1. Game Theory Flashcards
What is game theory?
The choices which motivated participants make in strategic decisions
What does motivated mean?
Participants have objectives and they can assess outcomes in terms of these objectives
When are situations strategic?
If achieving objectives depends on other players’ choices as well as one’s own
In a strategic situation what does a rational players choice depend on?
Her belief about another rational players choice
Pareto efficient
If there is no other strategy combination at which some player is better off and no one else is worse off
What is the dominance approach in the 2 player guessing half of the average game?
There is mutual knowledge of rationality ie I know that you know that I’m rational. Both choices converge to zero
What is the equilibrium approach of the 2 player guessing half of the average game
There is mutual knowledge of choices so we both end up at zero
What is an information set?
A collection of nodes representing what the player knows when making each decision
Terminal nodes
The final nodes in a decision tree which give a payoff
What is the normal form?
A contingent plan (strategy) specifying choice at every information set. Think of a strategy as instructions to an agent
What is extensive form?
All info necessary to represent how players see situations when making choices can be represented by direct answer in stage 1 or stage 2
If player choosing at info set knows history then how many nodes are in the info set?
Just one = singleton
If the info set isn’t a singleton what will the rational player do?
Form a belief about history to assess consequences of different available choices
What is a subgame root?
h is a subgame root if
-h singleton info set, and
-no path through h reaches another info set H which is reached by a path which doesn’t go through h
What does the normal form consist of?
The normal form consists of a list of
-players
-each players possible pure strategies which specifies her choice at every info set she controls even if an earlier choice precludes reaching that info set
-payoffs to each player after every strategy combination
What is a mixed strategy?
A random choice of a strategies e.g. player I chooses Si with probability P and chooses Ti otherwise
When does a rational player mix strategies?
When they are indifferent across those strategies
Reduced normal form
Normal form with all pairs of equivalent strategies merged
What are strategic situations?
Where you know consequences of feasible choices given other players’ choices. You may need to form beliefs about other players’ choices
Solution concept
Solution concept aims to accurately and correctly predict choices by rational players. This is useless if we can’t eliminate any strategy combinations
What is the dominance approach?
Eliminate strategies which are irrational regardless of what the other player chooses, and ignore strategies which are irrational for other players
Strict dominance
A strategy is strictly dominated if another strategy always provides a greater payoff regardless of the other players choice
Weak dominance
A strategy is weakly dominated if another strategy always provides the same or a greater payoff regardless of the other players choice
Inadmissible
Weakly dominated
Iterative admissibility
Eliminate some inadmissible strategies then eliminate some strategies which are now inadmissible after iteration 1
When is a game dominance solvable?
If some sequences of iterations lead to a unique strategy combination
Does the order of eliminations matter?
It can matter in weak dominance but not in strict dominance
When is backwards induction equivalent to iterative admissibility?
- Every info set is a singleton
- No two terminal nodes yield a given player the same payoff
When can a threat to a players own payoffs be useful to that player?
If a game requires coordination for the highest payoff, a threat can signal to the other player which move player 1 will make
Nash equilibrium
If an equilibrium is reached where no player has a profitable deviation
How many equilibrium outcomes does a generic finite game have?
An odd number of equilibrium outcomes. Generic means no player earns the same payoff at 2 distinct terminal nodes
How might players learn about other players strategies?
-past experience of playing similar games
-focality of belief- picking the more obvious or salient choice
How do we get to an SPE?
Truncate tree by sequentially eliminating strategy combinations which don’t form mash equilibrium in the last subgame(s)
What is the difference between SPE and NE
NE just requires that strategy combinations form equilibrium at every subgame reached on path. SPE required that threats be credible in every subgame so equilibriums are formed in every subgame
When are SPE and NE always the same?
In simultaneous move games since there are no proper subgames
How do contracts/ promises improve on play?
By restricting available choices (or at least making them more costly)
When isn’t commitment valuable?
-in a 1 person decision problem for a rational player with time consistent preferences
-in games conditional on other players’ choices
-when a threat is made but isn’t communicated which makes it incredible
What sort of events might a player be uncertain about?
Uncertainty about other players (endogenous) actions
Uncertainty about natures random (exogenous) choice
What is a lemon?
A low quality car/product
When will an owner of a used car sell it?
When the price is greater than the quality of the car
When does a consumer by a car?
When p< v x mu
What is a separating equilibrium?
Every type of player i chooses a different action. Players -i infer player i’s type exactly from action
What is a pooling equilibrium
Every type of give player chooses the same action. Players -i learn nothing about player i’s type from action so they retain their prior beliefs
What is the Spencer’s signalling game?
-players move simultaneously in double auction
-sender may choose signal to receiver
-receiver may learn about senders type from the signal
-in eq, receivers inferences from signals must be correct
What is the main lesson from Spence?
If costs of signalling differs across types then private info can be credibly conveyed
When does the separating equilibrium fail the intuitive criterion?
If y2>1
When does the pooling equilibrium fail the intuitive criterion?
Always
In Spence if different types have the same signalling cost what is the outcome?
Only pooling equilibria
In the cheap talk model, what does b=0 mean?
It means types have common interest
What happens in the case of the cheap talk model where types have monotonic preferences b>1?
The only result is pooling equilibrium because if any signal gives a better payoff then all types will switch to it
What happens in the cheap talk model where types have common preferences b=0?
There is a separating equilibrium and a pooling equilibrium
What happens in the cheap talk model when types have intermediate preferences? 0<b<1
Every type in an interval sends the same distribution of messages so R responds by choosing action equal to average type in that interval- pooling equilibrium
What is the field evidence on mandatory disclosure like?
There is little effect of financial disclosure because no one ever reads the small print. Mandatory reports of nutrition reduced high fat sales
What do players care about in a herding scenario?
The state and their own action
What do players observe in a herding scenario?
The actions of predecessors and private signals about natures choice of state before choosing sction
What is herding?
Rational imitation of others’ actions irrespective of own signal
In the simple herding model what does individual 1 do if she receives a signal that the state is good?
She acquires
What is the general result from our herding model?
Herding starts as soon as two consecutive individuals choose the same action
What is the probability of herding starting on the wrong decision?
Close to 3/8 when q is close to 1/2
How often does herding occur in the case where individual i can see all previous signals rather than previous actions?
Arbitrarily unlikely
What is bayes rule?
Conditional probability formula
P(A|B) = (P(B|A).P(A))/ P(B)
Intuitive criterion
Aims to reduce possible outcome scenarios by restricting the possible sender types to types who could obtain higher utility levels by deviating to off equilibrium messages, and to types for which the off equilibrium message is not equilibrium dominated
Utility function of car owner
U(n;q) = M+ qn
Budget constraint of car owner
Y= M+pn
What values does n take depending on whether the owner sells or keeps the car?
n=0 when car is sold at market price
n=1 when car is kept
Car consumer utility function
U(n;q) = M+ vqn
What is the equation for car consumer payoff?
Expected utility
Eu(n;q) = M+ vn x mu
