1 - Game theory Flashcards
What key assumptions makes a preference relation rational?
- completeness = 2 alternative outcomes can be ranked
- transitivity = can rank all outcomes and there is consistency in ranking
What makes a decision maker rational?
- there exists a utility function which represents a rational preference relation over outcomes (preference relation is represented by payoff function
- he chooses action a that maximises their utility
What are the rational choice assumptions?
- knows all possible actions
- knows all possible outcomes
- knows how each action affects which outcome will materialize
- has rational preferences over outcomes
what are uncertain outcomes?
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
What is the difference between certain and uncertain outcomes?
- probabilities to outcomes
- independence axiom
- continuity
what is the independence axiom?
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
what is the continuity axiom?
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
when does a utility function
if players preference satisfies completeness, transitivity, continuity, independence
What are the assumptions of a static game
- normal form game
- 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
what is a normal form game?
2 player game
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
what is a pure strategy?
a deterministic plan of action
what is a dominating strategy?
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
will a rational player ever play a strictly dominated strategy?
NO
when do you use iterated elimination of strictly dominated strategies?
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
what is a best response
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
what is nash equilibrium
- a stable state
- where each player is playing a best response
- and no individual has an incentive to change their behaviour
how to find NE?
holding fixed what the other person is doing is this my best choice - row and column
- collective BR
- survivor of IESDS
- strictly dominating
what is best response correspondence and how is it different to best response function
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
what is a MS?
players choose to randomise between several of their pure strategies = probability distribution over pure strategies
why do we use expected payoff
to calculate payoff in ms
what is NE in MS?
when the expected utility is highest playing this ms holding what everyone else is doing fixed = best responding
what are the 2 properties of ms NE?
property 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
what are the 2 properties of ms NE?
property 1
- pure strategies can be strictly dominated by a ms
when does a ms NE occur?
- when player 1s choices make player 2 indifferent between the pure strategies in their MS2
- player 2 makes player 1 indifferent between their pure strategies in their MS1
= everyone is BR to each other
- no incentive to change
what is a sequential game?
- 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
sequential game format
game tree
how do we find NE in sequential game ?
- find normal form representation
- find NE in normal form
why cant you use NE in sequential games ?
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
what is SPNE
a more refined equilibrium that incorporates the fact that player A can anticipate what player B will do - can look forward in the game
