GAME THEORY NOTES FOR ECONOMICS HONOURS FOR ALL UNIVERSITIES BY SOURAV SIRS CLASSES 9836793076

SOURAV SIR’S CLASSES (98367 93076)

GAME THEORY

What is Game?

A game is a formal description of a strategic situation.

Game theory

Game theory is the formal study of decision-making where several players must make choices that potentially affect the interests of the other players.

Some basic and important definitions

Mixed strategy

A mixed strategy is an active randomization, with given probabilities, that determines the player’s decision. As a special case, a

mixed strategy can be the deterministic choice of one of the given pure strategies.

Nash equilibrium

A Nash equilibrium, also called strategic equilibrium, is a list of strategies, one for each player, which has the property that no

player can unilaterally change his strategy and get a better payoff.

Payoff

A payoff is a number, also called utility, that reflects the desirability of an outcome to a player, for whatever reason. When the

outcome is random, payoffs are usually weighted with their probabilities. The expected payoff incorporates the player’s attitude

towards risk.

Perfect information

A game has perfect information when at any point in time only one player makes a move, and knows all the actions that have

been made until then.

Player

A player is an agent who makes decisions in a game.

Rationality

A player is said to be rational if he seeks to play in a manner which maximizes his own payoff. It is often assumed that the

rationality of all players is common knowledge.

Strategic form

A game in strategic form, also called normal form, is a compact representation of a game in which players simultaneously choose

their strategies. The resulting payoffs are presented in a table with a cell for each strategy combination.

Strategy

In a game in strategic form, a strategy is one of the given possible actions of a player. In an extensive game, a strategy is a

complete plan of choices, one for each decision point of the player.

Zero-sum game

A game is said to be zero-sum if for any outcome, the sum of the payoffs to all players is zero. In a two-player zero-sum game,

one player’s gain is the other player’s loss, so their interests are diametrically opposed

co-oprative and non co-operative games

Cooperative game theory

investigates such coalitional games with respect to the relative amounts of power held by various players, or how a successful

coalition should divide its proceeds. This is most naturally applied to situations arising in political science or international

relations, where concepts like power are most important. For example, Nash proposed a solution for the division of gains from

agreement in a bargaining problem which depends solely on the relative strengths of the two parties’ bargaining position.

The amount of power a side has is determined by the usually inefficient outcome that results when negotiations break down.

Nash’s model fits within the cooperative frame- work in that it does not delineate a specific timeline of offers and counteroffers,

but rather focuses solely on the outcome of the bargaining process. In contrast, no cooperative game theory

is concerned with the analysis of strategic choices. The paradigm of no cooperative game theory is that the details of the ordering


and timing of players’ choices are crucial to determining the outcome of a game. In contrast to Nash’s cooperative model, a

noncooperative model of bargaining would posit a specific process in which it is prespecified who gets to make an offer at a

given time. The term “noncooperative” means this branch of game theory explicitly models the process of players making

choices out of their own interest. Cooperation can, and often does, arise in noncooperative models of games, when players find it

in their own best interests.

Representation of game

The matrix below is a normal-form representation of a game in which players move simultaneously (or at least do not observe the

other player's move before making their own) and receive the payoffs as specified for the combinations of actions played. For

example, if player 1 plays top and player 2 plays left, player 1 receives 4 and player 2 receives 3. In each cell, the first number

represents the payoff to the row player (in this case player 1), and the second number represents the payoff to the column player

(in this case player 2).

Pure vs mixed strategy

A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation they could face. A player's strategy set is the set of pure strategies available to that player.

A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure

strategy. Since probabilities are continuous, there are infinitely many mixed strategies available to a player, even if their strategy set is finite.

Of course, one can regard a pure strategy as a degenerate case of a mixed strategy, in which that particular pure strategy is

selected with probability 1 and every other strategy with probability 0.

A totally mixed strategy is a mixed strategy in which the player assigns a strictly positive probability to every pure strategy.

(Totally mixed strategies are important for equilibrium refinement such as trembling hand perfect equilibrium.)

Two person zero sum game

A zero-sum game is a mathematical representation of a situation in which a participant's gain (or loss) of utility is exactly

balanced by the losses (or gains) of the utility of the other participant(s). If the total gains of the participants are added up, and the total losses are subtracted, they will sum to zero.

An example of Zero sum game

For example, consider the children's game of "Matching Pennies." In this game, the two players agree that one will be "even" and

the other will be "odd." Each one then shows a penny. The pennies are shown simultaneously, and each player may show either a

head or a tail. If both show the same side, then "even" wins the penny from "odd;" or if they show different sides, "odd" wins the penny from "even". Here is the payoff table for the game.

Odd

Head Tail

Even Head 1,-1 -1,1

Tail -1,1 1,-1

If we add up the payoffs in each cell, we find 1-1=0. This is a "zero-sum game."

http://en.wikipedia.org/wiki/Probability

http://en.wikipedia.org/wiki/Equilibrium_refinement

http://en.wikipedia.org/wiki/Trembling_hand_perfect_equilibrium

http://en.wikipedia.org/wiki/Mathematical_model

http://en.wikipedia.org/wiki/Utility


Another Example

Here is another example of a zero-sum game. It is a very simplified model of price competition. Like Augustin Cournot (writing

in the 1840's) we will think of two companies that sell mineral water. Each company has a fixed cost of $5000 per period,

regardless whether they sell anything or not. We will call the companies Perrier and Apollinaris, just to take two names at

random.

The two companies are competing for the same market and each firm must choose a high price ($2 per bottle) or a low price ($1 per bottle). Here are the rules of the game:

1) At a price of $2, 5000 bottles can be sold for a total revenue of $10000.

2) At a price of $1, 10000 bottles can be sold for a total revenue of $10000.

3) If both companies charge the same price, they split the sales evenly between them.

4) If one company charges a higher price, the company with the lower price sells the whole amount and the company with the

higher price sells nothing.

5) Payoffs are profits -- revenue minus the $5000 fixed cost.

Here is the payoff table for these two companies

Perrier

Price=$1 Price=$2

Apollinaris Price=$1 0,0 5000, -5000

Price=$2 -5000, 5000 0,0

(Verify for yourself that this is a zero-sum game.) For two-person zero-sum games, there is a clear concept of a solution. The

solution to the game is the maximin criterion -- that is, each player chooses the strategy that maximizes her minimum payoff. In

this game, Appolinaris' minimum payoff at a price of $1 is zero, and at a price of $2 it is -5000, so the $1 price maximizes the

minimum payoff. The same reasoning applies to Perrier, so both will choose the $1 price. Here is the reasoning behind the

maximin solution: Apollinaris knows that whatever she loses, Perrier gains; so whatever strategy she chooses, Perrier will choose the strategy that gives the minimum payoff for that row. Again, Perrier reasons conversely.

Linear Programming method of Game Theory

Two companies are competing for the same product. To improve its market share, company A decides to launch the following strategies.

A1 = give discount coupons

A2 = home delivery services

A3 = free gifts

The company B decides to use media advertising to promote its product.

B1 = internet

B2 = newspaper

B3 = magazine


Company B

Company A

B1 B2 B3

A1 3 -4 2

A2 1 -7 -3

A3 -2 4 7

Use linear programming to determine the best strategies for both the companies.

Solution.

Company B Minimum

Company A

B1 B2 B3

A1 3 -4 2 -4

A2 1 -7 -3 -7

A3 -2 4 7 -2

Maximum 3 4 7

Minimax = -2

Maximin = 3

This game has no saddle point. So the value of the game lies between –2 and +3. It is possible that the value of game may be

negative or zero. Thus, a constant k is added to all the elements of pay-off matrix. Let k = 3, then the given pay-off matrix becomes:

Company B

Company A

B1 B2 B3

A1 6 -1 5

A2 4 -4 0

A3 1 7 10

Let

V = value of the game

p1, p2 & p3 = probabilities of selecting strategies A1, A2 & A3 respectively.

q1, q2 & q3 = probabilities of selecting strategies B1, B2 & B3 respectively.


Company B Probability

Company

A

B1 B2 B3

A1 6 -1 5 p1

A2 4 -4 0 p2

A3 1 7 10 p3

Probability q1 q2 q3

Company A's objective is to maximize the expected gains, which can be achieved by maximizing V, i.e., it might gain more than V if company B adopts a poor strategy. Hence, the expected gain for company A will be as follows:

6p1 + 4p2 + p3 ≥ V

-p1 - 4p2 + 7p3 ≥ V

5p1 + 0p2 + 10p3 ≥ V

p1 + p2 + p3 = 1 and p1, p2, p3 ≥ 0

Dividing the above constraints by V, we get

6p1/V + 4p2/V + p3/V ≥ 1

-p1/V - 4p2/V + 7p3/V ≥ 1

5p1/V + 0p2/V + 10p3/V ≥ 1 p1/V + p2/V + p3/V = 1/V

To simplify the problem, we put p1/V = x1, p2/V = x2, p3/V = x3

In order to maximize V, company A can

Minimize 1/V = x1+ x2+ x3

subject to

6x1 + 4x2 + x3 ≥ 1

-x1 - 4x2 + 7x3 ≥ 1 5x1 + 0x2 + 10x3 ≥ 1

and x1, x2, x3 ≥ 0

Company B's objective is to minimize its expected losses, which can be reduced by minimizing V, i.e., company A adopts a poor strategy. Hence, the expected loss for company B will be as follows:

6q1 - q2 + 5q3 ≤ V

4q1 - 4q2 + 0q3 ≤ V

q1 + 7q2 + 10q3 ≤ V q1 + q2 + q3 = 1

and q1, q2, q3 ≥ 0

Dividing the above constraints by V, we get

6q1/V - q2/V + 5q3/V ≤ 1

4q1/V - 4q2/V + 0q3/V ≤ 1

q1/V + 7q2/V + 10q3/V ≤ 1 q1/V + q2/V + q3/V = 1/V


To simplify the problem, we put q1/V = y1, q2/V = y2, q3/V = y3

In order to minimize V, company B can

Maximize 1/V = y1+ y2+ y3

subject to

6y1 - y2 + 5y3 ≤ 1

4y1 - 4y2 + 0y3 ≤ 1 y1 + 7y2 + 10y3 ≤ 1

and y1, y2, y3 ≥ 0

Company B's problem is the dual of company A's problem.

To solve this problem, we introduce slack variables to convert inequalities to equalities. The problem becomes

Maximize y1 + y2 + y3 + 0y4 + 0y5 + 0y6

6y1 - y2 + 5y3 + y4 = 1

4y1 - 4y2 + y5 = 1

y1 + 7y2 + 10y3 + y6 = 1

Initial Basic Feasible Solution

y1 = 0, y2 = 0, y3 = 0, z = 0

y4 = 1, y5 = 1, y6 = 1

Table 1

cj 1 1 1 0 0 0

cB Basic Variables

B y1 y2 y3 y4 y5 y6

Solution values

b (=XB)

0 y4 6 -1 5 1 0 0 1

0 y5 4 -4 0 0 1 0 1

0 y6 1 7 10 0 0 1 1

zj-cj -1 -1 -1 0 0 0

Key column = y1 column

Minimum positive value = 1/6

Key row = y4 row

Pivot element = 6 y4 departs and y1 enters.


Table 2

cj 1 1 1 0 0 0

cB Basic Variable

B y1 y2 y3 y4 y5 y6

Solution values

b (=XB)

1 y1 1 -1/6 5/6 1/6 0 0 1/6

0 y5 0 -10/3 -10/3 -2/3 1 0 1/3

0 y6 0 43/6 55/6 -1/6 0 1 5/6

zj-cj 0 -7/6 -1/6 1/6 0 0

Final Table: Linear Programming Method

cj 1 1 1 0 0 0

cB Basic Variable

B y1 y2 y3 y4 y5 y6

Solution values

b (=XB)

1 y1 1 0 45/43 7/43 0 1/43 8/43

0 y5 0 0 40/43 -32/43 1 20/43 31/43

1 y2 0 1 55/43 -1/43 0 6/43 5/43

zj-cj

0 0 57/43 6/43 0 7/43

The values for y1, y2 & y3 are 8/43, 5/43 & 0 respectively.

I/V = y1 + y2 + y3 = 8/43 + 5/43 + 0 = 13/43 or V = 43/13

Company B's optimal strategy

q1 = V X y1 = 43/13 X 8/43 = 8/13

q2 = V X y2 = 43/13 X 5/43 = 5/13

q3 = V X y3 = 43/13 X 0 = 0

Hence, company B's optimal strategy is (8/13, 5/13, 0).

Company A's optimal strategy

The values for x1, x2 & x3 can be obtained from the final simplex table.

x1 = 6/43, x2 = 0 & x3 = 7/43

p1 = V X x1 = 43/13 X 6/43 = 6/13

p2 = V X x2 = 43/13 X 0 = 0 p3 = V X x3 = 43/13 X 7/43 = 7/13

Hence, company A's optimal strategy is (6/13, 0, 7/13).

Prisoners dilemma


There are two prisoners whose aim is to minimize the years of imprisonment. They have commit-ted a crime jointly. Each

prisoner is interviewed separately and there are not any contacts whatsoever between them. They decide individually to confess

or deny the crime taking into account possible decisions of the other prisoner (strategic game). Each prisoner chooses his

dominant strategy, that is the behaviour giving the best result regardless of the decision of the other prisoner.

PLAYER B

CONFESS DENY

Prisoner A CONFESS 3,3 1,4

DENY 4,1 2,2

The first number shows the years imprisonment of A, the second number of B. If for example A

confesses and B denies, A gets 1 year imprisonment and B 4 years (field at the top right

Basic terms

•

Players:

decision makers ("prisoner A or B")

•

Strategy:

behaviour of the players ("confess"/"deny")

•

Pay-off:

outcome (x years inmprisonment)

•

Dominant strategy:

the best outcome for a player regardless of the decision of the other player Which are the dominant strategies in this game? from

the point of view of prisoner A

•

if B confesses, I should also confess (3 years are less than 4 years)

•

if B denies, I should again confess (1 year is less than 2 years)

Ë

strategy of A:

I confess irrespective of the decision of B. "Confess" ist the dominant strategy

of A (3

years imprisonment) from the point of view of prisoner B

•

if A confesses, I should also confess (3 years are less than 4 years)

•

if B denies, I should again confess (1 year is less than 2 years)

Ë

strategy of B:

"Confess" is his dominant strategy

, too (3 years imprisonment)

If both prisoners could cooperate successfully, they would get a better outcome for both (2 years imprisonment). But they cannot

cooperate, thus, the dominant strategy is the best result which can be achieved when deciding individually. That is the dilemma

of the prisoners: By cooperation they could get a better result than by deciding individually

ANSWER THE FOLLOWING

1. WILL THE COMPANIES ADVERTISE OR NOT?


CIGARETTE

COMPANY B

CIGARETTE

COMPANY A

ADVERTISE NO ADVERTISE

\ ADVERTISE 30,30 50,20

NO ADVERTISE 20,50 40,40

2. Producing, yes or no?

COMPANY B

Producing NOT Producing

COMPANY A Producing -60,-85 60,0

NOT Producing 0,70 50,0

ANSWERS 1. for both cigarette companies "advertising" is the dominant strategy

(profit of 30 for each). The market shares do not change; the cost of advertising, hower, lowers the profits. By coordination they

could get a better outcome ("not advertising", profit of 40 for each). 2.. There is no dominant strategy for either firm, that's why

there is no prisoners' dilemma with possible rational decisions. from the point of view of A:

•

if B produces, I do not produce (0 > -60)

•

if B does not produce, then I produce (+60 > +50) from the point of view of B:

•

if A produces, I do not produce (0 > -85)

•

if A does not produce, then I produce (+70 > 0)

DOMINANCE PRINCIPLE

One move dominates another if all its payoffs are at least as advantageous to the player than the corresponding ones in the other

move. In terms of the payoff matrix, we can say it this way:

A. Row r in the payoff matrix dominates row s if each payoff in row r is ≥ the corresponding payoff in row s. B. Column r in the payoff matrix dominates column s if each payoff in row r is ≤ the corresponding payoff in column s.

Note that if two rows or columns are equal, then each dominates the other. A row or column strictly

dominates another if the one dominates the other and they are not equal.

Following the first principle of game theory, the move corresponding to a strictly dominated row or column

will never be played, and both players are aware of this by the second principle. Thus each player following

the principles of game theory will repeatedly eliminate dominated rows and columns as the case may be. (If

two rows or columns are equal, then there is no reason to choose one over the other, so either may be eliminated.) This process is called reduction by dominance.


AN EXAMPLE OF DOMINANCE

Example

Consider the above game once again.

Column Strategy

A B C

Row

Strategy

1

0 -1 1

2 0 0 2

3 -1 -2 3

Since the entries in Row 2 are ≥ the corresponding ones in Row 1, Row 2 dominates Row 1.

Since the entries in Column B are ≤ the corresponding ones in Column A, Column B dominates Column A.

Reducing the Above Game by Dominance

Since Row 2 dominates Row 1 we eliminate Row 1 to obtain

A B C

2

0 0 2

3 -1 -2 3

Since Column B now dominates both Columns A and C we eliminate both Columns A and C to obtain

B

2

0

3 -2

Since the top row now dominates the bottom row, we eliminate the bottom row, and we are reduced to the following 1×1 matrix

B

2 ( 0 )

In this case, we have solved the game by reduction by dominance: The row player should always play 2 and the column player

should always play B. Since the corresponding payoff is 0, we say that the game is fair (neither player has an advantage over the other).

Note that we were lucky here: Not all games can be reduced by dominance to a 1×1 game.

Saddle Point, Strictly Determined Game


A saddle point is a payoff that is simultaneously a row minimum and a column maximum. To locate saddle points, circle the row

minima and box the column maxima. The saddle points are those entries that are both circled and boxed.

A game is strictly determined if it has at least one saddle point. The following statements are true about strictly determined games.

A. All saddle points in a game have the same payoff value.

B. Choosing the row and column through any saddle point gives minimax strategies for both players. In other words, the game is solved via the use of these (pure) strategies.

The value of a strictly determined game is the value of the saddle point entry. A fair game has value of zero, otherwise it is unfair or biased.

AN EXAMPLE OF SADDLE POINT

Example

In the above game, there are two saddle points, shown in color.

A B C

1

0 -1 1

2 0 0 2

3 -1 -2 3

Since the saddle point entries are zero, this is a fair game.

Nash Equilibrium and Dominant Strategies

Nash Equilibrium is a term used in game theory to describe an equilibrium where each player's strategy is optimal given the

strategies of all other players. A Nash Equilibrium exists when there is no unilateral profitable deviation from any of the players

involved. In other words, no player in the game would take a different action as long as every other player remains the same.

Nash Equilibria are self-enforcing; when players are at a Nash Equilibrium they have no desire to move because they will be worse off.

Necessary Conditions

The following game doesn't have payoffs defined:

L R

T a,b c,d

B e,f g,h


In order for (T,L) to be an equilibrium in dominant strategies (which is also a Nash Equilibrium), the following must be true:

a > e

c > g

b > d

f > h

In order for (T,L) to be a Nash Equilibrium, only the following must be true:

a > or = e

b > or = d

SOME SPECIFIC EXAMPLES

Prisoners' Dilemma (Again)

If every player in a game plays his dominant pure strategy (assuming every player has a dominant pure strategy), then the

outcome will be a Nash equilibrium. The Prisoners' Dilemma is an excellent example of this. It was reviewed in the introduction, but is worth reviewing again. Here's the game (remember that in the Prisoners' Dilemma, the numbers represent years in prison):

Jack

C NC

Tom

C -10,-10 0,-20

NC -20,0 -5,-5

In this game, both players know that 10 years is better than 20 and 0 years is better than 5; therefore, C is their dominant strategy

and they will both choose C (cheat). Since both players chose C, (10,10) is the outcome and also the Nash Equilibrium. To check

whether this is a Nash Equilibrium, check whether either player would like to deviate from this position. Jack wouldn't want to deviate, because if he chose NC and Tom stayed at C, Jack would increase his prison time by 10 years.

Iterated Deletion of Dominated Strategies

Here's another game that doesn't have dominant pure strategies, but that we can solve by iterated deletion of dominated strategies.

In other words, we can eliminate strategies that are dominated until we come to a conclusion:

2

Left Middle Right

1

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Let's find the dominant strategies. The first strategy that is dominated, is Right. Player 2 will always be better off by playing

Middle, so Right is dominated by Middle. At this point the column under Right can be eliminated since Right is no longer an option. This will be shown by crossing out the column:

http://economics.fundamentalfinance.com/game-theory/introduction-game-theory.php


2

Left Middle Right

1

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Remember that both players understand that player 2 has no reason to play Right--player 1 understands that player 2 is trying to

find an optimum, so he also no longer considers the payoffs in the Right column. With the Right column gone, Up now

dominates Down for player 1. Whether player 2 plays Left or Middle, player 1 will get a payoff of 1 as long as he chooses Up. So now we no longer consider Down:

2

Left Middle Right

1

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

Now we know that player 1 will choose Up, and player 2 will choose Left or Middle. Since Middle is better than Left (a payoff of

2 vs. 0), player 2 will choose Middle and we have solved the game for the Nash Equilibrium:

2

Left Middle Right

1

Up 1,0 1,2 0,1

Down 0,3 0,1 2,0

To ensure that this answer (Up, Middle) is a Nash Equilibrium, check to see whether either player would like to deviate. As long

as player 1 has chosen Up, player 2 will choose Middle. On the other hand, as long as player 2 has chosen Middle, player 1 will choose up.

Multiple Nash Equilibria

Here's a game that demonstrates multiple Nash Equilibria: Two drivers are traveling towards each other on a road. Should they

drive on the left or the right side? They don't want to wreck...


Driver 2

Left Right

Driver 1

Left 1,1 -1,-1

Right -1,-1 1,1

Both (Left,Left) and (Right,Right) are Nash Equilibria. As long as they're on opposite sides of the road, the drivers are happy and

don't want to deviate. Games like this are often solved by social convention--beforehand all the players agree on a strategy so that

everyone is better off. Of course, everyone knows that the right side is the best side to drive on, so the game should look more like this:

Driver 2

Left Right

Driver 1

Left 1,1 -1,-1

Right -1,-1 2,2

In this case, the game itself gives the players a clue as to where the other player will be, even though there are two Nash

Equilibria.

Here's a game with three Nash Equilibria and no dominated strategies:

2

a b c

1

A 1,1 2,0 3,0

B 0,2 3,3 0,0

C 0,3 0,0 10,10

The Nash Equilibria are (A,a), (B,b), and (C,c).


An example of normal-form game

Player 1 \ Player 2 Player 2

chooses left

Player 2

chooses right

Player 1 chooses top 4, 3 −1, −1

Player 1 chooses bottom 0, 0 3, 4

Examples and exercises GAME THEORY

Procedure

Check each action pair to see if it has the property that each player's action maximizes her payoff given the other players' actions.

Example: coordination between players with different preferences

Two firms are merging into two divisions of a large firm, and have to choose the computer system to use. In the past the firms

have used different systems, I and A; each prefers the system it has used in the past. They will both be better off if they use the

same system then if they continue to use different systems.

1…We can model this situation by the following two-player strategic game.

Player 2

I A

Player 1

I 2,1 0,0

0,0 1,2

A

To find the Nash equilibrium, we examine each action profile in turn.

(I,I)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a

Nash equilibrium.

(I,A)

By choosing A rather than I, player 1 obtains a payoff of 1 rather than 0, given player 2's action. Thus this action profile

is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing I rather than A.]

(A,I)

By choosing I rather than A, player 1 obtains a payoff of 2 rather than 0, given player 2's action. Thus this action profile

is not a Nash equilibrium. [Also, player 2 can increase her payoff by choosing A rather than I.]

(A,A)

Neither player can increase her payoff by choosing an action different from her current one. Thus this action profile is a

Nash equilibrium.

We conclude that the game has two Nash equilibria, (I,I) and (A,A).

Example: players with opposing preferences

2…An established firm and a newcomer to the market of fixed size have to choose the appearance for a product. Each firm can

choose between two different appearances for the product; call them X and Y. The established producer prefers the newcomer's


product to look different from its own (so that its customers will not be tempted to buy the newcomer's product) while the

newcomer prefers that the products look alike.

We can model this situation by the following two-player strategic game.

Player 2

X Y

Player 1

X 2,1 1,2

1,2 2,1

Y

To find the Nash equilibrium, we examine each action profile in turn.

(X,X)

Firm 2 can increase its payoff from 1 to 2 by choosing the action Y rather than the action X. Thus this action profile is

not a Nash equilibrium.

(X,Y)

Firm 1 can increase its payoff from 1 to 2 by choosing the action Y rather than the action X. Thus this action profile is


(Y,X)

Firm 1 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. Thus this action profile is


(Y,Y)

Firm 2 can increase its payoff from 1 to 2 by choosing the action X rather than the action Y. Thus this action profile is


We conclude that the game has no Nash equilibrium!

Exercise

3…Find the Nash equilibrium of the following strategic game.

Player 2

L R

Player 1

T 2,2 0,0

0,0 1,1

B

Ans. (T,L)

Neither player can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.

(T,R)

Player 1 can increase her payoff from 0 to 1 by choosing the action B rather than the action T. Thus this action profile is


(B,L)


Firm 1 can increase its payoff from 0 to 2 by choosing the action T rather than the action B. Thus this action profile is


(B,R)

Neither firm can increase its payoff by choosing a different action, so this action profile is a Nash equilibrium.

We conclude that the game has two Nash equilibria, (T,L) and (B,R).

.

4…

Find all Nash equilibria in pure strategies in the following non-zero-sum games.

a

Column

Left Right

Row Up 2 , 4 1 , 0

Down 6 , 5 4 , 2

b

c

Column

Left Middle Right

Row

Up 0 , 1 9 , 0 2 , 3

Straight 5 , 9 7 , 3 1 , 7

Down 7 , 5 10 , 10 3 , 5

5…Problem

"If a player has a dominant strategy in a simultaneous-move game, then she is sure to get her best outcome." True or false?

Explain and give an example of a game that illustrates your answer.

Solutions

4

Column

Left Right

Row Up 1 , 1 0 , 1

Down 1 , 0 1 , 1


(a) Down is dominant for Row and Left is dominant for Column. Equilibrium: (Down, Left) with payoffs of (6, 5).

(b) Down and Right are weakly dominant for Row and Column, respectively, leading to a Nash equilibrium at (Down, Right). Brute force also shows another Nash equilibrium at (Up, Left).

(c) Down is dominant for Row; Column will then play Middle. Equilibrium is (Down, Middle).

5 False. A dominant strategy yields you the highest payoff available to you against each of your opponent's strategies. Playing a

dominant strategy does not guarantee that you end up with the highest of all possible payoffs. For example, in a prisoner's dilemma game, both players have dominant strategies but neither gets the highest possible payoff in the equilibrium of the game.

6…Dominant Strategy Rules (Dominance Principle)

If all the elements of a column (say ith column) are greater than or equal to the corresponding elements of any other

column (say jth column), then the ith column is dominated by the jth column and can be deleted from the matrix.

If all the elements of a row (say ith row) are less than or equal to the corresponding elements of any other row (say jth row), then the ith row is dominated by the jth row and can be deleted from the matrix.

Dominance Example: Game Theory

Player B

Player A

I II III IV

I 3 5 4 2

II 5 6 2 4

III 2 1 4 0

IV 3 3 5 2

Use the principle of dominance to solve this problem.

Solution.

Player B

Player A

I II III IV Minimum

I 3 5 4 2 2

II 5 6 2 4 2

III 2 1 4 0 0

IV 3 3 5 2 2

Maximum 5 6 5 4


There is no saddle point in this game.

7…Using dominance property

If a column is greater than another column (compare corresponding elements), then delete that column.

Here, I and II column are greater than the IV column. So, player B has no incentive in using his I and II course of action.

Player B

Player A

III IV

I 4 2

II 2 4

III 4 0

IV 5 2

If a row is smaller than another row (compare corresponding elements), then delete that row.

Here, I and III row are smaller than IV row. So, player A has no incentive in using his I and III course of action.

Player B

Player A

III IV

II 2 4

IV 5 2

Now you can use any one of the following to determine the value of game .

8…FIND NASH EQUILIBRIUM

Rock Paper Scissors

Rock 0, 0 -1, 1 1, -1

Paper 1, -1 0, 0 -1, 1

Scissors -1, 1 1, -1 0, 0


ANSWER.

There is no Nash Equilibrium of this game. Because for any pair of strategies, at least one player would wish to change

strategy in response to the other person’s strategy. For instance, if player 1 is playing Rock and player 2 is playing Paper

(the top middle cell), then player 1 wishes to switch to Scissors.

(Also, there is actually a Nash Equilibrium in “mixed strategies,” which means the players randomize over their strategies.

Specifically, each player plays each strategy with 1/3 probability. But you don’t have to know this.)

A)IS THERE ANY dominant strategy?

B) find the Nash equilibria.

C) Is there any first mover advantage?

Answer

9…(a) Here is the strategy-and-payoff matrix for “Chicken.” Spike picks the row and Biff picks the column. (“K” means 1000.)

Continue Stop

Continue -10K, -10K 1K, -1K

Stop -1K, 1K 0, 0

(b) There is no dominant strategy, because Continue is the best response to Stop, but Stop is the best response to Continue.

There is no dominant strategy equilibrium (DSE), because both players must have dominant strategies to have a DSE.

(c) There are two Nash Equilibria: (Continue, Stop) and (Stop, Continue). Why? Given that the other guy is going to

Stop, you want to Continue. Given that the other guy is going to Continue, you want to Stop.

(d) There can be a first-mover advantage if one player can commit in advance to playing Continue. For example, Spike

could throw his brake pedal out the window so that he’s incapable of stopping.

10…Consider the following game:

PLAYER 2

A B C D E F

A 2,1 2,1 2,1 4,0 4,0 4,0

B 2,1 2,1 2,1 4,0 4,0 4,0

C 2,0 3,2 1,2 2,0 3,2 1,2

D 2,0 3,2 0,3 2,0 3,2 0,3

(a) Does player 1 have a dominant strategy?

(b) Does player 2 have a dominant strategy?

(c) Is there a dominant-strategy equilibrium?

(d) Does player 1 have any dominated strategies?

(e) Does player 2 have any dominated strategies?

ANSWER


a) Player 1 does not have a dominant strategy.

(b) For Player 2 cis a weakly dominant strategy.

(c) There is no dominant strategy equilibrium.

(d) For Player 1 D is weakly dominated by C (and A and B are equivalent to each other).

(e) For Player 2 a is weakly dominated by b or c , b is weakly dominated by c , d is strictly dominated by b or c and weakly

dominated by a or e or f, e is weakly

dominated by c or b or f, f is weakly dominated by c .Thus the dominated strategies are: a, b, d,e and f(obviously, since c is a

dominant strategy!).

11…Find all the Nash equilibria of the following game.

P Q R

A 0,0 2,0 1,-1

B -1,0 1,2 2,1

C 0,1 2,1 3,2

D 0,2 2,3 3,3

ANSWER

THERE ARE MULTIPLE NASH EQUILIBRIA MARKED AS BOLD

12…The questions in this problem refer to the following game.

PLAYER 2

L M R

PLAYER 1 U 1,2 3,5 2,1

M 0,4 2,1 3,0

D -1,1 4,3 0,2

A. Determine if either player has any dominated strategies. If so, identify them.

b. Does either player have a dominant strategy? Why or why not?

c. Use iterated elimination of dominated strategies to solve this game. Be clear about the order in which you are eliminating

strategies. Also specify whether you are eliminating strictly or weakly dominated strategies

ANSWER

A)R for player 2 is dominated by M. For each player 1 strategy, M gives player 2 a higher

payoff than does R.

B)No. For either player to have a dominant strategy, 2 of her 3 strategies would need to be

dominated.

c)1. Eliminate R as above. (Strict) 2. In the 3 x 2 game, U strictly dominates M. 3. In the

2 x 2 game, M strictly dominates L. 4. In the 2 x 1 game, D strictly dominates U.

IEDS Solution = (D,M).


13. Is there a Mixed Strategy Nash Equilibrium?

a b

A [(-12, 1) (8, 8)]

B [(15, 1), (8,-1)]

(15, 1) and (8,8) are Nash Equlibria. However, could you still mix between (8,8) and (15,1)? For example, for P2 (column player)

to make P1 indifferent he could play b with a probability of 1. And player 1 could make player 2 indifferent with another probability mix.

14..Question 1

A dominant strategy is one which:

a. is best for a player no matter what strategy the other player chooses.

b. is optimal given the other player's strategy, but may not be optimal should the other player switch strategies.

c. will be chosen by the second player in a sequential game.

d. will be chosen by the first player in a sequential game.

Question 2

Select the correct statement:

a. Every dominant strategy equilibrium is a Nash equilibrium and every Nash equilibrium is a dominant strategy equilibrium

b. Every dominant strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium is a dominant strategy equilibrium

c. Not every dominant strategy equilibrium is a Nash equilibrium, every Nash equilibrium is a dominant strategy equilibrium

d. Some dominant strategy equilibriums are a Nash equilibrium, but not every Nash equilibrium is a dominant strategy

equilibrium

Company AAA

Left Right

Company BBB Up -1, -1 -10, 0

Down 0, -10 -8, -8

Question 3

In the above table, which of the following is true?

a. Company BBB has no dominant strategy.

b. Company BBB dominant strategy is Up.

c. Company BBB dominant strategy is Down.

Question 4

In the above table, which of the following is true?

a. Company AAA has no dominated strategies.

b. Company AAA strategy of Left is dominated.

c. Company AAA strategy of Right is dominated.

Question 5

Company AAA

price =1 price =2 price=3

Company BBB price =1 0, 0 50, -10 40, -20

price =2 -10, 50 20, 20 90, 10

price =3 -20, 40 10, 90 50, 50

In the table above, what is the Nash Equilibrium?

http://mathoverflow.net/questions/89346/game-theory-is-there-a-mixed-strategy-nash-equilibrium


14…Answers:

Question 1:

a. is best for a player no matter what strategy the other player chooses.

Question 2:

b. Every dominant strategy equilibrium is a Nash equilibrium, but not every Nash equilibrium is a dominant strategy equilibrium

Question 3:

c. Company BBB dominant strategy is Down.

Question 4:

b. Company AAA strategy of Left is dominated.

Question 5:

Both competitors choose a price of $1

GAME THEORY NOTES FOR ECONOMICS HONOURS FOR ALL UNIVERSITIES BY SOURAV SIRS CLASSES 9836793076

Education

Transcript of GAME THEORY NOTES FOR ECONOMICS HONOURS FOR ALL UNIVERSITIES BY SOURAV SIRS CLASSES 9836793076