Question 1

In the Centipede game there are two players. They take it in turns to choose Stop (S) or continue (C), starting with player 1. When a player chooses Stop the game ends and the payoff vector is either \((t,t)\) if player 1 chose Stop on their \(t\)th turn, or \((t-1,t+2)\) if player 2 chose Stop on their \(t\)th turn. After \(n\) turns if no-one has chosen Stop then the game ends with payoff vector \((n+1,n+1)\).

  1. Plot the extensive form for this game when \(n=3\).

  2. Use backward induction to determine the result of this game. Is this result surprising?

Question 2

Consider the following game. First player 1 chooses either \(A\) or B$. If player 1 chooses \(A\) then players 1 and 2 play a simultaneous game with payoffs

C D
C \((3,1)\) \((0,0)\)
D \((0,0)\) \((3,1)\)

If player 1 chooses \(B\) then player 2 chooses \(L\) with payoff \((1,1)\) or \(R\) with payoff \((2,2)\).

Find all the subgame perfect Nash equilibria for this game.

Question 3

Consider the modified Prisoner’s Dilemma game that we saw in lectures

A B
A \((3,3)\) \((0,5)\)
B \((5,0)\) \((1,1)\)

Let \(s_A\) be the strategy which always plays \(A\) and \(s_B\) be the strategy which always plays \(B\). We define two new strategies, \(s_T\) and \(s_C\). The strategy \(s_T\) is called the Tit-for-Tat strategy, which begins by playing \(A\) and then copies whatever the other player did in the previous stage. The strategy \(s_C\) is called the cautious Tit-for-Tat strategy, which begins by playing \(B\) and then copies whatever the other player did in the previous stage.

We assume that both players restrict themselves to this set of four pure strategies.

  1. What condition does the discount factor \(\delta\) have to satisfy for \((s_T, s_T)\) to be a Nash equilibrium?

  2. Show that \((s_T,s_T)\) is not a subgame perfect Nash equilibrium if \(\delta>2/3\).

Question 4

Consider the game

A B
A \((1,2)\) \((3,1)\)
B \((0,5)\) \((2,3)\)

Let \(s_A\) be the strategy which always plays \(A\) and \(s_B\) be the strategy which always plays \(B\). Define \(s_C\) to be the strategy where you being by playing \(B\) and continue playing \(B\) until your opponent plays \(A\), after which time you play \(A\) forever.

We assume that both players restrict themselves to this set of three pure strategies. Find the condition on the discount factor \(\delta\) so that \((s_C,s_C)\) is a Nash equilibrium.

Question 5

Consider the game

C D
C \((4,4)\) \((0,5)\)
D \((5,0)\) \((1,1)\)

Let \(s_P\) be the strategy called which starts by playing \(C\) and then plays \(C\) if both players played the same strategy in the previous stage, and plays \(D\) if they played different strategies.

Use the one-stage deviation principle to find a condition for \((s_P,s_P)\) to be a subgame perfect Nash equilibrium.

Question 6

Consider the two stage game analysed in Example 5.17 in lectures.

Let \(s_1\) be the strategy where a player plays \(F\) in stage 1, and in the second stage plays \(L\) if \((F,M)\) was played in stage 1 but plays \(G\) otherwise.

Let \(s_2\) be the strategy where a player plays \(M\) in stage 1, and in the second stage plays \(L\) if \((F,M)\) was played in stage 1 but plays \(G\) otherwise.

Show that \((s_1,s_2)\) is a subgame perfect Nash equilbrium, provided that \(\delta\geq\frac{2}{3}\).

Question 7

Consider the two stage game where stage 1 consists of the Centipede game with \(n=2\), and stage 2 consists of the cooperation game

A B
A \((1,1)\) \((0,0)\)
B \((0,0)\) \((3,3)\)

  1. If the discount factor is \(\delta=1\) then find a subgame perfect equilibrium in which the players received the payoff vector \((2,2)\) after the first stage.

  2. What is the smallest value of \(\delta\) for which the equilibrium you have just found is still valid?