01 - Static Games of Complete Information Flashcards
What is normal-form representation?
display a game as static game in matrix. some information-loss compared to extensive-form representation.
What does complete information mean?
all players’ available actions and utilities from various outcomes are common knowledge
What is the problem for players in normal-form representation?
strategic uncertainty - problem of predicting how others will behave, in order to figure out what is best
What is strategy space Sᵢ?
set of strategies available to player i
What is a strategy profile?
a combination of strategies
s = (s₁, . . . , sₙ) ∈ S,
where S := S₁ × · · · × Sₙ
What is a payoff function uᵢ?
uᵢ: S -> R specifies player i’s utility uᵢ(s₁, . . . , sₙ) for
every possible strategy profile (consequence).
What is the prisoner’s dilemma?
2 players, finite strategy spaces
Bonnie & Clyde committed crime, face prison sentence
best pay-off for both together: when both do not confess
individually best pay-off: confess while other does not confess
bad pay-off: when both confess
What is the outcome of the prisoner’s dilemma?
both will confess, because for both player it is individually strictly better to confess than not to confess (rationality & strict dominance)
What is a strictly dominant strategy?
a strategy sᵢ ∈ Sᵢ is a strictly dominant strategy for player i if for all sᵢ′ ≠ sᵢ,
uᵢ(sᵢ, s₋ᵢ) > uᵢ(sᵢ′, s₋ᵢ) for all s₋ᵢ ∈ S₋ᵢ.
What is a strictly dominated strategy?
a strategy sᵢ ∈ Sᵢ is a strictly dominated strategy for player i if there is another strategy sᵢ′ ∈ Sᵢ,
uᵢ(sᵢ’, s₋ᵢ) > uᵢ(sᵢ, s₋ᵢ) for all s₋ᵢ ∈ S₋ᵢ.
What is IESDS?
Iterated Elimination of Strictly Dominated Strategies
What is the idea behind IESDS?
- rational player never plays strictly dominated strategy
- all players know this and can thus eliminate those strategies from the game
- repeat reasoning on resulting smaller game
What are the epistemic foundations of IESDS?
- all players are rational
- rationality is common knowledge (i.e. everyone knows that they know etc etc that all players are rational)
Do you need specific epistemic foundations for strictly dominant strategies?
No, because they do not require any knowledge about the other players, only about their own payoffs
What is weak domination?
a strategy sᵢ ∈ Sᵢ is a weakly dominated strategy for player i if there is another strategy sᵢ′ ∈ Sᵢ, such that
uᵢ(sᵢ’, s₋ᵢ) >= uᵢ(sᵢ, s₋ᵢ) for at least one s₋ᵢ ∈ S₋ᵢ.
What could be problematic with weak dominance?
Strategies that are only weakly dominated cannot be ruled out based solely on rationality.
What is problematic with IEWDS?
outcome depends on order of elimination and on whether all or only some weakly dominated strategies are eliminated in each step
What is a Nash equilibrium?
A Nash equilibrium is a strategy profile with the property that no player can gain by unilaterally deviating from it.
When must an outcome be a Nash equilibrium?
If a game has a unique predicted outcome. Otherwise rational players (knowing the prediction) would deviate.
What does “Nash equilibrium as a self-enforcing agreement” mean?
nonbinding communication among players prior to playing until they can all trust it
What does “Nash equilibrium as a stable social convention” mean?
• If game is played repeatedly, a particular way of playing may arise over
time,
• Nash equilibrium as steady state of dynamic adjustment process
What is BRᵢ(s₋ᵢ)?
the set of all best responses against s₋ᵢ
How do IESDS and Nash equilibrium relate?
If IESDS results in a unique strategy profile, then this is the unique Nash equilibrium of the game.
How do IEWDS and Nash equilibrium relate?
Weakly dominated strategies can be part of a Nash equilibrium. There are Nash equilibria that do not survive IEWDS.
How do you practically do IESDS?
look for strategies that a player would never play (which are strictly dominated) and eliminate them from the game. turn to the other player and do the same.
What if there are two solutions in strictly or weakly dominant strategies?
Then there is no solution (because not unique)
Can there be two solutions in IESDS?
yes, but they would be called “strategies surviving IESDS”
Does a Nash equilibrium need to be strictly dominant?
No, it is sufficient if there is no incentive to deviate (i.e. weakly dominant)
What needs to be included in normal-form representation?
define:
- players N = {player 1, player 2}
- strategy spaces Sᵢ
- payoffs uᵢ
