2018/03/06 18:27

# 博弈论 斯坦福game theory stanford week 3-1

## 习题

### 第 1 个问题

We say that a game is dominance solvable, if iterative deletion of strictly dominated strategies yields a unique outcome. True or false: The above game si dominance solvable. Hint: Consider both pure strategies and mixed strategies to do your domination.

a) True;

正确
(a) is correct, the game is dominance solvable.

For player 2, R is dominated by a mixed strategy between L and M (for example, play L with probability 1/2 and M with probability 1/2).
Once R is removed, for player 1 D is dominated by U.
Finally, once R and D are removed, M is dominated by L.
Since the process of iterative deletion of strictly dominated strategies yields the unique outcome (U,L), the game is dominance solvable.


b) False.

### 第 2 个问题

True or false: in the above game the order of elimination of weakly dominated strategies does not matter (that is, the final outcome is the same regardless of the order in which weakly dominated strategies are eliminated.).

[Hint: start the process of iterative elimination of weakly dominated strategies by eliminating different strategies at the beginning of the process.]

a) True;

b) False.

正确
(b) is correct, in the previous game the order of elimination of weakly dominated strategies does matter.

Consider the following attempts:
If we start by eliminating the weakly dominated strategy L:
For 2, L is weakly dominated by R.
Once L is removed, for 1 U is weakly dominated by D.
Once L and U are removed, M is strictly dominated by R.
The outcome of this process of elimination is (D,R).
If we start by eliminating the weakly dominated strategy M:
For 2, M is weakly dominated by R.
Once M is removed, for 1 D is weakly dominated by U.
Once M and D are removed, L is strictly dominated by R.
The outcome of this process of elimination is (U,R).
Then, the order of elimination of weakly dominated strategies does affect the outcome of the process.


### 3。第 3 个问题

Consider the matching pennies game:

Which is a maxmin strategy for player 1:

a) Play Left.

b) Play Right.

c) Play Left and Right with probability 1/2.

正确
(c) is true.

Recall from lecture: S1=argmaxs′1∈S1mins2∈S2u1(s′1,s2)
Given a strategy of 1: play Left with probability p and Right with 1−p (0≤p≤1):
If p>1/2, s2=Right leads 1 to earn (−2)p+2(1−p)<0;
If p<1/2, s2=Left leads 1 to earn 2p+(−2)(1−p)<0;
If p=1/2, then regardless of 2's strategy 1 earns 0.
Thus p=1/2 is the maxmin strategy.


d) It doesn't exist.

### 第 4 个问题

Apply the Minimax theorem presented in lecture 3-4 to find the payoff that any player must receive in any Nash Equilibrium:

a) 2;

b) -2;

c) 1;

d) 0.

正确
(d) is true.

Since the previous game is a (finite) two-player, zero-sum game, by the theorem presented in the lecture we know that in any Nash equilibrium each player receives a payoff that is equal to both his maximin value and his minimax value.
From the previous question we know that each player's maximin strategy is to play Left and Right with probability 1/2, which gives an expected payoff (maximin value) of 0.


### 第 5 个问题

Consider the following assignment device (for example a fair coin):

With probability 1/2 it tells players 1 and 2 to play B, and with probability 1/2 it tells them to play F. Both players know that the device will follow this rule. What is the expected payoff of each player when both players follow the recommendations made by the device? If one of players follows the recommendation, does the other player have an incentive to follow the recommendation as well?

a) Expected payoff =2; player has an incentive to follow the recommendation.

b) Expected payoff =1; player does not an incentive to follow the recommendation.

c) Expected payoff =1.5; player has an incentive to follow the recommendation.

正确
(c) is true.

If both players follow the recommendation of the device, they will play (B,B) with probability 1/2 and (F,F) with probability 1/2. Then, the expected payoff is 1/2∗2+1/2∗1=1.5.
It is easy to check that if one of the players is following the recommendation, then the other player has an incentive to do the same:
Suppose that player 1 follows the recommendation of the device.
When player 2 is told to play B, he/she knows that player 1 was also told to play B (and that is the strategy that he/she will play).
Player's 2 best response to player 1 playing B is to also play B.
The same holds when player 2 is told to play F. Therefore, player 2 will follow what the device tells him/her to do.


d) Expected payoff =1.5; player does not have an incentive to follow the recommendation.

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