Game Theory

Eduardo Costa

Contents

• What is game theory?• Representation of games• Types of games• Applications of game theory• Interesting Examples

What is game theory?

• Game theory is the study of strategies for decision making.

• Formally: “the study of mathematical models of conflict and cooperation between intelligent rational decision-makers” (Roger B. Myerson, 1991)

Representation of games

• Extensive Form.• Game is represented

by a series of moves on a tree.

• Each node is a choice by a player. Lines out of a node is a possible action. Payoffs are specified at the bottom.

Representation of games (2)

• Normal-form game• Game represented by a matrix, with

every possible combination of actions.

• For this representation, players act simultaneously or without knowing the other player actions.

Representation of games (3)

• Characteristic function form.• Used for cooperative games.• A characteristic function is seen as

(N,v)– With N being the number of agents;– With v: 2N R

• It defines the payoff for a certain coalition of players.

Type of games

• There are many type of games, depending on what real-life situation they are trying to mimmic.

• Games can be cooperative or non-cooperative.

• In asymmetric games, the payoff is not the same for all players.

Type of games(2)

• In zero-sum games, the payoff in every combination of strategies always adds to 0.

• Poker, Go or Chess are real examples of zero-sum games.

• In non-zero-sum games, a gain by one player does not imply a loss by another.

Types of games(3)

• In a perfect information game, all players know the moves previously made by others. Only applies to sequential games.

• A game can also have incomplete information. Complete information requires a player to know all strategies and payoffs available to the other players.

Applications of game theory

• Game theory was initially developed in the studies of economics, to understand behaviors of firms, markets and consumers.

• Now used in many other fields as political science, sociological and psychological behaviors.

Applications of game theory – description and modelling• Game theory can be used to describe

and model how human populations behave.

• But humans do not always play in a way to maximize their wins.

• Example: guess 2/3 of average game– Each person tries to guess what 2/3 of

the average of their guesses will be (between 0 and 100).

– Everyone guess 0 is the perfect strategy.

Applications of game theory – Prescriptive analysis• Game theory may not be used as a

prediction of human behavior, but it can be used a suggestion of what actions should people take.

• Playing a strategy that is part of Nash Equilibrium – best response to actions of other players – should be the best strategy.

Applications of game theory – Prescriptive analysis (2)• But other players may not take the

best possible actions, and in these cases the player should also play non-equilibrium strategy.

• Example: Prisioner’s dilemma

Applications of game theory – economics and business• Game theory is an important method

in mathematical economics and business.

• Useful to model behavior of competing or cooperative agents.

• A wide list of economic situation can be model by game theory:– Auctions, bargaining, oligopolies, voting

systems, industrial organization and political economy.

Applications of game theory – economics and business (2)• Example: oligopoly – market

dominated by small number of sellers.

• Bertrand competition:– Firm A and B sell the same

product;– Customers always buy the

cheaper product;• Nash equilibrium is when

price of product for both firms equals the cost of producing it (no profit).

Applications of game theory – economics and business (3)• Second example: auction.• For 2 bidders:

– v – value of the item– other bidder uses strategy β(v2) = av2– if you bid b:

(v − b)prob(you win) = (v − b)prob(b > av2)= (v − b)prob(v2 < b/a)= (v − b) b/a

• Maximizing: b = v / 2

Applications of game theory – biology• In biology application, the focus is less

on equilibrium strategies but on ones maintained by evolutionary forces.

• Usually Evolutionary Game Theory (EGT) is used instead of classic game theory.

Applications of game theory – biology (2)• Example: game of chicken or hawk-

dove game.• Used to analyze fighting behavior and

territoriality.• First presented by John Maynard Smith

and George Price in a 1973 Nature paper.

Applications of game theory – philosophy• Game theory has several uses in

philosophy, specially in questions of morality and self-interest.

• In the prisoner’s dilemma game, the rational decision would be to betray the other. In real life, there is a bias towards stay silent.

Game Theory – Examples(Volunteer’s dilemma)• N players have to decide to make a

small sacrifice for all to benefit, or wait for another one to do it.

• Scenario example: electricity goes out in a neighborhood. At least one person has to contact the electricity company for the problem to get fixed.

Game Theory – Examples(Pirate game)• 5 pirates – A, B, C, D, E. • 100 gold coins• The first pirate – by alphabetical order

– proposes a distribution.• All pirates vote.• If distribution is not accepted, the

proposer is thrown into the sea and another round happens – with 1 less pirate.

Game Theory – Examples(Pirate game) (2)

• What is your choice as pirate A? Give many coins to the others so you at least survive?

• Actually, pirate A may take 98 coins in his first proposal and it will be the best for pirate C and E.

Game Theory – Examples(Pirate game) (3)

• Imagine when there is only D and E left. D will give himself 100 coins (tie in voting is acceptance).

• If C, D and E are left:– E knows that in next round he will get

nothing, so he will accept only 1 coin.– C may offer 99 coins for himself.

Game Theory – Examples(Pirate game) (4)• If B, C, D and E are left:

– D knows he will get nothing in the next round, so he may accept only 1 coin.

– B may offer 99 coins for himself.• So in the first round, if the pirates

think of all this:– C and E know that they will get nothing in

the next round, so they may accept only 1 coin.

– A may offer himself 98 coins.

Game Theory

Thank You For Your Attention

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