Flashcards in 02 - Static Games of Complete Information Deck (21)
What is a mixed strategy?
a probability distribution σᵢ = (σᵢ₁, ..., σᵢₖ₁) over Sᵢ
-> each strategy is being played with a certain probability
What is a pure strategy?
where σᵢₗ = 1 and σᵢₖ = 0 for all k ≠ l
-> one strategy is played with p = 1, other strategies are not played
What is a mixed strategy profile?
σ = (σ₁, ..., σₙ)
-> the set of the set of strategies of each player
What is player i's expected payoff from a mixed strategy profile in a normal-form game with finite strategy spaces?
vᵢ (σ₁, ..., σₙ) := the sum of each outcome's utility times that outcome's probability (=each player's probability of playing that specific strategy multiplied)
What are possible interpretations of mixed strategies?
- in repeated games, players randomize strategies
- uncertainty about other players
- population shares when randomly choosing individuals
What is Theorem 1 in relation to the Existence of Nash Equilibrium?
Any normal-form game with finite strategy spaces has a Nash equilibrium in mixed strategies
What is Theorem 2 in relation to the Existence of Nash Equilibrium?
A normal-form game with
- a convex and compact strategy space
- and continuous, quasi-concave payoff function
has a Nash equilibrium in pure strategies
What are the assumptions about mixed strategy choices?
- Players’ randomizations are independent,
- Players evaluate mixed strategy profiles by their expected payoffs
What is BRᵢ(σ₋ᵢ)?
the set of all best responses against σ₋ᵢ
What are the requirements for a mixed strategy profile to be a N.E.?
The mixed strategy profile σ* = (σ*₁, . . . , σ*ₙ) is a Nash equilibrium of G if for each player i:
- vi(σ*ᵢ, σ*₋ᵢ) >= vi(σᵢ, σ*₋ᵢ)
- iff σ*ᵢ ∈ BRᵢ(σ*₋ᵢ)
How can you find the mixed strategy equilibrium?
In a mixed N.E., each of the players needs to be indifferent between her pure strategies given the mixed strategy of the other player
When is a mixed strategy strictly dominated?
if it assigns positive probability to a strictly dominated pure strategy.
Can we lose any mixed-strategy equilibria when eliminating strictly dominated pure-strategies from a game?
Can a mixed strategy strictly dominate a pure strategy?
YES, even if that pure strategy is not dominated by any other pure strategy.
What does a game being finite tell us about Nash equilibria?
That at least one Nash equilibrium exists - if not in pure, then in mixed strategies.
How can you find a mixed strategy that dominates another pure strategy x?
1) give the strategies to be mixed probability p and 1-p
2) set inequalities for each possible strategy the other player could play:
payoffs for mixing p and 1-p > payoffs for x
3) solve the inequalities such that they provide a range for p that dominates x
What is the indifference principle?
- in mixed strategies, this principle helps us to maximize probabilities such that a nash equilibrium is found
- indifference = other player has no incentive to switch from either strategy
How can you show that there is no pure strategy Nash equilibrium?
1) suppose there is a pure strategy N. E. x̂ = ( x̂₁, x̂₂)
2) go through all cases of relations in strategies and payoffs
3) show that in each, there is an incentive to deviate
How to argue that a Nash equilibrium does exist?
1) How does Sᵢ look like? interval = compact + convex
2) is the utility continuous and quasi-concave? (i.e. no weird payoff jumps)
-> if yes + yes, according to Second Existence Theorem, a Nash equilibrium exists
How can you find the symmetric Nash equilibrium in a game with i players and a utility function uᵢ?
for each player, uᵢ needs to be maximized (FOC = 0, solve for relevant variable)