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Game Theory Game theory was developed by John Von

Neumann and Oscar Morgenstern in 1944 - Economists!

One of the fundamental principles of game theory, the idea of equilibrium strategies was developed by John F. Nash, Jr. (A Beautiful Mind), a Bluefield, WV native.

Game theory is a way of looking at a whole range of human behaviors as a game.

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Components of a Game

Games have the following characteristics: Players Rules Payoffs

Based on Information Outcomes Strategies

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Types of Games

We classify games into several types. By the number of players: By the Rules: By the Payoff Structure: By the Amount of Information

Available to the players

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Games as Defined by the Number of Players:

1-person (or game against nature, game of chance)

2-person n-person( 3-person & up)

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Games as Defined by the Rules:

These determine the number of options/alternatives in the play of the game.

The payoff matrix has a structure (independent of value) that is a function of the rules of the game.

Thus many games have a 2x2 structure due to 2 alternatives for each player.

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Games as Defined by the Payoff Structure: Zero-sum Non-zero sum (and occasionally Constant sum)

Examples: Zero-sum

Classic games: Chess, checkers, tennis, poker. Political Games: Elections, War , Duels ?

Non-zero sum Classic games: Football (?), D&D, Video games Political Games: Policy Process

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Games defined by information In games of perfect information, each

player moves sequentially, and knows all previous moves by the opponent. Chess & checkers are perfect information

games Poker is not In a game of complete information, the rules

are known from the beginning, along with all possible payoffs, but not necessarily chance moves

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Strategies We also classify the strategies that

we employ: It is natural to suppose that one

player will attempt to anticipate what the other player will do. Hence Minimax - to minimize the maximum

loss - a defensive strategy Maximin - to maximize the minimum

gain - an offensive strategy.

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Iterated Play

Games can also have sequential play which lends to more complex strategies. Tit-for-tat - always respond in kind. Tat-for-tit - always respond

conflictually to cooperation and cooperatively towards conflict.

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Game or Nash Equilibria Games also often have solutions or

equilibrium points. These are outcomes which, owing to

the selection of particular reasonable strategies will result in a determined outcome.

An equilibrium is that point where it is not to either players advantage to unilaterally change his or her mind.

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Saddle points

The Nash equilibrium is also called a saddle point because of the two curves used to construct it:

an upward arching Maximin gain curve and a downward arc for minimum loss. Draw in 3-d, this has the general shape of

a western saddle (or the shape of the universe; and if you prefer). .

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Some Simple Examples

Battle of the Bismark Sea Prisoner’s Dilemma Chicken

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The Battle of the Bismarck Sea

Simple 2x2 Game US WWII Battle

Japanese Options

Sail North

Sail South

US Options

Recon North

2 Days 2 Days

Recon South

1 Day 3 Days

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The Battle of the Bismarck Sea

Japanese Options

Sail North

Sail South

Minima of Rows

US Options

Recon North

2 Days 2 Days

2

Recon South

1 Day 3 Days

1

Maxima of Columns

2 3

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The Battle of the Bismarck Sea - examined This is an excellent example of a two-

person zero-sum game with a Nash equilibrium point.

Each side has reason to employ a particular strategy Maximin for US Minimax for Japanese).

If both employ these strategies, then the outcome will be Sail North/Watch North.

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Decision Tree

Japanese

Sail North

Sail South

SearchNorth

2

SearchSouth

1

SearchNorth

2

SearchSouth

3

Decision Tree Version of Battle of Bismark Sea

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The Prisoners Dilemma

The Prisoner’s dilemma is also 2-person game but not a zero-sum game.

It also has an equilibrium point, and that is what makes it interesting.

The Prisoner's dilemma is best interpreted via a “story.”

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A Simple Prisoner’s Dilemma

Prisoner A

~ Confess

Confess

Prisoner B

~ Confess

-1-1

0-10

Confess -100

-5-5

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Alternate Prisoner’s Dilemma Language

Prisoner A

Cooperate

Defect

Prisoner B

Cooperate

-1-1

0-10

Defect -100

-5-5

Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.

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What Characterizes a Prisoner’s Dilemma

Prisoner A

Cooperate

Defect

Prisoner B

Cooperate

RewardReward

TemptSucker

Defect SuckerTempt

PunishPunish

Uses Cooperate instead of Confess to denote player cooperation with each other instead of with prosecutor.

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What makes a Game a Prisoner’s Dilemma? We can characterize the set of choices in a

PD as: Temptation (desire to double-cross other player) Reward (cooperate with other player) Punishment (play it safe) Sucker (the player who is double-crossed)

A game is a Prisoner’s Dilemma whenever: T > R > P > S Or Temptation > Reward > Punishment > Sucker

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What is the Outcome of a PD? The saddle point is where both

Confess This is the result of using a Minimax

strategy. Two aspects of the game can make

a difference. The game assumes no communication The strategies can be altered if there is

sufficient trust between the players.

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Solutions to PD?

The Reward option is the joint optimal payoff.

Can Prisoner’s reach this? Minimax strategies make this

impossible Are there other strategies?

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Iterated Play

The PD is a single decision game in which the Nash equilibrium results from a dominant strategy.

In iterated play (a series of PDs), conditional strategies can be selected

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Chicken

The game that we call chicken is widely played in everyday life bicycles Cars

James Dean – variant Mad Max

Interpersonal relations And more…

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The Game of Chicken

Driver A

~ Swerve

Swerve

Driver B

~ Swerve

11

24

Swerve 42

33

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Chicken is an Unstable game

There is no saddle point in the game.

No matter what the players choose, at least one player can unilaterally change for some advantage.

Chicken is therefore unstable. We cannot predict the outcome

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Chicken is Nuclear Deterrence