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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)


How to find pareto-optimal (=efficient) symmetric allocation in a game with i players and a utility function uᵢ?

maximize aggregate welfare n•uᵢ (FOC = 0, solve for relevant variable z*)