Mobile Computing Group An Introduction to Game Theory part I Vangelis Angelakis 14 / 10 / 2004.

Mobile Computing GroupMobile Computing GroupMobile Computing GroupMobile Computing Group

An Introduction to Game TheoryAn Introduction to Game Theorypartpart II

Vangelis AngelakisVangelis Angelakis

14 / 10 / 200414 / 10 / 2004

IntroductionIntroduction

TNL - Mobile Computing Group Angelakis Vangelis 14/10/2003

Game theory is the mathematically foundedstudy of conflict and cooperation.

Game theoretic concepts apply when the decisions/actions of independent agents affect the

interests of others

Agents may be individuals, groups, firms, “intelligent” devices

Game theory provides them with a methodology for structuring and analyzing problems of strategic choice.

By formally modeling a situation as a game requires to enumerate the players, their preferences and their

strategic options,

The decision maker is given a clearer and broader view of the situation in hand.

Game theory is the mathematically foundedstudy of conflict and cooperation.

Game theoretic concepts apply when the decisions/actions of independent agents affect the

interests of others

Agents may be individuals, groups, firms, “intelligent” devices

Game theory provides them with a methodology for structuring and analyzing problems of strategic choice.

By formally modeling a situation as a game requires to enumerate the players, their preferences and their

strategic options,

The decision maker is given a clearer and broader view of the situation in hand.

A little history lessonA little history lesson


1838: The first formal game theoretic analysis –the study of a duopoly by A. Cournot.1921: E. Borel suggests a formal theory of games1928: “Theory of parlour games” by von Neumann1944: “Theory of Games and Economic behaviour” by von Neumann & O. Morgenstern –basic terminology & problem setup is standardized in this book.1951: J. Nash proves that finite games always have an equilibrium…1950’s-60’s: applications to war and (international) politics1970’s: revolutionalized the economic theory, applications in sociology,psychology, evolutionary biology1990’s: E/M spectrum auctions design1994: Nash is awarded the Nobel prize in economics

1838: The first formal game theoretic analysis –the study of a duopoly by A. Cournot.1921: E. Borel suggests a formal theory of games1928: “Theory of parlour games” by von Neumann1944: “Theory of Games and Economic behaviour” by von Neumann & O. Morgenstern –basic terminology & problem setup is standardized in this book.1951: J. Nash proves that finite games always have an equilibrium…1950’s-60’s: applications to war and (international) politics1970’s: revolutionalized the economic theory, applications in sociology,psychology, evolutionary biology1990’s: E/M spectrum auctions design1994: Nash is awarded the Nobel prize in economics

Defining GamesDefining Games


A Game is a formal model of an interactive situation

The formal definition of a game declares:• The players• Their preferences• Their information• The strategic actions that are available to them• How these actions influence the outcome

Games with single players are characterized decision problems

Two “schools” of game theory are formed depending on the focus of games and the granularity of the game description

A Game is a formal model of an interactive situation

The formal definition of a game declares:• The players• Their preferences• Their information• The strategic actions that are available to them• How these actions influence the outcome

Games with single players are characterized decision problems

Two “schools” of game theory are formed depending on the focus of games and the granularity of the game description

Cooperative GamesCooperative Games


A high level description declaring:-payoffs of each potential coalition that can play a game-not the formation process of the coalition

Hence, it is not defined as a game in which players actually do cooperate, but as a game in which any cooperation is enforced by an outside party.This outside party involvement and the possible bargaining process that takes place do not belong to the cooperative game description.

Cooperative game theory investigates coalitional games with respect to relative power held by the players and how a successful coalition should divide its gains.

A high level description declaring:-payoffs of each potential coalition that can play a game-not the formation process of the coalition

Hence, it is not defined as a game in which players actually do cooperate, but as a game in which any cooperation is enforced by an outside party.This outside party involvement and the possible bargaining process that takes place do not belong to the cooperative game description.

Cooperative game theory investigates coalitional games with respect to relative power held by the players and how a successful coalition should divide its gains.

Non-cooperative GamesNon-cooperative Games


Players make choices of their own interest

Order and timing may be crucial to determine the outcome of a game

In a non-cooperative game players are unable to make enforceable contracts outside of those specifically modelled in the game.

Hence, it is not defined as games in which players do not cooperate, but as games in which any cooperation must be self-enforcing.

Cooperation arises, when it is in the best interest of players

Players make choices of their own interest

Order and timing may be crucial to determine the outcome of a game

In a non-cooperative game players are unable to make enforceable contracts outside of those specifically modelled in the game.

Hence, it is not defined as games in which players do not cooperate, but as games in which any cooperation must be self-enforcing.

Cooperation arises, when it is in the best interest of players

AssumptionsAssumptions


Strategic BehaviorBeing aware of your opponents existence and

act trying to anticipate/counter their moves is called strategic behavior

Rationalityrational players always chose actions that give

them an outcome they most prefer, given what their opponents will do.Predict how a game will be played orAdvice how to best play in a game against rational opponents

Rationality is an assumption under common knowledge

Some define games as: The interaction among a group of rational agents who behave strategically

Strategic BehaviorBeing aware of your opponents existence and

act trying to anticipate/counter their moves is called strategic behavior

Rationalityrational players always chose actions that give

them an outcome they most prefer, given what their opponents will do.Predict how a game will be played orAdvice how to best play in a game against rational opponents

Rationality is an assumption under common knowledge

Some define games as: The interaction among a group of rational agents who behave strategically

Strategic & Extensive Form GamesStrategic & Extensive Form Games


Strategic form (normal form)A strategic form game can be a single round of a repeated game

Time invariant (no turns) game is played in the blink of an eyeList each players strategiesList outcomes that result from each possible combination of moves

The outcome of a game is the payoff each player getsThe Payoff is a numerical value, also called Utility.

Extensive form (game tree)The complete description of how a game is played as time goes by+ Order of players+ Information of players at each point in time

More on extensive forms in part II

Strategic form (normal form)A strategic form game can be a single round of a repeated game

Time invariant (no turns) game is played in the blink of an eyeList each players strategiesList outcomes that result from each possible combination of moves

The outcome of a game is the payoff each player getsThe Payoff is a numerical value, also called Utility.

Extensive form (game tree)The complete description of how a game is played as time goes by+ Order of players+ Information of players at each point in time

More on extensive forms in part II

DominanceDominance


Assume a rational player having strategies A and B.

If A has a higher utility than B no mater what strategic combination other players make, then strategy A is said to dominate strategy B.

Rational players do not play dominated strategies.

In some games examining available strategies and eliminating dominated strategies results in only one credible strategy for a rational player.

Assume a rational player having strategies A and B.

If A has a higher utility than B no mater what strategic combination other players make, then strategy A is said to dominate strategy B.

Rational players do not play dominated strategies.

In some games examining available strategies and eliminating dominated strategies results in only one credible strategy for a rational player.

Prisoners’ DilemmaPrisoners’ Dilemma


A bank is robbed and the robbers escape the police…

A pair of suspects are arrested later in the day and are held in separate cells.

Each is told that:

-if one alone confesses the robbery he will be granted pardon and the other goes in prison for a long time.

-if both confess then both go in prison for a short time

-if both deny then both go in prison for a very short time (let’s say they will be charged only with arms possession)

What action will each prisoner take?

A bank is robbed and the robbers escape the police…

A pair of suspects are arrested later in the day and are held in separate cells.

Each is told that:

-if one alone confesses the robbery he will be granted pardon and the other goes in prison for a long time.

-if both confess then both go in prison for a short time

-if both deny then both go in prison for a very short time (let’s say they will be charged only with arms possession)

What action will each prisoner take?



II

I

Deny Confess

Deny

2

2

3

0

Confess

0

3

1

1

Elimination of dominated strategiesElimination of dominated strategies



The individually rational outcome is worse for both players…

Arms races, environmental pollution, etc are modeled by prisoners’ dilemmas…

The solution to a single game is a max-min strategy, a better-play-it-safe strategy

In a repeated game patterns for cooperation among the players arise (“suspects” will actually play the “deny-deny” move…)

A repeated game allows for a strategy to be dependent on past moves, thus allowing for reputation effects and retribution.

In infinitely repeated games, trigger strategies such as tit-for-tat encourage cooperation.

The individually rational outcome is worse for both players…

Arms races, environmental pollution, etc are modeled by prisoners’ dilemmas…

The solution to a single game is a max-min strategy, a better-play-it-safe strategy

In a repeated game patterns for cooperation among the players arise (“suspects” will actually play the “deny-deny” move…)

A repeated game allows for a strategy to be dependent on past moves, thus allowing for reputation effects and retribution.

In infinitely repeated games, trigger strategies such as tit-for-tat encourage cooperation.

Quality SelectionQuality Selection


CustomerVoIP

Provider

Buy Don’t buy

High

2

2

1

0

Low

0

3

1

1

The Nash EquilibriumThe Nash Equilibrium


A set of strategies such that no player has incentive to unilaterally change her action.

Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less

than if she remained with her current strategy.

A set of strategies such that no player has incentive to unilaterally change her action.

Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less

than if she remained with her current strategy.

Revisiting Quality SelectionRevisiting Quality Selection


CustomerVoIP

Provider

Buy Don’t buy

High

2

2

1

0

Low

0

3

1

11

No dominant Strategies ! – Two Nash EquilibriaNo dominant Strategies ! – Two Nash Equilibria

Equilibrium selectionEquilibrium selection


In games with multiple Nash equilibria a theory of strategic interaction should guide players to the “most reasonable” equilibrium

Large number of papers focused with equilibrium refinementsthat attempt to derive conditions making an equilibrium more convincing than the another.

In games with multiple Nash equilibria a theory of strategic interaction should guide players to the “most reasonable” equilibrium

Large number of papers focused with equilibrium refinementsthat attempt to derive conditions making an equilibrium more convincing than the another.

Customer

VoIP

Provider

Buy Don’t buy

High

2

2

1

0

Low

0

1

1

1

For example, in the previous game the “High-buy” equilibrium yields higher utility for both players, so it can easily be argued that they will be self-coordinated to it.

For example, in the previous game the “High-buy” equilibrium yields higher utility for both players, so it can easily be argued that they will be self-coordinated to it.

Prisoners’ Dilemma Nash EquilibriumPrisoners’ Dilemma Nash Equilibrium


II

I

Deny Confess

Deny

2

2

3

0

Confess

0

3

1

1

Single Strategy combination rising from the process of dominated strategies elimination is a Nash EquilibriumSingle Strategy combination rising from the process of dominated strategies elimination is a Nash Equilibrium

Mixed StrategiesMixed Strategies


So far a player chose deterministically among one of her strategies. Such selections are named selections from pure strategies.

Randomizing one’s choice, by selecting among her pure strategies with a certain probability is called a mixed strategy.

Equilibrium is now defined by a mixed strategy for each player so that none can gain on average by unilaterally deviating.

Nash proved (1951) that under mixed strategies any game in strategic-form has an equilibrium

Such is therefore the game theorists recommendation in the case when an equilibrium in pure strategies does not exist.

So far a player chose deterministically among one of her strategies. Such selections are named selections from pure strategies.

Randomizing one’s choice, by selecting among her pure strategies with a certain probability is called a mixed strategy.

Equilibrium is now defined by a mixed strategy for each player so that none can gain on average by unilaterally deviating.

Nash proved (1951) that under mixed strategies any game in strategic-form has an equilibrium

Such is therefore the game theorists recommendation in the case when an equilibrium in pure strategies does not exist.

Compliance InspectionsCompliance Inspections


II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6



II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6

Mixed strategy for I: chose Inspect with a probability.

(giving a sufficient change to getting caught should deter II from choosing Cheat)

Mixed strategy for I: chose Inspect with a probability.

(giving a sufficient change to getting caught should deter II from choosing Cheat)• Inspect with p = 0.01 then II receives a utility of:

• 0 under Comply • 0.99 x 10 + 0.01 x (-90) = 9 under Cheat(has incentive to Cheat –inspection not too often)

• Inspect with p = 0.2 then II receives a utility of:• 0 under Comply • 0.8 x 10 + 0.2 x (-90) = -10 under Cheat(incentive to always Comply –inspections too

often)

• Inspect with p = 0.01 then II receives a utility of:• 0 under Comply • 0.99 x 10 + 0.01 x (-90) = 9 under Cheat(has incentive to Cheat –inspection not too often)

• Inspect with p = 0.2 then II receives a utility of:• 0 under Comply • 0.8 x 10 + 0.2 x (-90) = -10 under Cheat(incentive to always Comply –inspections too

often)



II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6

So if player I randomizes poorly she leads player II to selecting a pure strategy.Player I must make player II indifferent to achieve equilibrium

So if player I randomizes poorly she leads player II to selecting a pure strategy.Player I must make player II indifferent to achieve equilibrium

• Inspect with p = 0.1 then II receives a utility of:• 0 under Comply • 0.9 x 10 + 0.1 x (-90) = 0 under Cheat

Player II is rational so he knows player I will mix strategies only if I is indifferent too.

How can II make I indifferent?• Cheat with p = 0.2 then I receives a utility of:

• 0.8 x 0 + 0.2 x (-10) = -2 under Don’t Inspect• 0.8 x (-1) + 0.2 x (-6) = -2 under Inspect

• Inspect with p = 0.1 then II receives a utility of:• 0 under Comply • 0.9 x 10 + 0.1 x (-90) = 0 under Cheat

Player II is rational so he knows player I will mix strategies only if I is indifferent too.

How can II make I indifferent?• Cheat with p = 0.2 then I receives a utility of:

• 0.8 x 0 + 0.2 x (-10) = -2 under Don’t Inspect• 0.8 x (-1) + 0.2 x (-6) = -2 under Inspect



II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6

Equilibrium yields an expected payoff of 0 for player II-2 for player I…

Equilibrium yields an expected payoff of 0 for player II-2 for player I…

Does randomizing make sense for II ???

“ok, why don’t I just always chose comply since I got nothing to win anyways?”

Let’s assume player II always chooses Comply under this rationale, then I needs not randomize, just purely chose Don’t Inspect so player II’s strategy becomes sub-optimal (irrational).

With no incentive to select one strategy over the other a player can mix strategies and only thus is the equilibrium reached

Does randomizing make sense for II ???

“ok, why don’t I just always chose comply since I got nothing to win anyways?”

Let’s assume player II always chooses Comply under this rationale, then I needs not randomize, just purely chose Don’t Inspect so player II’s strategy becomes sub-optimal (irrational).

With no incentive to select one strategy over the other a player can mix strategies and only thus is the equilibrium reached



II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6

We saw that the probabilities for mixing one’s strategies depend on opponent's payoffs…

We saw that the probabilities for mixing one’s strategies depend on opponent's payoffs…

It could be reasoned by I that :

“We just have the penalty for Cheating disgustingly high and II will be deterred to select Cheat”

We saw that II’s payoffs determine the probabilities that will make I be indifferent…

It could be reasoned by I that :

“We just have the penalty for Cheating disgustingly high and II will be deterred to select Cheat”

We saw that II’s payoffs determine the probabilities that will make I be indifferent…



II

I

Comply Cheat

Don’t Inspect

0

0

10

-10

Inspect

0

-1

-90

-6

One could see this game as an evolutionary game:It is the interaction between an organization who chooses Don’t Inspect and Inspect for certain fractions of a large number of people.

One could see this game as an evolutionary game:It is the interaction between an organization who chooses Don’t Inspect and Inspect for certain fractions of a large number of people.

Player II’s actions Comply and Cheat are each chosen by a certain fraction of people involved in these interactions.

If these fractions would deviate from the equilibrium probabilities, the group whose strategies do better would increase.

For example: if player I chooses Inspect too often (relative to the penalty for a cheater who is caught), the fraction of cheaters will decrease, which in turn makes Don’t Inspect a better strategy

In this dynamic process, the long-term averages of the fractions approximate the equilibrium

probabilities

Player II’s actions Comply and Cheat are each chosen by a certain fraction of people involved in these interactions.

If these fractions would deviate from the equilibrium probabilities, the group whose strategies do better would increase.

For example: if player I chooses Inspect too often (relative to the penalty for a cheater who is caught), the fraction of cheaters will decrease, which in turn makes Don’t Inspect a better strategy

In this dynamic process, the long-term averages of the fractions approximate the equilibrium

probabilities


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Mobile Computing Group An Introduction to Game Theory part I Vangelis Angelakis 14 / 10 / 2004.

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