INSTRUCTIONS TO CANDIDATES
This paper contains FOUR questions, all carrying equal weight. Full marks will be awarded for complete solutions to THREE questions. Candidates may attempt all questions, but only the best THREE solutions will be taken into account.
By submitting solutions to this assessment you affirm that you have read and understood the Academic Integrity Policy detailed in Appendix L of the Code of Practice on Assessment.
1.a) A rustic poker game for two players, 𝐴 and 𝐵, begins with both players staking £1 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 one 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 “Raise” or “Stick”. If he chooses to “Raise” he must place a further £3 in the kitty and can then take a second card from the hat. If he chooses to “Stick”, he does not have to pay in any extra money into the kitty, but he cannot choose a further card. Now it is Player 𝐵’s turn again. He too can “Raise” – pay £3 into the kitty and select a second card from the hat – or he can “Stick” and do nothing further. At this point the game concludes, the players turn over the cards they hold in their hands and the money they each paid into the kitty is returned to them. 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 difference in the scores gives the monetary payoff for each player, e.g. if Player 𝐴 scores 3 and Player 𝐵 scores 1, then Player 𝐴 wins £2 and Player 𝐵 loses £2. If the scores are level there is no gain or loss for either player.
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 𝐴 chooses to call “Raise” only when holding a card numbered 4, whilst Player 𝐵 adopts a strategy of calling the exact opposite of Player 𝐴. Calculate (and show all your working) the expected payoffs for the two players if they play off these two contrasting strategies against one another. [4 marks]
2.a) Solve the following 4 × 2 zero-sum game, with payoff matrix (for the row player) given by
For full marks you must compute the value of the game and the optimum strategies of both players. [9 marks]
b) Solve the following player) given by 4 × 4 zero-sum game, with payoff matrix (for the row
(4 7 13 1
5 1 5 3
0 2 3 4
2 6 2 0) .
Once again for full marks you must compute the value of the game and the optimum strategies of both players. [11 marks]
3.a) Brian and Samantha are two fine art collectors specialising in decorative pieces of Chinese porcelain. They are particularly interested in porcelain manufactured during the eras of the Tang (618-907) and Ming (1368-1644) Dynasties respectively. Currently Brian owns 20 pieces of Ming porcelain and 14 pieces manufactured during the Tang Dynasty. By contrast Samantha owns 21 items of Tang porcelain and 15 dating to the Ming era. The two collectors meet together to discuss the possibility of trading pieces, to better satisfy their respective utility functions. Brian’s utility function for Ming (𝑚) and Tang (𝑡) porcelain is given by
𝑈𝐵(𝑚,𝑡) = 5𝑚𝑡/7𝑚+ 2𝑡,
whilst Samantha’s corresponding utility function (𝑚′ and 𝑡′ for Ming and Tang porcelain respectively) is given by
𝑈𝑆(𝑚′,𝑡′) = 10𝑚′𝑡′/2𝑚′ + 7𝑡′ .
Find the contract curve for this trading relationship in terms of (𝑚, 𝑡). Determine the coordinates of the endpoints of the contract curve, where it crosses the initial indifference curves of the two players. [12 marks]
b) Suppose the current market prices for Ming and Tang porcelain pieces are 𝑝𝑚 and 𝑝𝑡 respectively. Neither Brian or Samantha wants to make a trade unless the value of their new collection matches that of their original. Write down Brian’s budget constraint in terms of (𝑚, 𝑡). Assuming ideal market conditions, find a unique point on the contract curve (between the two endpoints) about which a trade will take place, and the number of porcelain items the two collectors will exchange in order to satisfy this valuation constraint. (Porcelain pieces are obviously indivisible, so round your figures up or down to the nearest integers to find the number of items they exchange.) [8 marks]