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(Microeconomics- Game theory) (Page # 1)

Q1. (a). Players: The players are two firms, Firm-1 and Firm-2 N= {1, 2}

Strategies: The strategy set for firm-1 is F1{F, A} in which F denotes that the firm decides to spend 1 million for developing new electronic hardware and A is vice-versa.

Similarly strategy set for firm-2 is F2 {F,A} where firm-2 decides whether to spend 1 million for development of software or not..

F2

F A

F

10 , 5

-1 , 0

0 , -1

0 , 0

F1

A

The cells of the above matrix show the payoffs for both the companies under different circumstances.

(b). A pure strategy is a term used to refer to strategies in Game theory. Each player is given a set of strategies, if a player chooses to take one action with probability 1 then that player is playing a pure strategy. This is in contrast to a mixed strategy where individual players choose a probability distribution over several actions.

Let (S, f) be a game, where S is the set of strategy profiles and f is the set of payoff profiles. When each player chooses strategy resulting in strategy profile x = (x1,…, xn) then player i obtains payoff fi(x). A strategy profile is a Nash equilibrium (NE) if no deviation in strategy by any single player is profitable, that is, if for all i

The pure strategy Nash Equilibrium cells for the above matrix would be {F, F} and {A , A}. (Ref. 1)

(Microeconomics- Game theory)

If a game has a unique Nash equilibrium and is played among players with certain characteristics, then it is true (by definition of these characteristics) that the NE strategy set will be adopted. The necessary and sufficient conditions to be met by the players are:

Each player believes all other participants are rational.

The game correctly describes the utility payoff of all players.

The players are flawless in execution.

The players have sufficient intelligence to deduce the solution.

Each player is rational.

These conditions signifies that there is a unique equilibrium. The game signifies that the unique playoff for all the players as seen above in the payoff matrix and other conditions are to be assumed for the unique equilibrium. Since each player is rational and have sufficient intelligence to deduce the solution, I expect that the two players would be satisfied with the outcome since either both the players would not invest in developing their products before prior agreement with each other or they will invest 1 million only after ascertaining that the other will also invest in building the complimentary product. If they both invest in their respective products then they will be able to earn huge profits. Moreover the double profit of firm 1 will not in any way hamper the growth of the firm 2 as they are not competitors and are in different branches of manufacturing.

There are two equilibria from the firms prospective, one is {F, F} where 10 and 5 signify the payoff that firm 1 and firm 2 receive if they both agree to invest 1 million for manufacturing of their product. The other equilibria is {A, A} where 0 is the payoff for both the fiems if they donot invest in their new product. Hence is the worst equilibrium from the firm’s perspective because they are not earning any profits and are wasting an opportunity which can be utilized to earn huge profits by both the firms through cooperation and coordination. This bad outcome is still an equilibrium because if any of the firms decides not to invest in developing new product then it will restrict the path of profit for the other firm and in order to avoid loss its option for optimum payoff would be not to spend for the manufacturing of new product either.

(c). FIRM 1

Suppose first firm 1 decides whether or not to develop new hardware then firm 2 decides the same then the tree diagram is as follows:

The sub-game perfect Nash equilibrium for the sub-game where firm 1 decides to develop (i.e. strategy F) its new hardware, would be {F,F} with payoff (10, 5) and the sub game perfect Nash equilibrium for the sub-game where firm 1 decides against (i.e. strategy A) developing new hardware, would be {A,A} with payoff (0, 0) respectively.

(d). Suppose Firm 2 decides whether to develop or not to develop its new software then the tree for this game would be:

The sub-game perfect Nash Equilibrium if firm 2 decides to invest would be {F, F} with payoff (10,5) for firm 1 and firm 2 respectively and if firm 2 decides against investing it would be {A, A} with payoff (0,0) for firm 1 and firm 2. (Ref. 2)

(e). In the question (b) there are two Nash equilibriums due to the uncertainty of the decisions that either of the firm takes. Since both the firms act simultaneously the fate of either of the firms depend upon the decision taken by the other firm. If any of the firm chooses against spending then the best strategy for the other firm would also be not to invest since both cannot survive without each other. But when either of the firm chooses to invest in their new product the Nash equilibrium changes as the optimum strategy for the other is also to spend to earn profits. In question (c) since Firm 1 decides first whether to develop or not to develop hardware, there are two sub-games that originate depending upon the decision taken by the firm 1, if firm 1 chooses to develop its product there is a new sub-game where firm 2 has to decide the same but the most favorable decision would be to develop the product so as to earn profit which is better than earning nothing. So the sub-game perfect Nash equilibrium would be {F, F} only. If firm 1 decides against developing then in this sub-game, sub-game perfect Nash equilibrium would be {A, A} only. Since the decision of firm 1 is known to firm 2 before taking any decision firm 2 has advantage over firm 1 as there is no risk for firm 2 but firm 1 is at great risk and is now dependent upon firm 2 for profits. In question (d) the sub-game Nash equilibrium remains the same in both the sub-game situation but the situation is just opposite as here firm 2 has taken the first decision and then firm 1 takes the decision, hence firm 1 plays its part after firm 2 makes its decision. Hence in this situation the firm who decides first is at disadvantage because it will risk its future at the hands of other firm.

Threats and promises are the basic conditional rules of this kind of game. They require a player to change the subsequent game so that he has the second move in it, and

( Microeconomics- Game theory) (Page # 5)

to commit in advance to how he will make that move in all conceivable eventualities. All strategic moves – commitments, threats, and promises – must be credible. The modern formal notion of credibility is sub-game perfectness; the outcome that the player making the strategic move wishes to achieve must be the result of a sub-game perfect equilibrium in the enlarged game. The strategic move that the player wishes to achieve must be the result of a sub-game perfect equilibrium in the enlarged game.

(f). If we consider the sequential structure in part (c) taking into consideration that firm 1 will go bankrupt if it spends 1 million for its product development but does not earn any profit. In part c the sequence is such that firm 1 decides first whether to invest or not to invest in its new product then firm 2 decides depending upon the decision taken by firm-1. If the firm 1 decides against investing for the new product then firm 2 can only save itself from loss by also not investing. But if firm 1 decides to take this risk and spend 1 million then it will become a puppet in the hands of firm 2 since its future existence solely depends upon firm 2 whether it invests or not. Firm 2 if it wants can extract whole 10 million from firm 1’s payoff and make it bankrupt through extortion. The firm 1 has no choice but will have to function under the terms and conditions of firm 2 and is at the mercy of firm 2. (Ref. 3, Dixit)

(g). If two firms are market leaders, one in software and the other in hardware, then also their profitability would be consistent with the above analysis since both of them are manufacturing the products that are useless without the other or both are complimentary to each other. A computer needs both hardware and software for functioning and one is useless without the other. The other aspects of relationship between the two companies could be that they could enter into long term contracts with each other regarding production, there could also be advertising relationship where both the products would be featured together in computer advertisements, both firms could tie up before entering a new market. But a software company can also produce many soft wares, since a hardware can play many soft wares, but that does not mean that it can survive without a hardware.

QUESTION 2

(a). Players: The players are two men X and Y respectively.

Strategy: Let S1(A,O) be the strategy of X and S2(A,O) be the strategy of Y where A denotes that the man chooses woman A and O denotes that the man chooses other women.

The payoff matrix game for this situation is as follows

( Microeconomics- Game theory) (Page # 6)

Y

-1, -1

1, 0

0, 1

0,0

A O

A

X

O

(b). If X and Y both ignore strategic interactions and each chooses the action that would be the best if he disregards the competition from other men then they will choose any other woman other than A. If both men behave this way they both will choose among the other women present in the pub.

In his first book, The Theory of Moral Sentiments, Smith argued that behavior was determined by the struggle between what Smith termed the “passions” and the “impartial spectator.” The passions included drives such as hunger and sex, emotions such as fear and anger, and motivational feeling states such as pain. Smith viewed behavior as under the direct control of the passions, but believed that people could override passion-driven behavior by viewing their own behavior from the perspective of an outsider—the impartial spectator—a “moral hector who, looking over the shoulder of the economic man, scrutinizes every move he makes” (Ref. 4, Grampp, 1948, p. 317). Here both the men disregard competition which effects their behavior and actions.

An outcome of a game is Pareto optimal if there is no other outcome that makes every player at least as well off and at least one player strictly better off. (Ref. 5) Non-strategic optimization for the two men does not lead to Pareto optimality because though both the men are well off with their respective women, the woman not being A then they are well off but neither of the man is better off since they have not chosen A.

(c). In part (a) the pure strategy Nash equilibria are {O, A} and {A, O} respectively. The situation in part (a) is such that if both the men choose to approach A then both will face disappointment since they are risk averse and payoff will be -1 for both. But if one of them approaches one of the other women and other approaches A or vice versa the payoff for the man who approaches A would be 1 while for the other would be 0. If both approach other women then both would be satisfied and have utility normalized to 0. According to me the one player must approach A while other can approach one of the other women. Thus both will be well off and their will be no conflict.

(d). When there are more players I think that most of the players would choose from the other women since there is no risk and men are risk averse. When every other women are

( Microeconomics- Game theory) (Page # 7)

occupied the man that is left will have no option but to approach A. One method that would be adopted by men could be to wait for sometime and observe whether any other man approaches A and approach A if he finds A alone. In non- strategic approach from part (b) it is clear that the men would first choose one of the other women as they are overcome by passion and they aren’t aiming for optimum, if their passion of disregarding the competition is discarded.

(e).Considering part (a), if X approaches A first then Y will have to choose from the other women so here the mixed strategy Nash equilibrium payoff for Y would be 0. If X chooses other woman, Y can choose either A or other woman, but most favorable would be A, hence mixed strategy Nash equilibrium payoff for Y would now be 1. Similarly if Y creates a situation for X the X will receive the same payoff as above.

There are three situations from where the players can derive utility. When a player chooses A alone his utility is 1, if he chooses other woman his utility is 0 and if he faces regection in meeting A his utility is -1. The expected utility of players X and Y facing uncertainity would be the weighted mean avg. of the utility in every situation that he faces.

X = Σx/n

= 1+0-1/3

= 0

Y = Σy/n

= 1+0-1/3

= 0

Yes, the cell {O,O} in the matrix yields the same expected utilities which gives the payoff 0 for both X and Y and is equal to the calculated each player’s expected utilities. (Ref. 6) This utility is the most probable utility that each player will achieve in the given situation.

(Microeconomics- Game theory) (Page # 8)

Bibliography:

1. http://en.wikipedia.org/wiki/Nash_equilibrium

2. http://en.wikipedia.org/wiki/Subgame_perfect_nash_equilibrium

3. Avinash Dixit, http://www.princeton.edu/~dixitak/home/Schelling.pdf, p: 7

4. Grampp, William.1948 “Adam Smith and the Economic Man. “Journal of Political Economy. p: 315

5. http://www.gametheory.net/dictionary/ParetoOptimal.html

6. http://en.wikipedia.org/wiki/Expected_utility

7. http://www.cdam.lse.ac.uk/Reports/Files/cdam-2001-09.pdf (Not in the text)

8.http://malroy.econ.ox.ac.uk/ccw/Games.shtml (Not in the text)