Question

Question 1 - Game Theory

Alice and Bob are playing a game of simultaneous moves where they are deciding which

concert they will attend on Saturday night. Two musicians are performing that night locally,

Artist 1 and Artist 2.

Alice prefers to go to Artist 1 with Bob rather than Artist 2 with Bob, and she prefers going

to Artist 2 with Bob rather than going to Artist 2 alone. Moreover, she prefers going to

Artist 1 alone rather than going to Artist 2 with Bob.

Bob prefers going to Artist 2 with Alice rather than Artist 1 with Alice and he prefers going

to Artist 1 with Alice rather than going to Artist 1 alone. Moreover, he prefers going to

Artist 1 with Alice rather than going to Artist 2 alone.

a) Write down the payoff matrix for the above game. Use payoffs (utility levels) that reflect

the above assumptions. You can use any numbers such that the above assumptions are

not violated. (1 mark)

b) Find the best response functions for Alice and for Bob. Does Alice have a dominant

strategy? What about Bob? (1 mark)

c) Find all the Nash Equilibria in the game. (1 mark) Which of these lead to Pareto

efficient outcomes? Is there an outcome that Alice would prefer that is not a Nash

equilibrium? Is there an outcome that Bob would prefer that is not a Nash equilibrium?

(1 mark)

d) Suppose that the game is played sequentially and Alice moves first. Write the game in

its extensive form and find all subgame-perfect Nash equilibria. (1 mark)

e) Suppose that the game is played sequentially and Bob moves first. Write the game in

its extensive form and find all find all subgame-perfect Nash equilibria. (1 mark)

f) Are any of the Nash equilibria that you found in part c) based on empty threats in the

sequential games in part d) or in part e)? Explain your answer briefly. (1 mark)

Answer 1

a) So, let us suppose this is the payoff matrix as per the given conditions:

	Artist 1	Artist 2
Artist 1	(3,2)	(2,0)
Artist 2	(0,1)	(1,3)

- Payoff matrix where Alice chooses a row and Bob choose a column. In each box, the first number represents Alice's payoff and the second represents Bob's payoff.

b) Best response functions are the strategy(s) that yield the most favourable outcome for the player, supposing the other player's strategies as given.

Now, if we denote Alice's strategy as 'p' and Bob's strategy as 'q'. 'p' and 'q' is the probability of Alice and Bob choosing Artist 1 and 1-p and 1-q are their probabilities of choosing Artist 2.

Alice's best response:

Expected payoff of Artist 1 is 3q

Artist 2 is 2(1-q ), now, if 3q> 2-2q , or q > 2/5, best response is Artist 1. (p = 1)

If 3q<2-2q or q <2/5 , best response is Artist 2 (p=0)

And if 3q = 2-2q, q = 2/5, Alice is indifferent.

Bob's best response:

Expected payoff of Artist 1 is 2p.

Artist 2 is 3(1-p)

Now, if 2p>3-3p , or p > 3/5, best response is Artist 1, (q=1)

If 2p <3-3p or p<3/5, best response is Artist 2 (q=0)

If 2p=3-3p, or p = 3/5, Bob is indifferent.

A dominant strategy is a strategy which alwasy yields a better result for a player, irrespective of what the other player does.

So, in this case, if Bob chooses artist 1, Alice will choose artist 1 as 3> 0. And if Bob chooses artist 2, alice will choose artist 1 again as 2>1. So, whatever be the action of Bob, Alice will choose artist 1. This is Alice's dominant strategy.

Now, in case of Bob, if Alice choose artist 1, Bob wil choose artist 1 as 2> 0. If Alice chooses artist 2, Bob will choose artist 2 as 3>1. So, Bob's strategy will depend on and change with Alice's choice. So, Bob doesnt have a dominant strategy.

c) Nash equlibrium is an optimal solution, in which a player doesnt gain from a deviation from their initially chosen strategy, provided the strategies of the other players remain unchanged. In this case there are two nash equilibriums, one is where both go to artist 1 and the other is where both go to artist 2, i.e. with the payoffs (3,2) and (1,3) are the nash equilibriums.

A Pareto efficient outcome implies that there is no other outcome possible for a player to increase his/her payoff without hurting the other player at the same time. In this case, the nash equilibriums are both pareto efficient outcomes as there are no other outcomes where either of the player could increase his/her payoff without hurting the other person. No, there is no other outcome that Bob might prefer which is not the nash equilibrium. However, for Alice, as the dominant strategy is artist 1, she might prefer the payoff (2,0) when Alice goes to artist 1 and Bob goes to Artist 2, but its not a nash equilibrium.

d) We could write it ina sequential, extensive form in the form of a game tree as described in fig 1. Here, Alice is player 1 and Bob is player 2. The respective payoffs are mentioned. Now, we find the subgame perfect nash equilibria using backward induction: There are three subgames, which are the nodes of Alice and the two seperate nodes of Bob, as none of these decision nodes are connected to other decision nodes. So, we start with the two subgames at the bottom. Using backward induction, Bob is a rational player, Bob is the second player so knows what decision Alice has taken. Bob will maximize his utility. So, he will choose artist 1. Now, Alice knows that Bob is a rational player and wil always choose a best response to any action she chooses in the first stage. so she will choose artist 1 as that gives her a higher payoff. So, going to artist 1 together or (3,2) is the perfect subgame nash equilibria in this game.

Have only answered the first 4 parts as per rules. Thank you!