

a pure (Bayesian) strategy is a function si : Θi → A Si , the set of pure Bayesian strategies of i

Σi = ∆(Si ) the set of mixed Bayesian strategies σi (a | θi ) the probability under σi that i plays a, given his type θi Definition:

Player i’s (ex ante) expected utility ui∗ : Σ → R: ui (σ) = ∗

X θ∈Θ

Definition:

  X Y    ρ(θ) σj (aj | θj ) ui (a , θ) a ∈A

j ∈N

A Bayes-Nash equilibrium is a strategy profile σ such that for all i and all σ0 : ui∗ (σ1 , . . . , σi , . . . , σn ) ≥ ui∗ (σi , . . . , σ0i , . . . , σn )

Fact:

Every Bayesian game has at least one Bayes-Nash equilibrium.




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Game Theory

Conclusion

Other Solution Concepts

Refinements of Nash equilibrium, e.g., quasi-strict equilibrium Coarsenings of Nash equilibrium, e.g., correlated equilibrium Deviations by coalitions: strong and coalition proof equilibrium Trembling hand perfect equilibrium Rationalizability ...




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Social Choice

Social Choice Theory

1

Game Theory

2

Social Choice Introduction Majority Voting Condorcet Impossibility Theorems Circumventing Impossibility Theorems

3


4





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Social Choice

Introduction

What Is Social Choice Theory Trying to Accomplish?

Goal: aggregate preferences of agents (e.g. voting)

In this tutorial: Elementary concepts Impossibility results




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Social Choice

Introduction

Social Choice Rules

Definition: Be N a set of players, X a set of alternatives, and Θ a class of preference relations over X . • Social Choice Function (scf): f : ΘN → X

• Social Choice Correspondence (scc):

F : ΘN → 2X

• Social Welfare Function (swf):

φ : ΘN → Θ

• Social Welfare Correspondence (swc): Φ : ΘN → 2Θ Remark: For simplicity we assume that Θ ⊆ L (X ), i.e. the class of linear preference orders.




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Social Choice

Introduction

Social Choice Example

Given alternatives X = {a , b }, and the the following preferences (each column is a preference order ∈ Θ, the first row indicates the number of players with that preference order): 3 7 a

b

b

a

Question:

Which alternative (a or b) is preferred?

Question:

Formulate a social choice function f : ΘN → X .




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Social Choice

Majority Voting

Majority Voting with Two Alternatives

Two alternatives a and b, three possible preference relations: ab

a∼b

ba

Majority voting: Order the two candidates proportional to the number of “votes” they obtain. Social choice function f selects the candidate with the most votes. Social choice correspondence F selects a subset of candidates that have the most votes. Social welfare function φ defines the social order proportional to the number of votes.




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Social Choice

Majority Voting

Anonymity and Neutrality

Intuition: Anonymity: The names of the players do not matter: if two players exchange types, the outcome is not affected. Neutrality: The names of the alternatives do not matter: if we exchange a and b in the preference profile of each agent, then the outcome is affected accordingly.




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Social Choice

Majority Voting

Anonymity and Neutrality

Definitions: Be φ a social welfare function, x , y ∈ X , (θ1 , . . . , θn ) ∈ ΘN

φ is anonymous if for every permutation π of N: φ(θ1 , . . . , θn ) = φ(θπ(1) , . . . , θπ(n) ) φ is neutral if for every permutation π of X : φ(π(θ1 ), . . . , π(θn )) = π(φ(θ1 , . . . , θn )) (where a π(θi ) b iff π(a ) θi π(b ), for all a , b ∈ X )




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Social Choice

Majority Voting

Choice on Two Alternatives: May’s Theorem Definition: A social welfare function φ on two alternatives a and b is positive responsive if for all players i and all θ = (θ1 , . . . , θi , . . . , θn ) and θ0 = (θ1 , . . . , θi0 , . . . , θn ): if a φ(θ) b, b %θi a and a %θi0 b, then a φ(θ0 ) b if a ∼φ(θ) b, a ∼θi b and a θi0 b, then a φ(θ0 ) b if a ∼φ(θ) b, b θi a and a %θi0 b, then a φ(θ0 ) b Intuition: If alternative a wins or is equal to alternative b, and one player strictly increases a in its ordering ceteris paribus, then alternative a wins. Theorem (May, 1952): If |X | = 2, majority voting defines the only social welfare function satisfying anonymity, neutrality and positive responsiveness.




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Social Choice

Majority Voting

Majority Rule on More than Two Alternatives

Question:

3

5

7

6

a

a

b

c

b

c

d

b

c

b

c

d

d

d

a

a

Who should be the winner according to the majority rule?




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Social Choice

Condorcet

Condorcet Winner

Definition: An alternative a is a Condorcet winner if against any other alternative b there is a majority preferring a to b.

Question:

3

5

7

6

a

a

b

c

b

c

d

b

c

b

c

d

d

d

a

a

Who is the Condorcet winner?




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Social Choice

Condorcet

Condorcet Paradox

The Condorcet Paradox: A Condorcet winner does not always exist.

θ1 a b c

θ2 c a b


θ3 b c a

b

a


c


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Social Choice

Condorcet

Examples of Social Choice Methods

Majority Voting Condorcet Consistent Method Borda Protocol




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Social Choice

Condorcet

The Borda Protocol Definition (The Borda Protocol): Given a finite set of alternatives X and strict individual preferences, for each ballot, each alternative is given one point for every other alternative it is ranked above. The alternatives are then ranked proportional to the number of points they aggregate. Example: 1

2

x

count

a

d

a

3+0=3

b

c

b

2+2∗1=4

c

b

c

1+2∗2=5

d

a

d

0+2∗3=6

In this case the Borda ranking is d c b a.




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Social Choice

Condorcet

The Borda Protocol Definition (The Borda Protocol): Given a finite set of alternatives X and strict individual preferences, for each ballot, each alternative is given one point for every other alternative it is ranked above. The alternatives are then ranked proportional to the number of points they aggregate. Example: 1

5

5

3

2

a d

a

c

b

b

d

b

a

d

b

c

a

d

c

c

b

d

c

a

Question:

Who is the Borda winner?




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Social Choice

Condorcet

Borda cannot always select one winner Example:

θ1 θ2 a b b c c a

Question:

θ3 c a b

x

count

a

2+0+1=3

b

1+2+0=3

c

0+1+2=3


Remark: The Borda protocol does not always provide a social choice function, but a social choice correspondence.




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Social Choice

Condorcet

Borda cannot always select one winner Example:

θ1 θ2 a b b c c a

Question:

θ3 c a b

x

count

a

2+0+1=3

b

1+2+0=3

c

0+1+2=3


Remark: The Borda protocol does not always provide a social choice function, but a social choice correspondence.




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Social Choice

Impossibility Theorems

A Trivial Impossibility Result

Proposition:

There is no anonymous and neutral social choice function.




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Social Choice


Anonymity and Neutrality (repeated)

Intuition: Anonymity: The names of the players do not matter: if two players exchange types, the outcome is not affected. Neutrality: The names of the alternatives do not matter: if we exchange a and b in the preference profile of each agent, then the outcome is affected accordingly.




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Social Choice


Anonymity and Neutrality (repeated)


φ is anonymous if for every permutation π of N: φ(θ1 , . . . , θn ) = φ(θπ(1) , . . . , θπ(n) ) φ is neutral if for every permutation π of X : φ(π(θ1 ), . . . , π(θn )) = π(φ(θ1 , . . . , θn )) (where a π(θi ) b iff π(a ) θi π(b ), for all a , b ∈ X )




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Social Choice


A Trivial Impossibility Result Proposition:

There is no anonymous and neutral social choice function.

Proof. Assume scf is anonymous and neutral. Consider θ, θ0 , θ00 :

θ1 a b c

θ2 θ3 b c c a a b

θ10 b c a

θ20 θ30 c a a b b c

θ100 θ200 a b b c c a

θ300 c a b

W.l.o.g., f (θ) = a. For π(a ) = b, π(b ) = c, and π(c ) = a, θ0 = π(θ). With neutrality, f (θ0 ) = π(f (θ)) = π(a ) = b . With anonymity, f (θ0 ) = f (θ00 ) = b. However θ = θ00 , a contradiction, since f (θ) = a.




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Social Choice


Properties of Social Welfare Functions

Intuition: Pareto optimality: If alternative a is unanimously preferred to alternative b, b should not be elected. Non-dictatorship: There is no player whose preference profile determines the strict preferences of the social welfare function. Unrestricted Domain: The social welfare function should define a social preference order for any given set of preference profiles.




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Social Choice


Properties of Social Welfare Functions


φ has the Pareto property if x θi y for all i ∈ N implies x φ(θ1 ,...,θn ) y φ is dictatorial if for some i ∈ N, for all (θ1 , . . . , θn ) ∈ ΘN : x θi y implies x φ(θ1 ,...,θn ) y φ has an unrestricted domain iff for every θ ∈ ΘN it holds that φ(θ) ∈ Θ.




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Social Choice


Independence of Irrelevant Alternatives

1

5

5

3

2

a

a

c

b

b

d

d

b

a

d

b

c

a

d

c

c

b

d

c

a

Ordering according to Borda scores: a b c d Ordering according to Borda scores: c b a




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Social Choice



1

5

5

3

2

a

a

c

b

b

d

d

b

a

d

b

c

a

d

c

c

b

d

c

a

Ordering according to Borda scores: a b c d Ordering according to Borda scores: c b a




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Social Choice



Definition (Independence of Irrelevant Alternatives): θ, θ0 ∈ ΘN and alternatives x , y ∈ X :

For preference profiles

x φ(θ) y ⇔ x φ(θ0 ) y, whenever x θi y ⇔ x θi0 y, for all i ∈ N

Intuition: The social preference of two alternatives only depends on the relative ordering of these two alternatives in the individual preference relations. Remark: IIA captures a consistency property of social choice rules. Lack of such consistency enables strategic manipulation.




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Social Choice


Arrow’s Impossibility Theorem

Theorem (Arrow, 1951): Let |X | ≥ 3 and |N | ≥ 2, then, any social welfare function with unrestricted domain satisfying the Pareto property (or unanimity) and Independence of Irrelevant Alternatives is dictatorial.




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Social Choice


Arrow’s Impossibility Theorem

Theorem (Arrow, 1951): Let |X | ≥ 3 and |N | ≥ 2, then, any social welfare function with unrestricted domain satisfying the Pareto property (or unanimity) and Independence of Irrelevant Alternatives is dictatorial.

Remark: No hope for general social welfare functions, but what about social choice functions?




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Social Choice


Muller-Satterthwaite Theorem Definition: Let θ, θ0 ∈ ΘN . θ0 is induced from θ through an improvement of a if for all i ∈ N and all x , y ∈ X with a , x and a , y both: a θi x implies a θi0 x, and x θi y iff x θi0 y Definition: A scf f is strongly monotonic if for all θ, θ0 ∈ ΘN and all a ∈ X such that θ0 is induced through some improvement of some a, f (θ0 ) = {a } or f (θ0 ) = f (θ). Intuition (strong monotonicity): Improving one alternative should not influence the relative ordering of other alternatives. Theorem (Muller-Satterthwaite, 1977): function is dictatorial.


Every strongly monotonic social choice



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Social Choice

Circumventing Impossibility Theorems

Circumventing the Impossibility Theorems

Argue against one of the axioms IIA (most usual in voting protocols) Pareto property Unrestricted domain, e.g., quasi-linear utility functions Strong monotonicity




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Social Choice


Single-peaked Preferences (I)

Definition: Provided that the preference domain has a one-dimensional ordering, a preference relation θi is single-peaked if there exists a point p ∈ X (its peak) such that for all x , y ∈ X such that p ≥ x > y or y > x ≥ p then x θi y. Intuition: Each player prefers points closer to their ideal points over points that are more distant. Definition: Given a set of single-peaked preference profiles (θ1 , . . . , θn ) with peaks (p1 , . . . , pn ), and a set of other alternatives y1 , . . . , ym the median voter rule selects the median of the union of the peaks and the other alternatives.




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Social Choice

Single-peaked Preferences (II) 1

1

2

2

3

3 a

Question:

b

c

b

a

c

Are all profiles in these two settings single-peaked?

θ1 c b a Harrenstein and De Weerdt (LMU and TU Delft)

θ2 a b c

θ3 b a c



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Social Choice

Single-peaked Preferences (II) 1

1

2

2

3

3 a

Question:

b

c

b

a

c

Are all profiles in these two settings single-peaked?

θ1 c b a Harrenstein and De Weerdt (LMU and TU Delft)

θ2 a b c

θ3 b a c



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Social Choice


Single-peaked Preferences (III) Example: p1

Question:

p2

p3

p4

p5

What time to go to the restaurant?




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Social Choice


Single-peaked Preferences (III) Example: p1

Question:

p2

p3

p4

p5

What time to go to the restaurant?

The median is p3 . Not dictatorial: no single player determines outcome Not strong monotonic: not always single-peaked after an improvement of a Incentive compatible (see next section)




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Introduction


1

Game Theory

2

Social Choice

3

Mechanism Design: Theory Introduction Strategic Voting Implementation Implementation in Dominant Strategies Implementation in Other Solution Concepts Implementation in Nash Equilibrium

4





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Introduction


Takahashi wishes to sell a precious Ming vase to either Ann or Bob, depending on who values it most. Yet, Takahashi does not know the valuations of either Ann or Bob. Hence, the goal he sets himself is to sell the vase to Ann if she values it most, and to Bob otherwise. (Social choice function). Then, he tries to construct a protocol (mechanism) that guarantees the vase to be sold to the one who values it most, for all possible valuations Ann and Bob may have. (Implement the social choice function).




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Introduction


Fix a set of possible outcomes X . Given a preference profile (θ1 , . . . , θn ) certain outcomes X ∗ ∈ X are more desirable than others from an outsider’s point of view (e.g., assigning an object to the agent that values it most). However, preferences of the players are unknown to the designer, or hard to obtain. Issue: Design a game (mechanism) such that the outcome given a particular (fixed) solution concept of this game generates one of the desired outcomes in X ∗ for all (relevant) preference profiles (θ1 , . . . , θn ).




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Strategic Voting

Strategic Behavior and the Importance of Truthfulness

Principles of voting make an election more of a game of skill than a real test of the wishes of the electors. My own opinion is that it is better for elections to be decided according to the wish of the majority than of those who happen to have most skill at the game. (C.L. Dodgson)




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Strategic Voting

Strategic Behavior and the Importance of Truthfulness

Consider the Borda rule.

θ1

θ2

θ3

θ1

θ2

θ30 b

a

a

d

a

a

b

b

c

b

b

c

c

d

b

c

d

d

d

c

a

d

c

a

Borda winner in (θ1 , θ2 , θ3 ) is a with 6 points Borda winner in (θ1 , θ2 , θ30 ) is b with 7 points Conclusion:

If θ represent the true preferences, Player 3 had better lie about them!




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Strategic Voting

Social Choice Functions as Strategic Game Forms

Preferences as strategies (lie or tell the truth) Combination of such strategies is a preference profile Outcomes are defined by a social choice function f Social choice functions reflect how to determine the desired (by the designer) outcomes relative to the agents’ preferences. E.g., assigning the object to the agent that values it most The true preferences of the agents are assumed to be unknown to the designer Each possible preference profile θ yields such a game




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Strategic Voting

Social Choice Functions as Strategic Game Forms

θ1

θ10


θ2

θ20

f (θ1 , θ2 )

f (θ1 , θ20 )

f (θ10 , θ2 )

f (θ10 , θ20 )



55 / 94


Strategic Voting

Social Choice Functions as Strategic Game Forms (θ1 , θ2 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ2 )


(θ1 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ20 )



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Strategic Voting


θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ2 )


(θ1 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ20 )



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Strategic Voting


θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ2 )


(θ1 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ20 )



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Strategic Voting


θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ2 )


(θ1 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ20 )



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Strategic Voting


θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ2 )


(θ1 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

θ2

θ20

θ1

f (θ1 , θ2 )

f (θ1 , θ20 )

θ10

f (θ10 , θ2 )

f (θ10 , θ20 )

(θ10 , θ20 )



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Strategic Voting

Incentive Compatibility of Social Choice Functions

Definition: A social choice function f is dominant strategy incentive compatible w.r.t. a set Θ of type profiles, if for all θ∗ ∈ Θ, all θ, θ0 ∈ Θ and all i ∈ N: f (θ1 , . . . , θi∗ , . . . , θn ) vi

⇓

Other bidders bid lower than you, but higher than your true valuation bi > b > vi

⇓

You don’t get the item.


You win the item, but have net utility: vi − b < 0.


Other bidders bid lower than your true valuation bi > vi > b

⇓ You win the item and have net utility: vi − b > 0.


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Second-price auctions

Bid lower than your true valuation, i.e. vi > bi

Other bidders bid higher than you and your true valuation b > vi > bi

Other bidders bid higher than you, but lower than your true valuation vi > b > bi

⇓

⇓

You don’t get the item.


You lose the item, while you could have positive net utility.


Other bidders bid lower than you vi > bi > b

⇓ You win the item and have net utility: vi − b > 0.


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Bid exactly your true valuation, i.e. bi = vi

Other bidders bid higher than you b > vi

Other bidders bid lower than you vi > b

⇓

⇓ You don’t get the item.

Question:

You win the item and have net utility: vi − b > 0.

Does this contradict Gibbard-Satterthwaite? Why (not)?




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Introducing payments

outcome x∈X

preference profile

θ = (θ1 , . . . , θn )

(f , p ) common knowledge

payment p = (p1 , . . . , pn )

Mechanism also includes a payment function p : ΘN → RN that defines the payment for each agent. Utility of each agent i depends linearly on payment: ui : X × ΘN → R, i.e. ui (x , θ) = vi (x , θ) − pi (θ) (and is therefore called a quasi-linear utility function) NB: In most current studies a preference profile θ consists just of the valuation functions vi of the agents.




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Truthful mechanisms

Definition: A direct-revelation mechanism (f , p ) is truthfully implementable or incentive compatible in dominant strategies iff for all θ∗ , θ, θ0 ∈ ΘN for all i it holds that ui (x , θ1 , . . . , θi∗ , . . . , θn ) ≥ ui (x , θ1 , . . . , θi0 , . . . , θn ). Intuition: A social choice function f is truthfully implementable iff for no player there are situations in which telling the truth can hurt.




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VCG Mechanisms

Vickrey-Clarke-Groves (I)

Definition: A mechanism (f , p ) is called a Vickrey-Clarke-Groves (VCG) mechanism if f (θ) ∈ arg maxx ∈X

P

i ∈N

vi (x , θ), i.e. f maximizes social welfare;

for some function h−i : ΘN have that for all valuations θ it holds that −i → R, we P pi (θ) = hi (θ1 , . . . , θi −1 , θi +1 , . . . , θn ) − j ,i vj (f (θ) , θ).




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VCG Mechanisms

Vickrey-Clarke-Groves (II)

Theorem (VCG): Every VCG mechanism for agents with quasi-linear preferences is incentive compatible (strategy-proof) and efficient. Intuition: The payment function aligns each agent’s utility with the social welfare and is independent of the agent’s declared type. Proof. The utility of agent i is vi (f (θ), θ) + j ,i vj (f (θ) , θ) − hi (θ−i ). Ignore hi , because does not depend on θi . P f (θ) ∈ arg maxx ∈X i ∈N vi (x , θ), so efficient by definition, and strategy-proofness follows, because other types θi0 , θi cannot improve the utility of agent i.

P




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VCG Mechanisms

Individual Rationality

Question: (f , p )?

When do agents want to take part in such a (VCG) mechanism




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VCG Mechanisms

Individual Rationality Definition: If every agent i ∈ N has a positive utility, i.e. if vi (f (θ−i , θi ), θ) − pi (θ−i , θi ) ≥ 0, we say that it is (ex post) individually rational (IR) for the agents to participate. Intuition: An agent can withdraw once the solution is known. Definition: If every agent i ∈ N has a positive expected utility given its type, i.e. if Eθ∈Θ [vi (f (θ−i , θi ), θ) − pi (θ−i , θi )] ≥ 0, we say that it is interim individually rational (IR) for the agents to participate. Intuition: An agent can withdraw once it knows its type. Definition: If every agent i ∈ N has a positive expected utility, i.e. if Eθ∈Θ [vi (f (θ), θ) − pi (θ)] ≥ 0, we say that it is ex ante individually rational (IR) for the agents to participate. Intuition: An agent can (only) withdraw before it knows its type.




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VCG Mechanisms

Clarke pivot rule

Definition: The Clarke pivot rule is the payment function where P hi (θ) = maxx ∈X j ,i vi (x , θ). Theorem: The VCG mechanism with the Clarke pivot payment is (ex-post) individually rational. Intuition: The utility of agent i with the Clarke pivot payment is P P vi (f (θ), θ) + j ,i vj (f (θ) , θ) − maxx ∈X j ,i vi (x , θ). This is its marginal contribution to the social welfare.




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VCG Mechanisms

Is VCG the Way to Go?

Theorem (Green and Laffont, 1977): VCG mechanisms are the only efficient and strategy-proof mechanisms for agents with quasi-linear preferences and general valuation functions, amongs all direct-revelation mechanisms. Remark: This has also been extended to hold for Bayesian-Nash mechanisms by Kirshna and Perry (1998) and Williams (1999).




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Shortcomings of VCG

Shortcomings of VCG

What to do with the payments? What if we cannot compute the optimal solution and the payments exactly?




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Shortcomings of VCG

Budget-balancedness (I)

Definition: A mechanism is (ex post) budget-balanced if

P

i

pi = 0.

Definition: A mechanism is (ex post) weakly budget-balanced if

P

i

pi ≥ 0.

Theorem: The VCG mechanism can be made (ex post) weakly budget-balanced. Remark: Ex ante budget-balancedness is defined on the expected sum of payments.




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Shortcomings of VCG

Budget-balancedness (II)

Theorem (Hurwucz, 1975 and Green-Laffont, 1979): No strategy-proof mechanism can implement an efficient (ex post) budget-balanced scf. However, sometimes part of the money can be redistributed.




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Shortcomings of VCG

Budget-balancedness (III)

Example: Bilateral trading one buyer, one seller, one good valuations v1 and v2 Theorem (Myerson-Satterthwaite, 1983): Even in bilateral trading no mechanism can implement an efficient (ex-post) weak budget-balanced and (interim) IR scf, even in Bayes-Nash equilibrium.




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Shortcomings of VCG

Computational properties

Theorem (Nisan-Ronen, 2007): VCG requires exact optimization (approximation loses IC) in general. However possible for some problems where approximation solution is maximal in range (MIR), i.e. max and not depending on agents’ types (eg multi-unit auctions) not possible for others, e.g., for any cost-minimization allocation problem any sub-optimal VCG-based mechanism is degenerate (i.e. can have solutions arbitrarily far from optimal).




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Shortcomings of VCG

Circumvent the shortcomings of VCG

In which situations can all/most of the payments be redistributed? Under which conditions can the optimal solution be computed efficiently? For which problems can we find MIR algorithms?




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Summary and conclusions

Future Work and Open Problems

Online mechanisms Distributed mechanisms Communicating agents’ types or preferences Other solution concepts Other desired properties, e.g. fairness




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Recommended reading We have selected one reference for each of the four parts. R. J. Aumann. Game theory. In J. Eatwell, M. Milgate, and P. Newman, editors, Game Theory, The New Palgrave, pages 1–54. Macmillan, London and Basingstoke, 1987. H. Moulin. Axioms of Cooperative Decision Making. Cambridge University Press, 1988. J. Moore. Implementation, contracts, renegotiations in environments with complete information. In J. Laffont, editor, Advances in Economic Theory, chapter 5, pages 182–282. Cambridge University Press, 1992. N. Nisan. Introduction to mechanism design (for computer scientists). In N. Nisan, T. Roughgarden, E. Tardos, and V. Vazirani, editors, Algorithmic Game Theory, chapter 9, pages 209–242. Cambridge University Press, 2007.




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Other References R. D. Luce and H. Raiffa. Games and Decisions. Introduction and Critical Survey. John Wiley & Sons, New York, 1957. A. Mas-Colell, M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford University Press, New York and Oxford, 1995. H. Moulin. The Strategy of Social Choice. North-Holland, 1983. M. J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, Cambridge, Mass., 1994. D. Parkes. Iterative Combinatorial Auctions: Achieving Economic and Computational Efficiency. PhD thesis, Department of Computer and Information Science, University of Pennsylvania, 2001. Y. Shoham and K. Leyton-Brown. Multiagent Systems: Algorithmic, Game-Theoretic, and Logical Foundations. Cambridge University Press, Cambridge, 2008. forthcoming August 2008. J. von Neumann and O. Morgenstern. Theory of Games and Economic Behavior. Princeton U.P., Princeton, N.J., 1944.




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Introduction to Game Theory and Mechanism Design - Userweb

Introduction to Game Theory and Mechanism Design - Userweb

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