Find the best state each player can induce with moving power » Full Grade

Assignment ID: FG133082131

Consider the following three games:

For each game:

1. Find the best state each player can induce with moving power.
2. Determine whether moving power is effective, irrelevant, or ineffective.
3. Determine whether the game is strongly, moderately, or weakly cyclic

2. The X-uel Problem (duel with X number of players): There are X rational players in a row, from X (in front) to 1(last).  Each is the enemy of the player behind him/her (e.g., 5 is the enemy of 4, 129 is the enemy of 128, etc.).  Each has only one bullet, and is a perfect shot.  Each may fire only at the person immediately in front of him/her (i.e., his/her enemy), but not vice versa, and only if that person is unarmed, having already fired – (e.g., if 5 has fired, 4 may shoot 5, but 5 may never shoot 4.) Therefore, if a player shoots, s/he becomes vulnerable to being shot.  Of course, each wants most to survive; after that, each would like to shoot his/her enemy.

Player X (at the front of the row) accidentally fires his gun into the air (which is not an option for the other players). Then what happens?

3. The Pirates Problem:   A certain pirate ship has a crew of five, ranked from 5 (high) to 1  (low).  The highest ranking surviving pirate is the captain.  These pirates have a unique way of dividing treasure. When treasure is found, the captain proposes an allocation of the treasure (exactly how much each pirate will receive); each pirate (including the captain) then votes for or against the plan.  The majority decides the fate of the plan; in case of a tie, the captain casts a deciding vote.  If a plan is rejected, the captain is forced to walk the plank (at which point s/he is eaten by sharks), and the next highest ranking pirate becomes captain and must suggest an allocation plan.  This continues until a plan is accepted by the voting rules described.  Of course, these pirates are intelligent, rational actors, whose dominating motivation (other than not being eaten by sharks) is greed.  But they are also mean, so if it will not decrease a pirate’s share, s/he votes to kill (i.e., s/he vote against the captain’s plan).

One fine day, the pirates discover a treasure chest with 12 gold coins in it.  What happens?