这是一个美国的博弈论限时测试代写

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

receive.

[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