Problem 1
Consider the following game:
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A, B
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l
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r
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L
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(1,0)
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(5,-1)
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R
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(1,2)
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(3,1)
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(a) Find all pure strategy Nash equilibria.
(b) For each player, find the best response to (each) mixed strategy of the opponent.
(c) Draw the best responses of both players that you have found in (b), as we have done on lecture and tutorial.
(d) Find all mixed strategy Nash equilibria.
Problem 2
Consider the following auction that most of you must have encountered before.
There is a prize (e.g. a chocolate rabbit). Each player buys tickets, worth 1 cent each, write his or her name on it, and put them in a box. Then a ticket is drawn at random and the person whose name is on the ticket receives the chocolate rabbit.
Assume, for all parts of this problem, that there are two players and that the rabbit is worth $1 for both. The auction I am describing is usually used for charity; ignore all aspects of charity and assume that the only thing players care about is getting the rabbit.
Note: In this problem, sometimes it may be more convenient to assume that tickets are perfectly divisible and sometimes it may be more convenient to assume that there is a minimal unit (one ticket). You can switch between these two assumptions in sub-problems as you wish, but please indicate which one you use.
For parts (a)-(c) assume that the box is black and players do not see how many tickets the other player has bought.
(a) Model this situation as a formal game (that is, specify strategies, payoffs and draw either a matrix or a tree). You need to explain all your choices.
(b) Find the best response of player 1.
(c) Does pure strategy Nash equilibrium exist in this game? Find one if your answer is yes and explain why not if you answer is no.
(d) Suppose that the rules change: Player 1 decides how many lottery tickets to buy and then Player 2 observes the decision of player 1 and decides how many lottery tickets to buy. Model this new situation as a formal game.
(e) Find the subgame perfect Nash equilibrium of this game. (Hint: although this game may look slightly different to what we have seen so far, approach it in the same way as other games: find that Player 2 will do given Player 1's choice; then find what Player 1 will do knowing the choice of Player 2 that will follow. You will need to write maximisation problems; pay attention what is the decision variable in these problems and what is not.)
Let us now consider a hybrid of (a) and (d). Let us suppose that Player 1 buys tickets and put them in the box, and Player 2 can observe whether the box is empty (Player 1 bought nothing) or has tickets, but cannot observe how many tickets there are inside.
(f) Model this situation as a formal game.
(g) Give an example of the strategy profile in this game.
(h) Explain why the strategy profile in (g) is / is not a Nash equilibrium.
(i) Does pure strategy Nash equilibrium exist in this game? Give an example if it does or explain if it does not.
(j) Suppose we are looking for a subgame perfect Nash equilibrium. How does you answer to (i) changes?