Question

Two friends Aditi and Raju are deciding independently whether to watch a movie or go to a music concert that evening. Both friends would prefer to spend the evening together than apart. Aditi would prefer that they watch a movie together, while Raju would prefer that they go to the concert together. The payoff matrix arising from their actions is presented below. p and (1 - p) are the probabilities that Aditi will decide in favour of the movie and concert, respectively. Similarly, q and (1 - q) are the probabilities that Raju will decide in favour of the movie and concert, respectively. Which one of the following options correctly contains all the Nash Equilibria ?

		Raju
Aditi		Movie	Concert
	Movie	2,1	0,0
	Concert	0,0	1,2

• A. \((p=0,q=0);(p=1,q=1);(p=\frac{2}{3},q=\frac{1}{3})\)
• B. \((p=0,q=1);(p=1,q=0);(p=\frac{2}{3},q=\frac{1}{3})\)
• C. \((p=0,q=0);(p=1,q=1);(p=\frac{1}{3},q=\frac{2}{3})\)
• D. \((p=0,q=1);(p=1,q=0);(p=\frac{1}{3},q=\frac{2}{3})\)

Answer 1

The Correct Option is A

To find the Nash Equilibria in the problem presented, we analyze the payoff matrix for Aditi and Raju. Below is the payoff matrix corresponding to their strategy choices:

		Raju
Aditi		Movie	Concert
	Movie	2,1	0,0
	Concert	0,0	1,2

A Nash Equilibrium occurs when no player can benefit by changing their strategy unilaterally. We check each possible strategy combination:

• (Movie, Movie): Aditi receives 2 and Raju 1. Neither can increase their payoffs by changing strategy alone since they would receive 0 otherwise (0,0 or 0,0). Hence, (0,0) is a Nash Equilibrium.
• (Concert, Concert): Aditi receives 1 and Raju 2. Switching would give them both a payoff of 0, hence (1,1) is a Nash Equilibrium.

Next, we find the mixed strategy Nash Equilibrium by setting up the expected payoffs:

• Aditi's expected payoff: Aditi will choose "Movie" with probability \(p\), and Raju's choice affects her payoff. Therefore, we express her expected payoff from "Movie" as:

  \(E_{Aditi}(Movie) = q \times 2 + (1-q) \times 0 = 2q\)

  From "Concert":

  \(E_{Aditi}(Concert) = q \times 0 + (1-q) \times 1 = 1-q\)

  For equilibrium, set equations equal:

  \(2q = 1-q \Rightarrow 3q = 1 \Rightarrow q = \frac{1}{3}\)

• Raju's expected payoff: Similar expressions apply for Raju. Choosing "Movie" with probability \(q\):

  \(E_{Raju}(Movie) = p \times 1 + (1-p) \times 0 = p\)

  From "Concert":

  \(E_{Raju}(Concert) = p \times 0 + (1-p) \times 2 = 2 - 2p\)

  Equilibrium when:

  \(p = 2 - 2p \Rightarrow 3p = 2 \Rightarrow p = \frac{2}{3}\)

Thus, the mixed strategy equilibrium is \((p=\frac{2}{3}, q=\frac{1}{3})\).

Conclusively, the Nash Equilibria are at \((p=0,q=0)\), \((p=1,q=1)\), and the mixed strategy \((p=\frac{2}{3},q=\frac{1}{3})\).