Test 1
First Version
- (3 points) Consider the following sequential game, and perform backward induction analysis.
- At the beginning, which one of moves A1 or A2 will Ann choose according to backward induction?
- When Beth decides between B1 and B2, which one will she choose according to backward induction?
- What payoffs will Ann and Beth get in this game if both play according to backward induction?
- Ann will choose A2.
- Beth will chose B1.
- Ann gets 7 and Beth gets 6.
- (5 points) Consider the following two-player game with players Ann and Beth, where
Ann has three choices, A1, A2, or A3, and Beth has the three choices,
B1, B2, and B3.
| B1 | B2 | B3 |
| A1 | 3, 4 | 5, 4 | 5, 3 |
| A2 | 5, 2 | 2, 1 | 6, 3 |
| A3 | 2, 3 | 5, 1 | 1, 5 |
- What is the payoff for Beth if Ann plays A2 and Beth plays B1?
- Find the Maximin moves for both Ann and for Beth.
- Which moves are dominated by which moves?
In each case tell whether the domination is weak or strict.
- Describe all Nash equilibria.
- 2
- A1 for Ann, B3 for Beth.
- A1 weakly dominates A3, B1 weakly dominates B2.
-
There are two Nash equilibria: (A1,B2) and (A2,B3).
- (1 point) What is a zero-sum game?
In a zero-sum game, the sum of all payoffs of all players
for every outcome equals 0.
- (1 point) Which of the following features would imply incomplete information?
- The game contains some random moves.
- The game is sequential, and at some point a player has to move without knowing all decisions (moves)
the other players moving earlier had done.
- Some player does not know the payoff of some other player for some outcome.
- The players are not allowed to talk about the game before they start.
- A player has different payoffs in different outcomes.
It is option (3). Complete Information means both players know the structure of the game,
including the payoffs for all players for all outcomes.
- (1 point) Assume a 3-person simultaneous game has a Nash equilibrium of Ann playing move A2,
Beth playing B3, and Cindy playing C2. Assume the three players talk before playing and agree playing these
moves. Remember that this is still "cheap talk", we have a noncooperative game with all agreements nonenforcable.
Why is it still unlikely then that any players plays something different than agreed?
Since if only one player deviates, the payoff for that player is less or equal to the payoff
in case of playing the agreed move. To get more, more than one player has to deviate.
Second Version
- (3 points) Consider the following sequential game, and perform backward induction analysis.
- At the beginning, which one of moves A1 or move A2 will Ann choose according to backward induction?
- When Beth decides between B3 and B4, which one will she choose according to backward induction?
- What payoffs will Ann and Beth get in this game if both play according to backward induction?
- Ann will choose A2.
- Beth will chose B3.
- Ann gets 7 and Beth gets 4.
(5 points) Consider the following two-player game with players Ann and Beth, where
Ann has three choices, A1, A2, or A3, and Beth has three choices,
B1, B2, and B3.
| B1 | B2 | B3 |
| A1 | 1, 2 | 6, 4 | 5, 6 |
| A2 | 3, 4 | 5, 2 | 6, 2 |
| A3 | 2, 3 | 5, 1 | 1, 6 |
- What is the payoff for Beth if Ann plays A2 and Beth plays B3?
- Find the Maximin moves for both Ann and for Beth.
- Which moves are dominated by which moves?
In each case tell whether the domination is weak or strict.
- Describe all Nash equilibria.
- 2
- A2 for Ann, B1 and B3 for Beth.
- A2 weakly dominates A3, B3 weakly dominates B2.
-
There is one Nash equilibrium: (A2,B1).
(1 point) Assume a 3-person simultaneous game has a Nash equilibrium of Ann playing move A2,
Beth playing B3, and Cindy playing C2. Assume the payoff for Ann for that outcome equals 3.
- True or false: The payoff for Ann in any outcome is at most 3.
- True or false: If Ann plays A3, Beth plays B3, and Cindy plays C2,
then the payoff for Ann must be at most 3.
False. True.
(1 point) Describe the difference between outcome and payoff, maybe using
an existing game?
An outcome is any end position of the game, payoffs are numbers attached to each outcome
(for each player). In Rock-Scissors-Paper, for instance, and of the nine possible combination of moves
could be considered to be an outcome, but the payoffs are just 1 for Ann and -1 for Beth,
or -1 or Ann and 1 for Beth.
(1 point) Which of the following features would imply imperfect information?
- The game contains some random moves.
- The game is sequential, and at some point a player has to move without knowing all decisions (moves)
the other players moving earlier had done.
- Some player does not know the payoff of some other player for some outcome.
- The players are not allowed to talk about the game before they start.
- A player has different payoffs in different outcomes.
It is option (b).
A sequential game has perfect information if every player, when about to move,
knows all moves done by the other players (including the random player) so far.
..