Note: In some exercises below you will find the expression“pure strategy”. By that we mean the kind of strategies that we see in the first week (particularly, in parts 1 and 2 of Week 1 Lecture Video). Later we will introduce concept of mixed strategies and the terminology will become clear.

Exercise 1.  Stag Hunt Game [3 marks]

Two individuals 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 four for successfully hunting a stag. A hunter will get a payoff equal to two for hunting a hare. Finally, a hunter will get a payoff equal to zero if she does not succeed in hunting anything.

(i) Identify the set of pure strategies for each player.

(ii) Write down the normal form game (including payoff/utility function) and the associated payoff ma- trix.

(iii) Identify the set of pure strategy Nash equilibria. Choose a Nash equilibrium (NE) and explain why it is a NE.

Exercise 2.  Strategic Subsidies [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 $15 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 $100 million.

(i) Identify the set of pure strategies for each player.

(ii) Write down the payoff matrix associated with the normal form game.

(iii) Compute the set of pure strategy Nash equilibria.

Now suppose the European government announces and credibly commits to $16 million subsidy for Airbus, contingent on Airbus not exiting. That is, Airbus will receive $16 million as long as it remains in the market[note: this means $16 million gets added to Airbus’s profit/loss figures stated above provided it stays]. How does this affect your answers to (ii) and (iii)?

Exercise 3. Beyond two players [3 marks]

Consider the following 3-player normal form game