Game theory and strategy - completo

*I

at least as well as s against every strategy of the other players, and against

*I

some it does strictly.

CHAPTER 4

In this chapter we’re going to study a second solution concept form strategic

form games: iterated elimination of dominated strategies (IEDS).

DEFINITION: a strategy s is dominated by another strategy s if the latter

*i ’I

does at least as well as s against every strategy of the other players, and

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against some it does strictly better.

If a strategy is not dominated by any other, it’s called an undominated

strategy.

7 PLAYER 1 \ PLAYER 2 NORTH SOUTH

HIGH p1 (H , N), p2 (H , N) p1 (H , S), p2 (H , S)

LOW p1 (L , N), p2 (L , N) p1 (L , S), p2 (L , S)

Consider the same example from chapter 3: in this strategic form “low” is

dominated by “high” if and only if (H , N) (L , N) and (H , S)

π ≥ π π

1 1 1

(L , S) with at least one of those inequalities being strict.

≥ π 1

Generally consider a game in which player i has many strategies: if there is a

dominant strategy, all the remaining strategies are dominated; if there is not a

dominant strategy there may not be any best strategy. Now using the IEDS let’s

find the solution of the following problem:

PLAYER 1 \ PLAYER 2 LEFT RIGHT

UP 1 , 1 0 , 1

MIDDLE 0 , 2 1 , 0

DOWN 0 , -1 0 , 0

Let’s start with player 1: we can easily see that “down” is dominated by “up”

and “middle” because she gets always a lower payoff, no matter what the

other player will choose. It’s important to remember that for a player is useless

to play a dominated strategy because there will be always a strategy that gets

a higher payoff, so we can “eliminate” “down”. Now it’s player 2 turn and he

has to choose between “left” and “right” related to the two other strategies

remained. “Left” will always offer a higher payoff than “right” for player 2 so, in

order to reach the final result, we have to consider a third stage in which player

1 have to choose between “up” and “middle”. For him, the best strategy is

always “up” rather than “middle”. The chosen strategy (UP , LEFT) is said to be

reached by iterated elimination of dominated strategies (IEDS), the

game itself is said to be dominance solvable.

The advantage of this concept is inherent in the simplicity of the dominance

concept: in fact if a player is convinced that a strategy always does worse than

some alternative strategy, then he will never use it. The disadvantages of this

concept are the layers of rationality. Let’s assume that the strategy (UP , LEFT)

costs a fee for player 1: he will not want to choose it anymore, so the game is

not anymore rational.

CHAPTER 5

In this chapter we will look to the most famous solution concept for strategic

form games: Nash equilibrium.

Suppose that you have a strategy B that is dominated by another strategy, say

A. We have seen that the player will always choose strategy A due to the

different and higher payoff than strategy B. Now suppose that the player has

some ideas about the other player’s intentions. In that case he would choose A

8

provided it does better than strategy B, given what the other player is going to

do. Hence, you do not know that strategy A is always better than strategy B

against all the enemy’s strategies, what is important to know is that strategy A

performs better than strategy B against a specific strategy. Indeed strategy A is

called best response against the known opponent’s strategy. Typically you will

not have the certain strategy of the enemy, you will have a guess about his

possible strategy choice. Of course, the guess might be wrong. If it is correct

and also the other player plays a best response, you will be in a Nash

equilibrium.

Two questions about Nash equilibrium arise:

1- Do we know that every game has a Nash equilibrium? Recalling the

dominant strategy solution or the IEDS, we can easily see that these

concepts yield no solution in many games.

2- Do we know that a given game will have exactly one Nash equilibrium?

We know that in many games there is more than one Nash equilibrium,

but the question is which one of them is the most reasonable?

Let us now show several examples:

EXAMPLE 1: BATTLE OF THE SEXES

HUSBAND \ WIFE FOOTBALL OPERA

FOOTBALL 3 , 1 0 , 0

OPERA 0 , 0 1 , 3

We can easily see that using the concept of best response, in green we might

see the best responses of player 1 against the two opponent’s strategies and in

red we might find the best responses of player 2 against the two opponent’s

strategies. In this game (FOOTBALL , FOOTBALL) and (OPERA , OPERA) are both

a Nash equilibrium.

EXAMPLE 2: PRISONERS’ DILEMMA

CALVIN \ KLEIN CONFESS NOT CONFESS

CONFESS 0 , 0 7 , -2

NOT CONFESS -2 , 7 5 , 5

Without using the concept of best response, we may use the previously stuff

about this game: we know that confess is a dominant strategy, but this is the

same thing as saying that both players are choosing the best response. Hence,

the only solution of this game is (CONFESS , CONFESS).

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The general relation between Nash equilibrium and IEDS can be summarized by

the following result: every IEDS solution must be a Nash equilibrium but vice

versa, every Nash equilibrium isn’t an IEDS solution. For instance:

FELIX \ OSCAR 3 HOURS 6 HOURS 9 HOURS

3 HOURS -13 , -8 -1 , -4 7 , -4

6 HOURS -4 , -1 4 , -1 4 , -4

9 HOURS 1 , 2 1 , -1 1 , -4

Using the concept of best response we should find three Nash equilibrium: (9H ,

3H); (6H , 6H) and (3H , 9H). Using the IEDS concept, “6 HOURS” and “3

HOURS” are dominated by “9 HOURS” for player one and is the same for player

2. Hence, the IEDS solution is only one: (9H , 3H) that is also a Nash

equilibrium.

CHAPTER 6

In this chapter, an application of the Nash equilibrium concept to a model of a

duopolistic market, the Cournot model, will be discussed.

Historically speaking, economists has spent the greater amount of time and

energy available studying two extreme forms of markets: monopoly, where

there is a single firm, and a perfectly competitive market, where there are

many firms in perfect competition. The reason for which we’ve made this

distinction is that typically a monopoly has no reason to take into consideration

the other competitor’s choices, because there are no competitors in this kind of

market. However, in reality, the most prevalent scenario is one in which there

are many firms in each given market. In this case, a company might guess the

action of the competitors in order to anticipate them and choose the best

response against them. So what should do the company? If they rise up the

prices, the market will match perfectly this delta or they are going to lose

customers? The Cournot model gives some answers to this problems.

In the model proposed by Cournot, two firms compete in a market for a

homogeneous product. In other words, consumers are unable to distinguish

from which firm the product that they have bought comes. Therefore, the two

firms have only one demand curve in the market that is given:

Q=α−βP

>

Say that , and Q = Q1 + Q2 that represent the sum of the

α 0 β> 0 α Q

−

P=

quantity produced by the two firms; the price will be: . If we use

β β

α 1

A= B=

and , the new price equation is: .

P= A−BQ

β β

10

Now we’re going to explain the Cournot model with real numbers: A = 10 and B

= 1, the inverse demand will now be P = 10 -Q.

Let’s assume now that the cost function does not vary with the numbers of

units produced and it’s the same for each firm. Formally, the cost of producing

quantity is Qi equal to cQi, in which c > 0 and i = 1,2. Now the question is: how

much would each firm produce? To answer at the question the firm has to take

two steps:

- Make a guess about the other firm’s production. This step will reveal at

the firm an idea about the likely market price: for instance, if he thinks

that the other firm is going to produce a lot, the price will be very low no

matter how much it produces.

- Determine the quantity to produce. In order to determine the quantity to

produce the firm has to weigh the benefits from increasing production.

A Nash equilibrium will be obtained if both issues are satisfied.

Let’s now consider firm 1: if it was the only firm of the market, it decisions are

able to determine the market price. It could compute the profits from selling

different quantity levels and pick the quantity that maximizes profits. As a

start, what firm 1 might do is ask, if firm 2 is going to produce Q , what

2

quantity should it produces in order to maximize profits.

What we should know form the above assumptions is that the price market, in

this case, is equal to: A – B(Q – Q ), say that the total production is Q = Q +

1 2 1

Q .

2

The revenues of firm 1 are therefore [A – B(Q + Q )]Q . Since costs are cQ , the

1 2 1 1

total profits are given by [A – B(Q + Q )]Q – cQ . Hence, the quantity that

1 2 1 1

maximizes the profits in this precise case is: Max [A – B(Q + Q ) – c]Q .

Q1 1 2 1

There is a maximum profit quantity, say Q , which we can compute with first-

*1

order condition to the problem. As a result, we have that Q is a best response

*1

A−c−B Q 2

of firm 1 to a quantity given produced by firm 2: Q = . Now let’s

*1 2 B

describe the reaction function R (Q ) that will denote the best response of both

1 2

firms.

11 A−c−B Q A−c

1 Q ≤

, if 1 B

2 B A−c

>

Q

0, if 1 B

Using the same logic, we can observe also the reaction function R (Q ) for firm

2 1

2.

In the above graph, we can find both best responses of the firms. Hence, this is

a pair of quantities for which:

- R (Q ) = Q

*1 *2

2

- R (Q ) = Q

*2 *1

1

In other words, this pair is a Cournot Nash equilibrium of this game.

From the above graph is easy computable the quantity, the price and the profit

for each firm: A−c

- Per-firm quantity: 3 B

1 2

+

A c

- Price: 3 3 2

( A−c)

- Per-firm quantity: 9 B

Let’s now assume that the two firms are operating as a cartel. This is possible

only if they coordinate their production decisions. As a cartel, their aim is to

maximize the total profits. In this case, the production function is: Max = [A

Q1,Q2

– B(Q + Q ) – c][Q + Q ]. The difference between these two cases is that in

1 2 1 2

the cartel case, the firms acknowledge explicitly that their prof