1. Game Theory

Question 1

Rational decision maker

Answer 1

Someone who’s decisions can be accurately modelled

Question 2

Actions

Answer 2

The alternatives from which a person can choose

Question 3

Outcomes

Answer 3

The consequences that result from each possible action

Question 4

Preferences

Answer 4

The player’s personal ranking over possible outcomes

Question 5

What does X represent?

Answer 5

Outcome space

Question 6

What makes a preference rational?

Answer 6

Completeness and transitivity

Question 7

Completeness

Answer 7

When any two outcomes can be ranked by the preference relation

Question 8

Transitivity

Answer 8

If a>b and b>c then a>c

Question 9

When does a player follow a possible choice paradigm?

Answer 9

1. They know all possible actions, A
2. They know all possible outcomes, X
3. They know the relationship between actions and outcomes
4. They have rational preferences over outcomes

Question 10

Lottery

Answer 10

A finite set of outcomes with an associated probability to each outcome

Question 11

Independence axiom

Answer 11

States if x1>x2 then px1 + (1-p)x3 > px2 + (1-p)x3 for all 0<p></p>

Question 12

Continuity axiom

Answer 12

If x1>x2>x3 then there exists p such that px1 + (1-p)x3 = x2

Question 13

What is a normal game?

Answer 13

Step 1: each players chooses a strategy simultaneously and independently
Step 2: conditional on players choices, payoffs are distributed to each player

Question 14

What is a payoff function?

Answer 14

It maps every combination of pure strategies to a player n’s payoff

Question 15

Strategy profile

Answer 15

A possible combination of all players strategies

Question 16

Dominant strategy

Answer 16

When one strategy produces a higher payoff regardless of the strategies chosen by other players

Question 17

What is IESDS?

Answer 17

Iterated Elimination of Strictly Dominating Strategies. Where we remove dominating strategies from the game

Question 18

Nash equilibrium

Answer 18

A strategy profile where no individual has a unilateral incentive to change their behaviour. It is a concept of stability

Question 19

Best response correspondence

Answer 19

A mapping from strategies for all players other than n into the subsets of Sn

Question 20

How do you find a players best response when given a utility function

Answer 20

• Maximising for each s2
• FOC sets this equal to 0
• since the best responses are unique, a NE is formed and the best responses found

Question 21

If there is no pure strategy how can you best play the game?

Answer 21

Randomise between strategies

Question 22

What will a rational decision maker do when facing randomness in others choices?

Answer 22

Pick his strategy to maximise his expected payoff

Question 23

When will players mix between pure strategies?

Answer 23

When they are indifferent between them

Question 24

What is a function?

Answer 24

Something that maps one input to one output

Question 25

What is a correspondence?

Answer 25

Something that maps one input to multiple outputs

Question 26

Sequential games

Answer 26

Games that unfold over time

Question 27

How is a nash equilibrium sometimes too general for extensive games?

Answer 27

* some equilibrium may not be self enforcing | * players may be able to anticipate other players’ moves

Question 28

What is a subgame perfect Nash equilibrium?

Answer 28

It must be a best response at each node, given the strategies of other players. It is a refinement of Nash Equilibrium

Question 29

What is a sub game?

Answer 29

It consists of a single decision node and all of its successors in a game

Question 30

How are SPNE solved?

Answer 30

With backwards induction.

Question 31

Pareto improvement

Answer 31

Everyone is at least as well off as before and someone is better off

Question 32

When do SPNE exist?

Answer 32

They always exist

Question 33

One deviation unimprovable

Answer 33

If there is no decision node where changing the strategy improves the outcomes of the game

Question 34

One deviation principle

Answer 34

A one deviation unimprovable strategy is optimal

Question 35

How can we check an SPNE quickly?

Answer 35

Using one deviation principle. You only need to consider one change at a time at each node

Question 36

What is the discount rate?

Answer 36

A way of measuring future payoffs compared to present payoffs. It is a value between 0 and 1

Question 37

What is a repeated game?

Answer 37

A normal form game that is played multiple times in succession by the same players. The players roles stay the same as do their strategy spaces

Question 38

Conditional strategy

Answer 38

Assigns a strategy at period t according to what happened in every previous round

Question 39

What is grim trigger?

Answer 39

It is a strategy where p1 will cooperate until p2 defects, then p1 will defect for every subsequent round

Question 40

What is a sub game for a repeated game?

Answer 40

A sub game of a repeated game is any number of repetitions which is smaller than T

Question 41

How can we find out if a SPNE is satisfied?

Answer 41

We check to see if at any period t there exists a single profitable deviation. If there is then it isn’t an SPNE

Question 42

What is the relationship between the NE of a normal form game and a repeated game?

Answer 42

In any repeated game, if the normal form game has a unique NE, then G has a unique SPNE where the NE of g is played in every period

Question 43

How do we check if we have a SPNE in an infinitely repeated game?

Answer 43

All sub games are equivalent to the total game so we just need to check if any player wants to deviate in period 1

Question 44

What is a convex Hull?

Answer 44

It is the set of all convex combinations of the given vectors