1. Game Theory Flashcards
Rational decision maker
Someone who’s decisions can be accurately modelled
Actions
The alternatives from which a person can choose
Outcomes
The consequences that result from each possible action
Preferences
The player’s personal ranking over possible outcomes
What does X represent?
Outcome space
What makes a preference rational?
Completeness and transitivity
Completeness
When any two outcomes can be ranked by the preference relation
Transitivity
If a>b and b>c then a>c
When does a player follow a possible choice paradigm?
- They know all possible actions, A
- They know all possible outcomes, X
- They know the relationship between actions and outcomes
- They have rational preferences over outcomes
Lottery
A finite set of outcomes with an associated probability to each outcome
Independence axiom
States if x1>x2 then px1 + (1-p)x3 > px2 + (1-p)x3 for all 0<p></p>
Continuity axiom
If x1>x2>x3 then there exists p such that px1 + (1-p)x3 = x2
What is a normal game?
Step 1: each players chooses a strategy simultaneously and independently
Step 2: conditional on players choices, payoffs are distributed to each player
What is a payoff function?
It maps every combination of pure strategies to a player n’s payoff
Strategy profile
A possible combination of all players strategies
Dominant strategy
When one strategy produces a higher payoff regardless of the strategies chosen by other players
What is IESDS?
Iterated Elimination of Strictly Dominating Strategies. Where we remove dominating strategies from the game
Nash equilibrium
A strategy profile where no individual has a unilateral incentive to change their behaviour. It is a concept of stability
Best response correspondence
A mapping from strategies for all players other than n into the subsets of Sn
How do you find a players best response when given a utility function
- 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
If there is no pure strategy how can you best play the game?
Randomise between strategies
What will a rational decision maker do when facing randomness in others choices?
Pick his strategy to maximise his expected payoff
When will players mix between pure strategies?
When they are indifferent between them
What is a function?
Something that maps one input to one output
What is a correspondence?
Something that maps one input to multiple outputs
Sequential games
Games that unfold over time
How is a nash equilibrium sometimes too general for extensive games?
- some equilibrium may not be self enforcing
* players may be able to anticipate other players’ moves
What is a subgame perfect Nash equilibrium?
It must be a best response at each node, given the strategies of other players. It is a refinement of Nash Equilibrium
What is a sub game?
It consists of a single decision node and all of its successors in a game
How are SPNE solved?
With backwards induction.
Pareto improvement
Everyone is at least as well off as before and someone is better off
When do SPNE exist?
They always exist
One deviation unimprovable
If there is no decision node where changing the strategy improves the outcomes of the game
One deviation principle
A one deviation unimprovable strategy is optimal
How can we check an SPNE quickly?
Using one deviation principle. You only need to consider one change at a time at each node
What is the discount rate?
A way of measuring future payoffs compared to present payoffs. It is a value between 0 and 1
What is a repeated game?
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
Conditional strategy
Assigns a strategy at period t according to what happened in every previous round
What is grim trigger?
It is a strategy where p1 will cooperate until p2 defects, then p1 will defect for every subsequent round
What is a sub game for a repeated game?
A sub game of a repeated game is any number of repetitions which is smaller than T
How can we find out if a SPNE is satisfied?
We check to see if at any period t there exists a single profitable deviation. If there is then it isn’t an SPNE
What is the relationship between the NE of a normal form game and a repeated game?
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
How do we check if we have a SPNE in an infinitely repeated game?
All sub games are equivalent to the total game so we just need to check if any player wants to deviate in period 1
What is a convex Hull?
It is the set of all convex combinations of the given vectors
