2/26/2015

1

Game Theory

Introduction to non-cooperative

games

Unit 3.

Plan

1. Non-cooperative game Theory Game concepts: Rules of the Game Game concepts: Common Knowledge 2 players games

2. Strategic games Representation Solution concept

Dominant Strategy Nash Equilibrium

Pure strategies Mixed Strategies

3. Extensive Games Subgame perfect Nash Equilibrium

Learning Objectives

2/26/2015

2

1. Non-Cooperative Game Theory

A game (dictionary definition) is a A competitive activity in which players contend with

each other according to a set of rules

Strategic Behavior:

Conscious behavior arising among competitors or players, in a situation where all are aware of their

conflicting interests and interdependence of their

payoffs.

Non-Cooperative Game theory provides the tools and techniques to analyze strategic

situations in which decision-makers interact.

A Game is a stylized model that depicts situations of strategic behavior.

Every game is played by a set of rules which have to specify four elements

Game Concepts: Rules of the Game

Who is playing: a set of players

What they are playing with- a set of alternative actions or choices, the strategies, that each player has

available.

strategies (rules telling each player which action to choose at each point of the game.

When each player gets to play

How much they stand to gain (or lose) from the choices made in the game.

Payoffs (usually consist of the profit or expected profit the players receive after the game have been played out)

2/26/2015

3

Game Concepts: Common Knowledge

We assume that the rules of the game are Common Knowledge

who, what, when, and how much constitutes the rules of the game.

That is, If you ask any two players in the game about the rules of the game they will give you the

same answer.

Every player knows the rules, and knows that other players knows that he knows the rules,,

ad infinitum.

2 players game

Player 1

(i) knows the rules of the game, and knows that player 2 knows the rules of the game, also

(ii) Knows that player 2 knows that player 1 knows (i), also

(iii) knows that player 2 knows (ii)

Ad infinitum

This is important because if at some point player 1 knows that he will never choose some actions (because it is not the

best available), he must know that player 2 knows that too.

Other information structures

Information structure of the game refer to what players know and when they know it during the

game.

Perfect information: each player knows all the history of the game before taking any action.

Games where players move simultaneously are games of imperfect information.

Incomplete information: some players have more information than others, at the beginning of the game.

Symmetric information: all players have exactly the same information when each player moves.

2/26/2015

4

Mathematical representation

Two principal mathematical representations of a game:

Strategic (strategic) form of a game which consider situations in which actions are chosen

one and for all.

Extensive form of a game which allows for the possibility that plan may be revised as they are

carried out. (dynamic games)

STRATEGIC GAMES

Examples

Representation of the game

Player 1 and player 2

Player 1 available actions 1, 1

Player 2 2, 2

Payoffs depends on own action and the other players action.

These games are games of imperfect information because both players move

simultaneously.

2/26/2015

5

For any game we need to know what actions will be chosen by the players? We also take into account that the best action of any

given player depends, in general, on the other players actions. The player must form a belief about the other players actions: when choosing an action a player must have in mind the

actions the other players will choose.

When we analyze a game we assume that Each player (rational decision maker) chooses the best

available action, and

Every players belief about other players actions is correct

Solution concept

A solution concept is a formal rule for predicting how a game will be played.

Predictions (called "solutions), describe which strategies will be adopted by players and, therefore, the result of the game. Iterated dominance

The most commonly used solution concepts are equilibrium concepts, Examples: Nash equilibrium.

In many cases different solution concepts result in more than one solution. We may apply a refinement to eliminate implausible

equilibria in richer games. Example: Backward induction,

Strategic Game: Representation

P2 2 2

P1

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

Players

2/26/2015

6

P2 2 2

P1

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

Players

Actions

P2 2 2

P1

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

1

2(1,2)

2(1,2)

1(1,2)

1(1,2)

Players

Actions

Payoffs

Alternatively

P2 2 2

P1

1 1

(1,2), 2

(1,2) 1

(1,2), 2

(1,2)

1 1

(1,2) , 2

(1,2)

1(1,2)

, 2(1,2)

2/26/2015

7

Zero Sum Games

Games in which one players gain is always matched by another players loss.

Typical example is the matching pennies game

Such games are called Strictly Competitive

Head Tail

Head 1, 1 1, 1

Tail 1, 1 1, 1

+ =

Sales of Ice Cream

Beginning Middle End

B 50,50 25,75 50, 50

M 75,25 50,50 75,25

E 50,50 25,75 50,50

Information Structures

Game of imperfect information

Both players move simultaneously to pick a location

Game of symmetric information

Both players have the same information when they play

Also, this is a zero-sum game

Because total sales (100%) will be divided between the two players any gain for a player is exactly

marched by a loss for the other player.

2/26/2015

8

We are looking for the solution(s) for this game

We need to introduce the concepts of

Dominant strategies

Nash Equilibrium

Mini-max strategy

Dominant Strategy

A dominant strategy is a strategy that outperform any other strategy no matter

what strategy the opponent selects.

Beginning Middle End

B 50,50 25,75 50, 50

M 75,25 50,50 75,25

E 50,50 25,75 50,50

Dominant Strategy

A dominant strategy is a strategy that outperform any other strategy no matter

what strategy the opponent selects.

Beginning Middle End

B 50,50 25,75 50, 50

M 75,25 50,50 75,25

E 50,50 25,75 50,50

2/26/2015

9

Dominant Strategy

A dominant strategy is a strategy that outperform any other strategy no matter

what strategy the opponent selects.

Beginning Middle End

B 50,50 25,75 50, 50

M 75,25 , 75,25

E 50,50 25,75 50,50

Equilibrium

Each player will play his dominant strategy and the solution is:

, the payoff attached with this equilibrium is 50% of the sales for each.

Iterated Dominance

In some games after eliminating a dominated strategy, one (or more) other strategies

become dominated in their turn.

A rational player will proceed the same way and eliminate them too.

The process of elimination of dominated strategies is called iterated dominance.

2/26/2015

10

Iterated Dominance

Iterated Dominance

2/26/2015

11

Nash Equilibrium

Definition: A Nash equilibrium is an action profile with the property that no player can do better by choosing an action different from

, given that every other player adheres to .

Recall the Ice cream trucks game: the solution obtained by eliminating strictly dominated action is also a Nash Equilibrium.

Consider the strategy profile , player 1 has no incentive to change and play Beginning

or End given that player 2 chooses Middle.

the same is true for player 2.

Which is not the case for any other action profile.

Consider whether the strategy profile , is a Nash Equilibrium

Player 1 has incentive to play instead Middle and increase his market share to 75%. (the same is true for

player 2).

Hence the strategy profile , is not a Nash equilibrium.

You might do the same with any other possible strategy profiles in this game. How many are they?

2/26/2015

12

Matching Pennies Game

Two players each must secretly turn a penny to heads or tails. The players reveal their

choices simultaneously. If the penny match

player1 keeps both pennies. If not player 2

keeps both. Head Tail

Head +1, 1 1, +1

Tail 1, +1 +1, 1

Matching Pennies Game

Consider {H,H}. Player 1 would like to change and play T instead of H, if player 2 adhere to H.

Because at least one player can do better by choosing T, {H,H} is not an equilibrium.

We can check that there is no equilibrium using the strategies T or H (with certainty).

We say there is no equilibrium in pure strategies for this game

But Nash Theorem says that there is at least one equilibrium for any finite game!

Mixed Strategies

A mixed strategy is an assignment of a probability to each pure strategy.

Since probabilities are continuous, there is an infinitely many mixed strategies available to each player.

A player rationally picks the probabilities that maximize his Expected payoff.

In the marching Pennies game let player 1 plays Head with probability p, and player 2 plays head with probability q, then the expected payoffs functions are the following:

2/26/2015

13

1 = +1 + 1 1 + 1

1 + 1 +1

After simplifications

1 = 2 1 + 1 + 2 1= 2 1 2 1

Provided an interior solution exists, expected profit is maximized if 1

= 2 2 1 = 0, =

1

2

1-p

p

1-q q

2 = 1 + 1 1 + 1

+1 + 1 1

After simplifications

2 = 2 1 2 1

Provided an interior solution exists, expected profit is maximized if 2

= 2 2 1 = 0, =

1

2

1-p

p

1-q q

Notice that the expected profit for player 1, if player 2 plays the

mixed strategy q = , is

1 = 2 1 21

2 1 = 0, for any value of p. Hence the

value of p is irrelevant: whether player 1 uses H or T all the time or

any of the two randomly (at any p different from ) his expected

payoff will not be different, as long as the opponent uses a mixed

strategy q=1/2.

The same is true for player 2, if player 1 plays the mixed strategy p =

, his expected payoff will be always 2 = 2 1

21

2 1 = 0, for any value of q.

This is a characteristic of optimal mixed strategies.

1-p

p

1-q q

2/26/2015

14

(intuitively) optimal mixed strategy makes your opponents selection of a strategy irrelevant to its

outcome.

We can use this intuition to solve for a mixed strategy without using calculus:

Player 1: the optimal strategy is the one (p) that result in equal payoff for player 2 whether heplays

always H or T

+ 1 = 1 1 2 = 2 1

2 = 4, = 1/2

1-p

p

1-q q

Similarly, we can prove that the optimal mixed strategy for player 2 is q=1/2.

It is clear that strategy , = (1

2,

1

2) is a

Nash Equilibrium since no player can improve

on the outcome by playing a different strategy.

Notice that a pure strategy is a special case of mixed strategies (as it is a degenerate mixed

strategy (attaching a probability 1 to one of

the available strategies).

Any equilibrium using a combination of strategies in some proportion is called a

mixed strategy Nash Equilibrium.

A mixed strategy Nash Equilibrium is unstable because payoffs are the same regardless of the

strategy adopted by the players, so they have

little incentives to maintain the equilibrium.

2/26/2015

15

Ice Cream Trucks Game again

Another concept of solution for Zero-Sum games is the mini-max strategy: each player

chooses the strategy that minimizes the

maximum possible outcome for other player.

Prisoners Dilemma Games

Two suspects in a major crime are held in separate cells. (players)

There is no enough evidence to convict either of them unless one of them acts as an informer against the other.

Players: two suspects

Actions: Each players set of actions is {Quiet, Fink}

Preferences : Suspect 1s ordering from best to

This a game of imperfect information because both suspects move simultaneously.

It is also a static game (because both suspects move simultaneously)

Prisoners Dilemma

Solution:

1. Check whether there is a dominant strategy

Fink is a dominant strategy for Suspect 1(no matter what strategy suspect 2 adopts) because 3>2 and 1>0. Likewise, Fink is a dominant strategy for suspect 2.

The equilibrium is {Fink, Fink} and the payoffs (1,1).

2/26/2015

16

We can check that the strategy profile {Quiet, Quiet} is not a Nash Equilibrium. (because at least one of the suspects may improve on the payoff of 2 by playing (another strategy, i.e. Fink)

Upon checking the payoff matrix it is clear that if both suspects cooperate (i.e. choose

{Quiet, Quiet}), they will be both better of

than with the no-cooperation outcome .

Prisoners Dilemma and Coordination

Upon checking the payoff matrix it is clear that if both suspects cooperate (i.e. choose

{Quiet, Quiet}), they will be both better of

than with the no-cooperation outcome .

Read the Press

This game has been used to model the problem of deployment of taxi cab over the

territory of Grand Tunis.

Le problme du service de taxis individuels dans le Grand-Tunis - Les

jaunes...partout et nulle part (Premire partie)

La Presse de Tunisie April 18, 2013. http://www.lapresse.tn/05022015/65903/le-probleme-du-service-de-taxis-individuels-dans-le-grand-

tunis.html

2/26/2015

17

EXTENSIVE GAMES

Many games are sequential, in which player 1 moves, then player 2 respond, then player 1 respond to player 2s responseThese games are dynamic (as opposed to static).

They are represented by game trees. The tree is defined by nodes and branches.

Nodes represent places where something happens in the game (such as a decision by one of the players), and

branches indicate the various actions that players can choose.

We represent nodes by solid circles and branches by arrows connecting the nodes.

A properly constructed tree is called an extensive-form representation

Example: Price competition

Perfect information

Find Nash Equilibria.

The rule look ahead and reason back.

2/26/2015

18

Player 2 optimal strategy

(L,L) because 2>1 and >0

Player 1 figure that out so if he plays H he earn 0, and if he plays L he earns million,

hence he plays L

The (unique) Nash Equilibrium is (L,(L,L))

Imperfect information

The two firms typically select their prices simultaneously and independently.

Neither firms observes the others move before making its choice.

CREDIBLE THREATSSUBGAME

PERFECT NE

Example: Entry Game

2/26/2015

19

2/26/2015

20

2/26/2015

21

2/26/2015

22

Summary

Game theory is the study of how interdependent decision makers make choices.

A game must include players, actions, information, strategies, payoffs, outcomes, and equilibria.

In simple zero-sum games, the mini-max strategy is a dominant solution to the game.

At least one Nash Equilibrium (in pure or random strategies) exists for all finite games.

Games comes with different information structures: Perfect Information (every player knows the history of

play before taking any action)

In games with complete information, nature does not move first, or natures move is observed by all

players.

In games of certain information, nature never moves after another player moves.

In games of symmetric information, all players have exactly the same information when each moves.

A dominant solution to the classical prisoners dilemma game results in a non-optimal

solution for the players.