Stable Nash Equilibrium In Pure Strategies Economics Essay

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COMMENT ON THE RELATIVE PAYOFFS OF BOTH PLAYERS From the table above, the university has two strategies, that is; to offer support and not offer support. The student also has two strategies, that is; study to pass and do not study to pass. The university is the row player while the student is the column player.

However, if the university chooses to offer support, she gets a Marginal Utility of 6 if the student chooses to study and the student gets a payoff of 4. It is the best payoff to the university considering that she has offered support and the student painstakingly decides to study. At this point, the university is fulfilled. But if the student chooses not to study, when the university chooses to offer support, the university gets a negative payoff (-2) and the student gets a payoff of 6. The negative marginal utility to the university suggests that as a result of the student not studying, it impacts negatively on the university in the sense that the student may choose to defer, or worse still, become a dropout of the university and may even become a nuisance to the society. All of this does not portray the university in a good image. Also, as a result of the student dropping out of school, the university's expenditure will increase with no equal matching increase in revenue accruing to the university.

On the other hand, if the university chooses not to offer support, she gets a payoff of -2 if the student chooses to study to pass while the student gets a payoff of 2. In this case, the university will be disappointed by not offering support when the student chooses study to pass. The University chose not offer support forming the belief that the student will choose not to study to pass. If the University chooses not offer support and the student chooses not to study, they get a payoff of 0 each; implying that nobody wins or looses or regrets his course of action.

(B)

DEMONSTRATION THAT THERE IS NO STABLE NASH EQUILIBRIUM IN PURE STRATEGIES

Nash equilibrium can be seen as a strategy in which there is no incentive for the players in a game to deviate. This means that deviation to another strategy will not be profitable. Therefore, it can be seen as a strategy that maximises a player's payoff given the other player's strategy (Rasmusen 2007:26-27). In other words, in a nash equilibrium, it will pay a player to maintain his current strategy given that the other player does not change his strategy.

By pure strategies, it is meant that each player is choosing a strategy once and for all. That is to say that the player sticks to his choice (Varian 2010:524-525). An unstable nash equilibrium implies that there is more than one nash equilibrium outcome in a game or that no one situation is more preferable for all players (Emelichev etal 2003). As noted by Varian 2010, having an unstable nash equilibrium is one of the problems of the nash equilibrium notion

To demonstrate that there is no stable nash equilibrium in pure strategies for this game, we consider the decisions to both players below:

PAYOFF TO UNIVERSITY:

Offer support, student study = 6

Not offer support, student study = -2

Offer support if student studies

Offer support, student do not study = -2

Not offer support, student do not study = 0

Not offer support if student does not study

PAYOFF TO STUDENT:

Study, University offers support = 4

Do not study, University offers support = 6

Do not study if University offers support

Study, University does not offer support = 2

Do not study, University does not offer support = 0

Study if University does not offer support.

From the above, it can be seen that there is no convergence of at least one strategy combination by both players. If the University chooses to 'offer support if the student studies', then the student will not study. If the University chooses 'not to offer support if the student does not study', then the student will choose to study. Also, if the student chooses 'not to study if the University offers support', the University will not offer support. And if the student chooses to 'study if University does not offer support', then the University will offer support.

Therefore, there is no stable nash equilibrium in this game.

(C)

MIXED STRATEGY EQUILIBRIUM FOR BOTH PLAYERS

Mixed strategy, according to Varian (2010:526) can be seen when agents are allowed to randomize their strategies. In mixed strategy, probabilities are assigned to each choices and the choices are actually played with the assigned probabilities.

The probability of a player is gotten, taking into account the payoffs of the other player in the game.

Let the probability of the University offering support and the probability of the university not offering support be Ps and (1-Ps) respectively. And the probability of the student studying and not studying to pass be Pq and (1-Pq) respectively. Therefore, adopting the payoff-equating method noted in Rasmusen (2007:74), the probabilities of the mixed strategy is calculated as follows:

Probabilities for the University taking into account the student's payoff:

4Ps + 2(1-Ps) = 6Ps + 0(1-Ps)

Moving like terms to one side and collecting them,

-2Ps = -2(1-Ps) - - - - - - - - - - (1)

But Ps + (1 - Ps) =1 - - - - - - - -(2)

-2Ps = -2 + 2Ps

-4Ps = -2

Ps = -2/-4

Ps = ½ or 0.5

Putting Ps = 0.5 into (2),

0.5 + (1-Ps) = 1

(1-Ps) = 1 - 0.5

(1-Ps) = 0.5 or ½

Probabilities for the student taking into account the University's payoff:

6Pq +[ -2(1-Pq)] = -2Pq + 0(1-Pq)

Moving and collecting like terms,

8Pq = 2(1-Pq) - - - - - - - - - - - - -(3)

But Pq + (1-Pq)=1 - - - - - - - - - - (4)

8Pq = 2 - 2Pq

10Pq = 2

Pq = 2/10 =1/5 or 0.2

Substituting Pq = 0.2 into (4),

0.2 + (1-Pq) = 1

(1-Pq) = 1 - 0.2

(1-Pq) = 0.8

We can now determine the mixed strategy equilibrium outcome for both players, thus:

university = [0.5(0.2*6) + (0.8 *- 0.2)] + 0.5(0.2 * -2) + (0.8 * 0)

=0.5( 1.2 - 1.6) + 0.5(-0.4 + 0)

0.5(-0.4) + 0.5(-0.4)

=-0.2 -0.2

=-0.4(<0)

student = [0.2(0.5 * 4) + (0.5 * 2) + 0.8(0.5 * 6) + (0.5 * 0)

=0.2(2 + 1) + 0.8(3 + 0)

=0.2(3) + 0.8(3)

=0.6 + 2.4

=3 (>0)

This means that with the use of mixed strategy, both players (university and student) will have an outcome of-0.4 and 3 on average respectively regardless of which strategy they use. The mixed strategy nash equilibrium which cannot be 'exploited' by either player are:

University: 0.5Ps + 0.5(1-Ps);

Student: 0.2Pq + 0.8(1-Pq)

In summary, the probabilities of a mixed strategy are represented as:

with outcomes

(D)

CRITIQUE OF THE MIXED STRATEGY NASH EQUILIBRIUM

(i)Mixed strategy nash equilibrium poses a problem by assuming that both players have the same outcome no matter what strategy they choose. The question arises as to why and how players randomize their decisions. It is intuitively problematic.

(ii) People are also unable to generate random outcomes without the aid of a random or pseudo-random generator. Players who are inexperienced with randomizing may never attain equilibrium if they are unable to hire or pay for the services of a random generator.

QUESTION 2

Given the parameters = 359, , Ca =109 and Cb = 105

Where P =

P= The market clearing price of the product in the market

Q= The total output for both firms to the market

and are parameters

The market inverse demand is given as:

P = 359 - 0.10 (qa + qb)

And Marginal cost of firm a (MCa) = 109

Marginal cost of firm b (MCb) = 105

General assumptions of the model:

(i)The firms have homogenous products

(ii)Each firm has a constant but different marginal cost, i.e 109 and 105 for firms a and b respectively.

(iii)There are two firms in the industry i.e. Firm a and firm b.

(A)

PROFIT FUNCTIONS FOR BOTH FIRMS.

Profit can simply be seen as the difference between the income(revenue) of a firm and its cost. Where revenue or income is price multiplied by quantity.

The generalised profit function is given of the form:

Max P = (P - i)qi

Where p= price

i = Marginal cost

qi = quantity

Given P = 359 - 0.10(qa + qb)

the profit function for firm a is:

a = ( P - a­ )qa

a = [359 - 0.10 (qa + qb) - 109]qa

a = 250qa - 0.10qa2 - 0.10qaqb - - - - - - - - - - - - - - -(1)

The profit function for firm b is:

b = ( P - b)qb

b = [359 - 0.10(qa + qb) - 105]qb

b = 254 - 0.10qb2 - 0.10qaqb - - - - - - - - - - - - - - - - - (2)

(B)

NASH EQUILIBRIUM OUTCOME IN PROFITS IN A COURNOT GAME

In a cournot model, no one producer is able to determine the price of its product alone, because total quantity of the product is determined by all the players [1] in the game. According to Bierman and Fernandez, 'the market price per unit of output, P, received by all firms is a decreasing function of the total output produced by all firms'.

In a cournot game, equilibrium is determined at the intersection of both firms' reaction functions [2] . In other words, the output chosen by one firm, (say firm 1) which maximizes his profit, given his beliefs or expectation of the other firm's (say firm 2) choice is correct with the other firm's (firm 2) choice of output given his expectation of firm 1's choice [Pindyck et al (2009:4540] and [Varian (2010:508).

P = 359 - 0.10(qa + qb)

a = ( P - a­ )qa

a = [359 - 0.10 (qa + qb) - 109]qa

a = 250qa - 0.10qa2 - 0.10qaqb - - - - - - - - - - - - - - (1)

The reaction function for firm a is:

a/qa = 250 - 0.2qa - 0.10qb = 0

qa* = 1,250 - 0.5qb - - - - - - - - - - - - - - -(2)

The profit function for firm b is:

b = ( P - b)qb

b = [359 - 0.10(qa + qb) - 105]qb

b = 254qb - 0.10qb2 - 0.10qaqb - - - -- - - - - - - - - - - - - -(3)

The reaction function for firm b is:

b/qb = 254 - 0.2qb - 0.10qa = 0

qb* = 1,270 - 0.5qa - - - - - - - - - - - - - - - - - - -(4)

Solving the reaction functions given in (2) and (4) simultaneously, the equilibrium quantities for both firms can be determined thus:

qa* = 1,250 - 0.5qb

qb* = 1,270 - 0.5qa

qa + 0.5qb = 1,250 - - - - - - - - - - - - - - - - -(5)

qb + 0.5qa = 1,270 - - - - - - - - - - - - - - - - - -(6)

to eliminate qb, multiply (5) by 2 and (6) by 1

2qa + qb = 2,500 - - - - - - - - - - - - - - - - - - -(7)

qb + 0.5qa = 1270 - - - - - - - - - - - - - - - - - - - (8)

subtracting (8) from (7),

1.5qa = 1,230

qa = 1,230/1.5

qa = 820 units of the product

Substitute qa = 820 into (8) to get qb;

qb + 0.5qa = 1,270

qb + 0.5(820) = 1,270

qb + 410 = 1,270

qb= 1,270 - 410

qb = 860 units of the product

But industry output (Q)= qa + qb = 820 + 860= 1680 units of the product.

And Cournot price = 359 - 0.10(qa + qb) = 359 - 0.10(820 + 860) = £191

Cournot nash equilibrium outcome in profits can be gotten by substituting both equilibrium quantities into the profit functions in (1) and (3).

Profit for firm a = 250qa - 0.10qa2 - 0.10q­aqb

= 250(820) - 0.10(820)2 - 0.10(820)(860)

= 205,000 - 67,240 - 70,520

= £67,240

Profit for firm b = 254qb - 0.10qb2 - 0.10qaqb

= 254(860) - 0.10(860)2 - 0.10(820)(860)

= 218,440 - 73,960 - 70,520

= £73,960

Industry profit () = a + b = £67,240 + £73,960 = £141,200.

GRAPHICAL REPRESENTATION OF REACTION FUNCTIONS IN A COURNOT MODEL

qb

2500 qa*(qb) = Firm a's reaction function

1270 A cournot equilibrium

860 qb*(qa)= Firm b's reaction function

820 1250 2540 qa

FIGURE 1

The diagram above (figure 1) shows the cournot equilibrium and the reaction function curves of both firms. qa*(qb) shows the output of firm a(qa) in terms of firm b while qb*(qa) shows the output of firm b in terms of firm a. Point 'A' represents the cournot nash equilibrium (quantity).

If firm 'a' is rational and believes that firm 'b' is also rational, then he (firm a) will never expect firm 'b' to produce more than 1,270 units of the product while firm 'b' will never expect firm 'a' to produce more than 1,250 units of the product.

In summary, given the market demand function, P=359 - 0.10(qa + qb) and marginal costs equal to £109 and £105 to firm a and b respectively, the cournot nash equilibrium outcome in profits ( ) where 𝛱 =( a + b) =£67,240 + £73,960 = £141,200.

(c)

NASH EQUILIBRIUM OUTCOME IN PROFITS IN A STACKLEBERG GAME

This is also a quantity based model which adopts the leader-follower mode. This means that the leader moves first while the follower reacts to this first move (the output decision of the leader ).

Given P= 359 -0.10Q

P = 359 - 0.10 (qa + qb )

And profit function (p -ci)qi

For firm 'a'

a= 359 -[0.10(qa + qb) - 109] qa

= 250qa -0.10qa2 - 0.10qaqb - - - - - - - - - (1)

While that of firm be is given as; b = 359 -[0.10(qa + qb ) - 105] qb

= 254qb - 0.10qaqb - 0.10qb2 - - - - - - - - - (2)

reaction function of firm 'b'

b/qb= 254 - 0.10qa - 0.2qb

= 1,270 - 0.5qa - - - - - - - - - -(3)

The stackleberg nash equilibrium is gotten by substituting the reaction function of the follower (firm 'b'- (3) above) into the profit function of the leader (firm 'a'-(1) above) and differentiating with respect to the quantity of the leader (qa). Thus;

a = 250qa - 0.10qa2 -0.10qa (1,270 - 0.5qa)

a = 250qa - 0.10qa2 - 127qa + 0.05qa2

a = 123qa - 0.10qa2 + 0.05qa2

a = 123qa - 0.05qa2 - - - - - - - - - - - - - - - - - - - - - - - - - - - - -(4)

Therefore, stackleberg quantity is:

as/ qa = 123 - 0.1qa = 0

qa = 123/0.1 = 1,230 units of the product

Firm 'b' reacts to this first move with:

qb = 1,270 - 0.5qa (as in 2* above)

= 1,270 - 0.5(1230)

= 1,270 - 615

qb = 655 units of the product

But stackleberg price is given as: P = 359 - 0.10(qa + qb )

P = 359 - 0.10 (1230 + 655)

P = 359 - 188.5

P = £170.5

Profit of both firms is gotten by substituting the relevant quanties in the their profit functions stated in (2) and (4) above. Thus:

For firm 'a'

a = 123qa - 0.005qa2

= 123(1,230) - 0.05(1,230)2

= 151,290 - 75,645

a = £75,645

For firm 'b':

b =254qb - 0.10qaqb - 0.10qb2

= 254(655) - 0.10(1230)(655) - 0.10(655)2

=166,370 - 80,565 - 42,903

=£42,902

Therefore, stackleberg profit = £75,645 + £42,902 = £118,547.

GRAPHICAL REPRESENTATION OF REACTION FUNCTIONS IN A STACKLEBERG MODEL

qb

2460

Qa*(qb) = Firm a's reaction function

1230

Stackleberg's equilibrium

655 B Qb*(qa) = Firm b's reaction function

1230 2460 qa

FIGURE 2

The diagram above (figure 2) shows the stackleberg equilibrium and the reaction function curves of both firms. qa*(qb) shows the output of firm a(qa) in terms of firm b while qb*(qa) shows the output of firm b in terms of firm a. Point 'B' represents the stackleberg nash equilibrium (quantity).

If firm 'a' is rational and believes that firm 'b' is also rational, then he (firm a) will never expect firm 'b' to produce more than 1,230 units of the product while firm 'b' will never expect firm 'a' to produce more than 1,230 units of the product.

COMPARISON OF STACKLEBERG AND COURNOT OUTCOMES IN PROFIT

As demonstrated, the cournot profit of £141,200 is greater than the stackleberg profit of £118,547. Also, the leader gains by choosing output first, (i. e. 1,230 > 820) while the follower is worseoff(i. e. 655 < 860). As a result of choosing ouput first, player a's profit is £57,645(up by £8,405) while that of player b is £42,902(down by £31,058) and industry profit is down by a net of £22,653 approximately. Therefore, it will pay the industry for the players to move simultaneously with no leader since profit with moving simultaneously is higher than that gotten from moving sequentially.

(D)

A cartel is the coming together of a group of firms to behave like a single firm and maximize the sum of their profits. According to Varian (2010:513), 'when firms get together and attempt to set prices (or) output so as to maximize total industry profit, they are known as a cartel.' With collusion, the firms will be better off compared to if they operated as single firms.

OUTCOME IN PROFIT FOR A CARTEL

Given the inverse demand function:

P = 359 -0.10(Q) and the profit function facing the cartel:

𝛱 = (P - c)Q = [359 - 0.10(Q) - c]Q

In going ahead to ascertain the outcome, the marginal cost for the merged entity has to be determined because the outcome is dependent on the marginal cost. Since both firms are merged, it means their marginal cost will lie somewhere in between their former marginal cost. (Slightly more or less than their individual marginal costs). Assuming marginal cost is 107 [3] , (using the weighted mean approach which is 109+105 ) .

2

𝛱 =[359 - 0.10(Q) - c] Q = [359 - 0.10(Q) - 107]Q

𝛱 = 252Q -0.10Q2 - - - - - - - - - - - - - (1)

𝛱 = 252 - 0.20Q

Q

Q =1,260 units of the product

But the cartel price is given as: P= 359 - 0.10(Q)

P= 359 - 0.10(1260)

= 359 - 126

= £233

But outcome in profit is gotten by substituting the quantity (1,260) in the profit function stated above in (1). That is:

𝛱 = 252Q - 0.10Q2 = 252(1260) - 0.10(1260)2

= 317,520 - 0.10(1,587,600)

= 317,520 - 158,760

𝛱 = £158,760

(E)

Tit for tat is a form of punishment for a repeated non-cooperative game. If either opponents defect in any period, then the loyal player will defect in the next period. As noted by [Bierman & Fernandez (1998:194)], a firm could adopt a tit for tat policy in period two, dependent on the action (cooperate or defect) of the other firm in period one (the previous period). By cooperating it is meant that firms produce the cartel quantity and by defection, the cournot quantity is produced (in this model).

Assuming the cartel fixes the production quota of each firm at say 630 units of the product, both firms would understand that this maximizes their combined profits and both will have an incentive to cheat. Assuming the monopoly profit is £158,760 for the industry, then it means each players payoff will be £79,380 (i. e. £158,760/2)(assuming profit is shared equally). Defection by either player, with the other sticking to the production quota, will generate a profit higher than the monopoly profit (to the deviant) while the profit of the loyal player will be reduced, even below the cournot equilibrium profit. In the words of [Varian 2010: 517], 'if you produce more output deviating from your quota, you make profit 𝛱d, where 𝛱d > 𝛱m'. 𝛱m here denotes individual profit in a monopoly (collusive) situation.

NUMERICAL ILLUSTRATION:

Assuming cartel quantity = 1260 units of the product

Each player's quantity = 630 units of the product

Cournot quantity (firm a) = 820

Cournot quantity ( firm d) = 860

ROUND 1 - BOTH COOPERATE

a= 250(630) - 0.10(630)2 - 0.10(630)(630)

= 157,500 - 39,690 - 39,690

= £78,120

b= 254(630) - 0.10(630)2 - 0.10(630)(630)

= 160,020 - 39,690 - 39,690

= £80,640

Industry profit (𝛱) = a + b = £78,120 + £80,640 = £158,760

ROUND 2 - B DEFECTS

a = 250(630) - 0.10(630)2 - 0.10(630)(860)

= 157,500 - 39,690 - 54,180

= £63,630

b = 254(860) - 0.10(860)2 - 0.10(630)(860)

= 218,440 - 73,960 - 54,180

= £90,300

Industry profit (𝛱)= a + b = £63,630 + £90,300 = £153,930

ROUND 3 - BOTH DEFECT

a = 250(820) - 0.10(820)2 - 0.10(820)(860)

= 205,000 - 67,240 - 70,520

= £67,240

b = 254(860) - 0.10(860)2 - 0.10(860)(820)

= 218,440 - 73,960 - 70,520

= £73,960

Industry profit (𝛱)= a + b = £67,240 + £73,960 = £141,200

ROUND 4 - BOTH DEFECT

a = 250(820) - 0.10(820)2 - 0.10(820)(860)

= 205,000 - 67,240 - 70,520

=£67,240

b = 254(860) - 0.10(860)2 - 0.10(860)(820)

= 218,440 - 73,960 - 70,520

= £73,960

Industry profit (𝛱)=a + b = £67,240 + £73,960 = £141,200

ROUND 5 - BOTH REPENT

a = 250(630) - 0.10(630)2 - 0.10(630)( 630)

= 157,500 - 39,690 - 39,690

= £78,120

b = 254(630) - 0.10(630)2 - 0.10(630)( 630)

= 160,020 - 39,690 - 39,690

=£80,640

Industry profit (𝛱)= a + b = £78,120 + £80,640 = £158,760

PRICES AT EACH ROUND

P= 359 - 0.10(Q)

P1 = 359 - 0.10(1260)

=£233

P2 = 359 - 0.10(1490)

=£210

P3 = 359 - 0.10(1680)

=£191

P4 = 359 - 0.10(1680)

=£191

P5 = 359 - 0.10(1260)

=£233

From the illustration above, as a result of the actions of the players, increased output (from round 1 to round 4) will drive price down and also industry profit. That is, at the end of round 4, expected profit would have been £635,040 but actual profit is given as £595,090. This will continue until both firms 'repent' and produce the fixed quantity, say in round 5.

In the case of a 'grim strategy', when one firm defects, the loyal one goes grim. That is, it retaliates by defecting and will never 'repent'. Suppose player b defects, say in round 3, he will earn a profit higher than that of the loyal firm. But since both firms defect from round 4 onwards, the profit will be constant and lower than the cartel profit till the end of time.

The industry profit when both firms cooperate will always be greater than that gotten if one or both players defected at any round of the game. Therefore, in the long run, defecting is not profitable.

In summary, the cartel will continue to operate because deviant behaviour by one or both players will always result in an industry profit less than that of the cartel.

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