Zero Sum Games

Question 1

What is a game?

Answer 1

Any situation where:

1) There are at least two players
2) Each player has strategies
3) For each choice of strategy, each player receives a payoff.

Question 2

What is a strategy?

Answer 2

An option for how a player can behave during the game.

Question 3

What are the underlying assumptions of games?

Answer 3

1) That everything a player cares about is summarizes in the players playoffs.
2) That each player knows everything about the structure of the game.
3) That each player chooses a strategy to maximize her how payoff given her beliefs about the strategy used by the other player.

Question 4

What is rationality?

Answer 4

The idea that a player will always:

1) Want to maximize her own payoff
2) Actually succeed in selecting the optimal strategy.

Question 5

What is a strictly dominate strategy?

Answer 5

A strategy that is will always be better than all the other options regardless of what the other player does.

Question 6

What is a best response?

Answer 6

The best choice of one player, given a belief about what the other player will do.

Question 7

How does decision theory relate to game theory?

Answer 7

Decision theory is about how you make decisions against "nature" instead of against another rational player.  You can think of it as a spectrum. 
Decision making under...
...certainty
...risk (known probability) 
...uncertainty (unknown probability)
...competition (This is game theory)

Question 8

What is a pessimistic decision making strategy?

Answer 8

When you try to maximize the minimum payoff that you could get (maximin).

Question 9

What is an optimistic decision making strategy?

Answer 9

When you are willing to risk the worst for a chance for the best. You chose the strategy that has the best possible outcome (maximax).

Question 10

What is a regretist decision making strategy?

Answer 10

When you try to minimize the regret you’ll feel from you choice. For each situation, you find the maximum amount of regret you might for each potential situation. Then you choose the strategy with the least amount of regret.

Question 11

What is a zero sum game?

Answer 11

A game where the sum of what both players get add up to zero (i.e. what player one gets is what player two looses).

Question 12

In a zero sum game, what decision making strategy is considered the solution, if a solution exists?

Answer 12

Maxmin (pessimistic strategy)

Question 13

What is a saddle point?

Answer 13

When the minimum of one player and the maximum of the other is the same. This means that even if your opponent knew what strategy you were going to play ahead of time, they would still pick the same strategy.

Question 14

When does a game have a solution?

Answer 14

When it has a saddle point.

Question 15

What is the value or outcome of a game?

Answer 15

It’s saddle point. It represents what will happen if both player follows the pessimistic maximin strategy.

Question 16

How do you make a game a “fair game”?

Answer 16

Subtract the solution from the values.

Question 17

How do you solve a game that does not have a certain solution?

Answer 17

You use a “mixed maximin” or “mixed strategy”.

Question 18

When does a game not have a saddle point?

Answer 18

When you might want to change your strategy if you knew what strategy the other player was going to do.

Question 19

What is the thinking behind a mixed maximin strategy?

Answer 19

Since you can’t know what strategy the other player will pick, your best option is to randomize your choices slightly so that you can’t be exploited if you play over and over.

Question 20

How do you determine the percentages you should play each strategy?

Answer 20

1) Remove any dominated strategies.

2) Select probabilities that give you the same expected value no matter what the other player does.

Question 21

What three strategies would a rational player chose in a game?

Answer 21

1) A strictly dominate strategy if available
2) A maximin if not
3) A mixed maximin if there is no saddle point.

Question 22

How do you find the expected value of a game with no saddle point?

Answer 22

1) Find the probabilities for the mixed maximin strategy by setting the two expected value equations equal to each other.
2) Plug those value back into on of the equations.