Background information for Q7 Two cyclists of equal ability

Background information for Q7:

Two cyclists of equal ability are competing in the Tour de Chance for a prize of $10,000. A banned substance is available. If one athlete takes the substance and the other does not, the one who takes it wins the prize. If both or neither takes the substance, they tie and split the prize. Taking the substance, however, imposes health risks as well as the possibility of testing positive and being banned from the sport for life. Assume these costs can be quantified and have a combined value of   X   dollars.

Q7 - i)

If X equals $2500 , the Nash equilibrium for this game would be:

No Nash equilibrium exists in this game if X = $2500.

Q7 - ii)

If   X equals $7500 , the Nash equilibrium for this game would be:

No Nash equilibrium exists in this game if X = $7500.

Additional information for next 2 questions:

Now suppose that the two cyclists have different levels of risk aversion and thus do NOT have the same value of X in this case. Assume that Player 1 has an X value (X1) of $2500 , while Player 2 has an X value (X2) of $7500.

Q7 - iii)

With these values of X1   and X2 , the Nash equilibrium for this game would be:

No Nash equilibrium exists in this game with the values of X1 and X2 in this case.

Q7 - iv)

Which of the following observations might be consistent with your findings in this question:

Solution

We shall represent this given info into Payoff Matrix

Player 2

Take Substance Donot Take

Player 1 Take Substance (2500-x,2500-x) (5000-x,0)  

Donot Take (0,5000-x) (2500,2500)

If X=2500 then there is No Nash Equilibrium in this game

Ans 7/ii)

   Player 2

Take Substance Donot Take

Player 1 Take Substance (-5000,-5000) (-2500,0)  

Donot Take (0,-2500) (2500,2500)

then NAsh Equilibrium is (Dont Take,Dont Take)

Ans 7/iii)

We have X1=2500 and X2=7500  

Player 2

Take Substance Donot Take

Player 1 Take Substance (0,5000) (2500,0)  

Donot Take (0,-2500) (2500,2500)

We have 2 Nash Equilibrium here those are (Take Substance, Take SUbstance) & (Dont Take, DOnt Take)

Ans 7/iv)

I think option B is correct as if number of Banned players increases Equality decreases and vice versa