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ECON 2112 pure strategy
Creation date：2024-06-11 16:59:00
Belonging category：经济Economics
pure strategy

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ECON 2112. PROBLEM SET 

Notes
• Please remember to write (a) your full name, (b) student ID, (c) name of your tutor, and (d)
your scheduled tutorial time.
• This problem set is worth 12 points. Submit your answers to Exercises 1, 2, and 3 only.
• In some exercises below you will find the expression “pure strategy”. By that we mean the kind
of strategies that we have seen in the first week. In the second week we will introduce concept
of mixed strategies and the terminology will become clear.
Exercise 1. [6 marks]
Two individuals A and B go out on a hunt. Each can individually choose to hunt a stag or hunt a hare.
Each hunter must choose one of these two actions without knowing the choice of the other. The only
way of hunting a stag is that both hunters choose to hunt it. An individual can always get a hare by
himself, but a hare is worth less than a stag. To be precise, each hunter will get a payoff equal to three
for successfully hunting a stag. A hunter will get a payoff equal to one for hunting a hare. Finally, a
hunter will get a payoff equal to zero if she does not succeed in hunting anything.
(i) [2 marks ]Write down the set of pure strategies and the utility functions for each player. (To write
down the utility functions, you will need to write down the payoff for each player associated with every
possible strategy profile).
(ii) [1 mark] Write down the normal form game as a matrix.
(iii) [1 mark] Identify two strategy profiles, one of which is a Nash equilibrium while the other one is
not. Provide explanation for any one of them.
Now suppose a third individual C enters the picture. Each individual can always get a payoff equal to
one for hunting a hare irrespective of what others are doing. If two individuals hunt a stag each get three.
If three individuals hunt stag, each gets two.
(iv) [2 marks] Redo parts (ii) and (iii) for this 3-player game. Note: doing nothing (i.e., not hunting) is
not an option for any player.
Exercise 2. [4 marks]
Airbus (a European firm) and Boeing (an US firm) are competitors in the market for aircrafts. The
demand for aircrafts has gone down significantly in recent times - in fact, so much so that only one
player can profitably operate in the market. If both stay in the market, each loses $25 million. If both
exit the market each gets 0, where as if one stays and one exits, the one who exits gets 0 while the one
who stays bags $150 million.
(i) [0.5 marks] Write down the set of pure strategies for each player.
(ii) [0.5 marks] Write down the normal form game as a matrix.
1
2(iii) [1.5 marks] Suppose Airbus chooses stay and Boeing chooses exit? Is that a Nash equilibrium? Is
that the only pure strategy Nash equilibrium?
Now suppose the European government announces and credibly commits to $26 million subsidy for
Airbus, contingent on Airbus not exiting. That is, Airbus will receive $26 million as long as it remains in
the market (note: this means $26 million gets added to Airbus’s profit/loss figures stated above provided
it stays).
(iv) [1.5 marks] How does this affect your answers to (ii) and (iii)?
Exercise 3. [2marks]
Bloom and other tools which use generative AI can sometimes get confused and make errors when trying
to find the Nash equilibria of a game. Explain the error made by the student in the following transcript
Student: Solve the following game for pure strategy Nash equilibria.
A B
X 2, 2 1, 2
Y 2, 1 1, 1
Bloom: Thank you for providing the specifics. Remember that a Nash equilibrium occurs when no
player can unilaterally improve their payoff by changing their strategy, given the other player’s strategy.
Let’s start with identifying best responses for each player. Can you tell me what Player 1’s best response
is if Player 2 plays Strategy A? What about if Player 2 plays Strategy B?
Student: If player 2 plays A, the best response is X. If player 2 plays B, the best response is X. Similarly
for player 2, if player 1 plays X or Y the best response is A. So the unique Nash equilibrium is (X,A).
Bloom: That’s absolutely correct! Well done! To summarise, the Nash equilibrium in this game is
indeed at (Player 1: Strategy X, Player 2: Strategy A). This is a situation where neither player can
improve their payoff by unilaterally changing their strategy, given the other player’s strategy
3Please do not submit answers for the ones below. These are ungraded problems. Try these your-
selves. Tutors might go through some of these in tutorials if time permits.
Exercise 4. Consider the following 3-player normal form game
L R
T 8, 8, 8 7, 7, 7
B 7, 7, 7 8, 8, 8
W
L R
T 0, 0, 9 1, 1, 8
B 1, 1, 8 0, 0, 9
E
(i) For each player, write his set of best responses against every strategy profile of his opponents.
(ii) Is (T, L,W) a Nash equilibrium? Explain.
Exercise 5. Provide an example of a 2-player normal form game where each player has 3 (pure) strate-
gies such that:
(i) There is no pure strategy Nash equilibrium.
(ii) There are exactly three pure strategy Nash equilibria. Identify those three.
Exercise 6. Consider the following two-player normal form game.
L R
T 2, 2 0, 0
B 0, 0 x, x
Try different values of x and work out Nash equilibria in pure strategies. Does this game always admit
at least one Nash equilibrium in pure strategies (irrespective of the value of x)? Now replace (2, 2) with
(y, y) where y , x. Is there still at least one NE in pure strategies?
Case category
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SVM. (10 points) Consider an SVM classifier using RBF kernel, and finish the following python code to search for the optimum kernel parameter γ. You are required to calculate the RBF function and search yourself, instead of using functions in Scikit-Learn (sklearn). def parameter search(gamma list ,Xtr,ytr,Xts,yts ,...): """ Parameters gamma list: A list of RBF kernel parameter gamma to search for Xtr: Training data, shape (ntr,nrow,ncol) ytr: Training labels, shape (ntr,) Xts: Test data, shape (nts,nrow,ncol) yts: Test labels, shape (nts,) ... Add other parameters as needed Returns: gamma optimum: The gamma with minimum error on Xts, yts """ ... return gamma optimum
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