1. a) A rustic poker game for two players, 𝐴 and 𝐵, begins with both players
staking £2 into the kitty. In this game there is a hat containing four cards, two
marked with the number ‘2’ and two marked with the number ‘4’. The game
starts with Player 𝐴 selecting (at random) a card from the hat (both players are
careful not to show their cards to their opponent). Player 𝐵 then selects a card
from those remaining in the hat. Now Player 𝐴 must decide to call either
“Raise £3” or “Stick”. However, in this poker game the players are only
allowed make a call of “Raise £3” if they can either

i) show, by turning it over, that they currently hold a number ‘2’ card,
or

ii) in response to a previous Raise call from their opponent.
So in order to make a raise, Player 𝐴 must first turn over his original card to
show the number ‘2’. Then he pays £3 into the kitty and selects one of the two
remaining cards from the hat. Alternatively Player 𝐴 may call “Stick”, without
having to reveal the card he holds, but then he is not allowed to select a second
card. Now it is Player 𝐵’s turn. He too can call “Raise £3” or “Stick” as above.

Again if he chooses to Raise then he pays £3 into the kitty and selects a second
card from the hat.

At this point the game concludes, the players turn over all the cards they hold
in their hands. The winner of the game is the player with the higher score. A
player’s score is given by the total numbers on their respective card/s, minus
the money they paid into the kitty to obtain those cards. The player with the
higher score wins all the money in the kitty. If the scores are equal, the kitty is
shared.

Draw a game tree for this game, including the information sets, the relevant
probabilities of the possible scenarios and the monetary payoffs the players
[11 marks]

b) Compute the total number of playing strategies for each player. Write out
all of Player 𝐴’s playing strategies in full. Do the same for any three of Player
𝐵’s playing strategies.
[5 marks]

c) Suppose Player 𝐵 adopts a strategy of calling “Raise £3” whenever, he has
the opportunity to do so. Calculate the expected payoffs he can expect to
receive if he plays this strategy against all the possible playing strategies of
Player 𝐴.
[4 marks]

2. a) In a 2 × 2 strategic game, the payoff bi-matrix for the two players A (row)
and B (column) is given by

Using the swastika method, or otherwise, find the Nash equilibria of the game.
[8 marks]

b) A more general version of this particular 2 × 2 strategic game, is defined by
the following payoff bi-matrix