Probability of both head and tail in a fair coin is 0.5.
As we know that two fair coins are tossed, the out comes could be HH, HT, TH and TT
where H=heads and T=tails
a) Extensive form of the game
As player one has to make the next make the next move that wheather he has to fold or stay. game tree should be drawn in his respect. Kindly refer to below tree to understands more.
Player I has two choices, either to fold or stay. If he folds, in all the outcomes he has to pay 2$ to player II.
If he stays, Player II can guess 0 tails, 1 tail or 2 tails.
Probability of each outcome i.e. HH, HT, TH and TT is 0.25 each
Thus if Player I stays and Player II guesses 0 tails. So according to diffenrent outcomes, the payoff to player I is explained below:
IF HH comes, then Player II is correct in guessing, so player I must pay him 2$ for each tail. But as no tail showed up, he has to pay 0$
IF HT comes, then Player II guessed wrongly, so player I gets 2$ for each head. Thus player I gets 2$
IF TH comes, then Player II guessed wrongly, so player I gets 2$ for each head. Thus player I gets 2$
IF TT comes, then Player II guessed wrongly, so player I gets 2$ for each head. Thus player I gets 0$ has no heads showed up
Thus the net payoff taking probability into consideration is 0.25*0+0.25*2+0.25*2+0.25*1= 1$
Similarly the entire table can be filled up.
https://d2vlcm61l7u1fs.cloudfront.net/media%2F9d9%2F9d930ecd-2955-4b8d-b4b2-06e742af6f63%2FphpOAF90j.png
b) A pure strategy is all the available moves that a player can adopt during the game.
Pure strategy for Player I: Stay or fold ( 2 pure strategies)
Pure strategy for Player II: Guess 0 tails, 1 tail or 2 tails (3 pure strategies)
It is depicted in table below:
https://d2vlcm61l7u1fs.cloudfront.net/media%2Fd73%2Fd733a575-e070-466d-bf3b-f987af2f0727%2Fphp6zRn40.png
c) Finding the Nash equilibrium in table below:
https://d2vlcm61l7u1fs.cloudfront.net/media%2Fd2b%2Fd2b1b331-52ff-4fd7-865b-c21a11fe242a%2Fphpl5jSqX.png
We see that we find a solution to game i.e. Player I should Stay in the game and Player II should guess 1 tail.
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