ECON 2112. PROBLEM SET 1

DUE: 5 PM, SEPTEMBER 20 (FRIDAY)

Note: In some exercises below you will find the expression “pure strategy” .  By this we mean the kind of strategies that we saw in parts I and II of the Week 1 Lecture Videos. In III we informally introduced the idea of mixed strategies, which we discuss in detail in Week 2.

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 worthless than a stag. To be precise, each hunter will get a payof equal to three for successfully hunting a stag.  A hunter will get a payof equal to one for hunting a hare.  Finally, a hunter will get a payof equal to zero if she does not succeed in hunting anything.

(i) 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 payof for each player associated with every possible strategy profile).

(ii) Write down the normal form game as a matrix.

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

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) Write down the set of pure strategies for each player.

(ii) Write down the normal form game as a matrix.

(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 afect your answers to (ii) and (iii)?

Exercise 3. Beyond two strategies [2 marks]

Provide an example of a 2-player normal form game where each player has 3 (pure) strategies such that:

(i) There is no pure strategy Nash equilibrium.

(ii) There are exactly three pure strategy Nash equilibria. Identify those three.

Exercise 4. Can AI do game theory? [3 marks]

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.

(i)  Consider the output below which is a variant of a game generated by an AI tool.

AI: Scenario: Two neighbouring restaurants, Bistro A and Cafe B, are deciding whether to invest in outdoorseating or enhance their indoor ambiance.

• If both restaurants choose outdoor seating, they each achieve a payof of 6, benefiting from the extra ambience created by the other restaurant, but sharing the increased summer foot traffic.

• If Bistro A chooses outdoor seating while Cafe B enhances indoor ambiance, Bistro A gets 7 and Cafe B gains less with 4, attracting less summer diners.

• If Cafe B chooses outdoor seating while Bistro A enhances indoor ambiance, Cafe B gets 7 and Bistro A gets 4.

• If both choose to enhance indoor ambiance, they each get a payofof 5.

Is the game the AI has generated an example of a prisoners dilemma? Why/why not?

(ii) Explain the error made by the student in the following transcript and rewrite the final response from Bloom in a way that corrects the student’s mistake.

Student: Solve the following game for pure strategy Nash equilibria.

Bloom: Thank you for providing the specifics. Remember that a Nash equilibrium occurs when no player can unilaterally improve their payof 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 Xor 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 payof by unilaterally changing their strategy, given the other player’s strategy.

The problems below are only for discussion in tutorials (if time permits); they will not be graded

Exercise 5. Consider the following two-player normal form game.

Try diferent 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?

Exercise 6. Definition

Provide a formal definition for the following concepts:

(i) n-player normal form game.

(ii) Nash equilibrium.

(iii) A player’s best response.

Exercise 7. Beyond two players

Consider the following 3-player normal form. game

(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.