ECO 3101 Intermediate Microeconomics

Game theory

About two months ago we moved to a new house; I would always go out over the weekends to explore the neighborhood.  One of the weekends I  met my long time friend, we had separated five years as he had moved after being  recruited by Non-governmental organization to attend to the needs of a group of special need children.  We had been walking and sharing our moments over the past years, when he received a call to attend to an urgent issue at his office. He left while we promised each other to meet at the same place the next weekend and possibly go for some leisure activity. 

Over the week we did not communicate since Nick had changed his telephone number and we both had also changed our physical addresses. Therefore we only could talk when we meet again the following weekend for some leisure activity. In the past we would spend time either  riding bicycle or going to the cinema halls . I particularly liked riding bicycle and wished to spend the coming weekend up and down the streets riding, however my friend liked spending time in the cinema halls. Hence it was expected that we would either go to the cinema hall or spend time riding and rekindling the experience we had five years ago.

Translation Into Formal Game

The situation of choice on what activity to undertake during the weekend could be represented as a formal game with my friend and I being the set of players. Either of us having two make a choice between either riding or  going to the cinema halls. Our separate actions must have a common outcome based on trade offs that offer the best response to the other players actions. In achieving the outcome of the game, I would either lose or win as same as my friend.  The strategic interaction could be assumed to have been undertaken without prior knowledge of the other party’s choice on the activity to engage in; making the game to be a static game.

Solution Concepts

The solution to the game is either we both go for ridding or we both visit the cinema hall .

We shall assume that the payoff for me if we went for riding is 10 while the payoff for nick, my friend, is 10 if we went for the cinema halls and if we choose to agree with the other persons choice the payoff would be 5 for each of others. It therefore implies that  if we disagreed on the activity to engage in, then  the payoff would  be 0. A 0 pay  off implies that the leisure activity will not be undertaken and that we would only go back to our own engagements. The game is illustrated below, with the column representing Nick’s utility and the rows representing my utility.  The payoffs would therefore be as shown

The action space A₁ for both us as players (Nick, me) was (cinema hall, riding) and A₂ = (Cinema hall, riding).  The game lacks a dominant strategy because no single choice could be the best no matter what nick or I chose. If I chose riding I risk nick deciding to chose cinema hall which would cancel the much awaited leisure weekend. The game  was therefore a coordination game since there were no dominant actions and neither could we eliminate dominated actions. More rationality was hence required to ensure that the outcome of the game favored the desired leisure time; and therefore it was necessary to think of the best response.

If I were to go first on choosing the activity to engage in nick’s best response would be to agree  with me and go for the riding, however if nick went first in choosing the game to be played the best response would again be agreeing with him.

Solving For The Mixed Strategy Between Me And Nick

Let my pay off be represented by M

And Nick’s pay off be represented by N

Nick’s  pay off from visiting the cinema hall

U_c2=10.M + (1-N)0 = 10M

Nick’s pay if we went riding

U_R2= 0.M + (1-N)5 = 5-5M

Equating

U_C2=U_R2

10M = 5-5M

15M=5

M=1/3

My pay off from visiting cinema hall

U_C1= 5.N + (1-N)0 =5N

My pay off if we went riding

U_R2= 0.N + (1-N). 10 =10-10N

Equating

U_C2=U_R2

5N=10-10N

15N=10

N=2/3

Equilibria of the game

There were two equilibria in pure actions; equilibria existed if we both chose to go to the cinema which represents the first equilibria (cinema hall, cinema hall) and the second equilibria would occur if we both chose to go riding (riding, riding). The mixed strategies game also presented a third equilibrium in which either of us had a probability of either 2/3 or 1/3 as shown above. Nick played to win a visit to the cinema hall with a probability of 1/3 and a probability of going for a ride with 2/3 probability. On my side, I played to win a visit to the cinema with a probability of 2/3 and a probability to go for riding of 1/3. This equilibria points represent the Nash equilibrium for the game of choosing the appropriate leisure activity. As earlier stated, the best response to any of us in the game was agreeing to the choice of the  other player as a deviation would not be profitable.

From the discussion it can be easily be depicted that (cinema hall, cinema hall) or (riding, riding) were plausible solutions. During the weekend each of us strongly supported the most liked activity not withstanding the other party’s option. However, we resolved the conflict and went for cinema hall. The outcome was not inline with the two possible solutions because we both did not have full information on the choice of the other party and the associated utilities.

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