CDS270: Optimization, Game and Layering in Communication Networks

Lecture 1: Static Games and Classical Mechanism Design

Lijun Chen

10/04/2006

Agenda

2

Strategic games and their solution conceptsStrategic form games and dominated strategiesNash equilibrium and correlated equilibrium

Classical mechanism design Incomplete information gamesIncentive-compatible mechanismVCG mechanism

Strategic game

3

Def: a game in strategic form is a triple

is the set of players (agents)is the player strategy space

is the player payoff function Notations

: the set of all profiles of player strategies

: profile of strategies: the profile of

strategies other than player

},,{ NiNi uSNG ∈∈=

NiS i

RSui → : i

NSSSS ×××= L21

),,,( 21 Nssss L=

),,,,,,,( 1121 Niii ssssss LLL +−− =i

4

Implicitly assume that players have preferences over different outcomes, which can be captured by assigning payoffs to the outcomesThe basic model of rationality is that of a payoff maximizerFirst consider pure strategy, will consider mixed strategy later

Example: finite game

5

4,3

2,1

3,0

5,1

8,4

9,6

6,2

3,6

2,8

L M

U

R

M

D

column

row

Example: Continuous strategy game

6

Cournot competitionTwo players: firm 1 and firm2Strategy : the amount of widget that firm produces The payoff for each firm is the net revenue

where is the price, is the unit cost for firm

],0[ ∞∈is

iiii scsspsssu −+= )(),( 2121

p ici

i

Dominated strategies

7

How to predict the outcome of a game?Prisoner’s Dilemma

Two prisoners will play (C,C)Def: a strategy is (weakly) dominated for player if there exists such that

-2,-2

-1,-5

-5,-1

-4,-4

D C

D

C

is

i ii Ss ∈′

iiiiiiii Ssssussu −−−− ∈≥′ allfor ),(),(

Iterated elimination of dominated strategies

8

Iterated elimination of dominated strategies

However, most of games are not solvable by iterated elimination of dominated strategies

2,8

4,3

2,1

3,0

5,1

8,4

9,6

6,2

3,6

L M

U

R

M

D

Nash equilibrium

9

Def: a strategy profile is a Nash equilibrium, if for all ,

For any , define best response function

Then a strategy profile is a Nash equilibriumiff

*s

i

iiiiiiii Ssssussu ∈≥ −− allfor ),(),( ***

ii Ss −− ∈

}. ),( ),(|{)( iiiiiiiiiiii SsssussuSssB ∈′∀′≥∈= −−−*s

).( **iii sBs −∈

Examples

10

Battle of the Sexes

Soccer

2,1

0,0

0,0

1,2

Ballet

Ballet

Soccer

Two Nash equilibria (Ballet, Ballet) and (Soccer, Soccer)

Cournot Competition

11

Suppose a price function Suppose cost Then, the best response function

Nash equilibrium satisfies , i.e.,

)}(1,0max{)( 2121 ssssp +−=+

10 21 ≤==≤ ccc

2/)1()(2/)1()(

112

221

cssBcssB

−−=−−=

)()(

122

211

sBssBs

==

)1()1(

2

1

cscs

−=−=

3/3/

Second price auction

12

An object to be sold to a player inEach player has a valuation of the object. We further assume The players simultaneously submit bids,The object is given to the player with highest bid. The winner pays the second highest bid.The payoff of the winner is his valuation of the object minus the price he pays. All other players’ payoff is zero.

N

i iv021 >>>> Nvvv L

Nbb ,,1 L

13

is Nash equilibriumPlayer 1 receives the object and pay , and has payoff . Player 1 has no incentive to deviate, since his payoff can only decrease.For other players, the payoff is zero. In order to change his payoff, he needs to bid more than

, but that will result in negative payoff. So, no player has incentive to change.

Question: are they more Nash equilibria?

),,(),,( 11 NN vvbb LL =

2v021 >−vv

1v

14

Not all games have (pure) Nash equilibriumMatching Pennies

-1,1

1,-1 -1,1

1,-1

Heads Tails

Heads

Tails

Mixed strategies

15

Let denote the set of probability distribution over player strategy spaceA mixed strategy is a probability mass function over pure strategiesThe payoff of a mixed strategy is the expected value of the pure strategy profiles

iΣ

iSi

ii Σ∈σ

ii Ss ∈

∑ ∏∈ ∈

=Ss

iNj

jji susu )())(( σ

Mixed strategy Nash equilibrium

16

Def: a mixed strategy profile is a (mixed strategy) Nash equilibrium if for all

A mixed strategy profile is a (mixed strategy) Nash equilibrium if for all

The payoff is the same for allThe payoff for each is not larger

*σ

i

iiiiiiii uu Σ∈≥ −− σσσσσ allfor ),(),( ***

*σ

i

iiiiiiii Sssuu ∈≥ −− allfor ),(),( *** σσσ

),( *iii su −σ )(supp *

iis σ∈

),( *iii su −σ )(supp *

iis σ∉

Example

17

Assume row (column) player choose “ballet” with probability ( ) and “soccer” with probability ( )

Mixed strategy Nash equilibrium is

Soccer

2,1

0,0

0,0

1,2

Ballet

Ballet

Soccer

pp−1 q−1

q

)1(20)1(01)1(10)1(02ppppqqqq−×+×=−×+×−×+×=−×+×

⎩⎨⎧

==

3/13/2

qp

Existence of Nash equilibrium

18

Theorem (Nash ‘50): Every finite strategic game has a mixed strategy Nash equilibrium.Example: Matching Pennies game has a mixed strategy Nash equilibrium (½, ½; ½, ½)

Proof: using Kakutani’s fixed point theorem. See section 1.3.1 of the handout for details

1,-1 -1,1

1,-1

Heads TailsHeads

Tails -1,1

Continuous strategy game

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Theorem (Debreu ’52; Glicksberg ’52; Fan ’52): Consider a strategic game with continuous strategy space. A pure strategy Nash equilibrium exists if

is nonempty compact convex setis continuous in and quasi-concave in

Theorem (Glicksberg ’52): Consider a strategic game with nonempty compact strategy space. A mixed strategy Nash equilibrium exists if is continuous.

},,{ NiNi uSN ∈∈

iS

iSiu S

},,{ NiNi uSN ∈∈

iu

Correlated equilibrium

20

In Nash equilibrium, players choose strategies independently. How about players observing some common signals? Traffic intersection game

Two pure Nash equilibria: (stop, go) and (go, stop)One mixed strategy equilibrium: (½, ½; ½, ½)If there is a traffic signal such that with probability ½ (red light) players play (stop, go) and with probability ½ (green light) players play (go, stop). This is a correlated equilibrium.

0,0

2,2 1,3

Stop GoStop

Go 3,1

21

Def: correlated equilibrium is a probability distribution over the pure strategy space such that for all

A mixed strategy Nash equilibrium is a correlated equilibriumThe set of correlated equilibria is convex and contains the convex hull of mixed strategy Nash equilibria

)(⋅pi

iiis

iiiiiiii Stsstussusspi

∈≥−∑−

−−− , allfor 0] ),(),()[,(

Dynamics in games

22

Nash equilibrium is a very strong concept. It assumes player strategies, payoffs and ration-ality are “common knowledge”.“Game theory lacks a general and convincing argument that a Nash outcome will occur”.One justification is that equilibria arise as a result of adaptation (learning).

Consider repeated play of the strategic gamePlayers are myopic, and adjust their strategies based on the strategies of other players in previous rounds.

23

Best response

Fictitious play, regret-based heuristics, etcMany if not most network algorithms are repeated and adaptive, and achieving some equilibria. Will discuss these and networking games later in this course.

))(()1( tsBts iii −=+

Classical mechanism design (MD)

24

Mechanisms: Protocols to implement an outcome (equilibrium) with desired system-wide properties despite the self-interest and private information of agents.Mechanism design: the design of such mechanisms.Provide an introduction to game theoretic approach to classical mechanism design

Game theoretic approach to MD

25

Start with a strategic model of agent behaviorDesign rules of a game, so that when agents play as assumed the outcomes with desired properties happen

Incomplete information games

26

Players have private type Strategy is a function of a player’s type Payoff is a function of player’s typeAssume types are drawn from some objective distributionDef: a strategy profile is a Bayesian-Nash equilibrium if every players plays a best response to maximize expected payoff given its belief about distribution , i.e.,

Θ∈),,,( 21 Nθθθ L

iii Ss ∈)(θ

Rsu ii ∈)),(( θθ

),,,( 21 Np θθθ L

*s

)),(,()|(maxarg)( ** ∑−

−−−∈i

iiiiiiiisii ssups

θ

θθθθθ

i

)|( iip θθ−

Example: variant of Battle of the Sexes

27

Two types: either wants to meet the other or does notAssume row player wants to meet column player, but not sure if column player want to meet her or not (assign ½ probability to each case); and column player knows row player’s typeIf column player want to meet row player, the payoffs are Soccer

2,1

0,0

0,0

1,2

Ballet

Ballet

Soccer

28

If column player does not want to meet row player, the payoffs are

The Bayesian-Nash equilibrium is (Ballet, (Ballet, Soccer))

E[Ballet, (Ballet, Soccer)]= ½x2+ ½x0=1E[Soccer, (Ballet, Soccer)]= ½x0+ ½x1= ½

Soccer

2,0

0,1

0,2

1,0

Ballet

Ballet

Soccer

Stronger solution concepts

29

Def: a strategy profile is Ex post Nash equilibrium if every player ‘s strategy is best response whatever the type of others

Def: a strategy profile is dominant strategy equilibrium if every player ‘s strategy is best response whatever the type and whatever strategy of others

*s

i

iiiiiisii ssusi

−−−∈ θθθθ allfor )),(,(maxarg)( **

i

iiiiiisii ssusi

−−−∈ θθθθ ,s allfor )),(,(maxarg)( i-*

*s

Example: second price auction

30

The type is player valuationEach player submit bid A dominant strategy is to bidPlayers don’t need to know valuations (types), or strategies of others.

iv)( ii vb

iii vvb =)(*

Model of MD

31

Set of alternative outcomesPlayer has private information (type) Type defines a value function for outcome for each player Player payoff for outcome and paymentsThe desired properties are encapsulated in the social choice function

e.g., choose to maximize social welfare, i.e.,

O

i iθ

Rov ii ∈);( θOo∈ i

iiiii povou −= );();( θθ o

ip

Of →Θ:

o∑

∈=

iiOoouf );(maxarg)( θθ

32

The goal is to implement social choice function

A mechanism is defined by an outcome rule and a payment ruleA mechanism M implements social choice function if , where the strategy profile is an equilibrium solution of the game induced by M.

)(θf

…

)( 11 θs

)( NNs θ

Mechanism )(sgo =

)(),,( 1 sppp N =L},{ pgM =

OSg →:nRSp →:

)(θf)())(,),(( *

11* θθθ fssg NN =L

),,( *1

*Nss L

33

Properties of social choice functions and mechanisms

Pareto optimal:

Efficient:

Budget-balance:

A mechanism that implements the corresponding social choice functions is called Pareto optimal, efficient or budget-balanced mechanisms, respectively.

)),((),( j )),() ),every for if θθθθθθ scfuau(scfu(a,θuf(a jjii <∃⇒>≠

∑∈i

iia

avf ),(maxarg)( if θθ

∑ =i

ip 0)( if θ

Incentive-compatible mechanism

34

Revelation principle: any mechanism can be transformed into an incentive compatible, direct-revelation mechanism that implements the same social choice functionDirect-revelation mechanism is a mechanism in which player strategy space is restricted to their types

…

1θ

Nθ

Mechanism )(θgo =

)(),,( 1 θppp N =L},{ pgM =

35

Incentive-compatible means the equilibrium strategy is to report truthful information about their types (truth-revelation).

First price auction is not incentive-compatible. In first price auction, the buyer with highest bid gets the object and pays his bid. The second price auction is incentive compatible, direct-revelation mechanism.

Truthful mechanism

36

Truthful (aka “strategy-proof”) mechanism: truth-revelation is a dominant strategy equilibrium.

Very robust to assumption about agent rationality and information about each otherAn agent can compute its optimal strategy without modeling the types and strategies of others

Vickrey-Clarke-Groves mechanisms

37

VCG mechanism:Collect from agents

: select an outcome: agent pays ,

where

),,,( 21 Nθθθθ L=

)(θg ∑∈

∈i

iiOoovo );(maxarg* θ

)(θp i ∑∑≠≠

− −ij

jjij

ji

j ovov );();( * θθ

∑≠∈

− ∈ij

jjOo

i ovo );(maxarg θ

38

Theorem: VCG mechanism is efficient and truthful.Proof: VCG mechanism is the only mechanism that is efficient and strategy-proof amongst direct-revelation mechanisms.

∑ ∑≠ ≠

−− −+=

ij ijj

ijjjiiiii ovovovu );();();(),( ** θθθθθ

Combinatorial auction

39

Goods Outcomes: allocations , where and s are not overlapped.Agent valuation forGoal: allocate goods to maximizeApplications: wireless spectrum auction, course scheduling, …

P

),,( 1 NAAA L= PAi ⊆

iA

);( iii Av θ PAi ⊆

∑i

iii Av );( θ

40

Two items A and B; 3 agents (taken from Parkes)Valuation

Agent 3 wins AB and pays 10-0=10.

1

A B AB

32

5

00

05

0 12

5

5

41

Another valuation

Agents 1 and 2 win and each pays 7-5=2

1

A B AB

32

5

00

05

0 7

5

5

Remarks

42

Only consider the incentive issue: to overcome the self-interest of agentsNot discuss computational and informational issues. Will discuss these in distributed mechanism design and its applications in networking.