3.1.4 types of games. strategic behavior in business and econ outline 3.1. what is a game ? 3.1.1....

3.1.4 Types of Games

Strategic Behavior in Business and Econ

Outline

3.1. What is a Game ?3.1.1. The elements of a Game3.1.2 The Rules of the Game: Example3.1.3. Examples of Game Situations3.1.4 Types of Games: Complete and Incomplete Information

3.2. Solution Concepts3.2.1. Static Games of complete information: Dominant

Strategies and Nash Equilibrium in pure and mixed strategies3.2.2. Dynamic Games of complete information: Backward

Induction and Subgame perfection

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Games can be classified into different categories according to different criteria:

Number of players Number of strategies Simultaneous moves or sequential moves Zero-sum games or not Informational structure Nature of the game: conflict, coordination, . . . etc

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From a practical point of view, a meaningful classification is based on:

Number of players Number of strategies Simultaneous moves or sequential moves Zero-sum games or not Informational structure Nature of the game: conflict, coordination, . . . Etc

They determine the analysis tools (techniques) to use

Strategic Behavior in Business and Econ

In this sense we can find, basically, four different types of games

Complet e Inf ormat ion Incomplet e Inf ormat ion

S t at ic

Dynamic

Player s have all r elevant

inf ormat ion, and move

simulat enously

Not all player s have all t he

r elevant inf ormat ion, and

move simulat enously

Player s have all r elevant

inf ormat ion, and move

sequent ially

Not all player s have all t he

r elevant inf ormat ion, and

move sequent ially

All games in a given category are represented and solved alike

Strategic Behavior in Business and Econ

Examples:

Complet e Inf ormat ion Incomplet e Inf ormat ion

S t at ic

Dynamic

Pr isoner s' Dilemma,

S ummer pr icing

FCC auct ions

(sealed- bid auct ion)

Chess,

Market E nt r y

Collect ibles auct ions

(E nglish auct ions)

Strategic Behavior in Business and Econ

Static Games of Complete Information

They are, in general, the most simple games. Their main featuresare:

All the players choose their strategies simultaneously. This does not mean “at the same time” but “without knowing the choice of others”

Because of this simultaneity they can be represented by means of a table (payoff matrix)

They are “one-shot games”, that is, they are played only once All the players have all the information regarding who are the other

players, what are the own strategies and the strategies of the others, what are the own payoffs and the payoffs of the others, and what are the rules of the game

Strategic Behavior in Business and Econ

Static Games of Incomplete Information

They are like the previous ones, only that the informational structureis different

Not all the players have all the information regarding who are the other players, what are the strategies of the others, or what are the payoffs of the others.

The typical case is when the players don't know the payoffs of the other players

Example: Sealed-bid Auctions: You don't know what is the value of the item to other players

Strategic Behavior in Business and Econ

Dynamic Games of Complete Information

Their main features are:

Players choose their strategies sequentially, one after the other This means some players take actions knowing what other have done

Because of this sequentiality they must represented using trees They are also “one-shot games”, that is, they are played only once Nevertheless, some special dynamic games consist of the repetition

of a one-shot game played several times All the players have all the information regarding who are the other

players, what are the own strategies and the strategies of the others, what are the own payoffs and the payoffs of the others, and what are the rules of the game

Strategic Behavior in Business and Econ

Dynamic Games of Incomplete Information

Again, they are like the previous ones, only that the informational structure is different

Not all the players have all the information regarding who are the other players, what are the strategies of the others, or what are the payoffs of the others.

The typical case is when the players don't know the payoffs of the other players

Example: English Auctions: You don't know what is the value of the item to other players

3.2 Solution Concepts

Strategic Behavior in Business and Econ

Outline

3.1. What is a Game ?3.1.1. The elements of a Game3.1.2 The Rules of the Game: Example3.1.3. Examples of Game Situations3.1.4 Types of Games

3.2. Solution Concepts3.2.1. Static Games of complete information: Dominant

Strategies and Nash Equilibrium in pure and mixed strategies3.2.2. Dynamic Games of complete information: Backward

Induction and Subgame perfection

Strategic Behavior in Business and Econ

Solution Concepts

A solution to a game consists of a prediction of whatwill be the logical outcome of the game (what will be thestrategies chosen by the players and the correspondingpayoffs) based on the rationality and common knowledgeassumptions of Game Theory

Depending on the type of the game, the representationwill be different and hence the analysis and the method tofind the solution (the solution concept) won't be the same.

Strategic Behavior in Business and Econ

Solution Concepts

Complet e Inf ormat ion Incomplet e Inf ormat ion

S t at ic Bayesian Nash E quilibr ium

Dynamic

E quilibr ium in Dominant S t rat egies,

E liminat ion of Dominat ed S t rat s.

Nash E quilibr ium

Nash E quilibr ium,

Backward Induct ion,

S ubgame Per f ect E quilibr ium

Bayesian Per f ect Nash E quilibr ium,

S equent ial E qulibr ium

A solution of a game is called an Equilibrium of the game

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Equilibrium

The basic idea behind any equilibrium concept is that itcorresponds to an outcome of the game (a choice of strategies by all the players) that is stable, in the sense that

Given what the other players are doing, nobody has any reason to change his or her own strategy

Therefore, since an equilibrium describes a choice ofstrategies for the players that nobody wants to change, itseems logical to think that it is a “good” prediction.

(Basic) Game Theory, though, does not explain how this equilibrium is reached

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$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Suppose that Bar 1 chooses $5 and Bar 2 chooses $4

Strategic Behavior in Business and Econ

$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Suppose that Bar 1 chooses $5 and Bar 2 chooses $4Then, the payoffs are: 15 to Bar 1

28 to Bar 2

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$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Suppose that Bar 1 chooses $5 and Bar 2 chooses $4Then, the payoffs are: 15 to Bar 1

28 to Bar 2But if Bar 2 is charging $4 then Bar 1 wants to charge $4(to earn a payoff to 20 instead of 15)

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$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Hence, Bar 1 choosing $5 and Bar 2 choosing $4 is NOTstable, is NOT an equilibrium

Strategic Behavior in Business and Econ

$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Consider Bar 1 charging $4 and Bar 2 charging $4 instead

Strategic Behavior in Business and Econ

$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Consider Bar 1 charging $4 and Bar 2 charging $4Now the payoffs is 20 to each Bar

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$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Consider Bar 1 charging $4 and Bar 2 charging $4Now the payoff is 20 to each Bar Do they want to change the strategy ?

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$2 $4 $5

Bar 1$2 10 , 10 14 , 12 14 , 15$4 12 , 14 20, 20 28 , 15$5 15 , 14 15 , 28 25, 25

Bar 2

Example: Summer Pricing

Consider Bar 1 charging $4 and Bar 2 charging $4Now the payoff is 20 to each Bar Do they want to change the strategy ?

NO !Bar 1 charging $4 and Bar 2 charging $4 is stable, is an Equilibrium of the game

Strategic Behavior in Business and Econ

Good news !

The equilibrium tells us what seems to be the logical (stable)outcome of the game

Bad news !

Game Theory does not tell us how the equilibrium is reached,how players “discover” that they should be playing the equilibriumstrategies. This is specially important when there is more thanone equilibrium (Ex: Antena 3 vs. Telecinco)

(Learning ?, Trial-and-error ?, Evolution ?, Hint ?)

Strategic Behavior in Business and Econ

Example: The Driving Game

Paul and Mary are driving in opposite directions on a twolane road.What should they do when they cross ?

Drive on the right lane ?Drive on the left lane ?

Strategic Behavior in Business and Econ

Mary

Right Left

PaulRight 0 , 0 -50 , -50

Left -50 , -50 0 , 0

Example: The Driving Game

If both drive on “their” right, then there is no accident If they drive on different lanes, then there is an accident

that has a cost of -$50,000 to each of them

Strategic Behavior in Business and Econ

Mary

Right Left

PaulRight 0 , 0 -50 , -50

Left -50 , -50 0 , 0

Example: The Driving Game

There are 2 equilibrium:Both players driving on the rightBoth players driving on the left

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Mary

Right Left

PaulRight 0 , 0 -50 , -50

Left -50 , -50 0 , 0

Example: The Driving Game

There are countries stabilized on the Drive on the right equilibriumThere are countries stabilized on the Drive on the left equilibriumThe question is: How did they reach one or the other ?

Strategic Behavior in Business and Econ

Best Reply

All the solution methods that we are going to study arebased on the prior concept of Best Reply

For each player, a best reply to the strategy(ies) chosen bythe other player(s) is, simply, the strategy that gives the highest payoff

In all the previous examples, and all the examples to come, the red circles indicate the best replies in each case

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Documentary

Match

Antena 3(player 1)

Telecinco (player 2)

1, 1

DocumentaryMatch

2, 1.5

0.75, 0.751.5, 2

Example: The Battle for TV share

Strategic Behavior in Business and Econ

For Antena 3,

If Telecinco chooses Match, my Best Reply is Documentary• If Telecinco chooses Documentary, my Best Reply is Match

For Telecinco,

If Antena 3 chooses Match, my Best Reply is Documentary• If Antena 3 chooses Documentary, my Best Reply is Match

Strategic Behavior in Business and Econ

Employees can work hard or shirk

Salary: $100K unless caught shirking

Cost of effort: $50K

Managers can monitor or not

Value of employee output: $200K

Profit if employee doesn’t work: $0

Cost of monitoring: $10K( Monitoring will catch a shirking worker for sure)

Example: A (simple) Principal-Agent game

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ManagerMonitor No Monitor

EmployeeWork 50 , 90 50 , 100

Shirk 0 , -10 150 , -100

Example: A (simple) Principal-Agent game

What are the Best Replies ? (The Red Circles)

Strategic Behavior in Business and Econ

ManagerMonitor No Monitor

EmployeeWork 50 , 90 50 , 100

Shirk 0 , -10 150 , -100

Example: A (simple) Principal-Agent game

Notice that there is no equilibrium !!