Pure Strategy

Question 1

What is the specific definition of a NE in a simultaneous game?

Answer 1

Each player is simultaneously playing a mutual best response.

Question 2

What assumption does Nash Equilibrium fundamentally rely on? What would make it a stronger equilibrium?

Answer 2

That players form correct beliefs about the other players’ actions/strategies.
A stronger solution would rely only on rationalisability - i.e actions/strategies that apply regardless of what the other person plays, so do not require perfect beliefs.

Question 3

What is the specific definition of a best response?

Answer 3

An action/strategy for which
ui(ai,aj) >= ui(a’i, aj) for all possible actions of a’i that are a member of the action set A, holding the action of the other player(s) j fixed

Question 4

What does Cournot model and what are the equilbria of the standard Cournot Model?

Answer 4

Models quantity-setting competition in oligopoly. Unique NE where quantity is an interior solution: q1=q2= a-c/3
[p(q) = a -q1-q2]

Question 5

What is the difference between a best response correspondence and a function?

Answer 5

Correspondence - there may not always be a best response to an action
Function - always a defined best response for an action
[an input value (i.e action) can have only 1 outcome (does it have to be bijective?***)]

Question 6

What does Bertrand model and what are the NE outcome(s)?

Answer 6

Price-setting competition in oligopoly. The only NE is where pi =pj =c. So profits = 0 in equilibrium.

Question 7

Difference between Cournot and Bertrand?

Answer 7

• EQM Profits in Cournot are positive, in Bertrand they are equal to 0.
• Cournot models strategic substitutes, Bertrand strategic complements
• whilst Cournot has a unique best response for every case, Bertrand can produce either NO best response or MULTIPLE best responses

Question 8

Similarity(ies) between Cournot and Bertrand model?

Answer 8

In monopoly, produces the same level of profit

Question 9

What are strategic substitutes?

Answer 9

Strategies that move in the opposite directions to each other. e.g in Cournot, when one firm selected higher q, the other wanted to select lower q

Question 10

What are strategic complements?

Answer 10

Strategies that move in the same direction as each other. e.g in Bertrand competition, each firm wanted to set prices close to the other, but slightly undercutting

Question 11

Definition of a strictly dominant action

Answer 11

A ‘superior’ action that delivers strictly higher utility to a player, NO MATTER WHAT THE OTHER PLAYER PLAYS

Question 12

Mathematical definition of strict dominance

Answer 12

ai’’ strictly dominates ai’ iff

ui(ai’’,aj) > ui(a’i, aj) for every list of actions j can take

Question 13

Why wouldn’t a strictly dominated action appear in a NE?

Answer 13

The player playing this action would NOT be playing a best response. There is another action that delivers strictly higher utililty, for ALL actions of the other player.

Question 14

Definition of weakly dominant action

Answer 14

The action is as least as good as the other action no matter what the other player plays, and is strictly better than the other action for some specific actions of the other player

Question 15

Mathematical definition of weakly dominant actions

Answer 15

ai ‘’ weakly dominated ai’ iff

ui(ai’’,aj) >= ui(ai’, aj) for all aj

Question 16

What is a strict NE?

Answer 16

Where no players’ equilibrium action can be weakly dominated by another action.

Question 17

What is a non-strict NE ?

Answer 17

Where a player’s equilibrium can be weakly dominated by another action.

Question 18

What is dominance solvability?

Answer 18

Where we have a unique solution following iterated deletion of strictly dominated actions

Question 19

**How do we show that Cournot is dominance solvable?

Question 20

What conditions do we need to guarantee the existence of a NE?

Answer 20

1. Closed and bounded set of actions each player can take e.g [0,1]
2. Continuous best response functions - i.e a unique best action that a player can take in response to another’s action

Question 21

What lemma do we use to prove dominance solvability?

Answer 21

Suppose qj ∈ [q, q], then all qi > Bi(q) are strictly dominated by qi = Bi(q) and all qi < Bi(q)
are strictly dominated by Bi(q)

Question 22

What is the essence of proving dominance solvability?

Answer 22

Start at the boundaries and work inwards,

Question 23

What is the D(qj) function? (Dominance solvability)? What property does it have when we study the upper bound of a set?

Answer 23

The difference in firm i’s profits, when i selects a quantity q lower than qhat (where qhat is the best response to the bound)

D(qj ) = πi(ˆqi, qj ) − πi(qi, qj
D(qbar i) > 0 , as qhat is unique best response to the bound

Question 24

What do we need to prove to show dominance solvability in Cournot?

Answer 24

That D(qj) > 0 for all qi < qˆi and all qj ∈ [q(Lbound), q (Ubound)]

Question 25

How do we prove that D(qj) > 0 in the dominance solvability model?

Answer 25

Differentiate D(qj) wrt to qi 
get -qi(hat) + qi 
Decreasing function - therefore for qj < qbar, D(qj)>0
(as D(qbar) is already  positive)
qhat dominates all qi < qihat

Question 26

In simple words explain Dominance solvability Lemma>

Answer 26

If j takes the lower bound value of a set, then i’s choice to set qi higher than their best response to j is strictly dominated by playing the best response
If j takes the upper bound value of a set, then i’s choice to set qi lower than their best response to j is strictly dominated by playing the best response.

Question 27

What is the t’th stage best response to j playing qj = 0 in cournot (simultaneous)

Answer 27

B^t(0)= qm* [SUM of k=1 to t of: (-1/2)^k-1]

Question 28

State the best response of firm i in a Bertrand model:

Answer 28

• Any pi > pj if pj < c
• Any pi ≥ c if pj = c
• Does not exist if pj ∈ (c, pM]
• pM if pj > pM