Flashcards in 02 - Static Games of Complete Information Deck (21)

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1

## What is a mixed strategy?

###
a probability distribution σᵢ = (σᵢ₁, ..., σᵢₖ₁) over Sᵢ

-> each strategy is being played with a certain probability

2

## 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

3

## What is a mixed strategy profile?

###
σ = (σ₁, ..., σₙ)

-> the set of the set of strategies of each player

4

## 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)

5

## What are possible interpretations of mixed strategies?

###
- in repeated games, players randomize strategies

- uncertainty about other players

- population shares when randomly choosing individuals

6

## 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

7

## 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

8

## What are the assumptions about mixed strategy choices?

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- Players’ randomizations are independent,

- Players evaluate mixed strategy profiles by their expected payoffs

9

## What is BRᵢ(σ₋ᵢ)?

### the set of all best responses against σ₋ᵢ

10

## 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)

11

## 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

12

## When is a mixed strategy strictly dominated?

### if it assigns positive probability to a strictly dominated pure strategy.

13

## Can we lose any mixed-strategy equilibria when eliminating strictly dominated pure-strategies from a game?

### NO

14

## Can a mixed strategy strictly dominate a pure strategy?

### YES, even if that pure strategy is not dominated by any other pure strategy.

15

## 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.

16

## 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

17

## 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

18

## 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

19

## 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

20

## 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)

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