ECOS3012 Strategic Behaviour
Tutorial 4
Q1. The money-grenade game with commitment
Consider the money-grenade game covered in Lecture 4. Now, suppose that Player 2’s
grenade is synced with a deposit machine that has the following four modes:
(i) ignite the grenade in 30 seconds unless a deposit of $1000 is made to Player 2,
(ii) ignite the grenade in 30 seconds if a deposit of $1000 is made to Player 2; otherwise,
don’t ignite,
(iii) always ignite the grenade in 30 seconds, and
(iv) never ignite the grenade
Before Player 1 decides whether to give $1000 to Player 2 or to keep the money, Player 2
switches the machine to one of these modes. Once switched, the mode cannot be changed.
a. Write down the extensive-form representation of this game (i.e., the game tree).
b. Find the subgame perfect equilibrium using backward induction.
c. Is there any Nash equilibrium in which Player 1 keeps the money?
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Q2easier Consider the following entry deterrence game, where an entrant decides whether to enter
a market or not, and the incumbent decides whether to fight or accommodate the entrant if
he enters. When the entrant stays out, the payoff of the entrant is 0 and the payoff of the
incumbent is 2. When the entrant enters the market, if the incumbent fights with a price war
then both firms gets -1; if the incumbent accommodates then both get 1.
a. How many subgames are there? What are they?
b. Find the subgame perfect equilibrium using backward induction.
c. Is there any other Nash equilibrium?
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Q3. Three-period sequential bargaining (Gibbons 2.1.D)
Player 1 and 2 are bargaining over one dollar. They alternate in making offers:
1. At the beginning of the first period, player 1 proposes to take a share s1 of the dollar,
leaving 1− s1 for player 2. Player 2 either accepts the offer (in which case the game
ends) or rejects the offer (in which case play continues to the second period).
2. At the beginning of the second period, player 2 proposes that player 1 take a share s2 of
the dollar, leaving 1− s2 for player 2. Player 1 either accepts the offer (in which case
the game ends) or rejects the offer (in which case play continues to the third period).
3. In the third period, the dollar is split evenly between the players, i.e., each gets 50 cents
and the game ends.
In addition, assume that
(i) a player accepts an offer when he is indifferent between acceptance and rejection, and
(ii) the players are impatient and they discount payoffs received in later periods by a factor
of 0.9 per period. E.g., at the beginning of period 1, player 1’s utility is 0.9 for the event
“receiving one dollar in period 2” and 0.81 for the event “receiving one dollar in period 3”.
Find the subgame perfect equilibrium by backward induction:
a. If the players enter the second period, what offer should player 2 propose? Does player 1
accept the offer?
b. What offer should player 1 propose in the first period? Does player 2 accept the offer?
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