Game Theory Flashcards
Perfect information game
At each decision node, the players know perfectly what has happened until that point
Definition of a strategy
Complete contingent plan to specify what to do at each of the player’s information sets
Eg. play H if A plays H, play H if A plays T is one possible strategy
Definitions of actions and a condition they must satisfy
An action leads to any non-initial node from its predecessor
It must satisfy that if you are at node x not node y but x and y share a predecessor then the actions to get to x must have been different to get to y (or the action at x must be different to y)
Mixed strategy
Assigns each pure strategy a probability
Behavioural strategy
A player mixing between the possible actions at each information set
Same distribution over outcomes can be achieved with a mixed strategy
Two assumptions
- Players are rational: maximising expected utility if uncertainty
- Common Knowledge of Rationality (CKR): All players know all players know all players are rational
Strictly dominant strategy
A is a strictly dominant strategy if for all opponents’ strategies A has a strictly higher payoff for the player than any other strategy
A is strictly dominated if
for player i there exists another possible strategy that yields strictly higher payoff for all possible strategies by opponents
Weakly dominant strategies
When played?
same as strict domination but greater than or equal to for payoffs
Only potentially if very strong (certain) beliefs about what opponent(s) will play
Can a mixed strategy dominate or be dominated
Yes, both
A strategy is rationalisable if
it survives the iterated deletion of strategies that are never a best response
A strategy is never a best response if
There is no strategy by the opponent for which it is a best response
i.e. if it is never the case that that strategy yields a payoff higher than or equal to any other strategy
Can iteratively delete these like dominated
How does the set of rationalisable strategies relate to the set of those which survive iterated deletion of strictly dominated strategies
They are the same if there are only 2 players
If more than 2 players, the set of rationalisable strategies contains the non-dominated survivors ones
Assumptions needed for Nash Equilibrium
Common Knowledge of Rationality Common beliefs (correct)
When would a Nash Equilibrium be an obvious was of playing the game
If not obvious, can be very poor prediction
- Focal point
- Pre-play communication
- Stable social convention
- Learning
Trembling hand perfect strategies
Essentially removes NE of weakly dominated strategies
See if best response to a mixed strategy of some kind (even with small probabilities)
Risk domination
If the product of the deviation payoffs (usually negative for a NE, possibly always) is greater than for another NE then the first risk dominates the 2nd
One deviation principle
If there is a one-shot profitable deviation, the strategy is not subgame perfect (if not then is)
A payoff is individually rational for a player if
that payoff is higher than the minimax payoff
A strategy profile is sequentially rational if
given a system of beliefs, the player plays the strategy which leads to their highest expected payoff
Consistency with strategies
Strategies are consistent with beliefs (or vise versa) if the believed probability of playing x at information set I is equal to the probability of playing x at all divided by the probability of reaching that information set (conditional probability rule)
Weak Perfect Bayesian Equilibrium
- the strategy profile is sequentially rational given beliefs
- the system of beliefs is consistent with the strategy profile
can involve forming beliefs about information sets not reached and best responding to those
Sequential equilibrium
Very similar to trembling hand perfection in dynamic sequential games with imperfect
information
Is a WPBE and SPNE
Intuitive Criterion
If you deviate, you must be doing so to get a higher payoff therefore do not believe a deviator is a type who could get higher payoff by not deviating
