1 - Game theory

Question 1

What key assumptions makes a preference relation rational?

Answer 1

1. completeness = 2 alternative outcomes can be ranked
2. transitivity = can rank all outcomes and there is consistency in ranking

Question 2

What makes a decision maker rational?

Answer 2

1. there exists a utility function which represents a rational preference relation over outcomes (preference relation is represented by payoff function
2. he chooses action a that maximises their utility

Question 3

What are the rational choice assumptions?

Answer 3

1. knows all possible actions
2. knows all possible outcomes
3. knows how each action affects which outcome will materialize
4. has rational preferences over outcomes

Question 4

what are uncertain outcomes?

Answer 4

when outcomes are uncertain = payoffs defined over random outcomes, you dont know what outcome will prevail

• if a roll a dice - uncertain what the outcome will be

Question 5

What is the difference between certain and uncertain outcomes?

Answer 5

1. probabilities to outcomes
2. independence axiom
3. continuity

Question 6

what is the independence axiom?

Answer 6

I weakly prefer X to Y if I weakly prefer a lottery with p chance of getting pX and 1-p of getting z –> to a lottery with p chance of getting Y and 1-p chance of getting Z for all p between 0 and 1

= you will always prefer a gamble of x over y

Question 7

what is the continuity axiom?

Answer 7

if you prefer outcome x1 to x2 and you prefer x2 to x3 then there exists a lottery with p chance of x1 and 1-p chance to x3 which you like exactly as much as getting x2 with certainty

Question 8

when does a utility function

Answer 8

if players preference satisfies completeness, transitivity, continuity, independence

Question 9

What are the assumptions of a static game
- normal form game

Answer 9

• players independently choose once and for all actions
• conditional on players choices their payoffs are distributed
• outcome = intersection of payoffs
• everybody knows all possible actions of each player
• all possible outcomes
• preferences of every player over outcomes
• payoffs

Question 10

what is a normal form game?

2 player game

Answer 10

has finite players
set of pure strategies
payoff functions for each player that give a payoff value to each combi of the players choices

• choose independently and simultaenously
• matrix represents strategies and their outcomes

Question 11

what is a pure strategy?

Answer 11

a deterministic plan of action

Question 12

what is a dominating strategy?

Answer 12

the best choice regardless of the other persons choice

• s2 is strictly dominated by s1 if for any possible combination of the other players strategies payoff from s1 is greater than s2
• no matter what the other person does s1>s2

Question 13

will a rational player ever play a strictly dominated strategy?

Answer 13

NO

Question 14

when do you use iterated elimination of strictly dominated strategies?

Answer 14

it is common knowledge that no player will play a strictly dominated strategy
- so can use IESDS to get rid of dominated plays
- creating smaller restricted game

Question 15

what is a best response

Answer 15

the strategy that produces the most favourable outcome for a player given what the other player is doing

• the strategy pays a higher payoff than any other strategy conditional on the other persons choice

Question 16

what is nash equilibrium

Answer 16

• a stable state
• where each player is playing a best response
• and no individual has an incentive to change their behaviour

Question 17

how to find NE?

Answer 17

holding fixed what the other person is doing is this my best choice - row and column

• collective BR
• survivor of IESDS
• strictly dominating

Question 18

what is best response correspondence and how is it different to best response function

Answer 18

function = 1 unique strategy that maximises your payoff given what everyone else is doing

correspondence = given what everyone is doing there are multiple strategies that maximise my payoff = BR includes multiple strategies

Question 19

what is a MS?

Answer 19

players choose to randomise between several of their pure strategies = probability distribution over pure strategies

Question 20

why do we use expected payoff

Answer 20

to calculate payoff in ms

Question 21

what is NE in MS?

Answer 21

when the expected utility is highest playing this ms holding what everyone else is doing fixed = best responding

Question 22

what are the 2 properties of ms NE?

property 1

Answer 22

1. if a player is randomising between 2 pure strategies in their ms then they must be indifferent between them
   - otherwise would not play the one with lower payoff and only play 1 strategy - so would not be BR since you can change to get higher payoff

Question 23

what are the 2 properties of ms NE?

property 1

Answer 23

2. pure strategies can be strictly dominated by a ms

Question 24

when does a ms NE occur?

Answer 24

1. when player 1s choices make player 2 indifferent between the pure strategies in their MS2
2. player 2 makes player 1 indifferent between their pure strategies in their MS1

= everyone is BR to each other
- no incentive to change

Question 25

what is a sequential game?

Answer 25

• order of moves
• different actions are available at different times
• payoffs are dependent on history of actions
• earlier players take into account the rationality of later players can predict what they will do

Question 26

sequential game format

Answer 26

game tree

Question 27

how do we find NE in sequential game ?

Answer 27

1. find normal form representation
2. find NE in normal form

Question 28

why cant you use NE in sequential games ?

Answer 28

because allows for equilibria that arent self enforcement - need to get rid of non credible threats from normal form NE by using backward induction SPNE
- NF NE treats all choices as once for all when they arent

Question 29

what is SPNE

Answer 29

a more refined equilibrium that incorporates the fact that player A can anticipate what player B will do - can look forward in the game