# A slave has just been thrown to the lions in the Roman Colosseum. Three lions are chained down in a.

A slave has just been thrown to the lions in the Roman
Colosseum. Three lions are chained down in a line, with Lion 1 closest to the
slave. Each lion’s chain is short enough that he can only reach the two players
immediately adjacent to him. The game proceeds as follows. First, Lion 1
decides whether or not to eat the slave. If Lion 1 has eaten the slave, then
Lion 2 decides whether or not to eat Lion 1 (who is then too heavy to defend
himself). If Lion 1 has not eaten the slave, then Lion 2 has no choice: he
cannot try to eat Lion 1, because a fight would kill both lions. Similarly, if
Lion 2 has eaten Lion 1, then Lion 3 decides whether or not to eat Lion 2. Each
lion’s preferences are fairly natural: best (4) is to eat and stay alive, next
best (3) is to stay alive but go hungry, next (2) is to eat and be eaten, and
worst (1) is to go hungry and be eaten.

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(a) Draw the game tree, with payoffs, for this
three-player game.

(b) What is the rollback equilibrium to this game? Make
sure to describe the strategies, not just the payoffs.

(c) Is there a first-mover advantage to this game?
Explain why or why not.

(d) How many complete strategies does each lion have?
List them.