Algorithmic Game Theory

Algorithmic GameTheory

Karel Ha

Motivationalexamples

The Prisoner’sDilemma

Tragedy of theCommons

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Correlated equilibrium

Finding Equilibria

Algorithmic Game TheoryBasic Solution Concepts and Computational Issues

Karel Ha

Spring School of Combinatorics 2014

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Outline

Motivational examplesThe Prisoner’s DilemmaTragedy of the CommonsCoordination games

Definitions

Solution conceptsDominant strategyNash equilibriaCorrelated equilibrium

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Outline

Definitions

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Outline

Definitions

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Outline

Definitions

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The Prisoner’s Dilemma

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

ISP routing game

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Pollution gamemultiplayer version of Prisoner’s dilemma

I n countries

I to control pollution or not to

I to control 7→ cost of 3

I not to control 7→ adds 1 to the cost of all countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I n countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I n countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I n countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I n countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I n countries

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Routing congestion game

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Simultaneous move game

I set N of players {1, 2, . . . , n}I each player i ’s own set of possible pure strategies Si ,

from which a strategy si ∈ Si is selectedI the outcome determined by the vector of selected

strategies s ∈ S1 × · · · × Sn =: SI preference ordering on these outcomes: a complete,

transitive, reflexive binary relation (on the set of allstrategy vectors)

I the payoff (utility) ui : S → R to each player i , or costci : S → R in other games

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

I set N of players {1, 2, . . . , n}

I each player i ’s own set of possible pure strategies Si ,from which a strategy si ∈ Si is selected

I the outcome determined by the vector of selectedstrategies s ∈ S1 × · · · × Sn =: S

I preference ordering on these outcomes: a complete,transitive, reflexive binary relation (on the set of allstrategy vectors)

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

from which a strategy si ∈ Si is selected

I the outcome determined by the vector of selectedstrategies s ∈ S1 × · · · × Sn =: S

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

strategies s ∈ S1 × · · · × Sn =: S

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Representations of games

Standard (matrix) form

I given by the list of all possible strategy combinationstogether with respective payoffs

Compactly represented game

I a succinct formulation rather than the explicit one

I for example, the formula in the Pollution game

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Dominant strategyIt is the strategy vector s ∈ S such that for each player i andeach alternate strategy vector s ′ ∈ S , we have that

ui (si , s ′−i ) ≥ ui (s ′i , s ′−i ),

where s−i denotes the strategy vector s without the i-thstrategy si .

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Pure Nash equilibrium

It is the strategy vector s ∈ S such that for each player i andeach alternate strategy s ′i ∈ Si we have that

ui (si , s−i ) ≥ ui (s ′i , s−i ).

Clearly, every dominant strategy is a pure Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Pure Nash equilibrium

It is the strategy vector s ∈ S such that for each player i andeach alternate strategy s ′i ∈ Si we have that

ui (si , s−i ) ≥ ui (s ′i , s−i ).

Clearly, every dominant strategy is a pure Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Matching pennies

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Mixed strategyA mixed strategy mi : Si → [0, 1] of player i is aprobabilistic distribution over the set Si .

Each player i thus aims to maximize the expected payoffui (mi ,m−i ) under this distribution.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Mixed strategyA mixed strategy mi : Si → [0, 1] of player i is aprobabilistic distribution over the set Si .Each player i thus aims to maximize the expected payoffui (mi ,m−i ) under this distribution.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Mixed strategyA mixed strategy mi : Si → [0, 1] of player i is aprobabilistic distribution over the set Si .Each player i thus aims to maximize the expected payoffui (mi ,m−i ) under this distribution.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Mixed Nash equilibrium

A vector m = (m1, . . . ,mn) of mixed strategies of all playersis called mixed Nash equilibrium if for each player i andeach alternate mixed strategy m′i we have that

ui (mi ,m−i ) ≥ ui (m′i ,m−i ).

Theorem (Nash, 1951)

Any game with a finite set of players and finite set ofstrategies has a mixed Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Mixed Nash equilibrium

A vector m = (m1, . . . ,mn) of mixed strategies of all playersis called mixed Nash equilibrium if for each player i andeach alternate mixed strategy m′i we have that

ui (mi ,m−i ) ≥ ui (m′i ,m−i ).

Theorem (Nash, 1951)

Any game with a finite set of players and finite set ofstrategies has a mixed Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Traffic light

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

An external correlation device suggests a strategy vector swith the probability p(s) = p(si , s−i ).

A correlated equilibrium is a probability distribution s overstrategy vectors such that for each player i and eachalternate strategy s ′i we have that∑

p(si , s−i )ui (si , s−i ) ≥∑s−i

p(s ′i , s−i )ui (s ′i , s−i ).

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

An external correlation device suggests a strategy vector swith the probability p(s) = p(si , s−i ).A correlated equilibrium is a probability distribution s overstrategy vectors such that for each player i and eachalternate strategy s ′i we have that∑

p(si , s−i )ui (si , s−i ) ≥∑s−i

p(s ′i , s−i )ui (s ′i , s−i ).

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

PPAD complexity classPolynomial Parity Argument (Directed case)

Examples of problems

I cutting Ham SandwichesI Given n sets of 2n points each in n dimensions, find a

hyperplane which, for each of the n sets, leaves n pointson each side.

I finding Brouwer and Borsuk-Ulam fixpoints

I finding Arrow-Debreu equilibria in markets

See Papadimitriou (1994) for details.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Examples of problemsI cutting Ham Sandwiches

I Given n sets of 2n points each in n dimensions, find ahyperplane which, for each of the n sets, leaves n pointson each side.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Complexity of finding Nash equilibrium

Theorem

Finding Nash equilibrium is PPAD-complete.

Proof.Chapter 2 of Nisan, Roughgarden, Tardos, Vazirani:Algorithmic Game Theory.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Theorem

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Theorem

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Two-Person Zero-Sum GamesThe sum of the payoffs of the two players is zero for any choice ofstrategies.

I payoff matrix A (for the row player)

I Nash equilibrium: row vector p∗, column vector q∗

I expected payoff equals v∗ = p∗Aq∗

Proposition

Two-Person Zero-Sum Games can be solved via the linearprogramming.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Proposition

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Proposition

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Proposition

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Proposition

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode I

I In a Nash equilibrium, even if a player knows thestrategies of others, he cannot be better off bydeviating.

I If the row player’s strategy p was publicly known, thecolumn player would want to minimize his loss.

I Hence, he would choose the minimum entries in pA.

I So the best publicly announced strategy (for the rowplayer) is to maximize this minimum value.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode I

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode I

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode I

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode I

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode II

vr : = max v∑i

pi = 1

p ≥ 0

(pA)j ≥ v for all j

I The row player can guarantee to win at least vr in anyequilibrium, therefore vr ≤ v∗.

I An equilibrium is stable even if known to the opponent,therefore v∗ ≤ vr .

I Thus v∗ = vr .

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode II

vr : = max v∑i

pi = 1

p ≥ 0

I Thus v∗ = vr .

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode II

vr : = max v∑i

pi = 1

p ≥ 0

I Thus v∗ = vr .

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode II

vr : = max v∑i

pi = 1

p ≥ 0

I Thus v∗ = vr .

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

The proofEpisode III

Similarly, for the linear program (of the column player)

vc : = min v∑j

qj = 1

q ≥ 0

(Aq)i ≤ v for all i

we have that v∗ = vc , and therefore, vr = vc .

Optimal solutions pr and qc to the two linear programs forma Nash equilibrium.

Note that the programs are duals of each other.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

vc : = min v∑j

qj = 1

q ≥ 0

we have that v∗ = vc , and therefore, vr = vc .Optimal solutions pr and qc to the two linear programs forma Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

vc : = min v∑j

qj = 1

q ≥ 0

we have that v∗ = vc , and therefore, vr = vc .Optimal solutions pr and qc to the two linear programs forma Nash equilibrium.

Karel Ha

Coordination games

Definitions

Solution concepts

Dominant strategy

Nash equilibria

Finding Equilibria

Thank you!

Karel Ha

Games with turns

Ultimatum game

I seller S , buyer BI 2 steps:

I S offers price pI B reacts

I S has payoff p if the sale occurs, and 0 otherwise.

I B has a value v for the good and he has payoff v − p ifhe buys, and 0 otherwise.

I S is aware of B’s value v (full information game).

Karel Ha

Games with turns

Ultimatum game

I seller S , buyer B

I 2 steps:I S offers price pI B reacts

Karel Ha

Games with turns

Ultimatum game

Karel Ha

Games with turns

Ultimatum game

Karel Ha

Games with turns

Ultimatum game

Karel Ha

Games with turns

Ultimatum game

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