CS711: Introduction to Game Theory and Mechanism Design · Domination Normal form game hN;(S i)...

CS711: Introduction to Game Theoryand Mechanism Design

Teacher: Swaprava Nath

Domination, Elimination of Dominated Strategies, Nash Equilibrium

Domination

Normal form game 〈N, (Si)i∈N , (ui)i∈N 〉

Definition (Dominated Strategy)

A strategy s′i ∈ Si of player i is strictly dominated if there exists anotherstrategy si of i such that for every strategy profile s−i ∈ S−i of the other players

ui(si, s−i) > ui(s′i, s−i).

A strategy s′i ∈ Si of player i is weakly dominated if there exists anotherstrategy si of i such that for every strategy profile s−i ∈ S−i of the other players

ui(si, s−i) > ui(s′i, s−i),

and there exists some s̃−i ∈ S−i such that

ui(si, s̃−i) > ui(s′i, s̃−i).

1 / 12 Game Theory and Mechanism Design Domination, Elimination of Dominated Strategies, Nash Equilibrium

Domination

Normal form game 〈N, (Si)i∈N , (ui)i∈N 〉

Definition (Dominated Strategy)

A strategy s′i ∈ Si of player i is strictly dominated if there exists anotherstrategy si of i such that for every strategy profile s−i ∈ S−i of the other players

ui(si, s−i) > ui(s′i, s−i).

A strategy s′i ∈ Si of player i is weakly dominated if there exists anotherstrategy si of i such that for every strategy profile s−i ∈ S−i of the other players

ui(si, s−i) > ui(s′i, s−i),

and there exists some s̃−i ∈ S−i such that

ui(si, s̃−i) > ui(s′i, s̃−i).

Domination (Contd.)

Definition (Dominant Strategy)

A strategy si is strictly (weakly) dominant strategy for player i if si strictly(weakly) dominates all other s′i ∈ Si \ {si}.

Definition (Dominant Strategy Equilibrium)

A strategy profile (s∗i , s∗−i) is a strictly (weakly) dominant strategy equilibrium

(SDSE (WDSE)) if s∗i is a strictly (weakly) dominant strategy for every i, i ∈ N .

Examples

Neighboring kingdoms’ dilemma

A\B Agriculture DefenseAgriculture 5,5 0,6

Defense 6,0 1,1

Do the players have a dominant strategy? Which kind?

Examples

Neighboring kingdoms’ dilemma

A\B Agriculture DefenseAgriculture 5,5 0,6

Defense 6,0 1,1

Examples (contd.)

One indivisible object for sale

Two players, each having a value for the object, v1 and v2

Rules:

I each player can choose a number in [0,M ] which reflects her willingness tobuy the object

I player quoting the larger number (tie broken in favor of player 1) winsI pays the losing players chosen numberI utility of the winning player is = her value - her paymentI utility of the losing player is zero

What is the normal form representation of this game?

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

Rules:

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

Rules:

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

Rules:I each player can choose a number in [0,M ] which reflects her willingness to

buy the object

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

buy the objectI player quoting the larger number (tie broken in favor of player 1) wins

I pays the losing players chosen numberI utility of the winning player is = her value - her paymentI utility of the losing player is zero

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

buy the objectI player quoting the larger number (tie broken in favor of player 1) winsI pays the losing players chosen number

I utility of the winning player is = her value - her paymentI utility of the losing player is zero

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

buy the objectI player quoting the larger number (tie broken in favor of player 1) winsI pays the losing players chosen numberI utility of the winning player is = her value - her payment

I utility of the losing player is zero

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

buy the objectI player quoting the larger number (tie broken in favor of player 1) winsI pays the losing players chosen numberI utility of the winning player is = her value - her paymentI utility of the losing player is zero

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise;

u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise; u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Examples (contd.)

N = {1, 2}, S1 = S2 = [0,M ]

u1(s1, s2) =

{v1 − s2 if s1 > s2

0 otherwise; u2(s1, s2) =

{v2 − s1 if s1 < s2

0 otherwise

Iterated Elimination of Dominated Strategies

Rational players do not play dominated strategies

To find a reasonable outcome, eliminate the dominated strategies

For strictly dominated strategies, the order of elimination does not matter –always reaches the same residual game

For weakly dominated strategies, the order matters

It can also eliminate some reasonable outcomes1\2 L C RT 1,2 2,3 0,3M 2,2 2,1 3,2B 2,1 0,0 1,0

Order: T, R, B, C, Outcome: ML, Payoff: 2,2

Order: B, L, C, T, Outcome: MR, Payoff: 3,2

DSE: Does it always exist?

Coordination game: drive to the left of right?

1\2 L RL 1,1 0,0R 0,0 1,1

Football or Cricket game

1\2 F CF 2,1 0,0C 0,0 1,2

Dominance cannot explain a reasonable outcome in this game – then what?

Refine the equilibrium concept

1\2 L RL 1,1 0,0R 0,0 1,1

1\2 F CF 2,1 0,0C 0,0 1,2

1\2 L RL 1,1 0,0R 0,0 1,1

1\2 F CF 2,1 0,0C 0,0 1,2

1\2 L RL 1,1 0,0R 0,0 1,1

1\2 F CF 2,1 0,0C 0,0 1,2

Nash EquilibriumThis is a strategy profile from where no player gains by a unilateral deviation

Definition (Pure strategy Nash equilibrium)

A strategy profile (s∗i , s∗−i) is a pure strategy Nash equilibrium (PSNE) if ∀i ∈ N

and ∀si ∈ Si

ui(s∗i , s∗−i) > ui(si, s

∗−i).

1\2 F CF 2,1 0,0C 0,0 1,2

A best response view

Definition (Best response set)

A best response of agent i against the strategy profile s−i of the other players is astrategy that gives the maximum utility against the s−i chosen by other players,i.e.,

Bi(s−i) = {si ∈ Si : ui(si, s−i) > ui(s′i, s−i), ∀s′i ∈ Si}.

and ∀si ∈ Si

∗−i).

1\2 F CF 2,1 0,0C 0,0 1,2

and ∀si ∈ Si

∗−i).

1\2 F CF 2,1 0,0C 0,0 1,2

and ∀si ∈ Si

∗−i).

1\2 F CF 2,1 0,0C 0,0 1,2

and ∀si ∈ Si

∗−i).

1\2 F CF 2,1 0,0C 0,0 1,2

PSNE: Best Response View

Definition (PSNE)

A strategy profile (s∗i , s∗−i) is a pure strategy Nash equilibrium if

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Stability – a point from where no player would like to deviate

Self-enforcing agreement among the players

this still assumes players to be fully rational

multiplicity of equilibria – which one should players coordinate to

Definition (PSNE)

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Definition (PSNE)

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Definition (PSNE)

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Definition (PSNE)

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Definition (PSNE)

∀i ∈ N, s∗i ∈ Bi(s∗−i).

Properties of PSNE

Risk averse playersrisky equilibrium

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

player 1 may choose T, since that is least riskyplayer 2 should then choose Lanother aspect of rationality – where players make pessimistic estimate aboutothers’ play (instead of utility maximization)this worst case optimal choice is known as max-min strategy

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

ui(s∗i , t−i) > vi, ∀t−i ∈ S−i

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

player 1 may choose T, since that is least risky

player 2 should then choose Lanother aspect of rationality – where players make pessimistic estimate aboutothers’ play (instead of utility maximization)this worst case optimal choice is known as max-min strategy

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

player 1 may choose T, since that is least riskyplayer 2 should then choose L

another aspect of rationality – where players make pessimistic estimate aboutothers’ play (instead of utility maximization)this worst case optimal choice is known as max-min strategy

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

player 1 may choose T, since that is least riskyplayer 2 should then choose Lanother aspect of rationality – where players make pessimistic estimate aboutothers’ play (instead of utility maximization)

this worst case optimal choice is known as max-min strategy

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

s∗i ∈

argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

s∗i ∈ argmaxsi∈Si

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

mins−i∈S−i

ui(si, s−i)

maxmin value

vi = maxsi∈Si

mins−i∈S−i

ui(si, s−i)

Max-min and DominanceRelationship of max-min strategies and dominant strategies

TheoremIf s∗i is a dominant strategy for player i, then it is a max-min strategy for player ias well, for all i ∈ N . Such a strategy is a best response of player i to anystrategy profile of the other players.

Proof sketch: [for strictly dominant strategies]

Let s∗i is the strictly dominant strategy of player i

ui(s∗i , s−i) > ui(s

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

holds for every ss′i−i ∈ argmins−i∈S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

The second part of the theorem follows from definition

Exercise: finish the proof for weakly dominant strategies

Proof sketch: [for strictly dominant strategies]Let s∗i is the strictly dominant strategy of player i

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

The second part of the theorem follows from definition�

′i, s−i), ∀s′i ∈ Si \ {s∗i }, s−i ∈ S−i

ui(s′i, s−i)

ui(s∗i , s

s′i−i) > ui(s

′i, s

s′i−i), ∀s

′i ∈ Si \ {s∗i }

mins−i∈S−i

ui(si, s−i)

The second part of the theorem follows from definition�

Exercise: finish the proof for weakly dominant strategies10 / 12 Game Theory and Mechanism Design Domination, Elimination of Dominated Strategies, Nash Equilibrium

More results

TheoremIf every player i ∈ N has a strictly dominant strategy s∗i , then the strategy profile(s∗1, . . . , s

∗n) is the unique equilibrium point of the game and also the unique

profile of max-min strategies.

Proof: exercise (can use the previous result)

Relationship with pure strategy Nash equilibrium

Theorem

For every PSNE s∗ = (s∗1, . . . , s∗n) of a normal form game satisfies ui(s

∗) > vi, forall i ∈ N .

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

More results

Theorem

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

More results

Theorem

1\2 L RT 2,1 2,-20M 3,0 -10,1B -100,2 3,3

Proof:

ui(si, s∗−i) > min

s−i∈S−i

ui(si, s−i), ∀si ∈ Si, by definition of min

Now, ui(s∗i , s∗−i) > ui(si, s

∗−i), ∀si ∈ Si, by the best response definition

Hence, ui(s∗i , s∗−i) = max

si∈Si

ui(si, s∗−i) > max

si∈Si

mins−i∈S−i

ui(si, s−i) = vi

Proof:

s−i∈S−i

si∈Si

mins−i∈S−i

ui(si, s−i) = vi

Proof:

s−i∈S−i

si∈Si

mins−i∈S−i

ui(si, s−i) = vi

CS711: Introduction to Game Theory and Mechanism Design · Domination Normal form game hN;(S i)...

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