Question

Consider the following market entry game: two entrepreneurs are trying to decide if they want to start a business and enter the market for selling protective facemasks, but the market is very saturated. If both enter the market, for instance, they will both lose their initial investment of $50,000. If they both stay out of the market, they lose nothing. However, if only one enters the market, the one who enters will net $100,000 in profits while the one who didn’t enter will lose nothing.

a. Draw the game matrix (for simplicity, feel free to use 100 and 50 instead of 100,000 and 50,000).

b. List any pure strategy Nash Equilibria, if they exist.

c. Define the mixed strategy Nash Equilibrium for this game.

Answer 1

a. Figure-1 in the document attached below illustrates the  table representing the payoff matrix for both the entrepreneurs with regards to their respective decisions to whether enter the facemask market or not. The first entrepreneur has been denoted as E-1 and the second entrepreneur as E-2 and the respective numerical or mathematical payoffs of both E1 and E2 under different circumstances or contingencies are in thousands of dollars.

b. Now, based on figure-1 in the attached document note that for E-1, if he or she decides to enter the facemask market and E-2 also prefers the same or enter then E-1 looses $50,000 and if E-2 decides not to enter the market then E-1 would obtain a payoff of $100,000. Alternatively, if E-1 decides not to enter the market and E-2 also decides the same then E-1 gets nothing and looses nothing and if E-2 decides to enter the market then E-1 also gets nothing and looses nothing. Hence, considering that E-1 is a risk aversive individual, he or she would preferably better off choosing not to enter as there is a possibility of loosing $50,000 by entering the market contingent on the decision taken by E-2 and the payoff from not entering the market is constant at 0 or loosing nothing and obtaining nothing regardless of the decision of E-2. Similarly, if E-2 chooses to enter the facemask market and E-1 decides to enter then E-2 looses $50,000 and if E-1 decides not to enter the market then E-2 gets a payoff of $100,000 and if E-2 decides not to enter the market and E-1 enters the market then E-2 gets nothing and loosing nothing with a payoff of 0 and if E-1 chooses to not to enter the market then E-2 obtains payoff of 0 as well implying that he or she gets nothing and looses nothing at the same time. Hence, assuming that E-2 is also a risk aversive individual, he or she is better off not entering the market in which case, the payoff of 0 is guaranteed regardless of the decision of E-1 as opposed to bearing the risk of loosing $50,000 by entering the market contingent on the decision of E-1. Therefore, in this case, a pure strategy Nash equilbrium would be both entreprenuers deciding not to enter the facemask market with the respective payoffs of (0,0).

c. Now, let's suppose that E-1 chooses to enter the market with probability p and E-2 decides to enter the market with a probability q. Hence, based on the possible payoffs of E-1 from choosing to enter the market based on the responsive decision of E-2, the expected payoff for E-1 from choosing to enter the market= p*(-50)+(1-p)*(100)=-50p+100-100p=100-150p and his expected payoff from not entering the market=p*(0)+(1-p)*0=0. Therefore, to find the expected payoff at which E-1 would be indifferent about entering or not entering the market requires the equality between the expected payoffs under both situations.

Therefore, based on the condition to establish indifference of E-1 under both possible circumstances, it can be stated:-

100-150p=0

-150p=-100

p=-100/-150

p=0.66

Thus, E-1's probability of entering the market would be 0.66 and not entering the market would be (1-0.66)=0.34 approximately.

Now, the expected payoff of E-2 from entering the market=q*(-50)+(1-q)*(100)=-50q+100-100q=100-150q and his or her expected payoff from not entering the market=q*(0)+(1-q)*(0)=0. Again, to make E-2 indifferent between choosing both the options his or her expected payoffs under both situations have to be equal.

Hence, based on the condition to establish the indifference of E-2 about both decision options, it can be stated:-

100-150q=0

-150q=-100

q=-100/-150

q=0.66

E-2's probability of entering the market is 0.66 and not entering the market=(1-0.66)=0.34.

Therefore, the joint or combined probability of both E-1 nd E-2 choosing to enter the market=(0.66)*(0.66)=0.4356 and the combined probability of both E-1 and E-2 not entering the market=(0.34)*(0.34)=0.1156 and combined probability of E-1 entering and E-2 not entering the market=(0.66)*(0.34)=0.2244 and the joint probability of E-1 not entering the market and E-2 entering=(0.34)*(0.66)=0.2244. Hence, based on the mixed strategy Nash equilibrium the joint probability of both E-1 and E-2 entering the facemask market is the highest, in this case.