algorithmic game theory
TRANSCRIPT
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Outline
Motivational examplesThe Prisoner’s DilemmaTragedy of the CommonsCoordination games
Definitions
Solution conceptsDominant strategyNash equilibriaCorrelated equilibrium
Finding Equilibria
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Outline
Motivational examplesThe Prisoner’s DilemmaTragedy of the CommonsCoordination games
Definitions
Solution conceptsDominant strategyNash equilibriaCorrelated equilibrium
Finding Equilibria
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Outline
Motivational examplesThe Prisoner’s DilemmaTragedy of the CommonsCoordination games
Definitions
Solution conceptsDominant strategyNash equilibriaCorrelated equilibrium
Finding Equilibria
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Outline
Motivational examplesThe Prisoner’s DilemmaTragedy of the CommonsCoordination games
Definitions
Solution conceptsDominant strategyNash equilibriaCorrelated equilibrium
Finding Equilibria
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
The Prisoner’s Dilemma
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
ISP routing game
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Routing congestion game
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 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)
I the payoff (utility) ui : S → R to each player i , or costci : S → R in other games
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 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)
I the payoff (utility) ui : S → R to each player i , or costci : S → R in other games
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 =: S
I 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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 .
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Matching pennies
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Traffic light
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Correlated equilibrium
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∑
s−i
p(si , s−i )ui (si , s−i ) ≥∑s−i
p(s ′i , s−i )ui (s ′i , s−i ).
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Correlated equilibrium
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∑
s−i
p(si , s−i )ui (si , s−i ) ≥∑s−i
p(s ′i , s−i )ui (s ′i , s−i ).
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
PPAD complexity classPolynomial Parity Argument (Directed case)
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.
I finding Brouwer and Borsuk-Ulam fixpoints
I finding Arrow-Debreu equilibria in markets
See Papadimitriou (1994) for details.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
PPAD complexity classPolynomial Parity Argument (Directed case)
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.
I finding Brouwer and Borsuk-Ulam fixpoints
I finding Arrow-Debreu equilibria in markets
See Papadimitriou (1994) for details.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
PPAD complexity classPolynomial Parity Argument (Directed case)
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.
I finding Brouwer and Borsuk-Ulam fixpoints
I finding Arrow-Debreu equilibria in markets
See Papadimitriou (1994) for details.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
PPAD complexity classPolynomial Parity Argument (Directed case)
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.
I finding Brouwer and Borsuk-Ulam fixpoints
I finding Arrow-Debreu equilibria in markets
See Papadimitriou (1994) for details.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 .
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 .
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 .
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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 .
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
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.
Algorithmic GameTheory
Karel Ha
Motivationalexamples
The Prisoner’sDilemma
Tragedy of theCommons
Coordination games
Definitions
Solution concepts
Dominant strategy
Nash equilibria
Correlated equilibrium
Finding Equilibria
Thank you!
Algorithmic GameTheory
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).
Algorithmic GameTheory
Karel Ha
Games with turns
Ultimatum game
I seller S , buyer B
I 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).
Algorithmic GameTheory
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).
Algorithmic GameTheory
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).
Algorithmic GameTheory
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).
Algorithmic GameTheory
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).