Game Theory By: Ali Farahani Rad Benjamin Ghassemi

Game Theory

By

Ali Farahani Rad Benjamin Ghassemi

2

What is Game Theory Game theory is a branch of applied

mathematics that is often used in the context of economics It studies strategic interactions between agents

In economics an agent is an actor in a model that (generally) solves an optimization problem In this sense it is equivalent to the term player which is also used in economics but is more common in game theory

3

What is Game Theory In strategic games agents choose

strategies that will maximize their return given the strategies the other agents choose

The essential feature is that it provides a formal modeling approach to social situations in which decision makers interact with other agents Game theory extends the simpler optimization approach developed in neoclassical economics

4

What is Game Theory Neoclassical economics refers to a general

approach in economics focusing on the determination of prices outputs and income distributions in markets through supply and demand These are mediated through a hypothesized maximization of income-constrained utility by individuals and of cost-constrained profits of firms employing available information and factors of production

Antonietta Campus (1987) marginal economics The New Palgrave A Dictionary of Economics v 3 p 323

5

Applications of Game Theory

Mathematics Computer Science Biology Economics Political Science International

Relations

Psychology Law Military Strategy Management Sports Game Playing Philosophy

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Representation of games The games studied by game theory are well-defined mathematical objects A game consists of a set of players a set of moves (or strategies) available to those players and a specification of payoffs for each combination of strategies

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Extensive form Games here are often presented as trees Here each vertex (or node) represents a point of choice for a player The player is specified by a number listed by the vertex The lines out of the vertex represent a possible action for that player The payoffs are specified at the bottom of the tree

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Extensive form

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Normal form The normal (or strategic form) game is usually

represented by a matrix which shows the players strategies and payoffs More generally it can be represented by any function that associates a payoff for each player with every possible combination of actions

Player 2chooses Left

Player 2chooses Right

Player 1chooses Up

4 3 ndash1 ndash1

Player 1chooses Down

0 0 3 4

Normal form or payoff matrix of a 2-player 2-strategy game

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Normal form When a game is presented in normal form

it is presumed that each player acts simultaneously or at least without knowing the actions of the other If players have some information about the choices of other players the game is usually presented in extensive form

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Types of games Cooperative or non-cooperative

Symmetric and asymmetric

Zero sum and non-zero sum

Simultaneous and sequential

Perfect information and imperfect information

Infinitely long games

Discrete and continuous games

Meta games

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Cooperative or non-cooperative A game is cooperative if the players are able to form binding

commitments For instance the legal system requires them to adhere to their promises In non-cooperative games this is not possible

Often it is assumed that communication among players is allowed in cooperative games but not in non-cooperative ones This classification on two binary criteria has been rejected (Harsanyi 1974)

Of the two types of games non-cooperative games are able to model situations to the finest details producing accurate results Cooperative games focus on the game at large Considerable efforts have been made to link the two approaches The so-called Nash-program has already established many of the cooperative solutions as non-cooperative equilibrium

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Symmetric and asymmetric A symmetric game is a game where the

payoffs for playing a particular strategy depend only on the other strategies employed not on who is playing them If the identities of the players can be changed without changing the payoff to the strategies then a game is symmetric

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Zero sum and non-zero sum Zero sum games are a special case of

constant sum games in which choices by players can neither increase nor decrease the available resources In zero-sum games the total benefit to all players in the game for every combination of strategies always adds to zero (more informally a player benefits only at the expense of others)

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Simultaneous and sequential Simultaneous games are games where both players move

simultaneously or if they do not move simultaneously the later players are unaware of the earlier players actions (making them effectively simultaneous) Sequential games (or dynamic games) are games where later players have some knowledge about earlier actions This need not be perfect information about every action of earlier players it might be very little knowledge For instance a player may know that an earlier player did not perform one particular action while he does not know which of the other available actions the first player actually performed

The difference between simultaneous and sequential games is captured in the different representations discussed above Normal form is used to represent simultaneous games and extensive form is used to represent sequential ones

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Perfect information and imperfect information An important subset of sequential games consists

of games of perfect information A game is one of perfect information if all players know the moves previously made by all other players Thus only sequential games can be games of perfect information since in simultaneous games not every player knows the actions of the others

Perfect information is often confused with complete information which is a similar concept Complete information requires that every player know the strategies and payoffs of the other players but not necessarily the actions

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Infinitely long games Games as studied by economists and real-world

game players are generally finished in a finite number of moves Pure mathematicians are not so constrained and set theorists in particular study games that last for infinitely many moves with the winner (or other payoff) not known until after all those moves are completed

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Discrete and continuous games Most of the objects treated in most branches of

game theory are discrete with a finite number of players moves events outcomes etc However the concepts can be extended into the realm of real numbers This branch has sometimes been called differential games because they map to a real line usually time although the behaviors may be mathematically discontinuous Much of this is discussed under such subjects as optimization theory and extends into many fields of engineering and physics

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Meta games These are games the play of which is the

development of the rules for another game the target or subject game Meta games seek to maximize the utility value of the rule set developed The theory of meta games is related to mechanism design theory

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Key Elements of a Game

Players Who is interacting

Strategies What are their options

Payoffs What are their incentives

Information What do they know

Rationality How do they think

21

Cigarette Advertising on TV

All US tobacco companies advertised heavily on

television

Surgeon General issues official warning

bullCigarette smoking may be hazardous

Cigarette companiesrsquo reaction

bullFear of potential liability lawsuits

Companies strike agreement

bullCarry the warning label and cease TV advertising in exchange for

immunity from federal lawsuits

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Strategic Interactions Players Reynolds and Philip Morris Strategies Advertise Do Not Advertise Payoffs Companiesrsquo Profits

Each firm earns $50 million from its customers Advertising costs a firm $20 million Advertising captures $30 million from competitor How to represent this game

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Payoff Table

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Best responses

Best response for Reynolds bullIf Philip Morris advertises advertise bullIf Philip Morris does not advertise advertise

Advertise is dominant strategy This is another Prisonersrsquo Dilemma

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What Happened

After the 1970 agreement cigarette advertising decreased by $63 million

Profits rose by $91 million

Whyhow were the firms able to escape from the Prisonerrsquos Dilemma

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Changing the Game through Government-Enforced Collusion

The agreement with the government forced the firms not to advertise

The preferred outcome (No Ad No Ad) then was all that remained feasible

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Rationality

Most economic analysis assumes ldquorationalityrdquo of decision-makers that you make decisions by

1forming a belief about the world

2choosing an action that maximizes your welfare given that belief

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And Common Knowledge of Rationality Most game-theoretic analysis makes the

further assumption that playersrsquo rationality is common knowledge

bullEach player is rationalbullEach player knows that each player is rationalbullEach player knows that each player knows that each

player is rationalbullEach player knows that each player knows that each

player knows thateach player is rationalbullEach player knows that each player knows that each

player knows that each player knows that each player is rational

bullEtc etc etc

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And Correct Beliefs

Nash equilibrium assumes that each player has correct beliefs about what strategies others will follow

Implicitly this is saying that in novel strategic situations each player knows what the other believes

Requires all players to thoroughly understand each other

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Nash Equilibrium Nash Equilibrium

bullA set of strategies one for each player such that each playerrsquos strategy is a best response to othersrsquo strategies

Best Response

bullThe strategy that maximizes my payoff given othersrsquo strategies

Everybody is playing a best response

bullNo incentive to unilaterally change my strategy

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Dominant StrategiesRecall Cigarette Ad Game

Reynoldsrsquo best strategy is Ad regardless of what Philip Morris does 1048774Ad is ldquodominant strategyrdquo

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Dominant Strategies and Rationality

If you are rational you should play your dominant strategy Period

No need to think about whether others are rational etc

Rationality + dominant strategies implies Nash equilibrium bullno need for common knowledge or correct beliefs

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Dominant Strategies and Rationality

Nash equilibrium is not the right concept for some strategic situations Real players make mistakes or for other reasons may

fail to be ldquorationalrdquo

Yet dominant strategies give a clear prescription of what to do regardless

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Example SUV Price WarsldquoGeneral Motors Corp and Ford Motor Co

slapped larger incentives on popular sport-utility vehicles escalating a discounting war in the light-truck categoryhellip Ford added a $500 rebate on SUVs boosting cash discounts to $2500 The Dearborn Mich auto maker followed GM which earlier in the week began offering $2500 rebates on many of its SUVsrdquo

--Wall Street Journal January 31 2003

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SUV Price Wars The Game

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SUV Price Wars Outcome

Each firm has a unilateral incentive to discount but neither achieves a pricing advantage

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Prisonersrsquo DilemmaSUV Price War is a ldquoprisonersrsquo dilemmardquo game

1 Both firms prefer to Discount regardless of what the other does (Discount is a dominant strategy)

2 But both firms are worse off when they both Discount than if they both Donrsquot

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Prisonersrsquo Dilemma Game

Key features

bullBoth players have a dominant strategy to Confess

bullBUT both players better off if they both donrsquot

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Prisonersrsquo Dilemma Game

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Reaction Curves in Prisonersrsquo Dilemma

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Evolution in Prisonersrsquo Dilemma (One Population)

Row and Col players are drawn from the same population

Those who Confess get higher payoff so Confess dominates the population

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Loyal Servant Game

Key featuresbullOne player (Master) has dominant strategybullOther player (Servant) wants to do the same

thing as Master

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Loyal Servant Game

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Reaction Curves in Loyal Servant Game

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Evolution in Loyal Servant Game (Two Populations)